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Plastic Anisotropy of Hexagonal Closed Packed Metals

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PLASTIC ANISOTROPY OF HEXAGONAL CLOSED PACKED METALS By BRIAN W. PLUNKETT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Brian W. Plunkett

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iii ACKNOWLEDGMENTS The author thanks each member of his supervisory committee for their suggestions and support. Special appreciation is extended to Dr. Oana Cazacu for her enthusiasm and support throughout the course of this study. The author also wishes to thank Dr. JeongWhan Yoon, Alcoa Technical Center, for advice on the implementation of material models into finite element code. The author sincerely appreciates the US Air Force Air Armament Center, Engineering Directorate (AAC/EN) for th eir financial support and for providing the opportunity to pursue this study, and to the Air Force Research Lab, Munitions Directorate, Computationa l Mechanics Branch (AFRL/MNAC) for employment and future research opportunities. Of course, this degree would not have been possible without the moral support of fellow graduate students Mike Nixon and Stef an Soare. Saving the most important for last, the author also wishes to thank his wife Tiffany Plunkett for her love and support during the duration of this degree program, and God for all things.

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iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES..........................................................................................................vii ABSTRACT......................................................................................................................xii CHAPTER 1 INTRODUCTION...........................................................................................................1 2 STATE OF THE ART IN MODELI NG PLASTICITY OF METALS...........................8 2.1 Description of Initial Yielding...............................................................................8 2.1.1 Isotropic Yield Criteria................................................................................8 2.1.2 Orthotropic Yield Criteria.........................................................................10 2.1.3 Modeling Asymmetry Between Te nsile and Compressive Yield.............17 2.2 Hardening Laws...................................................................................................18 2.3 Survey of Mesoscale Plasticity Modeling (Polycrystalline Models)...................20 2.3.1 Kinematics of a Polycrystal.......................................................................22 2.3.2 Taylor Model.............................................................................................24 2.3.3 Visco-Plastic Self-Cons istent (vpsc) Model..............................................26 3 PROPOSED YIELD CRITERION................................................................................34 3.1 Proposed Isotropi c Yield Criterion......................................................................34 3.1.1 Comparison to Polycrystal Simulation......................................................45 3.1.2 Yield Surface Derivatives and Convexity.................................................47 3.2 Extension of the Proposed Isotropic Yield Criterion to Include Orthotropy.......52 3.2.1 Application of the Proposed Cr iterion to the Description of Yielding of Hcp Metals............................................................................60 3.2.1.1 Magnesium Alloys..........................................................................60 3.2.1.2 Titanium Alloys...............................................................................66 3.2.2 Derivatives of the Or thotropic Yield Function..................................67 4 PROPOSED HARDENING LAW................................................................................72

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v 4.1 Introduction..........................................................................................................72 4.2 Elasto-Plastic Problem.........................................................................................74 4.3 Proposed Anisotropic Hardening Law.................................................................77 4.4 Application for an Isotropic Material..................................................................82 4.5 Alternate Method for Anisotropic Hard ening Implementation Interpolation...86 4.6 Application to Zirconium....................................................................................94 4.7 Application to Magnesium Alloys.....................................................................113 4.7.1 Application To Mg-Th............................................................................113 4.7.2 Application to AZ31B.............................................................................113 5 INCORPORATING THE EFFECTS OF STRAIN-RATE AND TEMPERATURE.121 5.1 Introduction........................................................................................................121 5.2 Elasto-Viscoplastic Theory................................................................................122 5.2.1 Perzynas Viscoplastic Approach............................................................122 5.2.2 Consistency Approach..............................................................................123 5.3 Energy Balance..................................................................................................124 5.4 Proposed Anisotropic Elastic/Viscoplastic Theory...........................................124 5.4.1 Using the Perzyna Method......................................................................124 5.4.2 Using the Consistency Method................................................................129 5.5 Numerical Examples..........................................................................................131 5.5.1 Using the Perzyna Method......................................................................131 5.5.2 Using the Consistency Method................................................................137 5.6 High-Strain Rate Modeling of the Be havior of Zirconium in Compression.....143 5.7 High-Strain Rate Modeling of the Be havior of Tantalum in Compression.......155 6 MODIFICATION OF THE PROPOSED YIELD CRITERION COMPARISON TO 2090-T3 ALUMINUM.......................................................................................166 7 SUMMARY AND FUTURE WORK.........................................................................174 LIST OF REFERENCES.................................................................................................177 BIOGRAPHICAL SKETCH...........................................................................................183

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vi LIST OF TABLES Table page 3.1 MG-TH yield surface coefficients..............................................................................63 3.2 MG-LI yield surface coefficients................................................................................64 3.3 MG yield surface coefficients.....................................................................................65 3.4 4Al-1/4O2 coefficients................................................................................................68 4.1 Voce hardening parameters for zirconium for a strain rate of 2110 s.......................95 4.2 Zirconium coefficients corresponding to the yield surface e volution depicted in Figure 4.7.................................................................................................................97 4.3 Zirconium coefficients correspondi ng to the yield surface for in-plane compression............................................................................................................110 4.4 Zirconium coefficients correspondi ng to the yield surface for in-plane compression............................................................................................................111 4.5 Voce hardening parameters for AZ31B magnesium................................................115 4.6 AZ31B coefficients corresponding to the yield surface evolution depicted in Figure 4.19.............................................................................................................117 5.1 Parameters used for numerical simulation................................................................131 5.2 Zirconium coefficients correspondi ng to the yield surface for in-plane compression............................................................................................................146 5.3 Tantalum coefficients for the proposed orthotropic criterion...................................156 6.1 2090-T3 aluminum coefficients................................................................................169 6.2 Tool dimensions used for cup drawing simulations.................................................170

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vii LIST OF FIGURES Figure page 1.1 Schematic of (a) slip and (b) deformation twinning...................................................2 1.2 Typical deformation systems for hexagonal closed packed metals...........................3 2.1 Orientation of test specimen w ith the rolling direction of the sheet........................12 2.2 Pictorial description of various hardening rules.......................................................18 2.3 Notations for (a) slip and (b) defo rmation twinning in a single crystal...................23 2.4 Geometry of a slip system within a single crystal....................................................25 3.1 Plane stress yield loci fo r different values of the ratio C T between the yield stress in tension and compression, in comparison with the von Mises locus...........37 3.2 Plane stress yiel d loci corresponding to /TC = 1.13 (k = 0.2) and /TC = 1/1.13 ( k = -0.2)......................................................................................................38 3.3 The influence of th e value of the parameter k on the ratio /TC of the uniaxial yield stress in tension and compression, for various values of the exponent a........39 3.4 Projection in the deviatoric plane of the yield loci for a = 2 and various values of k in comparison to von Mises and Tresca loci..........................................40 3.5 Projections in the tension-torsion pl ane of the proposed yi eld loci for various k values and a = 2 (fixed), in comparison with Tresca and von Mises (k=0, a=2) loci........................................................................................................................... .43 3.6 x f vs. k for the case of pure shear (a = 2)............................................................44 3.7 Comparison between the vpsc yield lo cus for randomly oriented fcc and bcc polycrystals deforming solely by twinning and the predictions of the proposed criterion: (a) for plane stress ( xy = 0) (b) on the -plane........................................46

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viii 3.8 Comparison between the VPSC yield locus for randomly oriented hcp zirconium polycrystals deforming solely by twinning and the predictions of the proposed criterion: (a) plane stress ( xy = 0) (b) on the -plane..............................................47 3.9 Comparison between the plane stress yield loci (0xy ) for a Mg-0.5% Th sheet predicted by the proposed theory and experiments.........................................63 3.10 Comparison between the plane stress yield loci (0xy ) for a Mg-4% Li sheet predicted by the proposed theory and experiments..................................................64 3.11 Comparison between the plane stress yield loci ( 0xy ) for a pure Mg sheet predicted by the proposed theory and experiments..................................................65 3.12 Comparison between the plane stress yield loci ( 0xy ) for a 4Al-1/4O2 sheet predicted by the proposed theory and experiments..................................................68 3.13 Comparison between the plane stress yield loci ( 0xy ) for a 4Al-1/4 O2 sheet predicted by Hosfords 1966 modified Hill criterion and experiments...................69 4.1 Evolution of the yield surface for varying k.............................................................84 4.2 Results of single element compression tests for a = 2.............................................85 4.3 Yield surface evolution for a cold ro lled sheet of mg-th us ing the interpolation method......................................................................................................................89 4.4 Discrete yield loci and yield st resses used for the in terpolation method.................92 4.5 Results of the inter polation method compared to th e continuous method from a single element compression test...............................................................................93 4.6 Stress-strain response for a clock -rolled plate of zirc onium for in-plane compression, in-plane tension, and through-thickness compression.......................96 4.7 Yield surface evolution for a clock-rolled plate of zirconium.................................97 4.8 Comparison between experimental data and simulation results using the proposed model coupled with VPSC and us ing an isotropic hardening law for a clock-rolled zirconium plate.....................................................................................99 4.9 Schematic of the four-point bend test....................................................................100 4.10 Comparison of the experimentally meas ured strain distributions with the results of finite element simulations usi ng the proposed model and VPSC linked directly to EPIC for the case C0.............................................................................103

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ix 4.11 Comparison of the experimentally meas ured strain distributions with the results of finite element simulations usi ng the proposed model and VPSC linked directly to EPIC for the case C90...........................................................................104 4.12 Comparison of the experimentally meas ured plastic strains distributions and the ABAQUS finite element predictions us ing the proposed model and the proposed yield criterion with isotropi c hardening for the C0 case........................................105 4.13 Comparison of the experimental plas tic strains distributions and the ABAQUS finite element predictions using the proposed model and the proposed yield criterion with isotropic hardening for the C90 case...............................................106 4.14 Comparison of experimentally photogra phed x-z cross-section of the bent bars versus the predictions of VPSC/EPIC and the proposed model; (a) and (c) correspond to the case C0 while (b) and (d) correspond to the case C90..............107 4.15 Yield surface evolution for a zirc onium clock-rolled plate during in-plane compression............................................................................................................110 4.16 Yield surface evolution for a zi rconium clock-rolled plate during through thickness compression............................................................................................111 4.17 Comparison of the final sections of th e zirconium cylinders after: a,b) in-plane compression, c,d) through-thickness compression.................................................112 4.18 Comparison between experimental data and simulation results using the proposed hardening law and using isotropic hardening for a cold rolled sheet of MG-TH alloy..........................................................................................................114 4.19 Comparison between the plane stress yield loci ( 0xy ) for a AZ31B magnesium cold rolled sh eet predicted by the proposed theory and yield strengths calculated using the vpsc polycrystal model...........................................117 4.20 Comparison between calculated data using a VPSC model and simulation results using the proposed hardening law and using isotropic hardening for a cold rolled sheet of AZ31B magnesium...................................................................................119 4.21 Axial stress distribution along the beams center cross-sections...........................120 5.1 Simulation results using the Pe rzyna method for various strain rates corresponding to uniaxial tension fo r loading and unloading conditions..............134 5.2 Simulation results using the Pe rzyna method for various strain rates corresponding to uniaxial compression for different variations of the strength differential coefficient k.........................................................................................135 5.3 Effect of strain rate and the size of the strain in crement on the accuracy of the Perzyna Method at 5% levels of viscoplastic strain...............................................136

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x 5.4 Effect of convergence tolerance an d the size of the strain increment on the accuracy of the Perzyna method at 5% levels of viscoplastic strain......................137 5.5 Simulation results using the consis tency method using the assumptions from Heeres et al. (2002) for va rious strain rates correspondi ng to uniaxial tension for loading and unloading conditions..........................................................................139 5.6 Comparison between the Perzyna method and consistency method using the assumptions from Heeres et al. (2002) corresponding to uniaxial tension for loading and unloading conditions..........................................................................140 5.7 Simulation results using the consis tency method without us ing the assumptions from Heeres et al. (2002) for various strain rates corres ponding to uniaxial tension for loading and unloading conditions........................................................141 5.8 Effect of convergence tolerance an d the strain rate on the accuracy of the consistency method at 5% levels of viscoplastic strain..........................................142 5.9 Schematic of the Taylor impact test........................................................................144 5.10 Yield surface evolution for a clock -rolled plate of zirconium subjected to inplane compression about the x-axis fo r 0.2%, 1%, 5%, 25%, 35%, 45% and 60% levels of effectiv e plastic strain..............................................................................146 5.11 In-plane compression simulation resu lts using the both the Perzyna method and the consistency method with the proposed yield criter ion and hardening law in comparison with experimental data........................................................................149 5.12 Comparison of predicted and experiment al logarithmic strain profile for the post test zirconium Taylor impact specimen.................................................................150 5.13 Comparison of the simulated and expe rimental cross-sections of the post-test zirconium Taylor impact experiment for (a) the major profile, (b) the minor profile, and (c) the footprint...................................................................................151 5.14 Comparison of predicted (assuming isotropic hardening) and experimental logarithmic strain profile for the post test zirconium Taylor impact specimen.....152 5.15 Comparison of the simulated (assumi ng isotropic hardening) and experimental cross-sections of the post-test zirconiu m Taylor impact experiment for (a) the major profile, (b) the minor pr ofile, and (c) the footprint.....................................153 5.16 Effect of mesh density a nd time step on the final solution....................................154 5.17 Yield surface corresponding to a rolled sheet of tantalum.....................................156 5.18 Uniaxial simulation results for various strain rates at 25C in comparison with experimental data...................................................................................................158

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xi 5.19 Uniaxial simulation results for va rious strain rates and temperatures in comparison with experimental data........................................................................159 5.20 Schematic of the orientation of cylin drical specimens cut from a tantalum plate.160 5.21 (a) Major and (b) minor profiles for the tantalum Taylor impact specimen..........162 5.22 Footprint for the tantalum Taylor impact specimen...............................................163 5.23 Visual comparison between simulated and experimental tantalum Taylor impact specimens for the (a) major side profile and (b) footprint.....................................164 5.24 Calculated temperature (degrees K) contours for the post te st tantalum Taylor impact specimen.....................................................................................................165 6.1 Comparison between predicted and expe rimental variation of yield stress and rvalues with sheet orientation..................................................................................168 6.2 Predicted initial yield surface for 2090-T3 aluminum...........................................169 6.3 Schematic of circular cup drawing.........................................................................170 6.4 Finite element mesh used for c up drawing simulation of 2090-T3 aluminum......171 6.5 Predicted and experimentally determ ined earing profile for a drawn cup of 2090T3 aluminum..........................................................................................................172 6.6 Predicted and experimentally determ ined earing profile for a drawn cup of 2090T3 aluminum..........................................................................................................173

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xii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PLASTIC ANISOTROPY OF HEXAGONAL CLOSED PACKED METALS By Brian W. Plunkett December 2005 Chair: Oana Cazacu Major Department: Mechanic al and Aerospace Engineering Due to the effects of twinning and textur e evolution, the yield surface for hexagonal closed packed (hcp) metals displays an asymmetry between the yield in tension and compression, and significantly changes its shape with accumulated plastic deformation. Traditional initial yield criteria or hardeni ng assumptions such as isotropic or kinematic hardening cannot accurately model these phenom ena. In this dissertation, a macroscopic anisotropic model that can desc ribe both the initial yielding and influence of evolving texture on the plastic response of hexagonal metals is propos ed. Initial yielding is described by a newly devel oped macroscopic yield crite rion that accounts for both anisotropy and asymmetry between yieldi ng in tension and compression. The coefficients involved in this proposed yield criterion as well as the size of the elastic domain are then considered to be func tions of the accumulated plastic strain. Viscoplastic self-consistent pol ycrystal simulations and a ne wly developed interpolation technique are then used to determine the evolution laws. The proposed model was

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xiii implemented into the implicit finite element code ABAQUS and used to simulate the three-dimensional deformation of a pure zirc onium beam subjected to four-point bend tests along different directio ns with respect to the texture orientation. Comparison between predicted and measured macroscopic st rain fields and beam sections shows that the proposed model describes very well the co ntribution of twinning to deformation. The proposed model is then extended to incl ude the effects of strain rate and the temperate increase within the material due to mechanical work. The proposed rate and temperature dependent model was implemen ted into ABAQUS/EXPLICIT and used to simulate the three-dimensional Taylor impact experiment for speci mens made from pure zirconium and from a tantalum alloy. The post experiment data and simulation results are shown to be in very good agreement.

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1 CHAPTER 1 INTRODUCTION Weight reduction while maintaining functi onal requirements is one of the major goals of engineering design and manufacturing so that materials, energy, and costs are saved and environmental damage reduced. Because of their low density, thermal properties, damping capacity, fatigue propertie s, dimensional stability, and machinability, hexagonal closed packed (hcp) metals such as magnesium and titanium alloys offer great potential to reduce weight and thus replace th e most commonly used materials, i.e., steel and polymers, plastics. Currently, the use of hcp metal sheets is re stricted because of a lack of fundamental understanding of th eir three-dimensional flow behavior. Plastic deformation of polycrystalline metals occurs by either s lip or twinning (see Figure 1.1). Whether slip or twinning is the dominant deformation mechanism depends on which mechanism requires the least stress to initiate and sustain plastic deformation. Metals with cubic crystal symme try have many slip systems, so twinning is usually not a significant deformation mechanism at ambien t temperatures, but may become important as the temperature decreases or the strain rate increases (Blewitt et al., 1957, Huang et al., 1996). In low symmetry materi als such as hcp metals, whic h have too few slip systems to accommodate any shape change, twinning may become a dominant mechanism. Twinning, unlike slip, is sensitive to the sign of the applied stress; i.e., if a particular twin can be formed under a shear stre ss, it will not be formed by a shear stre ss of opposite sense. Because of the polar nature of twi nning, hcp materials disp lay a strong asymmetry between the yield in tension and compression.

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2 Figure 1.1 Schematic of (a) slip and (b) deformation twinning Due to the strong crystallogr aphic texture induced by the rolling process, the yield loci for cold rolled sheets of hcp metal may also exhibit a pr onounced anisotropy (Hosford, 1993). Texture refers to the non -uniform distribution of crystallographic orientations in a polycrystalline aggregate. Most cold-rolled hcp alloy sheets have basal or nearly basal textures, i.e., the basal planes of the grains are aligned with the sheet, with a degree of spread from this ideal texture by up to 20 about the transverse direction for (a) (b)

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3 magnesium while for alpha titanium and zirconium alloys the spread is up to 40 (see Hosford, 1966). The easiest slip directions are the closed-packed 0 2 11 directions which are normal to the c-axis, but slip on these systems does not produce any elongation or shortening parallel to the c-axis (see Fi gure 1.2). Thus, only twinning or pyramidal slip can allow inelastic shape changes in the c direction. For most hcp metals, the most easily activated twinning mode is the tensile twin 10121011 which is activated by compression in the plane of the sheet or tens ion in the normal direction of the sheet, i.e., through-thickness tension. Since the pyra midal slip and compression twinning are much harder than the primary deformation modes of basal slip and tension twinning, most hcp sheets display a resistance to thinning, and a very pronounced difference between the stress-strain behavior in tens ion and compression is observed (see for example, Tom et al., 2001). Figure 1.2 Typical deformation systems for hexagonal closed packed metals. The correct modeling of this strong asym metry between tension and compression due to deformation twinning remains a challenge. As discussed by Van Houtte (1978) and Tome et al. (1991), a major obstacle in extending the crystal plasticity framework to include deformation twinning is the diffi culty in handling the large number of orientations created by twinned regions. Twin ning activity plays an important role in the

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4 evolution of hardening by cr eating barriers to the propagati on of dislocations or other twin systems. In addition, twinning influe nces anisotropy evolutio n by reorienting the grains (Kaschner et al., 2001) Although progress has been made and models that track the evolution of the twinned regions in th e grain and account for predominant twin reorientation (e.g. Tome and Lebensohn, 2004 b) or intergranular mechanisms (e.g., Staroselsky and Anand, 2003) have been propo sed, the use of such models for forming analyses is still limited because of rather large computation time. Unlike the recent progress in the formul ation, numerical implementation, and validation of macroscopic plasticity models fo r cubic materials, macroscopic modeling of hcp materials is less developed. Due to the l ack of adequate macroscopic criteria for hcp materials, hcp sheet forming finite element simulations are still performed using classic anisotropic formulations for cubic metals such as Hill (1948) (see for example, Takuda et al., 1999; Kuwabara et al., 2001). General presentation of the dissertation. This dissertation is a contribution to modeling and simulation of plastic anisotropy and strength differential effects in hcp metals. Chapter 2 consists of a survey of ma jor contributions to the description of plastic behavior of metals at different length scal es. Macroscopic plasticity models will be discussed including isotropic and orthotropic yield criteria that describe the onset of plastic behavior, and hardening laws which m odel subsequent plastic deformation. Then, two of the most widely used mesoscopic plas ticity models, the Taylor-Bishop-Hill model and the viscoplastic self-consistent (vpsc) model, will be described. Chapter 3 is devoted to the development of a new yield criterion for hcp metals. This yield function is capabl e of describing both the tens ion/compression asymmetry and

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5 the anisotropic behavior of hcp metals and all oys. The approach used in constructing this criterion was first to develop an isotropic yi eld criterion that can capture the asymmetry between tension and compression, and then extend this criterion to include orthotropy. The expression of the isotropic criterion was based on numerical tests using the vpsc model. Specifically, since there are no isot ropic pressure insensitive materials that exhibit tension/compression asymmetry, the vpsc model was used to obtain information concerning the shape of yield loci for ra ndomly oriented polyc rystals (isotropic) deforming by twinning (directional shear mech anism). The proposed isotropic criterion involves only two parameters k and a, where a represents the degree of homogeneity of the yield function. For a fixed, the parameter k is expressible solely in terms of the ratio between the yield stress in tension and the yield stress in compression. Comparisons with the results of polycrystalline simulations show that the proposed macroscopic criterion describes very well the strength differential e ffect due to twinning in body centered cubic (bcc), face centered cubic (fcc), and hcp polycr ystals. An orthotropic extension of the isotropic yield criterion is developed. Orthotropy is introduced through a linear transformation applied to the deviator of the Cauchy stress tensor. Then, the yield loci obtained using the proposed or thotropic criterion are compar ed with experimental yield loci for sheets of textured polycrystalline binary Mg-Th, Mg-Li alloys, pure Mg (data after Kelley and Hosford, 1968), and Titanium (data after Lee and Backofen, 1966). Very good agreement between theoretical and e xperimental yield loci is obtained. In addition, we compare the yield loci predicte d by the proposed orthot ropic criterion with the calculated yield loci obtained using the vpsc model for AZ31B magnesium and pure zirconium.

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6 Next, in chapter 4 a new and rigorous me thod for describing an isotropic hardening due to evolving texture during plastic de formation is proposed. The anisotropy coefficients as well as the size of the elastic domain are considered to be functions of the accumulated plastic strain. An interpolation technique is introduced to determine the evolution laws based on the results from mechanical tests and/or numerical tests performed with polycrystal models in the absence of experimental data for the corresponding strain paths. The proposed mo del was implemented into the implicit finite element code ABAQUS and used to simulate the three-dimensional deformation of pure zirconium and magnesium alloys subjected to different loading conditions. Validation of the model for strain paths that have not been used for parameter identification is given for zirconium (data after Kaschner et al., 2001). Comparison be tween predicted and measured macroscopic strain fields and beam sections for zirconium beams subjected to 4-point bending experiments shows that the proposed model describes very well the contribution of twinning to deformation. The difference in response between the tensile and compressive fibers and the shift of the neutral axis is particularly well captured. In chapter 5, an extension of the propos ed elasto-plastic an isotropic model that includes the effect of strain-rate and temperatur e is presented. To introduce rate effects in the inviscid (elasto-plastic) model, two different modeling approaches are considered: the Perzyna overstress method (Perzyna, 1966) and the consistency method (Wang et al., 1997). Both approaches will be used to simulate the high strain-rate Taylor impact test. The simulated results will be compared to experimental Taylor impact tests for pure zirconium (data after Kaschner et al., 1999). For comparison purposes, we also simulated the high strain rate behavior of a metal w ith bcc structure (tanta lum) by setting the

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7 strength differential parameter k to zero and using isotropic hardening. The influence of the crystallographic structure on the high stra in rate response was clearly demonstrated for both zirconium and tantalum. In th e case of zirconium, the model reproduces correctly the very high hardening rate which is due to twinning; thus the profile of the zirconium post-test specimen displays a much less pronounced mushrooming effect. For tantalum which deforms only by sl ip, mushrooming is significant. Chapter 6 is devoted to the description of plastic anisotropy in metals having cubic crystal structure. Focus is on modeling of the behavior of an aluminum alloy that exhibits strength differential effects as well as orthotropy. The proposed model is an extension to orthotropy of th e isotropic yield criterion presented in Chapter 3. Two linear transformations were introduced to capture bot h the anisotropy in the yield stresses for tensile and compressive loadings and the Lank ford coefficients (r-values). Thus, the number of independent coefficients invol ved in the formulati on was doubled. The proposed modified criterion was then applied to the prediction of th e earing profile of a circular cup drawn from 2090-T3 aluminium. Conclusions and future research directions opened by this research are presented in Chapter 7. It is also worth mentioning that this dissertation has thus far resulted in three journal articles that have been submitted or accepted for publication in International Journal of Plasticity and Acta Materialia.

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8 CHAPTER 2 STATE OF THE ART IN MODELING PLASTICITY OF METALS 2.1 Description of Initial Yielding 2.1.1 Isotropic Yield Criteria For sufficiently small values of stress and strain, a metal will reassume its original shape upon unloading. When loaded beyond th is reversible (elastic) range, the specimen will not reassume its original shape upon unlo ading, but will exhibit a permanent (plastic) deformation. In the plastic range, it is typical for metals to work harden; i.e., the flow stress monotonically increases with accumula ted plastic strain. After a specimen has been subjected to a stress exceeding the yiel d limit which separates the elastic and plastic ranges, the current stress becomes the new yiel d limit if the material is unloaded. For a multi-axial state of stress, a materials yield limit is mathematically described by a yield criterion. The oldest yield criterion was proposed by Tresca in 1864. According to Trescas criterion, the material transitions to a plas tic state when the maximum shear stress reaches a critical value. The Tresca criterion is given by Y 1 3 3 2 2 1, max (2.1) where 1, 2, and 3 are the principal stresses, and Y is the yield stress in uniaxial tension. The projection of Trescas yield surface on the -plane (the plane which passes through the origin and is perpendicula r to the hydrostatic axis) is a hexagon centered on the origin whose size depends on the magnitude of Y.

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9 Possibly the most widely used isotropi c yield criterion is the one proposed independently by Huber in 1904, von Mises in 1913, and Hencky in 1924. This criterion is usually referred to as the von Mises crite rion. The von Mises crite rion is based on the observation that a hydrostatic pressure cannot cause yielding of the material. The plastic state corresponds to a critical value of the elastic energy of distortion, i.e. 2 2k J (2.2) where k is a constant and J2 is the second invariant of th e Cauchy stress deviator given by 2 1 3 2 3 2 2 2 1 26 1 J (2.3) or alternatively, ) ( 2 12 3 2 2 2 1 2S S S J (2.4) where S1, S2, and S3 are the principal values of the Cauchy stress deviator, defined as ij kk ij ijS 3 1 ( i,j,k = 1,2,3) (2.5) The projection of the von Mises yield locus on the -plane is a circle that circumscribes the Tresca hexagon. Experimental evidence has shown that the yi eld loci for most isotropic metals with cubic crystal structure lie between the yield loci pred icted by the Tresca and the von Mises criteria (see Taylor an d Quinney, 1931). In order to represent the behavior of certain metals (e.g., aluminum alloys) for which the yield loci are located between the Tresca and von Mises yield loci, Drucke r (1949) proposed the following criterion: 32 23JcJF (2.6)

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10 where J3 is the third invariant of th e Cauchy stress deviator, and c and F are material constants. In order to ensure that the yield surface is convex, c is limited to a given numerical range, 278,2.25 c Unlike the isotropic yield criteria menti oned so far, which were postulated on the basis of macroscopic experiments, Hers hey in 1954 and then Hosford in 1972 used Taylor-Bishop-Hill polycrystalline simulations (for an explanation see section 2.3.2) to arrive at the following macroscopic yield criterion. m m m mY21 3 3 2 2 1 (2.7) In (2.7), Y is the uniaxial yield stress and m is the degree of hom ogeneity which can vary between 1 and Equation 1.7 reduces to the Tresca criterion for m = 1 or m = and to the von Mises criterion for m = 2 or m = 4. It has been shown that the yield loci of fcc and bcc metals are best represented with m = 8 and m = 6 respectively (Logan and Hosford, 1980, Hosford, 1993, Hosford, 1996). 2.1.2 Orthotropic Yield Criteria Due to thermo-mechanical processing, me tal sheets exhibit or thotropic symmetry with the axes of orthotropy being aligned wi th the rolling direct ion, the transverse direction, and the norm al direction to the plane of the sheet ( x y and z respectively). In 1948, Hill proposed a generalization of th e von Mises isotropic yield criterion to orthotropy. Thus, this yiel d criterion is expressed by a qua dratic function of the form: 1 2 2 22 2 2 2 2 2 xy zx yz y x x z z yN M L H G F (2.8) where F, G, H, L, M, and N are anisotropy constants, and x y and z are the orthotropy axes (axes perpendicular to the 3 mutua lly orthogonal planes of symmetry of the

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11 material). When F = G = H = 26 1 k and L = M = N = 22 1 k equation (2.8) reduces to the von Mises criterion (2.2). The coefficients involved in the expres sion of Hills yield criterion can be determined from simple mechanical tests. Denoting the tensile yield stresses for the x y and z directions as X Y and Z respectively, it can be sh own that according to Hills theory 11 XY GHHF 1 Z FG thus, 2 2 21 1 1 2 X Z Y F ; 2 2 21 1 1 2 Y X Z G ; 2 2 21 1 1 2 Z Y X H (2.9) Denoting the shear yield stresses as R S and T corresponding to the yz zx and xy directions respectively, then: 21 2 R L ; 21 2 S M ; 21 2 T N (2.10) The thinning resistance of metal sheets is generally characterized by the Lankford coefficients, commonly referred to as r -values. The Lankford coefficients are defined as the width-to-thickness strain ra tio during a uniaxial test. In classic plasticity theory, the plastic strain increments are derived from a plastic potential, which for metals is generally supposed to coincide with the yiel d function (associated flow rule) such that p f dd (2.11) where pd is the plastic strain increment, is a scalar variable, and f is the yield function. Therefore, the strain increment vector is orthogonal to the yield surface. Assuming

PAGE 25

12 plastic incompressibility, the Lankford coeffi cient for a uniaxial tensile loading in the x direction can be written as yy xx yy yy xx yy zz yy xxf f f d d d d d r (2.12) Likewise, r the strain ratio corresponding to a uniaxial tensile loading at an arbitrary angle, to the x-direction (see Figure 2.1) is given by yy xx yy xy xxf f f f f r 2 2cos 2 sin sin (2.13) Figure 2.1 Orientation of te st specimen with the rol ling direction of the sheet

PAGE 26

13 According to Hills criterion, the Lankford coefficients can be expressed in terms of the anisotropy coefficients as G H r 0, F H r 90; 2 145 G F N r (2.14) Thus, the anisotropy coefficients can be expresse d in terms of yield stre sses in the rolling, at 45 degrees to the rolling, and in the tran sverse directions, or as functions of the r values. In genera l, the yield loci obtained based on r -values differ from those obtained based on the yield stresses (Barlat et al., 2005). Hills yield criterion (2.8) is the most wi dely used criterion for describing yielding of textured metals. However, (2.8) can not adequately repr esent the behavior of certain aluminum alloys which have an average value of the Lankford coefficients less than 1 and the yield stress in biaxial tension is great er than the yield stress in uniaxial tension (Banabic et al., 2000). Therefore, in order to better represent yielding of aluminum alloys, Hill developed another yiel d criterion in 1979 of the form: m m m m m m mY N M L H G F 2 1 3 1 3 2 3 2 1 2 1 1 3 3 22 2 2 (2.15) This yield criterion has a major limitation sinc e it is written in terms of the principal Cauchy stresses. In order for this criterion to be valid, the principal stress axes and anisotropy axes must be superimposed thus any state involving sh ear stresses cannot be accounted for. Barlat et al. (1991) proposed a six-com ponent yield criterion denoted Yld91, that extends the isotropic Hershey and Hosford criterion (see eq 2.7) to orthotropy. The extension to orthotropy is accomplished by repl acing the principal values of the Cauchy stress tensor in the expressi on of the isotropic criterion by those of a transformed stress

PAGE 27

14 tensor. The transformed stress tensor is obt ained from the Cauchy st ress tensor modified with weighting coefficients. This procedure is equivalent with the a pplication of a fourth order linear transformation operator on the Cauchy stress tensor. The orthotropic criterion is written as 1223312mmm mY (2.16) where L (2.17) 1, 2, and 3 are the principal values of the tensor is the Cauchy stress tensor, and L is a fourth order tensor of orthotr opic symmetry. Thus, with respect to (x,y,z) the symmetry axes, 2332 3311 2112 4 5 61 3 3 3 3 cccc cccc cccc c c c L (2.18) where c1, c2, c3,, and c6 are constants. Note that plastic anisotropy is represented by the same number of coefficients as Hills criterion (2.8). In order to improve the accuracy of Yl d91, Barlat et al. (1997) proposed a new orthotropic criterion (de noted Yld96) of the form: 3121232312mmm mY (2.19) where 1, 2, 3 are functions of the pr incipal directions of and are defined as 2 3 2 2 2 1k z k y k x kp p p (2.20)

PAGE 28

15 where pij are the ith component of the kth principal direction of the tensor with respect to the anisotropy axes of the material. Additionally, x, y, and z are three functions given by 1 2 1 1 2 02 sin 2 cos x x x 2 2 1 2 2 02 sin 2 cos y y y (2.21) 3 2 1 3 2 02 sin 2 cos z z z where 1, 2, and 3 represent the angle between the major principal directions of and the axes of anisotropy, and the quantities x0, x1, y0, y1, z0, z1 are anisotropy coefficients. The yield funtion (2.19) reduces to the yield function (2.16) if each value of is set to unity, and further reduces to th e Hershey and Hosford criterion (2.7) when L is the identity tensor. Barlat et al. (2005) showed that any pressure independe nt isotropic yield function written in terms of the principal values of the Cauchy stress deviator can be generalized to anisotropy through a linear transformati on acting on the Cauchy stress tensor. The principal values of the transformed tensor (2 .22) will then replace the principal values of the Cauchy stress deviator from the isotropic yield criterion. CsCT L (2.22) Here, is the transformed tensor, C is an anisotropic linear tensor, and T transforms the Cauchy stress tensor into its deviator s. Barlat et al. (2005) also recently proposed an orthotropic yield criterion (d enoted Yld2004-18p) (2.23) wh ich involves 18 anisotropy coefficients. This orthotropic yield criter ion is a generalization of the Hershey and Hosford criterion in which anisotropy is introduced through two linear transformations each containing 9 independent coefficients. The expression of this yield criterion is

PAGE 29

16 a a a a a a a a a aY 4" 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 (2.23) where '''' CsCT L (2.24) and """" CsCT L (2.25) When C = C (or L = L), and the number of independent coefficients is imposed to be 6, Yld2004-18p (2.23) reduces to Yld91 (2.16). This criterion represents yield loci of aluminum alloys with increased accuracy. Another approach to extend any isotropi c criterion to anis otropy is through generalized invariants (Cazacu and Barl at, 2001 and 2003) using the theory of representation of tensor functions (e.g. Boehler, 1978 and Liu, 1982). Using the generalized invariants approach, Cazacu and Barlat (2001) extended Druckers isotropic yield criterion to orthotropy as follows: 32 00 23JcJF (2.26) where 2 6 2 5 2 4 2 3 2 2 2 1 0 26 6 6yz xz xy z x z y y xa a a a a a J (2.27) and

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17 z y x yz x y z xy x z y xz z y x y x z x z y z y x z yz xz xy y xb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b J 7 6 7 6 2 5 10 5 10 2 8 9 8 9 2 2 4 3 1 4 2 1 2 4 3 2 2 1 4 1 3 3 2 4 1 11 3 4 3 3 2 1 0 33 2 2 3 2 2 3 9 1 9 1 9 1 9 2 2 27 1 2 27 1 27 1 (2.28) All of the yield criteria discussed, bot h isotropic and anisotropic, make the assumption that yield in tension and compre ssion coincide. This basic assumption makes these yield criteria inadequate for modeling hcp metals. 2.1.3 Modeling Asymmetry Between Tensile and Compressive Yield To describe the asymmetry in yielding due to twinning, Cazacu and Barlat (2004) proposed an isotropic yield criterion of the form: F cJ J f 3 2 3 2 (2.29) where 33 3333 2TC TCc (2.30) T and C being the uniaxial yield stresses in tension and compression, respectively. Note that for equal yield stresses in tension and compression: c = 0 hence the proposed criterion reduces to the von Mi ses yield criterion. For the isotropic yield function (2.29) to be convex, the constant c is limited to a given numerical range: [33/2,33/4] c For any 0 c, the yield function is homogeneous of degree 3 in stresses and equation (2.29) represents a triangle with rounded corn ers. This isotropi c yield criterion was extended to include orthotropy using the generalized invarian ts approach and applied to the description of magnesium and its alloys Having a fixed degree of homogeneity (of

PAGE 31

18 order 3), this criterion is not flexible enough to represent yielding of certain hcp alloys which have nearly elliptical yield loci. A yield criterion capable of capturing both anisotropy and yield asymmetry between tensio n and compression of such materials is needed. 2.2 Hardening Laws The plastic behavior of metal for a multi-axia l state of stress is described by a yield criterion, a flow rule, and a hardening law. The two most common hardening laws are isotropic hardening and kinematic hardening (see Figure 2.2). Figure 2.2 Pictorial descripti on of various hardening rules Isotropic hardening assumes that the yi eld surface maintains its shape, but it expands with accumulated plastic deformation. Generally, the size of the yield locus is given in terms of a scalar hardening variable such as the effective plastic strain in equation (2.31) ()()0pfY (2.31)

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19 where () f is the yield function that de pends on the Cauchy stress, and () p Y is the hardening function that depends on the effective plastic strain, p The effective plastic strain is an invariant of the plastic strain tensor, i.e., 2 pptr where is a constant that is determined such that p reduces to the strain in the loading direction for a uniaxial test. Due to its simplicity, isotropi c hardening is the mo st common method used in sheet metal forming simulations (Yoon et al., 1999, 2000, and 2004). Since the yield loci expand without changing shap e or the location of their cen ter, isotropic hardening is a good approximation for monotonic loading along a certain strain path for materials deforming by slip. This model can not represent phenomena such as the Bauschinger effect or different strain paths hardening at different rates due to deformation twinning. The Baushcinger effect is a common phenomena in metals, and occurs when a material is deformed up to a given plastic strain, then unlo aded and loaded in the reverse direction. The yield strength after the strain reversal is lower than it would have been before the first deformation step. Introduced by Prager in 1955, ki nematic hardening allows for a translation of the yield surface without changing its size or sh ape. Thus, if the initial yield surface ()0 f then for a given plastic state, the yield condition is given by ()0 f (2.32) where (a second order tensor call ed the backstress tensor) de fines the updated center of the yield surface. This model can be used to represent phenomena like the Bauschinger effect due to load reversals. Kinematic hardening can be used in combination with isotropic hardening to describe both expansi on and translation of th e yield surface during

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20 plastic deformation. Models such as those proposed by Teodosiu et al. (1995) and Li et al. (2003) take into account other microstructu ral phenomena associated with changes in strain path, such as the evol ution of dislocation structur e in cubic metals, through the addition of other tensorial hardening variables to the kinematic hardening model. As previously mentioned, twinning produces a major reorientation of the grains. The blockage of further slip or twinning due to grain reorientation re sults in higher work hardening rates for strain paths where twi nning is the dominant mechanism as compared to strain paths that would involve only slip. Therefore, due to the fact that the hardening rate and the evolution of the texture depend on the strain pat h, the evolution of the yield surface for an hcp metal is highly anisotropic. In order to correctly model hcp metals, a hardening rule would need to allow for th e distortion of the yield surface due to the evolving texture with accumula ted plastic deformation. 2.3 Survey of Mesoscale Plasticity Modeling (Polycrystalline Models) Crystalline structure is a key factor in th e mechanical response of a metal since the individual crystals are anisot ropic in both their elastic and plastic behaviors. Plastic deformation mechanisms within a single crystal, such as slip and twinning, are linked to crystallographic planes and dir ections, making the crystal strengt h inherently anisotropic. If a polycrytalline material contains a large number of grains whos e lattice orientations are randomly distributed, the strength of th e polycrystal would exhibit little if any anisotropy. However, as a result of thermo mechanical processes such as cold-rolling, most metals display crystallographic textur e, i.e. a patterned or a non-random lattice orientation. Therefore, the anisotropy of the polycrystal is dir ectly related to the anisotropy of the single crystal and the dist ribution of the crysta l lattice orientation.

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21 Polycrystalline models that derive the polycrystal behavior from that of its constituents using homogenization techniques have been proposed. The first step in constructing such models is defining a repr esentative volume element (rve). This volume element must be small enough to be regarded as having uniform pr operties (including orientation) such that stress and strain di stribution within this volume be treated as homogeneous. For metallic materials, a single cr ystal is considered as an rve. The next step is to give a description of the behavior at the rve level and the interaction laws between each grain and its su rroundings. Polycrystal models typically ne glect elastic strains. The constitutive law of the single crystal consists of a kinematical relationship and an energetic assumption. The kinematical relationship, further described in the next section, relates the velocity gradient with the deformati on rates of all acti ve slip or twin systems within the crystal. There are several versions of the energetic assumption, including those of the Taylor model and th e vpsc model which will be discussed in sections 2.3.2 and 2.3.3, respectively. Assuming that the rate of deformation and stress distribution is known for each single crystal, the rate of deformation and the stress for the macroscopically homogeneous polycrystal is given by a volume average over each crystal. c D D and c (2.33) While equation (2.33) is a straightforward probl em to solve, its inve rse of partitioning a given macroscopic component into the grain-level components, depends on the assumptions made about the interaction of the grain with its su rroundings. The most common assumption is that of Taylor (1938) fo r which each grain in the polycrystal is subject to the same strain rate as that appl ied to the overall polyc rystal, which enforces

PAGE 35

22 compatibility but results in an upper bound approximation of the stress. In Selfconsistent models such as the vpsc mode l the interaction of each grain with its surroundings is based on the geometry of each grain and the average properties of the polycrystal. The self-consiste nt models do not require that the strain distribution is constant within the polycrystal which is a much more accurate assumption than the Taylor assumption. Furthermore, the Taylor model assumes that a fixed number of slip systems are active for a given plastic deform ation, while rate-depe ndent models like the vpsc model do not. Therefore, the vpsc model can better represent materials that possess a limited number of available deform ation systems such as hcp metals. Since polycrystal models can track the latt ice rotation of each individual grain, the material anisotropy is naturally evolutional, which makes this approach very attractive. These models can also be very useful for providing information about yield behavior for stress paths for which no experimental data is available. For example, if the single crystal properties and initial texture of a given material ar e known, the effect of texture evolution caused by plastic deformations on yielding can be studied. Furthermore, the effects that certain mechanisms such as twi nning have on the yield behavior of materials can be explored. In the fo llowing, we present the Tayl or model followed by the vpsc model that will be further used to obtain in formation about the evolution of yield loci. 2.3.1 Kinematics of a Polycrystal Within a single crystal, slip (see Figure 1.1) along a given plane with a unit normal vector n causes a displacement of th e upper half of the crystal with respect to the lower half in a certain direction of a unit vector b (commonly referred to as the burgers vector ) (see Figure 2.3). Deformation twinning is simila r in its kinematic asp ects (Figure 2.3) in that it acts on a given plane in a particular direction. The processes of both slip and

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23 twining are considered to be simple shears not pure shears since th ey correspond to a displacement in the direction of b on one side of a plane perpendicular to n, but do not result in an equal displacement in the direction of n on the plane perpendicular to b. Simple shears involve rotations, which cause the evolution of th e texture during the plastic deformation of a polycrystal. Figure 2.3 Notations for (a) slip and (b) de formation twinning in a single crystal. The velocity gradient for a crystal is gi ven by the sum of the shear rates from each slip or twin system. In the crystal coordina te system which is parallel with a set of orthogonal crystal axes, the velo city gradient is given by csss ijij sLbn (2.34) In (2.34) s is the shear rate for a given slip or twin system (s). Thus, the rate of deformation tensor, Dc, which is the symmetric part of Lc, and the spin tensor, Wc, which is the antisymmetric part of Lc are given by s s ij s c ijm D (2.35) b n b n (a) (b)

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24 s s ij s c ijq W (2.36) where m (commonly referred to as the Schmid tensor) and q are defined as i j j i s ijn b n b m 2 1 (2.37) i j j i s ijn b n b q 2 1 (2.38) When solving a problem incrementally, the incremental strain and incremental rotation for a single crystal become: s s ij s c ijm d d (2.39) s s ij s c ijq d dw (2.40) After each deformation step, the texture of the material is updated by equations (2.39) and (2.40), thus the model is capable of cap turing the effects of evolving texure. 2.3.2 Taylor Model Assuming that plastic deformation only occu rs by slip, the resolved shear stress acting on a slip system (s) due to a general state of st ress acting on a single crystal (c), is given by s cs ijijm (2.41) where sm is defined in equation (2.37). In part icular, for a single cr ystal subjected to uniaxial tension, the tensile stress is = F/A. The force acting along the direction of slip is cosF and the area of the slip plane is cos A (see Figure 2.4). Therefore, the resolved shear stress on a single slip plane is cos cosc s (2.42)

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25 The activation criterion for a given slip system is given by Schmids law which states that slip will occur when the resolved shear stress on the slip system reaches a critical value s cr s (2.43) where s cr is the critical resolved shear stress for the slip system. Figure 2.4 Geometry of a slip sy stem within a single crystal When a polycrystal deforms, the shape change in each crystal must be compatible with that in the neighboring crys tals. In order to satisfy th is requirement, Taylor (1938) assumed that all grains undergo the same shape change as the entire polycrystal. It has been shown that to accommodate the five independent strain components necessary for plastic deformation in a pressure independent material, five independent slip systems are generally required (von Mises, 1928). Tayl or then hypothesized that among all possible combinations of five slip systems capabl e of accommodating the imposed strains, the active combination is the one that would require the minimu m amount of plastic work.

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26 The plastic work per volume (neglecting elastic strains), dW, that is expended by the active slip systems within a single crystal is: cccsss ijijijijcr ssdWddmd (2.44) Implicit in (2.44) is the assumption that the crit ical stress required for slip is the same for all slip systems. Thus, the five active slip systems within the single crystal are the five that give the minimum value of s sd corresponding to the strain increment applied to the crystal (Hosford, 1993). Once the active slip systems in each crysta l have been identified, it is possible to determine the strains, stresses, and latti ce rotations by equations (2.39), (2.42), and (2.41), respectively, and thus update the texture. The assumpti on of uniform strain in all grains irregardless of orientation leads to stress discontinuities at the grain boundaries. However, this model has been used with success at predicting textures for large deformations. Bishop and Hill (1951) later proposed a similar, mathematically equivalent approach. Thus th e polycrystalline model given by equations (2.41) (2.44) is often referred to as the Taylor-Bishop-Hill model. The original formulation did not include twinning, however, Chin et al. (1969) and Hosford (1973) inco rporated twinning in the Taylor-Bishop-Hill model. These au thors assumed that twinning is analogous to slip with the exception that twinning is direct ional, i.e. twinning onl y occurs if a positive shear acting on a given twin syst em reaches a critical value. 2.3.3 Visco-Plastic Self-Consistent (vpsc) Model In the vpsc polycrystal formulation, orig inally proposed by Molinari et al. (1987), the polycrystal is represented as an aggregate of orientations with weights that represent volume fractions chosen to reproduce the initia l texture. Each grain is treated as an

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27 anisotropic, visco-plas tic, ellipsoidal inclusi on embedded in an anisot ropic, visco-plastic, homogeneous effective medium (HEM) with the stress applied at the boundary of the medium. The method is called self-consistent because the overall properties for the HEM are determined from the known properties of th e grains. The followi ng description of the vpsc model is based on the following refere nces: Kocks et al. (2000), Molinari (1997), Tome et al. (2001), Tome and Lebe nsohn (2004a and 2004b), and Asaro (1983). The plastic flow on a slip system (s) within a particular grain is governed by the rate sensitive law 0:n sc s s c mS (2.45) where s is the rate of shearing on the slip system, 0 is a material parameter, Sc is the deviatoric stress, ms is the Schmid tensors for the grain, and n is the strain rate sensitivity. s c is the critical resolved sh ear stress for the slip system and may be represented by a Voce-type hardening law such as s s s s s s c 1 0 1 1 0exp 1 (2.46) where s 0, s 1, s 0, and s 1 are material parameters, and is the accumulated plastic strain for the deformation system. In ra te-dependent formulations such as the vpsc model, all available deformation systems are c onsidered active. However, in practice, for large values of n (1 n), yielding would appear to occur abruptly as s s c in equation (2.45). For s s c the corresponding s is very small when 1n. The plastic strain rate for the grain is given by the sum of the shears contributed by all systems (assuming that elastic deformations are negligible)

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28 s ij s s c ijm D (2.47) which can be combined with (2.45) to give 1 (sec) 00::n n ss scsc ijkl csccc ijijklijklkl sss ss cccmm DmSPS mSmS (2.48) where 1 (sec) 0:n ss sc ijkl c ijkl ss s ccmm P mS (2.49) here Dc and Pc(sec) are the strain rate and the visco-plastic secant compliance for the grain, respectively. Pc(sec) is not a material property of the crystal since it depends on the stress state, except for when n = 1. When the stress is uniform within the grain, (2.48) is exact, however, a linear relation valid in the vicinity of a point 0 S can be obtained through a first order Taylor series expans ion of (2.48) about the point 0 S 000()()|()cc ij cccc ijij c ijD DD S cc S S SSS (2.50) which can be rewritten as (tan)0()cccc ijijklklijDPSD cS (2.51) where the tangent modulus is defined as 0(tan)(sec)|cc ij cc c ijD n S SPP (2.52) and the back extrapolated term 0(sec)(tan)0()(1)cccccnDPPSD (2.53) The secant approximation (2.48) has been proven to be too stiff, giving results close to the upper bound results. The tangent approxima tion (2.51) gives a much more compliant

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29 response. Similarly, at the polycrystal level, the overall strain rate and stress are related through either a secant or a tangent relationship. sec: D PS (2.54a) or, tan0sec0:: n DPSDPSD (2.54b) and the macro-scale polycrystal compliance is a function of the overall stress. Within the HEM, the local response of the medium is al so governed by the macro-scale polycrystal compliance such that sec():() D xPSx (2.55a) or sec0():()n D xPSxD (2.55b) such that x represents the physical coordinate syst em of the polycrystal. Self-consistent models impose equilibrium on th e grain-to-HEM interaction, but not on grain-to-grain interaction. Therefore, solving for equilibri um for a grain whose constitutive response is given by (2.48), embedded in an effective medium (response given by 2.55) leads to the interaction equation ():()ccc D DMSS (2.56) where 1sec()::ceffnMIEEP (2.57) Here, cM is the accommodation tensor and E is the Eshelby tensor which is a function of the overall compliance of the polycrystal an d of the geometry of the ellipsoid that

PAGE 43

30 represents the grain. The parameter effn depends on the interaction between the grains and the HEM, in particular effn is related to the interactions indicated in (2.58). 0Taylor 1Secant Tangent 1EffectiveInteractioneff effn n nn (2.58) When effn= 0, the strain rate in the grain equals the strain rate in the polycrystal, therefore the Taylor interacti on is recovered. The fourth case allows for an effective interaction between the secant and the tangent interactions. In particular, for the simulations used in this disse rtation research, n = 20 while effn= 10. Combining equations (2.48), (2.54a), and (2.56), the stresses in a grain and the stresses for polycrystal are related by :cc S BS (2.59) where (sec)1sec1sec1sec(()::):(()::)cceffeffnnBPIEEPPIEEP (2.60) Now, the condition that the weighted average of stress and strain rate over the grains must equal the corresponding macroscopic ma gnitudes provides an expression from which the macro-scale polycrystal secant comp liance and macro-scale back extrapolated term can be calculated. sec(sec):cc P PB (2.61) 0(sec)0:ccDP D (2.62) where (sec)1sec100(()::):()ceffcn PIEEPDD (2.63)

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31 and refers to the weighted average. Equa tions (2.60) and (2.61) are valid when effn is constant for each grain within the polycry stal, and each grain has the same shape. Equation (2.61) indicates that the polycrystal compliance is given by a weighted average of the single crystal complian ces and the localization tensor, cB. However, cB is a function of sec P (see equation 2.60), therefore, equati on (2.60) represents an implicit equation from which sec P must be obtained iteratively. Twinning is incorporated into the vpsc model by assuming that it is analogous to slip, in that a twin system has a critical resolved shear stress that will activate deformation. However, twinning differs from slip in its directionality, since twinning will only be activated by a positive shear. The fact that twinned regions contribute to the texture of the aggregate, and more importantly, act as effective barriers for further slip and twinning is also taken into account by the vpsc model through latent hardening coefficients coupled with the Voce harden ing law for each deformation system. The critical resolved stress for each system is updated by (2.64) using the latent hardening coefficients ('ssh) which account for the effect of dislo cations caused by other slip or twin systems (s) on the current system (s). ' s s ss s c s ch (2.64) A predominant twin reorientation scheme proposed by Tome et al. (1991) is also incorporated into the model that selectively reorients certain grains affected by twinning. Under this scheme, the shear strain contribute d by each twin system within each grain is tracked, and the sum of all twin systems over all grains associated with a given twin

PAGE 45

32 mode are calculated. Some grains are fully reoriented during an incremental step once a threshold value is accumulate d for a given twin system. Since the overall properties of the polyc rystal are not known a priori, the vpsc model must be solved iteratively as follows: Given: D and a time step t Outer loop: Estimate an initial stress in each grai n by enforcing a Taylor interaction Calculate (sec) c P (2.49) and 0 c D (2.53) for each grain Estimate the sec P and 0 D as the averages of the corresponding grain values Inner loop: Calculate the Eshelby tensor based on the estimated sec P and the current shape of the ellipsoidal grain. Calculate cM (2.57), cB (2.60), and (2.63) Calculate sec P (2.61) and 0 D (2.62) and compare to the estimation from the outer loop. If they are within a tolerance, exit this loop. If not, use this updated sec P and 0 D to restart the loop and iterate unt il the tolerance has been met. End of inner loop Use sec P and 0 D from the inner loop to calculate S (2.54) Calculate cM (2.57) Using (2.59) and (2.48) calculate c D (2.51) Calculate the weighted average stress and strain rate from each grain Calculate the overall stress (2.55) Compare the average stress and strain ra te with the overall stress and imposed strain rate. Compare the previous stress in each grain to the recalculated stress in the grain.

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33 If all three are within a to lerance then exit th e outer loop, if not use the new value of the stress in each grain and return to the beginning of the outer loop End of Outer Loop. Once convergence has been obtained for the st ress and strain rate in each grain, the hardening for each deformation system is updated using the Voce hardening law (2.40), and the grain orientation and grain shape are updated as we ll. The process is then repeated for the next strain increment. The vpsc model can better model low-symmetry materials such as hcp metals that are characterized by a variety of active defo rmation modes present in each grain, nonnegligible twinning activity, and significantly anisotropic single crysta ls (Kocks et al., 2000). Therefore, this model was used in this dissertat ion to determine the yield properties for both isotropic and anisotropic materials.

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34 CHAPTER 3 PROPOSED YIELD CRITERION 3.1 Proposed Isotropic Yield Criterion Twinning and martensitic sh ear are directional deformation mechanisms, and if they occur, yielding will depend on the si gn of the stress (Hosford, 1993). Early polycrystal simulation results by Chin et al. (1969), who analyzed deformation by mixed slip and twining in fcc crystals, predicted a yield stress in uniax ial tension 25% lower than that in uniaxial compression. Hosford and Allen ( 1973) extended the ca lculations to other types of loading. Based on the simula tion results they concluded that yield loci with a strong asymmetry between tension and compression should be expected in any isotropic pressure insensitive material that deforms by twinning or directional slip. An isotropic yield criterion capable of describing strength differentials between tension and compressive yield is proposed of the form F kS S kS S kS Sa a a 3 3 2 2 1 1 (3.1) where iS, i = 1 are the principal values of th e Cauchy stress deviat or. At difference with the yield criterion (2.29), the propos ed yield function (3.1) is a homogeneous function in stresses of degree a, which could range from 1 to Also, in (3.1) k is a material constant, while F gives the size of the yield lo cus and depends on the chosen hardening rule. The physical significance of the material parameter k may be revealed from uniaxial tests. Indeed, according to the proposed criterion (3.1), the ratio of tensile to compressive uniaxial yield stress is given by

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35 121 121 33 21 121 33aa a T aa Ckk kk (3.2a) or 1 122 1 22 22 1 22a a a TC a TC a a a TC a TCk (3.2b) Hence, for a fixed, the parameter k is expressible solely in terms of the ratio TC (see 3.2b). Note that for an y value of the exponent a if k = 0, there is no difference between tension and compression. In particular, for k = 0 and a = 2, the proposed criterion reduces to von Mises yield cr iterion. From (3.2b) follows that for a given exponent a , for the parameter k to be real, TC should belong to 1122aa aa TC (3.3) Specifically, For 121a a TC 10k For 112a a TC 01k As an example, in Figure 3.1 are shown the representation in the plane stress yield loci (3.1) corresponding to a =2 (fixed) and /TC = 2, 1.26, 1.13, and 1 (von Mises), respectively (i.e ., corresponding to k =1; 0.4; 0.2; 0, respectivel y). Note that the highest the ratio between the yield stress in tension and compression, the greater is the departure

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36 from the von Mises ellipse; for the highest admissible value for k the yield function (3.1) represents a triangle wi th rounded corners. Furthermore, TCCTkk (see equation 3.2b). To illustrate this property of the proposed yield function, in Figure 3.2 are represented the plane stress yield loci (3.1) corresponding to C T = 1.13 ( k = 0.2) and C T = 1/1.13 ( k = 0.2). It is clearly seen that a change in the sign of k results in a mirror image of the yield surface. The variation of C T with k is illustrated in Figure 3. 3 for different values of the exponent a If k =1 then 12aa TC, so for a = 1, TC while for a 2 C T If k = -1 then 12aa TC, so for a = 1 there is no difference between tension and compression, while if a then 12TC For any value of the exponent a and for 11k the yield function (3.1) is convex (for the proof, see section 3.1.2). Figure 3.4 shows the represen tations in the deviatoric -plane of the proposed yield loci (3.1) for various values of the coefficient k between 0 and 1 and a = 2 (fixed) along with the von Mises and Tresca yield loci for comparison. As k increases, the ratio /TC is increasing and the yield loci depart drastically from the circular von Mises locus.

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37 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5-1-0.500.511.5 k = 1 k = 0.4 k = 0.2 k = 0 (von Mises) 2/ 1/a = 2 Figure 3.1 Plane stress yield loci for different values of the ratio C T between the yield stress in tension and compressi on, in comparison with the von Mises locus (1 and 2 are the principal values of the Cauchy stress).

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38 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5-1-0.500.511.5 k = 0.2 k = -0.2 2/T 1/Ta = 2 Figure 3.2 Plane stress yi eld loci corresponding to /TC = 1.13 ( k = 0.2) and /TC = 1/1.13 ( k = -0.2).

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39 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 -1-0.500.51Ck a=100 a=5 a=3 a=2 Figure 3.3 The influence of the value of the parameter k on the ratio /TC of the uniaxial yield stress in tension and co mpression, for various values of the exponent a

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40 k=0 (von Mises) k=0.2 k=0.4 k=1 Tresca 3 Figure 3.4 Projection in the deviatoric plane of the yield loci (3.1) for a = 2 and various values of k in comparison to von Mises and Tresca loci.

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41 For combined tension and torsion conditions where the uniaxial te nsile stress is set equal to the shear stress is set equal to and all other stress components are zero, the proposed yield criterion becomes 22 22111 64643aa a aaakkkF (3.4) Figure 3.5 shows the representation in the tension-torsion plane T T / /of the proposed yield loci corresponding to a fixed value of a ( a = 2) and several different values of k Note the clear deviation from both Tresca and Mises criteria for k different from zero. It is also worth noting that the proposed yi eld criterion (3.1) is capable of predicting ratcheting due to shear loading reversal. In deed, inspection of Figur e 3.5 indicates that a loading in pure shear could produce plastic stra ins along the axial direction. In order to predict such a phenomena 0xf where f is defined by equation (3.1), when all stresses equal zero except the shear stress The principal values of the deviator of the Cauchy stress tensor are defined as 3 ) cos( 22 1 1J S 2 212 2cos 33 J S (3.5) 2 312 2cos 33 J S where 1 is the angle satisfying 103 and whose cosine is given by

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42 3 2 3 1 23 cos3 2 J J (3.6) where 2J and 3J are the second and third invariants of the stress deviator (see Malvern, 1969). Therefore, 232 232ii x ixxx f fSJJSJ SJJJ (3.7) but for the case when only one sh ear stress component is non-zero, 02 xJ, so equation (3.7) reduces to the following form 3 3 i x ix f fSJ SJ (3.8) After substitution of (3.1), (3.5), and (3.6), equation (3.8) reduces to the form given by equation (3.9) when all stresses equal ze ro except for one sh ear stress component, 11 66xakkakk f (3.9) Figure 3.6 shows the plot of the variation of x f with k given by equation (3.9). Note that according to the propos ed criterion, pure shear will result in axial plastic strains for any case other than k = 0. The direction of the axial strains is independent of the sign of the applied shear stress, thus the proposed criterion (3.1) can predict a ratcheting effect due to shear loading reversals in the presence of a strength differential between tension and compression.

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43 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 00.20.40.60.81k=0 (von Mises) k=0.2 k=0.4 k=1 Tresca Figure 3.5 Projections in the tension-torsion plane of the pr oposed yield loci (3.1) for various k -values and a = 2 (fixed), in comparison with Tresca and von Mises (k=0, a=2) loci.

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44 -1.5 -1 -0.5 0 0.5 1 1.5 -1-0.500.51 k Figure 3.6 x f vs. k for the case of pure shear (a = 2). x f

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45 3.1.1 Comparison to Polycrystal Simulation Since no data is available on the yield behavi or for an isotropic pressure-insensitive material, the vpsc model was used to calculate the initial yield loci for randomly oriented fcc polycrystals deforming solely by 2 11 111 twinning, bcc polycrystals deforming solely by 1 1 1 112 twinning, and hcp polycrystals deforming solely by tensile twinning 1 1 10 2 1 10 and compressive twinning 3 2 11 2 2 11 Due to the polarity of twinning, this type of simula tion will produce an isotropic yi eld locus that has different yield strengths in tension and compression. The orient ation distribution function describing a random orientation of the crys tallographic texture was constructed by varying the Euler angles which descri be the orientation of each crystal by 2 1, cos , using the notation of Bunge (see Kocks et al., 2000). To demonstrate the predictive capabilities of the proposed isotro pic criterion, we compare the yield loci obtained using the propos ed criterion (3.1) with the isotropic yield loci calculated using the vpsc polycrystal mo del described in the section 2.3.3. The proposed yield condition (3.1) involves 2 parameters: the exponent a and the parameter k, which for a fixed is expressible solely in terms of the /TC ratio (see equation 3.2). The vpsc model predicted a ratio of 0.83 be tween the yield stress in tension and compression for the randomly oriented fcc polycrystal deformi ng only by twinning. Assuming a = 2, we obtain k = -0.3098 for the proposed criterion. Figures 3.7 (a) and (b) show the yield stresses (open circles) obtained using the vps c model and the projection of the yield locus predicted by the proposed criterion (3.1) for a = 2, k = -0.3098 (solid line) for plane stress ( xy = 0) and on the -plane, respectively. It is clearly seen that the

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46 proposed isotropic criterion describes very well the asymmetry in yielding due to activation of twinning. On the same figures are shown the comparison between the yield loci obtained with the VPSC model for ra ndomly oriented bcc polycrystals deforming solely by twinning (solid circles) and the yi eld loci according to the proposed criterion (3.1) with a = 2 and for k = 0.3098 (which correspond to a ra tio between the yield stress in tension and compression of 1.20, which is the reciprocal of the value corresponding to fcc polycrystals). Figures 3.8 (a) and (b) show a comparison between the yield loci obtained using the proposed criterion (for a = 3 and k = -0.0645) with the yield loci for randomly oriented hcp zirconium polycry stals deforming solely by tensile and compressive twinning calculated using the VPSC model. Ag ain, the strength differential effect is very well captured. Figure 3.7 Comparison between the vpsc yield locus for randomly oriented fcc (open circles) and bcc (closed ci rcles) polycrystals deformi ng solely by twinning and the predictions of the proposed criterion: (a) for plane stress ( xy = 0) (b) on the -plane. xx zz yy -1.5 -1 -0.5 0 0.5 1 1.5 -1.5-1-0.500.511.5 yy/ xx/a) b)

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47 Figure 3.8 Comparison between the VPSC yield locus for randomly oriented hcp zirconium polycrystals deforming sole ly by twinning (open rectangles) and the predictions of the proposed cr iterion (3.1): (a) plane stress ( xy = 0) (b) on the -plane. 3.1.2 Yield Surface Derivatives and Convexity The associated flow rule used to obtain the plastic strain increments is given by equation (2.11), but restated here as p f dd Therefore, it is necessary to determine the fi rst derivatives of the yield criterion. The proposed isotropic yield criter ion (3.1) is of the form F kS S kS S kS Sa a a 3 3 2 2 1 1 where for the general 3-dimens ional case, the principal valu es of the deviator of the Cauchy stress tensor can be defined as 3 ) cos( 22 1 1J S 2 212 2cos 33 J S xx zz yy -1.5 -1 -0.5 0 0.5 1 1.5 -1.5-1-0.500.511.5 yy/ xx/a) b)

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48 2 312 2cos 33 J S where 123SSS and 1 is the angle satisfying 103 and whose cosine is given by 3 2 3 1 23 cos3 2 J J where 2J and 3J are the second and third invariants of the stress deviator. Note that for 0 6 and 3 the state of stress corresponds to uniaxial tension, pure shear, and uniaxial compression, respectively. Similar to equation (3.7), the general form of the first derivative of the proposed isot ropic yield criterion is ij k ij ij k k ijJ J S J J J J S S F F 2 2 3 3 2 2 (3.10) where, k S S kS S a S Fk k a k k k 1, 3 sin 22 1J S 222 2sin 33 SJ 322 2sin 33 SJ 2 1 2 2 13 cos 3 1 J J S, 1 2 22 212 cos 333 SJ J 1 2 32 212 cos 333 SJ J 3 sin 1 2 32 5 2 3 2J J J 2 3 2 33 3 sin 6 1 J J ij ijS J 2, where ij kk ij ijS 3

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49 ij kj ik ijJ S S J 2 33 2 Clearly, singularities exist when 0 and 3 in the terms 2J and 3J When using shell elements in finite elem ent models or for calculating the r-value expressions (2.13), plane stress conditions (023 13 33 ) can be assumed. For plane stress, the computation of the first deriva tives is simplified. Now, instead of using relations (3.12) and (3 .13), the following relations may be used to determine the principal values of the deviator of the Cauchy stress tensor 2 1 13 1 3 2 S 1 2 23 1 3 2 S 2 1 33 1 3 1 S where, 2 12 2 22 11 22 11 14 1 2 2 12 2 22 11 22 11 24 1 2 The general form of the first derivative of th e proposed isotropic criterion for plane stress conditions becomes ij l l k k ijS S F F (3.11) for which no singularities exist. In order for a material to be a stable plastic material, a yield surface must be convex. The convexity of a yield surface al so guarantees a unique relationship between the stresses and plastic strain increments assuming an associated flow rule (Malvern,

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50 1969). For the yield function to be convex, its Hessian matrix must be positive semidefinite. Let H be the Hessian matrix, i.e. 2ij ij f H (3.12) where i j =1 and i are the principal stresses. We shall prove that for 1,1k and any integer1a, the proposed yield functi on (3.1) is convex. Isot ropy dictates three fold symmetry of the yield surface, thus it is suffici ent to prove its convexity for stress states in terms of the principa l stresses corresponding to 123 For 10/6 the principal values of the devi ator of the Cauchy stress tensor are 1230,0,0 SSS and 222 11123(1) 4(1)(1)1() 9a aaaaaaa HkSkSS 222 22123(1) (1)(1)1(4) 9a aaaaaaa HkSkSS 222 33123(1) (1)(1)1(4) 9a aaaaaaa HkSkSS 222 12123(1) 2(1)(1)1(2) 9a aaaaaaa HkSkSS 222 13123(1) 2(1)(1)1(2) 9a aaaaaaa HkSkSS 222 23123(1) (1)(1)1(22) 9a aaaaaaa HkSkSS (3.13) Note that 3 10ij jH, for any i =1, 2, or 3. Thus, the determinant of H is zero and its principal values are 1 2 and 3 = 0. Furthermore,

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51 222 121236(1) ()11(1)() 9aa aaaaaa trkSkSS H 22 2222 222 2 2 1212 212 22222 2 1323111 1 9 3131aaa aaaa a aaaaakSkSkSS aa tr kSSkSS HSince 1230,0,0SSS it follows that for 1,1 k and any integer 1a : tr(H) = 12 0 and 2120 tr H, i.e., the Hessian is always positive semidefinite. For the case 1/6/3 222 11123(1) (1)(4)(1)1 9a aaaaaaa HkSSkS 222 22123(1) (1)(4)(1)1 9a aaaaaaa HkSSkS 222 33123(1) (1)()4(1)1 9a aaaaaaa HkSSkS 222 12123(1) 2(1)()(1)1 9a aaaaaaa HkSSkS 222 13123(1) (1)(2)2(1)1 9a aaaaaaa HkSSkS (3.14) 222 23123(1) (1)(2)2(1)1 9a aaaaaaa HkSSkS It follows that 3 10ij jH, for any i =1. Thus, the determinant of H is zero and 222 1236(1) ()1()1(1) 9aa aaaaaa trkSSkS H

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52 22 2222 222 2 2 2212 212 222222 2313111 1 9 3131aaa aaaa aaaaaakSkSkSS aa tr kSSkSS H Since 1230,0,0 SSS it follows that for 1,1 k and any integer 1 a : tr( H ) 0 and 20 tr H. Thus, for 1,1 k and any integer 1a the yield function is convex. 3.2 Extension of the Proposed Isotropic Yield Criterion to Include Orthotropy To describe both the asymmetry between yield in tension and compression and the anisotropy observed in hcp metal sheets, we extend the proposed isotropic criterion (3.1) to orthotropy. For the description of incompressible plastic anisotropy, Cazacu and Barlat (2001, 2003) introduced a general and rigorous method which is based on the theory of representation of tensor functions (s ee equation 2.26). It consists in substituting in the expression of any give n isotropic criterion, the 2nd and 3rd invariants of the stress deviator with generalizations of these invariants compatible with the symmetry group of the material considered. However, with th is approach, convexity is reinforced only numerically. For this reason, a particular case of this general theory, which is based on applying a fourth-order linear transformation operator on the Cauchy stress tensor or its deviator, has received more attention (Sobodka 1969; Barlat et al., 1991; Karafillis and Boyce, 1993; etc.). It is worth noting that by using the linear transformation approach, the convexity of the resulting anisotropi c extension is automatically satisfied (Rockafellar, 1974). Following Barlat et al. (2005), orthotr opy is introduced by means of a linear transformation on the deviator of the Cauchy st ress tensor, i.e., in the expression of the

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53 isotropic criterion (3.1), the principal values of the Cauchy stress deviator are substituted by the principal values of the transformed tensor defined as kl ijkl ijS L (3.15) where L is a 4th order tensor whose coefficients assign a weight to different stress components. Thus, the proposed orth otropic criterion is of the form F k k ka a a 3 3 2 2 1 1 (3.16) where 1,2,3 are the principal values of In the absence of any shear stresses, the values of xx, yy, and zz are the principal values of However, if the shear stresses are present and 03 the principal values of are the roots of the 3rd order algebraic equation 32 1230 XHXHXH (3.17) where, zz yy xxH 1, ) (2 2 2 2xy zx yz yy xx xx zz zz yyH and 2 2 2 32xy zz zx yy yz xx zz yy xx xy zx yzH The tensor L satisfies the major and minor symmetr y and the requirement of invariance with respect to the orthotr opy group. Thus, for 3-D stre ss conditions th e orthotropic criterion involves 9 independent anisotropy coefficients, a nd reduces to the isotropic criterion (3.1) for L equal to the identity tensor. Let ( x y z ) be the reference frame associated with orthotropy. In the case of a sheet, x y, and z represent th e rolling direction, the long tran sverse direction, and the short

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54 transverse direction or the through thickness direction, respectivel y. Relative to the orthotropy axes ( x y ,z), the tensor (in vector form) is represented by 111213 122223 132333 44 55 66 x xxx y yyy zzzz y zyz zxzx x yxyS LLL S LLL S LLL S L S L S L or in terms of the Cauchy stresses 111213 122223 132333 44 55 66211 333 121 333 112 333 1 1 1 x x xx y y yy zz zz y z yz zx zx x y xyLLL LLL LLL L L L Taking into account that S is traceles s, (3.15) can also be written as 12111311 12222322 13332333 44 55 660 0 0 x x xx y y yy zz zz y z yz zx zx x y xyS LLLL S LLLL S LLLL S L S L S L

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55 In the case of a thin sheet, plane stress conditions can be assumed. Using these assumptions, the only non-zero stress co mponents are the in-plane stresses ,, x xyyxy and the principal values of are 2 2 11 4 2 x xyyxxyyxy (3.18) 2 2 21 4 2 x xyyxxyyxy zz 3 where yy xx xxL L L L L L 13 12 11 13 12 113 1 3 2 3 1 3 1 3 1 3 2 (3.19) yy xx yyL L L L L L 23 22 12 23 22 123 1 3 2 3 1 3 1 3 1 3 2 yy xx zzL L L L L L 33 23 13 33 23 133 1 3 2 3 1 3 1 3 1 3 2 xy xyL 66 If 0T and 0C define the yield stress in tens ion and compression along the rolling direction x according to the proposed orthotropi c criterion (3.16) it follows that a a a a Tk k k F1 3 3 2 2 1 1 0 (3.20) a a a a Ck k k F1 3 3 2 2 1 1 0

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56 where 13 12 11 13 1 3 1 3 2 L L L (3.21) 23 22 12 23 1 3 1 3 2 L L L 33 23 13 33 1 3 1 3 2 L L L Similarly, if T90 and C90 are tensile and compressive yi eld stresses in the transverse direction, y then a a a a Tk k k F1 3 3 2 2 1 1 90 (3.22) a a a a Ck k k F1 3 3 2 2 1 1 90 where 13 12 11 13 1 3 2 3 1 L L L (3.23) 23 22 12 23 1 3 2 3 1 L L L 33 23 13 33 1 3 2 3 1 L L L Yielding under pure shear parallel to the orthotropy axes occurs when x y is equal to

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57 a a akL C kL L F1 66 66 66 66 0 (3.24) Yielding under equibiaxia l tension occurs when x x and yy are both equal to a a a T bk k k F3 3 2 2 1 1 (3.25) while yielding under equibiaxia l compression occurs when x x =yy = C b a a a C bk k k F3 3 2 2 1 1 (3.26) where, 13 12 11 13 2 3 1 3 1 L L L (3.27) 23 22 12 23 2 3 1 3 1 L L L 33 23 13 33 2 3 1 3 1 L L L Furthermore, we assume that the plastic pot ential coincides with the yield function. According to the proposed or thotropic criterion, the Lankford coefficients which are defined as the width to thickness strain ratio s in a uniaxial loading (see equation 2.13), become ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) 1 (3 2 3 1 3 2 1 2 1 1 1 1 3 1 3 2 1 2 1 1 1 0 a a a a a a a a a a a a Tk k k k r (3.28) ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) 1 (3 1 3 1 3 1 1 1 2 2 1 2 3 1 3 1 1 1 2 1 2 90 a a a a a a a a a a a a Tk k k k r

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58 ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) 1 (3 2 3 1 3 2 1 2 1 1 1 1 3 1 3 2 1 2 1 1 1 0 a a a a a a a a a a a a Ck k k k r ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) 1 (3 1 3 1 3 1 1 1 2 2 1 2 3 1 3 1 1 1 2 1 2 90 a a a a a a a a a a a a Ck k k k r with 1 to 3 given by (3.7), 1 to 3 given by (3.9), and th e superscripts T and C designating tensile and compre ssive states, respectively. Using equations (3.20) (3.28), the coeffici ents for the yield criterion (3.16) can be determined by minimizing an error function of the form m m data m predicted n n data n predictedr r weight weight Error2 21 1 (3.29) where the index n represents the number of experime ntal yield stresses available, and m represents the number of experi mental r-values. Each term has an assigned weight which can be used to distinguish yield stresses fr om r-values. A better accuracy for the yield stresses is usually required because a differe nce of a few percent in the flow stress is much more significant than in the r-values (Barlat et al., 2005). Although the transformed tensor is not deviatoric, th e proposed orthotropic criterion is nevertheless inde pendent of hydrostatic pressure In order to prove this concept, we shall show that the derivative of the proposed orthotropi c yield function, F, with respect to the hydrostatic pressure, p is zero. The proposed orthotropic yield condition is 123112233(,,)aaafkkk whose derivative with respect to hydrostatic pressure is

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59 ij m mijff p p (3.30) where i are the principal values of the transformed stress tensor We shall prove that ij p =0, hence f p = 0, i.e., the condition of plastic incompressibility is satisfied. Indeed, the transformed stress tensor can be expressed as LSLT (3.31) where T denotes the 4th order deviatoric projec tion that transforms a 2nd order tensor in its deviator. Thus, ij ijklklijkk B B p i,j, k = 1 3, (3.32) where B = L T is the 4th order orthotropic tensor that relates the transformed tensor to the Cauchy stress Relative to ( x y z ), the tensor L is represented by 111213 122223 132333 44 55 66LLL LLL LLL L L L L (3.33) while T is given by 211 121 112 1 3 3 3 3 T

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60 In (3.33) we used the simplified contract ed indices convention of Voigt was adopted (111111121122131133;;defdefdef L LLLLL etc.). It follows that the non-zero components of the 4th order tensor B are 111112132/3 BLLL 121112132/3 BLLL 131112132/3 BLLL 211222232/3 BLLL 221222232/3 BLLL 313231332/3 BLLL 323132332/3 BLLL 333133322/3 BLLL Hence, we obtain 111213 212223 3132330 0 0 BBB BBB BBB (3.34) Thus, ij p =0 and f p = 0 satisfying the condition of plastic incompressibility. 3.2.1 Application of the Proposed Criterion to the Description of Yielding of Hcp Metals 3.2.1.1 Magnesium Alloys Kelley and Hosford (1968) reported the resu lts of an experimental investigation into the anisotropy and asymmetry in yielding of textured polycrystalline pure Mg and

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61 binary Mg-Th (0.5 % Th) and Mg-Li (4% Li) all oys. The data consists of the results of uniaxial compression tests in the rolling, transv erse, and normal directions, respectively, uniaxial tensile tests in the rolling and transv erse directions, as well as plane-strain compression tests. Based on these data, the experimental yield lo ci corresponding to several constant levels (1, 5, and 10%) of th e effective plastic strain were reported (see Figures 3.9, 3.10, and 3.11 where experiment al data are represented by symbols). Due to the mechanical processing of a co ld rolled sheet, magnesium alloy sheets have a strong basal pole alignment in the th ickness direction. Therefore, the easily deformed 1 1 10 2 1 10 tensile twin system about magne siums c-axis is activated by compression in the plane of the sheet. Howeve r, this twin system is not active due to tension within the plane of the sheet. The effect of 1 1 10 2 1 10 twinning is clearly evident by the initially low compressive strengths with respect to tensile strengths at 1% effective plastic strain. By 10% strain, th e third quadrant strengths are comparable to those in the first quadrant owing to the extra barriers to further deformation processes due to the reoriented twins created by lo adings in the third quadrant. Figure 3.8 shows the section of the theoretical plane stress yield loci (equation 3.16) with xy = 0 for Mg-Th together with the expe rimental data reported in Hosford and Kelley (1968). The constant a was set to 2 while the anisot ropy coefficients involved in the expression of the theoretical yield loci for biaxial stress states as well as the constant k were determined using equations (3.20) to (3.29) and the da ta corresponding to the given strain level. The obtained values of thes e parameters corresponding to the 1%, 5%, and 10% effective plastic strain surfaces are given in Table 3.1. Note that the proposed theory reproduces very well the ob served asymmetry in yielding.

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62 The experimental yield loci for the Mg-Li alloy sheets are similar in shape to those for the Mg-Th alloy, but with much reduced yield stresses due to the occurrence of prism slip and to the weaker crystallogr aphic texture. The effect of 1 1 10 2 1 10 twinning is evident in the low compressive strengths at 1% and 5% strains. Figure 3.10 shows the theoretical yield loci for Mg-Li along with th e data reported by Ke lley and Hosford. The constant a was chosen to be 2 for this material. The coefficients involved in the expressions of the biaxial yield loci are given in Table 3.2. The yield locus for the textured pure ma gnesium has a highly asymmetrical shape for the 1% and 5% yield locus due to twinning. Note the much greater strength in tension than in compression and the higher tensile streng th in the transverse direction than in the rolling direction. The yield locus at 5% strain shows asymmetry similar to that of the locus at 1% strain. At 10% strain, the third quadrant strengths are comparable to the first quadrant strengths due to the hardening effects of 1 1 10 2 1 10 twinning. Figure 3.11 shows the yield loci of the pres ent theory with the 5 data po ints for each level of strain given by Kelley and Hosford. The constant a was chosen to be 3 for the 1% and 5% locus due to the asymmetry. This constant was chosen to be 2 for the 10% locus since the yield locus becomes more elliptical. The yield locus generated by the present theory is in good agreement with the published da ta. The coefficients involved in the expressions of the biaxial yiel d loci are given in Table 3.3.

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63 -300 -200 -100 0 100 200 300 -300-200-1000100200300 yy xx1% 5% 10% Figure 3.9 Comparison between th e plane stress yield loci (0xy ) for a Mg-0.5% Th sheet predicted by the proposed theory (solid lines) and experiments (symbols). Data after Kelly and Hosford (1968). Stresses in Mpa Table 3.1 MG-TH yield surface coefficients a k L11 L12 L13 L22 L23 L33 1% 2 0.3539 1.0 0.4802 0.2592 0.9517 0.2071 0.4654 5% 2 0.2763 1.0 0.3750 0.0858 0.9894 0.0659 0.1238 10% 2 0.0598 1.0 0.6336 0.2332 1.4018 0.5614 0.7484

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64 -200 -100 0 100 200 -200-1000100200 yy xx1% 5% 10% Figure 3.10 Comparison between th e plane stress yield loci ( 0xy ) for a Mg-4% Li sheet predicted by the proposed theory (solid lines) and experiments (symbols). Data after Kelly and Hosford (1968). Stresses in MPa Table 3.2 MG-LI yield surface coefficients a k L11 L12 L13 L22 L23 L33 1% 2 0.2026 1.0 0.5871 0.6975 0.9783 0.2840 0.1497 5% 2 0.2982 1.0 0.6103 0.8056 1.0940 0.5745 0.1764 10% 2 0.1763 1.0 0.5324 0.8602 1.0437 0.8404 0.2946

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65 -200 -150 -100 -50 0 50 100 150 200 -200-150-100-50050100150200 yy xx1%5% 10% Figure 3.11 Comparison between th e plane stress yield loci ( 0xy ) for a pure Mg sheet predicted by the proposed theory (solid lines) and experiments (symbols). Data after Kelly and Hosford (1968). Stresses in MPa Table 3.3 MG yield surface coefficients a k L11 L12 L13 L22 L23 L33 1% 3 0.6293 1.0 0.4349 -0.05131.0178 0.1294 0.3417 5% 3 0.4320 1.0 0.4123 -0.01190.8617 0.0570 0.2024 10% 2 0.0776 1.0 0.2807 -0.03380.9916 0.1219 0.6874

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66 3.2.1.2 Titanium Alloys In the following, we apply the proposed orthotropic criterion (3.16) to the description of the anisotropy and tens ion-compression asymmetry of 4Al-1/4 O2 textured (hcp) titanium alloy (data after Lee and Backofen, 1966). True stress-strain curves were reported for different loading paths: uniaxial tension in the x direction (rolling direction), uniaxial compression in the zdirection (through-thic kness compression), and plane strain compression in the z and y (transve rse) directions. The material had nearly ideal basal texture with a devi ation of about 25 degrees from the sheet normal toward the transverse direction. Based on these data, the experimental yield loci corresponding to several constant levels of the largest prin cipal strain were re ported (see Figure 3.12, experimental data are represented by symbols) Due to the strong basal pole alignment in the direction of the normal to the sheet, 1012twinning was activate d by compression perpendicular to this direction, but is no twinning was revealed in tension testing within the plane (see Lee and Backof en, 1966). The effect of 1012twinning is clearly evident in the low compressive strengths in the rolling and transverse directions. Figure 3.4 also shows the theoretical yield loci along with the experimental data. The coefficients involved in the expressions of the theoretical yield loci are given in Table 3.4. Note the ability of the proposed criterion to correctly describe the asymmetry in yielding of 4Al-1/4 O2. In order to account for the eccentricity of the yield surfaces of titanium and its alloys, Hosford (1966) proposed a modification of the Hill criterion to include terms linear in stress 1 ) ( ) ( ) ( ) (2 2 2 y x x z z y z y xH G F A B B A (3.35)

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67 The linear terms in equation (3. 35) allow for the center of the yield surface to shift, thus allowing for different yield strengths in te nsion and compression. However, by adding these terms, the criterion is no longer pressure -independent, nor is it as accurate as using the proposed yield criterion. Hosfords (1966) yield functi on given by Eq. (3.35) was applied to the same 4Al-1/4 O2 textured titanium alloy (see Fi gure 3.13). Comparison between theoretical and experimental yield lo ci show that the pr oposed criterion (3.16) describes with greater accu racy the yield behavior of the titanium alloy. 3.2.2 Derivatives of the Orthotropic Yield Function The proposed orthotropic yield crit erion (3.16) is of the form F k k ka a a 3 3 2 2 1 1 where 1,2,3 are the principal values of the transformed tensor kl ijkl ijS L For a general 3-dimensional problem, it is necessary to develop an expression between the principal values of and its components in order to determine the derivatives of the yield function. This expression can be devel oped through the use of the deviator of denoted by since the cubic equation (3.17) woul d then lack the quadratic term upon substitution of for Once an expression is obtained between the principal values of and its components, the spherical component of can be added to obtain a relationship between the principal values of and its components. Substituting ( 3ijijij I where I is the first invariant of ) into equation (3.17) yields a re lationship whose roots are the principal values of 3 230 XJXJ (3.36) where 2J and 3J are the second and third scalar invariants of Solving for the roots

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68 -1500 -1000 -500 0 500 1000 1500 -1500-1000-500050010001500 yy xx4% 0.2% 1% Figure 3.12 Comparison between th e plane stress yield loci (0xy ) for a 4Al-1/4O2 sheet predicted by the proposed theory (solid lines) and experiments (symbols). Data after Lee and Back ofen (1966). (Stresses in MPa) Table 3.4 4Al-1/4O2 coefficients a k L11 L12 L13 L22 L23 L33 0.2% 2 0.1556 1.0 0.2285 0.0374 1.2967 0.2439 0.3244 1% 2 -0.1868 1.0 0.0431 0.3369 0.9562 0.3139 1.0861 4% 2 -0.2577 1.0 0.2178 0.3635 1.0422 0.3754 0.8825

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69 -1500 -1000 -500 0 500 1000 1500 -1500-1000-500050010001500 0.2% data 1% data 4% data yy xx4% 0.2% 1% Figure 3.13 Comparison between th e plane stress yield loci (0xy ) for a 4Al-1/4 O2 sheet predicted by Hosfords 1966 modified Hill criterion (solid lines) and experiments (symbols). Data after Lee and Backofen (1966) (Stresses in Mpa)

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70 of equation (3.36) and adding the spherical portion of yields an expression for the principal values of 2 112cos() 33 J I 2 212 2cos 333 J I (3.37) 2 312 2cos 333 J I where 123 and 1 is the angle satisfying 103 and whose cosine is given by 3 2 3 1 23 cos3 2 J J (3.38) Now, the first derivatives of the proposed orthotropic yield crit erion (3.16) can be written as 3 22 232 qqq klmn ijqklklklklmnijJ JJ FFIS JJJIS (3.39) where, k k a Fq q a q q q 1, 1 22sin 3 J 2 22 2sin 33 J 3 22 2sin 33 J

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71 1 2 1 2 21 cos 33 J J 1 2 2 2 212 cos 333 J J 1 2 3 2 212 cos 333 J J 1 3qI 3 5 2 2 2331 4sin3 J J J 3 2 3213 6sin3 JJ 2 ij ijJ where 3ijijijI 3 22 3ikkjij ijJ J ij ijI kn nn kkL S 44 23 23L S 55 31 31L S 66 12 12L S all other 0 mn klS 3 2 ii iiS, 3 1 jj iiS for ij 1 ij ijS for ij all other 0 ij mnS. As was the case for the isotropic criterion (3 .1), singularities exis t for the orthotropic criterion when 0 and 3 in the terms 2J and 3J

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72 CHAPTER 4 PROPOSED HARDENING LAW 4.1 Introduction Characterization of a metal s plastic response requires th e specification of a yield function and a flow rule by which subsequent inelastic deformation can be calculated for specified loadings and displacements. Traditio nally, the evolution of the yield surface is described by a combination of isotropic and kinematic hardening laws. Isotropic hardening implies a proportional expansion of the surface, without any changes in shape or position. An isotropic hardening mode l is only truly valid for monotonic loading along a given strain path assuming that every strain path hardens at the same rate. In the case of simulation of sheet forming operations of cubic metals (both fcc and bcc), the assumption of isotropic hardening is r easonably adequate (Yoon et al., 2004). Pure translation of the initial yield surf ace could be described by the classic linear kinematic hardening laws. To better mode l the smooth elastic-plastic transition upon reverse loading, multi-surface models as well as non-linear kinematic hardening models have been proposed. Recently, physically-based models of the evolution of the anisotropic work-hardening of bcc material s (mild steel) under arbi trary strain path changes that involve several tensorial hardening variables have been proposed (e.g. Teodosiu et al., 1995 and Li et al., 2003). It is to be noted that none of these models account for the evolution of texture during work-hardening. Due to non-negligible twinning activity accompanied by grain reor ientation and highly directional grain interactions, the influence of the texture evolution on work-harde ning of hcp materials

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73 cannot be neglected even for the simplest m onotonic loading paths. Existing macro-scale phenomenological plasticity models cannot desc ribe the experimentally observed change in shape of the yield loci with accumulated plastic deformation. A general framework for the description of yielding anisotr opy and its evolution with accumulated deformation for both quasi -static and dynamic loading conditions is offered by polycrystal plasticity. Recently many efforts have been undertaken to incorporate anisotropy due to crystallographic te xture into finite element simulations (e.g. Tome et al., 2001). Direct im plementation of polycrystal vi scoplasticity mode ls into FE codes, where a polycrystalline aggregate is asso ciated with each finite element integration point, has the advantage that it follows the evolution of anisotropy due to texture development. However such finite elem ent calculations are computationally very intensive, thus limiting the applicability of th is approach to problems that do not require a fine spatial resolution. The objective of the present chapter is to propose a macroscopic anisotropic model that can describe the influence of evolving te xture on the plastic re sponse of hexagonal metals. Initial yielding is described by the pr oposed orthotropic yield criterion (3.16) that accounts for both anisotropy and asymmetr y in yielding between tension and compression. Experimental measurements of the crystallographic texture for a given material are used to calculate the flow stre ss in a finite number of loading directions using the vpsc model. Then an interpolation technique is used to construct the evolution of the yield surface. The anisotropy coefficients as well as the size of the elastic domain are considered to be functions of the accumula ted plastic strain. The proposed model was implemented into the implicit FE code ABAQUS (2003) and used to simulate the three-

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74 dimensional deformation of pure zirconium and AZ31B magnesium specimens subjected to various loading conditions. Comparis on between predicted and experimentally measured macroscopic strain fields for pur e zirconium shows that the proposed model describes very well the contributi on of twinning to deformation. 4.2 Elasto-Plastic Problem Some assumptions of rate-i ndependent plasticity theory include a yield function or stress potential, Q which separates elastic and plastic states, QY Equation Chapter 4 Section 4 (4.1) the additive decomposition of the total strain increment into an elastic and plastic part, epddd (4.2) an associated flow rule relating the pl astic strain-rate to the stress potential, pQ (4.3) and Hookes law in incremental form edd C (4.4) where ~ is the scalar effective stress, Y represents the materials hardening, is the scalar plastic multiplier and is equivalent to the effective plastic strain, an overhead dot represents the time derivative, and C is the elastic stiffness tensor. An effective stress must be a homogeneous functi on of degree 1, and be able to reduce to the hardening relationship (i.e., uniaxial tension for a given direction) under that state of stress. For example, the effective stress based upon the proposed isotropic crit erion (3.1) assuming that the hardening relationship, Y, is base d upon uniaxial tension about the x-direction is given by

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75 1 112233(,)a aa a pASkSSkSSkS (4.5) where, a a ak k A13 1 3 1 2 3 2 3 2 1 (4.6) and for the proposed orthotropic criterion (3.16) 1 112233(,)aaa a pBkkk (4.7) where, 1 1122331a aaaB kkk (4.8) and, 13 12 11 13 1 3 1 3 2 L L L (4.9) 23 22 12 23 1 3 1 3 2L L L 33 23 13 33 1 3 1 3 2 L L L. Note that for the case of a = 2, k = 0, A reduces to 2 3 which is the constant associated with the von Mises effective stress. The basic problem in elasto-plasticity is to obtain stresses that fulfill both the Kuhn-Tucker conditions and the consistenc y condition. The Kuhn-Tucker conditions

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76 require that 0 0 Q and 0 Q while the consistency condition requires that the stress state remain on the yield surface during plastic loadings. The usual starting point for the elasto -plasticity problem is the non-linear differential equation epdd C (4.10) whose solution is given by 111n nep nnnnd C (4.11) where n is a counter and stands fo r a certain time step of the deformation process. epC is the elasto-plastic tangent modulus, and depe nds on the updated stat e of the problem. Since epC is unknown for the updated state, an it erative scheme must be applied. The first step in an iterative scheme is to choose a starting point, and in elastoplasticity the usual starting point is to assu me the stress state is purely elastic, i.e., 1 trial nnC (4.12) If this starting stress, commonly referred to as the trial elastic stress, satisfies 0 Q then the trial stress is accepted as th e current stress state. If 0 Q then the stress state must be returned to the yield surface. The stress st ate is returned to the yield surface through a plastic corrector step in which the yield surf ace also expands due to hardening. This method is an implicit and direct method since the resulting equations become implicit in the unknown variables, and the consistency cond ition is directly used to determine the increment of effective plastic strain. This approach is also referred to as a returnmapping method since the increment of the effec tive plastic strain is adjusted such that the stress is returned to the yield surface.

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77 4.3 Proposed Anisotropic Hardening Law The proposed hardening law allows for the ch ange in shape of the yield loci during the deformation process by letting the yield criterion coefficients be functions of the hardening parameter. Thus, the yi eld function would take the form (,)(,)()0pppQY (4.13) where ~ is the effective stress based on the stress potential, is the Cauchy stress tensor, Y is the effective stress-effective plastic stra in relationship in a gi ven direction (e.g. the tensile rolling direction), and p is the effective plastic strain which will be used as the hardening parameter. If 11(,)0trialtrial npnQ, the effective plastic strain increment for global step n+1 must be determined to bring the stress stat e back to the yield surface through a local iterative process (see Simo and Hughes, 1998) Denoting the elastic trial state according to equation (4.12) as iteration 0 i ( i being the local iterat ion counter), the stress increment update for iterations 0i take the effects of the plastic strains into account according to (4.14) using the flow rule (4.3). 1 111 11111 1i iiiii nnnnn n C (4.14) In equation (4.14) C is the stress correction due to the plastic strains, and denotes the variation of the va riable between local iterations i +1 and i i.e., 11 111iii nnn The gradient of the stress potential based upon the unknown updated state (iteration i +1) may be approximated using a Taylor series expansion about the previous state a nd the variation of during the previous iteration (iteration i ) by

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78 1 22 2 11 1 1i i ii n nnp n n (4.15) When =0, the stress state is returned to th e yield surface along a vector normal to the current state, and when =1 the stress state is returned normal to the updated state. The method is fully implicit when =1. After combining equations (4.14) and (4.15), the stress correction can be approximated as 1 22 1111 11111 2 iiiii nnnnn p C (4.16) where all derivatives are evaluated at the curre nt state. The increm ental variation of the consistency parameter, or effective plastic strain, 1 1 i n may be obtained through a Taylor expansion of the yield crite rion about the current state 1111 111111 1 1(,)(,)0i i iiiiii npnnpnnn n p nQQ QQ (4.17) Substituting equation (4.16) into (4.17) and realizing that all derivatives are evaluated for the previous step (step j ) yields 2 111 111111(,)0iiiiii npnnnnn ppY Q H (4.18) thus 11 1 1 2 1(,)ii npn i n i n pppQ Y H (4.19) where,

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79 1 2 1 1 2 i n HC (4.20) The stresses and plastic strains are then updated through and the yield criterion 11 11(,)ii npnQ is checked to within a specified tolerance. If the tolerance has not been met, the plastic corrector step will be repe ated until convergence has been obtained. Once convergence is obtained, the updated stresses and strains are accepted as the current state. Note that if equals zero, then this model reduces to that of isotropic hardening. The consistent tangent modulus relates the current stress increment to the current total strain increment, and is used to pred ict the total strain increment for the next iteration. Taking the derivative of (4.13) yields 0 QQ dQdd (4.21) which leads to the following relationship in incremental form after the substitution of equation (4.16) with 0 since we are using the current step. 0 Y CC (4.22) Using (4.22) to solve for the effective plastic strain increment based on the current state gives currentY C C (4.23)

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80 The combination of (4.23), (4.2) (4.4), and (4.10) yields a relationship between the stress increment and total strain increment from which the consistent tangent modulus can be found. Y CC C C (4.24) Here the term in brackets is the consistent tangent modulus and represents the tensor product. The procedures for implementing th e proposed hardening model using 0 are summarized: 1. Given: ,,npn where n represents the previous time step and is the total strain increment for the current time step. 2. Calculate the trial state (i = 0): 1 trial nn C and 0 1 p npn 3. Check for consistency: If 0 11(,)trial npnQtolerance elastic stress state. Accept the trial stress state as the current state and the to tal strain increment as an elastic strain increment and exit. Else continue 4. Determine the starting values for the iteration (i = 0) a) 00 11 npnYY b) 00 11 npnkk here k represents all yield criterion coefficients c) 0 10 1|pnn pY h d) 0 10 1|pnn pk r e) 00 111(,)trial nnpn

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81 f) 0 110 1 ,|trial nnn k q g) 0 1 1, 0 1| ~ n trial nk nk p 5. Begin Iterati on loop (i = 0N) a) 1 11 1 11111 ii i nn n iiiii nnnnnY hpr qCq b) 11 1111 iiii nnnn Cq c) 11 111 iii p npnn 6. Check for consistency: If 11 11(,)ii npnQtolerance Accept the current state of stress and strain (i = i+1) and go to step 9 then exit. Else continue. 7. Continue with Iteration loop a) 11 11 ii npnYY b) 11 11 ii npnkk c) 1 11 1|i pni n pY h d) 1 11 1|i pni n pk r e) 111 111(,)iii nnpn f) 11 111 1 ,|ii nni n k q g) 11 111 1 ,|ii nni n kp k 8. Go to step 5 9. Calculate the elasto-p lastic tangent modulus a) 11 1 11111 T ii nn ep n iiiii nnnnnhpr CqCq CC qCq

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82 In order to improve the stability of the return-mapping algorithm such as the one just described, Yoon et al. (2004) proposed a mu lti-step procedure to be used if the strain increment is very large. This procedure re quires that the consiste ncy condition be solved in several steps when the initial consistency check yields a value large compared to the initial yield stress. During the first step, th e convergence tolerance would be on the order of the magnitude of the initial consistency check minus the yield stress. Subsequent steps would converge towards progressively smaller magnitudes until the convergence tolerance equals the original tolerance value near zero. Then the state of stress and strain would be accepted as the current stat e, and the iteration loop exited. 4.4 Application for an Isotropic Material Let us use the numerical procedure describe d in the previous section to model the response of an isotropic material obeying an isotropic yield crit erion of the form ,ppY where ,p is the effective stress a ssociated with the proposed isotropic criterion (3.1). The evolutio n of the yield surface is dictated by ()pkk thus the effective stress is dependant upon the effec tive plastic strain. Allowing a variation of the coefficient k with p captures the change in the ratio between the yield in tension and compression with accumulated plastic deform ation. We assume a law of variation of k of the form p kAB for 0.0critical pp (4.25) 0k for critical pp where A and B are material constants. It mean s that above a critical level of p the yield in tension is equal to the yield in compressi on, i.e., no strength diffe rential effects exist

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83 above a critical level of p For illustration purpos es, let us assume that 0.4A 0.4 0.05 B and 0.05critical p with a constant degr ee of homogeneity of 2a For 4 0 0 k) a nearly linear relationship exists between k and /TC when 2a (see Figure 3.3). Thus, equation (4.25) with the assumed constant values will give a nearly linear response between k and /TC Figure 4.1 shows the ev olution of the shape of the yield surface for k varying according to (4.25) with the assumed constant values. A power hardening law will also be used ()()m ppYED (4.26) with assumed values for illustration purposes of E = 650, D = 0.0463, and m = 0.227. Uniaxial compression tests were carried out for k varying according to (4.25), as well as for k held constant at k = 0, k = 0.4 for comparison. Note that the cases for which k is held constant, the proposed hardening law reduces to isotropic hardening. The response for uniaxial compression for k = 0 is identical to that of tensile yield, since for a = 2, k = 0, the yield criterion (3.1) redu ces to von Mises. For the case of k held constant at 0.4, the material has an initial compressi ve yield stress lower than the tensile yield stress, with the hardening rate being the same for both loading paths. In the case when k varies according to (4.25), the material initially yields at the same level of compressive stress as in the case when k is fixed at 0.4, but then hardens at a much higher rate than uniaxial tension until p = 0.05 when k becomes 0, and the yield stress in tension and compression become equal. These results are shown in Figure 4.2.

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84 -500-2500250500 -500 -250 0 250 500 a = 2k = 0.4 k = 0 Figure 4.1 Evolution of the yield surface for varying k

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85 250 300 350 400 00.010.020.030.040.050.060.07 k = 0 k = 0.4 k varies from 0.4 to 0StressPlastic Strain Figure 4.2 Results of single element compression tests for a = 2.

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86 4.5 Alternate Method for Anisotropic Ha rdening Implementation Interpolation By applying a linear transformation to exte nd the isotropic yield criterion (3.1) to orthotropy, 9 additional coefficients are added. It is not trivial to determine analytical expressions for all of the coefficients in te rms of the hardening variable. The available experimental data provides information about the shape of the yield surfaces corresponding to different given levels of e ffective plastic strain based on results of monotonic loading tests (see Kelley and Hosford, 1968 and Lee and Backofen, 1966). However, even if the expressions of Lij, k and a for a given level of strain can be determined based on the data, establishing Lij( p ) requires a large amount of data. Therefore, an alternative approach to the hardening law is proposed. From experimental and/or numerical result s from polycrystal calculations, we can identify the coefficients involved in the pr oposed orthotropic yield criterion (3.16) for a set of values of equivalent plastic strain, say 1 p < 2 p << m p and calculate the effective stress {,(),k(),()}jjjj pppa L, as well as jj p YY() corresponding to each of the individual levels of effective plastic strain j p j = 1m. Then, an interpolation procedure can be used to obtain the yield su rfaces corresponding to any given level of accumulated strain. Thus, for a given arbitrary p the anisotropic yi eld function is of the form p ppQ(,)(,)() (4.27) with jj1 pp()(1()) (4.28) and

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87 jj1 pp()Y(1())Y (4.29) for any jj1 ppp j = 1m-1. For linear interpolation, the weighting parameter p () appearing in equations (4.27) and (4.28) is defined as j1 p p p j1j p p() (4.30) such that j p()1 and j1 p()0 By considering that the anisotropy coefficients Lij, the strength differential parameter k and the homogeneity parameter a evolve with the plastic deformation, the observed distortion and change in shape of the yield loci of hcp materials could be captured. Obviously, if these coefficients are taken constant, the proposed hardening law reduces to the classic isotropic hardening law. The derivatives for and needed for stress integration become, 1()(1())jj pp (4.31) 1 1 jj jj p pp (4.32) 1 1 jj jj p ppYY (4.33) 2221 222()(1())jj pp (4.34) and, 1 2 1 jj jj ppp (4.35)

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88 The values of , p p 22 and 2p will replace the values of ~, Y, p and p Y 22 and 2p respectively, in the stress integration algorithm. To illustrate how the yield surface will evol ve using the interpolation method, this method will be applied to the yield surfaces fo r the cold rolled plate of the magnesium alloy mg-th displayed in Figure 3.1. Using the three discrete yield loci corresponding to 1%, 5%, and 10% levels of effective plasti c strain (represented by solid lines) as 1 p 2 p and 3 p respectively, intermediate yield surf aces (represented by dashed lines) for 1 p < p < 2 p and 2 p < p < 3 p can be calculated according to equation (4.27) by varying the interpolation parameter p () (see equation 4.30). The results are plotted in Figure 4.3. The procedures to implement the altern ate interpolation me thod of the proposed hardening model assuming 0 are as follows: 1. Given: ,,npn where n represents the previous time step and is the total strain increment for the current time step. 2. Calculate the trial state (i = 0): 1trial nn C and 0 1 p npn 3. Check for consistency a) Determine which levels of effectiv e plastic strain the current state lies within 01 1 jj p pnp i.e., determine the appropriate j and j+1 (Caution, j and j+1 represent discrete levels of the yield criterion, not to be confused with i and i+1 which represent time steps) b) 10 1 0 1 1 j ppn n jj pp c) If 0 11(,)trial npnQtolerance elastic stress state. Accept the trial stress state as the current state and the total strain increment as an elastic strain increment and exit. Else continue

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89 -300 -200 -100 0 100 200 300 -300-200-1000100200300 yy xx 1% 5% 10% Figure 4.3 Yield surface evolut ion for a cold rolled sheet of mg-th using the interpolation method. Solid lines represent calculated yield loci from ex perimental data. Dashed lines represent yield loci determined by interpolation.

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90 4. Determine the starting values for the iteration (i = 0) a) 0001 111()(1)()jj nnpnpYY b) 1 0 1 1()()jj p p n jj pppYY h c) Determine the appropriate ()j p L ()j p k and ()j p a for j and j+1 d) 0, 11{,(),(),()}jtrialjjj nnpppka L e) 0,1111 11{,(),(),()}jtrialjjj nnpppka L f) 000,00,1 11111(1)jj nnnnn g) 10, 1{,(),(),()} |trial njjj ppp j nka L t h) 1111 0,1 1{,(),(),()} |trial njjj ppp j nka L t i) 000,00,1 11111(1)jj nnnnn qtt j) 0,10, 0 11 1 1 jj nn n jj pppv 5. Begin Itera tion loop (i = 0N) a) 1 11 1 1111 ii i nn n iiii nnnnhv qCq b) 11 1111 iiii nnnn Cq c) 11 111 iii p npnn 6. Check for consistency: If 11 11(,)ii npnQtolerance Accept the current state of stress and strain (i = i+1) and go to step 9 then exit. Else continue. 7. Continue with Iteration loop a) Determine which levels of effectiv e plastic strain the current state lies within 11 1 jij p pnp i.e., determine the appropriate j and j+1. (Caution, j and j+1 represent discrete levels of the yield criterion, not to be confused with i and i+1 which represent time steps)

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91 b) 11 1 1 1 1 ji ppn i n jj pp c) 1111 111()(1)()iijij nnpnpYY d) 1 1 1 1()()jj p p i n jj pppYY h e) Determine the appropriate ()j p L ()j p k and ()j p a for j and j+1 f) 1,1 11{,(),(),()}ijijjj nnpppka L g) 1,11111 11{,(),(),()}ijijjj nnpppka L h) 1 1 1 0 1 1 1 0 1 1 1~ ) 1 ( ~ j i n n j i n n i n i) 1 11, 1{,(),(),()} |i njjj ppp ij nka L t j) 1 1111 1,1 1{,(),(),()} |i njjj ppp ij nka L t k) 111,11,1 11111(1)iiijiij nnnnn qtt l) 1,11, 1 11 1 1 ijij i nn n jj pppv 8. Go to step 5 9. Calculate the elasto -plastic tangent modulus a) 11 1 11111 T ii nn ep n iiiii nnnnnqq hpr CC CC qCq The interpolation approach was also impl emented for the isotropic yield criterion (3.1) in order to compare the results directly with the continuous approach described in the previous section. Three di screte surfaces were used to represent the evolution of the yield surface from 00.05p in particular 1 p = 0, k = 0.4; 2 p = 0.025, k = 0.2; and 3 p = 0.05, k = 0 while the homogeneity coefficient was set equal to 2. For p > 0.05 the surface was allowed to harden isotropically wi th k = 0, a = 2. These yield loci, along

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92 with the interpolation of the yield stresses corresponding to th e three levels of effective plastic strain from (4.26) are shown in Figure 4.4. The results of a single element compression simulation using the interpolati on method are compared with the results from the continuous method in Figure 4.5. The results are very close since the linear interpolation scheme is compared to a linear law of variation for yi eld surface coefficient k with a nearly linear hardening law. Clearly, if the law of variation for k was not linear, the accuracy of a linear interp olation scheme would increase as more interpolation points are used. Figure 4.4 Discrete yield lo ci and yield stresses used fo r the interpolation method. 320 330 340 350 360 370 380 390 00.010.020.030.040.05 Continuous Yield Stress Linear Interpolation of Yield StressUniaxial Tensile Yield StressPlastic Strain -500 -250 0 250 500 -500-2500250500 k = 0.4 k = 0.2 k = 0

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93 250 300 350 400 00.010.020.030.040.050.060.07 Interpolation Method Continuous MethodStressPlastic Strain Figure 4.5 Results of the interpolation met hod compared to the continuous method from a single element compression test

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94 4.6 Application to Zirconium Kaschner et al. (2000), Kasc hner et al. (2001), and To m et al. (2001) have conducted experimental studies on the anis otropy of the deformation of textured polycrystalline pure zirconium. This material is highly anisotropi c both at the single crystal and polycrystal level. It was pro cessed through a series of clock-rolling and annealing cycles to produce a plate with str ong basal texture (-a xes of the crystals predominantly oriented along the plate normal di rection). The process of multiple rolling passes while rotating the plate between passe s was used in order to obtain a nearly isotropic in plane texture. Right-circular cylindrical test specimens were sectioned from both the through-thickness and in-plane plate directions. Quasi-static (3110s ) compression tests were conducted on these sample s, while quasi-static tension tests were conducted only in the in-plane direction. Wh ile the experiments repor ted in Kaschner et al. (2001) and Tom et al. (2001) correspond to two differe nt temperatures (room and liquid nitrogen), in what follows we will use only the room temperature results to validate our approach. The tests have shown that th e mechanical response is strongly dependent on the predominant orientation of the -axes with respect to the loading direction. To obtain the yield stresses for through-th ickness tension and pure shear load paths for the identification of the anisotropy coeffici ents involved in the proposed orthotropic yield criterion, numerical tests with the vps c polycrystal model were performed using the reported initial texture, defo rmation systems operational at room temperature (i.e., prismatic -slip, pyramidal -slip, and tensile twinning), and the values for the crystallographic coefficients for the single crys tal, see Tom et al. (2001). Voce hardening parameters (see equation 2.46) we re adjusted for each active deformation

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95 system for a strain rate of 3110s until the predicted loading from the vpsc model reproduced the experimental response of the cl ock-rolled plate of zirconium from Tome et al. (2001). The set of parameters that gave the best fit to the data are given in Table 4.1. A comparison between the calculate d response using the vpsc model and the experimental data for uniaxial tension and compressi on in the plane of the sheet and for through-thickness compression is presented in Figure 4.6. Table 4.1 Voce hardening parameters for zirconium for a strain rate of 2110 s 0 1 0 1 prismatic slip 5 30 1500 50 pyramidal slip 60 280 3500 45 tensile twin 50 0 100 100 Using the yield stress data from mechanical tests and the results of the numerical tests, the yield surfaces for five different levels of accumulated plastic strain: 1 p =0.002, 2 p =0.01, 3 p =0.05, 4 p = 0.1 and 5 p = 0.15 were identified. The numerical values for the coefficients for each level of accumulated plas tic strain are given in the Table 4.2. For each level of plastic strain the coefficient L11 was equal to 1.0, therefore, L11 was not listed in Table 4.2. Next, fo r each individual strain level j p j =1, jj p YY() was calculated using the experimental in -plane tension loading curve, and jjjj ppp{,(),(),()}ka L Finally, the yield surface co rresponding to any given level of accumulated plastic deformation (between 0 and 0.15) can be obtained using the interpolation technique described in secti on 4.5. Figure 4.7 shows the biaxial plane xxyy, projections of the five individual yiel d surfaces (solid lines) using equation

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96 (3.16), and of several yield loci (dashed lin es) obtained using the proposed interpolation technique (see equations (4.27)(4.30)). 0 100 200 300 400 500 600 00.050.10.150.20.25 TTC IPC IPTStress (MPa)Strain Figure 4.6 Stress-strain res ponse for a clock-rolled plate of zirconium for in-plane compression (IPC), in-plane tens ion (IPT), and through-thickness compression (TTC). Solid lines repr esent vpsc calculations. Symbols represent experimental data (after Tome et al. 2001)

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97 -1000 -800 -600 -400 -200 0 200 400 600 -1000-800-600-400-2000200400600 yy xx 0.2% 1% 5% 10% 15% Figure 4.7 Yield surface evolu tion for a clock-rolled plate of zirconium. Predicted yield surfaces using the proposed criteri on are represented by solid lines. Interpolated intermediate yield surfaces are represented by dashed lines. Vpsc calculations represented by symbols. Table 4.2 Zirconium coefficients correspondi ng to the yield surface evolution depicted in Figure 4.7 (L11 = 1.0 for each case) a k L12 L13 L22 L23 L33 L44 L55 L66 0.2% 2 -0.0017 3.7403 2.1468 0.9926 2.0845 0.5393 1.2958 1.4085 5.1832 1% 2 0.2756 2.7390 2.0252 0.9413 1.9900 1.2506 0.4906 0.5560 1.9010 5% 2 -0.1621 3.2358 1.6811 0.8142 1.5974 1.3113 0.6567 0.6290 2.7404 10% 2 -0.1659 3.2749 1.6275 0.6188 1.5191 1.3190 0.7695 0.7021 3.0585 15% 2 -0.1828 3.1351 1.6353 0.72011.5212 1.1806 0.7549 0.6858 2.9195

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98 Calculations were carried out using the above interpolat ion model to simulate the response of zirconium at room temperature, fo r the cases of in-plane tension, in-plane compression, and through-thickness compressi on. For comparison purposes, we have also performed simulations for the same material assuming fixed values of the anisotropy coefficients (these values correspond to yi eld stress data at 0.002 equivalent plastic strain) which is equivalent to isotropic ha rdening. Figure 4.8 shows the stress-strain curves obtained using the proposed model according to the yield surface evolution depicted by Figure 4.7 (solid lines), togeth er with those obtained by means of the proposed orthotropic yield criterion but assumi ng isotropic hardening (dashed lines), and the data from mechanical tests (symbols). Note that the proposed model captures well the experimental trends. Obvi ously, since isotropic hardeni ng implies that the material hardens at the same rate in every testing direction, it cannot adequately describe deformation that involves the activation of deformation mechanisms different from the ones operational during in-plane tension (i.e., the test used to adjust the values of () j j p YY ). The proposed model will be used to simulate a series of four-point bending tests at room temperature reported in Kaschner et al (2001). The experiments were carried out on rectangular bars of square section cut from the same clock-rolled zirconium. Before loading, the beams were aligned in one of th e two possible orientatio ns with respect to the main texture component: with the main texture component contained in the bending plane, i.e., -axes mostly aligned with the z-axis of the beam (case C0), and perpendicular to it, i.e., -axes mostly ali gned with the x-axis of the beam (case C90) (see also Figure 4.9 for a schematic of the test). The initial dimensions of the beams were

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99 Figure 4.8 Comparison between experimental data (solid rectangles) and simulation results using the proposed model coupled with VPSC (solid lines) and using an isotropic hardening law (dashed lines) for a clock-rolled zirconium plate. Data after Tome et al. (2001). 6.35 x 6.35 x 50.8 mm. The beams were bent as the upper dowel pins were displaced downwards by 6mm and the lower pins were held rigid. Special experimental techniques were developed to map and measure the lo cal strain field. Detailed information concerning the variation of each strain compone nt as a function of the location along the width of the specimen were re ported. Also, a detailed finite element analysis of the bending tests using the explicit finite elemen t code EPIC coupled with the vpsc model was performed in Kaschner et al. (2001) assu ming the presence of a polycrystal at each integration point. For a detailed descripti on of the linkage between the finite element code and the polycrystal mode l, see Tome et al. (2001). ABAQUS finite element simulations of the zi rconium bent beam tests using the proposed model were performed. Due to the symmetry of the problem, only half of the beam was analyzed using 2916 three-dimensional lin ear brick elements. Free-surface boundary 30 60 90 120 150 180 210 00.050.10.15 In-Plane CompressionStress (Mpa)Plastic Strain 0 100 200 300 400 500 600 00.050.10.15 Through-Thickness CompressionStress (Mpa)Plastic Strain 30 60 90 120 150 180 00.050.10.15Stress (Mpa)Plastic Strain In-Plane Tension

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100 Figure 4.9 Schematic of the four-point bend test. conditions were imposed on the beam except at the nodes that coinci de with the contact points of the dowel pins. The results from these simulations, along with the experimental data, and the VPSC/EPIC predictions reported in Kaschner et al. (2001) are shown in Figures 4.10 (case C0) and 4.11 (case C90). Inspection of Figure 4.10 (case C90) reveals that the simulation results using either VPSC/EPIC or the proposed model are reasonabl y close to the experimental data. Both models capture very well the rigidity of the beam response along the hard-to-deform axes preferential orientation, which in this case is parallel to the z-axis. Also, both models capture the asymmetry between tensio n and compression (i.e., the differences in yield values and hardening rates) and thus co rrectly predict an upward shift of the neutral plane. The deformation along the beam axis is better predicted by the proposed model, since the VPSC/EPIC model unde r predicts the deformation in the lower half of the beam. Also, the proposed model gives a more accurate prediction for the final location of the neutral plane. For the case when the ma jor texture component is aligned with the xaxis of the beam (C90, Figure 4.11), the VPSC/EPIC overpredicts the strains for the Y X Z

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101 upper half of the beam. Also, the predicted VPSC/EPIC neutral plane remains at the center of the beam. The proposed model, on the other hand, pred icts more accurate results for this case, including the experimentally observed upward shift of the neutral plane. Figures 4.12 and 4.13 present the calculate d strain distributi ons using the yield surface evolving according to isotropic hardening adjusted to the in-plane tensile loading response of the original zircon ium plate for the cases C0 and C90, respectively. Since an isotropic hardening law cannot capture the di fference in hardening rates between tension and compression, the simulations are less accu rate than those obtained when the yield surface evolves according to the propos ed anisotropic hardening model. Figure 4.14 shows the final configurations for the photographed experimental x-z cross-section of the bent beams superimposed to the predictions obtained with VPSC/EPIC (white dots), as reported in Kasc hner et al. (2001). The figure also shows the calculated x-z cross-sections of the be nt beams obtained using the proposed model (Figures 4.14a and 4.14b). Note that both m odels predict wedged cross-sections only for the case C0 (Figs. 4.14a and 4.14c) when th e hard-to-deform -axes are predominantly parallel to the z-axis. For the case C90 (Figures 4.14b and 4.14d) when -axes is aligned with the x-axis, both mode ls describe correctly the rigidity in the hard direction, and the final cross-section remains rectangular. In Figure 4.7, the yield surface for the zirc onium plate associated with a given level of plastic deformation is represented by yi eld stresses corresponding to uniaxial tests conducted for different loading directions. This yield surface representation is capable of accurately predicting whether or not a given state of stress will produce plastic

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102 deformation, however, it does no t guarantee that th e r-values correspond ing to that given state of stress will be correctly predicted. Initially, the r-values for the zirconium plate due to in-plane loadings are very high (>>1 ) indicated by the steep slope of the yield surface. Due to the initial hardness of the through thickness direction as compared to the in plane directions, and to the high hardening rate observed in th e through thickness direction, the very high predicted r-values remain for all in-plane loading directions at each level of plastic strain according to Figur e 4.7. However, twinning acts to randomize the materials texture. This effect becomes significant for in plane compression at higher levels of accumulated plastic strain (greater than 15%), and conseq uently the r-values begin to decrease with accumulated deformation. Therefore, while the yield surface evolution according to the method shown in Figure 4.7 is a good approximation for low to moderated levels of plastic strain for a ny given loading directi on, the true yield surface corresponding to the updated texture of the material needs to be used to accurately simulate deformations involving tw inning at higher levels of deformation for the zirconium plate. For example, in order to determine the t rue yield surface for an evolved texture due to plastic deformation along a given stra in path, the material must first be prestrained along that strain path. Then, specime ns cut from the pre-strained material should be tested to determine the yield strength unde r different loading c onditions to construct the updated yield surface. After repeating this procedure for di fferent levels of pre-strain, the evolution of the yield surface is determined for the given strain path. This type of experimental characterization would take numerous experiments, which may not even be possible in compression for high levels of pre-strain due to buckling.

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103 0 1 2 3 4 5 6 -0.2-0.15-0.1-0.0500.050.10.150.2 Eyy data Ezz data VPSC / EPIC Proposed modelBeam Height (mm)Plastic Strain Eyy Ezz Exx Figure 4.10 Comparison of the experimentally measured strain distributions (symbols) with the results of finite element simulations using the proposed model (solid lines) and VPSC linked direc tly to EPIC (dashed lines ) for the case C0 (i.e., when the -axes are predominantly c ontained in the bend ing plane). Data and VPSC/EPIC simulations results reported in Kaschner et al. (2001).

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104 0 1 2 3 4 5 6 -0.2-0.15-0.1-0.0500.050.10.150.2 Eyy data Ezz data VPSC / EPIC Proposed modelBeam Height (mm)Plastic Strain Eyy Ezz Exx Figure 4.11 Comparison of the experimentally measured strain distributions (symbols) with the results of finite element simulations using the proposed model (solid lines) and VPSC linked directly to EPIC (dashed lines) for the case C90 (i.e., when the -axes are predominantly perpendicular to the bending plane). Data and VPSC/EPIC simulations results reported in Kaschner et al. (2001).

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105 0 1 2 3 4 5 6 -0.2-0.15-0.1-0.0500.050.10.150.2 Eyy data Ezz data Proposed model Isotropic hardeningBeam Height (mm)Plastic Strain Eyy Ezz Exx Figure 4.12 Comparison of the experimental ly measured plastic strains distributions (symbols) and the ABAQUS finite elem ent predictions using the proposed model (solid lines) and the proposed yiel d criterion with isotropic hardening (dashed lines) for the C0 case. Da ta after Kaschner et al. (2001).

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106 0 1 2 3 4 5 6 -0.2-0.15-0.1-0.0500.050.10.150.2 Eyy data Ezz data Proposed model Isotropic hardeningBeam Height (mm)Plastic Strain Eyy Ezz Exx Figure 4.13 Comparison of the experimental plastic strains distributions (symbols) and the ABAQUS finite element predicti ons using the proposed model (solid lines) and the proposed yield criterion w ith isotropic harden ing (dashed lines) for the C90 case. Data after Kaschner et al. (2001).

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107 Figure 4.14 Comparison of experimentally photographed x-z cross-section of the bent bars versus the predictions of VPSC /EPIC (white dots) and the proposed model (Exx contours); (a) and (c) corres pond to the case C0 (-axes mostly parallel to the z-axis) while (b) and (d ) correspond to the case C90 (-axes mostly aligned with the x-axis of the beam, respectively). The orientation of the basal poles is indicated by the arrows (data and VPSC/EPIC simulations after Kaschner et al., 2001). (a) (b) (c) (d)

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108 Alternatively, the vpsc model can be us ed to determine such a yield surface evolution. Specifically, using the vpsc mode l, the evolving yield function for in plane compression was determined by pre-strainin g the polycrystal to a given level of deformation for in plane compression then numerically probing the polycrystal along different loading directions. In Figure 4. 15, the yield points obt ained with the VPSC model are represented by symbols. For each prestraining level, these data points are further used to determine the coefficients involved in the proposed orthotropic yield criterion. The values of the coefficients fo r each surface are listed in Table 4.3. Then, using the interpolation procedur e described in Section 4.5, the anisotropic yield function (4.27) associated with the e volution of texture during in pl ane compression is determined. This procedure was repeated for through thickness compression as illustrated in Figure 4.16 and Table 4.4. This yield function is in turn used in ABAQUS simu lations of the in-plane compression of right cylinders of circular cros s section to 28% strain, and compared with the experimental results repor ted in Tome et al. (2001) (s ee Figure 4.17). The cylinders were initially 17.7 mm long and 2.25 mm in diam eter. One was machined with its axis parallel to the plane of the plate, and the other with its axis parallel to the through thickness direction of the sheet. For the in-plane compression cylinder, the measured post-test in-plane expansion was of 22% while the out-of-plane expansion was of 6% and the model prediction using the yield surface evolution according to Figure 4.15 agrees exactly with the experimental results. Fi gure 4.17a shows the final FE mesh. For comparison, the photograph of the IP compre ssion sample cross-sect ion, along with the VPSC/EPIC simulation results (dashed line) are shown in Figure 4.17b (after Tome et al.,

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109 2001). The comparison demonstrates that a ve ry good agreement with the experiments is obtained when using an appropr iate yield surface representa tion and anisotropic evolution laws that take into account geometrical and m echanical hardening. On the other hand, if the proposed orthotropic yield surface is used but its evolution is adjusted using an isotropic hardening model, the predicted expa nsions were 27% and 1% for the in-plane and out-of-plane directions, respectively, i. e., the ovalization is clearly over predicted using isotropic hardening. Since the ovali zation of the cylinder due to a compressive loading is determined by the r-values, the ovalization is similarly over predicted using the method to determine the yield surface evolu tion according to Figure 4.7 due to the very high r-values. In through-thickness compression, compression takes place along the axis of symmetry of the sample, which remains cylindrical in section. Since twinning due to compression along the -axis was not reported as a significant mode of deformation at room temperature (Tome et al., 2001), the materials texture does not significantly change due to compression along the thickness dir ection of the plate. Thus, there is not a considerable change in the r-values due to deformation for this loading direction. Therefore, using the yield surface evolution according to Figure 4.16 or Figure 4.7 will correctly predict the final circular cross se ction experimentally observed (see Figures 4.17c-4.17d).

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110 -800 -400 0 400 800 -800-4000400800 yy xx 0.2% 1% 5% 35% 25% Figure 4.15 Yield surface evolution for a zirc onium clock-rolled plate during in-plane compression. Symbols represent yield points calculated us ing the vpsc code; solid lines represent yield surfaces using the proposed yield criterion; dashed lines represent the yield surface evolu tion using the interpolation method. Table 4.3 Zirconium coefficients corres ponding to the yield surface for in-plane compression a k L12 L13 L22 L23 L33 L44 L55 L66 0.2% 2 0.0706 2.8831 1.8747 1.2517 1.9649 1.2040 0.9890 1.0750 3.9560 1% 2 0.2242 2.1498 1.4068 1.1789 1.4383 0.8922 0.6041 0.6167 1.6914 5% 2 0.5001 2.9227 1.8231 0.8649 1.7452 0.4181 0.9844 1.1450 2.9802 25% 2 0.2632 2.7041 1.7271 1.8211 2.0558 0.8686 0.9493 0.7906 1.4031 35% 2 0.1738 2.5052 1.6829 2.3196 2.0535 1.0118 0.8888 0.6942 1.0977

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111 -1000 -500 0 500 1000 -1000-50005001000 yy xx 0.2% 1% 5% 35% 25% Figure 4.16 Yield surface evolution for a zi rconium clock-rolled plate during through thickness compression. Symbols represen t yield points calculated using the vpsc code; solid lines represent yiel d surfaces using the proposed yield criterion; dashed lines represent the yield surface evolution using the interpolation method. Table 4.4 Zirconium coefficients corres ponding to the yield surface for in-plane compression a k L12 L13 L22 L23 L33 L44 L55 L66 0.2% 2 0.0706 2.8831 1.8747 1.2517 1.9649 1.2040 0.9890 1.0750 3.9560 1% 2 0.2370 2.9643 1.5555 0.8027 1.4756 0.4332 1.1522 1.1726 3.3126 5% 2 0.4062 2.6516 1.6192 0.8131 1.5413 0.5825 0.9879 1.1270 2.8578 25% 2 0.1983 3.3478 1.5745 1.0389 1.5247 0.6025 1.8829 2.0812 3.3910 35% 2 0.0839 3.2715 1.5235 1.1200 1.4708 0.6405 2.1223 2.3116 3.8467

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112 (a) (b) (c) (d) Figure 4.17 Comparison of the final sections of the zirconium cylinders after: a,b) inplane compression, c,d) through-thic kness compression. The photographs of the experimental shapes and simulated final shapes using VPSC/EPIC (dotted lines superimposed on the photographs) ar e taken from Tome et al. (2001).

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113 4.7 Application to Magnesium Alloys 4.7.1 Application To Mg-Th The proposed hardening law and the propos ed orthotropic yield criterion (3.16) were used to model the response of a co ld-rolled sheet of magnesium alloyed with thorium (see Figure 4.3). Uniaxial tension, uniaxial compression, and balanced biaxial tension simulations were carried out using the proposed method for the cold rolled sheet of Mg-Th. The same simulations were run us ing an isotropic hardening rule based on the coefficients determined for the 1% effectiv e plastic strain yield surface. For modeling purposes, the yield surface corresponding to 1% effective plastic strain was assumed to be the initial yield surface. The results using the proposed hardening law (solid lines) are plotted along with the results using the isotropic hardening law (dashed lines), and the experimental data (symbols) in Figure 4.18. The proposed method captures the data for each loading direction very well. Since is otropic hardening assumes that the yield surface expands without any change to its shape, the material hardens at the same rate for each strain path. Therefore, isotropic hardenin g is not able to represent the behavior for loading directions other than that on which the hardening la w was based, i.e., tension in the rolling direction of the sheet. 4.7.2 Application to AZ31B Agnew et al. (2001) reports that the ma jor deformation systems active at room temperature for AZ31B magnesium are basal sl ip, pyramidal slip, and tensile twinning (see Figure 1.2). The reported Voce hardeni ng parameters for each system are given in Table 4.5.

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114 Figure 4.18 Comparison between experiment al data (symbols) and simulation results using the proposed hardening law (solid lines) and using isotropic hardening (dashed lines) for a cold rolled sheet of MG-TH alloy. Data after Kelley and Hosford (1968). 185 190 195 200 205 210 215 00.020.040.060.080.1 Uniaxial Tension Rolling DirectionStress (Mpa)Plastic Strain 80 100 120 140 160 180 200 220 00.020.040.060.080.1 Uniaxial Compression Rolling DirectionStress (Mpa)Plastic Strain 160 170 180 190 200 210 220 230 00.020.040.060.080.1 Uniaxial Tension Transverse DirectionStress (Mpa)Plastic Strain 140 150 160 170 180 190 200 210 220 00.020.040.060.080.1 Balanced Biaxial TensionStress (Mpa)Plastic Strain 80 100 120 140 160 180 200 00.020.040.060.080.1 Uniaxial Compression Transverse DirectionStress (Mpa)Plastic Strain

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115 Table 4.5 Voce hardening parameters for AZ31B magnesium 0 1 0 1 basal slip 30 30 300 0 pyramidal slip 90 80 800 0 tensile twin 15 0 30 30 Styczynski et al. (2004) pres ents a metallurgical evaluati on of cold rolled sheets of AZ31 magnesium. From the resu lts of metallurgical studies, the authors suggest that the texture of the sheet can be de scribed by varying equally wei ghted Euler angles using two components such that 00 122,,90,15, and 00 122,,90,15, Using a vpsc model along with the single crystal properties for AZ31B magnesium and the orientation distribution function desc ribing the initial texture for a cold-rolled sheet of AZ31B yield strengths were calculate d for different levels of effective plastic strain corresponding to uniaxia l tension and uniaxial comp ression in th e rolling, transverse, and through-thickness directions of the sheet, as well as in pure shear for a strain rate of 1 310 s. Just like for zirconium, anisot ropy coefficients for the proposed yield criterion (3.16) were calculated for the different levels of effective plastic strain (listed in Table 4.6) using thes e calculated data points. The resulting yield loci shown in Figure 4.19 represent yielding for monotonic lo ading along a given path. Intermediate yield surfaces can be calculated using the interpolation method (4.27). Although AZ31B magnesium has been reported to fail due to in-plane compression at room temperatures at approximately 14%p (Agnew et al., 2005), in-plane tensile loadings at room temperatur e do not fail until well after 50%p (Agnew et al., 2001). Therefore, the evolution of the yield surface was determined using vpsc calculations for

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116 strain levels of up to 50%p In order to predict failure in this material, a proper failure criterion must be estab lished since different load paths cause the material to fail at different levels of stress and strain. Furthermore, since the sheet of AZ31B magnesium does not exhibit an extremely hard through thickness directi on like the plate of zirconium, the predicted r-values for AZ31B are not so high when using monotonic lo ading yield stresses to approximate the yield surface. Therefore, using this appr oach may not be as inaccurate for simulating deformations at higher levels of st rain for AZ31B as it was for zirconium. Uniaxial tension and compression simulati ons were carried out in ABAQUS using the interpolation method (4.27) for cold rolled sheets of AZ31B magnesium. The same simulations were run using an isotropic hardening rule based on the coefficients determined for the 0.2% effective plastic st rain yield surface. The results from the interpolation method of the propos ed model (solid lines) are pl otted along with the results from the isotropic hardening m odel (dashed lines), and the calculated data points from the vpsc model (symbols) in Figure 4.20. The proposed method captures the data for each loading direction very well. Similar to the previous section, isot ropic hardening is not able to give an adequate fit to the yield stresses in any direction except for the direction that the hardening was based upon, i.e., tensi on in the rolling directi on of the sheet. Strain paths in AZ31B magnesium wh ich deform by significant amounts of {10 12} <10 11> tensile twinning, i.e., in plane compression and thru-thickness tension, initially yield at lower stress es than strain paths that do not involve tensile twinning. However, the material hardens at a much faster rate for these strain paths than for strain

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117 -800 -600 -400 -200 0 200 400 600 -800-600-400-2000200400600 yy xx 0.2% 1% 5% 10% 50% Figure 4.19 Comparison between the plane stress yield loci (0xy ) for a AZ31B magnesium cold rolled sheet predicte d by the proposed theory (solid lines) and yield strengths calculated using the vpsc polycrystal model (symbols). (Stresses in Mpa) Table 4.6 AZ31B coefficients corresponding to the yield surface evolution depicted in Figure 4.19 a k L12 L13 L22 L23 L33 L44 L55 L66 0.2% 2 0.2941 0.1724 -0.0701 1.0503 0.0552 0.7723 0.7547 1.2899 1.0641 1% 2 0.2290 0.0091 -0.1160 1.1598 0.0585 1.2048 1.6770 2.1879 1.2230 5% 2 0.3575 -0.0014 0.0342 1.1384 0.1493 1.2361 1.9628 2.2370 1.5302 10% 2 0.4097 -0.0551 0.0177 1.1270 0.1506 1.2578 1.9373 2.0409 1.5667 50% 2 0.2995 -0.2387 0.4351 1.2409 0.6143 2.2740 2.2647 2.6521 1.6595

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118 paths that primarily involve slip such as in plane tension. Theref ore, if an isotropic hardening model utilizing the initial yield loci for AZ31B magnesium were used to simulate the deformation of a bending beam, one would expect to see an increased amount of compressive deformation along the axis of the beam compared to the proposed model which can account for the increased hardening rates. Finite element simulations were performed in ABAQUS for a three-dimensional beam made from a cold rolled sheet of AZ 31B magnesium subjected to a four-point bending test. This beam was initially 100mm l ong with a square cross-section of 3mm, however only half of the beams length was modeled due to symmetry, and 1872 linear brick elements were used to simulate the beam. The beams axis was assumed to be aligned with the sheets x-axis (rolling di rection). Free-surface boundary conditions were imposed for the beam except at the points of the pins. The upper pins were located at mm and the lower pins were located at m m from the center of the beam. The beam bends as the upper pins were displa ced down by a magnitude of 8mm. As expected, the proposed model predicted much higher compressive axial stresses and lower compressive axial strains than the isotropic hardening m odel. The proposed model also predicts a further shift in the neut ral axis of the beam in order to balance the tensile and compressive stresses. In Figur e 4.21 the axial stresses are plotted as a function of beam height for a path along the ce nter of the beams cen ter cross-section. Although no experimental data exists for sp ecimens made from AZ31B to validate the proposed model for AZ31B, the uniaxial and beam bending simulations clearly demonstrate the ability of the proposed m odel to capture the an isotropic hardening behavior of such a material, while an isotropic hardening assumption could not.

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119 Figure 4.20 Comparison between calculated data using a VPSC model (symbols) and simulation results using the proposed ha rdening law (solid lines) and using isotropic hardening (dashed lines) for a cold rolled sheet of AZ31B magnesium. 200 250 300 350 400 450 500 00.10.20.30.40.5 TD TensionStressPlastic Strain 100 200 300 400 500 600 00.10.20.30.40.5 RD CompressionStressPlastic Strain 100 200 300 400 500 600 00.10.20.30.40.5 TD CompressionStressPlastic Strain 100 150 200 250 300 350 400 00.10.20.30.40.5 Thru-Thickness CompressionStressPlastic Strain 100 200 300 400 500 600 700 800 00.10.20.30.40.5 Thru-Thickness TensionStressPlastic Strain 200 250 300 350 400 00.10.20.30.40.5StressPlastic Strain RD Tension

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120 -600 -400 -200 0 200 400 600 -1.5-1-0.500.511.5 Isotropic Hardening Proposed ModelAxial Stresses (MPa)Beam Height (mm) Figure 4.21 Axial stress di stribution along the beams center cross-sections.

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121 CHAPTER 5 INCORPORATING THE EFFECTS OF STRAIN-RATE AND TEMPERATURE 5.1 Introduction Thus far, we have considered only the qua si-static deformation of metals for which an appropriate modeling framework is that of the plasticity theory. Experimental evidence suggests that the plastic deformati on of metals depends on the applied strain rate and is also sensitive to temperature (K aschner et al., 2000, Maudlin et al., 1999a and 1999b, Perzyna, 1966). The main goal of this chapter is to develop a model capable of describing the dynamic and temperature dependent anisotropic plastic response of hcp textured metals. We begin with a presentation of Perzynas approach (Perzyna, 1966) and the consistency approach (Wang et al., 1997) which are the tw o most widely used methods for extending rate-independent elasto-plastic models such as to include the effect of strain rate. Both approaches are used in the current work to incorporate strain rate effects in the anisotropic elasto-plastic models developed for hcp metals (see Chapter 4). Next, the algorithmic aspects related to the finite element implementation of the rate dependent anisotropic formulations developed will be presented along with the stress-strain response for various loading paths and strain rate conditions. Validation of the proposed anisotropic rate dependent formulations is provided by comparing simulation results to Taylor impact data on pure zirconium (d ata after Maudlin et al., 1999b) and on a tantalum alloy (data after Maudlin et al., 1999a). The very good agreement between the simulated and experimental post-test geometries in terms of major and minor side profiles

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122 and impact-interface footprints shows the ab ility of the proposed model to describe the texture evolution due to deformation twinning. 5.2 Elasto-Viscoplastic Theory 5.2.1 Perzynas Viscoplastic Approach Perzynas approach (Perzyna, 1966) is the most widely used method for introducing strain rate effects into elasto-plas tic models. The basic assumption is that the viscous properties of materials become manifest only after the passage to the plastic state. Thus, the strain rate can be decomposed additively into an elastic E and an inelastic VP part. EVP (5.1) The inelastic portion of the stra in rate represents combined viscous and plastic effects, therefore it is called viscoplastic. The evolution of the viscoplastic strain rate is defined as (Perzyna, 1966) ()VPg f (5.2) with a viscosity parameter, a function that depends on the rate-independent yield function,f being the internal variable, while ,g is the rate-independent plastic potential. In equation (5.2), are the McCauley brackets such that 0 for 0, () () for 0.f f ff (5.3) The function must fulfill the following conditions: (1) to be continuous and convex in 0, and (2) 00 The following expressions are generally used for the function () f (see Perzyna, 1966)

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123 (), 1n f fn or ()exp()1 ff (5.4) Note that equation (5.2) involves the assumption that there is flow (i.e., 0VP ) only if the current state of st ress is outside the yield surface, the rate of increase of the viscoplastic strain being a function of the excess stresses above th e yield criterion. Because of this feature, this viscoplastic theory is commonly called overstress law. 5.2.2 Consistency Approach An alternative approach to model strain rate effects on the inelastic deformation which was proposed recently by De Borst and co-workers (see Wang et al., 1997; Heeres et al., 2002). These authors introduced a ra te-dependent yield surface and imposed that during viscoplastic flow the stress state rema in on the rate-dependent yield surface. The rate-dependent yield function rd f is expressed as (see Wang et al., 1997) (,,)0rdrdff (5.5) where and incorporates rate effects. As in elasto-plasticity, the viscoplastic strain evolves according to the flow rule rd vp f (5.6) with a positive scalar called viscoplastic multiplier or consistency parameter which can be estimated from the Kuhn-Tuckner conditions (5.7). 0, 0, 0rdrdff (5.7) Since the consistency is reinforced through the Kuhn-Tuckner conditions, this model is called the consistency viscoplastic model.

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124 5.3 Energy Balance The energy equation is a consequence of the first law of thermodynamics, and involves an additional quantity, the internal energy of the mate rial. This equation can be used to relate the temperature rise to the mechanical work ap plied to the material. Within a dynamic deformation event (<< 1 second) the effect of heat flux through the material becomes negligible. Furthermore, for a solid body subjected to a plastic deformation, the effects of an internal heat source may not be relevant. Therefore, the energy equation reduces to the following form (Malvern, 1969) :u t D (5.8) with being the density of the material, u the internal energy, and D the rate of deformation tensor. From thermodyna mics, the internal energy is given by pucT (5.9) where pc is the specific heat of the material, and T is the temperature change. Combining equations (5.8)-(5.9) and writing the equations in incremental form yields a relationship from which a rise in temperature is related to the current value of stress and the strain increment. :pT c (5.10) 5.4 Proposed Anisotropic El astic/Viscoplastic Theory 5.4.1 Using the Perzyna Method Our goal is to develop a macroscopic anisot ropic elasto-viscoplastic model that can describe simultaneously the influence of st rain rate and temperature along with the evolving texture on the inelastic response of hexagonal metals.

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125 To this end, we use Perzynas overstress approach to incorporate strain rate and temperature effects in the rate-independent elastic/plastic model developed for hcp materials (see Chapter 4). The yield function in this rate-independent formulation is given by (,,)1 ,vp vp vpfT YT (5.11) where vp is the effective viscoplastic strain which will be used as the hardening parameter. When using the interpola tion method described in section 4.5, 1(,)()(1()) j j vpvpvp (5.12) 1 1 j vpvp vp jj vpvp (5.13) and, 1 112233,(),(),()aaa a jjjj vpvpvpkaBkkk L (5.14) for 1p< 2p< j vp< 1 j vp< m p with : LS L being a fourth-order orthotropic tensor which reflects the plastic anisotropy of the material, S the deviator of the Cauchy stress tensor, k is the strength differential parameter, B (see equation 4.8) is the constant which allows the yield criterion to reduce to the uniaxial loading directi on hardening is based upon, and a is a homogeneity constant. ,vpYTis the rate independent hardening law that depends on the hardening variable, vp and with the temperat ure, T. The rate independent plastic potential is assumed to coincide with the yield function, i.e.,

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126 ,,g. We take m f f, where m is a constant. Thus, the viscoplastic law (5.2) specifies to 1 ,m vp vp vpYT (5.15) with a viscosity parameter and m a constant. In displacement-based FE formulations stress updates take place at the Gauss points for a prescribed nodal displacement. We start from time t, with the known converged state ,,,,ttttt vpvpT and calculate the corresponding values at time tt : ,,,,tttttttttt vpvpT In this incremental proce ss, the total strain increment is decomposed into an elastic E part and a viscoplastic part vp according to E vp (5.16) The stress increment is related to the elastic strain by Hookes law E vp C C (5.17) where C is the fourth-order stiffness tensor. Next, a trial elastic stress: ttt trial C is calculated. If (,)0tt trialvpf then tttt trial however, if (,)0tt trialvpf there is viscoplastic flow, and the viscoplastic strain increment vp must be determined (see Simo and Hughes, 1998). The viscoplastic strain increment vector can be separated into a scalar (which is equal to an increment of effective viscoplastic strain), and direction given by the gradient of the plastic potential. Thus, we need to determine to update both the viscoplastic strain increment and the hardening parameter, vp

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127 tt tttt vpvpvpvp (5.18) ttt vpvp (5.19) Therefore, combining equations (5.18) and (5.15) and assuming t yields tttt tt vpff t (5.20) From (5.20), the viscoplastic multiplier may be determined by requiring the residual (see equation 5.21) to approach zero during a local iterative procedure. 0ttr t (5.21) Currently, the values of tt and are unknown, but may be determined through a limited Taylor series expansion of the residu al about the previous step of the local iterative procedure (i.e., step n). 11110tt tttt tttt nnnnn nn nrrr rrT T (5.22) where n is a counter for th e local iteration (n=0 denotes the trial elastic state), and all derivatives in equation (5. 22) are evaluated at step n. In equation (5.22), denotes the variation of the variable between increments n+1 and n, i.e., 11nnn 11nnn and 11nnnTTT The stress update for the trial state (n=0) is found by assuming that the total strain increment was elastic, however, the stress update for subsequent steps (n>0) are determined by considering the effect of the viscoplastic strain on the state of stress.

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128 111 1nnnnn nC for n>0 (5.23) The gradient of the stress potential based upon the unknown updated state (iteration n+1) may be approximated by a limited Taylor series expansion about the current state using the variation of during the iteration n (see equation 5.24). 22 2 1 n n nnvp n n (5.24) In equation (5.24), (01 ) is an interpolation parameter. For 0 the gradient is determined solely from the current stat e, thus approximating the direction of the updated viscoplastic strain increment from the current yield surface. Combining equations (5.23) and (5.24) yields 1 22 1 111 2 nnnnn vpC (5.25) Plugging (5.25) into (5.22) yi elds an expression from whic h the increment of effective viscoplastic strain may be found. 1 21n n n vpvprT T H t (5.26) where, 1 2 1 2 nHC (5.27) 11 1mm YY (5.28)

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129 1 21 1m vpvpvpY m YYY (5.29) 1 21mY m TYYT (5.30) and all derivatives are evaluated at the previous iteration (iteration n). The state of stress and strain are updated through equation (5.26) until the residual (5.21) meets a specified tolerance for convergence. 5.4.2 Using the Consistency Method The consistency method (Wang et al., 1997) enforces the stress state for a viscoplastic loading to lie on the yield surface similar to rate-independent plasticity. Using the proposed anisotropic hardening law, the onset of viscoplastic behavior is governed by a scalar rate and temperature dependant yield condition of the form (,)(,,)0rdvpvpvpfYT (5.31) where is the effective stress which depends on the state of stress and the effective viscoplastic strain (see equations 5.12 5.14), Y is the hardening relationship based on a given loading direction such as uniaxial tension and is a function of the effective viscoplastic strain, viscoplastic strain rate, and temperat ure. Here the effective viscoplastic strain is equal to the viscoplastic multiplier or consistency parameter defined by the flow rule (5.6). For quasi-static conditions (1 ), the proposed viscoplastic model (equation 5.31) reduces to the rate-independent model proposed in Chapter 4. Like the Perzyna method, if the trial stress state (ttt trial C ) lies outside of the yield surface viscoplastic flow occurs. At difference with the Perzyna method

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130 consistency must be restored using the pr esent method, thus th e viscoplastic strain increment must be determined such that the state of stress is returned to the yield surface. The effective viscoplastic strain incremen t (which is equal to the viscoplastic multiplier from the flow rule (5.6)) needed to update the hardening, and from which the viscoplastic strain incr ement is determined, can be approximated through a limited Taylor series expansion of equatio n (5.31) about the current state (n=0 denotes the trial state) 11(,,,) (,,,)0rdn+1vpnvpn rdrdrdrd rdnvpnvpnn+1n+1n+1n+1 vpvpfT ffff fTT T (5.32) where denotes the variation of the variable between increments n+1 and n (i.e., 11 nnn ) and all derivatives are evaluate d at the current state. After introducing the stress variation (see equation 5.25) and replacing with t equation (5.32) can be rearrange d to give an expression for 1 n 1 2(,,) 1vpnvpn n n vpvpvpvpY fT T YY t n H (5.33) where, 1 2 1 2 nHC (5.34) The stresses are then updated by equation (5.23) and the effective viscoplastic strain is updated by equation (5.19). Iterations will co ntinue until the yield criterion (5.26) has been satisfied to within a given tolerance.

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131 5.5 Numerical Examples 5.5.1 Using the Perzyna Method In this section, we test the finite element implementation by simulating the response of a single integration point subjec t to uniaxial loading conditions at constant deformation rate. To simplify calculations, the effects of anisotropy on the pl astic response were neglected (i.e., in equation (5.14) the tensor L was set equal to the fourth-order identity tensor). The specific form of the law of va riation of the strength differential coefficient k was chosen such that initially the value of uniaxial compressive yiel d stress is less than the tensile uniaxial yield stress, but the yiel d in uniaxial tension and compression become equal at the critical level of effective viscoplastic strain, critical vp, i.e., for 0 0 for critical vpvpvp vp critical vpvpAB k (5.35) with A B being constants. The rate-independe nt constitutive hardening law was considered to be of the form ()()F vpvpYED (5.36) with E, D, and F are constants. The numeri cal values for the parameters involved in the model are given in Table 5.1. Table 5.1 Parameters used for numerical simulation A B critical vp E (Mpa) D F (1s ) m a E (Youngs Modulus) (Gpa) 0.4 8 0.05 650 0.04630.227 2000 2 2 100

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132 Figure 5.1 depicts the simulated stress-vi scoplastic strain response corresponding to uniaxial tension for both lo ading and unloading at the constant deformation rates 1 110.001 s, 110 s, and 1100 s respectively. The simulations were carried out for a total strain increment of 5 11110 x and a time step of 2110 txs, 6110 txs, and 7110 txs, respectively. The iterative procedure was considered to be converged if the square of the residual (5. 21) becomes less or equal than 610 According to the Perzyna method, the viscopl astic strain rate is determined by the overstress, i.e., the stress state outside of the static yield surface. At the onset of viscoplastic flow, the state of stress lies on the static yield surface. For a viscoplastic deformation involving very high strain rates, th e stress moves away from the static yield surface to account for the high rates of strain. As the stress moves aw ay from the surface, the viscoplastic strain rate increases fr om zero to the final value depending on the constitute law. Consequently, during unloa ding, viscoplastic strains continue to accumulate until the state of stress decrea ses below the static yield surface when the viscoplastic strain rate finally decreases b ack to zero. During reloading at the same applied total strain rate, viscoplastic stra ins begin accumulating once the stress exceedes the static yield stress (see Figure 5.1). Figure 5.2 shows the simula tion results for uniaxial co mpression under constant axial deformation rate of 1 110.001 s 110 s and 1100 s respectively. Since the hardening relationship is base d on the uniaxial stress-str ain curve in tension and the material displays yielding asymmetry between tension and compression, the stress-strain response curves for uniaxial compressi on depend on the law of variation of k Simulations were carried out us ing the law of variation for k given by equation (5.35) as

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133 well as for k held constant at k = 0 (saturation value) and k = 0.4 (initial value), respectively. Note that k = constant corresponds to isotropi c hardening. As expected, for the case when k varying according to (5.35) the material first yields at the same as the same stress level as in the case when k is held constant at 0.4, but then hardens at a much higher rate until critical vpvp when k becomes 0, and the yield stress in tension and compression become equal. Next, we have studied the effect of the strain rate and the size of the strain increment on the accuracy of the numerical re sults using the Perzyna method. For this purpose, simulations were conducted using various strain rates and strain increments for uniaxial tension to 5% viscoplastic strain. The results were compared to the onedimensional (1-D) theoretical solution accordin g to equation (5.15) by assuming uniaxial tension about the x-direction. 1 0,1m vpx x vpxvpxvpxY (5.37) For a very small strain rate (the ratio of the strain increment and the time increment) (i.e., 0.001s-1) the terms used in the residual (5.15) become very small in magnitude, thus a given tolerance can be sati sfied with a larger percent error than for larger strain rates which result in larger resi dual terms. Furthermore, a very small change in the viscoplastic strain can lead to large changes in the stress, th us the accuracy of the Perzyna method is sensitive to the strain rate as well as the size of the strain increment as indicated by Figure 5.3. In Figure 5.3, per cent error refers to the comparison between the theoretical and numerical stress predictions after 5% accumulated viscoplastic strains, i.e., %numericaltheoreticaltheoreticalerror.

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134 0 100 200 300 400 500 00.0050.010.0150.02Stress (Mpa)Total Strain strain rate = 100 s-1 strain rate = 10 s-1 strain rate = 0.001 s-1 Figure 5.1 Simulation results using the Perzyna method for various strain rates corresponding to uniaxial tension for loading and unloadi ng conditions.

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135 200 250 300 350 400 00.010.020.030.040.050.06Stress (Mpa)Total Strain k = 0 k varies 0.4 0.0 k = 0.4 strain rate = 0.001 s-1 250 300 350 400 450 00.010.020.030.040.050.06Stress (Mpa)Total Strain k = 0 k varies 0.4 0.0 k = 0.4 strain rate = 10 s-1 300 350 400 450 500 00.010.020.030.040.050.06Stress (Mpa)Total Strain k = 0 k varies 0.4 0.0 k = 0.4 strain rate = 100 s-1 Figure 5.2 Simulation results using the Perzyna method for various strain rates corresponding to uniaxial compression for different variations of the strength differential coefficient k.

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136 -0.5 0 0.5 1 1.5 2 2.5 3 10-610-510-4Percent ErrorStrain Increment Size 0.001 s-1 10 s-1 100 s-1 convergence tolerance = 1e-6 Figure 5.3 Effect of strain ra te and the size of the strain increment on the accuracy of the Perzyna Method at 5% levels of viscoplastic strain. The state of stress and strain within a given increment will be accepted as the current state once the square of the residua l becomes less than a specified convergence tolerance. Therefore, the accuracy of the Perzyna method is also dependent on the tolerance for convergence since, as previ ously mentioned, a small change of the viscoplastic strain leads to a large change in stress. This is illustrated in Figure 5.4 for different strain increment size s at a strain rate of 0.001s-1 for simulations conducted to 5% viscoplastic strain. Choosing an appropriate stra in increment size and converg ence tolerance is a tradeoff between accuracy and computational efficiency. A strain increment of 5 11110x with a convergence tolerance of 6110x was chosen for numerical simulation because this combination produced an acceptable level of accur acy for each level of strain rate while not being overly computationally expensive.

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137 0 0.5 1 1.5 2 2.5 3 10-810-710-610-510-4Convergence Tolerance strain increment = 1e-4 strain increment = 5e-5 strain increment = 1e-5Percent Error strain rate = 0.001 s-1 Figure 5.4 Effect of convergence tolerance and the size of the strain increment on the accuracy of the Perzyna method at 5% levels of viscoplastic strain. 5.5.2 Using the Consistency Method At first sight, the consistency model and Perzynas model appear to be quite different. Recently, Heeres et al. (2002) compared both approaches by assessing the evolution of the viscoplastic multiplier and th e evolution of the internal variable for different loading/unloading condi tions. It was shown that for progressive viscoplastic loading, the stress-strain response accordi ng to Perzynas overstress theory and the consistency theory are identical. However, for unloading, due to th e assumption that the authors made regarding the viscoplastic stra in rate the two theories give different responses. The assumption made by Heeres et al. (2002) was that the viscoplastic strain rate could only monotonically increase, so that upon unloading the viscoplastic strain rate would not decrease to zero. Thus, the yield point was fixed according to (,)ultimateY.

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138 Therefore, the material woul d transition to a pu rely elastic state immediately upon the onset of unloading. Furthermore, upon reloadin g, the material would remain elastic until the fixed yield point wa s reached. This differs from th e Perzyna method as described in the last section. Figure 5.5 shows the results of uniaxial lo ading and unloading simulations carried out using the consiste ncy method with the Heeres et al. (2002) assumption and the same material parameters used for the Perzyna method. In order to compare the two methods a rate dependent hardening relations hip based on uniaxial tension about the materials x-direction equiva lent to equation 5.37 was used to define (,)Y (see equation 5.31). Figure 5.6 shows a comparison between the two methods in a blown up view of the region in which they differ. When the consistency method was first introduced (see Wang et al., 1997), the assumption that the viscoplastic strain ra te can only monotonica lly increase was not stipulated. In fact, it was stated that at th e viscoplastic strain rate was to be calculated during each time increment based upon the vi scoplastic strain increment. This assumption is more physically reasonable sin ce a non-zero viscoplastic strain rate can not be realized without a non-zer o viscoplastic strain incremen t. When using the method described in Wang et al. (1997) to determine th e viscoplastic strain rate without using the assumptions described by Heeres et al. (2002 ) to fix the viscoplastic strain rate upon unloading, the consistency method and the Perzyna method predict exactly the same results. In fact, using the sa me material parameters as before, uniaxial tensile simulations were carried out using the consistency method assuming that the viscoplastic strain rate was not fixed upon unloading. Th e results are presented in Fi gure 5.7 and are identical to

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139 0 100 200 300 400 500 00.0050.010.0150.02Stress (Mpa)Total Strain strain rate = 100 s-1 strain rate = 10 s-1 strain rate = 0.001 s-1 Figure 5.5 Simulation results using the cons istency method using the assumptions from Heeres et al. (2002) for various strain rates corresponding to uniaxial tension for loading and unloading conditions.

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140 300 350 400 450 0.0060.0080.010.0120.014 Consistency method assuming that the viscoplastic strain rate is fixed for unloading and reloading Perzyna MethodStress (Mpa)Total Strain Static Hardening Curve strain rate = 100 s-1 Figure 5.6 Comparison between the Perzyna method and consistency method using the assumptions from Heeres et al. (2002) corresponding to uni axial tension for loading and unloading conditions.

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141 0 100 200 300 400 500 00.0050.010.0150.02Stress (Mpa)Total Strain strain rate = 100 s-1 strain rate = 10 s-1 strain rate = 0.001 s-1 Figure 5.7 Simulation results using th e consistency method without using the assumptions from Heeres et al. (2002) for various strain ra tes corresponding to uniaxial tension for loading and unloading conditions.

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142 the results predicted using th e Perzyna method. The uniaxial compression results using the material parameters described by equatio ns (5.35) (5.36) and Table 5.1 with the consistency method were identical to the re sults shown in Figure 5.2 for the Perzyna method. The state of stress and stra in for a given elastic-viscoplastic increment using the consistency method is accepted as the current st ate once consistency is restored. Thus, at difference with the Perzyna method, the terms used in the convergence criterion are of stress. Since small differences in stress result in very small differences in strain, the consistency method is not as se nsitive to the step size or convergence tolerance as the Perzyna method (see Figure 5.8). Therefore, a strain increment size of 110.001 with a convergence tolerance of 4110x was used. 0 0.02 0.04 0.06 0.08 0.1 0.12 10-410-310-210-1100Convergence TolerancePercent Error 0.001 s-1 10 s-1 100 s-1 strain increment = 0.001 Figure 5.8 Effect of convergence tolerance and the strain rate on the accuracy of the consistency method at 5% leve ls of viscoplastic strain.

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143 5.6 High-Strain Rate Modeling of the Be havior of Zirconium in Compression Kaschner et al. (2000) repor ted results of uniaxial compression tests on a highpurity crystal-bar zirconium. The materi al was clock-rolled (see section 4.6 for discussion). Mechanical tests were carried out at 76 Kelvin (K) and 298 K (i.e., -197 Celsius (C) and 25 C, respectively), however, we will use only the results for 298 K to validate our approach. Cylindrical compre ssion specimens were machined from the zirconium plate such that the cylindrical axis was originally in the plane of the plate, and tests were carried out at strain rates of 0.001 s-1, 0.1 s-1 and 3500 s-1. The tests at the strain rate of 3500 s-1 were performed using a split Hopki nson pressure bar. The results of these tests have shown the importance of twinning on the mechanical response of zirconium deformed at high rates of strain. In fact, metallurgical evidence showed an increase in the amount of twinning in the zirconium at the stra in rate increased. The Taylor cylinder impact test (see Figure 5.9) was developed during World War II by G. I. Taylor (Taylor, 1948) to screen materials for use in ballistic applications. The test involves firing a small cylin drical rod at a high velocity against a massive and rigid target producing strain-r ates on the order of 4511010s The impact plastically deforms and shortens the rod by causing material at the impact surface to flow radially outward relative to the rod axis. The Taylor impact test involves gradients of stress, strain, and strain rate to produce the final strain distribution. Therefor e, a model used to simulate such an event must be able to capture the materials behavior as a function of accumulated viscoplastic strain and strain-ra te. Simulations involving the high speed impact of metals typically assumes isotr opy using the von Mises yield criterion with a rate-dependent, isotropic hardening relationshi p (Maudlin et al., 1999a). However, such

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144 assumptions would be incapable of capturing the initial or plastic deformation induced anisotropy of the material, and thus would onl y be capable of predicting a circular cross section for the post test specimen. The propos ed model, on the other hand, would be able to capture the initial anisotropy of the speci men, as well as the anisotropy evolution due to the plastic deformation during the high strain-rate deformation. Figure 5.9 Schematic of the Taylor impact test Taylor impact experiments were conducte d using zirconium specimens made from a clock-rolled plate of high purity crystalbar zirconium (see Maudlin et al., 1999b). The cylindrical specimens (50.8 mm long, 7.62 mm diameter) were machined such that the specimens cylindrical axis would have original ly been in the plane of the plate. The specimen was fired from a gas driven gun and im pacted a steel anvil target at 243 m/s. Photographs of the post-test specimens and th e strain profile as a function of specimen height were presented. In this section, we attempt to model th e high strain-rate behavior of zirconium by introducing both anisotropy and the anisotropic evolution of the yield surface behavior of zirconium using the proposed model. Due to the capability of both methods to predict the same material response, only the Perz yna method was used for simulation. V0 before after

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145 Before the aforementioned rate-dependent m odel can be utilized, a description of a materials yield behavior must be obtained. Due to deformation twinning, hexagonal closed packed metals such as zirconiu m exhibit a strong reorientation of the crystallographic texture with accumulated pl astic deformation (Tome et al., 2001), and thus a pronounced evolution of the shape of its yield surface. This concept was earlier discussed in Chapters 1 and 4, with the repr esentation of the yield surface evolution for a clock-rolled plate of pure zirc onium subjected to in-plane compression out to levels of 35% effective plastic strain presented in Chap ter 4 for static analysis. During the Taylor impact experiment, plastic strains of greater than 50% are expected. Thus, the yield surface evolution for a clock-rolled plate of zirconium subjected to in-plane compression out to levels of 60% effective plastic stra in were determined using the vpsc model by prestraining the material to a given level of plastic deformation. The polycrystal was then numerically probed along different strain paths to determine the yield surface corresponding to the updated texture. Furtherm ore, the materials yield behavior due to applied shear stresses was determined using the vpsc model. These results are presented in Figure 5.10, with vpsc calculations re presented by symbols, the yield surface determined by the proposed orthotropic yield criterion (3.16) repr esented by solid lines, and intermediate yield surfaces representing the evolution determined using the interpolation method (4.27) re presented by dashed lines. It is evident from Figure 5.10 that at high levels of plastic deformation, the materials texture is significantly changed by twinning. It will be nece ssary to capture this behavior when simulating the high deformation, high strain-rate Taylor impact te st discussed later in this section. The

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146 coefficients used by the propos ed yield criterion for the su rfaces depicted in Figure 5.10 are listed in Table 5.2. -800 -400 0 400 800 -1200-800-4000400800 yy xx Figure 5.10 Yield surface evolution for a cloc k-rolled plate of zirconium subjected to inplane compression about the x-axis for 0.2%, 1%, 5%, 25%, 35%, 45% and 60% levels of effective plastic strain. Symbols represent calculations using the vpsc model. Solid lines represent yield surface representation using the proposed yield criterion. Stresses in Mpa. Table 5.2 Zirconium coefficients corres ponding to the yield surface for in-plane compression a k L12 L13 L22 L23 L33 L44 L55 L66 0.2% 2 0.0706 2.8831 1.8747 1.2517 1.9649 1.2040 0.9890 1.0750 3.9560 1% 2 0.2242 2.1498 1.4068 1.1789 1.4383 0.8922 0.6041 0.6167 1.6914 5% 2 0.5001 2.9227 1.8231 0.8649 1.7452 0.4181 0.9844 1.1450 2.9802 25% 2 0.2632 2.7041 1.7271 1.8211 2.0558 0.8686 0.9493 0.7906 1.4031 35% 2 0.1738 2.5052 1.6829 2.3196 2.0535 1.0118 0.8888 0.6942 1.0977 45% 2 -0.2692 0.8528 1.4444 1.1429 0.6238 1.0072 0.6379 0.4607 0.6646 60% 2 -0.2200 0.7655 0.8583 0.6371 0.8999 0.6435 0.2985 0.1888 0.2547

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147 In addition to the yield surface evolution due to plastic deformation, the effect of strain rate on the constitutive behavior on the material must al so be characterized in order to model the material at high rates of strain. Such information is presented in Kaschner et al. (2000), and is necess ary to determine the strain rate coefficients and m (5.15) used in the Perzyna method. The coefficients 12500s and 7.0m were found to give the best fit for the in-plane comp ression data (see Figure 5.11). The constitutive behavior for the in-plane compression corr esponding to strain s beyond the experimental data was estimated based upon vpsc calculations. Since the Taylor impact experiment (Kaschner et al., 1999b) primarily involves in-plane co mpression, the coefficient B (see equation 5.14) was chosen such that the effectiv e stress reduces to in-plane compression. The proposed model using the Perzyna method was implemented into an ABAQUS/EXPLICIT user mate rial subroutine using the yield surface evolution according to in-plane compression (see Figure 5. 10) to model the Taylor impact test for the zirconium specimen. Due to the lack of experimental data to characterize the zirconium at high temperatures, the effect of temperature increase resulting from mechanical work was neglected. As a comp arison to isotropic hardening, a simulation was also carried out using th e Perzyna method assuming isotropic hardening based on the 0.2% yield surface of zirconium shown in Fi gure 5.10. Due to the orthotropic symmetry of the material, only a quarter of the cyli nder was modeled. The zirconium cylinder was modeled using 2117 ABAQUS C3D8R linear brick elements with free boundary conditions and an initial velocity of 243 m/s. The step size used to simulate the event was t=2e-8 s and was conducted until 90 s. At 90 s the specimen had rebound off of

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148 the target, and all plastic deformation had ceas ed. The anvil target was modeled as an analytical rigid surface. The predicted loga rithmic strain profile along the major and minor axes of the post-test specimen is pres ented in Figure 5.12 and compared to the experimental data. Figure 5.13 display a visual compar ison between the simulated and experimental major and minor profiles and foot print of the post-test specimen. Figures 5.14 5.15 present the same information except the simulation results were obtained using isotropic hardening based on the 0.2% yield surface. Figures 5.12 and 5.13 reveal that the re sults using the proposed rate dependent model are in good agreement with the experime ntal data. The diffe rences between the experimental data and the simulation results are most likely related to the increase of temperature due to the large viscoplastic de formations, and a possible difference in the shape of the yield surface at high strain-rates due to the higher le vels of twinning and the shape of the yield surfac e for static conditions used for simulation. Initially, the yield behavior for the zirc onium specimen is highly anisotropic with a very strong through thickness direction as compar ed to the in plane directions. However, as the Taylor specimen deforms by in-plane compression at very high strain rates along the axis of the rod, twinning acts to change the texture produ cing a more isotropic texture as the deformation progresses. While the proposed method is capable of capturing this phenomena, an isotropic hardening assumption can not. This is evident in figures Figures 5.14 and 5.15, which illustrate that an isotropic hardening assumption is not a good approximation for modeling the high strain -rate, large deformation behavior of zirconium.

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149 A study of the mesh density and the step size on the accuracy of the solution was also conducted using the Perzyna method. The mesh density was increased by approximately 40% with very litt le effect on the final solution. Similarly, the time step was cut in half with little to no effect on the solution. Therefore it was determined that using 2117 elements with a time step of t=2e-8 s was a good choice (see Figure 5.16). 0 100 200 300 400 500 600 00.050.10.150.20.25 Experimental Data Predicted ResultsStress (Mpa)Total Strain 3500 s-1 0.1 s-1 0.001 s-1 Figure 5.11 In-plane compression simulation results us ing the both the Perzyna method and the consistency method with the proposed yield criterion and hardening law (solid lines) in comparison wit h experimental data (symbols).

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150 Figure 5.12 Comparison of predicted (Perzy nas method) and experimental (symbols) logarithmic strain profile for the post test zirconium Taylor impact specimen. The simulation results were obtained us ing 2117 linear brick elements and a step size of t=2e-8 s. Data after Maudlin et al. (1999b). 0 5 10 15 20 25 30 35 40 00.150.30.45Specimen Height (mm)Logarithmic Strain (ln(r/r0)) Minor Profile 0 5 10 15 20 25 30 35 40 00.150.30.45Specimen Height (mm)Logarithmic Strain (ln(r/r0)) Major Profile

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151 (a) (b) (c) Figure 5.13 Comparison of the simulated (P erzynas method) and experimental crosssections of the post-test zirconium Tayl or impact experiment for (a) the major profile, (b) the minor profile and (c) the footprint. Data after Maudlin et al. (1999b).

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152 Figure 5.14 Comparison of predicted (assumi ng isotropic hardening) and experimental (symbols) logarithmic strain profile for the post test zirconium Taylor impact specimen. The simulation results we re obtained using 2117 linear brick elements and a step size of t=2e-8 s. Data after Maudlin et al. (1999b). 0 5 10 15 20 25 30 35 40 00.20.4Specimen Height (mm)Logarithmic Strain (ln(r/r0)) Minor Profile 0 5 10 15 20 25 30 35 40 00.20.40.60.8Specimen Height (mm)Logarithmic Strain (ln(r/r0)) Major Profile

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153 (a) (b) (c) Figure 5.15 Comparison of the simulate d (assuming isotropic hardening) and experimental cross-sections of the post-test zirconium Taylor impact experiment for (a) the major profile, (b) the minor profile, and (c) the footprint. Data afte r Maudlin et al. (1999b).

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154 Figure 5.16 Effect of mesh density and ti me step on the final solution. Black lines represent 2117 elements and 28tes red lines represent 2117 elements and 18tes and green lines represent 2975 elements and 28tes Data after Maudlin et al. (1999b). 0 5 10 15 20 25 30 35 40 00.150.30.45Logarithmic Strain (ln(r/r0))Specimen Length (mm) Minor Axis 0 5 10 15 20 25 30 35 40 00.150.30.45Logarithmic Strain (ln(r/r0))Specimen Length (mm) Major Axis

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155 5.7 High-Strain Rate Modeling of the Be havior of Tantalum in Compression Although the proposed model was develo ped to capture the anisotropic, asymmetric hardening behavior of hcp metals, in this section the versatility of this model is demonstrated by modeling the high stra in rate defomation of bcc tantalum. A material characterization and the resu lts from Taylor impact tests using specimens cut from a rolled plate of tantalum were reported (see Maudlin et al., 1999a). Tantalum is a bcc metal and metallurgical evidence shows negligible texture evolution due to the large viscoplastic deformations of the Taylor impact test. Therefore, isotropic hardening can be assumed while modeling this material. The proposed modeling approach can easily be used to model tantalum by fixing (see eq 4.31), thus the proposed model reduces to isotropic hardening. Furthermore, since tantalum is a cubic metal, the yield in tension and compression is equal, and the proposed yield criterion can be used to represent the yiel d behavior of tantalum by se tting the strength differential coefficient k to zero. Based upon the initial texture of the ta ntalum plate, a Taylor-Bishop-Hill polycrystal model was used to construct a yield surface by probing the material along different loading directions (see Maudlin et al., 1999a). Using the information from the polycrystal analysis, coefficients were de termined to best re produce the yield surface using the proposed orthotropic yield criter ion. Figure 5.17 shows the polycrystal calculations from Maudlin et al. (1999a) as symbols and the co rresponding yield surface using the proposed orthotropic criterion as a solid line on the biaxial plane. The coefficients used by the proposed criterion to co nstruct this surface are listed in Table 5.3. Due to a lack of information regarding the yield behavior due to shear stress, yield in

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156 shear can be assumed to be isotropic. For instance, according to the von Mises isotropic yield criterion, yield due to pure shear is 0.577 times that due to uniaxial tension. -1.5 -1 -0.5 0 0.5 1 1.5 -1.5-1-0.500.511.5 yy xx Figure 5.17 Yield surface corresponding to a rolled sheet of tantalum. Polycrystal calculations (symbols). Yield surface us ing the proposed yiel d criterion (solid line). Data after Maudlin et al. (1999a). Table 5.3 Tantalum coefficients fo r the proposed orthotropic criterion a k L11 L12 L13 L22 L23 L33 2 0 1.0000 -0.1911 -0.0687 1.0411 0.0067 1.1366

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157 In the same manner as for zirconium, the uniaxial stress strain curve for tantalum for static conditions was represented usi ng the interpolation method. Based on experimental data published from Maudlin et al (1999a), the rate coefficients of equation (5.15) were chosen to be 11500s and 5m The predicted results for uniaxial compression simulations using the Perzyna met hod are compared to e xperimental data in Figure 5.18. Note the change in the hardening rate as the strain rate increases. In order to better model the very high st rain-rate Taylor impact test, the strain hardening rate was chosen to give a better fit to the higher stra in-rate data to model the constitutive response of the tantalum specimen. In addition to the high stra in rate data collected for room temperature conditions, Maudlin et al. (1999a) also presents stress strain data collected for various high temperatures. In order to model the effect of temperature on the constitutive response of tantalum, a temperature dependa nt stress strain curve of the form shown in equation (5.33) was used in the ex pression of equation (5.15). 1,()(1())1h jj rm vppp meltrmTT YTYY TT (5.33) In equation (5.33), Trm is the room temperature (typically 25 degrees C), Tmelt is the materials melting temperature, and h is a material parameter. The temperature portion of equation (5.33) is the temperature contri bution from the Johns on-Cook model (Johnson and Cook, 1983). Choosing 0.42h to give a best average f it to the experimental data and using 3250meltTK for tantalum along with the strain rate parameters previously mentioned for tantalum, uniaxial compression simulations were carried out using the Perzyna method and are compared to experimental data in Figure 5.19. Due to the

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158 similarity between the resu lts using the Perzyna method and the consistency method, tantalum was modeled only using the Perzyna method. 0 200 400 600 800 1000 00.050.10.150.20.25 Experimental Data Predicted ResultsStress (Mpa)Total Strain 1300 s-1 0.1 s-1 0.001 s-1 Figure 5.18 Uniaxial simulation results (solid lines) for various strain rates at 25C in comparison with experimental data (symbols). Data after Maudlin et al. (1999a).

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159 0 100 200 300 400 500 600 700 00.050.10.150.20.25 Experimental Data Predicted ResultsStress (Mpa)Total Strain 2800 s-1, 200C 3900 s-1, 800C 3000 s-1, 1000C Figure 5.19 Uniaxial simulation results (s olid lines) for vari ous strain rates and temperatures in comparison with experi mental data (symbols). Data after Maudlin et al. (1999a).

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160 Using the yield surface, temperature, and st rain rate parameters for tantalum along with equation (5.10) to calculate the rise in temperature due to mechanical deformation ( 316640kgm, 140pcJkgK ), the proposed model was implemented into an ABAQUS/EXPLICIT user material subroutine to model the Taylor impact experiment reported in Maudlin et al. (1999a ) for tantalum. The experime nt consisted of a specimen cut from a tantalum plate such that the cyli ndrical axis of the sp ecimen was either from the 1 or the 2 direction of the plate (see Fi gure 5.20). The cylindrical specimens were 7.62 mm in diameter with a length of 38.1 mm. Figure 5.20 Schematic of the orientation of cylindrical specimens cut from a tantalum plate Due to the nearly isotropic yield behavior in the plane of the plate, simulations were only carried out assuming that the cylindr ical axis was along the 1 direction of the plate. A quarter section of the cylindrical specimen was modeled due to the ortotropic symmetry of the material using 8170 ABAQUS C3D8R linear brick elements with free boundary conditions and an initial velocity of 175 m/s. The step size used to simulate the event was t=2e-8 s and was conducted until 100 s. At 100 s the specimen had

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161 rebound off of the target, and all plastic defo rmation had ceased. The anvil target was modeled as an analytical rigid surface. The higher number of elements used for the tantalum simulations as compared to the zi rconium simulations was due to the larger deformations of the tantalum specimen. The simulated and experimental profiles of the major and minor axis of the specimens post test elliptical cross section are compared in Figure 5.21 with excellent agreement. The si mulated and experimental footprints of the post test specimen are compared in Figure 5. 22. The data shown in Figures 5.21 5.22 are the results from three se parate experiments conducted for different specimens under the same conditions. The simulated and expe rimental post test specimens are visually compared in Figure 5.23. The calculated post test temperature distribution is shown in Figure 5.24 In Maudlin et al. (1999a), the tantalum Taylor impact test was simulated by introducing anisotropy through using a di rect link between th e Taylor-Bishop-Hill polycrystal model and the finite element code EPIC, assuming that each integration point was a polycrystal. Their results were also in very good agreement with the experimental data. However, Figures 5.21 5.23 clearly demonstrate that introducing anisotropy into the high strain rate simulati ons using the proposed method can also produce ve ry accurate results, but at a fraction of the computational expense.

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162 (a) (b) Figure 5.21 (a) Major and (b) minor profiles for the tantalum Taylor impact specimen. Solid lines represent simulation results, symbols represent experimental data. Data after Maudlin et al. (1999). 0 5 10 15 20 25 30 -10-50510Specimen Height (mm)Radius (mm) 0 5 10 15 20 25 30 -10-50510Specimen Height (mm)Radius (mm)

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163 -10 -5 0 5 10 -10-50510y (mm)x (mm) Figure 5.22 Footprint for the tantalum Tayl or impact specimen. Solid lines represent simulation results, symbols represent experimental data. Data after Maudlin et al. (1999).

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164 Figure 5.23 Visual comparison between simu lated and experimental tantalum Taylor impact specimens for the (a) major side profile and (b) footprint. (a) (b)

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165 Figure 5.24 Calculated temperature (degrees K) contours for the post test tantalum Taylor impact specimen.

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166 CHAPTER 6 MODIFICATION OF THE PROPOSED YI ELD CRITERION COMPARISON TO 2090-T3 ALUMINUM Barlat et al. (2005) demonstr ated that when large amounts of data are available for yield surface representation, anistotropy coefficients from one linear transformation may not be adequate. In fact, as discussed in section 1.2, Barlat et al. (2005) introduced Yld2004-18p as a modification of Yld91 to include two linear transformations with a total of 18 anisotropy coeffici ents. The proposed orthotropi c yield criterion (3.16), can be extended to include two linear transformations. F k k k k k ka a a a a a 3 3 2 2 1 1 3 3 2 2 1 1 (6.1) This modified yield criterion includes 18 anisotropy coefficients from the two transformed tensors L S and L S along with two strength differential coefficients k and k. Equation (6.1) reduces to equation (3.16) when and kk An asymmetry between tensile and compre ssive yield has been experimentally observed in the aluminum alloy 2090-T3 (Y oon et al., 2000). In Yoon et al. (2000), uniaxial tensile and compressive yield strengths and Lankford coefficients for 7 different orientations with respect to th e rolling direction of the sheet, along with the yield strength for balanced biaxial tension were presented to describe the initial yield surface. The tensile data and Lankford coeffi cients were used as input fo r Yld96 (2.19). Since (2.19) requires that yield in tensi on is equal to yield in comp ression, it was not able to accurately represent the compressive data. As a remedy, Yoon et al. (2000) translated the

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167 stress axes until the uniaxial yield stresses for 0 and 90 to the rolling di rection of the sheet for both tension and compression could simultaneously be captured. While translating the stress axes allowed the uniaxia l predictions for 0 a nd 90 to the rolling direction of the sheet to capture the respective data, Yld96 (2.19) could not accurately represent the intermediate orientations between 0 and 90. It is more appropriate to represent the yield data for a metal that exhibits an asymmetry between tension and compression us ing a yield criterion that can allow for such phenomena. Since the published data in Yoon et al. (2000) in cludes 22 data points to describe the initial yield surface, the modified orthotro pic yield criterion (6.1) was used. The ability for (6.1) to represent th is data is shown in Figure 6.1, and the corresponding yield surface is given in Figure 6.2. The coefficients for the criterion (6.1) calculated from the experimental data and used in Figures 6.2 and 6.3 are presented in Table 6.1. For cubic metals such as aluminum, it is typical to simulate forming operations such as cup drawing using an isotropic hardenin g law as describe in section 2.2. Yoon et al. (2000) presents the earing profile for an experimentally formed cup along with the simulated earing profile using Yl d96 (1.19) with and without the translation of the stress axes assuming isotropic hardening. The specif ic dimensions of the tools as specified in Yoon et al. (2000) are given in Table 6.2. The schematic view of the cup drawing process is shown in Figure 6.3.

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168 Figure 6.1 Comparison between predicted (solid line) and experimental (symbol) variation of yield stress and r-v alues with sheet orientation. 0.8 0.85 0.9 0.95 1 0153045607590Tensile Yield Stress (normalized)Angle (degrees) from rolling direction 0.8 0.85 0.9 0.95 1 0153045607590Compressive Yield Stress (normalized)Angle (degrees) from rolling direction 0 0.5 1 1.5 2 0153045607590Tensile r-valuesAngle (degrees) from rolling direction

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169 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5-1-0.500.511.5 Balanced Biaxial Tension Data Point yy xx Figure 6.2 Predicted initial yi eld surface for 2090-T3 aluminum. Table 6.1 2090-T3 aluminum coefficients k L11 L12 L13 L22 L23 L33 L44 L55 L66 0.1553 1.0 0.0323 -0.6781 0.9958 -0.0085 0.7699 -0.7470 -0.7470 -0.7470 k L11 L12 L13 L22 L23 L33 L44 L55 L66 -0.0536 1.0 -0.0705 0.0789 1.2900 -0.2759 0.1314 1.7340 1.7340 1.7340 a 5.0

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170 Table 6.2 Tool dimensions used for cup drawing simulations Punch diameter Dp = 97.46 mm Punch profile radius rp = 12.70 mm Die opening diameter Dd = 101.48 mm Die profile radius rd = 12.70 mm Blank diameter Db = 158.76 mm Blank thickness t0 = 1.6 mm Figure 6.3 Schematic of circular cup drawing. The proposed modified criterion (6.1) and an isotropic hardening law were implemented into an ABAQUS user material subroutine to simulate the experimental cup from Yoon et al. (2000). Only a quarter se ction of the cup was analyzed due to the orthotropic material symmetry of the aluminum sheet. The finite element mesh was composed of 1407 nodes and 780 three-dimensi onal linear brick elements (see Figure 6.4). The blank holding force used in the si mulation was 22.2 kN (5.55 kN for a quarter cup section), and the Coulomb coefficient of friction was 0.1. The resulting earing profile using the proposed yiel d criterion (6.1) along with th e experimental data and the results from Yld96 (2.19) are given in Figure 6.5. The proposed extended criterion (6.1)

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171 predicts the height of the bottom of the ears much much more accurately than Yld96 especially for 0 and 180, while both criteria predict the height of the top of the ear to within the spread of the data, and similar ear widths. The same results using the proposed criterion (6.1) are compared to Yld96 (2.19) with translation of the stress axes in Figure 6.6. The proposed criterion (6.1) and Yld96 (2.1 9) with translation of the stress axes both predict similar earing heights, however, (6.1 ) more accurately captu res the width of the ear. Figure 6.4 Finite element mesh used for cup drawing simulation of 2090-T3 aluminum

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172 36 38 40 42 44 46 48 50 090180270360 Yld 96 Experimental Data Proposed CriterionCup Height (mm)Angle from rolling direction (deg) Figure 6.5 Predicted and experimentally de termined earing profile for a drawn cup of 2090-T3 aluminum.

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173 36 38 40 42 44 46 48 50 090180270360 Yld96 with surface translation Experimental Data Proposed CriterionCup Height (mm)Angle from rolling direction (deg) Figure 6.6 Predicted and experimentally de termined earing profile for a drawn cup of 2090-T3 aluminum.

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174 CHAPTER 7 SUMMARY AND FUTURE WORK An isotropic criteri on that can describe the asymme try in yielding between tension and compression of pressure insensitive meta ls was proposed. This criterion is expressed in terms of the principal values of the st ress deviator and involves two parameters: the parameters a which gives the degree of homogeneity of the yield function and the parameter k, which for a fixed va lue of the parameter a depends only on the ratio between the tensile and compressive yiel d strengths. The yi eld function was proven to be convex when the constant k belongs to a given numerical ra nge: [-1, 1], for any value of a. The macroscopic yield locus was demonstrated to be capable of capturing the asymmetric yield locus for randomly oriented bcc, fcc, and hcp polycrystals deforming solely by twinning as computed using the vpsc model. The proposed isotropi c criterion reduces to von Mises when k = 0 and a = 2. The proposed isotropic criter ion was extended such as to describe orthotropy by using a linear transformation on the deviator ic stress tensor. The proposed criterion involves 11 parameters, in cluding 9 anisotropy co efficients along with k and a from the isotropic criterion. The proce dure for identification of these parameters from simple tests was outlined. The orthotropic criterion was th en used to describe the strong asymmetry and anisotropy observed in text ured binary Mg-Th and Mg-L i alloy sheets (data after Kelley and Hosford, 1968) and for 4Al-1/4 O2 titanium sheet (data after Lee and Backofen, 1966). Very good agreement between theoretical and experimental yield loci corresponding to different levels of total strain was obtained.

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175 A macroscopic anisotropic ha rdening model that can de scribe the influence of evolving texture on the plas tic response of hexagonal metals was proposed. Initial yielding was described using the proposed yield criterion that accounts for both anisotropy and asymmetry between yielding in tension and compression. Yield stresses calculated using a polycrystal model were used to determine the evolution of the macroscopic yield surface with accumulated deformation. The proposed model was implemented into the implicit finite elemen t code ABAQUS. Simulations of the three dimensional deformation of pure zirconium b eams subjected to a four-point bend test were performed. Predicted and measured macr oscopic strain fields and beam sections are in very good agreement. Similar simulations were conducted for AZ31B magnesium which again demonstrated the effectiveness of the proposed model over using isotropic hardening, however no experimental data fo r AZ31B was available for comparison. The simulation results for zirconium and magnesium suggest that a computationally efficient macro-scale modeling, when used in conj unction with polycrystal line modeling, can accurately describe the strength differential effect and the anisotropic hardening observed in hcp metals. The proposed model was extended to include the effects of strain-rate and temperature using two common approaches: the overstress method of Perzyna, and a rate-dependant consistency method. Both methods were shown to produce identical or nearly identical results depending on the as sumptions made regarding unloading for the consistency model. The rate depende nt model was implemented into an ABAQUS/EXPLICIT user material subroutine to simulate high strain-rate events.

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176 Simulations of the Taylor impact test for zi rconium and tantalum were performed and the results were shown to be in good agreement with experimental data. For situations when a vast amount of e xperimental data is available for the determination of a materials yield surface, the proposed orthotropic yield criterion was modified through the addition of a second linear transformation, thus doubling the amount of available anisotropy coefficients. Using this m odified criterion, the yield surface was represented for an aluminum allo y for which strength differentials between tensile and compressive yield ha ve been experimentally observe d. Using this criterion, circular cup drawing simulations were carried out, with the results shown to be in good agreement with experimental data. In order to better captur e the true behavior of th e material, an appropriate anisotropic damage and anisotropic failure m odel need to be developed and incorporated into the proposed modeling approach. Furt hermore, a more generalized approach to simultaneously capture the evol ution of both yield stresses and r-values for any given strain path out to high levels of plastic deformation is needed.

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177 LIST OF REFERENCES ABAQUS Version 6.4 Reference Manuals, 2003. Pawtucket, RI. Agnew, S. R., Yoo, M. H., Tom, C.N., 2001. Simulation of texture simulation to understanding mechanical behavior of Mg and solid solutions alloys containing Li or Y, Acta mater.49, 4277-4289. Agnew, S. R., Ashutosh, J., 2005. Anisotro pic and asymmetric yield behaviour of magnesium alloy sheet assessed using a combined experimental and theorietical approach, Proceedings of 2005 NSF DM II Grantees Conf., Scottdale, AZ. Banabic, D., Bunge, H-J, Pohlandt, K., Te kkaya, A.E., 2000. Formability of metallic materials: plastic anisotropy, formability testing, forming limits, Springer, Berlin, pp. 119-172. Banabic, D., Cazacu, O., Barlat, F., Comsa, D.S., Wagner, S., Siegert, K., 2002. Description of anisotropic behaviour of AA3103-O aluminium alloy using two recent yield criteria. Journal de Physique IV 105, 297-304. Barlat, F., Lege, D.J., Brem, J.C., 1991. A six-component yield function for anisotropic materials. Int. J. Plasticity 7, 693-712. Barlat, F., Becker, R.C., Hayashida, Y., Maeda, Y., Yanagawa, M., Chung, K., Brem, J.C., Lege, D.J., Matsui, K., Murtha, S.J ., Hattori, S., 1997. Yielding description of solution strengthened aluminum alloys Int. J. Plasticity 13, 185-401. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheetPart I: Theory. Int. J. Plasticity 19, 1297-1319. Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 2005. Linear transformation-based anisotropic yield f unctions. Int. J. Plasticity 21, 1009-1039 Bishop, J. W. F, Hill, R. 1951. A theoretical deviation of the plastic properties of a polycrystalline face-centered metal. Phil. Mag. Ser. 7 42, 1298-1307. Blewitt T. H., Coltman R. R., Redman J. K. 1957. Low-temperature deformation of copper single crystals. J ournal of Applied Physics 28, 651. Boehler, J. P., 1978. Lois de comportement anisotrope des milieux continues. J. Mec 17, 153-190

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178 Bron, F., Besson, J. 2004. A yield function for anisotropic materials Application to aluminum alloys. Int. J. Plasticity 20, 937-963 Cazacu, O., Barlat, F., 2001. Generalization of Drucker's yield criterion to orthotropy. Mathematics and Mechanics of Solids 6, 613-630. Cazacu O. and Barlat, F. 2003. Application of representation theory to describe yielding of anisotropic aluminum all oys. Int. J. of Engng. Sci. 41, 1367-1385. Cazacu O. and Barlat, F. 2004. A criterion for description of anisotropy and yield differential effects in pressu re-insensitive metals. Int. J. Plasticity. 20, 2027-2045. Chin, G. Y., Mammel, W. L., Dolan, M.T., 1969. Taylor analysis for 111112twinning and 111110slip under conditions of axisymmetric flow. Trans. TMS-AIME 245, 383-388. Crisfield M.A., 1991. Non-linear finite element analysis of solids and structures. Wiley, West Sussex, England. Dafalias, Y.F., 1984. The plastic spin concept and a simple illustration of its role in finite plastic transformations. Mech anics of Materials 3, 223. Drucker, D.C., 1949. Relation of experiments to mathematical theories of plasticity. J. Appl. Mech. 16, 349-357. Drucker, D.C., 1950. Stress-strain relations in the plastic range, a survey of theory and experiment. Report to the Office of Naval Research NR-041-032. Fundenberger, J.J. ,Phillipe, M.J., Wagne r, F., Esling, C., 1997. Modelling and predictions of mechanical pr operties for materials with hexagonal symmetry (zinc, titanium, and zirconium alloys ). Acta Mater. 45, 4041-4055. Heeres, O. M., Suiker, A. S. J., de Bors t, R. 2002. A comparison between the Perzyna viscoplastic model and the Consistency viscop lastic model. European J. of Mech. 21, 1-12. Hershey, A. V. 1954. The plasticity of an is otropic aggregate of anisotropic face centered cubic crystals. J. Appl. Mech. Trans. ASME, 21, 241-249. Hill, R., 1948. A theory of the yielding and pl astic flow of anisotropic metals. Proc. Roy. Soc. London A193, 281. Hill, R., 1950. The Mathematical Theory of Plasticity. Oxford University Press, London. Hill, R., 1990. Constitutive modelling of orthotropic plasticity in sheet metals. J. Mech. Phys. Solids 38, 405.

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180 Lee, D., Backofen, W. A., 1966. An experime ntal determination of the yield locus for titanium and titanium-alloy sheet. TMS-AIME 236, 1077-1084 Li, S., Hoferlin, E., Van Bael, A., Van Hou tte, P. and Teodosiu, C., 2003. Finite element modeling of plastic anisotr opy induced by texture and stra in-path change. Int. J. Plasticity, 19, 647. Liu, S. I., 1982. On representation of anisotr opic invariants. Int. J. Eng. Science 20, 10991109 Logan, R. W., Hosford, W. F., 1980. Uppe r-bound anisotropic yield locus calculations assuming 111 pencil glide. Int J. Mech Sci. 22, 419-430 Malvern, L. E. 1969. Introduction to the m echanics of a continuous medium. PrenticeHall. Inc, Upper Saddle River, NJ. Maudlin, P. J., Bingert, J. F., House, J. W ., Chen, S. R., 1999a. On the modeling of the Taylor cylinder impact test for orthotropi c textured materials: experiments and simulations. Int J. Plas. 15, 139-166 Maudlin, P. J., Gray, T. G., Cady, C. M., Kaschner, C. G., 1999b. High rate material modeling and validation using the Taylor cylinder impact test. Phil Trans R. Soc. 357, 1707-1729 Molinari, A., Canova, G. R., Ahzi, S., 1987. A self consistent approach of the large deformation polycrystal viscoplas ticity. Acta Mater. 35, 2983-2994 Molinari, A., 1997. Self-consistent modeling of plastic and viscoplas tic polycrystalline materials. In: Large plastic deformation of crystalline aggregates, Springer, New York, pp. 173-246 Perzyna, P, 1966. Fundamental problems in viscoplasticity, in: Recent Advances in Applied Mechanics, vol. 9, Acad emic Press, New York, pp. 243377 Rockafellar, R.T., 1972. Convex Analysis. Prin ceton University Press, Princeton, NY. Simo, J. C., Hughes, T. J. R., 1998. Comput ational Inelasticity. Springer-Verlag, New York. Simo, J. C., Taylor, R. L., 1985. Consistent tangent operators for rate independent elastoplasticity. Comput Methods A ppl Mech Engrg. 48, 101-118. Sobotka, Z., 1969. Theorie des plastischen Fliessens von anisotropen Krpern. Zeit. Angew. Math. Mech. 49, 25. Spitzig, W.A, Sober, R.J., Richmond, O., 1976. The effect of hydrostatic pressure on the deformation behavior of Maraging an d HY-80 steels and its implication for plasticity theory. Metall. Trans. 7A, 1703-1710.

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182 Wang, W. M., Sluys, L. J., De Borst, R., 1997. Viscoplasticity for instabilities due to strain softening and strain-rate softening. Int. J. Num. Meth. Engrg. 40, 3839-3864. Yoon, J. W., Yang, D. Y., Chung, K., Barlat, F., 1999. A general elasto-plastic finite element formulation based on incremental de formation theory for planar anisotropy and its application to sheet metal fo rming. Int. J. Plasticity 15, 35-67 Yoon, J. W., Barlat, F. Chung, K., Po urboghrat, F., Yang, D. Y., 2000. Earing predictions based on asymmetr ic nonquadratic yield function. Int. J. Plasticity 16, 1075-1104. Yoon, J. W., Barlat, F., Dick, R. E., Chung, K., Kang, T. J., 2004. Plane stress yield function for aluminum alloy sheets part II: FE formulation and its implementation. Int. J. Plasticity 20, 495-522.

PAGE 196

183 BIOGRAPHICAL SKETCH Brian W. Plunkett was born in Hartsville, SC on July 3, 1974. He graduated with a Bachelor of Science in Mechanical Engin eering from the University of Alabama in December, 1996. Upon graduation, he was commissioned as an officer in the United States Air Force, and served from February 1, 1997, until January 31, 2001, as a mechanical engineer at Robins AFB, GA. Brian was then hired by the Air Force as a civilian weapons test engineer for the Munitions Test Division at Eglin AFB, FL. While working at Eglin AFB, he completed a Master of Engineering degree from the University of Florida in August of 2003. Following an Ai r Force fellowship as a full-time graduate student working on a PhD at the University of Florida, Brian transferred to the Air Force Research Laboratory at Eglin AFB, FL, in August of 2005.


Permanent Link: http://ufdc.ufl.edu/UFE0012500/00001

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Title: Plastic Anisotropy of Hexagonal Closed Packed Metals
Physical Description: Mixed Material
Copyright Date: 2008

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Holding Location: University of Florida
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Permanent Link: http://ufdc.ufl.edu/UFE0012500/00001

Material Information

Title: Plastic Anisotropy of Hexagonal Closed Packed Metals
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
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PLASTIC ANISOTROPY OF HEXAGONAL CLOSED PACKED METALS


By

BRIAN W. PLUNKETT













A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Brian W. Plunkett















ACKNOWLEDGMENTS

The author thanks each member of his supervisory committee for their suggestions

and support. Special appreciation is extended to Dr. Oana Cazacu for her enthusiasm and

support throughout the course of this study. The author also wishes to thank Dr. Jeong-

Whan Yoon, Alcoa Technical Center, for advice on the implementation of material

models into finite element code. The author sincerely appreciates the US Air Force Air

Armament Center, Engineering Directorate (AAC/EN) for their financial support and for

providing the opportunity to pursue this study, and to the Air Force Research Lab,

Munitions Directorate, Computational Mechanics Branch (AFRL/MNAC) for

employment and future research opportunities.

Of course, this degree would not have been possible without the moral support of

fellow graduate students Mike Nixon and Stefan Soare. Saving the most important for

last, the author also wishes to thank his wife Tiffany Plunkett for her love and support

during the duration of this degree program, and God for all things.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iii

LIST OF TABLES .............. ............................................. ......... vi

L IST O F FIG U R E S .... ...... ...................... ........................ .. ....... .............. vii

ABSTRACT ........ .............. ............. .. ...... .......... .......... xii

CHAPTER

1 IN T R O D U C T IO N ............................................................................... .............. 1

2 STATE OF THE ART IN MODELING PLASTICITY OF METALS ...........................8

2 .1 D description of Initial Y wielding ..............................................................................8
2.1.1 Isotropic Y ield C riteria.......................................... ........... ............... 8
2.1.2 O rthotropic Y ield C riteria .................... .. ........................ ... 10
2.1.3 Modeling Asymmetry Between Tensile and Compressive Yield ............17
2.2 H ardening L aw s .................. ........... ......... ...... ........................ .. ..... ........ 18
2.3 Survey of Mesoscale Plasticity Modeling (Polycrystalline Models)...................20
2.3.1 Kinem atics of a Polycrystal ......... .... .... ................... .............. 22
2.3.2 T aylor M odel ........................................... ....... ........... .... .......... 24
2.3.3 Visco-Plastic Self-Consistent (vpsc) Model............................................26

3 PROPOSED YIELD CRITERION .......................................... .......................... 34

3.1 Proposed Isotropic Yield Criterion................................................................... 34
3.1.1 Comparison to Polycrystal Simulation................... .................. ...............45
3.1.2 Yield Surface Derivatives and Convexity .............................................. 47
3.2 Extension of the Proposed Isotropic Yield Criterion to Include Orthotropy.......52
3.2.1 Application of the Proposed Criterion to the Description of
Y ielding of H cp M etals ......... .. ............ ............................... ...............60
3.2.1.1 M agnesium A lloys ........................................ ....... ............... 60
3.2.1.2 Titanium A lloys........................ .. ............. ............... ....66
3.2.2 Derivatives of the Orthotropic Yield Function ................................67

4 PROPOSED HARDENING LAW ........................................ .......................... 72









4 .1 Intro du action ............................................................................... 72
4.2 E lasto-Plastic Problem ................................................. ............. ............... 74
4.3 Proposed Anisotropic Hardening Law ..................................... ............... 77
4.4 Application for an Isotropic M material ....................................... ................. 82
4.5 Alternate Method for Anisotropic Hardening Implementation Interpolation... 86
4.6 A application to Zirconium ............................................................................. 94
4.7 Application to Magnesium Alloys....................... .......................................113
4.7.1 Application To Mg-Th ........................................... ............... 113
4.7.2 Application to AZ31B ........................................ .................. 113

5 INCORPORATING THE EFFECTS OF STRAIN-RATE AND TEMPERATURE. 121

5 .1 Intro du action ...................1...................2...................1.........
5.2 Elasto-V iscoplastic Theory........................................... .......... ............... 122
5.2.1 Perzyna's Viscoplastic Approach............................................... 122
5.2.2 C consistency A pproach........................................ .......................... 123
5.3 E energy B balance ............................ .. .... ..... ......................... ......... .. .... 124
5.4 Proposed Anisotropic Elastic/Viscoplastic Theory .................................. 124
5.4.1 U sing the Perzyna M ethod ........................................... ............... 124
5.4.2 U sing the Consistency M ethod..................................... ............... 129
5.5 N um erical Exam ples................................................. ............................. 131
5.5.1 U sing the Perzyna M ethod ........................................... ............... 131
5.5.2 Using the Consistency Method..................... ............... 137
5.6 High-Strain Rate Modeling of the Behavior of Zirconium in Compression .....143
5.7 High-Strain Rate Modeling of the Behavior of Tantalum in Compression....... 155

6 MODIFICATION OF THE PROPOSED YIELD CRITERION COMPARISON
TO 2090-T3 ALUM INUM ......................................................... .............. 166

7 SUMMARY AND FUTURE WORK ............................................................. 174

LIST OF REFEREN CES ........................................................... .. ............... 177

BIOGRAPHICAL SKETCH ............................................................. ............... 183
















LIST OF TABLES

Table page

3.1 M G -TH yield surface coefficients ........................................ ......................... 63

3.2 M G -LI yield surface coefficients.......................................... ........... ............... 64

3.3 M G yield surface coefficients.......... ................. .......... ............... ............... 65

3 .4 4A 1-1/40 2 coefficients ........................................................................ .................. 68

4.1 Voce hardening parameters for zirconium for a strain rate of 10 2s-1 .......................95

4.2 Zirconium coefficients corresponding to the yield surface evolution depicted in
F ig u re 4 .7 ......................................................................... 9 7

4.3 Zirconium coefficients corresponding to the yield surface for in-plane
com pression ............. ..... .. ......... .......... ............................110

4.4 Zirconium coefficients corresponding to the yield surface for in-plane
com pression ............. .. ..... ................... ..... ......... ............... 111

4.5 Voce hardening parameters for AZ31B magnesium ................... ... .............115

4.6 AZ31B coefficients corresponding to the yield surface evolution depicted in
F ig u re 4 .19 ......... .............................................. ................................... 1 17

5.1 Param eters used for numerical simulation..................................... ............... 131

5.2 Zirconium coefficients corresponding to the yield surface for in-plane
compression .............. ...... .. ........ ....... ...... .............. .......... 146

5.3 Tantalum coefficients for the proposed orthotropic criterion...............................156

6.1 2090-T3 alum inum coefficients...................... .............................. ............... 169

6.2 Tool dimensions used for cup drawing simulations .............................................170
















LIST OF FIGURES


Figure page

1.1 Schematic of (a) slip and (b) deformation twinning........................................2

1.2 Typical deformation systems for hexagonal closed packed metals .......................3

2.1 Orientation of test specimen with the rolling direction of the sheet ........................12

2.2 Pictorial description of various hardening rules...................................................18

2.3 Notations for (a) slip and (b) deformation twinning in a single crystal .................23

2.4 Geometry of a slip system within a single crystal.................................................25

3.1 Plane stress yield loci for different values of the ratio -, /o- between the yield
stress in tension and compression, in comparison with the von Mises locus...........37

3.2 Plane stress yield loci corresponding to I/ c = 1.13 (k = 0.2) and a, /c =
1/1.13 ( k = -0.2)................................ .. ................... ........ ............... 38

3.3 The influence of the value of the parameter k on the ratio a, /Ic0 of the uniaxial
yield stress in tension and compression, for various values of the exponent a........39

3.4 Projection in the deviatoric 7" plane of the yield loci for a = 2 and various
values of k in comparison to von Mises and Tresca loci................. ............. ...40

3.5 Projections in the tension-torsion plane of the proposed yield loci for various k -
values and a = 2 (fixed), in comparison with Tresca and von Mises (k=0, a=2)
lo c i ................... ...................4...................3..........

Qf
3.6 vs. k for the case of pure shear (a = 2) ............... .......... ................44


3.7 Comparison between the vpsc yield locus for randomly oriented fcc and bcc
polycrystals deforming solely by twinning and the predictions of the proposed
criterion: (a) for plane stress (oxy = 0) (b) on the 7t-plane.....................................46









3.8 Comparison between the VPSC yield locus for randomly oriented hcp zirconium
polycrystals deforming solely by twinning and the predictions of the proposed
criterion: (a) plane stress (Gxy = 0) (b) on the 7t-plane ............................................47

3.9 Comparison between the plane stress yield loci (o, = 0) for a Mg-0.5% Th
sheet predicted by the proposed theory and experiments.................. ...............63

3.10 Comparison between the plane stress yield loci (o, = 0) for a Mg-4% Li sheet
predicted by the proposed theory and experiments...............................................64

3.11 Comparison between the plane stress yield loci (o, = 0) for a pure Mg sheet
predicted by the proposed theory and experim ents ..................................................65

3.12 Comparison between the plane stress yield loci (o- = 0) for a 4A1-1/402 sheet
predicted by the proposed theory and experiments...............................................68

3.13 Comparison between the plane stress yield loci (o = 0) for a 4A1-1/4 02 sheet
predicted by Hosford's 1966 modified Hill criterion and experiments ...................69

4.1 Evolution of the yield surface for varying k .............. .........................................84

4.2 Results of single element compression tests for a = 2. ........................................85

4.3 Yield surface evolution for a cold rolled sheet of mg-th using the interpolation
m ethod ...............................................................................89

4.4 Discrete yield loci and yield stresses used for the interpolation method. ................92

4.5 Results of the interpolation method compared to the continuous method from a
single element compression test.................. ............... ......................93

4.6 Stress-strain response for a clock-rolled plate of zirconium for in-plane
compression, in-plane tension, and through-thickness compression. ....................96

4.7 Yield surface evolution for a clock-rolled plate of zirconium ...............................97

4.8 Comparison between experimental data and simulation results using the
proposed model coupled with VPSC and using an isotropic hardening law for a
clock-rolled zirconium plate ......... .............. ................................. ... ............ 99

4.9 Schematic of the four-point bend test. ........... ................... .............. 100

4.10 Comparison of the experimentally measured strain distributions with the results
of finite element simulations using the proposed model and VPSC linked
directly to EPIC for the case CO......... ........... .. ........ ...................... 103









4.11 Comparison of the experimentally measured strain distributions with the results
of finite element simulations using the proposed model and VPSC linked
directly to EPIC for the case C90 .......................................... ....... .............. 104

4.12 Comparison of the experimentally measured plastic strains distributions and the
ABAQUS finite element predictions using the proposed model and the proposed
yield criterion with isotropic hardening for the CO case. .......................................105

4.13 Comparison of the experimental plastic strains distributions and the ABAQUS
finite element predictions using the proposed model and the proposed yield
criterion with isotropic hardening for the C90 case. ............................................ 106

4.14 Comparison of experimentally photographed x-z cross-section of the bent bars
versus the predictions of VPSC/EPIC and the proposed model; (a) and (c)
correspond to the case CO while (b) and (d) correspond to the case C90. .............107

4.15 Yield surface evolution for a zirconium clock-rolled plate during in-plane
compression.................... ...... ..... ................................... ....... 110

4.16 Yield surface evolution for a zirconium clock-rolled plate during through
thickness com pression ...... .............................. ................... .............. ... 111

4.17 Comparison of the final sections of the zirconium cylinders after: a,b) in-plane
compression, c,d) through-thickness compression ...............................................112

4.18 Comparison between experimental data and simulation results using the
proposed hardening law and using isotropic hardening for a cold rolled sheet of
M G -TH alloy.......................................................................................................... 114

4.19 Comparison between the plane stress yield loci (o, = 0) for a AZ31B
magnesium cold rolled sheet predicted by the proposed theory and yield
strengths calculated using the vpsc polycrystal model ................ ................117

4.20 Comparison between calculated data using a VPSC model and simulation results
using the proposed hardening law and using isotropic hardening for a cold rolled
sheet of AZ31B magnesium ............... ....................... 19

4.21 Axial stress distribution along the beam's center cross-sections.........................120

5.1 Simulation results using the Perzyna method for various strain rates
corresponding to uniaxial tension for loading and unloading conditions. .............134

5.2 Simulation results using the Perzyna method for various strain rates
corresponding to uniaxial compression for different variations of the strength
differential coefficient k. .............................................. .............................. 135

5.3 Effect of strain rate and the size of the strain increment on the accuracy of the
Perzyna M ethod at 5% levels of viscoplastic strain......... ......... ..................136









5.4 Effect of convergence tolerance and the size of the strain increment on the
accuracy of the Perzyna method at 5% levels of viscoplastic strain......................137

5.5 Simulation results using the consistency method using the assumptions from
Heeres et al. (2002) for various strain rates corresponding to uniaxial tension for
loading and unloading conditions. .............................................. ............... 139

5.6 Comparison between the Perzyna method and consistency method using the
assumptions from Heeres et al. (2002) corresponding to uniaxial tension for
loading and unloading conditions. .............................................. ............... 140

5.7 Simulation results using the consistency method without using the assumptions
from Heeres et al. (2002) for various strain rates corresponding to uniaxial
tension for loading and unloading conditions. ....................................... .......... 141

5.8 Effect of convergence tolerance and the strain rate on the accuracy of the
consistency method at 5% levels of viscoplastic strain............... ... ...............142

5.9 Schem atic of the Taylor im pact test ...................................... ......... ............... 144

5.10 Yield surface evolution for a clock-rolled plate of zirconium subjected to in-
plane compression about the x-axis for 0.2%, 1%, 5%, 25%, 35%, 45% and 60%
levels of effective plastic strain. ........................................ ........................ 146

5.11 In-plane compression simulation results using the both the Perzyna method and
the consistency method with the proposed yield criterion and hardening law in
comparison with experim ental data ............................. .......... .... ............... 149

5.12 Comparison of predicted and experimental logarithmic strain profile for the post
test zirconium Taylor impact specimen. ..................................... ............... 150

5.13 Comparison of the simulated and experimental cross-sections of the post-test
zirconium Taylor impact experiment for (a) the major profile, (b) the minor
profile, and (c) the footprint. ..... ......................................................................151

5.14 Comparison of predicted (assuming isotropic hardening) and experimental
logarithmic strain profile for the post test zirconium Taylor impact specimen. ....152

5.15 Comparison of the simulated (assuming isotropic hardening) and experimental
cross-sections of the post-test zirconium Taylor impact experiment for (a) the
major profile, (b) the minor profile, and (c) the footprint...............................153

5.16 Effect of mesh density and time step on the final solution. ..................................154

5.17 Yield surface corresponding to a rolled sheet of tantalum................................... 156

5.18 Uniaxial simulation results for various strain rates at 250C in comparison with
experim mental data. ........................................... .. .... ........... ........ 158









5.19 Uniaxial simulation results for various strain rates and temperatures in
comparison with experim ental data............................................... ................... 159

5.20 Schematic of the orientation of cylindrical specimens cut from a tantalum plate .160

5.21 (a) Major and (b) minor profiles for the tantalum Taylor impact specimen. .........162

5.22 Footprint for the tantalum Taylor impact specimen ............................................163

5.23 Visual comparison between simulated and experimental tantalum Taylor impact
specimens for the (a) major side profile and (b) footprint. ................................... 164

5.24 Calculated temperature (degrees K) contours for the post test tantalum Taylor
im pact specim en. ...................... ........ ................ ... ...... ................165

6.1 Comparison between predicted and experimental variation of yield stress and r-
values w ith sheet orientation .................................. ............... ............... 168

6.2 Predicted initial yield surface for 2090-T3 aluminum. ........................................169

6.3 Schematic of circular cup drawing. ......................... .. .. ............. ................... 170

6.4 Finite element mesh used for cup drawing simulation of 2090-T3 aluminum ......171

6.5 Predicted and experimentally determined hearing profile for a drawn cup of 2090-
T 3 alum inum .......................................................................172

6.6 Predicted and experimentally determined hearing profile for a drawn cup of 2090-
T 3 alum inum .......................................................................173















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

PLASTIC ANISOTROPY OF HEXAGONAL CLOSED PACKED METALS


By

Brian W. Plunkett

December 2005

Chair: Oana Cazacu
Major Department: Mechanical and Aerospace Engineering

Due to the effects of twinning and texture evolution, the yield surface for hexagonal

closed packed (hcp) metals displays an asymmetry between the yield in tension and

compression, and significantly changes its shape with accumulated plastic deformation.

Traditional initial yield criteria or hardening assumptions such as isotropic or kinematic

hardening cannot accurately model these phenomena. In this dissertation, a macroscopic

anisotropic model that can describe both the initial yielding and influence of evolving

texture on the plastic response of hexagonal metals is proposed. Initial yielding is

described by a newly developed macroscopic yield criterion that accounts for both

anisotropy and asymmetry between yielding in tension and compression. The

coefficients involved in this proposed yield criterion as well as the size of the elastic

domain are then considered to be functions of the accumulated plastic strain.

Viscoplastic self-consistent polycrystal simulations and a newly developed interpolation

technique are then used to determine the evolution laws. The proposed model was









implemented into the implicit finite element code ABAQUS and used to simulate the

three-dimensional deformation of a pure zirconium beam subjected to four-point bend

tests along different directions with respect to the texture orientation. Comparison

between predicted and measured macroscopic strain fields and beam sections shows that

the proposed model describes very well the contribution of twinning to deformation.

The proposed model is then extended to include the effects of strain rate and the

temperate increase within the material due to mechanical work. The proposed rate and

temperature dependent model was implemented into ABAQUS/EXPLICIT and used to

simulate the three-dimensional Taylor impact experiment for specimens made from pure

zirconium and from a tantalum alloy. The post experiment data and simulation results

are shown to be in very good agreement.














CHAPTER 1
INTRODUCTION

Weight reduction while maintaining functional requirements is one of the major

goals of engineering design and manufacturing so that materials, energy, and costs are

saved and environmental damage reduced. Because of their low density, thermal

properties, damping capacity, fatigue properties, dimensional stability, and machinability,

hexagonal closed packed (hcp) metals such as magnesium and titanium alloys offer great

potential to reduce weight and thus replace the most commonly used materials, i.e., steel

and polymers, plastics. Currently, the use of hcp metal sheets is restricted because of a

lack of fundamental understanding of their three-dimensional flow behavior.

Plastic deformation of polycrystalline metals occurs by either slip or twinning (see

Figure 1.1). Whether slip or twinning is the dominant deformation mechanism depends

on which mechanism requires the least stress to initiate and sustain plastic deformation.

Metals with cubic crystal symmetry have many slip systems, so twinning is usually not a

significant deformation mechanism at ambient temperatures, but may become important

as the temperature decreases or the strain rate increases (Blewitt et al., 1957, Huang et al.,

1996). In low symmetry materials such as hcp metals, which have too few slip systems

to accommodate any shape change, twinning may become a dominant mechanism.

Twinning, unlike slip, is sensitive to the sign of the applied stress; i.e., if a particular twin

can be formed under a shear stress, it will not be formed by a shear stress of opposite

sense. Because of the polar nature of twinning, hcp materials display a strong asymmetry

between the yield in tension and compression.










Atomic
/ planes --- Shear stress



SSlip
E E. ~plane




(a) "-






(111) Twinning
Plone



(111) Twinning
Plane


Matrix

(b)




Figure 1.1 Schematic of (a) slip and (b) deformation twinning

Due to the strong crystallographic texture induced by the rolling process, the yield

loci for cold rolled sheets of hcp metal may also exhibit a pronounced anisotropy

(Hosford, 1993). Texture refers to the non-uniform distribution of crystallographic

orientations in a polycrystalline aggregate. Most cold-rolled hcp alloy sheets have basal

or nearly basal textures, i.e., the basal planes of the grains are aligned with the sheet, with

a degree of spread from this ideal texture by up to + 200 about the transverse direction for









magnesium while for alpha titanium and zirconium alloys the spread is up to 400 (see

Hosford, 1966). The easiest slip directions are the closed-packed (1120) directions

which are normal to the c-axis, but slip on these systems does not produce any elongation

or shortening parallel to the c-axis (see Figure 1.2). Thus, only twinning or pyramidal

slip can allow inelastic shape changes in the c direction. For most hcp metals, the most

easily activated twinning mode is the tensile twin {1012} <10 11>, which is activated by

compression in the plane of the sheet or tension in the normal direction of the sheet, i.e.,

through-thickness tension. Since the pyramidal slip and compression twinning are much

harder than the primary deformation modes of basal slip and tension twinning, most hcp

sheets display a resistance to thinning, and a very pronounced difference between the

stress-strain behavior in tension and compression is observed (see for example, Tome et

al., 2001).









(1010)<1120> (1011)<1123> (1012)<1011> (1122)<1123>
prismatic slip pyramidal slip tensile twin compressive twin

Figure 1.2 Typical deformation systems for hexagonal closed packed metals.

The correct modeling of this strong asymmetry between tension and compression

due to deformation twinning remains a challenge. As discussed by Van Houtte (1978)

and Tome et al. (1991), a major obstacle in extending the crystal plasticity framework to

include deformation twinning is the difficulty in handling the large number of

orientations created by twinned regions. Twinning activity plays an important role in the









evolution of hardening by creating barriers to the propagation of dislocations or other

twin systems. In addition, twinning influences anisotropy evolution by reorienting the

grains (Kaschner et al., 2001). Although progress has been made and models that track

the evolution of the twinned regions in the grain and account for predominant twin

reorientation (e.g. Tome and Lebensohn, 2004b) or intergranular mechanisms (e.g.,

Staroselsky and Anand, 2003) have been proposed, the use of such models for forming

analyses is still limited because of rather large computation time.

Unlike the recent progress in the formulation, numerical implementation, and

validation of macroscopic plasticity models for cubic materials, macroscopic modeling of

hcp materials is less developed. Due to the lack of adequate macroscopic criteria for hcp

materials, hcp sheet forming finite element simulations are still performed using classic

anisotropic formulations for cubic metals such as Hill (1948) (see for example, Takuda et

al., 1999; Kuwabara et al., 2001).

General presentation of the dissertation. This dissertation is a contribution to

modeling and simulation of plastic anisotropy and strength differential effects in hcp

metals. Chapter 2 consists of a survey of major contributions to the description of plastic

behavior of metals at different length scales. Macroscopic plasticity models will be

discussed including isotropic and orthotropic yield criteria that describe the onset of

plastic behavior, and hardening laws which model subsequent plastic deformation. Then,

two of the most widely used mesoscopic plasticity models, the Taylor-Bishop-Hill model

and the viscoplastic self-consistent (vpsc) model, will be described.

Chapter 3 is devoted to the development of a new yield criterion for hcp metals.

This yield function is capable of describing both the tension/compression asymmetry and









the anisotropic behavior of hcp metals and alloys. The approach used in constructing this

criterion was first to develop an isotropic yield criterion that can capture the asymmetry

between tension and compression, and then extend this criterion to include orthotropy.

The expression of the isotropic criterion was based on numerical tests using the vpsc

model. Specifically, since there are no isotropic pressure insensitive materials that

exhibit tension/compression asymmetry, the vpsc model was used to obtain information

concerning the shape of yield loci for randomly oriented polycrystals isotropicc)

deforming by twinning (directional shear mechanism). The proposed isotropic criterion

involves only two parameters k and a, where a represents the degree of homogeneity of

the yield function. For a fixed, the parameter k is expressible solely in terms of the ratio

between the yield stress in tension and the yield stress in compression. Comparisons with

the results of polycrystalline simulations show that the proposed macroscopic criterion

describes very well the strength differential effect due to twinning in body centered cubic

(bcc), face centered cubic (fcc), and hcp polycrystals. An orthotropic extension of the

isotropic yield criterion is developed. Orthotropy is introduced through a linear

transformation applied to the deviator of the Cauchy stress tensor. Then, the yield loci

obtained using the proposed orthotropic criterion are compared with experimental yield

loci for sheets of textured polycrystalline binary Mg-Th, Mg-Li alloys, pure Mg (data

after Kelley and Hosford, 1968), and a Titanium (data after Lee and Backofen, 1966).

Very good agreement between theoretical and experimental yield loci is obtained. In

addition, we compare the yield loci predicted by the proposed orthotropic criterion with

the calculated yield loci obtained using the vpsc model for AZ31B magnesium and pure

zirconium.









Next, in chapter 4 a new and rigorous method for describing anisotropic hardening

due to evolving texture during plastic deformation is proposed. The anisotropy

coefficients as well as the size of the elastic domain are considered to be functions of the

accumulated plastic strain. An interpolation technique is introduced to determine the

evolution laws based on the results from mechanical tests and/or numerical tests

performed with polycrystal models in the absence of experimental data for the

corresponding strain paths. The proposed model was implemented into the implicit finite

element code ABAQUS and used to simulate the three-dimensional deformation of pure

zirconium and magnesium alloys subjected to different loading conditions. Validation of

the model for strain paths that have not been used for parameter identification is given for

zirconium (data after Kaschner et al., 2001). Comparison between predicted and

measured macroscopic strain fields and beam sections for zirconium beams subjected to

4-point bending experiments shows that the proposed model describes very well the

contribution of twinning to deformation. The difference in response between the tensile

and compressive fibers and the shift of the neutral axis is particularly well captured.

In chapter 5, an extension of the proposed elasto-plastic anisotropic model that

includes the effect of strain-rate and temperature is presented. To introduce rate effects

in the inviscid (elasto-plastic) model, two different modeling approaches are considered:

the Perzyna overstress method (Perzyna, 1966) and the consistency method (Wang et al.,

1997). Both approaches will be used to simulate the high strain-rate Taylor impact test.

The simulated results will be compared to experimental Taylor impact tests for pure

zirconium (data after Kaschner et al., 1999). For comparison purposes, we also simulated

the high strain rate behavior of a metal with bcc structure (tantalum) by setting the









strength differential parameter k to zero and using isotropic hardening. The influence of

the crystallographic structure on the high strain rate response was clearly demonstrated

for both zirconium and tantalum. In the case of zirconium, the model reproduces

correctly the very high hardening rate which is due to twinning; thus the profile of the

zirconium post-test specimen displays a much less pronounced mushrooming effect. For

tantalum which deforms only by slip, mushrooming is significant.

Chapter 6 is devoted to the description of plastic anisotropy in metals having cubic

crystal structure. Focus is on modeling of the behavior of an aluminum alloy that

exhibits strength differential effects as well as orthotropy. The proposed model is an

extension to orthotropy of the isotropic yield criterion presented in Chapter 3. Two linear

transformations were introduced to capture both the anisotropy in the yield stresses for

tensile and compressive loadings and the Lankford coefficients (r-values). Thus, the

number of independent coefficients involved in the formulation was doubled. The

proposed modified criterion was then applied to the prediction of the hearing profile of a

circular cup drawn from 2090-T3 aluminium.

Conclusions and future research directions opened by this research are presented in

Chapter 7. It is also worth mentioning that this dissertation has thus far resulted in three

journal articles that have been submitted or accepted for publication in International

Journal of Plasticity and Acta Materialia.














CHAPTER 2
STATE OF THE ART IN MODELING PLASTICITY OF METALS

2.1 Description of Initial Yielding

2.1.1 Isotropic Yield Criteria

For sufficiently small values of stress and strain, a metal will reassume its original

shape upon unloading. When loaded beyond this reversible (elastic) range, the specimen

will not reassume its original shape upon unloading, but will exhibit a permanent (plastic)

deformation. In the plastic range, it is typical for metals to work harden; i.e., the flow

stress monotonically increases with accumulated plastic strain. After a specimen has

been subjected to a stress exceeding the yield limit which separates the elastic and plastic

ranges, the current stress becomes the new yield limit if the material is unloaded. For a

multi-axial state of stress, a material's yield limit is mathematically described by a yield

criterion.

The oldest yield criterion was proposed by Tresca in 1864. According to Tresca's

criterion, the material transitions to a plastic state when the maximum shear stress reaches

a critical value. The Tresca criterion is given by

maxci- "2 U 3 0-i= Y (2.1)

where GI, 02, and G3 are the principal stresses, and Yis the yield stress in uniaxial tension.

The projection of Tresca's yield surface on the t7-plane (the plane which passes through

the origin and is perpendicular to the hydrostatic axis) is a hexagon centered on the origin

whose size depends on the magnitude of Y.









Possibly the most widely used isotropic yield criterion is the one proposed

independently by Huber in 1904, von Mises in 1913, and Hencky in 1924. This criterion

is usually referred to as the von Mises criterion. The von Mises criterion is based on the

observation that a hydrostatic pressure cannot cause yielding of the material. The plastic

state corresponds to a critical value of the elastic energy of distortion, i.e.

J, = k2 (2.2)

where k is a constant and J2 is the second invariant of the Cauchy stress deviator given by

J2 = [(or1 U2 )2 + (02 C3)2 + (03 r' )2] (2.3)


or alternatively,

1
J2 = (S +S + ) (2.4)

where S1, S2, and S3 are the principal values of the Cauchy stress deviator, defined as


S, = C -Ikkj (ij,k= 1,2,3) (2.5)

The projection of the von Mises yield locus on the 7t-plane is a circle that circumscribes

the Tresca hexagon.

Experimental evidence has shown that the yield loci for most isotropic metals with

cubic crystal structure lie between the yield loci predicted by the Tresca and the von

Mises criteria (see Taylor and Quinney, 1931). In order to represent the behavior of

certain metals (e.g., aluminum alloys) for which the yield loci are located between the

Tresca and von Mises yield loci, Drucker (1949) proposed the following criterion:

J -cJ = F (2.6)









where Js is the third invariant of the Cauchy stress deviator, and c and F are material

constants. In order to ensure that the yield surface is convex, c is limited to a given

numerical range, c e [-27/8, 2.25].

Unlike the isotropic yield criteria mentioned so far, which were postulated on the

basis of macroscopic experiments, Hershey in 1954 and then Hosford in 1972 used

Taylor-Bishop-Hill polycrystalline simulations (for an explanation see section 2.3.2) to

arrive at the following macroscopic yield criterion.

(a1 2Y)m + (0 a3)m' + (-3 a)" = 2Ym (2.7)

In (2.7), Y is the uniaxial yield stress and m is the degree of homogeneity which can vary

between 1 and oo. Equation 1.7 reduces to the Tresca criterion for m = 1 or m = o, and to

the von Mises criterion for m = 2 or m = 4. It has been shown that the yield loci of fcc

and bcc metals are best represented with m = 8 and m = 6 respectively (Logan and

Hosford, 1980, Hosford, 1993, Hosford, 1996).

2.1.2 Orthotropic Yield Criteria

Due to thermo-mechanical processing, metal sheets exhibit orthotropic symmetry

with the axes of orthotropy being aligned with the rolling direction, the transverse

direction, and the normal direction to the plane of the sheet (x, y, and z, respectively). In

1948, Hill proposed a generalization of the von Mises isotropic yield criterion to

orthotropy. Thus, this yield criterion is expressed by a quadratic function of the form:

F( a a + G(a -z )2 + H(-x ) + 2Lr + 2Mrx + 2Nt2 = 1 (2.8)

where F, G, H, L, M, and N are anisotropy constants, and x, y, and z are the orthotropy

axes (axes perpendicular to the 3 mutually orthogonal planes of symmetry of the









material). When F = G = H = 1/6k2 and L = M = N = 1/2k2, equation (2.8) reduces to

the von Mises criterion (2.2).

The coefficients involved in the expression of Hill's yield criterion can be

determined from simple mechanical tests. Denoting the tensile yield stresses for the x, y,

and z directions as X, Y, and Z, respectively, it can be shown that according to Hill's

theory


XY= Z=
G+H H+F F+G

thus,

1 1 1 1 1 1 1 1 1
2F = + 2G = + 2H = + (2.9)
Y2 Z2 X2 z2 X2 Y2' X2 Y2 Z2

Denoting the shear yield stresses as R, S, and T corresponding to the yz, zx, and xy

directions respectively, then:

1 1 1
2L = 2M = ; 2N =- (2.10)
R2 S2 T2

The thinning resistance of metal sheets is generally characterized by the Lankford

coefficients, commonly referred to as r-values. The Lankford coefficients are defined as

the width-to-thickness strain ratio during a uniaxial test. In classic plasticity theory, the

plastic strain increments are derived from a plastic potential, which for metals is

generally supposed to coincide with the yield function (associated flow rule) such that


de' = dA (2.11)


where de' is the plastic strain increment, K is a scalar variable, andfis the yield function.

Therefore, the strain increment vector is orthogonal to the yield surface. Assuming






12

plastic incompressibility, the Lankford coefficient for a uniaxial tensile loading in the x-

direction can be written as


dE, dE, + dE,


Of +f
ou+J
ao- -o,


Likewise, ra, the strain ratio corresponding to a uniaxial tensile loading at an arbitrary

angle, a, to the x-direction (see Figure 2.1) is given by


sin2 ca
ta-


sin 2a cf + cos2 a f

Of Of
c o toy


Figure 2.1 Orientation of test specimen with the rolling direction of the sheet


(2.13)


(2.12)


Sheet Metal


: : oi ia : i i








rolling direction


:: :









According to Hill's criterion, the Lankford coefficients can be expressed in terms of the

anisotropy coefficients as

H H N 1 (2
ro= 0 r9 '45F (2.14)
F F+G 2

Thus, the anisotropy coefficients can be expressed in terms of yield stresses in the rolling,

at 45 degrees to the rolling, and in the transverse directions, or as functions of the r-

values. In general, the yield loci obtained based on r-values differ from those obtained

based on the yield stresses (Barlat et al., 2005).

Hill's yield criterion (2.8) is the most widely used criterion for describing yielding

of textured metals. However, (2.8) can not adequately represent the behavior of certain

aluminum alloys which have an average value of the Lankford coefficients less than 1

and the yield stress in biaxial tension is greater than the yield stress in uniaxial tension

(Banabic et al., 2000). Therefore, in order to better represent yielding of aluminum

alloys, Hill developed another yield criterion in 1979 of the form:

F 0- cm + G c,3 1- 0 + H 0-a2 + L21 2 32- (2.15)
+M20-2 0-3 -1 m +N23 -a1 -a21 = Jm

This yield criterion has a major limitation since it is written in terms of the principal

Cauchy stresses. In order for this criterion to be valid, the principal stress axes and

anisotropy axes must be superimposed thus any state involving shear stresses cannot be

accounted for.

Barlat et al. (1991) proposed a six-component yield criterion denoted Yld91, that

extends the isotropic Hershey and Hosford criterion (see eq 2.7) to orthotropy. The

extension to orthotropy is accomplished by replacing the principal values of the Cauchy

stress tensor in the expression of the isotropic criterion by those of a transformed stress









tensor. The transformed stress tensor is obtained from the Cauchy stress tensor modified

with weighting coefficients. This procedure is equivalent with the application of a fourth

order linear transformation operator on the Cauchy stress tensor. The orthotropic

criterion is written as


(,-2) +( )m + (3 1) = 2Ym (2.16)

where

S= L, (2.17)

E 2,, and C3 are the principal values of the tensor X, o is the Cauchy stress tensor,

and L is a fourth order tensor of orthotropic symmetry. Thus, with respect to (x,y,z) the

symmetry axes,

C2 + C3 -C3 -C2
-C3 C3 + C -C1
1 -C2 -C C + (2.18)
L= (2.18)
3 3c4
3c5
3c6

where cl, C2, c3,..., and c6 are constants. Note that plastic anisotropy is represented by the

same number of coefficients as Hill's criterion (2.8).

In order to improve the accuracy of Yld91, Barlat et al. (1997) proposed a new

orthotropic criterion (denoted Yld96) of the form:

a3 (-1 2)m + ( 3)m 2 (3 -)m = 2Ym (2.19)

where al, a2, a3 are functions of the principal directions of Z, and are defined as

+k = a + yP2k + pk (2.20)









where p, are the ith component of the kth principal direction of the tensor E with respect

to the anisotropy axes of the material. Additionally, ax, ay, and az are three functions

given by

ax = axO cos2 20, + ax, sin2 20,

a, = a,' cos2 20, + 'a, sin2 202 (2.21)

a, = a,0 cos2 203 + a, sin2 203

where 01, 02, and 03 represent the angle between the major principal directions of E and

the axes of anisotropy, and the quantities axo, axi, ayo, ayi, azo, azi are anisotropy

coefficients. The yield function (2.19) reduces to the yield function (2.16) if each value of

a is set to unity, and further reduces to the Hershey and Hosford criterion (2.7) when L is

the identity tensor.

Barlat et al. (2005) showed that any pressure independent isotropic yield function

written in terms of the principal values of the Cauchy stress deviator can be generalized

to anisotropy through a linear transformation acting on the Cauchy stress tensor. The

principal values of the transformed tensor (2.22) will then replace the principal values of

the Cauchy stress deviator from the isotropic yield criterion.

X = Cs = CTr = Lu (2.22)

Here, X is the transformed tensor, C is an anisotropic linear tensor, and T transforms the

Cauchy stress tensor o into its deviator s. Barlat et al. (2005) also recently proposed an

orthotropic yield criterion (denoted Yld2004-18p) (2.23) which involves 18 anisotropy

coefficients. This orthotropic yield criterion is a generalization of the Hershey and

Hosford criterion in which anisotropy is introduced through two linear transformations

each containing 9 independent coefficients. The expression of this yield criterion is









1'l- sl" + 'l- y" + Yl- 3Y +

y-2 Y;l + -2 '2' + -2 '3' + (2.23)

'3 1-l + '3 "2 + '3 3- = 4Y'

where

' = Cs = C'To = La (2.24)

and

X" = C"s = C"To = L" (2.25)

When C = (or L' = L"), and the number of independent coefficients is imposed to be

6, Yld2004-18p (2.23) reduces to Yld91 (2.16). This criterion represents yield loci of

aluminum alloys with increased accuracy.

Another approach to extend any isotropic criterion to anisotropy is through

generalized invariants (Cazacu and Barlat, 2001 and 2003) using the theory of

representation of tensor functions (e.g. Boehler, 1978 and Liu, 1982). Using the

generalized invariants approach, Cazacu and Barlat (2001) extended Drucker's isotropic

yield criterion to orthotropy as follows:


()- c(J=)2 F (2.26)

where

S0 2, a 32 C ( 2 C 2
J2= a (ox -- +ya2 a (0 -_ C) +a4 + a5, +2 Z7- (2.27)
6 6 6









1 1
Jo 1 (b, + b2)O + (b3 + b4)y3 + 2bl ,, ,xzy + (2(b, + b4)- b2 -b)
27 27 27



1-UX [2b b b (2b b8 ]- [b-3b -
9 [(b, -b2 +b4x +(b1, b +b)]2 [2b01 b4b (x9 b8)u]
9 3

[2boz b 05, (2bj b)x]- O 6( bb,)7 b6 y bg]
3 3

All of the yield criteria discussed, both isotropic and anisotropic, make the

assumption that yield in tension and compression coincide. This basic assumption makes

these yield criteria inadequate for modeling hcp metals.

2.1.3 Modeling Asymmetry Between Tensile and Compressive Yield

To describe the asymmetry in yielding due to twinning, Cazacu and Barlat (2004)

proposed an isotropic yield criterion of the form:

3
f (2 cJ3 =F (2.29)

where


c= ( ) 3 (2.30)
2(o +o-)

CT and oc being the uniaxial yield stresses in tension and compression, respectively.

Note that for equal yield stresses in tension and compression: c = 0 hence the proposed

criterion reduces to the von Mises yield criterion. For the isotropic yield function (2.29)

to be convex, the constant c is limited to a given numerical range: c e [-3,3/2, 3,/3/4].

For any c # 0, the yield function is homogeneous of degree 3 in stresses and equation

(2.29) represents a "triangle" with rounded corners. This isotropic yield criterion was

extended to include orthotropy using the generalized invariants approach and applied to

the description of magnesium and its alloys. Having a fixed degree of homogeneity (of









order 3), this criterion is not flexible enough to represent yielding of certain hcp alloys

which have nearly elliptical yield loci. A yield criterion capable of capturing both

anisotropy and yield asymmetry between tension and compression of such materials is

needed.

2.2 Hardening Laws

The plastic behavior of metal for a multi-axial state of stress is described by a yield

criterion, a flow rule, and a hardening law. The two most common hardening laws are

isotropic hardening and kinematic hardening (see Figure 2.2).


Isotropic Kinematic
hardening hardening





lI I









Figure 2.2 Pictorial description of various hardening rules

Isotropic hardening assumes that the yield surface maintains its shape, but it

expands with accumulated plastic deformation. Generally, the size of the yield locus is

given in terms of a scalar hardening variable such as the effective plastic strain in

equation (2.31)

f (7) Y(p) = 0 (2.31)









where f(a) is the yield function that depends on the Cauchy stress, and Y(sp) is the

hardening function that depends on the effective plastic strain, s The effective plastic


strain is an invariant of the plastic strain tensor, i.e., Ep = a tr where a is a

constant that is determined such that E reduces to the strain in the loading direction for a

uniaxial test. Due to its simplicity, isotropic hardening is the most common method used

in sheet metal forming simulations (Yoon et al., 1999, 2000, and 2004). Since the yield

loci expand without changing shape or the location of their center, isotropic hardening is

a good approximation for monotonic loading along a certain strain path for materials

deforming by slip. This model can not represent phenomena such as the Bauschinger

effect or different strain paths hardening at different rates due to deformation twinning.

The Baushcinger effect is a common phenomena in metals, and occurs when a material is

deformed up to a given plastic strain, then unloaded and loaded in the reverse direction.

The yield strength after the strain reversal is lower than it would have been before the

first deformation step.

Introduced by Prager in 1955, kinematic hardening allows for a translation of the

yield surface without changing its size or shape. Thus, if the initial yield surface

f(o) = 0, then for a given plastic state, the yield condition is given by

f( a) = 0 (2.32)

where a (a second order tensor called the backstress tensor) defines the updated center of

the yield surface. This model can be used to represent phenomena like the Bauschinger

effect due to load reversals. Kinematic hardening can be used in combination with

isotropic hardening to describe both expansion and translation of the yield surface during









plastic deformation. Models such as those proposed by Teodosiu et al. (1995) and Li et

al. (2003) take into account other microstructural phenomena associated with changes in

strain path, such as the evolution of dislocation structure in cubic metals, through the

addition of other tensorial hardening variables to the kinematic hardening model.

As previously mentioned, twinning produces a major reorientation of the grains.

The blockage of further slip or twinning due to grain reorientation results in higher work

hardening rates for strain paths where twinning is the dominant mechanism as compared

to strain paths that would involve only slip. Therefore, due to the fact that the hardening

rate and the evolution of the texture depend on the strain path, the evolution of the yield

surface for an hcp metal is highly anisotropic. In order to correctly model hcp metals, a

hardening rule would need to allow for the distortion of the yield surface due to the

evolving texture with accumulated plastic deformation.

2.3 Survey of Mesoscale Plasticity Modeling (Polycrystalline Models)

Crystalline structure is a key factor in the mechanical response of a metal since the

individual crystals are anisotropic in both their elastic and plastic behaviors. Plastic

deformation mechanisms within a single crystal, such as slip and twinning, are linked to

crystallographic planes and directions, making the crystal strength inherently anisotropic.

If a polycrytalline material contains a large number of grains whose lattice orientations

are randomly distributed, the strength of the polycrystal would exhibit little if any

anisotropy. However, as a result of thermomechanical processes such as cold-rolling,

most metals display crystallographic texture, i.e. a patterned or a non-random lattice

orientation. Therefore, the anisotropy of the polycrystal is directly related to the

anisotropy of the single crystal and the distribution of the crystal lattice orientation.









Polycrystalline models that derive the polycrystal behavior from that of its

constituents using homogenization techniques have been proposed. The first step in

constructing such models is defining a representative volume element (rve). This volume

element must be small enough to be regarded as having uniform properties (including

orientation) such that stress and strain distribution within this volume be treated as

homogeneous. For metallic materials, a single crystal is considered as an rve. The next

step is to give a description of the behavior at the rve level and the interaction laws

between each grain and its surroundings. Polycrystal models typically neglect elastic

strains. The constitutive law of the single crystal consists of a kinematical relationship

and an energetic assumption. The kinematical relationship, further described in the next

section, relates the velocity gradient with the deformation rates of all active slip or twin

systems within the crystal. There are several versions of the energetic assumption,

including those of the Taylor model and the vpsc model which will be discussed in

sections 2.3.2 and 2.3.3, respectively.

Assuming that the rate of deformation and stress distribution is known for each

single crystal, the rate of deformation and the stress for the macroscopically

homogeneous polycrystal is given by a volume average over each crystal.

D =(DC) and a =(c) (2.33)

While equation (2.33) is a straightforward problem to solve, its inverse of partitioning a

given macroscopic component into the grain-level components, depends on the

assumptions made about the interaction of the grain with its surroundings. The most

common assumption is that of Taylor (1938) for which each grain in the polycrystal is

subject to the same strain rate as that applied to the overall polycrystal, which enforces









compatibility but results in an upper bound approximation of the stress. In Self-

consistent models such as the vpsc model the interaction of each grain with its

surroundings is based on the geometry of each grain and the average properties of the

polycrystal. The self-consistent models do not require that the strain distribution is

constant within the polycrystal which is a much more accurate assumption than the

Taylor assumption. Furthermore, the Taylor model assumes that a fixed number of slip

systems are active for a given plastic deformation, while rate-dependent models like the

vpsc model do not. Therefore, the vpsc model can better represent materials that possess

a limited number of available deformation systems such as hcp metals.

Since polycrystal models can track the lattice rotation of each individual grain, the

material anisotropy is naturally evolutional, which makes this approach very attractive.

These models can also be very useful for providing information about yield behavior for

stress paths for which no experimental data is available. For example, if the single

crystal properties and initial texture of a given material are known, the effect of texture

evolution caused by plastic deformations on yielding can be studied. Furthermore, the

effects that certain mechanisms such as twinning have on the yield behavior of materials

can be explored. In the following, we present the Taylor model followed by the vpsc

model that will be further used to obtain information about the evolution of yield loci.

2.3.1 Kinematics of a Polycrystal

Within a single crystal, slip (see Figure 1.1) along a given plane with a unit normal

vector n causes a displacement of the upper half of the crystal with respect to the lower

half in a certain direction of a unit vector b (commonly referred to as the burgers vector )

(see Figure 2.3). Deformation twinning is similar in its kinematic aspects (Figure 2.3) in

that it acts on a given plane in a particular direction. The processes of both slip and









twining are considered to be 'simple shears' not 'pure shears' since they correspond to a

displacement in the direction of b on one side of a plane perpendicular to n, but do not

result in an equal displacement in the direction of n on the plane perpendicular to b.

Simple shears involve rotations, which cause the evolution of the texture during the

plastic deformation of a polycrystal.

















(a) (b)


Figure 2.3 Notations for (a) slip and (b) deformation twinning in a single crystal.

The velocity gradient for a crystal is given by the sum of the shear rates from each

slip or twin system. In the crystal coordinate system which is parallel with a set of

orthogonal crystal axes, the velocity gradient is given by

L, = Z bnj; (2.34)


In (2.34) Y" is the shear rate for a given slip or twin system (s). Thus, the rate of

deformation tensor, DC, which is the symmetric part of LC, and the spin tensor, W4, which

is the antisymmetric part of L are given by

D' = ZIm (2.35)
s









W, = s yq, (2.36)


where m (commonly referred to as the Schmid tensor) and q are defined as

s =1 (b,n + bn (2.37)

qs I = (bn, -bn,) (2.38)



When solving a problem incrementally, the incremental strain and incremental rotation

for a single crystal become:

dEc = dyn'm (2.39)


dwic = d dy"q, (2.40)


After each deformation step, the texture of the material is updated by equations (2.39)

and (2.40), thus the model is capable of capturing the effects of evolving texure.

2.3.2 Taylor Model

Assuming that plastic deformation only occurs by slip, the resolved shear stress

acting on a slip system (s) due to a general state of stress acting on a single crystal (c), is

given by

rz = cJcm (2.41)

where m" is defined in equation (2.37). In particular, for a single crystal subjected to

uniaxial tension, the tensile stress is a = F/A. The force acting along the direction of slip

is Fcos f and the area of the slip plane is A/cos a (see Figure 2.4). Therefore, the

resolved shear stress on a single slip plane is


rT = ac cosacos f


(2.42)









The activation criterion for a given slip system is given by Schmid's law which

states that slip will occur when the resolved shear stress on the slip system reaches a

critical value

r > r; (2.43)

where r' is the critical resolved shear stress for the slip system.


F
Slip plane
normal

SSlip direction














Figure 2.4 Geometry of a slip system within a single crystal

When a polycrystal deforms, the shape change in each crystal must be compatible

with that in the neighboring crystals. In order to satisfy this requirement, Taylor (1938)

assumed that all grains undergo the same shape change as the entire polycrystal. It has

been shown that to accommodate the five independent strain components necessary for

plastic deformation in a pressure independent material, five independent slip systems are

generally required (von Mises, 1928). Taylor then hypothesized that among all possible

combinations of five slip systems capable of accommodating the imposed strains, the

active combination is the one that would require the minimum amount of plastic work.









The plastic work per volume (neglecting elastic strains), dW, that is expended by the

active slip systems within a single crystal is:

dW = ojdc = o ldr"m = Tr, dy (2.44)
s s

Implicit in (2.44) is the assumption that the critical stress required for slip is the same for

all slip systems. Thus, the five active slip systems within the single crystal are the five

that give the minimum value of -dy, corresponding to the strain increment applied to
s

the crystal (Hosford, 1993).

Once the active slip systems in each crystal have been identified, it is possible to

determine the strains, stresses, and lattice rotations by equations (2.39), (2.42), and

(2.41), respectively, and thus update the texture. The assumption of uniform strain in all

grains irregardless of orientation leads to stress discontinuities at the grain boundaries.

However, this model has been used with success at predicting textures for large

deformations. Bishop and Hill (1951) later proposed a similar, mathematically

equivalent approach. Thus the polycrystalline model given by equations (2.41) (2.44) is

often referred to as the Taylor-Bishop-Hill model. The original formulation did not

include twinning, however, Chin et al. (1969) and Hosford (1973) incorporated twinning

in the Taylor-Bishop-Hill model. These authors assumed that twinning is analogous to

slip with the exception that twinning is directional, i.e. twinning only occurs if a positive

shear acting on a given twin system reaches a critical value.

2.3.3 Visco-Plastic Self-Consistent (vpsc) Model

In the vpsc polycrystal formulation, originally proposed by Molinari et al. (1987),

the polycrystal is represented as an aggregate of orientations with weights that represent

volume fractions chosen to reproduce the initial texture. Each grain is treated as an









anisotropic, visco-plastic, ellipsoidal inclusion embedded in an anisotropic, visco-plastic,

homogeneous effective medium (HEM) with the stress applied at the boundary of the

medium. The method is called self-consistent because the overall properties for the HEM

are determined from the known properties of the grains. The following description of the

vpsc model is based on the following references: Kocks et al. (2000), Molinari (1997),

Tome et al. (2001), Tome and Lebensohn (2004a and 2004b), and Asaro (1983).

The plastic flow on a slip system (s) within a particular grain is governed by the

rate sensitive law


=> = o -o (2.45)


where "s is the rate of shearing on the slip system, o0 is a material parameter, S' is the

deviatoric stress, ms is the Schmid tensors for the grain, and n is the strain rate sensitivity.

r- is the critical resolved shear stress for the slip system and may be represented by a

Voce-type hardening law such as


r: = + (r + 1 exp- Oo)l (2.46)


where rz, rl, 0o, and 06 are material parameters, and F is the accumulated plastic

strain for the deformation system. In rate-dependent formulations such as the vpsc

model, all available deformation systems are considered active. However, in practice, for

large values of n (n >> 1), yielding would appear to occur abruptly as rs > rf in equation

(2.45). For rs < i- the corresponding P" is very small when n >> 1.

The plastic strain rate for the grain is given by the sum of the shears contributed by

all systems (assuming that elastic deformations are negligible)









D' = "m (2.47)

which can be combined with (2.45) to give

's ms:SI m m mL :S,
D om = oom Km c {m S pc( se c)S (2.48)
Sc j c s c j

where

pc(sec) *,y^" f lf m:ST,14
k =0 ki M i: SC (2.49)


here Dc and PC(sc) are the strain rate and the visco-plastic secant compliance for the grain,

respectively. P(sec) is not a material property of the crystal since it depends on the stress

state, except for when n = 1. When the stress is uniform within the grain, (2.48) is exact,

however, a linear relation valid in the vicinity of a point So can be obtained through a

first order Taylor series expansion of (2.48) about the point So

5$DC
D((Sc) = D(Sco) + o- 'O (Sc Sco) (2.50)
Sc

which can be rewritten as

D (Sc tan)S DO (2.51)

where the tangent modulus is defined as

pctan) = Ic = nP(sec) (2.52)
Sc S

and the back extrapolated term

DCO = (pc(sec) pc(tan))ScO = (1- n)Dc (2.53)

The secant approximation (2.48) has been proven to be too stiff, giving results close to

the upper bound results. The tangent approximation (2.51) gives a much more compliant









response. Similarly, at the polycrystal level, the overall strain rate and stress are related

through either a secant or a tangent relationship.

D = Pse: S (2.54a)

or,

D = "tan :S +D= nPsec :S+ D (2.54b)

and the macro-scale polycrystal compliance is a function of the overall stress. Within the

HEM, the local response of the medium is also governed by the macro-scale polycrystal

compliance such that

D(x)= Pse : S(x) (2.55a)

or

D(x) = nPse : S(x) + D (2.55b)

such that x represents the physical coordinate system of the polycrystal. Self-consistent

models impose equilibrium on the grain-to-HEM interaction, but not on grain-to-grain

interaction. Therefore, solving for equilibrium for a grain whose constitutive response is

given by (2.48), embedded in an effective medium (response given by 2.55) leads to the

interaction equation

(DC D) = Mc : (S S) (2.56)

where

S= neff(I E) : E : Psec (2.57)

Here, MC is the accommodation tensor and E is the Eshelby tensor which is a function of

the overall compliance of the polycrystal and of the geometry of the ellipsoid that









represents the grain. The parameter nff depends on the interaction between the grains

and the HEM, in particular n is related to the interactions indicated in (2.58).


1 -> Secant
neff = (2.58)
n Tangent
1 < ne < n Effective Interaction

When n = 0, the strain rate in the grain equals the strain rate in the polycrystal,

therefore the Taylor interaction is recovered. The fourth case allows for an effective

interaction between the secant and the tangent interactions. In particular, for the

simulations used in this dissertation research, n = 20 while neff= 10.

Combining equations (2.48), (2.54a), and (2.56), the stresses in a grain and the

stresses for polycrystal are related by

Sc = Bc:S (2.59)

where

Bc (pc(sec) +n(e E) 1 : E: Psec) : (Psec + neff E)1 : E : psec) (2.60)

Now, the condition that the weighted average of stress and strain rate over the grains

must equal the corresponding macroscopic magnitudes provides an expression from

which the macro-scale polycrystal secant compliance and macro-scale back extrapolated

term can be calculated.

Pse = pc(sec) : B) (2.61)


D0 = PC(sec) : + DC) (2.62)

where

0 = (pc(sec) +neff( E)1 : E : Pse) 1 : (Do DCO) (2.63)









and ( ) refers to the weighted average. Equations (2.60) and (2.61) are valid when nef

is constant for each grain within the polycrystal, and each grain has the same shape.

Equation (2.61) indicates that the polycrystal compliance is given by a weighted average

of the single crystal compliances and the localization tensor, BC. However, BC is a

function of Psec (see equation 2.60), therefore, equation (2.60) represents an implicit

equation from which Psec must be obtained iteratively.

Twinning is incorporated into the vpsc model by assuming that it is analogous to

slip, in that a twin system has a critical resolved shear stress that will activate

deformation. However, twinning differs from slip in its directionality, since twinning

will only be activated by a "positive" shear. The fact that twinned regions contribute to

the texture of the aggregate, and more importantly, act as effective barriers for further slip

and twinning is also taken into account by the vpsc model through latent hardening

coefficients coupled with the Voce hardening law for each deformation system. The

critical resolved stress for each system is updated by (2.64) using the latent hardening

coefficients (h""') which account for the effect of dislocations caused by other slip or twin

systems (s') on the current system (s).


Ar C -h'A)Ay" (2.64)


A predominant twin reorientation scheme proposed by Tome et al. (1991) is also

incorporated into the model that selectively reorients certain grains affected by twinning.

Under this scheme, the shear strain contributed by each twin system within each grain is

tracked, and the sum of all twin systems over all grains associated with a given twin









mode are calculated. Some grains are fully reoriented during an incremental step once a

threshold value is accumulated for a given twin system.

Since the overall properties of the polycrystal are not known a priori, the vpsc

model must be solved iteratively as follows:

* Given: D and a time step At

Outer loop:

* Estimate an initial stress in each grain by enforcing a Taylor interaction

* Calculate pc(sec) (2.49) and DCO (2.53) for each grain

* Estimate the Psec and DO as the averages of the corresponding grain values

Inner loop:

* Calculate the Eshelby tensor based on the estimated Psec and the current shape of
the ellipsoidal grain.

* Calculate MC (2.57), BC (2.60), and 0 (2.63)

* Calculate Psec (2.61) and DO (2.62) and compare to the estimation from the outer
loop. If they are within a tolerance, exit this loop. If not, use this updated Psec and
DO to restart the loop and iterate until the tolerance has been met.

End of inner loop

* Use Psec and DO from the inner loop to calculate S (2.54)

* Calculate M (2.57)

* Using (2.59) and (2.48) calculate DC (2.51)

* Calculate the weighted average stress and strain rate from each grain

* Calculate the overall stress (2.55)

* Compare the average stress and strain rate with the overall stress and imposed
strain rate.

* Compare the previous stress in each grain to the recalculated stress in the grain.









* If all three are within a tolerance then exit the outer loop, if not use the new value
of the stress in each grain and return to the beginning of the outer loop

End of Outer Loop.

Once convergence has been obtained for the stress and strain rate in each grain, the

hardening for each deformation system is updated using the Voce hardening law (2.40),

and the grain orientation and grain shape are updated as well. The process is then

repeated for the next strain increment.

The vpsc model can better model low-symmetry materials such as hcp metals that

are characterized by a variety of active deformation modes present in each grain, non-

negligible twinning activity, and significantly anisotropic single crystals (Kocks et al.,

2000). Therefore, this model was used in this dissertation to determine the yield

properties for both isotropic and anisotropic materials.














CHAPTER 3
PROPOSED YIELD CRITERION

3.1 Proposed Isotropic Yield Criterion

Twinning and martensitic shear are directional deformation mechanisms, and if

they occur, yielding will depend on the sign of the stress (Hosford, 1993). Early

polycrystal simulation results by Chin et al. (1969), who analyzed deformation by mixed

slip and twining in fcc crystals, predicted a yield stress in uniaxial tension 25% lower

than that in uniaxial compression. Hosford and Allen (1973) extended the calculations to

other types of loading. Based on the simulation results they concluded that yield loci

with a strong asymmetry between tension and compression should be expected in any

isotropic pressure insensitive material that deforms by twinning or directional slip.

An isotropic yield criterion capable of describing strength differentials between

tension and compressive yield is proposed of the form

(S, -kS) + (S, -kS2) + (S3 -kS3) =F (3.1)

where S,, i = 1... 3 are the principal values of the Cauchy stress deviator. At difference

with the yield criterion (2.29), the proposed yield function (3.1) is a homogeneous

function in stresses of degree a, which could range from 1 to oo. Also, in (3.1) k is a

material constant, while F gives the size of the yield locus and depends on the chosen

hardening rule. The physical significance of the material parameter k may be revealed

from uniaxial tests. Indeed, according to the proposed criterion (3.1), the ratio of tensile

to compressive uniaxial yield stress is given by









1

(l+k) +2 (1 k)
33 3
CT ( -)a-(I -)a(3.2a)
c ((1-k) )a (1+,


or




k = (2-/-c) 2 (3.2b)

1+ 2a -2(c /1c)a a
(2 cT "- c)a -2


Hence, for a fixed, the parameter k is expressible solely in terms of the ratio o-T/oc (see

3.2b). Note that for any value of the exponent a, if k = 0, there is no difference between

tension and compression. In particular, for k = 0 and a = 2, the proposed criterion

reduces to von Mises yield criterion. From (3.2b) follows that for a given exponent "a",

for the parameter k to be real, -o, 1c should belong to

1-a a-1
2 a <: T /-c < 2 a (3.3)

Specifically,

l-a
* For 2
a- 1
* For 1 0< k<1

As an example, in Figure 3.1 are shown the representation in the plane stress yield loci

(3.1) corresponding to a =2 (fixed) and a /oac= -, 1.26, 1.13, and 1 (von Mises),

respectively (i.e., corresponding to k =1; 0.4; 0.2; 0, respectively). Note that the highest

the ratio between the yield stress in tension and compression, the greater is the departure









from the von Mises ellipse; for the highest admissible value for k, the yield function (3.1)

represents a triangle with rounded corners.

Furthermore, k(o-/o-) =-k(cr/or) (see equation 3.2b). To illustrate this

property of the proposed yield function, in Figure 3.2 are represented the plane stress

yield loci (3.1) corresponding to o,/cc = 1.13 (k = 0.2) and o,/loc = 1/1.13 ( k = -

0.2). It is clearly seen that a change in the sign of k results in a mirror image of the yield

surface.

The variation of o-, /uo with k is illustrated in Figure 3.3 for different values of the

exponent a. If k =1 then o,/loc = 2(" 1)a, so for a = 1, c, = ", while for a-> c,

o-c/c -2. Ifk= -1 then /lo-c = 2(1 ")", so for a = 1 there is no difference between

tension and compression, while if a-> oo, then /loc -> 1/2. For any value of the

exponent a and for -1
section 3.1.2).

Figure 3.4 shows the representations in the deviatoric 71 -plane of the proposed

yield loci (3.1) for various values of the coefficient k between 0 and 1 and a = 2 (fixed),

along with the von Mises and Tresca yield loci for comparison. As k increases, the ratio

o, /o c is increasing and the yield loci depart drastically from the circular von Mises

locus.










1.5


a=2






0.5 -
















k = 0.2

S k = 0 (von Mises)
-1. ---------------------------

-1.5
-1.5 -1 -0.5 0 0.5 1 1.5

1 T
Figure 3.1 Plane stress yield loci for different values of the ratio o-, /o between the
yield stress in tension and compression, in comparison with the von Mises
locus (o-1. and U2 are the principal values of the Cauchy stress).













a=2


-0.5


-1------------- ^^>--^------r---------------^----------------
-1


-1.5~~~~- --------- ----------

-1.5
-1.5 -1 -0.5 0 0.5 1


1 T


Figure 3.2 Plane stress yield loci corresponding to a, /C o
1/1.13 (k =-0.2).


1.13 (k= 0.2) and a, /ac


/ /


/










2.25

2.2 -------------------------------------------
2 a=100


a=5
1.75
a=3













S-.25- ---- 0-------- -----
0 .7 5 ^-- ^-------- ^ ^ ^/------------- - - - - - -







-1 -0.5 0 0.5 1

k
Figure 3.3 The influence of the value of the parameter k on the ratio c-, / ac of the
uniaxial yield stress in tension and compression, for various values of the
exponent a.












k=O (von Mises)


Tresca


Sk=0.2
k=0.4
k= 1











1 2






Figure 3.4 Projection in the deviatoric r" plane of the yield loci (3.1) for a= 2 and
various values of k in comparison to von Mises and Tresca loci.









For combined tension and torsion conditions where the uniaxial tensile stress is set

equal to o, the shear stress is set equal to r, and all other stress components are zero,

the proposed yield criterion becomes



6 4 4 3


Figure 3.5 shows the representation in the tension-torsion plane (o- / o-, r / ,)of the

proposed yield loci corresponding to a fixed value of a (a = 2) and several different

values of k. Note the clear deviation from both Tresca and Mises criteria for k different

from zero.

It is also worth noting that the proposed yield criterion (3.1) is capable of predicting

ratcheting due to shear loading reversal. Indeed, inspection of Figure 3.5 indicates that a

loading in pure shear could produce plastic strains along the axial direction. In order to

5f
predict such a phenomena 0, where f is defined by equation (3.1), when all


stresses equal zero except the shear stress r. The principal values of the deviator of the

Cauchy stress tensor are defined as


S = 2 cos(a1 )
V 3


S2 = 2cosL r-2 (3.5)



S3 =2cos a, + 2


where a, is the angle satisfying 0 < 3a, < and whose cosine is given by









3
J3 (3.6)
cos(3a,) = (3.6)
2 J2

where J2 and J3 are the second and third invariants of the stress deviator (see Malvern,

1969).

Therefore,

f =f as, [a J2 3a, a0 as, aJ2 (37)
a0x aS, a aJ2 ax aiJ aoJ aJ2 aO C


but for the case when only one shear stress component is non-zero, = 0, so equation


(3.7) reduces to the following form

af af aS, Oa J(3.8)
00o- as, Oa J,003 xo

After substitution of (3.1), (3.5), and (3.6), equation (3.8) reduces to the form given by

equation (3.9) when all stresses equal zero except for one shear stress component, r .

Of a(I-k)(r -kr) a(-l-k)(r +kr)
+ (3.9)
dc, 6 6

Figure 3.6 shows the plot of the variation of f/0o- with k given by equation (3.9).

Note that according to the proposed criterion, pure shear will result in axial plastic strains

for any case other than k = 0. The direction of the axial strains is independent of the sign

of the applied shear stress, thus the proposed criterion (3.1) can predict a "ratcheting"

effect due to shear loading reversals in the presence of a strength differential between

tension and compression.











k=0 (von Mises)
Sk=0.2
0.6 -V --
k=0.4

0 5 - -
0.5


0.4 k
k=l1


Tresca


0 0.2 0.4 0.6 0.8 1

T
Figure 3.5 Projections in the tension-torsion plane of the proposed yield loci (3.1) for
various k -values and a = 2 (fixed), in comparison with Tresca and von Mises
(k=0, a=2) loci.







44



1.5




1




0.5 -



O\f
0o" 0 -




-0.5 -
--------- -- -- -- -- -- --- -













-1




-1.5
-1 -0.5 0 0.5 1

k



Figure 3.6 vs. k for the case of pure shear (a = 2).
do,









3.1.1 Comparison to Polycrystal Simulation

Since no data is available on the yield behavior for an isotropic pressure-insensitive

material, the vpsc model was used to calculate the initial yield loci for randomly oriented

fcc polycrystals deforming solely by {111 112) twinning, bcc polycrystals deforming

solely by {l12_ l1) twinning, and hcp polycrystals deforming solely by tensile

twinning {1012}1011) and compressive twinning { 122}1123). Due to the polarity of

twinning, this type of simulation will produce an isotropic yield locus that has different

yield strengths in tension and compression. The orientation distribution function

describing a random orientation of the crystallographic texture was constructed by

varying the Euler angles which describe the orientation of each crystal by

A(, A cos 0, A(p, using the notation of Bunge (see Kocks et al., 2000).

To demonstrate the predictive capabilities of the proposed isotropic criterion, we

compare the yield loci obtained using the proposed criterion (3.1) with the isotropic yield

loci calculated using the vpsc polycrystal model described in the section 2.3.3. The

proposed yield condition (3.1) involves 2 parameters: the exponent a and the parameter k,

which for a fixed is expressible solely in terms of the cr, /Ir ratio (see equation 3.2).

The vpsc model predicted a ratio of 0.83 between the yield stress in tension and

compression for the randomly oriented fcc polycrystal deforming only by twinning.

Assuming a = 2, we obtain k = -0.3098 for the proposed criterion. Figures 3.7 (a) and (b)

show the yield stresses (open circles) obtained using the vpsc model and the projection of

the yield locus predicted by the proposed criterion (3.1) for a = 2, k = -0.3098 (solid line)

for plane stress (Gxy = 0) and on the 7t-plane, respectively. It is clearly seen that the










proposed isotropic criterion describes very well the asymmetry in yielding due to

activation of twinning. On the same figures are shown the comparison between the yield

loci obtained with the VPSC model for randomly oriented bcc polycrystals deforming

solely by twinning (solid circles) and the yield loci according to the proposed criterion

(3.1) with a = 2 and for k = 0.3098 (which correspond to a ratio between the yield stress

in tension and compression of 1.20, which is the reciprocal of the value corresponding to

fcc polycrystals). Figures 3.8 (a) and (b) show a comparison between the yield loci

obtained using the proposed criterion (for a = 3 and k = -0.0645) with the yield loci for

randomly oriented hcp zirconium polycrystals deforming solely by tensile and

compressive twinning calculated using the VPSC model. Again, the strength differential

effect is very well captured.

15
a) i b)


05 -

U







-1 5
5 -1 -05 0 05 1 5



Figure 3.7 Comparison between the vpsc yield locus for randomly oriented fcc (open
circles) and bcc (closed circles) polycrystals deforming solely by twinning and
the predictions of the proposed criterion: (a) for plane stress (cxy = 0) (b) on
the -plane.
Fiue37Cmaionbtentevs iedlcsfrrnolyoine e oe
cice) n cc(lse icls olcytasdfrmin soeyb winn n






47


15
a) b)







-05-
1 5







zirconium polycrystals deforming solely by twinning (open rectangles) and
the predictions of the proposed criterion (3.1): (a) plane stress ( 0) (b) on




the 7t-plane.

3.1.2 Yield Surface Derivatives and Convexity

The associated flow rule used to obtain the plastic strain increments is given by

equation (2.11), but restated here as
de dA
a o/o









Other e, it is necessary to determine the first derivatives of the yield criterion. The

proposed isotropic yield criterion (3.1) is of the form


S3.2 kSel + Su kSface D a and Convey F









where for the general 3-dimensional case, the principal values of the deviator of the

Cauchy stress tensor can be defined as

S1 = 2 cos()

ra3
S, = 2 cos(ar)J

23 3










S3=2cos al + -2 J


where S,> S > S3 and a, is the angle satisfying 0 < 3a, < z and whose cosine is given

by

3
/ ^ J3 3 i2
cos(3 ) =-
2 J2

where J, and J3 are the second and third invariants of the stress deviator. Note that for

a = 0, z/6, and z/3 the state of stress corresponds to uniaxial tension, pure shear, and

uniaxial compression, respectively. Similar to equation (3.7), the general form of the first

derivative of the proposed isotropic yield criterion is

aF OF OaS, [ aa aJ, + a aJ, +S, 8J, a (3.10)
00 ask Oa L 2 9OI J 00-y J2 9O-,

where,


O a=Sk k Ska- Sk k j


SS aS2 2z J2 8S3 2z 2
a, = -2 sin a J2 2 -2 sin a 3 -2sin a + ,
aa 3 da 3 3 a 3) 3

1 1 1
as, 1 J. 2 8S2 1 29 J2 2 OS3 1 29( J2 2


3
a V3 3 1 a 1 3 )
OJ2 2 2 sin 3a' aJ3 6 sin 3a J2


= S,, where S = -kk









aJ3 2
S= S kSk 2
00-, 3

Clearly, singularities exist when a = 0 and a = ;r/3 in the terms Oa/8J2 and a/8JJ3

When using shell elements in finite element models or for calculating the r-value

expressions (2.13), plane stress conditions (o,3 = o = 023 = 0) can be assumed. For

plane stress, the computation of the first derivatives is simplified. Now, instead of using

relations (3.12) and (3.13), the following relations may be used to determine the principal

values of the deviator of the Cauchy stress tensor

2 1 2 1 1 1
S1 = l 2, S2 =-U2 1 S3 1 2
3 3 3 3 3 3

where,

U -11 + 0U22 1 2 2
1 2 + (O- +022) +(012)


U 2 ( (011 +22)2 +(012)2
2 4


The general form of the first derivative of the proposed isotropic criterion for plane stress

conditions becomes

OF F Sk -u, (3.11)
, 1 Sk a0-, 0-,

for which no singularities exist.

In order for a material to be a stable plastic material, a yield surface must be

convex. The convexity of a yield surface also guarantees a unique relationship between

the stresses and plastic strain increments assuming an associated flow rule (Malvern,









1969). For the yield function to be convex, its Hessian matrix must be positive semi-

definite. Let H be the Hessian matrix, i.e.,


HY 2f (3.12)
d0, d-
where i,j =1...3 and o, are the principal stresses. We shall prove that for k e [-1,1] and

any integera > 1, the proposed yield function (3.1) is convex. Isotropy dictates three fold

symmetry of the yield surface, thus it is sufficient to prove its convexity for stress states

in terms of the principal stresses corresponding to o, > -2 > U- .

For 0 < a, < n / 6, the principal values of the deviator of the Cauchy stress tensor

are S, > 0, S, <0, S3 <0, and


H,, = a(a -1) 4(1-k) s- 2+(1+k)(-1) (S-2 +S 2)
9 2

H22a(a= 9 {(m-k)" So 2 +(I+ k)" (-)a (4S2 2+S2
22 9 2 1)


H33 a(a
9

12 a(a-
9

H13 a(a-
9

H23 = a(a-
9


S(1-k)0S 2+(1+k)_( 1)O(S +4S' )}
-1) {(1- k)Y S0-2 + (1 + k) (1)a (2S-2 + 4S3 2)


1){2(l-k)OS'- (l+k)(-l) (2S2 S:2)}


1){-2(1 -k) S10 (1+ k) (-1)a (-S2 + 2S2 )


){( 2 (1 k)"S2+(+k) (-1) (-2S2 2S 2)}


(3.13)


3
Note that Y H, = 0, for any i =1, 2, or 3. Thus, the determinant of H is zero and its
j=1


principal values are AZ, 12, and A3 = 0. Furthermore,









tr(H) = +1 +4- 6a
9


a2 (a 1)2 (1 k)2a (2)a-2 + (1+ k)2a (S)a-2 +(I- k2) (S)a-2 (-S2) a-2+
9 3- 2 3(- 2 )a a-2 (-S1a-2 + 3(1+k)2a (- 2 a-2 (-)a-2

Since S, > 0, S2 <0, S3 <0, it follows that for k e (-1, 1) and any integer a > 1:


tr(H) = i+A2 > 0 and tr2(H)

definite.



For the case ;- /6

H = a(a-
9
H22 = a(a

9

H33 a(a-
H,, =


9
12 a(a-

9

H13= a(a-
9
H23 a(a -
9


3
It follows that H,
j=1


{(1-k)_(-2Sr'


1) {(1- k) (S-


1A2 1> 0, i.e., the Hessian is always positive semi-


-2+S22) 2(1+ k) (-1)aS2}


-2S2-2) 2(1+k)" (1)" S2}


(3.14)


0, for any i =1... 3. Thus, the determinant of H is zero and


tr(H)=6a(a -1){(-k)a(Sa-2 +S -2)+(1+k)a(_)aSa-2
9


1){(1 k)0(4So2+S2 -2)+(l+k) (l1) S2

) {(1 k) (So2 +4S 2)+(1 + k)_ (-1) S 2}


-k) (So-2 + S-2 )+ 4(1+ k)_ (1)a S 2}


1){2(1 -k) (S-2 + S-2)+(1+ k)( (- 1) Sf2


k)a Sa-2 + (1 + k)a (-1)a (a-2 + Sa-2)}









a2 (_ 12 1 2 a S)2 a-2 + (- k)2a ( )a-2 + (1- k)2a (S)a-2 (S2)a-2
9 L3(1+ k)2a (S2-2+ 3)a(1+k)2a (Sa)-2 a-2 S3 a-2



Since S, > 0, S > 0, S3 < 0, it follows that for k e (-1,1) and any integer a > 1:

tr(H) > 0 and tr2 (H) > 0. Thus, for k e [-1,1] and any integer a > 1 the yield function

is convex.

3.2 Extension of the Proposed Isotropic Yield Criterion to Include Orthotropy

To describe both the asymmetry between yield in tension and compression and the

anisotropy observed in hcp metal sheets, we extend the proposed isotropic criterion (3.1)

to orthotropy. For the description of incompressible plastic anisotropy, Cazacu and

Barlat (2001, 2003) introduced a general and rigorous method which is based on the

theory of representation of tensor functions (see equation 2.26). It consists in substituting

in the expression of any given isotropic criterion, the 2nd and 3rd invariants of the stress

deviator with generalizations of these invariants compatible with the symmetry group of

the material considered. However, with this approach, convexity is reinforced only

numerically. For this reason, a particular case of this general theory, which is based on

applying a fourth-order linear transformation operator on the Cauchy stress tensor or its

deviator, has received more attention (Sobodka, 1969; Barlat et al., 1991; Karafillis and

Boyce, 1993; etc.). It is worth noting that by using the linear transformation approach,

the convexity of the resulting anisotropic extension is automatically satisfied

(Rockafellar, 1974).

Following Barlat et al. (2005), orthotropy is introduced by means of a linear

transformation on the deviator of the Cauchy stress tensor, i.e., in the expression of the









isotropic criterion (3.1), the principal values of the Cauchy stress deviator are substituted

by the principal values of the transformed tensor Y defined as

1, = LklASki (3.15)

where L is a 4th order tensor whose coefficients assign a weight to different stress

components. Thus, the proposed orthotropic criterion is of the form

(1 -k1) + 2 -k2),f + -k3)Y = F (3.16)

where E,, 2, 23 are the principal values of 1. In the absence of any shear stresses, the

values of Exx, Ey, and Ezz are the principal values of 1. However, if the shear stresses are

present and a3 # 0, the principal values of Y are the roots of the 3rd order algebraic

equation

X3 HX + HX H = 0 (3.17)

where,

H, = Y-" + EY + Y,

H, = 2 Y-Y- + 2XYX + Y Y (2 +y 2 + 2), and

H3 = 2Y, YX Z + Y, Zy yz 2 _y2_ zz,2.

The tensor L satisfies the major and minor symmetry and the requirement of invariance

with respect to the orthotropy group. Thus, for 3-D stress conditions the orthotropic

criterion involves 9 independent anisotropy coefficients, and reduces to the isotropic

criterion (3.1) for L equal to the identity tensor.

Let (xy,z) be the reference frame associated with orthotropy. In the case of a sheet,

x, y, and z represent the rolling direction, the long transverse direction, and the short






54

transverse direction or the through thickness direction, respectively. Relative to the

orthotropy axes (x,y,z), the tensor Y (in vector form) is represented by

xx l 142 L 3 Sxx
yy L12 L22 L23 S,
zz L3 L23 L33 Szz
Zyz L44 Syz
zzx L55 Szx
xy _L66 xy

or in terms of the Cauchy stresses

2 1 1

Zxx 11 12 13 3 3 3 cxx
Yyy L12 L22 L23 yy
3 3 3
zz 13 L23 L33 1 1 2
yz L44 3 3 3 ryz
zx L55 1 O"zx
xy L66_ 1 Oxy
1


Taking into account that S is traceless, (3.15) can also be written as

Sxx 0 12 1 3 Sxx
yy 12 L22 0 L23 L22 Sy
zz '3 L33 L23 L33 0 Szz
Yyz L44 Syz
zx L55 Szx
Zxy _L66 Sxy







In the case of a thin sheet, plane stress conditions can be assumed. Using these
assumptions, the only non-zero stress components are the in-plane stresses

(o-,o, o- and the principal values of Y are

1 = yx + + yYxx 2 + (3.18)




:3 = 2z

where

x = 11-L12 L13 + 11 + L12 L3 y (3.19)
3 3 3 3 3 3

y L12 L22 3 L23> x ( L12 + L22 L23 i yy

S2 1 1 2 1
L = L13 L- 23- L33 + L13 + L23 L 33 J y
3 3 3 3 3 3

1xy = L66 Cxy

If o-' and o-c define the yield stress in tension and compression along the rolling

direction x, according to the proposed orthotropic criterion (3.16) it follows that


Cr0 = 21 F1- ] (3.20)



0 [|,1 +kDI++ll+]k, 2 + 1(;3 +k( }) 1






56

where



21


(2LA1 LA1L
1 = L- L12 3 (3.21)








Similarly, if c9o, and ,,,o are tensile and compressive yield stresses in the transverse

direction, y, then


SF (3.22)



90 1 1 + k 1 + T2 + k 12 + \\3 + kV 1f

where





T2 I 1 L2 -123

-f( 1l 2 _L 1

Y u L13n 23 is e 33

Yielding under pure shear parallel to the orthotropy axes occurs when a is equal to









f F V
T =--- --- (3.24)
L66 +kL66 + C66 +k] (3.24)

Yielding under equibiaxial tension occurs when a, and a, are both equal to


{- = \ _- io+- F _+ -_ t (3.25)
0b 1 kQ, a+ Q, kM, a + 03 knM,


while yielding under equibiaxial compression occurs when o, =o = o ,C


o- = F +kQl-+2 F 1 +kQ31- (3.26)
b 1I + kI, + 2, + kaa + 03 +3 k3 j

where,


Q = I1 + L12- -13 (3.27)



2 = L12 + I L222 L23)



3 =(I 13 + L 23 33


Furthermore, we assume that the plastic potential coincides with the yield function.

According to the proposed orthotropic criterion, the Lankford coefficients which are

defined as the width to thickness strain ratios in a uniaxial loading (see equation 2.13),

become

(1 k)0I( D T + (-1 k)((D "' l2j +3 'I'3).
T= (1 k) + k)" ', +3 '3(3.28)

(1 k k) + (-1- k)0(Y' + ', + Y +
r9o (1 k ) i 2 1((I2 2) ( I k ) a( Wj 1I ,1 1(I3 __ la )










(k1 (k)D-(W r) + (1 k) (1(D -Y12 +(DY 3I

C= _(-1- k)" ', + ( k)a(('. +3T '3)
S((-1 1k Y (, + ,) + (1 k)(, + Y +
'(_1_ k) (1(D + 2) + (1_ k)"(W D1I1 + W -1I3 + Wj + 3)


with ,D to 03 given by (3.7), Y, to Y3 given by (3.9), and the superscripts T and C

designating tensile and compressive states, respectively.

Using equations (3.20) (3.28), the coefficients for the yield criterion (3.16) can be

determined by minimizing an error function of the form


Error = weight ed + weight 1 rpedicted
O data data (3.29)

where the index n represents the number of experimental yield stresses available, and m

represents the number of experimental r-values. Each term has an assigned weight which

can be used to distinguish yield stresses from r-values. A better accuracy for the yield

stresses is usually required because a difference of a few percent in the flow stress is

much more significant than in the r-values (Barlat et al., 2005).

Although the transformed tensor Y is not deviatoric, the proposed orthotropic

criterion is nevertheless independent of hydrostatic pressure. In order to prove this

concept, we shall show that the derivative of the proposed orthotropic yield function, F,

with respect to the hydrostatic pressure, p, is zero. The proposed orthotropic yield

condition is

f(1,,e ,3)= (, i- kr)es ( + k y) k- + k

whose derivative with respect to hydrostatic pressure is









f Of am j (3.30)
p am acYi O 'p

where Yz are the principal values of the transformed stress tensor 1. We shall prove that


S=0, hence 0, i.e., the condition of plastic incompressibility is satisfied. Indeed,
Op Op

the transformed stress tensor Y can be expressed as

2= LS=LTo, (3.31)

where T denotes the 4th order deviatoric projection that transforms a 2nd order tensor in its

deviator. Thus,

ax
S= -Bjkk =-Bukk i, k= 1... 3, (3.32)
ap

where B = L T is the 4th order orthotropic tensor that relates the transformed tensor Y to

the Cauchy stress U. Relative to (xy,z), the tensor L is represented by

L11 L12 L13
L12 L22 L23
L13 L23 L33
L = (3.33)
L44
L55
L66

while Tis given by

2 -1 -1
-1 2 -1
1 -1 -1 2
3 3
3









In (3.33) we used the simplified contracted indices convention of Voigt was adopted

def def def
(L = 11; L12 = L1122; L13 = L1133, etc.). It follows that the non-zero

components of the 4th order tensor B are

B,, = (2L, L2 L3) /3

B,2 = (-LI, +2L2 L13)/3

B,3 = (-L~ L12 + 2L13)/ 3

B,2 = (2L2 L22 L23)/ 3

B22 =(-I2 + 2L22- L23)/3

B31 = (-L32+ 2L31 L33)/ 3

B32 = (-L3 +2L32 L33)/3

B33 = (- L3+ 2L33 L32 )/3

Hence, we obtain

B,, + B12 + B13 = 0
B21 + B22 + B23 =0. (3.34)
B31 + B32 + B33 = 0


Thus, -=0 and 0 satisfying the condition of plastic incompressibility.
8p ap

3.2.1 Application of the Proposed Criterion to the Description of Yielding of Hcp
Metals

3.2.1.1 Magnesium Alloys

Kelley and Hosford (1968) reported the results of an experimental investigation

into the anisotropy and asymmetry in yielding of textured polycrystalline pure Mg and









binary Mg-Th (0.5 % Th) and Mg-Li (4% Li) alloys. The data consists of the results of

uniaxial compression tests in the rolling, transverse, and normal directions, respectively,

uniaxial tensile tests in the rolling and transverse directions, as well as plane-strain

compression tests. Based on these data, the experimental yield loci corresponding to

several constant levels (1, 5, and 10%) of the effective plastic strain were reported (see

Figures 3.9, 3.10, and 3.11 where experimental data are represented by symbols).

Due to the mechanical processing of a cold rolled sheet, magnesium alloy sheets

have a strong basal pole alignment in the thickness direction. Therefore, the easily

deformed {O12}1011) tensile twin system about magnesium's c-axis is activated by

compression in the plane of the sheet. However, this twin system is not active due to

tension within the plane of the sheet. The effect of {0 1210 11) twinning is clearly

evident by the initially low compressive strengths with respect to tensile strengths at 1%

effective plastic strain. By 10% strain, the third quadrant strengths are comparable to

those in the first quadrant owing to the extra barriers to further deformation processes due

to the reoriented twins created by loadings in the third quadrant.

Figure 3.8 shows the section of the theoretical plane stress yield loci (equation

3.16) with ,xy = 0 for Mg-Th together with the experimental data reported in Hosford and

Kelley (1968). The constant a was set to 2 while the anisotropy coefficients involved in

the expression of the theoretical yield loci for biaxial stress states as well as the constant k

were determined using equations (3.20) to (3.29) and the data corresponding to the given

strain level. The obtained values of these parameters corresponding to the 1%, 5%, and

10% effective plastic strain surfaces are given in Table 3.1. Note that the proposed

theory reproduces very well the observed asymmetry in yielding.









The experimental yield loci for the Mg-Li alloy sheets are similar in shape to those

for the Mg-Th alloy, but with much reduced yield stresses due to the occurrence of prism

slip and to the weaker crystallographic texture. The effect of {1012 1011) twinning is

evident in the low compressive strengths at 1% and 5% strains. Figure 3.10 shows the

theoretical yield loci for Mg-Li along with the data reported by Kelley and Hosford. The

constant a was chosen to be 2 for this material. The coefficients involved in the

expressions of the biaxial yield loci are given in Table 3.2.

The yield locus for the textured pure magnesium has a highly asymmetrical shape

for the 1% and 5% yield locus due to twinning. Note the much greater strength in tension

than in compression and the higher tensile strength in the transverse direction than in the

rolling direction. The yield locus at 5% strain shows asymmetry similar to that of the

locus at 1% strain. At 10% strain, the third quadrant strengths are comparable to the first

quadrant strengths due to the hardening effects of {0 12}(10 1 ) twinning. Figure 3.11

shows the yield loci of the present theory with the 5 data points for each level of strain

given by Kelley and Hosford. The constant a was chosen to be 3 for the 1% and 5%

locus due to the asymmetry. This constant was chosen to be 2 for the 10% locus since

the yield locus becomes more elliptical. The yield locus generated by the present theory

is in good agreement with the published data. The coefficients involved in the

expressions of the biaxial yield loci are given in Table 3.3.










300


200




100


10%


-100




-200


-300
-300


-200


-100


200


300


xx


Figure 3.9 Comparison between the plane stress yield loci (or = 0) for a Mg-0.5% Th
sheet predicted by the proposed theory (solid lines) and experiments
(symbols). Data after Kelly and Hosford (1968). Stresses in Mpa

Table 3.1 MG-TH yield surface coefficients
a k L1 L12 L13 L22 L23 L33

1% 2 0.3539 1.0 0.4802 0.2592 0.9517 0.2071 0.4654

5% 2 0.2763 1.0 0.3750 0.0858 0.9894 0.0659 0.1238

10% 2 0.0598 1.0 0.6336 0.2332 1.4018 0.5614 0.7484













200





100


10%


-100





-200


-200


-100


100


200


xx
Figure 3.10 Comparison between the plane stress yield loci (Uo = 0) for a Mg-4% Li
sheet predicted by the proposed theory (solid lines) and experiments
(symbols). Data after Kelly and Hosford (1968). Stresses in MPa

Table 3.2 MG-LI yield surface coefficients
a k L11 L12 L13 L22 L23 L33
1% 2 0.2026 1.0 0.5871 0.6975 0.9783 0.2840 0.1497

5% 2 0.2982 1.0 0.6103 0.8056 1.0940 0.5745 0.1764

10% 2 0.1763 1.0 0.5324 0.8602 1.0437 0.8404 0.2946










200


150


100


50


10%


13 0


-50


-100


-150


-200
-200 -150 -100


0 50 100 150 200


Figure 3.11 Comparison between the plane stress yield loci (cr = 0) for a pure Mg
sheet predicted by the proposed theory (solid lines) and experiments
(symbols). Data after Kelly and Hosford (1968). Stresses in MPa

Table 3.3 MG yield surface coefficients
a k L1 L12 L13 L22 L23 L33
1% 3 0.6293 1.0 0.4349 -0.0513 1.0178 0.1294 0.3417

5% 3 0.4320 1.0 0.4123 -0.0119 0.8617 0.0570 0.2024

10% 2 0.0776 1.0 0.2807 -0.0338 0.9916 0.1219 0.6874


-----------









3.2.1.2 Titanium Alloys

In the following, we apply the proposed orthotropic criterion (3.16) to the

description of the anisotropy and tension-compression asymmetry of 4A1-1/4 02 textured

a (hcp) titanium alloy (data after Lee and Backofen, 1966). True stress-strain curves

were reported for different loading paths: uniaxial tension in the x direction (rolling

direction), uniaxial compression in the z-direction (through-thickness compression), and

plane strain compression in the z and y (transverse) directions. The material had nearly

ideal basal texture with a deviation of about 25 degrees from the sheet normal toward the

transverse direction. Based on these data, the experimental yield loci corresponding to

several constant levels of the largest principal strain were reported (see Figure 3.12,

experimental data are represented by symbols). Due to the strong basal pole alignment in

the direction of the normal to the sheet, {1012} twinning was activated by compression

perpendicular to this direction, but is no twinning was revealed in tension testing within

the plane (see Lee and Backofen, 1966). The effect of {10o12 twinning is clearly evident

in the low compressive strengths in the rolling and transverse directions.

Figure 3.4 also shows the theoretical yield loci along with the experimental data.

The coefficients involved in the expressions of the theoretical yield loci are given in

Table 3.4. Note the ability of the proposed criterion to correctly describe the asymmetry

in yielding of 4A1-1/4 02.

In order to account for the eccentricity of the yield surfaces of titanium and its

alloys, Hosford (1966) proposed a modification of the Hill criterion to include terms

linear in stress

A, + Bay + (-B A)c, + F(cy a, )2 + G(o, Ox)2 + H(o, oy)2 = 1 (3.35)









The linear terms in equation (3.35) allow for the center of the yield surface to shift, thus

allowing for different yield strengths in tension and compression. However, by adding

these terms, the criterion is no longer pressure-independent, nor is it as accurate as using

the proposed yield criterion. Hosford's (1966) yield function given by Eq. (3.35) was

applied to the same 4A1-1/4 02 textured a titanium alloy (see Figure 3.13). Comparison

between theoretical and experimental yield loci show that the proposed criterion (3.16)

describes with greater accuracy the yield behavior of the titanium alloy.

3.2.2 Derivatives of the Orthotropic Yield Function

The proposed orthotropic yield criterion (3.16) is of the form

1 k1)a + Z k) + 3 kY3- = F

where 1, 2, 3 are the principal values of the transformed tensor Yj = LJklSkl. For a

general 3-dimensional problem, it is necessary to develop an expression between the

principal values of I and its components in order to determine the derivatives of the yield

function. This expression can be developed through the use of the deviator of 1, denoted

by 2, since the cubic equation (3.17) would then lack the quadratic term upon

substitution of 2 for 1. Once an expression is obtained between the principal values of

2 and its components, the spherical component of I can be added to obtain a

relationship between the principal values of and its components.

Substituting 2 (E, = -5 1,/3, where I. is the first invariant of 1) into

equation (3.17) yields a relationship whose roots are the principal values of I

X3 -J2ZX J3 =0 (3.36)

where J22 and J32 are the second and third scalar invariants of I.. Solving for the roots










1500


1000


500




0




-500




-1000


-1500
-1500


-1000


-500


500


1000


1500


XX
Figure 3.12 Comparison between the plane stress yield loci (o- = 0) for a 4A1-1/402
sheet predicted by the proposed theory (solid lines) and experiments
(symbols). Data after Lee and Backofen (1966). (Stresses in MPa)


Table 3.4 4A1-1/402 coefficients
a k L1 L12 L13 L22 L23 L33

0.2% 2 0.1556 1.0 0.2285 0.0374 1.2967 0.2439 0.3244

1% 2 -0.1868 1.0 0.0431 0.3369 0.9562 0.3139 1.0861

4% 2 -0.2577 1.0 0.2178 0.3635 1.0422 0.3754 0.8825


1










1500




1000 -
150---------------- ---------------------------------


500--








-500 --- ----------- ---------- --
4% /^ 1%






0.2% data
-1000 -
A 1% data

S 4% data

-1500
-1500 -1000 -500 0 500 1000 1500
CY
XX


Figure 3.13 Comparison between the plane stress yield loci (Ur = 0) for a 4A1-1/4 02
sheet predicted by Hosford's 1966 modified Hill criterion (solid lines) and
experiments (symbols). Data after Lee and Backofen (1966). (Stresses in
Mpa)









of equation (3.36) and adding the spherical portion of E yields an expression for the

principal values of E


S= 2 cos(a,) +
F3 3


S2 = 2cos a,



3 = 2 cos (a1


2 +
3 3


3 F3 3


(3.37)


where Y, > 2 2>3 and a is the angle satisfying 0 < 3a3 <, ~ and whose cosine is given

by
3
cos(3a)=J-i 3-73 (3.38)


Now, the first derivatives of the proposed orthotropic yield criterion (3.16) can be

written as


aF F I B, aq
800, ax fO da aOJ2,


QCa 8J1
+ 1+
0i 3Z OY


axq aJ2Z'
0iy dZYk


+ CI az amns" (3.39)
aI a I as a,, ,
2a/' kijmn


where,


ax a kq Y1
dC^


S= -2sina '12,
ca N 3 c


2;r .f -Y
3 )


-2 sin a'


2 sin a+


2;r i 2-Y
3j^









c J2Z 2 BE,2 1 2 o2 2 Z(J 2
cosa -- ---cos (a- ,


cos a + J 2
COS + 3 3 2 -
3


3J3 J3 1 a _
4 (j ) a sin3a' J3 z


E, where = 3 -y ,
3 1


3
1 3 2
6 sin 3a J2 2Z


J1z 2 2 BI -
a x k kJ 3 O2 y a'

kk Lk 23 L 31 = L 55 12 = L all other k = O,
S kn S 44' 5 5' BS O66' aS
asn 23 S31 S 12 asmn

aS,, 2 --S, 1 for i j, = 1 for i j, all other Smn = 0.
auc 3' au- 3 ,j a ,

As was the case for the isotropic criterion (3.1), singularities exist for the orthotropic

criterion when a = 0 and a = ;r/3 in the terms aalJa2 and aal/J3.


a 3
0J 2Z

axI
ai,


aJ2,
8 J2














CHAPTER 4
PROPOSED HARDENING LAW

4.1 Introduction

Characterization of a metal's plastic response requires the specification of a yield

function and a flow rule by which subsequent inelastic deformation can be calculated for

specified loadings and displacements. Traditionally, the evolution of the yield surface is

described by a combination of isotropic and kinematic hardening laws. Isotropic

hardening implies a proportional expansion of the surface, without any changes in shape

or position. An isotropic hardening model is only truly valid for monotonic loading

along a given strain path assuming that every strain path hardens at the same rate. In the

case of simulation of sheet forming operations of cubic metals (both fcc and bcc), the

assumption of isotropic hardening is reasonably adequate (Yoon et al., 2004).

Pure translation of the initial yield surface could be described by the classic linear

kinematic hardening laws. To better model the smooth elastic-plastic transition upon

reverse loading, multi-surface models as well as non-linear kinematic hardening models

have been proposed. Recently, physically-based models of the evolution of the

anisotropic work-hardening of bcc materials (mild steel) under arbitrary strain path

changes that involve several tensorial hardening variables have been proposed (e.g.

Teodosiu et al., 1995 and Li et al., 2003). It is to be noted that none of these models

account for the evolution of texture during work-hardening. Due to non-negligible

twinning activity accompanied by grain reorientation and highly directional grain

interactions, the influence of the texture evolution on work-hardening of hcp materials









cannot be neglected even for the simplest monotonic loading paths. Existing macro-scale

phenomenological plasticity models cannot describe the experimentally observed change

in shape of the yield loci with accumulated plastic deformation.

A general framework for the description of yielding anisotropy and its evolution

with accumulated deformation for both quasi-static and dynamic loading conditions is

offered by polycrystal plasticity. Recently, many efforts have been undertaken to

incorporate anisotropy due to crystallographic texture into finite element simulations (e.g.

Tome et al., 2001). Direct implementation of polycrystal viscoplasticity models into FE

codes, where a polycrystalline aggregate is associated with each finite element integration

point, has the advantage that it follows the evolution of anisotropy due to texture

development. However such finite element calculations are computationally very

intensive, thus limiting the applicability of this approach to problems that do not require a

fine spatial resolution.

The objective of the present chapter is to propose a macroscopic anisotropic model

that can describe the influence of evolving texture on the plastic response of hexagonal

metals. Initial yielding is described by the proposed orthotropic yield criterion (3.16) that

accounts for both anisotropy and asymmetry in yielding between tension and

compression. Experimental measurements of the crystallographic texture for a given

material are used to calculate the flow stress in a finite number of loading directions

using the vpsc model. Then an interpolation technique is used to construct the evolution

of the yield surface. The anisotropy coefficients as well as the size of the elastic domain

are considered to be functions of the accumulated plastic strain. The proposed model was

implemented into the implicit FE code ABAQUS (2003) and used to simulate the three-









dimensional deformation of pure zirconium and AZ31B magnesium specimens subjected

to various loading conditions. Comparison between predicted and experimentally

measured macroscopic strain fields for pure zirconium shows that the proposed model

describes very well the contribution of twinning to deformation.

4.2 Elasto-Plastic Problem

Some assumptions of rate-independent plasticity theory include a yield function or

stress potential, Q, which separates elastic and plastic states,

Q=d-Y (4.1)

the additive decomposition of the total strain increment into an elastic and plastic part,

de = dee + de (4.2)

an associated flow rule relating the plastic strain-rate to the stress potential,


e =A JQ (4.3)


and Hooke's law in incremental form

do = Cde, (4.4)

where & is the scalar effective stress, Y represents the material's hardening, X is the

scalar plastic multiplier and is equivalent to the effective plastic strain, an overhead dot

represents the time derivative, and C is the elastic stiffness tensor. An effective stress

must be a homogeneous function of degree 1, and be able to reduce to the hardening

relationship (i.e., uniaxial tension for a given direction) under that state of stress. For

example, the effective stress based upon the proposed isotropic criterion (3.1) assuming

that the hardening relationship, Y, is based upon uniaxial tension about the x-direction is

given by










(a,r,) = A(S kS,)+(S2 -kS2)a +(S3S -kS3) (4.5)

where,

1


A 1+271l+kD (4.6)
2 2 11
-_k +2 -+-k
_(3 3 J 3 3

and for the proposed orthotropic criterion (3.16)


(ao,,)= B [(1 -k1) +(S, -k2) +(Y k3)l,] (4.7)

where,


B-- (I0 1 k()o + (1(3 k3) (4.8)


and,


1 = L,, L12 L13 (4.9)



2 = L2 L22 L23
S=l -3L,, 3 3


3 = L13 3 L23 3 L33).


Note that for the case of a = 2, k =0, A reduces to 3/2 which is the constant associated

with the von Mises effective stress.

The basic problem in elasto-plasticity is to obtain stresses that fulfill both the

Kuhn-Tucker conditions and the consistency condition. The Kuhn-Tucker conditions









require that A > 0, Q < 0, and AQ = 0, while the consistency condition requires that the

stress state remain on the yield surface during plastic loadings.

The usual starting point for the elasto-plasticity problem is the non-linear

differential equation

do = CePd (4.10)

whose solution is given by

'n+1
n,+1 =- + CePde = a, + Aon+ (4.11)


where n is a counter and stands for a certain time step of the deformation process. C' is

the elasto-plastic tangent modulus, and depends on the updated state of the problem.

Since Cp is unknown for the updated state, an iterative scheme must be applied.

The first step in an iterative scheme is to choose a starting point, and in elasto-

plasticity the usual starting point is to assume the stress state is purely elastic, i.e.,

o"+1 = ao + CA (4.12)

If this starting stress, commonly referred to as the trial elastic stress, satisfies Q < 0 then

the trial stress is accepted as the current stress state. If Q > 0 then the stress state must be

returned to the yield surface. The stress state is returned to the yield surface through a

plastic corrector step in which the yield surface also expands due to hardening. This

method is an implicit and direct method since the resulting equations become implicit in

the unknown variables, and the consistency condition is directly used to determine the

increment of effective plastic strain. This approach is also referred to as a return-

mapping method since the increment of the effective plastic strain is adjusted such that

the stress is returned to the yield surface.









4.3 Proposed Anisotropic Hardening Law

The proposed hardening law allows for the change in shape of the yield loci during

the deformation process by letting the yield criterion coefficients be functions of the

hardening parameter. Thus, the yield function would take the form

Q(o, p)= ~(a,E)- Y(p) <0 (4.13)

where & is the effective stress based on the stress potential, ois the Cauchy stress tensor,

Y is the effective stress-effective plastic strain relationship in a given direction (e.g. the

tensile rolling direction), and sp is the effective plastic strain which will be used as the

hardening parameter.

If Q(E$t, n+1) > 0, the effective plastic strain increment AA for global step n+1

must be determined to bring the stress state back to the yield surface through a local

iterative process (see Simo and Hughes, 1998). Denoting the elastic trial state according

to equation (4.12) as iteration i = 0 ( i being the local iteration counter), the stress

increment update for iterations i > 0 take the effects of the plastic strains into account

according to (4.14) using the flow rule (4.3).

AC7 = A' +8 = Ac'7 8,C (4.14)
Ao-j Aofn+ 1 + 3u~j= B An+ 1 3Azc+1 (4.14)
L on+1

In equation (4.14) -8AC is the stress correction due to the plastic strains, and 3
ao-

denotes the variation of the variable between local iterations i+1 and i, i.e.,

AA1 = AAi + 8A'\. The gradient of the stress potential based upon the unknown

updated state (iteration i+1) may be approximated using a Taylor series expansion about

the previous state and the variation of AA during the previous iteration (iteration i) by









Si + l' +2 I 2 ~n
+ +L0 0 & 5 06 + 82\ R (4 .1 5 )
So0 So1n+1 &2 n0+1 1 1 +1

When 0=0, the stress state is returned to the yield surface along a vector normal to the

current state, and when 0=1 the stress state is returned normal to the updated state. The

method is fully implicit when 0=1. After combining equations (4.14) and (4.15), the

stress correction can be approximated as


ar+', C- + 1, L 1++ +1 d+0A+1 (4.16)


where all derivatives are evaluated at the current state. The incremental variation of the

consistency parameter, or effective plastic strain, 8Alj may be obtained through a Taylor

expansion of the yield criterion about the current state


0( <, i ) ,0(+ I, pn+ [ Q +11 = 0 (4.17)
nl n 1)1 l -1 P "'n+l

Substituting equation (4.16) into (4.17) and realizing that all derivatives are evaluated for

the previous step stepp) yields

Q(,,,, )-H +-- + 5n~n1 o 0 (4.18)


thus



HL +O)+la--- +g--
(cn+l 0 ( (4.19)
do LC asaac/ asg as


where,










H = C 1+ s3RA, 02 (4.20)


The stresses and plastic strains are then updated through RA, and the yield criterion

Q(aon+, n+1) is checked to within a specified tolerance. If the tolerance has not been

met, the plastic corrector step will be repeated until convergence has been obtained.

Once convergence is obtained, the updated stresses and strains are accepted as the current

state. Note that if c&-/lA equals zero, then this model reduces to that of isotropic

hardening.

The consistent tangent modulus relates the current stress increment to the current

total strain increment, and is used to predict the total strain increment for the next

iteration. Taking the derivative of (4.13) yields

dQ= = do + dA = 0 (4.21)


which leads to the following relationship in incremental form after the substitution of

equation (4.16) with 0 = 0 since we are using the current step.

ao 7 ad ad aY ad
CAs-AA C -AA +AA =0 (4.22)


Using (4.22) to solve for the effective plastic strain increment based on the current state

gives

CAe
AAcurrent o= c (4.23)
a A C-+---A
dr dr di d









The combination of (4.23), (4.2) (4.4), and (4.10) yields a relationship between the

stress increment and total strain increment from which the consistent tangent modulus

can be found.

06 CCO 06
CA+= C- 0 ---7 Ac (4.24)
o& CO& OY -o


Here the term in brackets is the consistent tangent modulus and 0 represents the tensor

product.


The procedures for implementing the proposed hardening model using 0 = 0 are

summarized:

1. Given: on, p, As where n represents the previous time step and As is the total
strain increment for the current time step.

2. Calculate the trial state (i = 0): otal = an + CAs and pn+ = n

3. Check for consistency: If Q(o 'l ,o pn+) <- tolerance :> elastic stress state.
Accept the trial stress state as the current state and the total strain increment as an
elastic strain increment and exit. Else -> continue

4. Determine the starting values for the iteration (i = 0)

a) Y0o = n+)

b) kl = k(~n+), here k represents all yield criterion coefficients

) +lo = Ik


o ak
d) r+ =- -o
e) 1 pn +1

e) (^i=((o-^ e+0






81


) n+l O trial ,+

g 0 O
g) 1+n1+l a ,kn+l



5. Begin Iteration loop (i = 0...N)

a) .A-' = -n+1 Y +1
aq+l nC+l + h'+1 p'+lr +l

b) O'n+ = n+l + An+lCqn+l
C) -1+1 -l l+1
C) p n+l 'p n+l + 'n+l

6. Check for consistency: If Q(al, ,E'+1) < tolerance > Accept the current
state of stress and strain (i = i+1) and go to step 9 then exit. Else >
continue.

7. Continue with Iteration loop
a) Y,'1 = Y (+I
+1 Ep n+l





d) l +1_ k+1
Y n+1 -- \pn+1


SSp pn+l

e) Z+1 ~ +1 -+1
S+1 o
d) q n+1 = 0 1+1 a 7" i+l
e n+l & cn+ ,cpn+]


g) q+1 a o

Sn+1 1,k~+1
g) n+ l n + 1

8. Go to step 5

9. Calculate the elasto-plastic tangent modulus

a) C )n+ 1 n1 .C
a) C, P = C -
qn+lCq1+l + h,+l p' + r+l









In order to improve the stability of the return-mapping algorithm such as the one

just described, Yoon et al. (2004) proposed a multi-step procedure to be used if the strain

increment is very large. This procedure requires that the consistency condition be solved

in several steps when the initial consistency check yields a value large compared to the

initial yield stress. During the first step, the convergence tolerance would be on the order

of the magnitude of the initial consistency check minus the yield stress. Subsequent steps

would converge towards progressively smaller magnitudes until the convergence

tolerance equals the original tolerance value near zero. Then the state of stress and strain

would be accepted as the current state, and the iteration loop exited.

4.4 Application for an Isotropic Material

Let us use the numerical procedure described in the previous section to model the

response of an isotropic material obeying an isotropic yield criterion of the form

(',s ) =Y(_ ) where &(a,s ) is the effective stress associated with the proposed

isotropic criterion (3.1). The evolution of the yield surface is dictated by k = k(ep), thus

the effective stress is dependant upon the effective plastic strain. Allowing a variation of

the coefficient k with sp captures the change in the ratio between the yield in tension

and compression with accumulated plastic deformation. We assume a law of variation of

k of the form

k = A Bsp for 0.0 < Ep < critical (4.25)

k=0 for Sp > critical

where A and B are material constants. It means that above a critical level of sE the yield

in tension is equal to the yield in compression, i.e., no strength differential effects exist









above a critical level of sp. For illustration purposes, let us assume that A= 0.4,

0.4 critical
B=- and cn =a 0.05 with a constant degree of homogeneity of a =2. For
0.05

k e [0,0.4]) a nearly linear relationship exists between k and o, / oc when a = 2 (see

Figure 3.3). Thus, equation (4.25) with the assumed constant values will give a nearly

linear response between k and o-, / o. Figure 4.1 shows the evolution of the shape of

the yield surface for k varying according to (4.25) with the assumed constant values. A

power hardening law will also be used

Y(Ep) = E(D + p)m (4.26)

with assumed values for illustration purposes ofE = 650, D = 0.0463, and m = 0.227.

Uniaxial compression tests were carried out for k varying according to (4.25), as

well as for k held constant at k = 0, k = 0.4 for comparison. Note that the cases for which

k is held constant, the proposed hardening law reduces to isotropic hardening. The

response for uniaxial compression for k = 0 is identical to that of tensile yield, since for a

= 2, k = 0, the yield criterion (3.1) reduces to von Mises. For the case of k held constant

at 0.4, the material has an initial compressive yield stress lower than the tensile yield

stress, with the hardening rate being the same for both loading paths. In the case when k

varies according to (4.25), the material initially yields at the same level of compressive

stress as in the case when k is fixed at 0.4, but then hardens at a much higher rate than

uniaxial tension until Ep = 0.05 when k becomes 0, and the yield stress in tension and


compression become equal. These results are shown in Figure 4.2.









500


a=2


k=0


250


k = 0.4


-250


-500
-500


-250


Figure 4.1 Evolution of the yield surface for varying k.


250


500









400








350


k=0


k varies from 0.4 to 0


300


k= 0.4


250


0 0.01


0.02 0.03 0.04 0.05 0.06 0.07
Plastic Strain


Figure 4.2 Results of single element compression tests for a = 2.









4.5 Alternate Method for Anisotropic Hardening Implementation Interpolation

By applying a linear transformation to extend the isotropic yield criterion (3.1) to

orthotropy, 9 additional coefficients are added. It is not trivial to determine analytical

expressions for all of the coefficients in terms of the hardening variable. The available

experimental data provides information about the shape of the yield surfaces

corresponding to different given levels of effective plastic strain based on results of

monotonic loading tests (see Kelley and Hosford, 1968 and Lee and Backofen, 1966).

However, even if the expressions of Ly, k, and a for a given level of strain can be

determined based on the data, establishing Ljy( ) requires a large amount of data.

Therefore, an alternative approach to the hardening law is proposed.

From experimental and/or numerical results from polycrystal calculations, we can

identify the coefficients involved in the proposed orthotropic yield criterion (3.16) for a

set of values of equivalent plastic strain, say -E < s <... < and calculate the effective

stress &' =={,L( p), k(p ),a(-pJ)}, as well as YJ = Y(8p), corresponding to each of

the individual levels of effective plastic strain Esp, j = 1...m. Then, an interpolation

procedure can be used to obtain the yield surfaces corresponding to any given level of

accumulated strain. Thus, for a given arbitrary p the anisotropic yield function is of

the form

Q(o, ) = F(o, p)- nI(p), (4.27)

with

F= (( ) +( -(p ))1+1 (4.28)

and









I = (gp )y + (1- (g P))YI1 (4.29)


for any j S p S ', j = 1...m-1. For linear interpolation, the weighting parameter

(p ) appearing in equations (4.27) and (4.28) is defined as


( ) p (4.30)
p p
--j+l -E

such that ((j) = 1 and -(p ') =0. By considering that the anisotropy coefficients Li,

the strength differential parameter k, and the homogeneity parameter a evolve with the

plastic deformation, the observed distortion and change in shape of the yield loci of hcp

materials could be captured. Obviously, if these coefficients are taken constant, the

proposed hardening law reduces to the classic isotropic hardening law.

The derivatives for F and n needed for stress integration become,

OT c(.7 c0 /j+l
= p) + (1 )) (4.31)


cY jj+1 oj
5 (4.32)
c c7+1 -1+ j
p p p
PP P -P

S-J+- J (4.33)
c c7+1 -1+ j
p p p
PP P -P

82r d2oJ 2 J+1
02 ( -2 +(1 ()) 2 (4.34)
8u2 P) OU2 +2P)) OC2

and,

r J+1 rF J
a2F O_ o (4.35)
8 -- -p
ao-p 81+ 81c