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Experimental Characterization and Modeling of Polycrystalline Molybdenum

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Title:
Experimental Characterization and Modeling of Polycrystalline Molybdenum
Creator:
Kleiser, Geremy J
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (153 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
CAZACU,OANA
Committee Co-Chair:
SANKAR,BHAVANI V
Committee Members:
KUMAR,ASHOK V
BOGINSKIY,VLADIMIR L
Graduation Date:
5/2/2015

Subjects

Subjects / Keywords:
Anisotropy ( jstor )
Axial strain ( jstor )
Compressive stress ( jstor )
Deformation ( jstor )
Impact velocity ( jstor )
Modeling ( jstor )
Molybdenum ( jstor )
Specimens ( jstor )
Strain rate ( jstor )
Structural strain ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
characterization -- impact -- mechanical -- mechanicalcharacterization -- molybdenum -- taylor -- taylorimpact
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, Ph.D.

Notes

Abstract:
This dissertation concerns the experimental characterization, modeling and simulation of the plastic anisotropy and tension-compression asymmetry of polycrystalline molybdenum. In addition, extensive cross-validation of the model was done for both quasi-static and high strain-rate deformation regimes encountered in impact. For the first time, it was established that polycrystalline molybdenum has ductility in tension for low strain rates and that the failure strain is strongly dependent on the orientation. To accurately quantify the anisotropy in plastic deformation digital image correlation techniques were used. While the current practice is to assume plastic incompressibility when evaluating the plastic strain ratios, a novel approach was taken using an orthogonal configuration of cameras to allow direct measurement of the thickness strain of the specimen. For the first time, the tension-compression asymmetry of polycrystalline molybdenum in yielding was determined (yield stress in compression larger than in tension for all orientations). Furthermore, evaluation of the ellipticity of the deformed compression specimens allowed uncovering that although the material exhibits strong strain anisotropy in tension, it has a weak strain anisotropy in uniaxial compression. For the first time, Taylor impact tests were successfully conducted on this material for impact velocities in the range 140-160 m/s. An elastic-plastic model that accounts for all the specificities of the plastic deformation of the material was developed. Key in the formulation was the use of a yield function that simultaneously accounts for anisotropy and tension-compression asymmetry. Validation of the model was done through comparison with test results on notched specimens for the quasi-static strain-rate regime and deformed Taylor impact specimens for the high strain-rate regime. Quantitative agreement between measured and predicted response was obtained. In particular the effect of loading orientation on the response was very well described. For the Taylor impact test the model was used to gain understanding of the dynamic deformation process of this material. It was thus shown not only the predictive capabilities of the model but also its potential for use in virtual testing of complex systems composed of the material. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: CAZACU,OANA.
Local:
Co-adviser: SANKAR,BHAVANI V.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-11-30
Statement of Responsibility:
by Geremy J Kleiser.

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UFRGP
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Applicable rights reserved.
Embargo Date:
11/30/2015
Classification:
LD1780 2015 ( lcc )

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EXPERIMENTAL CHARACTERIZATION AND MODELING OF POLYCRYSTALLINE MOLYBDENUM By GEREMY KLEISER 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 2015

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2015 Geremy Kleiser

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To my parents and grandparents for being excellent role models and setting in motion the person I have become

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ACKNOWLEDG MENTS I would like to thank my committee members for sharing their limited time to evaluate and provide pertinent comments on my research. Specifically, I would like to acknowledge my committee chair . Presumably, the pursuit of a doctoral degree begins wit h an idea and morphs into something bigger and more complete than initially envisioned as a result of acquiring skills and different perspectives throughout the process. Despite passionately pursuing her own research she always found time to discuss my own . She was my navigator throughout this process and I’m eternally grateful. I want to thank the members of my research group for our conversations that were at times technical and at times personal. They helped me along the way and made the process more enj oyable. I am obligated to express sincere thanks to the unsung heroes of experimental testing which are the technicians. Finally, I would like to thank my wife and daughter for their patience and understanding as I attempted to balance a demanding personal endeavor with my responsibilities toward them. 4

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TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 7 LIST OF FIGURES .......................................................................................................... 8 LIST OF ABBREVIATIONS ........................................................................................... 13 ABSTRACT ................................................................................................................... 14 CHAPTER 1 INTRODUCTION .................................................................................................... 16 Motivation ............................................................................................................... 16 State of the Art in Understanding the Mechanical Properties of Molybdenum ........ 17 Research Objectives ............................................................................................... 23 2 MATERIAL CHARACTERISTICS ........................................................................... 26 Introduction ............................................................................................................. 26 Chemical Composition ............................................................................................ 26 Texture .................................................................................................................... 26 Porosity ................................................................................................................... 29 3 MECHANICAL CHARACTERIZATION ................................................................... 33 Quasi Static Testing ................................................................................................ 33 Experimental Configuration ..................................................................................... 33 Uniaxial Tension Results ........................................................................................ 41 Effects of the Strain Rate on the Tensile Response in the Quasi Static Regime .......................................................................................................... 41 Yielding and Strain Hardening in Tension ........................................................ 42 Plastic Flow in Tension ..................................................................................... 47 Strain at Failure in Tension ............................................................................... 50 Uniaxial Compr ession Results ................................................................................ 51 Yielding and Strain Hardening in Compression ................................................ 51 Plastic Flow in Compression ............................................................................ 56 TensionCompression Asymmetry .......................................................................... 57 High Strain Rate Testing ......................................................................................... 60 Experimental Configuration .............................................................................. 62 Results ............................................................................................................. 64 Conclusions ............................................................................................................ 67 5

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4 CONSTITUTIVE MODEL F OR MOLYBDENUM ..................................................... 70 Elastic Plastic Model ............................................................................................... 70 Identification of the Parameters of the Model for Polycrystalline Molybdenum ....... 73 Finite Element Formulation and Implicit Time Integration Algorithm Used for Implementation of the Elastic Plastic Model. ....................................................... 76 General Form of Governing Equations for 3D problems ................................. 77 Time Integration Algorithm for the Orthotropic Elastic Plastic Model ............... 81 Comparison of Model Prediction with Measured Stress versus Strain .................... 87 5 MOLYBDENUM NOTCHED T ENSILE TESTING AND M ODEL VALIDATION ....... 89 Introduction ............................................................................................................. 89 Experimental Configuration and Results ................................................................. 90 Finite Element Simulations of the Notched Tensile Tests and Comparison with Measurements ................................................................................................... 101 Conclusions .......................................................................................................... 108 6 MOLYBDENUM TAYLOR IM PACT TESTING AND MODEL VALIDATION ......... 109 Introduction ........................................................................................................... 109 Taylor Impact Tests on Molybdenum .................................................................... 112 Finite Element Simulations of the Taylor Impact Test and Comparison with Measurements ................................................................................................... 122 Conclusions .......................................................................................................... 138 7 SUMMARY ........................................................................................................... 140 LIST OF REFERENCES ............................................................................................. 144 BIOGRAPHICAL SKETCH .......................................................................................... 153 6

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LIST OF TABLES Table page 2 1 Molybdenum plate chemical composition ........................................................... 26 2 2 In plane porosity – 0.00196 – 0.2% .................................................................... 31 3 1 Average yield stress at strain = 0.5% ................................................................. 46 3 2 Average yield stress in compression at strain = 0.5% ........................................ 56 4 1 Model parameters for a polycrystalline molybdenum. ......................................... 75 6 1 Molybdenum Taylor impact test data ................................................................ 118 6 2 Measured and Predicted Material Response .................................................... 125 7

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LIST OF FIGURES Figure page 1 1 Body centered cubic (bcc) unit cell showing a <111> slip direction and a (110) slip plane. .................................................................................................. 18 2 1 Rolling operation and resulting orthotropy, with the respective symmetry axes: rolling direction (RD), transverse direction (TD), and normal direction (ND). ................................................................................................................... 27 2 2 Important texture orientations for bcc metals ...................................................... 28 2 3 Orientation of Mo microstructure with max intensity occurring within the <110> direction indicating the plate has been subject ed to a rolling operation. .. 30 2 4 Images of the microstructure for a given direction and its corresponding porosity highlighted in red. .................................................................................. 32 3 1 Specimen orientation and reference frame relative to plate reference system. .. 35 3 2 Specimen geometry ............................................................................................ 35 3 3 Instron test frame with load cell. ......................................................................... 36 3 4 Digital Image Correlation (DIC technique). ......................................................... 37 3 5 Tensile test configuration. ................................................................................... 39 3 6 DIC strain maps. ................................................................................................. 40 3 7 Uniaxial compression testing. ............................................................................. 41 3 8 Stress strain curves for 90 specimen illustrating variation in ductile response with change in strain rate. ................................................................................... 43 3 9 Vari ation of stress strain curves for the specimens having an axial direction corresponding to RD. .......................................................................................... 44 3 10 Variation of stress strain curves for the specimens having an axial direction cor responding to 45. ......................................................................................... 44 3 11 In plane anisotropy of the initial yield stress in tension at = 0.5%.. .................. 45 3 12 Stress strain curves for polycrystalline Mo subjected to tensile loading at a strain rate of 105/s. ............................................................................................. 46 3 13 Comparison of predicted stress (solid line) using the power law relationship with data (circles) from three experimental RD tests. ......................................... 47 8

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3 14 Tests needed to determine the yield surface of the anisotropic plate. ................ 48 3 15 Strain ratios for loading orientations . .................................................................. 49 3 16 Potential error in r value determination ............................................................... 50 3 17 Varia tion of r values with increasing angle from rolling direction (RD). ............... 50 3 18 Change in ductility as angle between axial direction and rolling direction increases. ........................................................................................................... 52 3 19 Stress versus strain for RD specimen with corresponding strain maps at several average global strain values. . ................................................................ 52 3 20 Stress versus strain for 45 specimen with corresponding strain maps at several average global strain values. . ................................................................ 53 3 21 Stress versus strain for 90 specimen with corresponding strain maps at several average global strain values. . ................................................................ 53 3 22 Variation of stress strain curves for the RD compression specimens. ................ 54 3 23 Stress strain curves for polycrystalline Mo subj ected to compression loading at a strain rate of 105/s. . ..................................................................................... 55 3 24 Stress strain curve corresponding to the through thickness or ND directi on. ..... 55 3 25 In plane anisotropy of the initial yield stress in compression at = 0.5%. . ......... 56 3 26 Deformed cross section of an anisotropic material resulting in major and minor diameters. ................................................................................................. 57 3 27 Variation of cross section change with axial direction for compression specimens at = 0.25 shown in the form of aspect ratio. . .................................. 58 3 28 Tensioncompression asymmetry of the stress strain response ......................... 59 3 29 Sequence of events for the Split Hopkinson Pressure Bar (SHPB) test. ............ 61 3 30 Split Hopkinson Pressure Bar reference system for determining high strainrate mechanical properties. (Gray 2000) ............................................................ 63 3 31 Split Hopkinson Pressure Bar test specimens .................................................... 63 3 32 Images of th e Split Hopkinson Pressure Bar test configuration. ......................... 64 3 33 Evolution of the stress and strain for a RD specimen. ........................................ 65 9

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3 34 Flow stress for polycrystalline Mo subjected to compressive loading at a strain rate of 400/s. . ............................................................................................ 66 3 35 In plane anisotropy of Mo peak stress at a strainrate of 400/s. ......................... 66 3 36 Variation of cross section change with axial direction for compression specimens at = 0.25 shown in the form of aspect ratio. . .................................. 67 4 1 Comparison between the yield locus of polycrystalline molybdenum according to the orthot ropic yield criterion of Cazacu et al. (2006) (Eq.(47)) and data (symbols). . ........................................................................................... 75 4 2 Predicted and experimental evolution of the Lankford coefficients (r values) wi th the orientation for polycrystalline Mo in comparison with the experimental values (symbols). .......................................................................... 76 4 3 An updated Lagrangian formulation. ................................................................... 77 4 4 Predicted and measured stress vs strain response of the RD specimen subjected to quasi static tensile loading. . ........................................................... 88 5 1 Notched tensile specimen geometry. .................................................................. 91 5 2 Position of the cameras used and the axial strain maps acquired with each camera. .............................................................................................................. 91 5 3 Effect of specimen geometry on the response for the specimens oriented at = 0, 45, 90 to RD . .............................................................................................. 94 5 4 Axial an d width strain distribution for both camera views for the RD specimen. . .......................................................................................................... 96 5 5 Axial and width strain distribution for both camera views for the 45 specimen. ........................................................................................................... 97 5 6 Axial and width strain distribution for both camera views for the TD (90) specimen. ........................................................................................................... 98 5 7 Evolution of axial strain with displacement at the notch root cross section for the TD (90) specimen. . ...................................................................................... 99 5 8 Evolution of average strains (RD, 45, & 90) at the notch root acquired with 2 cameras with displacement at the notch root for both camera views. ............... 102 5 9 Evolution of the plastic strain ratios (width/thickness) as a function of the tensile loading orientation determined from tests on notched and smooth specimens. ....................................................................................................... 103 10

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5 10 Finite element mesh of the notched specimen. ................................................ 103 5 11 Average strain with displacement at the notch root for a specimen with RD loading orientation. ........................................................................................... 105 5 12 Average strain with displacement at the notch root for a specimen with 90 loading orientation ............................................................................................ 106 5 13 Comparison of DIC m easurements and model predictions for axial and width strain. ................................................................................................................ 107 6 1 Dimensions of the cylindrical specimens used for the Taylor impact tests. ...... 112 6 2 Cross sectional view of the Taylor impact test set up showing the specimen, the barrel, and the target (anvil). ....................................................................... 115 6 3 Photographs depicting the configuration of the Taylor impact test ................... 115 6 4 Failure processes for Taylor specimen . ............................................................ 116 6 5 Consequences of slight misalignment .............................................................. 117 6 6 Schematic rep resentation of the post test Molybdenum specimen. .................. 119 6 7 Experimental deformat ion of Mo Taylor RD, 45, and TD specimens after Taylor impact test at ~150 m/s . ........................................................................ 120 6 8 Comparison of the post test geometries corresponding to various impact velocities for the RD, 45, and TD specimens, respectively . .............................. 121 6 9 FE mesh of the quarter specimen with zoom of the impact surface. ................ 123 6 10 Ex perimental and predicted deformation profiles corresponding to the 90 and ND material axes for an impact velocity of 140m/s. ................................... 125 6 11 Experimental and predicted deformation profiles c orresponding to the 90 and ND material axes for an impact velocity of 151m/s. ................................... 126 6 12 Experimental and predicted deformation profiles corresponding to the 90 and ND material axes for an impact velocity of 165m/s. ................................... 126 6 13 Experimental and predicted deformation profiles corresponding to the RD and ND material axes for an impact velocity of 141m/s. ................................... 127 6 14 Experimental and predicted deformation profiles corresponding to the RD and ND material axes for an impact velocity of 151m/s. ................................... 127 6 15 Experimental and predicted deformation profiles corresponding to the RD and ND material axes for an impact velocity of 161m/s. ................................... 128 11

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6 16 Pressure evolution in time at v arious locations along the specimen centerline (impacted end corresponds to x=0). ................................................................. 129 6 17 Pressure oscillations at different locations along the specimen centerline. ...... 130 6 18 Variation of axial velocity in time at various locations along the centerline of the specimen. ................................................................................................... 131 6 19 Simulated evolution of the axial velocity in time at x= 3.2mm (solid lines) and x=19mm (interrupted lines), along the centerline for impact velocities v= 140 m/s (blue), v= 150m/s (red), and v= 165 m/s (bl ack). ....................................... 132 6 20 Isocontour of the plastic strain rate at different times from an impact velocity of 150m/s. ......................................................................................................... 134 6 21 Evolution of the plastic strainrate in time at various locations along the centerline of the specimen. ............................................................................... 135 6 22 Comparison between the plastic strain rate evolution at the centerline and outer radius of the cross section at x= 3.2 mm from the impacted end. ........... 135 6 23 Evolut ion of the equivalent plastic strain in time at various locations within the specimen. ......................................................................................................... 136 6 24 Comparison between two radial points of the equivalent plastic strain at 3.2mm from the impact surface. ....................................................................... 137 6 25 Distribution of the final equivalent plastic strain within the specimen. . .............. 137 12

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LIST OF ABBREVIATIONS ASTM American Society for Testing and Materials bcc body centered cubic DIC Digital Image Correlation EBSD Electron Back Scatter Diffraction LVDT Linear Variable Displacement Transducer Mo Molybdenum ND Normal Direction of plate ODF Orientation Distribution Function RD Reference Direction of plate SHPB Split Hopkinson Pressure bar TD Transverse Direction of plate 13

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Abstract of Dissertation Presented to the Graduate School of the Un iversity of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EXPERIMENTAL CHARACT ERIZATION AND MODELI NG OF POLYCRYSTALLIN E MOLYBDENUM By Geremy J. Kleiser May 2015 Chair: Oana Cazacu Major: Mecha nical Engineering This dissertation concerns the experimental characterization, modeling and simulation of the plastic anisotropy and tensi oncompression asymmetry of polycrystalline molybdenum. In addition, extensive cross validation of the model was don e for both quasi static and high strain rate deformation regimes encountered in impact. For the first time, it was established that polycrystalline molybdenum has ductility in tension for low strain rates and that the failure strain is strongly dependent on the orientation. To accurately quantify the anisotropy in plastic deformation digital image correlation techniques were used. While the current practice is to assume plastic incompressibility when evaluating the plastic strain ratios, a novel approach w as taken using an orthogonal configuration of cameras to allow direct measurement of the thickness strain of the specimen. For the first time, the tensioncompression asymmetry of polycrystalline molybdenum in yielding was determined (yield stress in compr ession larger than in tension for all orientations). Furthermore, evaluation of the ellipticity of the deformed compression specimens allowed uncovering that although the material 14

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exhibits strong strain anisotropy in tension, it has a weak strain anisotropy in uniaxial compression. For the first time, Taylor impact tests were successfully conducted on this material for impac t velocities in the range 140160 m/s. An elastic plastic model that accounts for all the specificities of the plastic deformation of t he material was developed. Key in the formulation was the use of a yield function that simultaneously accounts for anisotropy and tensioncompression asymmetry. Validation of the model was done through comparison with test results on notched specimens for the quasi static strain ra te regime and deformed Taylor impact specimens for the high strain rate regime . Quantitative agreement between measured and predicted response was obtained. In particular the effect of loading orientation on the response was very well described. For the Taylor impact test the model was used to gain understanding of the dynamic deformation process of this material . It was thus shown not only the predictive capabilities of the model but also its potential for use in virtual testing o f complex systems composed of the material. 15

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CHAPTER 1 INTRODUCTION Motivation There is growing demand for structural materials suitable for high temperature applications. The current materials used for these applications, notably nickel based super alloys, are reaching their technological limit and suitable replacements should be identified. A promising path for the development of a suitable alternative involves materials based on a refractory metal. A high temperature refractory metal is defined as having a melting temperature above 2,200C which would include metals such as niobium, molybdenum, tantalum, tungsten, and rhenium. Both niobium and molybdenum have significantly lower atomic masses compared to the other high temperature refractory m etals, thus making them and their alloys attractive candidates for applications where high temperatures and specific weight are of concern. Although niobium is the more abundant metal, m olybdenum is produced in far greater quantities and can be found in a wide variety of high temperature, corrosive environments such as electronic heat sinks , heat treating furnaces, and glass melting furnace components where long t erm creep resistance is needed ( Shields, 1999) . Because molybdenum’s ductile to brittle transit ion temperature is typically within approximately 100C of room temperature, critical components have not typically used molybdenum in this temperature range. In order, for molybdenum and molybdenum alloys to find a wider use and to be considered for appli cations that must exhibit fracture resistance, it is necessary to better understand its deformation and fracture properties in the temperature range between ambient and 1200C. 16

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State o f the A rt in U nderstanding the Mechanical Properties of M olybdenum Revie w of the literature indicates a strong need for a combined experimental and theoretical effort in order to fill the gap in knowledge of the mechanical response of molybdenum, particularly for polycrystalline Mo. Since the polycrystalline material is an aggregate of grains having the structure of a singlecrystal it is essential to have an understanding of the mechanical response of Mo singlecrystal. However, the singlecrystal mechanics of Mo is unclear, as it is the case with body centered cubic (bcc) cry stals. While it is well established that in bcc crystals plastic deformation occurs by slip and that the slip directions are the directions of closest packing, (111), determination of the planes on which slip occurs remains a very difficult issue and a ver y active area of research. It is known that slip typically occurs on one or a combination of the {110}, {112}, and {123} family of planes ( Figure 11 show s the unit cell of a bcc crystal and the (110) slip plane and the <111> slip direction). A slip system is defined by the pair (b,n), n indicating the direction of the normal to the slip system while b the slip direction. Due to the symmetry of the bcc unit cell, this yields 48 different potential slip systems. With the exception of the <110> family of planes, these slip systems involve slip directions associated with noncrystallographic or nonmirror planes of the crystal structure. As a result, there is no reason why the flow stress of a bcc crystal should remain the same when the direction of slip is rev ersed (backward instead of forward slip). Therefore, bcc crystals should in general display tensioncompression asymmetry in yielding. 17

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Figure 11 . Body centered cubic (bcc) unit cell showing a <111> slip direction and a (110) slip plane. The tens ion – compression asymmetry in flow stresses of bulk Mo single crystals (mm size) has been reported in several studies ( e.g. Hollang et al., 1997; Seeger and Hollang, 2000). Very recently, studies of the tensile and compressive behavior of <001> oriented bcc Mo micropillars of nanometer dimensions (smallest size ~200 nm diameter) have also been conducted (see Kim et al . ; 20102012). They reported flow stress asymmetry at the single crystal scale. Although the experimental evidence of the tensioncompression asymmetry of slip in bcc monocrystals is growing (e.g. Seeger and Hollang, 2000; Kim and Greer, 2009, 2010; Brown and Ghoniem, 2010; Kim et al . , 2012), its appropriate modelling remains a question of intense debate. At present, in the most advanced crystal plasticity models and codes, slip is considered to obey Schmid law (e.g. viscoplastic self consistent model (VPSC) of Lebensohn and Tome, 1993; crystal plasticity finiteelement (CPFEM) models such as ALAMEL developed by Van Houtte and collaborators, etc. ). Thus, in terms of modeling, at present the exception is considered to be “the rule”. Furthermore, twinning could also occur at low temperature (see Shaw, 1967) or for [ 111 ] ( 110 )> 18

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shockdeformation (see Wongwiwat and Murr, 1978) and thus contributing to drastic anis otropy evolution. While it is generally agreed that slip directions are the dense <111> directions, the planarity of slip is still intensely debated. Tremendous progress has been achieved both in terms of experimental characterization and modelling, however, the fundamental understanding is mature and predictive capabilities are well advanced only for fcc metals and alloys. The dislocation response to stress in these systems generally is well represented by the Schmid law. In a recent review by Cai et al. ( 2005), it is stated that existing perceptions about modeling the singlecrystal plasticity of bcc metals are heavily influenced by the views developed earlier for fcc metals. Dislocation cores in fcc materials tend to be planar and dissociate into partials that interact with each other and move by gliding on a slip plane. In contrast, dislocation mobility in bcc systems displays large deviations from Schmid law, the complexity of the crystalline mechanisms still poses tremendous challenges both in terms of identification of the dominant slip systems and modelling (e.g. plasticity criteria and hardening laws at the singlecrystal scale) despite intense research since the twenties (e.g. Taylor and Elam, 1926; Mathewson and Phillips, 1928; Chalmers, 1938; Gilman, 1953; Maddin and Chen, 1954; ReedHill and Robertson, 1957; Nakada and Key, 1966; Takeuchi, 1969a,b). The applicability of the Schmid law for the description of the yielding in bcc crystals is severely questioned both from a theoretical standpoint (e.g. Vitek et al ., 2004) and experimentally (example Hsiung and Lassila, 200 2 ; Hsiung, 2010; Kim et al . 20102012). As already mentioned, for symmetric planar slip on dense atomic planes and along dense atomic directions to be the unique plastic deformation mechanism it is 19

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rather exceptional. Under a wide range of conditions, <111> screw dislocations may crossslip easily, i.e. change frequently from one slip plane to another. For example, in Mo, Luft and Ritschel (1982) identified slip in the {112} and {110} planes of the <111> while Yoo et al., (1995) reported that up to a plastic strain of 0.09, the dominant slip system is the {123} <111> while {112}<111> and {211}<111> become active for plastic strain beyond this limit. Despite recent progress in understanding the nonplanar core structure of the mobile dislocations, the complex mechanisms of dislocation mobility (Chang et al . , 2001; Chaussidon et al . , 2006; Ventelon et al . , 2009; Gordon et al . , 2010) and dislocation interactions (Bulatov et al . , 2006; Madec and Kubin, 2008; Liu et al . , 2011) remain controversial. Specifically, as concerns the origin of the low mobility of screw dislocations until recently it has been an almost universal belief that the high Peierls stress and the limited mobility of the scr ew dislocation is a direct consequence of their nonplanar core structure and that the dislocation effectively anchors itself to the lattice, the anchoring being thought to be more pronounced when the core is polarized (e.g. atomistic simulations Yang et al., 2001 and highresolution electron microscopy experiments of Sigle, 1999 in highpurity Mo). This belief has been called into question by ab initio calculations ( Woodward et al., 20 0 2 which presented evidence that even nonpolarized screw dislocations i n Mo and Ta have high Peierls stress, of the order of 1GPa. On the other hand, it has also been observed that the Peierls stress is a sensitive function of core polarization (Cai et al., 2005). Nevertheless, since the elementary displacements of the disloc ations necessarily follow crystallographic paths, slip must always result in a succession of planar steps. However, the resulting slip on 20

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most of the nondense planes should be asymmetric with respect to the slip direction and will be highly dependent on t he loading type and crystal orientation. Furthermore, lattice friction stresses, which can be often considered as negligible in fcc metals may significantly contribute to plastic straining (e.g. see Wang et al . , 2004). As concerns hardening in bcc crystals , it is reported the typical threestage hardening process typical in fcc crystals is not observed (see for example, Arsenlis and Tang, 2003). The validity of the classic Taylor forest hardening model for bcc crystals is very difficult to ascertain because of the complexity of the dislocation patterns (type) and interactions that need to be accounted for. As concerns polycrystalline Mo, recently, a lot of attention has been devoted to understanding its macroscopic tensile properties. It was found that Mo an d its alloys display anisotropy in yielding (e.g. Zadari et al., 2005) and plastic flow (Walde, 2005) as revealed by the strong directionality of the Lankford coefficients (ratio between the width to thickness strains in tension along a given orientation). The significant influence of temperature on yielding and strain at failure, in particular the influence of interstitial content, impurities and alloying content, grain size, grain boundary character on the ductile to brittle transition at low temperatures have been the object of numerous studies (e.g. Sargent and Shaw, 1966, Wronski et al . , 1969, Thornley and Wronski, 1969, Furuya et al., 1981, Dobromyslov et al . , 1990). The ductileto brittle transition (DBTT) at low temperatures is generally attributed t o the inability of dislocations to multiply and/or move fast enough, at low temperatures, to relieve the stress concentrations that further cause cracking (see Cai et al., 2005). Yet, early experimental investigations by Ault and Spretnak (1965) indicate t hat fracture at room temperature 21

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(26C) is not always brittle. Fracture properties were evaluated by comparing the unnotched and notched tensile properties for various states of stress for wrought stress relieved Mo and also for recrystallized Mo in three different grain sizes. It was observed that for all the materials, the sample with the sharpest notch exhibits brittle fracture although specimens with smaller notch radii exhibit ductile fracture. Thus, a clear influence of the stress state (stress triaxi ality) on the ductility at room temperature is revealed. The yield stress was found to follow the Hall Petch dependence with grain diameter and that all detectable yielding occurs prior to crack initiation. Further evidence of the influence of the state of stress on the tensile strength, ductility, and DBTT has also been provided by more recent studies. For example, Cockeram (2010) has shown that rolling wrought molybdenum alloys to thinner sheet produces a finer grain size and finer distributions of second phases that results in slightly higher tensile strength values and lower DBTT values in some cases. The differences in tensile strength, ductility, and DBTT between the thicker starting materials and rolled sheet were shown to be related to the state of plane stress for the thin sheet specimens which results in a larger plastic zone size and greater amounts of ligament strain. While these studies provide valuable insights concerning the key role played by the stress triaxiality on the type of damage that o ccurs, fundamental questions concerning how the stress triaxiality affects the active deformation mechanisms in the fully dense material, damage nucleation, and the role of texture on subsequent damage evolution remain open. Systematic experimental investi gation at multiple length scales is needed to better understand the specific mechanisms. Furthermore, there is no clear explanation why there is only a slight influence of temperature on Lankford coefficients (Walde, 2005). Addressing this issue 22

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would not only advance our understanding of the role of temperature on texture evolution but will have direct impact on optimization of forming processes. The consequence of the limited understanding of Mo single crystal mechanics and the polycrystalline response is that finite element analyses of the mechanical behavior of refractory bcc polycrystals such as Mo are generally done using either empirical onedimensional laws for the strain at failure (e.g. Nemat Nasser et al., 1999, Cheng et al. , 2001) or crystal plas ticity models tailored for fcc metals (e.g. Walde, 2005). Clearly, a 1D formulation cannot capture the directionality of plastic deformation and damage. Since existing macroscopic models for deformation and ductile fracture were designed such as to account for the plastic deformation of the matrix by Schmid slip, which was shown to be invalid for most bcc (Seeger and Hollang, 2000), these models cannot capture the peculiar features of yielding and damage in certain bcc metals, in particular the vital role played by slip on planes other than {110}. In summary, at present yielding, hardening laws, and damage models at every length scale are poorly understood; the macroscopic response is not realistically modelled even for the simplest loading conditions. There is a critical need for a systematic experimental investigation of the tensioncompression asymmetry in flow stresses and hardening of polycrystalline Mo, quantification of the directionality of this asymmetry and its evolution with accumulated plastic d eformation for both quasi static and dynamic regimes. Research Objectives The central objective of this research is to fill the knowledge gap concerning the deformation and fracture of polycrystalline Mo for both quasi static and dynamic regimes. The outli ne of the dissertation is as follows. Chapter 2 presents the chemical 23

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composition of the polycrystalline plate material under study, determination of its initial texture and axes of material symmetry, and quantification of the initial porosity of the mater ial. Chapter 3 presents a comprehensive series of quasi static mechanical tests in uniaxial tension and compression that were carried out on smooth specimens. The need for measurements of local strain fields using digital image correlation (DIC) for accurate quantification of the plastic anisotropy of the material is emphasized. To characterize the influence of the strain rate on the material deformation and failure, highrate data using a Split Hopkinson Bar (S H PB) technique were also acquired. Based on th e key features of the response determined from these data an elastic plastic modeling framework was adopted for the description of the material’s response. The elastic plastic modeling approach, the key constitutive hypotheses, along with the numerical asp ects of the numerical implementation in a finiteelement framework is presented in Chapter 4. The identification of the model parameters is also outlined. Comparison between the theoretical and experimental description of yielding anisotropy and plastic fl ow anisotropy is also given. Finally the model is challenged by comparison between experimental and theoretical predictions of the stress strain response (global deformation) and also of the local deformation fields. Chapter 5 presents the investigation of the response of notched specimens. The combined influence of anisotropy and stress triaxiality on the mechanical response is thus assessed. Furthermore, the tests served for cross validation of the model predictions, Chapter 6 presents Taylor impact tests conducted on this material for impact velocities in the range 1401 6 0 m/s. These tests were also simulated with the proposed model. Comparison between measured and predicted geometry profiles is presented. 24

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Furthermore, the model is used to gain understanding of the dynamic deformation process of this material in terms of time evolution of pressures, when the transition to quasi stable deformation occurs, the extent of plastically deformed zone, distribution of the local plastic strain rates. A summary of t he findings is included in Chapter 7 along with an assessment of the predictive capability of the model. 25

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CHAPTER 2 MATERIAL CHARACTERISTICS Introduction Based on the literature, the mechanical behavior of polycrystalline molybdenum depends on the interstitial content and initial texture, therefore the first task of this research effort was to identify the composition of the plates and the axes of material symmetry. To this end, the two plates of molybdenum (Mo) used in this study were cha racterized in terms of their chemical composition, texture, and porosity. Chemical Composition The chemical composition was determined using a LECO combustion technique for carbon (C), oxygen (O), and nitrogen (N) content, and Glow Discharge Mass Spectrome try for the remaining elements. The quantity in the form of parts per million (ppm) of these elements are provided in Table 21 indicating that the two plates have nearly identical chemical composition The plates contained 0.99979% and 0.99981% Mo, respect ively. The maximum quantity for a nonreported element was 0.1ppm and any element with ppm less than 4 were summed and listed “as other”. Table 21 . Molybdenum p late c hemical composition PPM C O N Si K Cr Fe Ni W Other Plate 1 5 16 5 4.7 15 13 35 6.2 80 27.96 Plate 2 5 13 5 4.5 17 15 33 5 70 26.71 Texture Texture refers to the distribution of the orientations of the grain within the polycrystalline plate. A preferred orientation of grains results from the fabrication process. For example, rolling induces orthotropy of the plate or sheet, i.e. symmetry with respect to three mutually orthogonal planes. These three orthogonal planes have 26

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the normal along an orthogonal rolling or reference direction (RD), transverse direction (TD) (normal to RD in the plane of the plate), and the throughthickness or norma l direction (ND), to the plate ( Fig 2 1 and Kochs, 2000) . Figure 21 . Rolling operation and resulting orthotropy, with the respective symmetr y axes: rolling direction (RD), transverse direction (TD), and normal direction (ND). Molybdenum products such as sheets and bars are typically produced through compacting and sintering of Mo powder, operation, which is followed by hot or cold rolling (O ertel et al., 2007) . The texture generated within Mo during rolling is similar to other bcc metals and primarily results in {110} atomic planes aligning with the rolling direction ( Park et al., 1998) . The percentage of grains that align with the rolling di rection is dependent upon the extent of reduction in thickness through rolling. The fabrication history of the Mo plates investigated was unknown. To identify a rolling or reference direction and determine the texture of Mo we used the procedure described by Randle, 2000 for determining the texture of bcc polycrystalline materials. The texture was determined assuming that the fiber {110} is parallel to the rolling direction and the fiber has the <111> direction parallel with the normal to the rolling di rection ( Randle, 2000) . The and fibers in bcc crystals and the ideal cubic orientations for rolled bcc metals ( 2 = 45 crosssection) are reproduced in Figure 22 (A)(B) after (Park et al. , 1998) and (Kochs, 2000), respectively . ND RD TD Side End 27

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Fi gure 22 . Important texture orientations for bcc metals: A. Reduced Euler space with fibers and orientation relevant for bcc crystals ( Park et al., 1998) . B. Ideal cubic orien tations for rolled BCC metals. ( Kochs, 2000) A FEI Quanta 200 field emission gun was used to conduct electron back scatter diffraction (EBSD) scans of mechanically polished sections of the Mo plates to identify the texture. The orientation distribution function (ODF) was determined using the harmonic series expansion method with a maximum series expansion coefficient of 32. Multiple scans were conducted in various locations of the plates, with similar results. This indicates the uniformity of the texture. An ODF from one of the Mo plates is shown in Figure 23 represented by the 2 = 45 sectio n. As can be seen in the figure, the maximum intensity of 8.5 corresponds to rotated cube components (001)<110> indicating the plates had been indeed subjected to a rolling operation. The peak intensity of the ODF map corresponds to <110> but there are low er intensities near the fiber that are visible. There are grains within 10 of the (111)[011] orientation having an intensity of approximately 6. This means that t he texture of the plates is weak . Nevertheless, the directions of symmetry of the material (RD, TD, ND) were clear ly identified. A B 28

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Grain characteristics such as diameter, aspect ratio, and orientation were obtained from the EBSD scans. Multiple scans were conducted in different locations. The major axis of the majority of the grains (i.e.~70% of the grains within the pl ane of the plate) had an orientation within 30 of the rolling direction (RD). The orientation of the major axis for grains through the thickness of the plate were random. Let the aspect ratio of a grain be defined as the ratio of its minor axis to its major axis. The material has almost equiaxed grains, the average grain diameters along the inplane sections, RD and TD were approximately 50 m, with the average aspect ratio being of 0.5. The average grain diameter along ND was approximately 60 m with an as pect ratio of 0.55. Porosity A total of 12 specimens were used to determine the porosity of the plates used in this study. Two specimens for each direction (ND, RD, TD) were taken from each plate. These specimens were mechanically polished and 15 random i mages at 750X of the microstructure were obtained for each specimen. Voids approximately 2 m and larger were highlighted and the void area fraction of the image was determined. There was a distinct difference in porosity between ND and the inplane directions RD and TD. The average area fraction for the 8 specimens used to determine porosity for the in plane directions are given in Table 2 2 and indicate an average porosity of 0.2% with little variation between plates or location within plates. The inplane porosity was random and evenly distributed. The porosity along the thickness direction ND was significantly less and this can be attributed to the rolling process which flattened the voids. 29

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Figure 23 . Orientation of Mo microstructure with max intensity occurring within the <110> direction indi cating the plate has been subjected to a rolling operation. (001)[1 10] (112)[ 1 10] (111)[1 10] (110)[1 10] (111)[ 1 12] (001)[ 1 10] (110)[001] Calculation Method: Harmonic Series Expansion Series Rank (I): 32 Gaussian Smoothing: 2.0 Sample Symmetry: Orthotropic Representation: Euler Angles (Bunge) Rolling Direction 30

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Mechanical polishing of the ND surfaces reduced surface depth variation making the residual portion of the void difficult to distinguish. Examples of the microstructure and highlighted voids are s hown in Figure 2 4 . Table 22 . In plane porosity – 0.00196 – 0.2% Average Area Fraction Standard Deviation Maximum Area Fraction Minimum Area Fraction P1L1 0.002027 0.00048 0.0025 0.0012 P1L2 0.0021 0.000413 0.003 0.0014 P1L3 0.001993 0.000279 0.0024 0 .0015 P1L4 0.001673 0.000423 0.0029 0.0011 P2L1 0.00182 0.000354 0.0024 0.0012 P2L2 0.001853 0.000756 0.0034 0.0009 P2L3 0.002007 0.00057 0.0032 0.0012 P2L4 0.002213 0.000774 0.004 0.001 31

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Figure 24 . Imag es of the microstructure for a given direction and its corresponding porosity highlighted in red. ND ND R D R D T D T D 32

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CHAPTER 3 MECHANICAL CHARACTERIZATION Quasi S tatic Testing While the mechanical response of polycrystalline Mo in tension has been well documented (e.g. Oer tel et al . , 2007; Zaderii et al., 2005; Walde, 2008), the macroscopic response in compression has been largely unexplored. Furthermore, on the basis of tests p erformed at strain rates over 103 s1, currently, it is generally believed that at room temperat ure polycrystalline Mo has no ductility in tension. The prevalent view is that the brittle nature of its mechanical response is related to the inability of dislocations to multiply and/or move fast enough to relieve stress concentrations (e.g. Cai et al. , 2005 ). This is contrary to earlier experimental investigations conducted in the 60s (e.g. Ault and Spretnak, 1965), which indicate that fracture of Mo at room temperature is not always brittle, the type of failure being strongly influenced by the stress triaxiality. The reasons for the observed influence of the stress state (stress triaxiality) on the room temperature ductility are not understood and have not been further explored neither from a theoretical nor an experimental viewpoint. Moreover, in the literature there is no information concerning the influence of strain rate on the tensioncompression asymmetry. Therefore, there is a critical need for a systematic experimental investigation of the tensioncompressio n asymmetry for monotonic loadings, quantification of directionality of this asymmetry and its evolution with accumulated plastic deformation for both quasi static and dynamic regimes. Experimental Configuration The tensile and compression specimens were cut from the polycrystalline Mo plates by electrodischarge machining (EDM). To characterize the influence of the initial 33

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texture on the uniaxial tensile properties of the polycrystalline Mo plate, tests were conducted on specimens taken at an angle =0 , 15, 30, 45, 60, 75 and 90 to the r eference direction (RD) and in the normal direction (ND) of the plate (Fig. 31). The specimen geometry (Fig 32A) followed the ASTM PinLoaded Tension Test Specimen (ASTM E 8/E8M 09) with subsize specimen dimensions ( reduced overall by a factor of two in order to extract more samples from the plate and perform repeated tests for all orientations). As an example, the photograph of a pinconnected tension specimen showing its gage length and cross section is presented in Figure 33B. Compression tests were conducted according to ASTM E9 Standard Methods of compression testing. The specimens were taken at =0, 45, 60, 90, and in the ND orientation. The compression specimens were right circular cylinders as shown in Figure 3 2B with a diameter of 5.23 mm and length of 9.83 m resulting in an aspect ratio (length over diameter L/D ) of 1.82. An Instron Model 1332 mechanical test frame was used to conduct both tension and compression tests in conjunction with an Instron Model 3156115 load cell of capacity 22.7kN (50k lb). Th is test frame and load cell, shown in Figure 3 3 A, had just pas sed an annual calibration check. Displacement and corresponding average strain was a cquired by vertical translation of the piston below the specimen ; the displacement and rate of displacement w ere monitored using a linear variable displacement transducer (LVDT). The LVDT had a resolution of 0.00254 mm ( 0.0001 in) of piston movement and the rate of displacement was continuously monitored using an electronic feedback loop to ensure the desired rate of displacement and specimen strain rate. The output for the load cell and LVDT during the test was in the form of voltage 34

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Figure 31 . Specimen orientation and reference frame relative to pla te reference system. Figure 3 2 . Specimen geometry: A. Tensile specimen geometry; B. Compression specimen geometry 1.588mm 3.175mm 63.5mm 25.4mm 3.175mm 3.175mm A B 5.23mm 9.83mm T D R D N D Axial Thickness Width

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Figure 33 . Instron test frame with load cell. B. Pinconnected tension specimen prepared for loading. C Compression specimen prepared for loading. recorded using a 14bit Win600 Digital Oscilloscope with a resolution of approximately 1.2mV for the load cell ( approximately 0.5 N(1.2 lb ) force resolution) . The in situ strain fields were acquired using the Digital I mage C orrelation (DIC) method (Sutton 2009). Digital image correlation (DIC) is a non contact technique for measuring full field displacements. The technique works on the principle of discretizing a highcontr ast, random pattern across a gage length into a uniform subset of facets and tracking the displacement of the centroid of these facets relative to one another in order to calculate strain. Local strain is computed using an array of facets referred to as th e strain computation matrix. An example of the highcontrast, random pattern is shown in Figure 3 4A with a magnification of the pattern and facet representation in Figure 34B and the calculated strain for a RD specimen in Figure 34C. A B C 36

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Fig ure 34 . Digital Image Correlation (DIC technique) : A. RD tensile specimen with contrasting, random pattern covering gage section. B. Magnification of highlighted region in Figure 34A illustrating distribution of facets. C. Calculated strain map at 21% av erage true strain for gage section. Measurements using DIC require special care in specimen preparation, image acquisition, and analysis. Specimens were cleaned within the gage length to remove any debris or residue such as oil films. They were then alter nately painted using white and black spray paint to create a high contrast, random pattern of dots (Figure 34 & 3 7). Photron APX —RS high speed cameras having a 1024 x 1024 pixel sensor along with 105mm Fixed Focal Length Nikon lenses were used to acquire the images. The camera shutter speed was set for 1/200th of a second. A delay generator was used to control the acquisition timing of the sequence of images. For tension tests twodimensional (2D) DIC was employed since the gage length did not have curv ature. Initially, a single camera (camera 1) was used for the DIC image acquisition but this configuration was later modified as shown in Figure 35 to add camera 2 in order to directly measure the thickness strain. Due to the geometry of the compression s pecimens threedimensional (3D) DIC was employed with two cameras in stereo. The two acquisition techniques required different calibration methods. Once A B C 37

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calibrated, the 3D technique combined points from the two images to produce a common virtual point for analysis. Another reason for using 2D DIC for the tension tests is related to the stated objective of having deformation measurements up to fracture, and thus precisely determining the failure strain (Fig. 35A). Furthermore, use of two cameras enabled the independent measurement of all three strain components (axial, width, and thickness strains). This in turn will allow the determination with increased accuracy of plastic strain rate ratios (Lankford coefficients) for this material. The delay generator was set to instruct the camera to take images every minute for the first 15 30 minutes (i.e. until the specimen had reached the yield stress) and every five minutes (300 seconds) thereafter. After performing several tests, peculiar failure characteristics were observed for the specimens subjected to tensile loading. To capture the behavior close to failure in greater detail, for the same orientations additional tests were conducted with the timing on the delay generator changed such as to capture images ev ery 5 seconds near failure. For example, at a strainrate of 105 s1, the specimen gage length displaced an average of 1.3 m between frames taken every 5 seconds. ARAMIS v6.2.06 system, produced by GOM Optical Measuring Techniques, was used to measure facet displacement, and thus determine and track the evolution of the surface axial and transverse strains during testing. In some cases, the facet size used for the measurements was increased to 14x14 pixels with 7 pixel overlap such as to improve the grayscale variation, but for most tests a 10x10 pixel facet with 5 pixel overlap was employed. For each image, a time can be est ablished based on the time provided by the delay generator and this time was synchronized with the timing of the load cell in order to plot the stress versus axial strain. 38

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Figure 35 . Tensile test configuration: A. Pin connected tensile specimen and associated strain axes and corresponding cameras. B. Pinconnected tension specimen prepared for loading. As an example, for the uniaxial tension test at =90 on a smooth specimen, in Fig. 3 6 A B are shown the strain m aps obtained with the two cameras corresponding to a global average true strain of 10% measured over the gage length. Fig. 36 C D show A B 39

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for the uniaxial tension test at =90 on a notched specimen, the deformed specimen with the applied pattern, the contours plots of the experimental strain fields measured with each camera. The experimental set up with the DIC acquisition cameras for compression tests is shown in Fig. 37A while the specimen and the experimental axial strain map corresponding to a global av erage true strain of 10% obtained in a test at =RD is presented in Fig. 37B & C. Figure 36 . DIC strain maps: A. Axial strain map for a 90 smooth specimen using images from camera 1. B. Axial strain map for a 90 smooth specimen using images from camera 2. C. Axial strain map of a 90 notched specimen using images from camera 1. D. Axial strain map of a 90 notched specimen using images from camera 2. A B C D Camera View 1 Camera 1 Camera View 2 C amera 2 40

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Figure 37 . Uniaxial compression testing: A. Compression specimen prepared for loading and positioned for image acquisition by two cameras in stereo. B. Molybdenum compression specimen with stochastic pattern. C. Axial strain map within gage length of RD compression specimen at 10% average global true strain. Uniaxial Tension Results Effects of the Strain R ate on t he Tensile Response in the Q uasi S tatic R egime As mentioned earlier, on the basis of tests performed at strain rates over 103 s1, the prevalent view is that at room temperature Mo has no ductility in tension. Thus, the first tension test conducted was a uniaxial tensile test along 90 at 102 s1 to investigate whether the material studied displays this typically reported behavior. It was confirmed B C A 41

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that the specimen exhibited a brittle response and failed in the elastic regime as shown in Figure 38A. Gi ven the potential strainrate sensitivity associated with the material's bcc structure, a uniaxial tension test at a strain rate of 105/s (three orders of magnitude lower) was conducted. The true stress true strain curve is shown in Fig. 38B. Note that f or this strain rate the specimen exhibited ductility, the failure strain being nearly 0.14. Thus, for the first time it was shown that polycrystalline Mo has ductility at 105/s and it is expected a further increase in the failure strain when the strain rate becomes lower than 105/s. To investigate the effect of the loading orientation on the ductility in tension at 105/s, a systematic investigation was conducted. Yielding and Strain Hardening in T ension For each orientation, the tests were repeated at le ast three times. As an example, Figure 39 shows the results of five tests in the RD (true stress vs. true strain). Note the repeatability and consistency of the test results. The inset in Figure 49, shows that the most variability was observed in the ear ly phase of the deformation process, the yield stress (YS) in RD being 3088MPa. The YS with 0.5% offset determined for all seven inplane loading orientations tested are summarized in Table 31. Tests conducted using specimens with a 45 inplane axial or ientation resulted in the largest variation in YS, therefore additional tests were performed. A total of nine tests were conducted for this orientation, eight of those tests resulted in yield values within 5% of the 348MPa average; with one test resulting in a yield of approximately 8% higher than the average. The initial yield stress standard deviation for the 45 specimen was 16.3MPa. The results of all tests at 45 are shown in Fig. 310. 42

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Figure 38 . Stress strain curves for 90 specimen i llustrating variation in ductile response with change in s train rate: A . Stress strain response for a strain rate of 102 s1. B. Stress strain response for a strain rate of 105 s1. 0 100 200 300 400 500 0 0.04 0.08 0.12 0.16 0.2True Stress (MPa) True Strain 0 100 200 300 400 500 0 0.04 0.08 0.12 0.16 0.2True Stress (MPa) True Strain = Strain to Failure Tension = 10 / = 10 / A B Tension 43

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Figure 39 . Variation of stress strain curves for the specimens having an axial direction corresponding to RD. Figure 310. Variation of stress strain curves for the specimens having an axial direction corresponding to 45. 0 50 100 150 200 250 300 350 400 450 500 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2True Stress (MPa) True Strain RD-1 RD-2 RD-3 RD-4 RD-5 0 50 100 150 200 250 300 350 400 450 500 0 0.01 0.02 0.03 0.04 0.05 0.06True Stress (MPa) True Strain 1 2 3 4 5 6 7 8 9 225 250 275 300 325 350 375 0 0.005 0.01 0.015 0.02True Stress (MPa) True Strain 250 275 300 325 350 375 400 425 0 0.005 0.01 0.015 0.02 True Stress (MPa) True Strain 44

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In Fig. 3 11 it is plotted the variation of the YS with the loading direction. For each ori entation, the average value of the YS is represented by a symbol while the error bar indicates the range of variation determined on the basis of repeated tests. It can be concluded that the largest YS is at 4560, with ~14% increase with respect to the RD . As an example, in Figure 312 are shown the stress strain curves corresponding to 0, 45, 90; the inset highlighting the anisotropy in yielding of the material (the black vertical line indicates the offset). The bold black line within the inset of Fig ure 312 indicates = 0.5%. Note that although there is clear anisotropy in yield stresses, the response is very similar for larger deformation. Figure 311. In plane anisotropy of the initial yield stress in tension at = 0.5%. Closed circles represent average values and bars represent potential range for a given direction. 300 310 320 330 340 350 360 0 15 30 45 60 75 90 (MPa) Direction ( ) 45

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Table 31 . Average yield stress at strain = 0.5% Loading Orientation RD 15 30 45 60 75 90 Yield Stress (MPa) 308 324 335 348 350 337 347 Figure 312. Stress strain curves for polyc rystalline Mo subjected to tensile loading at a strain rate of 105/s. The inset highlights the initial yield region. To describe the strain hardening of the material, a power type law was used of the form given below: n 90 RD = 0 45 200 225 250 275 300 325 350 375 400 0 0.01 0.02 0.03True Stress (MPa) True Strain 46

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stressstrain curve in RD based on the hardening law (31) with A=300MPa, B=320M Pa, and n=0.3, along with the experimental stress strain curves obtained in three RD tests. These values of the coefficients are averages of the respective coefficients determined from each of these three experimental stress strain curves. Nevertheless, th e difference between predictions and data is within 2% of either of the experimental curves. Figure 313. Comparison of predicted stress (solid line) using the power law relationship with data (circles) from three experimental RD tests. Average coeffic ients were used and predicted stress values within 2% of any test for RD. Plastic Flow in Tension Lankford coefficients, also known as r values, are a measure of the formability of a metal sheet. The plastic strain ratio, r( ), is defined as the ratio of the width to thickness plastic strain increments measured in a uniaxial tension test along an inplane direction at an angle with respect to RD. For an isotropic material, r( ) = 1, irrespective 250 300 350 400 450 500 0 0.04 0.08 0.12 0.16 0.2True Stress True Strain 47

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of the tensile loading direction . The ratio of strains ar e related to the stress state and yield criterion by the principle of normality and define the slope of the yield locus, see Figure 3 14, at the point of loading ( Hosford, 1993) . Figure 314. Tests needed to determine the yield surface of the anisotropic plate. The ratio of plastic strains, r, defines the slope of the yield surface. ( Lee and Backofen, 1966) As can be seen in Figure 3 15 with width strain plotted against thickness strain, the relationship is linear for the duration of axial deformation indicating the r values remain constant, therefore the yield surface retains its original shape as it increases with strain. Specimens oriented 90 from the r eference direction had the highest 48

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average r value, 0. 93 , approaching an isotropic response where r = 1. Traditionally, the strain ratios are determined by measuring the axial and width strain and calculating the thickness strain using the plastic incompressibility assumption but this method was questioned and an alternative technique developed. A novel c onfiguration of cameras arranged orthogonally allowed the thickness strain to be measured directly. The difference in thickness strain acquired from the two methods is shown in Figure 316 highlighting a significant difference that increases with axial str ain. The average and range of values for the r values for specimens between 0 90 are shown in Figure 3 17 illustrating the difference in r value with loading orientation along with the accuracy of the two methods . Figure 315. Strain ratios for loading orientations: A . The r value for 90 specimen remains constant and is nearly an isotropic response. B. Evaluation of r values for directions 0, 45, and 90. -0.1 -0.08 -0.06 -0.04 -0.02 0 -0.1 -0.08 -0.06 -0.04 -0.02 0Width Strain Thickness Strain -0.1 -0.08 -0.06 -0.04 -0.02 0 -0.1 -0.08 -0.06 -0.04 -0.02 0Width Strain Thickness Strain 45 RD = 0 90 slope =1 (isotropic) 90 49

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Figure 316. Potential error in r value determination Figure 317. Variation of r values with increasing angle from rolling direction (RD). Strain at Failure in Tension At an early stage within the tension test series it became apparent that axial direction with respect to the r eference direction and its corresponding microstructur e -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0 0.05 0.1 0.15 0.2Thickness Strain Axial Strain 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 15 30 45 60 75 90R Direction ( Measured thickness Thickness strain using incompressibility assumption 50

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played a role in the ductility of the molybdenum tensile response. The variation in ductility is shown in Figure 3 18 and it is evident that a significant change in ductility occurs between 15 to 30 from the r eference direction. Specimens having an axi al direction of 15 or less did not fail by fracture and remained intact, yet specimens having an axial direction greater than 15 showed a significant loss in ductility and fractured at some magnitude of axial strain. The average values for strain corresponding to fracture are given by circles in Figure 3 18 and the range of strain values for a direction are provided by the vertical bars. The circles for 0 and 15 do not have vertical bars since they did not fracture. The tensile test was designed to yiel d 25% engineering strain which equates to 21% true strain for the tensile configuration. When testing at a strain rate of 105/s approximately 6 .5 hours were needed to complete the test which influenced the duration of the test. Typically, images for DIC analysis were acquired at 1 minute intervals during the initial yield region and then at every 5 minutes once the specimen had fully yielded. Once the failure pattern became evident the DIC timing was altered prior to fracture to take images every 5 seconds in an effort to see if molybdenum exhibited any unusual response prior to failure. Images shown in Figures 3 19 thru 21 show the axial strain maps prior to fracture along with an image of a fractured specimen indicating point of fracture along the gage length. Uniaxial Compression Results Yield ing and S train H ardening in Compression For each orientation, the tests were repeated at least three times. As an example, Figure 322 shows the results of three tests in the RD (true stress vs. true strain) indicati ng the mechanical response in compression was very repeatable. 51

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Figure 318. Change in ductility as angle between axial direction and rolling direction increases. Figure 319. Stress versus strain for RD specimen with corresponding strain maps at several average global strain values. Strain distribution is fairly uniform over the entire gage length. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0 15 30 45 60 75 90Failure Strain (True) Direction ( ) 0 50 100 150 200 250 300 350 400 450 500 0 0.05 0.1 0.15 0.2True Stress (MPa) True Strain Did not fracture 52

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Figure 320. Stress versus strain for 45 specimen with corresponding strain maps at several average global strain values. Strain distributi on is fairly uniform over the entire gage length. Figure 3 21. Stress versus strain for 90 specimen with corresponding strain maps at several average global strain values. Strain distribution is fairly uniform over the entire gage length. 0 50 100 150 200 250 300 350 400 450 500 0 0.05 0.1 0.15 0.2True Stress (MPa) True Strain 0 50 100 150 200 250 300 350 400 450 500 0 0.05 0.1 0.15 0.2True Stress (MPa) True Strain 53

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Compressi ve loading of Mo with respect to orientation shown in Figure 323 resulted in a similar response as observed in tension. The r eference directi on had an average yield of 319MPa with yield stress increasing with angle. The yield values in compression were larger than the values obtained in tension indicating a tensioncompression asymmetry is observed at the polycrystalline scale. Additionally, the compression yield of the ND specimen is shown in Figure 324. In Figure 325 it is plotted the variation of yiel d stress with loading orientation with average values represented by symbols while the error bar indicates variation determined on the basis of repeated tests. The average yield for the ND loading direction is provided at 90 and is distinguished as the re d diamond. The ND specimen yielded at 341.8MPa which was within 7% of the yield in compression for the r eference direction. The hardening response was similar to that observed in tension and failure was not observed for a loading orientation. Figure 322. Variation of stress strain curves for the RD compression specimens. 0 100 200 300 400 500 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2True Stress (MPa) True Strain RD-1 RD-2 RD-3 54

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Figure 3 2 3 . Stress strain curves for polycrystalline Mo subjected to compression loading at a strain rate of 105/s. The inset highlights the initial yield region. Figure 324. Stress strain curve corresponding to the through thickness or ND direction. 0 100 200 300 400 500 600 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2True Stress (MPa) True Strain 0 100 200 300 400 500 600 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2True Stress (MPa) True Strain 0 100 200 300 400 500 0 0.005 0.01 0.015 0.02 0.025 0.03True Stress (MPa) True Strain 55

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Figure 325. In plane anisotropy of the initial yield stress in compression at = 0.5%. Closed circles represent average values and bars represent potential range for a given direction. Table 32 . Average yield stress in compression at strain = 0.5% Loading Orientation RD 45 60 90 ND Yield Stress (MPa) 319 385 390 372 341 Pla stic Flow in Compression The r values were determined by the ratio of the width to thickness strains and for the Mo tensile data shown earlier, the thickness direction always corresponded to ND for the plate. Although the compression specimens were circular, the anisotropy of Mo would result in an elliptical cross section after deformation as shown in Figure 3.26. To determine the magnitude of cross section change in shape or ellipticity of the gage length when subjected to compressive loading, measurements of the major and minor diameter were made of the specimen after a final strain of = 0.25 was reached. After each compressive test the specimen was cleaned of paint and residue and the cross 310 330 350 370 390 410 0 15 30 45 60 75 90 (MPa) Direction ( ) 56

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section was measured. The scribe mark indicating the normal dir ection (ND) to the plate was preserved and used to relate the change in cross section to the original reference system of the plate. For all tests, the normal direction resulted in the largest strain and therefore was the major diameter. For comparison wit h the tensile data, the cross sectional change was determined in a similar manner as the r value by dividing the minimum diameter by the major diameter (ND). The variation of the aspect ratio (min/max diameter) with axial direction is shown in Figure 327 with specimens having an axial orientation of 0 resulting in an ellipticity of 0.967 and specimens with 90 an aspect ratio of 0.994. This trend was similar to the r values obtained under tensile loading where specimens having an axial orientation of 90 yielded the least difference in strain between directions and were the closest to an isotropic response. Figure 326. Deformed cross section of an anisotropic material resulting in major and minor diameters. TensionCompression Asymmetr y As reported in Chapter 1, molybdenum has a bcc structure and should exhibit tensioncompression asymmetry with regards to the flow stress. The asymmetry in flow stresses of bulk Mo single crystals (mm size) was reported to be large in one study Major Diameter Minor Diameter Initial Diameter Deformed Cross section 57

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Figure 3 27. Variation of cross section change with axial direction for compression specimens at = 0.25 shown in the form of aspect ratio. Aspect ratio: minor diameter/major diameter. since the flow stresses in compression were significantly greater than those in tension (Kim et al . , 2009). Flow stresses for single crystals have been reported but there is no data documenting the polycrystalline response. Comparison of the tension and compression mechanical response for the RD, 45, and 90 specimens are shown in Figure 328 and the asymmetry at the polycrystalline scale is much smaller than that o bserved at the single crystal scale. As evident in Figure 328 the RD and 90 specimens exhibit similar and consistent asymmetry in which the compression flow stress is on average 7% greater than the flow stress required to continue straining the specimen under tension. The asymmetry increases to 14% for the 45 loading orientation. 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 15 30 45 60 75 90Aspect Ratio Direction ( ) 58

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Figure 3 28. Tensioncompression asymmetry of the stress strain response: A. RD specimen. B. 45 specimen. C. 90 specimen 250 300 350 400 450 500 550 0 0.04 0.08 0.12 0.16 0.2True Stress (MPa) True Strain 250 300 350 400 450 500 550 0 0.04 0.08 0.12 0.16 0.2True Stress (MPa) True Strain 250 300 350 400 450 500 550 0 0.04 0.08 0.12 0.16 0.2True Stress (MPa) True Strain T ension Tension Tension Compression Compression Compression RD 45 90 A B C 59

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High StrainR ate Testing The Split Hopkinson Pressure Bar (SHPB) test is a method for generating uniform, uniaxial, stress loading of specimens at much higher strain rates relative to the mechanical test frame discussed earlier. Bertram Hopkinson laid the foundation for this tec hnique through exploration of the shape and evolution of stress wave propagation in long bars usi ng a variety of momentum traps ( see Hopkinson, 1914) . This work was followed by individual efforts by Kolsky and Davies which involved placing the specimen of interest between two long bars in series and measuring the elastic stress wave response in both bars using strain gages placed al ong the length of the rods ( Davies, 1948; Kolsky, 1949) . The sequence of events for a SHPB test is provided in Figure 3 29 below. An elastic stress wave is generated in the incident bar through impact by a striker bar which defines the wave or loading pulse by its length and impact velocity. After the wave has reached the specimen interface a portion of the wave continues to propagate through the specimen and into the transmitted bar and the remaining portion is reflected back into the incident bar. Propagation of the wave through the specimen and its resulting deformation alters the transmitted and reflected waves and the change i n wave shape provides data on the strain, strainrate, and stress of the specimen during its deformation. The specimen should be significantly smaller in length than the incident or transmitted bar since wave reflection is occurring locally within the spec imen during transmission and reflection of the original incident wave. If the specimen length is significantly shorter than the pulse duration then the frequency of reflection within the specimen will be high and the loading can be considered uniform. The length and diameter of the elastic bars are chosen to yield the desired strain and strain rate in the 60

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specimen yet the length to diameter ratio of the bars should exceed 20 to allow separation of the incident and refl ected waves for data reduction ( Gray , 2 000) . As the initial wave reaches the specimen interface there is a large oscillation in strain rate that occurs but quickly reduces to a constant strain rate. As a consequence, the initial yield stress for the material cannot be determined but the test pr ovides yield or flow stress with subsequent strain. Figure 329. Sequence of events for the Split Hopkinson Pressure Bar (SHPB) test. The stress, strain, and strainrate of the specimen can be determined using the measured elastic waves w ithin the bars. By using long bars, the waves can be treated as one dimensional with the objective of determining displacement of the specimen interfaces, u1 and u2, as shown in Figure 3 30 . A solution to the one dimensional wave propagation within each bar is given below: = 1 Transmitted Bar Incident Bar Striker Bar Time Specimen 61

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The strainrate in the specimen is then defined by the timederivative difference between u1 and u2 divided by the instantaneous length, Ls, of the specimen. After the initial ringing due to impact, the specimen is considered to be undergoing uniform deformation and is in load equilibrium which allows the strains in the two rods to be equated. The benefit of equating the strains is that the relationship for determining the strain rat e in the specimen can be reduced to a relationship containing just the measured reflected strain within the incident bar as shown below: ( ) = 2 ( ) Integration of the strain rate with respect to time provides the accumulated plastic strain for the specimen: ( ) = 2 ( ) The true stress of the specimen is then found using the reflected strain, reflected, which defines the instantaneous area along the length, Ls, and assuming that the volume is conserved. The instantaneous true stress is then provided by the relationship below: ( ) = ( ) Experi mental Configuration The specimens for the SHPB testing had a diameter of 5.08mm (0.2 inches) and a L/D ratio of 1 to minimize radial inertia. ( Davies and Hunter , 1963 ) The geometry of the specimens are shown in Figure 3 31. Multiple specimens were tested for the following directions: RD, 30, 45, 60, 90, and ND. A small film of MoSi2 industrial lubricant was used between ends to minimize frictional effects during radial expansion. 62

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Figure 330. Split Hopkinson Pressure Bar reference system for determining high strainrate mechanical properties . (Gray 2000) Figure 331. Split Hopkinson Pressure Bar test specimens For this series of testing, all three bars were composed of Inconel having a modulus of elasticity of 200GPa, and sound speed, cb, of 4892m/s. The diameter of the bars were 15.875mm and the lengths of the striker, incident, and transmitted rods were 1.372m, 6.536m, and 3.276m, respectively. The bars were aligned in series with the configuration shown in Figure 332, and held in place using Teflon bearings. The striker bar was accelerated using a torsion spring. Since the elastic wave duration is defined by twice the length of the striker bar, the pulse was approximately 560 s. Strain gages were placed 1.6m from each specimen interfa ce and sampled every 2 s using a Win600 digital oscilloscope. Compressive Split Hopkinson Pressure Bar (SHPB) tests were conducted for specimens having the following orientations: = RD, 30, 45, 60, Incident Bar Transmitted

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90, and ND to determine mechanical properties at inc reased strain rates, but tension tests using the SHPB configuration were not conducted since Mo exhibited no ductility in tension beyond a strainrate of 102/s. All tests were conducted using the same torsional strain energy for the spring resulting in a similar striker bar velocity and elastic wave magnitude within the incident bar. Figure 332. Images of the Split Hopkinson Pressure Bar test configuration. Results Flow Stress As the elastic wave within the incident bar reached the specimen interface the specimen began to deform at a much higher strainrate initially but quickly settled down to a relatively constant strainrate. The evolution of strainrate for the RD specimen reached a peak strain rate of 1130/s at approximately 28 s a nd 1.7 % true strain, yet 64

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settled to a strainrate of approximately 400/s by 80 s and 5% true strain. The flow stress for the specimen at 5% strain was 970MPa as shown in Figure 333 . The material continued to harden to approximately 14% strain reaching a peak st ress of 1063MPa. The tests conducted using the SHPB configuration were not isothermal. Due to the rapid deformation and corresponding rise in temperature within the specimen thermal softening occurred leading to a decline in yield stress with increased str ain. The stress decreased linearly to 906MPa at 29% strain. The stress strain curves for RD, 45, 90, and ND are shown in Figure 334 which includes an inset highlighting the thermal softening that occurs between 1114% accumulated strain. The peak stress was used to compare the variation of flow stress with angle from rolling direction. As seen in Figure 335, Mo remains anisotropic at high strain rates and exhibits a similar trend compared to quasi static strain rate with the exception of the 90 specime ns which had the lowest average peak stress. Figure 3 33. Evolution of the stress and strain for a RD specimen. 0 200 400 600 800 1000 1200 0 0.05 0.1 0.15 0.2 0.25 0.3True Stress (MPa) True Strain RD Compression 65

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Figure 334. Flow stress for polycrystalline Mo subjected to compressive loading at a strain rate of 400/s. Inset highlights thermal sof tening and corresponding reduction of flow stress. Figure 335. In plane anisotropy of Mo peak stress at a strainrate of 400/s. 0 200 400 600 800 1000 1200 0.05 0.1 0.15 0.2 0.25 0.3True Stress (MPa) True Strain 1000 1020 1040 1060 1080 1100 1120 1140 0 15 30 45 60 75 90Peak Stress (MPa) Direction ( 850 900 950 1000 1050 1100 1150 0.05 0.1 0.15 0.2 0.25 0.3True Stress (MPa) True Strain RD 45 90 ND 66

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Plastic Flow Recovered specimens were measured for change in cross section in a similar manner as the quasi static compre ssion specimens. Again, the ellipticity of the specimen cross section followed a similar trend as the specimens subjected to the quasi static strain rate. The major diameter corresponded to ND. The average aspect ratio (min/max diameter) for the RD shown i n Figure 336 was approximately 0.97 which decreased to 0.95 for 45 specimens since the ND direction increased at a faster rate. The response of the 90 specimens was consistent with the quasi static compression data and r value. Figure 336. Variation of crosssection change with axial direction for compression specimens at = 0.25 shown in the form of aspect ratio. Aspect ratio: minor diameter/major diameter. Conclusions The RD direction was identified and specimens were fabricated at various angles to RD according to ASTM standards for tension and compression loading. For the first 0.94 0.95 0.96 0.97 0.98 0.99 1 0 15 30 45 60 75 90Aspect Ratio Direction ( ) 67

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time, it was established that the material has ductility in tension at a strainrate of 105 s1. The material was observed to be anisotropic in stress and strain for both tension and compression tests with a strength differential (tensioncompre ssion asymmetry) between 7 – 14% with respect to loading orientation. The RD specimen was the softest loading direction for both tension and compression tests and the yield increased 14% and 22% to the hardest direction for tension and compression, respect ively. The material exhibited anisotropy in plastic flow under tensile loading which was quantified by the Lankford coefficient (r value). Initially, the r value was determined through 2D acquisition of the axial and width strains of the specimen and using the plastic incompressibility assumption to resolve the thickness strain. This procedure indicated the r value decreased with increasing angle from RD to 45 and then began to increase from 45 to 90. The plastic in compressibility assumption was challeng ed by using 2 orthogonally configured cameras for a 2D analysis on each side of the specimen which allowed the thickness strain to be measured directly. Although the trend of r values with respect to loading orientation was the same using 2 cameras instead of one, the data indicated the plastic incompressibility assumption for this material was invalid and the actual r values were significantly higher. A similar trend was observed for plastic flow in compression at both quasi static (105 s1) and high stra in rates (400 s1) Tensile ductility was sensitive to loading orientation with specimens having orientations between 15 75 of RD exhibiting drastic loss in ductility. Additional tests were conducted near the strain at failure to capture any response suc h as necking but analysis of the results of the highly resolved temporal resolution tests led to the conclusion that necking was not 68

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occurring. The flow stress of the material at a strainrate of 400 s1 was ~1000MPa which was approximately 3X larger than the initial yield obtained for quasi static testing. In conclusion, mechanical characterization of the material along the axes of symmetry indicated the material was anisotropic in stress and strain for both tension and compression loading tests, the asy mmetric response to tension or compression loading varied with loading orientation, drastic loss of ductility occurred in tension at various loading orientations, and the same trend in plastic flow was obtained in tension and compression across a broad range of strain rates. 69

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CHAPTER 4 CONSTITUTIVE MODEL F OR MOLYBDENUM ElasticPlastic Model To describe the combined effects of anisotropy and tensioncompression asymmetry observed experimentally as well as the strainrate influence on the plas tic deformation of the m olybdenum studi ed, a macroscopic level model developed within the framework of the mathematical theory of plasticity will be used. The governing equations are given in the following. The total strain rate e p )(:peC :eC e eC is the fourthorder stiffness tensor while “ :” denotes the doubled contracted product between the two tensors. Assuming small elastic strains and linear isotropic elasticity, with respect to any coordinate system jk il jl ik kl ij e ijkl C ij being the Kronecker unit delta tensor while eC can be represented by a 6x6 matrix, and the elastic stress strai n relation given by Eq. ( 4 2) can be rewritten as: 70

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e xx xx e yy yy e zz zz e xy xy e xz xz e yz yz 2 000 2000 2000 = 2 2 2 pppF(,)(,)Y()0 p pYY() =pF 0 pF(,)0 pF(,)0

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222 II IIII IIIIII=m where IIIIII : i.e. m I 3 / : mI I being the s econd order identity tensor ) . In Eq.( 7 ), L is a fourth order symmetric tensor that describes the plastic anisotropy of the material. Modeling the anisotropy by means of a 4th order symmetric and orthotropic tensor ensures that the material's response is i nvariant under any orthogonal transformation belonging to the symmetry group of the material (i.e. rotations along the symmetry axes). Furthermore, the minimum number of independent anisotropy parameters that is needed such as to satisfy these symmetry req uirements is introduced. For the orthotropic molybdenum material, in the coordinate system associated with the material symmetry axes (x, y, z ) ( for a plate material reference (RD), transverse (TD) and ND , respectively) and in Voigt notations the tensor L is represented by a 6x6 matrix given by: 111213122223132333445566000000000000000000000000LLLLLLLLLLLLL ( 4 9 ) The effective stress by Lij, being any positive number, the expression for the effective stress remains unchanged. Hence, the anisotropy coefficients can be 72

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scaled by L11, or equivalently, one can set L11 = 1. It is worth noting that although the transformed tensor see Eq 4 8) is not deviatoric, the orthotropic criterion is insensitive to hydrostatic pressure and thus the condition of plastic incompressibility is satisfied (for more details, see Cazacu et al. (2006)). In the expression of the effective stress k is a material parameter, while m is a constant defined such that the e ffective stress, x (or RD) direction. Thus, m is expressed in terms of the anisotropy c oefficients Lij , with i, j =1...3 and the material parameter k as follows: 222 1122331/ mkkk 1111213 2122223 3132333 k = 0, the constant , m , given by Eq. (410) becomes m= 2 3 , so the effective stress Identification of the Parameters of the Model for Polycrystalline M olybdenum The test results in uniaxial tension and compression at a strain rate of 105/s (see data presented in Chapter 3) will be used to identify the material parameters involved in the model, namely the parameters describing the anisotropic behavior Lij (see Eq. (4 9)) and the tensioncompression parameter k (see Eq. (47)). All the materials parameters are then determined by minimizing an error function defined as: 73

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22 exp exp 22 TT CC exp exp nexp nexp nexp nexp ii ii ii ii ij1 2 3 4 22 T2 T2 exp exp i=1 i=1 i=1 i=1 RD RD ii r-r e-e O(L,k)= re (4 12) In the above expression, expTi expCi expir the available Lankford coefficients in tension, exp iedenote the experimentally determined ellipticity of the deformed cross section in compression, T RD T i Ci i represent weighting factors. For molybdenum, the w eighting coefficients for the r values were double than the others which were set to unity. The objective function of Eq. 412 is minimized using the interior point method algorithm built in the Matlab software (2010b). The numerical values of the material parameters are given in Table 41. Note that according to the model, i n the plastic regime the material is almost of cubic symmetry (L11 xx , yy) (x being the r eference direction (RD) while y the transverse direction (TD) ) the theoretical yield surface for the Mo material according to the criterion (Eq. 47) for a strain level p

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The pred icted and experimental r values are given in Figure 42. It is worth noting that the model captures well both the anisotropy in flow stresses and r values of the material in tension, as well as the observed tensioncompression asymmetry. Table 41 . Model pa rameters for a polycrystalline m olybdenum. L11 L22 L33 L12 L13 L23 L44 L55 L66 k 1 0.997 1.02 0.93 0.93 0.94 0.64 0.64 0.64 0.055 Fig ure 4 1 . Comparison between the yield locus of polycrystalline molybdenum according to the orthotropic yield criter ion of Cazacu et al . (2006) (Eq.(4 7)) and data (symbols). Stresses are in MPa. -500 -400 -300 -200 -100 0 100 200 300 400 500 -400 -200 0 200 400TD (MPa) RD (MPa) 75

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Figure 42 . Predicted and experimental evolution of the Lankford coefficients (r values) with the orientation for polycrystalline Mo in comparison with the experimental values (symbols). Finite Element Formulation and Implicit Time Integration Algorithm U sed for I mplementation of the Elastic P lastic M odel . A user material subroutine (UMAT) was developed for the constitutive model described by Eqs. ( 4 1) ( 4 1 1 ) and implemented in the commercial implicit FE solver ABAQUS Standard (ABAQUS, 2009). A fully implicit integration algorithm was used for solving the governing equations that will be further used to simulat e both quasi static and dynamic deformation processes or quasi static response of a system. First, the governing equations for threedimensional problems and the strong and weak form of the balance of linear momentum and their finite element discretization are briefly 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 15 30 45 60 75 90R Value Direction ( 76

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reviewed. Next, the fully implicit time integrati on algorithm for solving the governing equations for the specific elastic/plastic model based on Cazacu et al. (2006) yield function is presented. General Form of Governing Equations for 3 D problems In this research the finite element problem will be sol ved using the implicit solver o f the commercial software ABAQUS (see ABAQUS, 20 09 ). Let X define the motion of a deformable solid from the reference to the current configuration. Abaqus software uses an updated Lagrangian formulati on, i.e. t+t t+t =+tttXXv . Fig 4 3 . An updated Lagrangian formulation. Thus, the deformation gradient, XF is calculated as t+ijtXFXij 77

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and in the governing equations, all the dependent variables i.e. the Cauchy stress X . Let D . To the governing equations are associated boundary conditions, such as tractions T on t v on v tv div= TnT t (4 13) In the above equations, is the current density, and a is the acceleration (i.e. =au a . It is to be noted that Eq. ( 4 13) depends only on the velocity field v , because the stresses can be expressed in terms of velocities by the constitutive model. Nevertheless, finding the exact solution to the set of differential equations given by Eq. 4 13 is difficult, if not impossible. For this reason, integral equations, called the weak form of Eq. ( 4 13 ) , are derived. The weak form of Eq. ( 4 13) promotes computing of approximate solutions by further discretization with finite element s. The weak form of the balance of linear momentum can be written as: 78

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tvvva v v v ::v 1 2Tvv D= , so the weak form can be written as: tvva ev which is an admissible trial function, i.e. it must at least be C0 and satisfy the boundary condition on v evv on v is discretized into a number of elements e with say, n nodes, so the approximate solution can be written as e kk 1,t=N()tn k vXXv , (4 16) where kN() X denote the shape functions, which have the desired level of continuity such that ev is admissible. In Galerkin finite elements, the trial functions, v

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of the same form as the trial functions with the additional restriction v v kk 1=N()tn kvvXX . Nodes remain coincident with material point labels at all times and kJkJN(X) = ea is given by the time derivative of the trial function, ev , so e kk1,t=N()tnkaXXa (4 17) The rate of deformation is the symmetric part of the spatial velocity gradient (i.e. derivatives are taken with respect to the spatial coordinate x and not X ), so in index notations : ij IiI,jIjI,i11D=vNvN2nek III eetee e e e a extintM[a]=ff M is defined as MeIJ IJNN

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while f dAetext IIeNT and fd etint k keN Time Integration Algorithm for the Orthotropic Elastic Plastic M odel Time integration algorithms are broadly characterized as implicit or explicit. In e xplicit algorithms the state variables at t+ t are updated based entirely on the available values at time t . The linear momentum equations are not solved. The central difference operator, which is the most commonly used explicit operator for stress analysis applications, is only conditionally stable, the stability lim it being approximately equal to the time for an elastic wave to cross the smallest element dimension in the model . Implicit schemes remove this upper bound on time step size by solving the equilibrium equations based not only on values at t , but also at t+ t. The elastic plastic model with yielding described by the orthotropic form of the Cazacu et al. (2006) yield criterion, associated flow rule and isotropic hardening (see Eq. ( 4 1 1 )) was implemented as a User Material Subroutine (UMAT). In ABAQUS, all stresses and strains are rotated by R , the proper orthogonal rotation corresponding to the polar decomposition of the gradient of deformation F (F = RU = VR) with U and V being symmetric and positivedefinite tensors, called the right stretch tensor and the left stretch tensor) before the UMAT is called. It is also important to note that all the mathematical form of the yield condition, flow rule is known in the coordinate frame associated to the axes of orthotropy of the material. Thus, the orientation of the material orthotropy axes must be known at any 81

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given time. In ABAQUS the orthotropy axes are transformed as well by the rotation R . Las tly, to ensure that the constitutive equations are objective (i.e. a superposed rigidbody motion rotation should not affect the material response), the TQ where D reduces to 0 0 F , and 0 F 0 F then the trial stress is accepted as the current stress state. 82

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If 0F 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 expands due to hardening. This approach is also referred to as a returnmapping method since the increment of the effective plastic strain is adjusted such that the stress is returned to the yield surface. The implicit time integration scheme includes an elastic predictor followed when necessary by a plastic cor rector (Hughes, 1984; Simo and Hughes, 1997). During a time step, 1 nnttt trial 11:e nnn trial 1,,0npnF trial 11 nn trial 1,,0npnF p 1 n 1n 11111,1 ,1,111,1,,,F: 0pn n pnnn pnnen nnpnFY C ( 4 21) Let m be the local iteration counter. If 0trial 11 nn 0 10n 0 m 11 111 11 111 mmm nnn mmm nnn

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where m and 1 m 11kn 11mn 2 11 11 2 11 11 11 11 1 1 11 11 11 11 1 11,, , , ,,0 : : :mm mm nn nn mm m m nn n n mmm m m mm nn nn nn m mm n nn nneeFF F FF F 0 11 11 11 1 1 1 11 11 11 1 11,, , , :: , :: ,m nn mm mm nn nn mm m nn n m n mm mm mm nn nn nnFF F FFF P P P denotes a fourth order tensor expressed as: 1 1 1 2 1 2[] [ , ]mm n e n m e nF IC C P ( 4 25) wi th I being the fourthorder identity tensor. The variation of the stress increment 11mn 111111111F:,: mmnnnnmmnmen PC ( 4 26) The stresses and the plastic strains are then updated until a specified tolerance is met. Once convergence is reached, the updated stresses and strains are accepted as 84

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the current state. The elastic plastic tangent modulus epC relates the current stress increment 1n 1 n 11 11 ep ep 11 11 11 11,, : CC ,,, :: : :; nn nn nn mm nn nn nnFF FFF PP P . However, the orthotropic form of the Cazacu et al . (2006) yield criterion (Eq. (48) is expressed in terms of the principal values of i ijmklmijklFF i , j , k,l, m =13 ( 4 28) where the convention of summation of repeated indices is adopted. Note thatThe transformed stress is expressed in terms of the Cauchy stress deviator T T denotes the 4th order deviatoric projection that transforms a 2nd order tensor in its deviator. In Voigt notations, T is represented by the 6x6 matrix: 2/3-1/3-1/3 -1/32/3-1/3 -1/3-1/32/3 T= 1 1 1

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ijijmpijmpmpklklmpklLT i , j , m,p,k,l =1 (4 31) To calculate m ij i 123 3JJ23332JJ233 where 1I=1/3 tr() 2 2J= tr( 33J= tr( ii32jjj=1iii32jjj=1 if if mFm i =13 The second derivatives of the yield function were calculated numerically based on: 2 jj ijFF F ii iicc

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Comparison of Model Pre diction with Measured Stress versus Strain We further apply the proposed model to simulate the plastic response of the smooth tensile specimen (for the geometry of the specimens, see Chapter 3) and compare with the experimental observations acquired using DIC. The FE mesh for the 1/8th section of the specimen analyzed consisted of 1827 linear hexahedral elements with reduced integration (A BAQUS C3D8R) ( Fig. 4 4). As discussed in Chapter 3, the mechanical response depends on the loading orientation, i.e. relative orientation between the r eference direction and the loading direction. With the UMAT for the constitu tive model given by Eq. (41) (4 11) simulations were conducted for specimens oriented along RD and TD, respectively. Fig.4 4 shows a comparison between the experimental and theoretical load vs axial displacement curves. The experimental axial displacement corresponds to the axial di splacement measured using DIC techniques. Note the excellent agreement in terms of description of the overall (global) response, in particular the accuracy with which it is described the influence of th e loading orientation. Furthermore, for each loading orientation, the F.E. predictions of the strain isocontours are compared to the DIC strain maps obtained experimentally for the same axial displacement. For the RD specimen, local strain maps extracted using DIC are compared to the predicted ones f or an axi al displacement of 5mm. It is to be noted the quantitative agreement between the numerical predictions and the experimental measurements of the local fields. Moreover, the shape of the isocontours corresponding to any level of strain conform with the exper imental ones. The predictive capabilities of the model will be further assessed by comparing the uniaxial tension response of notched specimens in Chapter 5. 87

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Figure 44 . Predicted and measured stress vs strain response of the RD specimen subjected to quasi static tensile loading. Inset highlights comparable strain magnitudes and distribution. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6Load (kN) Axial displacement (mm) RD FE DIC 88

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CHAPTER 5 MOLYBDENUM NOTCHED T ENSILE TESTING AND MODEL VALIDATION Introduction In the previous chapter, the mechanical behavior had been modeled using an anisotropic yield criterion associated to an isotropic hardening law. All the material parameters needed to describe the plastic behavior of the polycrystalline molybdenum has been identified using the experimental data presented in Chapter 3. In order to further assess the predictive capabilities of the plasticity model, in this chapter, uniaxial tension tests on notched specimens have been performed. The main objective is to further the understanding of the mechanical behavior of Mo when subject ed to non axisymmetric loadings. Indeed, introducing an imperfection such as a notch within the gage length section of the tensile specimen changes the stress state as compared to that experienced by a smooth specimen. Furthermore, the data generated will fill a gap in the knowledge of the mechanical response of Mo, since very limited studies of the influence of the loading on the response of Mo exist in the literature. Specifically, the influence of the specimen geometry on the mechanical response, i.e. on ductility (strain to failure), local strain fields (strain map obtained with DIC measurement ) , ratios between axial and transverse strains will be examined. Another objective is to assess the predictive capabilities of the elastic plastic model developed for Mo that was presented in Chapter 4. To this end, F.E. simulations using the model will be compared to global and local experimental data acquired using DIC techniques. 89

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Experimental Configuration and Results To improve the understanding of the plasti c behavior of molybdenum, uniaxial tension tests of notched specimens have been performed. Tests were conducted on specimens taken at an angle = 0, 45, and 90 to the r eference direction (RD). The same mechanical test frame, load cell, and DIC equipment p resented in Chapter 3 have been used to perform the uniaxial tension tests on the notched specimens. The geometry of the specimen is shown in Figure 51. The gage length cross section is a square of side of 1.5875 mm, and the notch radius is 3.175 mm. The evolution of local strain at the notch root was monitored with two cameras. C amera 1 monitor ed axial and width strain developing in the front face of the specimen, camera 2 monitored axial and thickness strain occurring in the side of the specimen. It is to be noted that for notched specimens the 2D DIC measurements require more precautions than these taken in the case of smooth specimens. While camera 1 can monitor the strain in the entire domain, due to the nonplanarity of the side surface of the specim en, camera 2 could only monitor the strains at the notch root ( Fig.5 2 ). Note that in order to determine accurately the axial displacement, instead of using machinebased displacements, DIC techniques were used. Machine based displacements were deemed less accurate because of the impossibility to eliminate the compliance of the Instron machine. Thus, the axial displacement was measured by following two points i nitially located outside of the notch region using camera 1. 90

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Figure 51 . Notched tens ile specimen geometry . Figure 52 . Position of the cameras used and the axial strain maps acquired with each camera. r= 3. 175mm 3.175mm 1.587mm 1.587mm 15.875mm 15.875mm 4.201mm Camera 1 Camera 2 Camera 1 Camera 2 A B C A B B C 10.599mm 10.599mm 91

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The introduction of the notch within the gage length of a tensile specimen creates a complex stress state , the ratios between the principal stresses depending on the radius of the notch. Furthermore, it is very interesting to assess how the presence of the notch affects the directionality of the mechanical response. The new data generated for the notched specimens is shown in Fig.5 3B while for comparison purposes the data on smooth specimens is shown in Fig.53A . First, irrespective of the loading orientation there is a large difference in loadcarrying capacity between the smooth and the notch ed specim en . This is explained by the fact that the cross section of the notched specimen is much smaller than that of the smooth specimen (Fig. 51 and Fig. 3 2) which provides the dimensions of the respective cross sections). Most interestingly, new and unexpect ed findings concerning the influence of the tensile loading orientation on the mechanical response for smooth vs. notched specimens are revealed. The presence of the notch doesn’t change the relative difference in strength between orientations , the RD dire ction remains the softest direction (lowest yiel d stress) , followed by the TD direction , and the 45 direction . However, as concerns the ductility (strain to failure), the trends are completely different. Indeed, for smooth specimens , a strong influence of the specimen orientation on ductility was observed, the RD specimen has strain at failure larger than 20% while for the TD specimen the strain at failure is of 13 %. On the contrary f or the RD notch ed specimen, the strain to failure is lower than that of the TD notched specimen. Specifically, failure of the RD notched specimen occurred at an axial displacement of 0.09mm while for the TD notched specimen failure occurred at an axial displacement of 0.18 mm. While, it is expected that the ductility of a notc h ed specimen to be lower than the ductility of a smooth specimen, for the 45 specimen the 92

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ductility of the notched specimen is almost the same as that of the smooth specimen. In conclusion, the effect of the notch on ductility depends on the orientation o f the specimen. The average axial stress at the notch root corresponding to an offset strain of 0.5% for the RD, 45, and TD specimens are of 341MPa, 408MPa, and 429MPa, respectively. The respective axial stresses were calculated as the ratio of the load over the smallest cross sectional area. In this research, efforts were made to take full advantage of the DIC techniques and capture the time evolution of the local strains in the notch area in 3 dimensions . Furthermore, this was done for different specimen orientations. The DIC strain maps in the notch region prior to failure acquired with camera 1 ( i.e. axial strain and width strain o n the front face of the specimen) and camera 2 (axial strain and thickness strain o n the side of the specimen) for the RD, 4 5, and TD specimens are shown in Figures 54 through 56. It is very important to point out that meaningful measurements of deformation can be taken only on planar surfaces, i.e. on the front face of the specimen (as taken with camera 1) and at the root of the notch , with camera 2 (along BC , Fig. 5 2). The fact that on the side meaningful data can be acquired only along BC is due to the curvature of the specimen (see also Fig. 51 for the geometry of the specimen). It is very important to note that irrespec tive of the orientation of the specimen, the axial strain that develops on the front face of the specimen differs from the axial strain that develops on the side . Specifically, the axial strain is larger on the side than on the front face. Furthermore, the strain is not homogenous in the cross section. 93

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Figure 53 . Effect of specimen geometry on the response for the specimens oriented at = 0, 45, 90 to RD: A . Smooth specimens. B. Notched specimens. 1400 1500 1600 1700 1800 1900 2000 2100 2200 0 1 2 3 4 5 6Load (Newtons) Axial Displacement (mm) 600 700 800 900 1000 1100 1200 1300 0 0.04 0.08 0.12 0.16 0.2Load (Newtons) Axial Displacement (mm) RD 90 45 N otched Un notched RD 90 45 B A 94

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Close examination of the strain fields within the notch region of the RD, 45, and 90 specimens indicates the influence of anisotropy of the material, differences in the maps of the thickness strains among the specimens being the most striking. For the RD and 45 specimens the zone where the thickness strain is maximum is extended over most of the notch region (maximum value is almost uniformly distributed), the maximum value for the 45 specimen being larger than that of the RD specimen. In contrast, for the TD specimen the zone of maximum thickness strain is more limited. For quantitative comparisons with simulation results , the average value of axial strain and width strain at the root of the notch were calculated by averaging the strains measured by DIC over the line AB (Fig.52) . Likewise, an average value of axial strain and thickness strain over the line BC (Fig.52) were also extracted from DIC measurements acquired with camera 2. As an example, in Figure 57 is shown the evolution of the axial strain along the line AB and the line BC, respectively for a TD specimen. The data shown were extracted from four images taken at increasing levels of the applied global displacement. Note that as the test progresses the more heterogeneous the axial strain at the root of the notch becomes . 95

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Camera 1 Camera 2 Axial Strain Axial Strain Width Strain Thickness Strain Figure 54 . Axial and width strain distribution for both camera v iews for the RD specimen. The images were taken prior to failure. Notch Root A B B C B B C A Notch Root 96

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Camera 1 Camera 2 Axial Strain Axial Strain Width Strain Thickness Strain Figure 55 . Axial and width strain distribution for both camera views for the 45 specimen. The images were taken prior to failure. Notch Root Notch Root 97

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Camera 1 Camera 2 Axial Strain Axial Strain Width Strain Thickness Strain Figure 56 . Axial and width strain distribution for both camera views for the TD (90) specimen. The images were taken prior to failure. Notch Ro ot Notch Root 98

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Camera 1 Camera 2 Camera 1 Camera 2 Camera 1 Camera 2 Camera 1 Camera 2 Figure 57 . Evolution of axial strain with displac ement at the notch root cross section for the TD (90) specimen. Last image was taken prior to failure. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.7 1.2 1.7 2.2 2.7Axial Strain Crosssection Width 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2Axial Strain Cross section Width 3 4 2 1 Frame 4 Displacement = 0.175mm Frame 3 Displacement = 0.1 13mm Frame 2 Displacement = 0. 065mm Frame 1 Displacement = 0. 010mm Camera 1 Camera 2 Cross section Width Cross section Width A B B C B A B C 99

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Based on the uniaxial test data on smooth specimens, the Lankford coefficients o r r values were determined ( Chapter 3). As mentioned, the anisotropy of r values, is indicative of the anisotropy in plastic flow of the material. It is interesting to evaluate the strain ratios for the notched specimens. This is because for each specimen the evolutions of the average strains along lines AB and BC with the g lobal axial displacement for the RD, 45, and 90 specimens are shown in Figure 58. It is to be noted that the average axial strain along BC (i.e. on the side, acquired with camera 2) increases faster than the average axial strain along line AB (i.e. on t he front face, acquired with camera 1). Specifically, for the RD and 45 specimens the average axial strain on the line BC ( camera 2) differs by 58% from the average axial strain on the line AB (camera 1) while for the TD specimen a difference of 40% was observed. It is to be noted that this very large difference observed depending on the camera views raises a key question regarding what is the local axial strain at failure that is reported in the literature. The major finding is that due to the anisotro pic plastic behavior of the material and the specific specimen geometry, the axial strain is no longer homogeneous in the cross section and the notion of reporting a unique "axial strain to failure" value is meaningless. As an example, for a RD, 45, and T D specimen, at failure, the axial strain differs of 2.3%, 1.8%, and 3.2%, respectively. This concept can be applied only in the case of smooth specimens, for which the axial strain acquired with either camera is the same ( Chapter 3). In Fig. 5 8 are shown the average width and thickness strains evolutions for each notched specimen. Given that all strains vary linearly with the applied load (global 100

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displacement), the ratios of the width to thickness strain are given as constant throughout the test. For each test, the width strain was plotted against the thickness strain and the respective slope was determined, thus determining the r value corresponding to the respective orientation. The variation of r values with orientation as determined from the tensile tes ts on the notched and smooth specimens, respectively is reported in Fig. 59. The data show that the introduction of the notch does not change the anisotropy in plastic flow. Finite E lement S imulations of the N otched T ensile T ests and Comparison with M easurements We apply the proposed anisotropic elastic plasti c model given by Eq. (41) (4 11) to simulate the mechanical response of the polycrystalline molybdenum notched specimens when subjected to uniaxial tension tests. Quantitative comparison between the simulated and measured local strain fields will be provided. The FE simulations were conducted using the UMAT developed (see chapter 4 for the details of the implicit integration algorithm). Due to symmetry of the problem only an eighth of the specimen w as modeled using 13056 hexahedral linear elements with reduced integration. The mesh was refined in the notch region as shown in Figure 510 in order to accurately capture the deformation at the notch root. Boundary conditions were applied such as to maint ain the symmetry of the problem, and an axial displacement was imposed to replicate the piston displacement of the mechanical test frame resulting in a strain rate of 105 s1. The elastic parameters values used were: E = 310 GPa, = 0.3. . 101

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Figure 58 . Evolution of average strains (RD, 45, & 90) at the notch root acquired with 2 cameras with displacement at the notch root for both camera views. 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.05 0.1 0.15 0.2Axial Strain Axial Displacement (mm) Camera 1 Camera 2 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0 0.05 0.1 0.15 0.2Strain Axial Displacement (mm) Camera 1 Camera 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.05 0.1 0.15 0.2 Axial Strain Axial Displacement (mm) Camera 1 Camera 2 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0 0.05 0.1 0.15 0.2Strain Axial Displacement (mm) Camera 1 Camera 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.05 0.1 0.15 0.2 Axial Strain Axial Displacement (mm) Camera 1 Camera 2 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0 0.05 0.1 0.15 0.2Strain Axial Displacement (mm) Camera 1 Camera 2 width thickness width width thickness thickness RD 45 90 102

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Figure 5 9 . Evolution of the plastic strain ratios (width/th ickness) as a function of the tensile loading orientation determined from tests on notched and smooth specimens. Figure 510. Finite element m esh o f the notch ed specimen. For the RD specimen, comparison between the simulated (solid line) and experimental measurements (symbols) of the evolution of the average axial strain and width strain occurring on the face of the specimen (line AB) with the axial displacement 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 15 30 45 60 75 90Plastic Strain Ratio Direction ( ) Un notched Notched 103

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is presented in Fig 5 11 along with the comparison of the evolution of the axial strain and thickness strain occurring at the side of the specimen. Note that the model correctly captures the evolution of all the local strains. Fig. 5 12 shows the comparison between simulated and experimental evolution of the strains that develop during a n uniaxi al tension test of a notch ed 90 ( TD ) specimen. The model predicts all the experimental trends. Note that for the 90 ( TD ) specimen the model predictions of the axial strain on the specimen side are in excellent agreement with the experimental measurement s extracted from the DIC measurements with camera 2, while over predicting the axial strains and width strains on the front side acquired with the camera 1. Similar conclusions can be drawn for the RD specimen by comparing predicted and experimental measur ements with each camera. Note also that the anisotropy in the mechanical response is correctly captured. Furthermore, the strain maps for both the RD and 90 specimens were compared to the model predictions. Since the entire surface of the notch region for camera 1 belong to a plane perpendicular to the camera, the local strain fields obtained by DIC measurement with the camera 1 prior to fracture for the RD and 90 specimens could be compared with the local field predicted by the mode. The specimen and loa ding is symmetric so in Fig. 5 13 the measured strain field is superposed on the top half the photograph of the specimen while on the bottom half are superposed the simulation results. Note that the simulated and measured axial strains isocontours for both RD and the 90 specimen are nearly identical. The strain magnitudes are similar for the width strain but the patterns are different since the simulation results are symmetric but the measured strains are not. 104

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Figure 511. Average strain with displacement at the notch root for a specimen with RD loading orientation. -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Average strain Axial displacement (mm) axial RD -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Average strain Axial displacement (mm) axial thickness Camera 1 Camera 2 105

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Figure 512. Average strain with displacement at the notch root for a specimen with 90 loading orientation -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Average Strain Axial Displacement (mm) -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Average Strain Axial Displacement (mm) axial axial width thickness 90 Camera 1 Camera 2 106

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Camera 1 Figure 51 3 . Comparison of DIC measurements and model predictions for axial and width strain. DIC FE DIC FE RD TD 107

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Conclusions Very careful DIC measurements of the local strain fields in notched specimens of different orientations were conducted. The influence of loading directions on t he load carrying capacity, overall ductility, and local strain fields was determined. New and unexpected findings resulted from this investigation as concerns the combined effects of stress triaxiality and anisotropy on the mechanical response. Comparison between the response of notched and unnotched specimens show that the type of anisotropy is not altered by the presence of the imperfection. However, the r values are much lower. It was established that for this material, how the notch affects the ductil ity is strongly dependent on the loading orientation. It was established that the concept of "axial strain to failure" cannot be used when analyzing/reporting data on notched specimens. Comparison between the anisotropic elastic plastic model predictions and local strain measurements of axial and width strain evolution show a very good agreement. Furthermore, it was shown that the model can predict with accuracy the distribution of the axial and width local strains. In particular, the influence of the loading orientation on the response was very well described. 108

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CHAPTER 6 MOLYBDENUM TAYLOR IM PACT TESTING AND MODEL VALIDATION Introduction As already mentioned, there is a lack of information in the open literature on the high strain rate mechanical behavior of Mo materials. The existing data are generally limited to either one loading direction or loading conditions. One of the objectives of this dissertation was to fill this knowledge gap by performing a systematic investigation of the effects of initi al texture on the mechanical response for a variety of loadings and strain rate conditions. The experimental technique used to characterize the behavior for strain rates of the order of 103104 s1 for the material under investigation was the Split Hopkins on Pressure Bar (SHPB) as described in Chapter 3. The SHPB (strain rate of 400 s1) stress strain curves in compression along the RD direction also served to quantify the combined effects of strain hardening and strain rate on hardening of the material and identify a JohnsonCook (1983) type law (Eq. 31). This hardening law in conjunction with the anisotropic criterion of Cazacu et al . (2006) was implemented as a user material subroutine (UMAT) in the commercial implicit FE solver ABAQUS Standard (ABAQUS, 2009). A fully implicit integration algorithm was used for solving the governing equations (see Chapter 4). T h e predictive capabilities of the model have already been assessed for quasi static loadings. Of particular importance is the investigation of the tensile mechanical response of the material under impact where the strainrates experienced by the material are greater than 102 s1. High strain rates of the order of 104105 s1 can be achieved in a Taylor impact test, which consists of launching a sol id cylindrical specimen at an elevated velocity of the order of 100300 m/s against a stationary rigid anvil (see Taylor, 1948; 109

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Cristescu, 1967). When the specimen impacts the rigid stationary anvil an elastic compressive wave generates at the impact inter face and travels through the specimen in the axial direction with a speed equal to the sound speed in the respective material . When this compressive wave reaches the other end of the specimen , it is reflected as a tensile wave. For a sufficiently high impact velocity , for which the magnitude of the compressive wave reaches the yield stress of the material , the impact end undergoes plastic deformation. The plastic front with maximum stress magnitude equal to this yield/ flow stress starts propagating from the impact interface at a much lower speed than the elastic wave. The travelling plastic front interacts with the reflected precursor elastic wave at some intermediate point along the length of the specimen . The e lastic wave then get s reflected at the elastic – plastic interface and travels towards the free end of the specimen . This back and forth movement of the elastic wave results in deceleration of the specimen. Taylor (1948), developed a one dimensional wave propagation analysis to estimate the dynamic (hi gh strainrate) yield strength. Later on, semi analytical models have been proposed to determine the yield strength as a function of density, impact velocity, original and final length, and length of the undeformed r egion of the retrieved specimen (see for example, Gilmore, 2002; Lu, 2001, Wang, 2003). Being onedimensional in nature, the abovementioned models are unable to accurately capture the threedimensional plastic deformation behavior of the material. Currently, the Taylor test is used more as a m eans of validating plasticity models and codes for the simulation of high rate phenomena such as impact and explosive deformation (see Zerilli and Armstrong (198 8 ); Maudlin et al. (1999), etc.) . 110

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It is important to note that to date Taylor impact data have been reported only for materials which exhibit tensile ductility ( ( e.g. for iron, see Zerilli and Armstrong, 1988; for copper , see House, 1989; for tantalum, see for example, Zerrilli and Armstrong, 1990; Maudlin et al.,1999a; for zirconium, see data repor ted by Maudlin et al. 1999b). For a material like the m olybdenum material investigated, which for quasi strain rates of order of 102/s shows no ductility, even the feasibility of a Taylor test is questionable. This is because the very l arge tensile stres s es generated would result in fracture /immediate disintegration of the projectile. To the best of my knowledge, Taylor impact test results on m olybdenum have not been reported. Thus, this is first attempted as part of this dissertation research. The outlin e of this chapter is as follows. First, it is report ed Taylor impact tests conducted on the Mo material. Next are presented FE simulations of the dynamic Taylor impact tests using the model presented in Chapter 4. The model predictions are compared with the measurements of the deformation of the impacted specimens. Furthermore, the model is used to gain insights into the physics of the process and provide predictions of local fields that are not accessible through experimental measurements. Indeed, images m ay be taken during the tests and measurements may be made of the post test geometry to determine the rate of deformation and post test extent of plastic deformation, but the state of the material during loading is not known. Therefore, it is crucial to hav e a reliable and accurate model in order to gain understanding of the dynamic deformation process of this material. The last section is devoted to such a discussion of the capabilities of the model and its potential to be used for "virtual testing". 111

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For th is effort, Mo cylinders were launched at a stationary anvil and the deformation of the recovered specimens was measured to quantify the response for a compressive loading orientation. The elastic plastic model discussed previously was used to gain insight into the dynamic loading event regarding stress, strain, and strainrate and validation of the model was achieved through a comparison between the measured and predicted geometry profiles. Taylor Impact Tests on Molybdenum All the tests were conducted at t he Air Force Research Laboratory, Eglin AFB, FL. For these tests, cylindrical specimens, having an initial diameter of 5.33mm and an initial length of 53.34mm (i.e. lengthto diameter ratio of 10) were used ( Figure 61), The cylinders were cut from the plane of the plate using an EDM process and followed by a turning operation with a lathe. Specimens with long axis either along RD, 45 , and 90, respectively were fabricated. For each specimen, the throughthickness direction of the plate (ND) was marked by an arrow on the cross section. This allowed the material axes to be tracked and document the deformation of the specimens in the different directions of symmetry and thus evaluate the effect of the inplane anisotropy of the material on its dynamic response. Figure 61 . Dimensions of the cylindrical specimens used for the Taylor impact tests. T he specimens were launched toward an anvil as shown in Figures 6 2 & 6 3 using a smooth bore ( Mann) barrel and propellant. The Mann barrel had a bore diamet er of 5.36mm. The anvil had a diameter of 190.5mm, thickness of 149.2mm and 53.34m 5 .3 3mm 112

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wa s made of steel 4340 alloy with a hardness of 58 HRC . During an y given experiment the specimen was placed inside the barrel near the breech, and a cartridge containing Red Dot propellant was placed in the breech behind the specimen . The quantity of propellant was chosen such as to acquire the desired impact velocity . The propellant was ignited using a percussion primer. A small polymer obt urator was placed between the specimen an d the cartridge to minimize gas slipping around the specimen during propellant by product expansion and acceleration through the barrel. Both impacting surfaces (i.e. anvil and specimen) were pol ished to minimize friction ( Fig. 6 3). Special care was taken such as to ensure that t he axis of the barrel bore is aligned at 90 to the impact surface of the anvil so normal impact of the specimen is achieved. Given that small indentations of the anvil’s polished surface are created during an impact test, after each experiment the anvil was rotated a few degrees to prevent the se indentations on the anvil from affecting the results obtained in a subsequent test . The distance between the end of the barrel and the anvil was approximately 76mm providing an opportunity to observe the dynamic deformation, yet minimize trajectory instability. Velocity measurements were acquired using a set of pressure transducers and an optical light detector connect ed to a highspeed camera. The p ressure transducers mounted in series at the exit end of the barrel were used to detect the leading edge of the pressure change due to the expansion of the propellant gas. A white light source was mounted beneath the table and illuminated a detector above the flight path of the specimen. As the specimen moved toward the anvil it block ed the light , which triggered the high speed camera. While b oth techniques were used to determine the specimen velocity , the measurements of the velocity using the high speed camera images w ere 113

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deemed to be the most a ccurate . S ince during any given test the specimen undergoes varying stress, strain, and strain rate, its geometry should change along its length (i.e. the deformation depends on the distance from the impact ed end) . The geometry profiles of the recovered sp ecimens were measured using a Deltronic DH214 optical comparator which was accurate to within 0.001mm. Due to the anisotropy of the material , the initial circular crosssections became elliptical. The major and minor axes of the cross section of the defor med specimen were recorded along its length, the respective faces of the deformed specimens being labeled major and minor profiles , respectively . As mentioned, due to Mo’s limited tensile ductility conducting Taylor tests and obtaining information on the dynamic deformation of Mo is particularly challenging. While it is desirable to have a high impact velocity such as to produce significant deformation that can be measured, in the case of Mo for an impact velocity greater than 150 m/s substantial failure of the specimen is likely to occur in the form of separation crack propagation along the centerline or radial cracks originating along the outer diameter and propagating toward the centerline. The photographs of the recovered specimens for velocities of appr oximately 180m/s are shown in Fig 6.4 A C. Furthermore, special care needs to be taken such as to prevent failure of the specimens caused by a slight misalignment. Typically, in a Taylor impact test the barrel is brought into contact with the anvil to ens ure alignment and then the barrel is retracted and fixed in position. For more ductile materials than Mo, this procedure would be sufficient. However, for the Mo tests, the same procedure was followed but a more rigorous assessment of the alignment using l evels had to be conducted in order to 114

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ensure normal impact. Otherwise catastrophic failure occurs as shown in Figure 6.5 (A C), which present the photographs of deformed specimens following impact at approximately 150m/s. Figure 62 . Cross sectional v iew of the Taylor impact test set up showing the specimen, the barrel, and the target (anvil). F igure 6 3 . Photographs depicting the configuration of the Taylor impact test. (A ) B arrel and anvil location with laser detection system and camera. ( B ) Close up view of the anvil and barrel reflection to show how well the respective surface s were polished in order to minimize frictional effects during impact. 115

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Figure 6 4 . Failure processes for Taylor specimen: A. Deformed region near impact surface exhibiting separation due to negative pressure (tensile loading) generated by impact at approximately 180m/s. B. Deformed region near impact surface exhibiting an excessive radial cracking response at approximately 180m/s. C. Corresponding impact surface for image 6 4B illustrating extent of radial propagation of cracks. A total of twelve Taylor impact tests at impact velocities in the range 140160 m/s were successfully conducted (specimens remained intact without excessive cracking). The specimen orientation with respect to the axes of symmetry of the material, the impact velocity, and information concerning the geometry of the post test specimen (change in length of the specimen, and the major diameter of the impacted end) are given in Table 61 . If possible, the impact v elocity was determined from both pressure transducer timing and from image analysis of the recordings with the highspeed camera. For the tests were it was not possible to obtain information with both diagnostics, it is indicat ed with a label NR (not recorded). A B C V=~180m/s V =~180m/s 116

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Figure 65 . Consequences of slight misalignment: A. The deformed region near the impact surface of an intact specimen with impact velocity of approximately 150m/s. B. Similar deformed region for a specimen having an impact velocity of 150m/s which failed during deformation attributed to misalignment. C. Image of both sections of specimen shown in Figure 65B. Upon impact, stress wave propagation is complex particularly during the first few microseconds of deformation and the material at the impact surface is subjected to the highest magnitude of stress and strainrate. Additionally, the impact surface is subjected to loading for a duration greater than any other cross section along the axial length of the s pecimen, therefore it should undergo the most radial expansion (see the schematic representation of a post test specimen shown in Fig. 66) . Indeed, t he attenuation of the strain rate and stress along the length of the cylinder results in a reduction of ra dial deformation with increasing distance from the impact surface. Nevertheless, due to the A B C V=~1 5 0m/s V=~1 5 0m/s V=~1 5 0m/s 117

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anisotropy of the material, any cross section is an ellipse (for example, see schematic representation of the impacted end after the test shown in Fig. 66). The major axis of the ellipse is along the ND direction of the plate. This indicates that the material's anisotropy remains the same irrespective of the strainrate experienced by the material. Indeed, both the quasi static data, the high strain rate data acquir ed with the Split Hopkinson Pressure Bar technique, and finally impact test data show that the hardest to deform direction for this material is 90 to the ND direction. Table 6 1 . Molybdenum Taylor impact test data Test Specimen orientation Impact Velocit y based on highspeed camera (m/s) Impact Velocity determined by Pressure transducers (m/s) Length of the post test specimen normalized by its initial length Maximum Radius of the impacted crosssection (mm) 1 RD 140 NR 0.923 3.539 2 RD 151 152 0.915 3 .750 3 RD 165 160 0.903 3.929 4 45 NR 148 0.923 3.708 5 45 NR 149 0.923 3.635 6 45 154 163 0.912 3.789 7 45 153 NR 0.919 3.834 8 90 141 140 0.930 3.542 9 90 145 145 0.923 3.872 10 90 NR 149 0.922 3.603 11 90 151 155 0.916 4.232 12 90 161 164 0.906 3.932 F or each specimen orientation, the measurements of the geometry of the recovered specimens following the Taylor tests at about the same impact velocity (~151 m/s) is provided in Fig. 67. Specifically, for RD, 45, and 90 specimens, th e final p rofiles are shown in Fig. 67A while the variation of the aspect ratio of the deformed crosssections as a function of the distance to the impacted end are presented in Fig. 67 B. 118

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Figure 66 . Schematic representation of the post test Molybdenum specimen: A. Deformed specimen. B. Ellipticity of the cross sections resulting from the anisotropy in deformation. Note that for the 45 specimen, the final length was of 49.23 mm, the major and minor diameters of the impacted cross section wer e of 3.635 mm and 3.489 mm, respectively. Plastic deformation extends to approximately 30mm from the impact surface. On the other hand, both the RD and the TD (90) specimen exhibited more axial deformation than the 45 specimen, and more radial expansion. However, for the TD specimen with the exception 1 – 3 m from the impact surface, the cross section remained practically circular (aspect ratio 1). For the RD specimen, the ellipticity of the crosssections was less important than for the 45 specimen. The major radii increase as expected with increasing impact velocity. The aspect ratio variation for RD and 90 specimens appears to be sensitive to impact velocity. As a general conclusion, irrespective of the specimen orientation, there is more deformation (change in length and radial expansion) with increasi ng impact velocity ( Table 61 and Fig. 68) . Major Diameter Minor Diameter Initial Diameter Deformed Cross s ection Anvil Radial Expansion During Deformation 119

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A B Figure 67 . Experimental deformation of Mo Taylor RD, 45, and TD specimens after Tayl or impact test at ~150 m/s : A. major and minor profil es. B. A spect ratio of the cross section as a function of the axial distance from the impacted end. 2.6 3 3.4 3.8 4.2 0 5 10 15 20 25 30Radius (mm) Axial Position (mm) Initial Major Minor 0.93 0.95 0.97 0.99 1.01 0 5 10 15 20 25 30Aspect Ratio Axial Position (mm) Initial 151m/s 2.6 3 3.4 3.8 4.2 0 5 10 15 20 25 30 Radius (mm) Axial Position (mm) Initial Major Minor 0.93 0.95 0.97 0.99 1.01 0 5 10 15 20 25 30Aspect Ratio Axial Position (mm) Initial 149m/s 2.6 3 3.4 3.8 4.2 0 5 10 15 20 25 30Radius (mm) Axial Position (mm) Initial Major Minor 0.93 0.95 0.97 0.99 1.01 0 5 10 15 20 25 30Aspect Ratio Axial Position (mm) Initial 151m/s RD – 151m/s RD – 151m/s 45 – 149m/s 45 – 149m/s 90 – 151m/s 90 – 151m/s 120

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A B Figure 68 . Comparison of the post test geometries corresponding to various impact velocities for the RD, 45, and TD specimens, respectively: A. D ef ormed shape (major profile). B . A spect ratio of the cross section as a function of the distance from the impacted end. 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 0 5 10 15 Major Radius (mm) Axial Position (mm) Initial 140m/s 151m/s 161m/s 0.93 0.95 0.97 0.99 1.01 0 5 10 15Aspect Ratio Axial Position (mm) Initial 140m/s 151m/s 165m/s 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 0 5 10 15 Major Radius (mm) Axial Position (mm) Initial 148m/s 149m/s 154m/s 0.93 0.95 0.97 0.99 1.01 0 5 10 15Aspect Ratio Axial Position (mm) Initial 148m/s 149m/s 154m/s 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 0 5 10 15Major Radius (mm) Axial Position (mm) Initial 141m/s 151m/s 161m/s 0.93 0.95 0.97 0.99 1.01 0 5 10 15Aspect Ratio Axial Position (mm) Initial 141m/s 151m/s 161m/s RD RD 45 45 90 90 121

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Finite Element S imulations of the Taylor Impact Test and Comparison with M easurements Although analytic al expressions are being developed to determine state variables within the Taylor impact specimen the complexity of wave mechanics limits their utility, therefore the finite element (FE) method and its ability to discretize the problem was chosen as the means to examine the highrate, inertial loading process. FE simulations using the constitutive model described in Chapter 4 in conjunction with the implicit solver were performed for the three impact velocities for both the RD and 90 specimens. For each si mulation, the geometry considered corresponded to the exact measured geometry of the specimen (initial diameter 5.33mm; initial length of 53.34mm). Due to the symmetry of the problem, only a quarter of the specimen was meshed with 6732 hexahedral linear el ements (ABAQUS C3D8R) with reduced integration. The mesh is refined in the impact zone ( zoom in Fig. 6 9 ) to accurately capture the deformation of the impacted surface of the specimen. Boundary conditions were applied such as to maintain the symmetry of th e problem. The simulations were performed in two phases, associated to the specimen launching and the impact, respectively . In the first phase, the impact velocity was applied to all of the nodes such as to represent the free flight or launch of the speci men. In the second phase, the impact of the specimen with the rigid anvil was reproduced by imposing a null velocity only to the nodes belonging to the impact surface while the other nodes were not constrained anymore. In this manner, the test conditions w ere reproduced with fidelity. Furthermore, it is assumed that the impact ends when the axial velocity at the end of the specimen becomes null. A density value of = 10,200 kg/m3 was used to account for 122

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inertia effects . To account for the strain rate effects a Johnson Cook isotropic hardening law mentioned in Chapter 4 and provided again below was used: n 0 = 105 s1. Given that an implicit time integration algorithm was used for solving the governing equations, the equilibrium equations were strictly satisfied at each time increment, thus ensuring high fidelity predictions of the stress field. Figure 69 . FE mesh of the quarter specimen with zoom of the impact surface. Analysis of the calculated deformation using the model and quantitative comparison with the experimental measurements of the deformation after impact are given in the following. As discussed in Section 6 2, irrespective of the orientation of the specimen, the geometry of the post impact specimen changes along its length or with the distance from the impact surface ( Fig. 6 6 and Table 61). Figures 610 to 612 show the strain pro files along the major ( y or ND) direction and minor (z or 90 = TD) Impact Surface 123

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direction of the RD post test specimen after impact at a velocity of 140, 151 and 165m/s, respectively. In these figures the predicted major and minor specimen profiles are represented by solid lines while the experimental measurements taken at several distances from the impacted end (impacted end corresponds to x = 0) are shown as symbols. Note the overall very good agreement between the numerical predictions and the experimental data for all impact velocities. The specimen was initially a cylinder with the axis along the RD (x direction). Due to the orthotropy of the material, the deformed cylinder cross sections are slightly elliptical in shape, with the maximum radius being along the ND direction and the minor radius along the 90 or TD plate direction ( z direction). It is also very worth noting that the very little ellipticity of the TD cross section is well captured by the model. Furthermore, the fact that there is very little difference in the ellipticity of the cross section along the specimen reinforces the remark made previously that the anisotropy of this material is very little influenced by the strain rate. The lesser accuracy in prediction of the ellipticity near the impact surf ace may be attributed to the fact that the slight radial cracking at the outer radius that was observed in the tests was not accounted for in the model. Comparison between the simulated outline of the deformed TD specimens and the experimental measurement s for all the three impact velocities tested are shown in Fig. 613 to Fig. 615. Note the overall very good agreement, also evidenced by comparing the measured and predicted axial deformation for each specimen reported in Table 62. 124

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Table 62 . Measured and Predicted Material Response Specimen Impact Velocity Experimental L Final /L Initial Numerical L Final /L Initial Difference RD 140 0.923 0.915 0.0086 RD 151 0.915 0.905 0.0109 RD 165 0.903 0.890 0.0143 90 141 0.930 0.920 0.0107 90 151 0.916 0.911 0. 0054 90 161 0.906 0.902 0.0044 Next, the model will be used to get insights into the deformation process. Analysis of the local state fields at different instances within the deformation process are provided next for the RD specimen impacted at a veloc ity v= 151 m/s. First the simulated pressure evolution in time at various locations along the specimen axis is discussed. The respective locations define the distance to the impacted end are marked by symbols in Fig. 6 16. Note that the model predicts a large amplitude oscillation in pressure (transient loading) which attenuates into a more stable uniform distribution within 15 s. Figure 610. Experimental and predicted deformation profiles corresponding to the 90 and ND material axes for an impact velocity of 140m/s. -5 -4 -3 -2 -1 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50Radius (mm) Deformed Length (mm) RD Impact Velocity: 140m/s Minor Profile Major Profile 125

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Figure 611. Experimental and predicted deformation profiles corresponding to the 90 and ND material axes for an impact velocity of 151m/s. Figure 612. Experimental and predicted deformation profiles corresponding to the 90 and ND material axes for an impact velocity of 165m/s. -5 -4 -3 -2 -1 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50Radius (mm) Deformed Length (mm) -5 -4 -3 -2 -1 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50Radius (mm) Deformed Length (mm) RD Impact Velocity: 165m/s RD Impact Velocity: 151m/s Minor Profile Minor Profile Major Profile Major Profile 126

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Figure 613. Experimental and predicted deformation profiles corresponding to the RD and ND material axes for an impact velocity of 141m/s. Figure 614. Experimental and predicted deformation profiles corresponding to the RD and ND material axes for an impact velocity of 151m/s. -5 -4 -3 -2 -1 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50Radius (mm) Deformed Length (mm) -5 -4 -3 -2 -1 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50Radius (mm) Deformed Length (mm) 90 Impact Velocity: 151m/s 90 Impact Velocity: 141m/s Minor Profile Minor Profile Major Profile Major Profile 127

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Figure 615. Experimental and predicted deformation profiles corresponding to the RD and ND material axes for an impact velocity of 161m/s. As it should be, at distances closest to the impact there is the largest oscillation in pressure, at a location on the centerline at x= 0.42mm from the impact surface the peak positive pressure predicted being of 7.5GPa while the peak negative pressure predicte d is of 2.2GPa. (tensile pressure according to the sign convention). The development of such very high tensile pressures is consistent with experimental results and the explanation of the lateral fracture that may occur/potential pulling apart of the spec imen is in agreement with published literature values (Chhabildas, 1989). The same oscillatory pressure is predicted at 3.2mm from the impact surface, but the magnitude has significantly reduced with the peaks being of 4 GPa and 1.5GPa, respectively. In F igure 617 (A), it can be seen that within15 s at locations beyond 3.2mm pressure oscillations are less than 0.5GPa. It is also important to note that within 3 s the pressure reaches about 35% of the length of the specimen, yet at the respective location (19 mm from impacted end) its amplitude is significantly smaller than closer to the impact surface and it is almost -5 -4 -3 -2 -1 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40 45 50Radius (mm) Deformed Length (mm) 90 Impact Velocity: 161m/s Minor Profile Major Profile 128

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exclusively compressive (positive value) (Fig 6.17a). At t= 20 s from the impact, the pressure at the location x=19mm is higher than that predicted at 0.42mm, 3.2mm, and 49mm and ranges between 225 and 375MPa. On the other hand, at x= 49mm, the pressure exhibits rapid oscillations between 100MPa to 350MPa with an average of approximately 125 MPa. Figure 616. Pressure evolution in time at various locations along the specimen centerline (impacted end corresponds to x=0). In conclusion, the model predicts large amplitude oscillation in pressure which attenuates dramatically within 15 s, thus marking the transition from transient to quasi stable deformation. -3000 -2000 -1000 0 1000 2000 3000 4000 5000 6000 7000 8000 0 5 10 15 20 25 30 35 40 45 50 55 60Pressure (MPa) Time (ms) Distance from Impact Surface 1 2 3 4 1 2 1: x= 0.42 mm 2: x= 3. 20 mm 3: x=19.0 mm 4: x= 49.0 mm 129

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Figure 617. Pressure oscillations at different locations along the specimen centerline: A. D uring the first 15 s after impact. B. 20 40 s after impact. It i s also worthwhile analyzing the predicted attenuation of the velocity and the distribution of the plastic strainrate in the specimen and relate to the predicted pressure distribution. In Fig. 6 18 is shown the evolution of the axial velocity as a function of time for the same centerline locations. Note that at the location near the impact surface (x= -3000 -2000 -1000 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Pressure (MPa) Time ( s) -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 20 22 24 26 28 30 32 34 36 38 40Pressure (MPa) Time ( s) Distance from Impact Surface 1: x= 0.42 mm 2: x= 3.20 mm 3: x=19.0 mm 4: x= 49.0 mm 1 2 3 1 , 2 3 A B 130

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0.42 mm), an extremely rapid deceleration occurs followed by a slight increase in velocity and a more gradual decrease towards zero velocity at t= 15 s. At the location x= 3.2mm velocity oscillation occurs during the first 5 s after impact, and there is a steady decline in velocity thereafter. Nevertheless, after 15 s, at both x= 0.42 mm and x= 3.2mm, the axial velocity is zero. At location x= 19 mm and x= 49 mm the decrease in velocity is more gradual and more similar, which is consistent with these material points experiencing quasi stable deformation. The time at which the velocity becomes zero at those locations is of 46 and 55 s, respectively. The same trends in velocity attenuation are predicted irrespective of the impact velocity, but with different magnitude of the oscillations and time at which the velocity becomes zero (Fig. 619). Figure 618. Variation of axial velocity in time at various locations along the centerline of the specimen. 0 20 40 60 80 100 120 140 160 0 5 10 15 20 25 30 35 40 45 50 55 60Axial Velocity (m/s) Time ( s) 131

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Figure 619. Simulated evolution of the axial velocity in time at x= 3.2mm (solid lines) and x=19mm (interrupted lines), along the centerline for impact velocities v= 140 m/s (blue), v= 150m/s (red), and v= 165 m/s (black). The predicted axial velocity evolution along the centerline indicates that the specimen underwent a rapid deceleration that needs to be consistent with the predicted evolution of the plastic strainrate. Fig. 6 20 shows the is ocontours of the equivalent plastic strain rate s very high plastic strain rates of the order of 104 s1are predicted, with the largest strain rates, of the order of 5.6 104 s1, being near the impact surface and along the centerline but only 1.6 104 s1 along the outer radius of the impacted surface. Furthermore, the largest strain rates were experienced by the material during the initial, transient phase of the deformation when nonzero plastic strain rates occur only in the vicinity of the impacted surface. At 8.4 s, the zone of peak plastic strain r ate along the centerline near the impact surface has grown but similar 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 30 35 40 45 50 55 60Axial Velocity (m/s) Time ( 132

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peak strain rates appear along the outer radius of the specimen ahead of the zone propagating along the centerline. By 10.3 s these regions along the outer radius with high plastic strain rate values have propagated toward the centerline and coalesced forming a circular region experiencing high plastic strainrate that is moving away from the impact surface. At the same time, the plastic strainrate of the material at the impact surface is rapidly diminishing. By 22.5 s, the material at the impact surface is no longer experiencing comparable strainrates and the zone of high plastic strainrate is propagating along the centerline with the specimen geometry exhibiting expansion in the region away from the impact surface. The sequence of images illustrate the non uniform loading response are consistent with the interpretation of the deformation of a Taylor impact specimen due to Taylor (1948) . As an example of the difference in magnitude in plastic strainrate experience d at different locations along the centerline Figure 621 shows the plastic strainrate evolution at x=0.42mm, 3.2 mm, and 19mm. The plastic strainrate at x= 0.42mm has a peak value of 15 104 s1 whereas at x= 3.2mm the peak value is of 6.5 104 s1. At x= 19mm the peak value is of only 0.8 104 s1 at approximately 45 s. At x= 49mm, plastic deformation doesn't occur. This is consistent with the pressure evolution predicted at these locations ( Fig. 6 16). In Fig. 6 22 is shown the evolution in time of the plastic strain rate at the centerline and outer radius of the cross section at x= 3.2 mm from the impacted end. The results are consistent with the isocontours shown in Fig. 6 20. 133

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Figure 620. Isocontour of the plastic strain rate at different times from an impact velocity of 150m/s . Time = 5.3 s Time = 8.4 s Time = 10.3 s Time = 22.5 s Regions of high plastic strain rate Coalescence of high plastic strain r ate 134

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Figure 621. Evolution of the plastic strainrate in time at various locations along the centerline of the specimen. Near the impact surface, there is an extremely rapid rise and fall in plastic strai n rate. Figure 6 22. Comparison between the plastic strain rate evolution at the centerline and outer radius of the cross section at x= 3.2 mm from the impacted end. 0 20000 40000 60000 80000 100000 120000 140000 160000 0 5 10 15 20 25 30 35 40 45 50 55 60Plastic Strain Rate (1/s) Time (ms) 0 5000 10000 15000 20000 25000 0 5 10 15 20 25 30 35 40 45 50Plastic Strain Rate (1/s) Time ( s) Centerline at x=3.2mm Outer radius at x=3.2mm 135

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The evolution in time of the equivalent plastic strain at the same center line loca tions is shown in Figure 623 and the difference in equivalent plastic strain between centerline and outer radius at 3.2mm over time is shown in Figure 624. As expected the strain near the impact surface (x=0.42mm) is the largest, reaching a peak of 0.57 at t=14 s. An interesting observation concerns the predicted final peak value of the plastic strain at x= 3.2 and 19mm respectively. While this value is the same at the end of impact the strain paths seen by these two points are completely different. The s ame level of plastic strain is reached (~ 0.14), at t= 15 s at the location x= 3.2 mm and only at t= 46 s at x=19mm. This is consistent with the evolution in axial velocity and pressure predicted at these locations and the time at which the transient phase of deformation ends. The final internal distribution of equivalent plastic strain for an RD specimen with impact velocity v=150m/s is shown in Figure 625 illustrating similar strain magnitudes despite different axial locations due to the nonuniform loading. Figure 623. Evolution of the equivalent plastic strain in time at various locations within the specimen. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 5 10 15 20 25 30 35 40 45 50 55 60Equivalent Plastic Strain Time (ms) 136

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Figure 6 24. Comparison between two radial points of the equivalent plastic strain at 3.2mm from the impact surface. Figure 62 5 . Distribution of the final equivalent plastic strain within the specimen. Note the two circled, light blue regions having similar strain values. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 5 10 15 20 25 30 35 40 45 50Equivalent Plastic Strain Time ( s) Centerline at 3.2mm Outer radius at 3.2mm 137

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C onclusions The Taylor impact test was chosen as means to subject Mo to high strainrate, nonuniform loading and compare the measured response against the predicted response from a model that can simultaneously acc ount for anisotropy and tensioncompression asymmetry. Tests were conducted using specimens having a length to diameter ratio of 10. Successful tests were conducted for an impact velocity ranging 140 – 165m/s. For impact velocities beyond this range, very high negative pressures (tensile pressures) are generated within the specimen immediately after impact exceeding the reported spall strength of Mo. The fact that for this range of impact velocity the tensile pressures that develop do not exceed the spall strength was also confirmed by the FE simulations conducted with the proposed model. The very important sensitivity to the slightest misalignment of t he specimen was also evidenced. Three successful tests per loading orientation (RD, 45, 90) were conducted having a velocity range between 140 165m/s and the post test geometry was measured to within 0.001mm accuracy. Similar to the quasi static result s and the results obtained through split Hopkinson pressure bar testing, the 45 loading orientation exhibited the largest plastic flow anisotropy followed by the RD orientation and the 90 orientation showing very little plastic flow anisotropy. Additionally, the anisotropy was not constant along the deformed length. Simulations were conducted using the previously discussed model to examine how pressure, strainrate, and strain evolved in time for a particular location within the specimen. A good agreement between numerical predictions of experimental results is reached by comparing the measured and predicted final profiles of the deformed specimen. The predictive tools (i.e. plasticity model accounting for key deformation 138

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features) developed as a result o f this effort combined with an implicit dynamic solver generate opportunities to improve the understanding of the molybdenum response under impact loading conditions. 139

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CHAPTER 7 SUMMARY Molybdenum (Mo) is a hightemperature, refractory metal (melting temperature > 2200 C) with a significantly lower atomic mass than the other refractory metals such as tantalum and tungsten. A review of the literature indicates a strong need for a combined experimental and theoretical effort that addresses the mechanical response of polycrystalline molybdenum. Current molybdenum research has been focused primarily at the single crystal scale indicating the material does not obey Schmid law during crystal slip and it has a tensioncompression asymmetry. Lim ited polycrystalline studies have reported anisotropy of the yield and plastic flow when subjected to tension but the data is very limited and compression is ignored. It was the objective of this effort to conduct a systematic experimental investigation of polycrystalline Mo in order to quantify the anisotropy and directionality of asymmetry and use this data to develop a constitutive model capable of predicting the response for more complicated loading scenarios. The research was conducted in a linear seq uential manner of determining the microstructural characteristics of the plates which included determining the axes of material symmetry. These axes of symmetry along with additional specimens at various angles to the axes were evaluated under both tension and compression loading configurations in order to quantify initial yield, strainhardening, and plastic flow for the respective loading configurations. The material was determined to be anisotropic under both tension and compression with the lowest yield or softest direction occurring for RD and the hardest direction occurring for loading orientations at 45 60 from RD. The initial yield stress taken at = 0.005 was 308MPa and 319MPa for tension and 140

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compression, respectively, for the RD specimens with the yield stress increasing to 348MPa and 385MPa for tension and compression of the 45 specimen. Asymmetry of yield at the polycrystalline scale was not as large compared to the single crystal scale. The asymmetry was determined to be approximately 7% (compression > tension) for the RD and 90 specimens and approximately 14% for the 45 specimen. Due to the interrelated nature of stress and strain, plastic flow of the material during yielding and strain hardening was of high interest. Under tension the strain ratio was 0.75 for the RD specimen which decreased to 0.67 for the 45 specimen, and increased again to 0.93 which was almost an isotropic response for the 90 specimen. Initially, the strain ratios were determined using the conventional method of monitoring the axial and width strain of a tension specimen and using the plastic incompressibility assumption to calculate the thickness strain. For comparison, a novel approach was taken using an orthogonal configuration of cameras to directly measure the thickness strain of the specimen and it was found to be smaller than if the incompressibility assumption was made therefore providing a more accurate determi nation of the plastic flow. The plastic flow in compression defined as the cross section ellipticity of the deformed specimen resulted in the same variation in strain ratio. Another interesting observation regarding the quasi static mechanical response und er tension was the strain to failure was high dependent upon loading orientation with significant loss in ductility for loading orientations between 30 75 with respect to RD. Compression tests were conducted at highstrain rates (400 s1) using the s plit Hopkinson pressure bar to evaluate strain rate sensitivity. The same trends such as yield/flow stress and plastic flow with respect to loading orientation were obtained but the magnitude of the flow stress increased to approximately 141

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1000MPa which was a 3X increase compared to the quasi static response. The data acquired during the mechanical characterization was used to develop a constitutive model for predicting the response of molybdenum. At the center of the elastic plastic constitutive model was a yield function that simultaneously accounted for anisotropy and tension compression asymmetry. A fourth order symmetric, orthotropic tensor was used to describe the plastic anisotropy and was applied to the deviator of the Cauchy stress tensor. The asymmet ry was accounted for using a material parameter denoted by k (Eq. 47). The parameters for the model (orthotropic tensor components and k) were identified by minimizing a weighted error function using the experimental quasi static data as input. The associ ated flow rule and isotropic hardening were assumed. The model was fully implicit requiring the equilibrium equations to be solved at every time step. Based on the experimental data acquired during mechanical characterization testing (Chapter 3) and the subsequent development of a constitutive model (Chapter 4) two validation efforts were pursued to evaluate the predictive capability of the model. The first effort examined the tensile response in the quasi static strain rate regime when a symmetric notch was introduced to the gage section and thereby altering the stress state. This effort had experimental merit since it had been reported in the literature that changes in the tensile stress state would alter the mechanical response. It was determined that duc tility was significantly reduced for all loading orientations with the introduction of the notch but the 90 specimen exhibited the greatest ductility. The strain ratios exhibited the same general trend but the respective magnitudes were significantly lowe red. Comparison between model predictions and local strain measurements 142

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showed a good agreement. In particular the effect of loading orientation on the response was very well described. The second effort utilized the Taylor impact test to explore the Mo response at very high strain rates and provide a rigorous challenge for model predictions. Despite the molybdenum’s limited tensile ductility that was highly sensitive to strainrate, specimens were successfully recovered intact and the deformation profiles were measured and compared with model predictions. Excellent agreement between model predictions and measurements were obtained with the exception of the impact surface which exhibited minor cracking which was not accounted for in the model. 143

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LIST OF REFERENCES 1. ABAQUS, 2009 . User’s Manual for Version 6.8, vol. I V. Dassault Systemes Simulia Corp., Providence, RI. 2. Arsenlis A., Tang M., 2003. Simulations on the growth of dislocation density during Stage 0 deformation in BCC metals . Model Simul M ater SC, 11. 251 – 264 3. Ault R., Spretnak J., 1965, Initial yielding and fracture in notched sheet molybdenum. Int. J. Mech. Sci. 7. 87102 4. Brown J., Ghoniem N., 2010. Reversible – irreversible plasticity transition in twinned copper nanopillars . Acta Materialia, 58 , 886 – 894 5. Belytschko, T., Liu, W. K., Moran, B., & Elkhodary, K., 2013. Nonlinear finite elements for continua and structures. John Wiley & Sons, New York 6. Bulatov V., Hsiung L., Tang M., Arsenlis A., Bartelt M., Cai W., Florando J., Hiratani M., Rhee M., Hommes G., Pierce T., de la Rubia T., 2006. Dislocation multijunctions and strain hardening . Nature letters, Vol. 440, 11741178 7. Burkhanov G., Kirillova V., Kadyrbaev A., Sdobyrev V., Dement’ev V., 2007. Peculiarities of tensile deformation of m olybdenum and molybdenum rhenium single crystals at room remperature. Metal 2007 2224, 17 8. Cai W., Bulatov V., Chang J., Li J., Yip S., 2004. Dislocation core effects on mobility . Chap. 64 in Dislocations in Solids. Nabarro F., Hirth J., vol. 12, 73106 9. Cazacu, O., Plunkett, B., Barlat, F., 2006. Orthotropic yield criterion for hexagonal close packed metals. Int. J. Plast. 22, 11711194. 10. Chalmers B., 1938. The specific resistance and the temperature variation of resistance in tin crystals . Philosophical magazine, 25 , 172 , 7th series, 110811 11. Chang J., Cai W., Bulatov V., Yip S., 2001. Dislocation motion in BCC metals by molecular dynamics . Mater Sci Eng A309– 310, 160– 163 12. Chaussidon J., Fivel M., Rodney D., 2006. The glide of screw dislocations in bcc F e: Atomistic static and dynamic simulations . Acta Materialia, 54, 3407– 3416 144

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13. Chauveau T ., Gerber P ., Bastie P ., Hamelin B ., Tarasiuk J ., Bacroix B ., 2002. Application of the refocused highenergy Laue method for analysis of recrystallization mechanisms after cold deformation of facecentered cubic metals . Journal de physique IV, 12 , PR6 , 107 114 14. Cheng J., Nemat Nasser S., Guo W., 2001. A unified constitutive model for strain rate and temperature dependent behavior of molybdenum . Mechanics of Materials Vol. 33, 603616 15. Chhabildas , L., Barker, L., Asay, J., Trucano, T., 1989. Spall strength measurements on shock loaded refractory metals. Shock Compression of Condensed M atter. American Physical Society Topical Conference. Albuquerque, NM, 429 432 16. Cockeram B., 2010. The role of stress state on the fracture toughness and toughening mechanisms of wrought molybdenum and molybdenum alloys , Mater Sci Eng A528, 288308 17. Cristes cu N., 1967. Dynamic Plasticity . North Holland Publishing Company Amsterdam , John Wiley & Sons, Inc. New York. 18. Davies R.M., 1948. A critical study of the Hopkinson pressure bar , Philos. Trans. R. Soc., 240, 375457 19. Davies E.D., Hunter S.C., 1963, The dynamic compression testing of solids by the method of the split Hopkinson pressure bar. J. Mech. & Physics of Solids, 11, 155179 20. Dobromyslov A., Dolgikh G., Talutz G., 1990. Orientational dependence of the temperature of ductileto brittle transition of molybdenum single crystals . Scripta Metallurgica et Materialia, Vol. 24, Is. 3, 543546 21. Furuya K., Nagata N., Watanabe R., Yoshida H., 1981. Effect of low cycle fatigue on the ductilebrittle transition of molybdenum . Journal of Nuclear Materials 104, 937 – 941 22. Gilman J. , 1953. Plastic deformation of rectangular zinc monocrystals . Transactions of the American institute of mining and metallurgical engineers, 197, 9 , 1217122 145

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23. Gilmore M. R., Foster J.C., Wilson L.L., 2002. Dynamic fracture studies using sleeved Taylor impact specimens , Shock Compression of Condensed Matter, 519522 24. Gordon P., Neeraj T., Li Y. Li J., 2010. Scr ew dislocation mobility in BCC metals: the role of the compact core on doublekink nucleation. Modelling and Simulation in Materials Science and Engineering, 18, 113 25. Gray, G.T., 2000, Classic split Hopkinson pressure bar testing, ASM Handbook Volume 8, M echanical Testing and Evaluation (ASM International) , 462 476 26. Green, A. E. and Naghdi, P.M., 1965. A general theory of an elastic plastic continuum. Archive Rational Mech. & Anal., 18, 251281. 27. Hollang L., Brunner D., Seeger A., 2001. Work hardening and flow stress of ultrapure molybdenum single crystals . Mater Sci Eng A319321, 233236 28. Hopkinson B., 1914. A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets , Philos. Trans. Royal Society A vol. 213, 437456 29. Hosford W.F., 1993. The mechanics of crystals and textured polycrystals. Oxford University Press, New York, 232234 30. House, J.W., 1989. Taylor Impact Testing, Technical Report, AFATLTR 8941, AD A215 018. 31. Hsiung L., 2010. On the mechanism of anomalous slip in bcc metals . Mater Sci Eng A528, 329 – 337 32. Hsiung, L., Lassila, D., 2002. Initial dislocation structure and dynamic dislocation multiplication in Mo single crystals. CMES, vol. 3, no.2, 185191 33. Hughes, T., 1984, Numerical implementation of constitutive models: rate independent deviatoric plasticity. in: S. Nemat Nasser et al. (Eds.), Theoretical Foundations for LargeScale Computations for Nonlinear Material Behavior Martinus Nijhoff, Dordrecht, 1984, 29– 57. 34. Johnson, G., Cook, W., 1983. A c onstitutive model and data for metals subjected to large strains, high strain rates, and high temperatures 146

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35. Kahloun C., Badji R., Bacroix B., Bouabdallah M., 2010. Contribution to crystallographic slip assessment by means of topographic measurements achieved with atomic force microscopy . Materials Characterization, 61, 9, 835 844 36. Kahloun C., Le L., Franciosi P., Chavane M H., Ait E., Monnet G., 2012. Topological analysis of (110) slip in alphairon crystal from in situ tests in Atomic Force Microscope. Acta Materialia, under review 37. Kim J., Greer J., 2009. Tensile and compressive behavior of gold and molybdenum single crystals at the nanoscale . Acta Materialia, 57, 52455253 38. Kim J., Jang D., Greer J., 2010. Tensile and compressive behavior of tungsten, mol ybdenum, tantalum, and niobium at the nanoscale . Acta Materialia, 58, 23552363 39. Kim J., Jang D., Greer J., 2012. Crystallographic orientation and size dependence of tension – compression asymmetry in molybdenum nanopillars . Int. J. Plasticity 28, 46 – 52 40. Ko cks U.F., 2000. Anisotropy and Symmetry, in: Kocks, U.F., Tome, C.N., Wenk, H.R., Texture and Anisotropy. Cambridge University Press, Cambridge UK, pp 30 41. Kolsky H., 1949. An investigation of the mechanical properties of materials at very high rates of loading . Proc. Phys. Soc. 62B, 676 700 42. Lebensohn R., Tome C., 1993. A self consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals. Application to zirconium alloys . Acta metal. Mater, 41, 26112624 43. L ee D., Backofen W., 1966. Yielding and Plastic Deformation in Textured Sheet of Titanium and Its Alloys. Transactions of the Metallurgical Society AIME, 236, 16961703 44. Liu B., Raabe D., Eisenlohr P., Roters F., Arsenlis A., Hommes G., 2011. Dislocation in teractions and low angle grain boundary strengthening . Acta Materialia, 59, 7125– 7134 147

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45. Lu G., Wang B., Zhang T. 2001. Taylor impact test for ductile porous materials part I: theory . International Journal of Impact Engineering, 25, 981991 46. Luft A., Ritsche l C., 1982. Deformation and Slip Behaviour of ColdWorked Molybdenum Single Crystals at Elevated Temperature I. The Effect of Crystals Orientation. Physica Status Solidi (a) 72, 225237 47. Maddin R., Chen N., 1954. Plasticity of Molybdenum Single Crystals at High Temperatures . Transactions AIME, Journal of metals, 280282 48. Madec R., Kubin L., 2008. Secondorder junctions and strain hardening in bcc and fcc crystals. Scripta Materialia 58, 767– 770 49. Malvern, L., 1969. Introduction to the Mechanics of a Continuo us Medium. PrenticeHall, Inc. Upper Saddle River, NJ 50. Mathewson C. , Phillips A., 1928. Twinning in beryllium, magnesium, zinc and cadmium . Transactions of the American institute of mining and metallurgical engineers, 78, 445 452 51. Maudlin, P.J., Bingert, J .F., House, J.W., Chen, S.R., 1999a. On the modelling of the Taylor cylinder impact test for orthotropic textured materials: experiments and simulations. Int. J. Plast. 15, 139166. 52. Maudlin, P.J., Gray, T.G., Cady, C.M., Kaschner, C.G., 1999b. High rate m aterial modelling and validation using the Taylor cylinder impact test. Philos. Trans. R. Soc. 357, 17071729 53. Mishra A., Kad B., Gregori F., Meyers M., 2007. Microstructural evolution in copper subjected to severe plastic deformation: Experiments and analysis . Acta Materialia, 55 , 1, 13 28 54. Nemat Nasser S., Guo W., Liu M., 1999. Experimentally based micromechanical modeling of dynamic response of molybdenum . Scripta Materialia, Vol. 40, No. 7, 859– 872 55. Nakada Y., Key A., 1966. Latent hardening in iron single crystals . Acta Metallurgica, 14 , 8 , 961 972 148

PAGE 149

56. Oertel C.G., Huensche I., Skrotzki W., Knabl W., Lorich A., Resch J., 2007, Plastic anisotropy of straight and cross rolled molybdenum sheets , Mater Sci Eng A483 484, 79 83 57. Park Y.B., Lee D.N., Gottstein G., 1998, The evolution of recrystallization textures in body centered cubic metals . Acta Metallurgica, 46, 33713379 58. Randle V., Engler O., 2000, Evaluation and Representation of Macrotexture Data, in: Randle V., Engler O., Introduction to Texture Analysis: Macrotexture, Microtexture, and Orientation Mapping, CRC Press, Boca Raton, pp 118 59. ReedHill R., Robertson W., 1957. Additional modes of deformation twinning in magnesium . Acta Metallurgica, 5, 717 727 60. Rollett A.D., Wright S.I., 2000. Typical Textures i n Metals, in: Kocks, U.F., Tome C.N., Wenk H.R., Texture and Anisotropy. Cambridge University Press, Cambridge UK, pp 196 61. Sargent G., Shaw B., 1966. Stress relaxation and the ductilebrittle transition temperature of molybdenum , Acta Metallurgica, Vol. 14 , Is. 8, 909 912 62. Seeger A., Hollang L., 2000. The flow stress asymmetry of ultrapure molybdenum single crystals . Materials Transactions, JIM, 41, 141151 63. Shields J., Lipetzky P., Mueller A., 1999. Fracture Toughness of 6.4mm arc cast molybdenum and moly bdenum TZM plate at room temperature and 300C . DE AC11 98PN38206 64. Sigle W., 1999. High resolution electron microscopy and molecular dynamics study of the (a/2)[111] screw dislocation in molybdenum . Philosophical Magazine A, Vol. 79., No. 5. 10091020 65. Sha w B., 1967. Twinning and the low temperature mechanical properties of polycrystalline niobium and molybdenum . Journal of the Less Common Metals, Vol. 13, Is. 3, 294 306 66. Simo, J. C., Hughes, T. J. R., 1998, Computational inelasticity. Springer Verlag New Y ork 149

PAGE 150

67. Sutton M., Orteu J., Schrier H., Image Correlation for Shape, Motion, and Deformation Measurements: Basic Concepts, Theory, and Applications, Springer 2009, 120137 68. Takeuchi T., 1969. Temperature Dependence of Work Hardening Rate in Iron Single Crystals . Journal of the Physical Society of Japan, 2 6 , 354 362 69. Takeuchi T., 1969. Orientation Dependence of Work Hardening in Iron Single Crystals . Japanese Journal of Applied Physics, 8 , 320328 70. Takeuchi T., Mano J., 1972. Latent hardening in iron single crystals with (110) extension axis . Acta Metallurgica, 20, 6, 809819 71. Taylor G., Elam C., 1926. The distorsion of iron crystals . Proceedings of the Royal Society London, A 112 ( 761), 337361 72. Taylor G.I., 1948, The use of flat ended projectiles for determining yield stress, Part I: theoretical considerations . Proc R. Soc. A, 194, 289 299 73. Thornley J., Wronski A., 1969. The relation between the ductilebrittle transition temperature and grain size in polycrystalline molybden um , Scripta Metallurgica, Vol. 3, Is. 12, 935938 74. Ungar T., Castelnau O., Ribarik G., Drakopoulos M., Bchade J., Chauveau T., Snigirev A., Schroer C., Bacroix B., 2007. Grain to grain slip activity in plastically deformed Zr determined by X ray micro diffraction line profile analysi s. Acta Materialia, 55 , 3 , 11171127 75. Van Houtte P., Li S., Engler O ., 2004. Taylor type homogenization methods for texture and anisotropy . Continuum Scale of Simulation of Engineering Materials. Wiley VCH. 466 469 76. Ventelon L., Willaime F., Leyronnas P., 2009. Atomistic simulation of single kinks of screw dislocations in alphaFe . Journal of Nuclear Materials, 386 – 388, 26– 29 77. Vitek V., Mrovec M., Grger R., Bassani J., Racherla V., Yin L., 2004. Effects of nonglide stresses on the plastic flow of single and polycrystals of molybdenum . Mater Sci Eng A387389, 138142 150

PAGE 151

78. Walde T., 2008. Plastic anisotropy of thin molybdenum sheets . Int. J. Refract Met H, 26, 396 403 79. Wang G., Strachan A., Cagin T., Goddard III W., 2004. Calculating the Peierls energy and Peierls stress from atomistic simulations of screw dislocation dynamics: a pplication to bcc tantalum . Model Simul Mater SC, 12. S371 – S38 80. Wang B., Zhang J., Lu G., 2003. Taylor impact test for ductile porous materials part I: experiments . International Journal of Impact Engineering, 28, 499511 81. Woodward C., Rao S., 2002. Flexib le ab initio boundary conditions: simulating isolated dislocations in bcc Mo and Ta. Physical Review Letters, Vol. 88, No. 21, 216402:14 82. Wongwiwat K., Murr L., 1978. Effect of shock pressure, pulse duration, and grain size on shock deformation twinning i n molybdenum . Mater Sci Eng, Vol. 35, Is. 2, 273285 83. Wronski A., Chilton A., Capron E., 1969. The ductilebrittle transition in polycrystalline molybdenum , Acta Metallurgica, Vol. 17, Is. 6, 751755 84. Yang L., Sderlind P., Moriarty J., 2001. Accurate atom istic simulation (a/2)<111> screw dislocations and other defects in bcc tantalum . Philosophical Magazine A, Vol. 81, No. 5, 13551385 85. Yoo M., Hiraoka Y., Choi J., 1995. Deformation characteristics at 300k of singlecrystalline molybdenum sheet prepared by secondary recrystallization. Scripta Metallurgica et Materialla, Vol. 33, No. 9, 14611467 86. Zaderii S., Kotenko S., Marinchenko A., Polishchuk E., Yushchenko K., 2005 Anisotropy of mechanical properties in deformed molybdenum alloy sheets . Strength of Mat erials, Vol. 37, No. 6, 566572 87. Zerilli, F.J., Armstrong, R.W., 1988. Dislocation mechanics based constitutive relations for material dynamics modelling: slip and deformation twinning in iron. In: Shock Waves in Condensed Matter. In: Smidth, S.C., Holmes, N.C. (Eds.), 1987. Elsevier, Amsterdam 151

PAGE 152

88. Zerilli, F.J., Armstrong, R.W., 1990. Description of tantalum deformation behavior by dislocation mechanics based constitutive relations. J. Appl. Phys. 68, 15801591. 152

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BIOGRAPHICAL SKETCH Ger emy received his Bachelor of Science degree in mechanical engineering in 2000 and Master of Science degree in mechanical engineering in 2002 from the University of Alabama at Birmingham , and received a Doctor of Philosophy degree in mechanical engineering from the University of Florida in the spring of 2015. The knowledge and skills acquired through pursuit of these degrees has allowed him to engage in stimulating work as an employee of the Air Force Research Laboratory. The remainder of his time is spent t ravelling with his wife and daughter , and reading monster stories to his young daughter. 153