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Stress-Gradient Failure Theory for Textile Structural Composites

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STRESS GRADIENT FAILURE THEORY FOR TEXTILE STRUCTURAL COMPOSITES By RYAN KARKKAINEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Ryan Karkkainen

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This document is dedicated to everyone cu rious enough to ask the questions, persistent enough to find the answers, and honest enough to respect the truth always.

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iv ACKNOWLEDGMENTS I offer my sincere thanks to my co mmittee and colleagues for their kind and invaluable assistance. Further thanks are due to the Army Research Office and Army Research Laboratory for their gracious funding of this project, and for their feedback and assistance.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES...........................................................................................................ix ABSTRACT....................................................................................................................... xi CHAPTER 1 INTRODUCTION........................................................................................................1 2 APPROACH...............................................................................................................10 Finite Element Micromechanical Method..................................................................10 Direct Micromechanics Method for Failure Analysis................................................19 Phenomenological Failure Criteria.............................................................................21 3 STIFFNESS AND STRENGTH DETERMINATION..............................................25 Stiffness Properties.....................................................................................................25 Strength Properties......................................................................................................28 Microstress Field Contour Plots.................................................................................34 Inspection of Failure Envelopes in Additional Stress Spaces....................................35 Design of Experimental Verification..........................................................................38 Chapter Summary.......................................................................................................40 4 PREDICTION OF FAILURE ENVELOPES.............................................................41 A Parametric Approach to Predicting Fa ilure Envelopes for a Given Stress Space..41 Development of a Quadratic Failure Criterion to Predict Failure for General Loading..................................................................................................................46 Optimization of Failure Coefficients..........................................................................53 Chapter Summary.......................................................................................................57 5 MULTI-LAYER ANALYSIS....................................................................................58 Stiffness Prediction of Multi-Layer Textile Composites............................................59 Strength Prediction of Multi-Layer Textile Composites............................................61

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vi Direct FEM Simulation of Multi-Laye r Textile Composites Using the DMM for Failure Analysis (DDMM).....................................................................62 Adaptation of the Single Layer DMM Results to Predict Strength for MultiLayer Textile Composites (ADMM)............................................................63 Implementation of the Quadratic Failure Theory to Predict Strength for Multi-Layer Textile Composites (QFT).......................................................65 Comparison of the Results of Mult i-Layer Failure Analysis Methods...............66 Practical Examples to Illustrate Stre ngth Prediction of a Two-Layer Textile Composite Plate...............................................................................................70 Chapter Summary.......................................................................................................78 6 CONCLUSIONS AND FUTURE WORK.................................................................80 APPENDIX: PERIODIC BOUNDARY CONDITIONS..................................................86 LIST OF REFERENCES...................................................................................................87 BIOGRAPHICAL SKETCH.............................................................................................91

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vii LIST OF TABLES Table page 2-1 RVE Dimensions......................................................................................................12 2-2 Periodic Displacement Boundary Conditions..........................................................17 3-1 Fiber Tow and Matrix Material Properties...............................................................27 3-2 Stiffness Properties for Plain Weave Textile Plate..................................................27 3-3 Fiber Tow and Matrix Failure Strength Properties (MPa).......................................28 4-1 Strength Values for Independent Load Conditions..................................................48 4-2 Normalized Failure Coefficients Cij for Quadratic Failure Equation. (Coefficient Cmn is in mth Row and nth Column)...........................................................................49 4-3 Normalized Failure Coefficients Di for Quadratic Failure Equation.......................49 4-4 Comparison of Quadratic Failure Equati on Predictions with DMM Results. Test Cases Include 4 Populated Load Terms Fi...............................................................51 4-5 Comparison of Quadratic Failure Equati on Predictions with DMM Results. Test Cases Include 5 Populated Load Terms Fi...............................................................52 4-6 Comparison of Quadratic Failure Equati on Predictions with DMM Results. Test Cases Include 6 (Fully Populated) Load Terms Fi...................................................53 4-7 Optimized Failure Coefficients Cij for Quadratic Failure Equation. (Coefficient Cmn is in mth Row and nth Column)...........................................................................54 4-8 Optimized Failure Coefficients Di for Quadratic Failure Equation.........................55 4-9 Data to Indicate Results of Coefficient Optimization..............................................55 5-1 Summary of the Various Methods Empl oyed in Multi-Layer Strength Analysis....61 5-2 Example Load Cases to Determine the Accuracy of Multi-Layer Analysis Methods. Accuracy is Indicated by a Ratio as Compared to DDMM Results........69

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viii 5-3 Further Example Load Cases (Including Moment Resultants) to Determine the Accuracy of Multi-Layer Analysis Met hods. Accuracy is Indicated by a Ratio as Compared to DDMM Results..............................................................................70 5-4 Geometry of the Simply Supported Textile Plate under Uniform Pressure.............71 5-5 Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 1 Geometry of Table 5-14 as Predicted from Va rious Multi-Layer Analysis Methods...............73 5-6 Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 2 Geometry of Table 5-14 as Predicted from Va rious Multi-Layer Analysis Methods...............73 5-7 Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 2 Geometry of Table 5-14 as Predicted from Va rious Multi-Layer Analysis Methods...............74 5-8 Case 1 and 2 Factor of Safety Across the Plate as Determined via the Conventional Approach............................................................................................75 5-9 Case 1 and 2 Factor of Safety Acro ss the Plate as Determined via the QFT...........76 5-10 Case 3 Factor of Safety Across the Plate as Determined via the Conventional Approach..................................................................................................................76 5-11 Case 3 Factor of Safety Across the Plate as Determined via the QFT.....................76 5-12 Geometry of the Textile Pressure Vessel.................................................................78 5-13 Maximum Pressure for the Textile Pr essure Vessel of Figure 5-3 with the Geometry of Table 5-12 as Predicted from Various Multi-Layer Analysis Methods....................................................................................................................78

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ix LIST OF FIGURES Figure page 1-1 A Schematic Illustration of Severa l Common Weave Patterns Employed with Textile Composites. A Box Indicati ng the Unit Cell Borders the Smallest Repeatable Geometry Element for Each Pattern........................................................3 1-2 A Schematic Illustration of Severa l Common Braid Patter ns Employed with Textile Composites. A Box Indicati ng the Unit Cell Borders the Smallest Repeatable Geometry Element for Each Pattern........................................................3 1-3 A Schematic Illustration of Seve ral Common 3D Weave Patterns Employed with Textile Composites.............................................................................................4 2-1 Example Load Cases to Illustrate the Importance of Including Moment Terms in the Analysis of a Textile RVE..................................................................................12 2-2 RVE Geometry of a Plain Weave Textile Composite..............................................13 2-3 Flowchart for Failure Analysis Us ing the Direct Micromechanics Method............20 3-1 Comparison of DMM Failure Envelopes with Common Failure Theories..............29 3-2 Effect of Bending Moment on the Failure Envelope...............................................31 3-3 Effect of Micro-Level Tow Failure Theory on the DMM Failure Envelope...........32 3-4 Effect of Changing th e Definition of Micro-Leve l Failure on the Failure Envelope...................................................................................................................33 3-5 Stress Contours for Plain-Weave Fi ber Tows in Uniaxial Extension......................34 3-6 DMM Failure Envelopes for Biaxial Loading with Multiple Constant Moment Resultants. For Illustration, an Ellipse is Fit to Each Data Set Using a LeastSquares Method........................................................................................................36 3-7 DMM Failure Envelopes for Biaxial Bending with Constant Applied Twisting Moment....................................................................................................................37 3-8 DMM Failure Envelopes with Force and Moment Resultants for Constant Applied Shear. .........................................................................................................37

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x 3-9 Schematic of the Specimen fo r Off-Axis Uniaxial Testing.....................................39 3-10 Critical Force Resultant under Uniaxial Loading as Calculated via Various Failure Theories........................................................................................................39 4-1 Effect of Applied Moment Resultant on the Major Axis of Elliptical Failure Envelopes.................................................................................................................42 4-2 Effect of Applied Moment Resultant on the Minor Axis of Elliptical Failure Envelopes.................................................................................................................43 4-3 Effect of Applied Moment Result ant on the Center Point Coordinates ( uo vo) of Elliptical Failure Envelopes.....................................................................................43 4-4 Failure Envelopes Predicted with the Parametric Approach as Compared to DMM Results (Applied Moment Resu ltant of 0.65 Critical Moment)....................45 4-5 Failure Envelopes Predicted with the Parametric Approach as Compared to DMM Results (Applied Moment Resu ltant of 0.9 Critical Moment)......................46 4-6 DMM Failure Envelopes for Biaxial Bending with Constant Applied Twisting Moment as Compared to the Quadra tic Failure Theory Predictions........................49 4-7 DMM Failure Envelopes with Force and Moment Resultants for Constant Applied Shear as Compared to the Qu adratic Failure Theory Predictions..............50 5-1 Schematic Illustration of the Single-La yer Strain and Curvature Stress Fields (As Found via the DMM) That Must Be S uperposed in Calculation of the Total Stresses Resulting from Multi-Layer Bending.........................................................64 5-2 Schematic of the Simply Supported Textile Plate under Uniform Pressure............71 5-3 Schematic of the Dual-Layer Textile Pressure Vessel.............................................77

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xi Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRESS GRADIENT FAILURE THEORY FOR TEXTILE STRUCTURAL COMPOSITES By Ryan Karkkainen May 2006 Chair: Bhavani Sankar Major Department: Mechanic al and Aerospace Engineering Micromechanical methods for stiffness and strength prediction are presented, the results of which have led to an effective failure theory for prediction of strength. Methods to account for analysis of multi-laye r textile composites are also developed. This allows simulation of a single representa tive volume element (RVE) to be applicable to a layup of an arbitrary number of layers eliminating the need for further material characterization. Thus a practical tool for failure analysis and design of a plain weave textile composite has been developed. Thes e methods are then readily adaptable to any textile microarchitecture of interest. A micromechanical analysis of the RVE of a plain-weave textile composite has been performed using the finite element metho d. Stress gradient effects are investigated, and it is assumed that the stress state is not uniform across the RVE. This is unlike most models, which start with the premise that an RVE is subjected to a uniform stress or strain. For textile geometries, non-uniform stress considerations are important, as the

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xii size of a textile RVE will typica lly be several orders of magni tude larger than that of a unidirectional RVE. The stress state is de fined in terms of the well-known laminate theory force and moment resultants [ N ] and [ M ]. Structural stiffness coefficients analogous to the [ A ], [ B ], [ D ] matrices are defined, and thes e are computed directly using the Direct Micromechanics Method (DMM), ra ther than making estimations based upon homogenized properties. Based upon these results, a robust 27-term quadratic failure criterion has been developed to predict failure under general lo ading conditions. For multi-layer analysis, the methods are adapted via three techniques: direct simulation of a multi-layer composite, an adjustment of the data output from single-layer FEM simulation, and an adjustment of the quadratic failure theory (without the requirement of determining a new set of failure coefficients). The adjusted single-layer data analysis and the adjusted quadratic failure criterion show 5.2% and 5.5% error over a variety of test cases. The entire body of work is then applied to several practical ex amples of strength prediction to illustrate their implementation. In many cases, comparisons to conventional methods show marked improvements.

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1 CHAPTER 1 INTRODUCTION Though composites in general are not a new material, c ontinuing advancements in constituent materials, manufacturing techni ques, and microstructures require that considerable amounts of research be devoted to the study of composite mechanics. The associated knowledge base is much sma ller when compared to more conventional materials, such as metals or ceramics. Composites have yet to be absorbed into widespread use across multiple industries, limiting their economy and familiarity. Much remains to be developed in the way of desi gn methodologies and effective employment in optimized structural applications. By far the two most commonly employed and studied composite materials are randomly oriented chopped-fiber composites, as well as laminated polymer composites with embedded unidirectional carbon, glass, or kevlar fibe rs. Such composites have received a great amount of treatme nt in the literature, and much exists in the way of stress analysis techniques, and effective prediction of stiffness, fatigue life, strength, and other such mechanical analyses. Textile composites are a subgroup of compos ite materials that are formed by the weaving or braiding of bundles of fibers (called tows or yarns), which are resinimpregnated and cured into a finished co mponent. Though heavily adapted, in some sense they draw upon traditional textile weavi ng processes and fabrication machines such as mandrels and looms, akin to those of th e textile clothing industries. Much of the terminology of structural textile composites draws upon this classical sense as well.

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2 The value of textile composites stems from many advantages, such as speed and ease of manufacture of even complex com ponents, consequent economy compared to other composite materials, and out-of-plane rein forcement that is not seen in traditional laminated composites. Further, textile com posites do not lose the classically valued advantage that composite materials possess over their metal or traditio nal counterparts, in that textile composites have an inherent capacity for the materi al itself to be adapted to the mechanical needs of the design. This is to say that the strength and stiffness of the material can be oriented in needed direct ions, and no material we ight is wasted in providing reinforcement in unnecessary dire ctions. For a conventional laminated composite, this is accomplished by oriented stacking of layers of unidirectional resinimpregnated fibers, such that fibers are aligne d with any preferred lo ading axes. A textile composite may also be so adapted by severa l methods, such as unbalanced weaves. The woven fiber tows in a preferred direction may be larger (containing more constituent fibers per tow) than in other directions. Also, an extremely diverse set of woven or braided patterns may be employed, from a si mple 2D plain weave to an eight-harness satin weave or a 3D or thogonal weave pattern, an y of which may exhibit a useful bias in orientation of material properties. Figure 1-1 through Figure 1-3 [1] illustrate some of the more common of these patterns. The economy of textile composites arises ma inly from the fact that manufacturing processes can be highly automated and rapi dly accomplished on loom and mandrel type machinery. This can lead to easier and quick er manufacture of a finished product, though curing times may still repres ent a weak link in the poten tial speed of manufacture.

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3 Figure 1-1. A Schematic Illustration of Se veral Common Weave Patterns Employed with Textile Composites. A Box Indicati ng the Unit Cell Borders the Smallest Repeatable Geometry Element for Each Pattern. Figure 1-2. A Schematic Illustration of Se veral Common Braid Patt erns Employed with Textile Composites. A Box Indicati ng the Unit Cell Borders the Smallest Repeatable Geometry Element for Each Pattern.

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4 Figure 1-3. A Schematic Illustration of Several Common 3D Weave Patterns Employed with Textile Composites. The out-of-plane reinforcement provided by textile composites comes by virtue of the fact that the constituent weave patterns l ead inherently to undul ation and interlacing of the woven fiber tows, which become oriented out of plane. A 2D weave will typically exhibit tow undulation varying from 0 to 15 degr ees out of plane. A 3D weave such as an orthogonal interlock may actually have fiber tows directly aligned in the out of plane direction. Naturally, there are also several tradeo ffs and disadvantages associated with laminated and textile composites. The most significant added disadvantage that textile composites possess may be the increased comple xity of mechanical analysis. This is directly owing to the undulati on inherent to the weave or br aid patterns, the complexity of microstructure, as well as the multiscale nature of the textile microstructure. The fiber tows woven together and embedded in a bulk pol ymer matrix are the most representative

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5 microstructure, but the tows themselves have an inherent microstructure of aligned fibers and intra-tow polymer matrix. Given this increased complexity of analys is, there are several outstanding issues with regards to textile composites. One of the most important issues, and the issue which is to be addressed in the cu rrent body of research, is a robust model for prediction of strength. Current failure theories are generally de veloped for unidirectional composites and do not capture the unique characteristics of textile composites. Though these theories may to some extent be applied in an adapted form to the analysis of textile composites, many inherent simplifying assumptions will no longer apply, and in general such techniques will not be suitable to the increased complexity intrinsic to textile geometry. Even at the micro scale, textile com posites maintain a relatively complicated microstructure. Even under simple loading c onditions, a textile micros tress state will be shown to be quite complex, and elastic cons tants are non-uniform due to the waviness of a woven fiber tow. Laminate analysis property homogenization, and other common approaches will no longer apply. Thus, current designs of textile structures will not be optimized for maximum damage re sistance and light-weight. Conventional micromechanical models for te xtile composites assume that the state of stress is uniform over a distance comparable to the dimensions of the representative volume element (RVE). However, due to comp lexity of the weave ge ometry, the size of the RVE in textile composites can be large comp ared to structural dimensions. In such cases, severe non-uniformities in the stress st ate will exist, and conventional models may fail. Such stress gradients also exist when th e load is applied over a very small region, as

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6 in static contact or foreign object impact loading, and when there are stress concentration effects such as open holes in a structure. Although micromechanical models have been successfully employed in predicting thermo-elastic constants of fiber-reinforced composite materials, their use for strength prediction in multiaxial loading conditions is not practical, as computational analysis must be performed in each loading case. Thus phenomenological failure criteria are still the predominant choice for design in indus try. There are three major types of engineering failure criteria for unidirectional composite materials: maximum stress criterion, maximum strain criterion, and quadrat ic interaction criteri on, such as the TsaiHill and Tsai-Wu failure theories [2]. Most of the micromechanical modeling work done thus far has focused on predicting thermo-mechanical properties [3-6]. To facilitate the use of textile composites in lightweight structures, it is required to have a lucid understanding of failure mechanisms, and design engineers must ha ve an accurate and practical model for prediction of failure stress. Most of the current analyti cal and numerical methodologies developed to characterize textile composite s [7-14] assume that the textile is a homogeneous material at th e macroscopic scale. Finite element analysis of initial failure of a plain weave composite [10] has shown that failure due to inter-tow normal stresses are the predominant mode of failure, and there is generally little or no damage volume of the bulk matrix between tows. This work is extended to a thorough investigation of progressive failure an alysis under axial extension using several different property knoc kdown schemes. This has shown stiffness losses on the order of 40% after initial failu re. More recently this work has been

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7 extended to include the capability for mo re detailed stress fields in the RVE under investigation [15], and techniques have been developed to minimize required computation times by employing boundary cond itions based on thorough exploitation of symmetry and periodicity of RVE geometry [16]. The Binary Model [17, 18] allows for quick and efficient analysis of any textile weave of interest. It has been shown to provide for accurate prediction of stiffness properties. Further, it is r obust and readily adaptable to pr ovide insight into effects of alteration of parameters such as tow wa viness, tow misalignment, varying weave architectures, etc. This technique does not yield a detailed map of RVE stress fields or allow for cross-sectional variation of tow geom etry, as the fiber tow is simulated as an embedded 1-D line element with representative material prope rties. Thus some microlevel detail is lost to provide for computati onal efficiency and macro-level representation. The Mosaic Model and its adaptations [ 7, 19, 20] represent a textile composite RVE as a collection of homogenized blocks each with unidirect ional composite or matrix properties. These blocks are then assembled to represent the weave geometry under consideration. In this way, classical lami nate plate theory can be used to determine the global stiffness matrix of the RVE. For macroscopically homogeneous load cases, good agreement has been shown with experi mental data, including three-dimensional weave geometries. Effective prediction of compressive streng th of braided textile composites using a detailed FEM micromechanical model ha s been performed [21], which shows good comparison to experimental results in a para llel study [22]. A deta iled 3D solid model was formed to exactly model a 2D triaxially braided composite RVE. Biaxial loading is

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8 considered in both the experimental and com putational analyses. Buckling analysis has been performed, and the effects of tow waviness and microarchitecture on the compressive strength are shown. For uniaxially loaded textile composites, consistent but optimistic strength estimates have been made by comparing the strength of fiber tows with the predicted stresses in the fiber yarns th at are aligned with the loadi ng axis [23-25]. The off-axis tows are given little consideration, without much effect on the outcome, as they play little part in such uniaxial loading cases. Multi-axi al loading presents an obvious escalation in modeling complexity. The failure envelope fo r combined transverse tension and in-plane shear has been presented as an ellipse [26], according to quadratic strength rules developed for unidirectional composites. Further, proposals for multiaxial loadings submit that axial strain in the textile geomet ry should be compared to a critical value of tow strain, analogous to a first-ply failure cr iterion for unidirectiona l composites [27]. A previous study [28] extended a method, known as the Direct Micromechanics Method [29] (DMM), to develop failure enve lopes for a plain-weave textile composite under plane stress in terms of applied macros copic stresses. In this study, it was assumed that the state of stress is uniform across the RVE. The micro scale stresses within the RVE were computed using finite element me thods. The relation between the average macrostress and macrostrains provides the constitutive relations for the idealized homogeneous material. The microstresses are us ed to predict the failure of the yarn or matrix, which in turn translates to failure of the textile composite. In the current research, micromechanical fi nite element analysis is performed to determine the constitutive relations and failu re envelope for a plai n-weave graphite/epoxy

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9 textile composite. The model is based upon the an alysis of an RVE, which is subjected to force and moment resultants of classical laminate theory. Thus there is no assumption about the uniformity of an a pplied load or strain, as any load can be represented by a combination of force and moment resultants. The micro-scale stresses within the RVE are computed using the finite element method. The relation between the average macrostress and macrostrains provides the cons titutive relations for the material. Thus constitutive characterization matrices [ A ] [ B ] [ D ] are found directly from micromechanics. The microstresses are also used to predict the failure of the yarn or matrix, which in turn translates to failure of the textile composite. Using the DMM, the failure envelope is developed for in-plane force resultants, with and without applied moment resultants. No currently accepted failure criteria exist that may be used explicitly for the analysis of textile composites. Thus the methods and results employed herein are used to develop phenomenological failure criteria for textile composites. The results are compared to conventional methods that are not specifically developed for the analysis of textile composites, as a basis for evaluation.

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10 CHAPTER 2 APPROACH Micromechanical finite element analysis is performed to determine the constitutive relations and failure envelope for a plainweave graphite/epoxy te xtile composite. The model is based upon the analysis of an RVE, which is subjected to force and moment resultants of classical laminate theory. T hus there is no assumpti on about the uniformity of an applied load or strain, as any load can be represented by a combination of force and moment resultants. The micr o-scale stresses within the RVE are computed using the finite element method. The relation between the average macrostress and macrostrains provides the constitutive relations for the ma terial. Thus constitutive characterization matrices [ A ] [ B ] [ D ] are found directly from micromechanics. The microstresses are also used to predict the failure of the yarn or matrix, which in turn translates to failure of the textile composite. Finite Element Micromechanical Method In the current study, stress gradient effect s are investigated, and it is assumed that the stress state is not uniform across the RVE. This represents an extension of the micromechanical models used to predict the strength of textile co mposites [28-32]. The stress state is defined in terms of the well-known laminate theory load matrices [ N ] and [ M ] which describe force and moment resultant s. Furthermore, structural stiffness coefficients analogous to the [ A ] [ B ] [ D ] matrices are defined. In this approach, these structural stiffness coefficien ts are computed directly from the micromechanical models, rather than making estimations based upon the homogeneous Youngs modulus and plate

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11 thickness. Accordingly, indivi dual unit strains and unit curvatures can be applied to the micromechanical finite element model, and the resulting deformations are used to define the stiffness coefficient matrices. Conventi onal models essentially neglect the presence of [ M ] terms that result from nonuniformity or gradients in a pplied force resultants, thus assuming a uniform stress state for which only the [ N ] matrix is populated. The additional analysis of the [ M ] term includes information about the distribution, or gradient, of a non-uniform load. This can greatly increase the ability of a failure model to accurately predict failure for load cases in which such effects may well be predominant, such as in thin plates, concentr ated loading, or impact loading. The significance of including the analysis of moment terms is further illustrated in Figure 2-1. The moment term describes the distribution, not only the magnitude of applied loading. Depending on the stress stat e of an RVE, an analysis incorporating stress gradient effects and in clusion of the consideration of applied moment could be of critical importance. In Figur e 2-1a, the force resultant ( N ) is non-zero, but the uniform loading results in zero moment resultant ( M ). However, in load cases such as Figure 2-1b and 2-1c, the non-uniformity of applied loading leads directly to an appreciable moment term, which must be included in the analysis. In fact, in some cases it is possible that the net force resultant is zero while the effectiv e moment resultant is non-zero, in which case conventional analysis techniques cannot be employed. The present micromechanical analysis of a plain-weave textile composite is performed by analyzing the representative volume element (RVE) using the finite element method. A typical weave-architecture ha s been selected and this RVE is detailed in Table 2-1 and also in Figur e 2-2. This architecture was ch osen from a literature source

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12 Figure 2-1. Example Load Cases to Illustrate the Importance of Including Moment Terms in the Analysis of a Textile RVE. Distributed Loads across the Large Dimensions of a Textile RVE Will Pr oduce Bending Moments That Must Be Incorporated into Stiffness and Strength Prediction. [33] that provided a complete and detaile d description of the needed geometrical parameters, such as those shown in Table 21. Given parameters ar e representative of microarchitectures as experimentally obser ved via SEM or standard microscope. Total fiber volume fraction, given these dimensions, will be 25%, incorporating the fact that the resin-impregnated tow itsel f has a fiber volume frac tion of 65% (this is calculated directly from ABAQUS software, which yields element volumes as outputs, thus the volume of all matrix elements can be compared to the volume of all tow elements). Though this volume fraction may seem low for structural uses, it can be representative of many signifi cant low-load, impact-resistant applications, such as automotive lightweight body panels. Table 2-1: RVE Dimensions Dimension a b c p t w Length (mm) 1.68 0.254 0.84 0.066 0.70

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13 Figure 2-2: RVE Geometry of a Plain Weave Textile Composite It should be noted that a nu mber of parameters are requi red to exactly specify the textile geometry. These specifications will have a significant effect on micromechanical modeling. Consequently, care should be take n when comparing the results of various studies that the textile geometries under comparison are truly equivalent. In order to evaluate the stiffness and stre ngth properties of the textile weave under consideration, the DMM is essentially employe d as an analytical laboratory that quickly and effectively replace s physical testing and experi mental procedures. Though experimental verification always provides a ba seline of veracity to FEM analysis, this procedure effectively overcomes the limitations of physical apparatus. Furthermore, as will be shown later, this allows for a dense population of analysis poi nts; thus a failure envelope may be quickly and fully constructe d with a large number of data points, and there is no need for interpolat ion of limited discrete experimental data points. Also, results achieved from the DMM are completely three-dimensional stress or strain fields. Thus, the results can be visualized thr oughout the thickness of the specimen. This overcomes the limitations inherent to physical application of experimental stress analysis techniques, which are labor-intensive and gene rally limited to surface visualizations.

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14 Specification of relative displacements on opposite faces of the RVE can represent any general macro deformation under investig ation. Displacements are applied using periodic boundary conditions. The periodic displacement boundary conditions isolate the mechanical effects of application of un it strains or curvatures, and ensure the repeatability of deformations. Thus, the RVE is not only rep eatable as a representative geometry, but is also mechanically repeatable in that each RVE has an identical response to strains and curvatures regardless of the location of that RVE in a textile plate or component, for example a x j i x j ix u x u a x xx w x w 2 2 2 2 (2.1) To ensure continuity of microstresses a nd compatibility of displacements across an RVE, periodic traction and displacement boundary conditions must be employed. A macroscopically homogeneous deformation can be represented as j ij M ix H u i,j = 1,2,3 (2.2) e.g. 23231(,,)(0,,)iiiuaxxuxxHa e.g. 13132(,,)(,0,)iiiuxbxuxxHb The derivation of the periodic boundary c ondition for unit curvat ure is presented below, and further examples are presented in the appendix. The periodic displacement boundary condi tion corresponding to unit curvature along the x -axis (x) will be derived. All other curvat ures will be zero. Curvatures are defined as follows 12 2 x wx 02 2 y wy 02 y x wxy (2.3)

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15 The periodic displacement boundary conditi on will be derived from the definition of curvature along the x -axis (x). Integrating once with respect to x yields ) ( y f x x w (2.4) where f ( y ) is an arbitrary function of y Differentiation of this expression with respect to y together with the requirement that xy is set as zero, indicates that 2()0xyw fy xy (2.5) Due to the above expression, f(y) must therefore be a constant, since xy = 0. Furthermore, this constant must be zero due to the specification that slope at the origin is zero, such that 00xw x Next, Equation 2.4 is integrated with respect to x giving rise to the following expression a nd an arbitrary function h ( y ) ) ( 22y h x w (2.6) Now coordinate values for opposite faces of the RVE (see Figure 2-2) may be substituted into Equation 2.6 ) ( ) , 0 ( ) ( ) , (2 2 1y h z y w y h a z y a w (2.7) These are then subtracted from each other, eliminating the unknown h ( y ), and effectively prescribing the relative displacement on opposite faces that can be used to apply unit curvature. 2 2 1) , 0 ( ) , ( a z y w z y a w (2.8)

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16 However, a further boundary condition is required to remove the transverse shear forces that will be present due to the applica tion of this displacement. In this way, the mechanical effect of curvature is isolated. Thus the requirement is that transverse shear strain is zero, which is defined as 0 x w z uzx (2.9) This can be rearranged as below, and the value of slope w x is known from Equation 2.4 above (where the arbitrary function has been shown to be zero). x x w z u (2.10) Similar to the above procedur e, this expression can then be integrated with respect to z, evaluated with coordinate values of oppos ite faces, which are then subtracted from each other to specify the relative displacement that must be proscribed. c zx u (2.11) za z y u z y a u ) , 0 ( ) , ( (2.12) The periodic boundary conditions as show n in Equations 2.8 and 2.12 must be simultaneously applied to isolate the e ffects of an applied unit-curvature. In order to satisfy equilibrium, tracti on boundary conditions are applied to ensure equal and opposite forces on opposite faces of the RVE. The traction boundary conditions for traction forces on th e lateral faces of the RVE are ) 0 , ( ) , ( ) 0 ( ) , ( ) , 0 ( ) , ( y x F c y x F z x F z b x F z y F z y a Fi i i i i i (2.13)

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17 In the Direct Micromechanics Method (DMM), the RVE is subjected to macroscopic force and moment resultants, wh ich are related to macroscopic strain and curvature according to ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ D B B A M N (2.14) Thus the constitutive matrices must be evaluated to determine this correlation. Once this has been determined, a macroscopi c deformation can be applied using an FEM code. In this way, the FEM results for st ress in each element yield the microstresses resulting from an applied force or moment resultant. The RVE is subjected to independent macr oscopic unit deformations in order to evaluate the stiffness matrices of Equation 2. 14. In each of the six cases shown in Table 2-2 below, a single unit strain or unit curvat ure is applied, and all other deformation terms are set to zero, and the appropriate periodic boundary conditions are applied. The four-node linear tetragonal elem ents in the co mmercial ABAQUS (Standard) FEM software package were used to model the yarn and matrix for all cases. An h refinement convergence study was performed in which analysis was performed for a progressively finer mesh of four-node linear te tragonal elements. For several reasons, the final mesh chosen employs 68,730 such elements. Table 2-2: Periodic Displacement Boundary Conditions u ( a,y ) u ( 0,y ) v(a,y)v(0,y) w(a,y)w(0,y) u(x,b)u(x,0) v(x,b)v(x,0) w(x,b)w(x,0) 1 x M = 1 a 0 0 0 0 0 2 y M = 1 0 0 0 0 b 0 3 xy M = 1 0 a/2 0 b/ 2 0 0 4 x M = 1 az 0 -a2/ 2 0 0 0 5 y M = 1 0 0 0 0 bz -b2/ 2 6 xy M = 1 0 az/ 2 -ay/ 2 bz/ 2 0 -bx/ 2

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18 This mesh refinement is significantly beyond the point of numerical convergence for which output element stresses can be assure d to be accurate. This also allows for a mesh that accurately covers the corners of an ellipsoidal tow cross-section without sacrificing element quality in such regions. Furthermore, a refined mesh can capture the intricacies of stress contours a nd stress gradients expected to be seen through the RVE. Note that the shared nodes are employed be tween each tow and its surrounding interstitial matrix. There are no tow-tow shared nodes, thus tows are not bound to each other, but only to the interstitial matrix. As a final note on the character and quality of mesh employed in this analysis, the following quality assurance metrics are indicated: fewer than 0.1% of elements have an interior angl e less than 20 degrees, fewer than 0.3% have an interior angle greater than 120 degrees, and less than 0.2% have an aspect ratio greater than 3 (the average aspect ratio is 1.66). The FEM results for each element yield the microstresses resulting from an applied macro-level strain and curvature. The corresponding macro-leve l force and moment resultants in each case can be computed by av eraging the microstresses over the entire volume of the RVE e e ij ab ijV N1 (2.15) e e ij ab ijV z M1 (2.16) where e denotes summation over all elements in the FE model of the RVE, Ve is the volume of the eth element, and a and b are the dimensions of th e RVE as shown in Figure 2-2. Thus the constitutive matrices of Equa tion 2.14 can be found by independently evaluating the six cases shown in Table 2-2, in tandem with Equations 2.15 and 2.16. By

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19 applying the appropriate displacements acco rding to Table 2-2 which correspond to a given unit strain or curvature case, the stiffn ess coefficients in a column corresponding to the non-zero strain can be eval uated directly from the force and moment resultant values as calculated from the finite element micro stresses via Equations 2.15 and 2.16. Thus the six load cases completely describe the six columns of the [ A ] [ B ] [ D ] matrix. This information having been determined, one is then able to evaluate the microstress field resulting from general loading cases via th e following steps: Step 1) Relate applied force and moment resultants to applied macro strain and curvature via the [ A ] [ B ] [ D ] matrices and Equation 2.14, Step 2) Apply this macro strain and curvature to the RVE using an FEM code, and Step 3) The element stresses from FEM results yield the microstress field in the yarn and matr ix. The present study assumes there are no residual stresses or pre-stresses in the co mposite. The significance of residual stresses would depend on the particular cure cycle em ployed in manufacturing the composite, as well as upon the weave pattern under investigation. For a pl ain-weave textile, it would be expected that residual pre-stresses w ould affect the center point of the failure envelope, given that 1) symmet ry of microarchitecture would lead to a level of symmetry of residual stresses, and 2) pre-stresses shift, rather than shrink, an existing failure envelope as the applied loads can either add to or be offset by the residual stresses. The magnitude of residual stresses could conceivabl y reach on the order of 10% of the failure strength. Direct Micromechanics Method for Failure Analysis The method described above can be used to predict strength by comparing the computed microstresses in each element agai nst failure criteria for the constituent yarn and matrix of the textile composite. Interfac e failure is not considered in the current

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20 study, but will be incorporated into future wo rk. The microstresses in each element can be extrapolated from the preliminary RVE anal ysis (described above) of each of the six linearly independent macrostrain components. The microstress state for a general applied force or moment resultant is obtained by superposing multiples of the results from the unit macrostrain analysis M M e eF ] [ (2.17) Where the 6 matrix [ Fe] contains the microstress in each element resulting from the unit strain and curvature analysis. For example, the microstress y in the RVE for x 0 =0.05 and y= 0.003 m-1 is calculated as y =0.05 F21+0.003 F25. Failure is checked on an element-by-element basis, and the failure criterion of each element can be selected appropr iately based upon whethe r it is a yarn or matrix element. It is assumed that the entire textile composite has failed, even if only one of the yarn or matrix elements has failed. Although this may be considered conservative, it is realistically representative of th e initial failure of the composite. Figure 2-3: Flowchart for Failure Analysis Using the Direct Micromechanics Method

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21 For the isotropic matrix elements, the Maxi mum Principal Stress criterion is used to evaluate element failure. For fiber tow elemen ts, the Tsai-Wu failure criterion is used. This criterion is more suitable to the ort hotropic nature of the fiber tow, which is essentially a unidirectional composite at the mi cro level. Microstresses in the yarn are transformed to local coordinates tangent to the path of the yarn and compared to strength coefficients for a unidirectional compos ite, using the Tsai-Wu criterion. A flow chart that describes the DMM pro cedure is shown in Figure 2-3. Failure envelopes are generated by first selecting a macrostress state to investigate. Then the macrostrains and curvatures resulting from this applied loading are calculated from Equation 2.14. The resulting stress field for the entire RVE is then calculated by Equation 2.17, based on the scaled superpositi on of the results from FEM analysis of the unit load cases shown in Table 2-2. Failure is then checked in each element against a given failure criterion. This cycle is th en repeated while progressively increasing a selected force or moment resultant and holdi ng all others constant until an element level failure criterion is exceeded. If a particular failure criterion is exceeded, the element and the RVE are considered to have failed, which then defines the threshold of the failure envelope at a given point. Thus failure envelopes for the textile composite can be generated in various force and moment resulta nt spaces. The scope of the current study considers analysis of in-plane force and moment resultants [ N ] and [ M ], though the methods are applicable to any general loading conditions. Phenomenological Failure Criteria In addition to being used to determine failure of the fiber tow or the matrix phase at the element or micro level, the Tsai-Wu phe nomenological failure cr iterion is used as a basis for comparison to the DMM for the macr oscopic failure of th e textile composite.

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22 Since the Tsai-Wu criterion was formulated in te rms of stresses, an adapted form of this criterion in terms of applied force resultants is used. 1 22 1 12 2 66 2 22 2 11 y x y x xy y xN F N F N N F N F N F N F (2.18) Employment of this criterion essentially represents fitting an ellipse of Tsai-Wu form to the DMM failure data. The failure coefficients ( Fij) that appear in Equation 2.18 are based on failure data from the DMM. These parameters are based on the failure strength of the material under various loadi ng conditions, and are typically determined by conducting physical tests on the specimen. For example, in order to obtain F11 and F1, a load of Nx is applied, and all other stresses are se t to zero. Then Equation 2.18 reduces to 11 2 11 x xN F N F (2.19) Since the DMM has been used to determine the maximum uniaxial tensile and compressive force resultant, each of which satisfy Equation 2.19, these two independent equations can be solved for F11 and F1 ) ( ) (1 1 1C TX XF (2.20) ) ( 1 11C TX XF (2.21) where XT and XC are the failure values of Nx for tension and compression, respectively. Using similar procedures, strength coefficients F22, F2, and F66 are evaluated. The resulting values are as follows ) ( ) (1 1 2C TY YF (2.22) ) ( 1 22C TY YF (2.23) 21 66 SF (2.24)

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23 In the above equations Y and S are the strengths in terms of the force resultants Ny and Nxy, and C and T denote compression and tension, resp ectively. In the literature, there exist many proposed methods for determining an appropriate F12. In this study, the coefficient F12 is determined by subjecting the unit-cel l to a state of biaxial stress such that Nx = Ny while Nxy=0, and then determining the maximum value of applied Nx = Ny = Nmax. The resulting coefficient takes the following form max 2 1 2 max 22 11 2 max 121 2 1 N F F N F F N F (2.25) Please note that for Equations 2.18 through 2.25, strength values must be in terms of force resultants ( N ), as noted in the nomenclature. Th is is different from the textbook definition of the Tsai-Wu failure theory, which is in terms of stresses. Also note that, as will be discussed later, the Tsai-Wu failure theory includes no provision for the incorporation of applied moment resultants into the prediction of failure. The Maximum Stress Theory is another fail ure theory to which the results of the DMM may be compared. Simply stated, failu re occurs when any single stress component exceeds an allowable level. S N Y N Y X N Xxy T y C T x C (2.26) And similarly, the Maximum Strain failure theory states that failure occurs when any single mid-plane strain compone nt exceeds an allowable level. 0 0 0CxT CyT xyLTee ee e (2.27)

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24 For example, the limiting mid-plane strain in a given direction is i t N iE emax (2.28) The bounds of the Maximum Strain Theory failure envelope ta ke the shape of a parallelogram whose sides are defined by lines of the form T x xy yS N N (2.29) As was mentioned regarding the Tsai-Wu failure theory, the Maximum Stress and Maximum Strain theories were developed in te rms of applied stresses, thus adapted forms of these criteria in terms of applied force resu ltants are used. Again, currently there is no provision for the incorporation of applied moments into the prediction of failure.

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25 CHAPTER 3 STIFFNESS AND STRENGTH DETERMINATION Using the DMM procedure as detailed in Chap ter 2, the failure envelope for a plain weave textile composite is developed for in -plane force resultan ts, with and without applied moment resultants. No currently accep ted failure criteria exist that may be used explicitly for the analysis of textile composites. Thus the methods and results employed herein are used to develop phenomenological failure criteria for textile composites. The results are compared to conventional failure en velopes that are not specifically developed for the analysis of textile composites, as a basis for evaluation. Stiffness Properties The fiber tow was assumed to have mate rial properties of a unidirectional composite (weave geometry is taken into account in the finite element model), in this case AS/3501 graphite-epoxy. The constitutive matrices relating macroscopic force and moment resultants to strains and curvatures were found using the aforementioned procedures and are found to be ) ( 10 18 0 0 0 0 14 4 52 0 0 52 0 14 4 ] [6m Pa A 0 ] [ B (3.1) ) ( 10 35 1 0 0 0 70 7 53 2 0 53 2 70 7 ] [3 3m Pa D

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26 The character of the constitutive matrices is analogous to an orthotropic stiffness matrix with identical elastic constants in the material principal direct ions (also referred to as a tetragonal stiffness matrix). Flexural stiffness values of the [ D ] matrix may seem slightly low, but it should be noted that the RVE under consideration is relatively thin at 0.254 mm. Also note that although the zero terms in the above matrices were not identically zero, they were several orders of magnitude below comparable matrix terms, and thus have been neglected with little or no effect on end results. The above constitutive matrices have b een calculated directly from the micromechanics model without any assumptions on the deformation of the composite such as plane sections remain plane etc. as in traditional plate theories. The results are quite different from commonly employed appr oximations. From classical laminate theory, a plane stiffness matrix is calcula ted from homogenized continuum stiffness properties (the familiar E G and ). The flexural stiffness matrix [ D ] is then calculated from this homogenized stiffness and the thickne ss of the textile RVE. By comparison to the direct micromechanics results of the DMM, these methods will misrepresent flexural stiffness values D11, D12, and D66 by as much as factors of 2.9, 1.1, and 0.7 respectively. The DMM results imply that there is no consis tent relation between in-plane and flexural properties, although the two properties are related. In-plane axial and shear stiffness values ( E, G ) are calculated directly from the [ A ] matrix. Compared to the bare properties [34] of the constituent fiber tows (Table 3-1), stiffness is lower by an order of magnitude But it must be noted that the textile composite under consideration here has on ove rall volume fraction of 25%, whereas the tow properties of Table 3-1 reflec t the 65% volume fraction as seen within the tow itself.

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27 As a basis for comparison, the in-plane homogenized stiffness properties are presented in Table 3-2, as cal culated directly from the [ A ] and [ D ] matrices above. Flexural moduli, which represent the bending st iffness of the textile composite, are also presented in Table 3-2 and are calculated from the relations shown in Equation 3.2 through Equation 3.5. 11 312 D t I M Ex yy x fx (3.2) 22 312 D t Efy (3.3) 66 312 D t Gfxy (3.4) 11 12D Dfxy (3.5) Flexural moduli are material-dependent properties, but are strongly geometrydependent as well. Similarly to what is seen for the in-plane stiffness, the flexural stiffness is shown to be much lower than th e constituent tow properties, for the relatively thin plate under considerati on here. The bending propertie s are strongly influenced by the plate thickness and the weave architecture. Table 3-1: Fiber Tow and Matrix Material Properties Material E1 (GPa) E2 (GPa) G12 (GPa) 12 AS/3501 Graphite/Epoxy (65% Fiber Volume) 138 9.0 6.9 0.30 3501 Matrix 3.5 3.5 1.3 0.35 Table 3-2: Stiffness Properties for Plain Weave Textile Plate Ex = Ey (GPa) Gxy (GPa) xy In-Plane Properties 16.0 0.71 0.13 Efx = Efy (GPa) Gfxy (GPa) f xy Flexural Properties 5.0 0.88 0.33

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28 Strength Properties Table 3-3 [34] shows the textbook values for failure strengths of the yarn and matrix materials. The subscripts L and T refer to the longitudinal and transverse directions, respectively. The superscripts + and - refer to tensile and compressive strength. These were used with the Tsai-Wu failure criterion to determine failure of the yarn at the micro level within an element. Figure 3-1 shows a comparison of the DMM failure envelope for the plain weave graphite/3501 textile composite with seve ral common failure theories: the Tsai-Wu, Maximum Stress, and Maximum Strain failure theories. Failure envelopes are shown in the plane of biaxial force resultants with no applied moment present. Since the DMM is used to define the macro level failure stre ngth, all theories share the same uniaxial strengths and are fit to these points. For the most part, the Maximum Stress Theo ry is much more conservative than all other theories. However, it is less conservative in Quadrant s II and IV, since this failure theory does not account for the interaction of biaxial stresses. The Maximum Strain and Tsai-Wu failure theories compare more closel y to the DMM failure envelope, especially the latter. For zero applied moment (Figure 3-1), th e DMM failure envelope follows closely with the form of a Tsai-Wu failure envelope. For the most part, the initial failure mode is transverse failure of the fiber tows. However, at the extremes of the major axis of the Table 3-3: Fiber Tow and Matrix Failure Strength Properties (MPa) SL (+) SL (-) ST (+) ST (-) SLT 3501 / Graphite Tow 1448 1172 48.3 60 62.1 3501 Matrix 70 70 70 70 70

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29 Figure 3-1: Comparison of DMM Failure Envelopes with Common Failure Theories. Adapted Forms of the Tsai-Wu, Maximum Strain, and Maximum Stress Failure Theories are Shown. Since the DMM is Used to Define the Macro Level Failure Strength, all Theories Sh are the Same Uniaxi al Strengths and are Fit to these Points. failure envelope (the outer corners of quadr ants I and III), the initial failure mode transitions to failure of the matrix material. Thus the DMM failure envelope is cut short at the ends (compared to the failure envelope that would exist if matrix failure were not considered) and is squared-off in these regi ons and resembles the maximum failure stress (force resultant) criterion. The DMM envel ope is more conservative than the Tsai-Wu type criterion in quadrant I, and slightly less conservative in quadrant III. Though failure loads are not generally re ported in terms of force and moment resultants, a translation of the failure envel ope force resultant values to traditional stress values shows that strength magnitudes are reas onable based upon comparison to literature

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30 and supplier-reported values fo r this material, geometry, a nd the relatively low fiber volume fraction (25%). Since the yarn takes much of the load, the matrix does not begin to fail until a much higher load than its bare tensile strengt h. Although the strength value is a fraction of the pure tow strength, the w oven tow is not completely aligned in the loading directions. Some of the tow is curved into the thickness direction, thus providing through-thickness reinforcement. Furthermore, af ter initial transverse failure of the fiber tow (indicative of the introduction of intra-to w micro-cracking), the structure will still maintain load-bearing capacity, though stress concentrations will begin to build up and part integrity will be degraded. Also note th at, due to symmetry of the textile RVE about the xand yaxes, the failure envelope exhibits this symmetry as well. The effect of an added moment Mx on the failure envelope is shown in Figure 3-2 with the applied moment Mx equal to half the critical valu e that would cause failure if it were the only applied load. The figure also includes results fro m Figure 3-1, the DMM results and quadratic failure envelope for the case of zero applie d bending moment. There is no Tsai-Wu, Maximum Stress, or Ma ximum Strain failure envelope to include applied moment resultants, as these theories ar e not developed to include such load types. As has been mentioned, strength estimates will be somewhat conservative given that failure is defined as failure of a single el ement that surpasses the maximum allowable microstress, but this presents a real istic definition of initial failure. Continuing to inspect Figure 3-2, an applied moment in the x -direction has the expectable effect of shrinking the failure enve lope in regions where tensile applied loads dominate. However, when only compressive loads are applied, an applied moment can actually increase the in-plane load capacity by offsetting some of the compressive stress

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31 with bending-induced tension. The magnitude of this load-capacity increase is limited, however, by the eventual failure of the matrix. As with the case of pure in-plane loading, the failure envelope at the outer corner of quadrant III is dominated by matrix failure. The effects of applied moment on the failu re envelope of the plain weave textile represents the importance of the consideration of stre ss gradients, or load nonuniformities. The appreciable difference that arises suggests that such consideration could be critical to the su ccessful design or optimization of a textile structural component. Failure Envelopes-1.50E+04 -1.00E+04 -5.00E+03 0.00E+00 5.00E+03 1.00E+04 1.50E+04 -1.50E+04-1.00E+04-5.00E+030.00E+005.00E+031.00E+041.50E+04Applied Force Resultant, Nx (Pa-m)Applied Force Resultant, Ny (Pa-m) Mx = 0 Tsai-Wu (1/2)Mx, crit Figure 3-2: Effect of Bendi ng Moment on the Failure Envelope. The Overall Envelope Decreases Significantly in Size. The Tsai-Wu Failure Ellipse is Unaltered when Moment is Applied, as the Theory Includes No Provision for this Load Type. The effect of changing the failure theory used to define failure of a tow element at the micro (elemental) level is shown in Figure 3-3. The DMM can easily be modified to employ any appropriate failure criterion for the constituent phases at the

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32 micromechanical level. Here, the Maximum Stress Failure Theory (MSFT) is used to replace the Tsai-Wu failure theory to determine first-element failure for the fiber tows. Though the overall character of the failure e nvelope is unchanged, the effect is quite significant in that the failure envelope become s roughly half as conser vative. It should be noted that, even in the case of simple uniax ial macro applied loads, the micro stress field that results is fully three-dimensional and non-homogeneous across the RVE. Thus, especially for the orthotropic fiber tows, a failure theory that includes multi-dimensional stress interaction effects (such as Tsai-Wu) should be more appropriate. Failure Envelopes-1.75E+04 -1.25E+04 -7.50E+03 -2.50E+03 2.50E+03 7.50E+03 1.25E+04 1.75E+04 -1.75E+04-1.25E+04-7.50E+03-2.50E+032.50E+037.50E+031.25E+041.75E+04Applied Force Resultant, Nx (Pa-m)Applied Force Resultant, Ny (Pa-m) DMM Tsai-Wu DMM MSFT Figure 3-3: Effect of Micr o-Level Tow Failure Theory on the DMM Failure Envelope. A Comparison is Shown between the Original DMM Using the Tsai-Wu Theory to Analyze Failure of a Tow Element and an Altered DMM in which the MSFT is Used As mentioned earlier, we have assumed that the entire textile composite has failed, even if only one of the yarn or matrix elemen ts has failed. It is possible to change the

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33 definition of failure by stating th at failure is considered to ha ve occurred when 1% of the total number of elements have failed. Fi gure 3-4 shows the effect of changing the definition of failure and consequently the macro failure envelope. Note that failure points are shown in the plane of uni axial applied force and moment resultants, in order to illustrate these effects under different loadi ng types. About a 30% increase in maximum allowable force or moment resultant results from changing the definition of initial failure. This is shown to present the possibility of a more stochastic or a less conservative approach to determining the point at which in itial failure occurs, t hus only a few points are presented. Failure Points-3.00E-04 -2.00E-04 -1.00E-04 0.00E+00 1.00E-04 2.00E-04 3.00E-04 -1.00E+04-8.00E+03-6.00E+03-4.00E+03-2.00E+030.00E+002.00E+034.00E+036.00E+038.00E+031.00E+04Applied Force Resultant, Nx (Pa-m)Applied Moment Resultant, Mx (Pa-m2) 1% Elems 1 Elem Figure 3-4: Effect of Changing the Definition of Micr o-Level Failure on the Failure Envelope. The Allowable Force or Mo ment is Shown to Increase by 30% if Failure is Defined as the Point at which 1% of Elements Have Exceeded Failure Criterion, as Compared to the Point at which One Single Element Has Failed.

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34 Figure 3-5: Stress Contours for Plain-Weave Fiber Tows in Uniaxial Extension. Detailed Microstress Field Results fr om the DMM Provide Valuable Insight into the Mechanical Response of an RVE under Any Loading Condition. Tow Elements or Matrix Elements Can Be Isolated to Provide Further Detail. Microstress Field Contour Plots Detailed stress-field contour plots are one output of th e DMM that provide great insight into the failure modes and points of maximum stress in the RVE. Tow or matrix portions of the RVE can be isolated for indi vidual scrutiny. Figur e 3-5 shows the stress field for the plain-weave fiber tows in uniaxial tension. Most of the load in this case is taken by the fiber tows aligned in the loading direction. However, the failure mode is transverse failure of the cross-axis tows, as th e strength is much lower in this direction. Tows aligned in the direction of loading tend to be pulled straight as load is applied. This has the secondary effect of applying bending to the cross-ax is tows, creating significant bending stresses. The maximum stress levels in this micro stress field tend to occur around the matrix pockets between tows, whic h tend to act as a micromechanical stress concentration. Similar inspection of the st ress field in the matrix surrounding the fiber

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35 tows (not shown here) shows that the maximum stress tends to occur along the relatively sharp edges of the lenticular fiber tows, which again tend to act as a micromechanical geometrical stress concentration. Further stress contour plots illustrating results and capabilities of the DMM show the RVE behavior in shear and bending load cases. For unit bending load cases, it is shown that the majority of the load is again taken by the fiber tows along the direction of curvature. Furthermore, just as there is a transition from tension to compression across the RVE in bending, a rapid stress gradient from tension to compression can be seen in an individual fiber tow. This again suggests the importance of the c onsideration of stress gradients in the micromechanical char acterization of a textile RVE. Inspection of Failure Envelopes in Additional Stress Spaces Based on an extension of the results presen ted also in [28] and shown earlier in Figure 3-2, failure envelopes in the space of Nx Ny Mx (a practical and useful failure space which illustrates the limits of biaxial lo ading and the importance of consideration of stress gradients across an RVE) are seen to be characteristically elliptical in nature. This is due to the prevalence of stress interaction effects, as well as the symmetry of the plain weave geometry under analysis. Figure 3-6 shows the discrete failure point s as determined from the DMM for cases of biaxial loading with constant applied moments of 0, 0.3, 0.5, and 0.8 times the critical moment that would cause failure if it were the only load present. Phenomenologically, it can be seen that the failure envelopes b ecome smaller as a larger moment (stress gradient) is applied. An applied moment has the expectable effect of shrinking the failure envelope, though in limited regions it has b een seen that the complexity of the superposed stress fields may have an offsetting or beneficial effect.

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36 Figure 3-6: DMM Failure Envelopes for Biaxial Loading with Multiple Constant Moment Resultants. For Illustration, an Ellipse is Fit to Each Data Set Using a Least-Squares Method. Also shown in this figure are elliptical fits to each failure envelope, which were computed by a Matlab based routine that was used to determine a least squares fit to the DMM data points. Regular analytical trends in these failure envelopes lead directly to the development of failure prediction me thods which will be detailed in Chapter 4. In general, failure envelopes in spaces other than those shown in Figure 3-6 will not necessarily be elliptical in nature, as has been observed fo r cases including shear, twist, and multiple moment loading terms. Figures 3-7 and 3-8 show failure envelopes in spaces of Mx My Mxy and Nx Mx Nxy respectively. Discrete points represent the failure envelopes as determined by the DMM. Figure 3-7 isolates the relative effect of mo ment resultants, or stress gradients, of varying types on the failure envelope, as well as the interaction of multiple moments. Each envelope is mapped out w ith a constant applied twist Mxy.

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37 Figure 3-7: DMM Failure Envelopes fo r Biaxial Bending with Constant Applied Twisting Moment. Figure 3-8: DMM Failure Envelopes with Fo rce and Moment Resultants for Constant Applied Shear.

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38 The overall character of Figure 3-7 is in some ways similar to that of Figure 3-6, in that the failure envelope symmetry reflects the symmetry of geometry and loading. Further, the strength limits in quadrants II and IV are lower due to stress interactions. Whereas for biaxial loading, strength is decreased when stresses act in the direction of the natural tendency of Poisson effects, in be nding the strength is decreased when biaxial bending acts in the direction of the natural tendency of anticlastic curvature. As is generally seen in all failure analyses that ha ve been performed in this study, the dominant mode of failure is transverse failure of the fiber tow. The Nx Mx Nxy failure envelope (Figure 3-8) provi des a visualization of the effect of the magnitude of stress gradient, or load ing non-uniformity, for a given force resultant. Each envelope is drawn for constant in-plane shear to further incorporate the effects of multiple loading types. The e nvelope is symmetric about the x -axis due to the mechanical equality of positive or negative bending moment. This symmetry is not seen about the y -axis since the carbon -epoxy plain weave responds differently when in tension or compression. Design of Experimental Verification A procedure is suggested here which may be used to illustrate the differences between DMM and conventional failure points (a point which will also be visited in Chapter 5). Figure 3-9 shows a schematic repr esentation of an off-axis test specimen. This specimen can be used unde r uniaxial tension to investig ate different stress states. The orientation angle ( ) represents the angle of the prin cipal material axes of the plainweave with respect to specimen bounds.

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39 Figure 3-9: Schematic of the Specime n for Off-Axis Uniaxial Testing. 0.00E+00 1.00E+03 2.00E+03 3.00E+03 4.00E+03 5.00E+03 6.00E+03 7.00E+03 8.00E+03 01020304050 Orientation Angle (degrees)Critical Force Resultant Nx (N/m) DMM Tsai-Wu MaxStress Figure 3-10: Critical Force Resultant under Un iaxial Loading as Calculated via Various Failure Theories. Figure 3-10 shows a comparison of the maximum allowable force resultant Nx that may be applied, as calculated via the DMM, Tsai-Wu Failure Theory, and the Maximum Stress Theory, for various orientation angles. Data collected for tests performed at 0.08 radians (4.5 degrees) will produce data that should illustrate the greatest disparity

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40 between DMM and the Tsai-Wu Failure Theory. At this point, the DMM predicts a failure load 17% less conservative than Tsai-W u. Thus this can serve as an effective illustration to compare and contrast the two theories. Similarly, tests performed for a 0.31 radians (17.8 degrees) specimen will provide a comparison point most useful to investigate differences between predictions from DMM vs. Max Stress failure theories. At this point, the DMM predicts failure levels 32% more conservative. Chapter Summary By analysis of the microstresses devel oped in a representative volume element (RVE), the Direct Micromechanics Method (D MM) has been used to construct failure envelopes for a plain weave car bon/epoxy textile composite in pl ane stress. To allow for the accommodation of stress gradients, or nonuniform applied loads, micromechanical analysis had been performed in terms of cl assical laminate theory force and moment resultants [ N ] [ M ] and constitutive matrices [ A ] [ B ] [ D ]. The predicted values of the stiffness matrices and ultimate strength values compare well to expectable magnitudes. The DMM failure envelope was shown to be la rgely elliptical of the form of a Tsai-Wu failure criterion and dominated by transverse fiber tow failure. But in cases of large biaxial tension or biaxial compression loads, the DMM failure envelope compared to the form of the Maximum Stress Criterion, and matrix failure was the mode of initial failure. The presence of applied moment resultants [ M ], as would exist in cases of non-uniform load across the RVE, was shown to have a si gnificant effect on the failure envelope. Thus its consideration, not covered in conve ntional failure models, can be critical. The diversity of the failure spaces seen he re is a harbinger of the further need to develop analytical methods to predict failure without pre-knowledge of the nature of the failure envelope. This will be discussed in Chapter 4.

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41 CHAPTER 4 PREDICTION OF FAILURE ENVELOPES The current chapter presents methods for u tilizing the re sults presented in Chapter 3 to develop a failure criterion for textil e composites. Using the DMM, the failure envelope is developed for in-plane force resu ltants, with and without moment resultants. No currently accepted failure criteria exist that may be used explicitly for the analysis of textile composites, or which include this ab ility to account for stre ss gradients at the micromechanical level. Thus the methods and results employed herein are used to develop phenomenological failure criteria for textile composites. Based on the DMM results, two methods are presented which may be used to predict failure of a textile composite. The first is a parametric method based on prediction of regular trends in the failure envelopes of a given 3D stress space. The second method represents the formulation of a 27-term quadratic failure e quation that can be evaluated to determine failure of the textile under any gene ral force and moment resultants. A Parametric Approach to Predicting Fa ilure Envelopes for a Given Stress Space Referring again to the results of Figure 3-6, the observation is repeated that, phenomenologically, the failure envelopes beco me smaller as a larger moment (stress gradient) is applied. An applied moment has the expectable effect of shrinking the failure envelope, though in limited regions it has b een seen that the complexity of the superposed stress fields may have an offsetting or beneficial effect. Analytically, it can be seen that there ar e definite trends in the axes and placement of the failure ellipses. Failure ellipses were then characterized with parameters such as

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42 major axis length, minor axis length, ellipse ax is orientation angle, and the ellipse center point. Any general ellipse in ( x y ) space can be represented by the expression 1 cos sin sin cos2 0 2 0 b v y x a u y x (4.1) where is the orientation angle of the major axis of the ellipse with respect to the x -axis, uo and vo are the coordinates of the ellipse center point, a is half the major axis length, and b is half the minor axis length. For each failure ellipse shown in Figure 3-6, these parameters were then plotted as functions of the moment applied for each case, in order to inspect parametric trends. For example, Figures 4-1 and 4-2 show the major and minor axis length of several Nx Ny Mx failure envelopes plotted ag ainst the moment resultant Mx applied in each case. A limited number of fitting cases were used, in order to reserve an adequate number of test cases and to prev ent over-fitting of the trends. Figure 4-1: Effect of Applied Moment Resultant on the Major Axis of Elliptical Failure Envelopes.

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43 Figure 4-2: Effect of Applied Moment Resultant on the Mi nor Axis of Elliptical Failure Envelopes. Figure 4-3: Effect of Applied Moment Resultant on th e Center Point Coordinates ( uo vo) of Elliptical Failure Envelopes. As mentioned earlier and as seen in these above figures a larger applied moment with a given failure envelope has the effect of shrinking the envelopes major and minor

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44 axes. These trends were regular enough to be closely approximated with a polynomial trend line. At the critical moment, the el lipse axes lengths become zero, as a load sufficient for failure is already applied, and no additional force resultants may be applied. In addition to shrinking the failure envel ope, larger applied moments also tend to cause a small but significant shift in the center po int of the failure ellipse. This is caused by the fact that an applied moment is stil l a directional loading, and thus produces a directional bias in the location of the failure envelope. These trends were also plotted (Figure 4-3) and approximated with polynomial trend lines. Ellipse orientation angle ( ) was found to be nearly constant, th us no trend plot is shown. Based on the above, an elliptical failure e nvelope of the form of Equation 4.1 can be predicted by evaluating the expressions fo r ellipse parameters in terms of applied moment resultant, as shown below 1 cos sin sin cos2 0 2 0 b v N N a u N Ny x y x (4.2) M M v M M u M M M b M M M M a M M Mcritical applied92 5 26 3 51 0 48 3 6 4 95 0 81 5 47 9 4586 1 73 0 85 5 44 11 77 72 0 2 0 2 3 2 3 4 (4.3) Thus any general Nx Ny Mx failure envelope can be predicted with the above procedure and expressions. For several test cases, a failure ellipse is predicted and

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45 compared to discrete failure points as cal culated directly from the DMM. These comparisons are shown in Figure 4-4 and Figur e 4-5. The average deviation between the failure envelope predicted from the parametric curve fitting as compared with the direct results of micromechanical modeling was less than 2%. The greatest value of this parametric a pproach to predicting failure is that it provides good insight into the exact nature of the failure space under consideration. Further, the results are quite accurate and could be useful for design purposes. However, the load cases are limited, and the methods pr esented here would have to be extended if more than three simultaneous loads were to be applied and analyzed. Figure 4-4: Failure Envelopes Predicted with the Parametric Approach as Compared to DMM Results (Applied Moment Resu ltant of 0.65 Critical Moment)

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46 Figure 4-5: Failure Envelopes Predicted with the Parametric Approach as Compared to DMM Results (Applied Moment Resu ltant of 0.9 Critical Moment) Development of a Quadratic Failure Criterion to Predict Failure for General Loading In order to bind together the failure spaces which can be quite different in nature, as can be seen in Figure 3-6 through Figure 38, methods of the previous section will not be readily applicable. Therefore, devel opment of an additional analytical method becomes necessary. Further, an analytical approach to binding the results of multiple failure envelopes as calculated via the D MM, along with the capacity to accommodate any general plate loading conditi on has a practical value. Gi ven the quadratic interactive nature of the stress state in determination of failure, an expression of the below form has been developed to predict failure. 1 i i j i ijF D F F C (4.4) where Fi represent general load terms ( Nx Ny Nxy Mx My Mxy) and Cij or Di represent 27 failure coefficients such that Equation 4.4 de fines failure when its magnitude exceeds 1.

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47 Failure coefficients Cij and Di can be solved given a sufficient amount of known failure points or failure envelopes. Howe ver, Equation 4.4 is numerically ill-conditioned given the great disparity in the magnitude of Nij and Mij loads which will cause failure. These values will differ by many orders of magni tude when typical units are utilized (Pam and Pa-m2). This makes accurate solution of fa ilure coefficients impossible when both such load types are present. Thus, E quation 4.4 must be defined in terms of Fi terms that are normalized with re spect to a critical Nmax or Mmax that would cause failure if it were the only load present. Coefficients Cii and Di can be solved by evaluating E quation 4.4 with a load of Fi and setting all other loads at zero. For example, in order to obtain C11 and D1 a load of F1 = Nx is applied, all others are set to zero, and Equation 4.4 reduces to 11 2 11 x xN D N C (4.5) Since the DMM has been used to determine the maximum tensile and compressive allowable Nx, each of which must satisfy Equation 4.5, two independent equations are generated which can be simultaneously solved to yield C11 and D1. Note that, as mentioned earlier, loads have been normali zed for numerical robustness (in this case, Nx / Nx,critical ). The Nx,critical for the plain weave carbon-epoxy in this study was found to be 6.40 x 103 Pa-m ; this was used to normalize all force resultant terms. The Mx,critical used to normalize all moment resultant terms was 1.85 10-4 Pa-m2. A complete table of strength values for the carbon-epoxy plain weav e textile composite is shown in Table 4-1. Remaining coefficients Cij can be solved by evaluating Equation 4.4 with the maximum possible Fi = Fj as determined by the DMM results. All other loads are set to zero. This reduction in terms, along w ith knowledge of previously determined

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48 Table 4-1: Strength Values for Independent Load Conditions Strength (+) Strength (-) Nx 6.40e3 Pa-m 5.86e3 Pa-m Ny 6.40e3 Pa-m 5.86e3 Pa-m Nxy 2.11e3 Pa-m 2.11e3 Pa-m Mx 1.85e-4 Pa-m2 1.85e-4 Pa-m2 My 1.85e-4 Pa-m2 1.85e-4 Pa-m2 Mxy 1.06e-4 Pa-m2 1.06e-4 Pa-m2 coefficients Cii and Di allows for the solution of all remaining coefficients. For example, C14 can be determined by applying the maximum possible F1 and F4 such that critical x critical xM M N N F F 4 1 (4.6) The failure coefficient is then solved from Equation 4.4 as 4 4 1 1 2 4 44 2 1 11 4 1 141 2 1 F D F D F C F C F F C (4.7) The full results of the above procedures are shown in Table 4-2 and Table 4-3 below. Note that coefficients D3 through D6 are equal to zero since positive and negative failure values are the same for any shear or moment loads. For complete evaluation of the 27 failure coefficients, it will generally not be necessary to complete 27 separate physical or simulated experimental evaluations. Exploitation of geometry can lead to a si gnificant reduction. The plain weave under investigation requires 13 evaluations to dete rmine all coefficients, and more complicated weaves will still often exhibit symmetry such that only this amount is required. For the most general case, 27 evaluations may be re quired. Twelve involve a single load Fi both in tension and compression. Fifteen eval uations will be needed that involve every combination of two loads applied equally (as normalized) until failure. These 27 evaluations may be performed either by physic al experiment or simulated via the DMM.

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49 Table 4-2: Normalized Failure Coefficients Cij for Quadratic Failure Equation. (Coefficient Cmn is in mth Row and nth Column) n = 1 2 3 4 5 6 m = 1 1.02 -0.81 2.45 0.15 0.15 -0.09 2 1.02 2.45 0.15 0.15 -0.09 3 9.29 0.15 0.15 -1.28 4 1.00 -0.65 0.29 5 1.00 0.29 6 3.05 Table 4-3: Normalized Failure Coefficients Di for Quadratic Failure Equation D1 -0.011 D2 -0.011 D3 0.000 D4 0.000 D5 0.000 D6 0.000 Figure 4-6: DMM Failure Envelopes for Biaxial Bending with Constant Applied Twisting Moment as Compared to the Quadratic Failure Theory Predictions. Referring to Figures 4-6 a nd 4-7, the failure spaces calculated via the DMM and previously shown in Figures 3-7 and 3-8 are compared to the elliptical curves labeled

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50 QFT, which represent the failure envelopes as predicted by the quadratic failure theory of Equation 4.4. The overall agreement in such test cases is seen to be quite suitable, but as seen in Figure 4-6, there will be corners or portions of the 6D failure space that will be missed with the essentially 6D ellipse space of the quadratic failure theory. In general, the QFT predictions tend to be cons ervative in areas of disagreement. Figure 4-7: DMM Failure Envelopes with Fo rce and Moment Resultants for Constant Applied Shear as Compared to the Quadratic Failure Theory Predictions. For further comparison and to incorpor ate more complex loading conditions, several test cases were computed to determin e the difference in solutions computed from Equation 4.4 as compared to the results of the DMM. In general, Fi may be populated by as many as 6 terms from among ( Nx Ny Nxy Mx My Mxy). For cases in which 1, 2, or 3 terms are populated, the solution is accurate to within a few percent. For test cases in which 4, 5, or 6 terms are populated, the average error was seen to be 9.3%, and several example test cases are tabulated below. Care was ta ken to select a broad spectrum of cases such

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51 that both positive and negative values are employed, and multiple load ratios and load types are employed. Load ratios ( ) are shown to characterize the test cases, defined as 1F Fi i (4.8) By maintaining the same load ratios, all pr edicted failure loads will maintain a single ratio with respect to DMM failure points. Thus one ratio can characterize the congruence of these two solutions. A ratio of one w ill imply complete agreement between the two solutions. A ratio less than one indicates a conservative failure pr ediction, and a ratio greater than one implies the converse. Ta bles 4-4 through 4-6 summarize the results. Table 4-4: Comparison of Quadratic Failure Equation Predictions with DMM Results. Test Cases Include 4 Populated Load Terms Fi F1 F2 F3 F4 F5 F6 1.00 0.00 0.92 6.67E-08 0.00 6.67E-08 DMM 1.20E+03 0.00E+00 1.10E+03 8.00E-05 0.00E+00 8.00E-05 Quadratic Theory 1.12E+03 0.00E+00 1.02E+03 7.44E-05 0.00E+00 7.44E-05 Ratio 0.93 1.00 1.83 0.00 6.67E-08 0.00 6.67E-08 DMM 1.20E+03 2.20E+03 0.00E+00 8.00E-05 0.00E+00 8.00E-05 Quadratic Theory 1.24E+03 2.27E+03 0.00E+00 8.24E-05 0.00E+00 8.24E-05 Ratio 1.03 1.00E+00 -0.87 0.00 3.91E-08 -1.40E-08 0.00 DMM -2.30E+03 2.00E+03 0.00E+00 -9.00E05 3.23E-05 0.00E+00 Quadratic Theory -2.55E+03 2.22E+03 0.00E+00 -9.99E05 3.59E-05 0.00E+00 Ratio 1.11 1.00 0.40 0.00 1.80E-08 -6.46E-09 0.00 DMM -5.00E+03 -2.00E+03 0.00E+00 -9.00E05 3.23E-05 0.00E+00 Quadratic Theory -5.50E+03 -2.20E+03 0.00E+00 -9.90E05 3.55E-05 0.00E+00 Ratio 1.1

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52 Table 4-4: Continued F1 F2 F3 F4 F5 F6 1.00 -0.50 0.00 -1.61E08 -8.08E-09 0.00 DMM -4.00E+03 2.00E+03 0.00E+00 6.45E-05 3.23E-05 0.00E+00 Quadratic Theory -3.72E+03 9.30E+02 1.53E+03 6.51E-05 6.51E-05 7.44E-05 Ratio 0.93 Table 4-5: Comparison of Quadratic Failure Equation Predictions with DMM Results. Test Cases Include 5 Populated Load Terms Fi F1 F2 F3 F4 F5 F6 1.00 1.00 1.00 6.25E-08 0.00 6.67E-08 DMM 1.20E+03 1.20E+03 1.20E+03 7.50E-05 0.00E+00 8.00E-05 Quadratic Theory 1.04E+03 1.04E+03 1.04E+03 6.53E-05 0.00E+00 6.96E-05 Ratio 0.87 1.00 1.67 1.20 4.67E-08 5.33E-08 0.00 DMM 1.50E+03 2.50E+03 1.80E+03 7.00E-05 8.00E-05 0.00E+00 Quadratic Theory 1.32E+03 2.20E+03 1.58E+03 6.16E-05 7.04E-05 0.00E+00 Ratio 0.88 1.00 -1.33 -1.07 -4.67E08 3.33E-08 0.00 DMM -1.50E+03 2.00E+03 1.60E+03 7.00E-05 -5.00E-05 0.00E+00 Quadratic Theory -1.32E+03 1.76E+03 1.41E+03 6.16E-05 -4.40E-05 0.00E+00 Ratio 0.88 1.00 1.33 1.00 -3.33E08 2.67E-08 0.00 DMM -1.50E+03 -2.00E+03 -1.50E+03 5.00E-05 -4.00E-05 0.00E+00 Quadratic Theory -1.32E+03 -1.76E+03 -1.32E+03 4.40E-05 -3.52E-05 0.00E+00 Ratio 0.88 1.00 -0.45 0.45 2.27E-08 0.00 3.18E-08 DMM -2.20E+03 1.00E+03 -1.00E+03 -5.00E05 0.00E+00 -7.00E-05 Quadratic Theory -2.16E+03 9.80E+02 -9.80E+02 -4.90E05 0.00E+00 -6.86E-05 Ratio 0.98

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53 Table 4-6: Comparison of Quadratic Failure Equation Predictions with DMM Results. Test Cases Include 6 (Fully Populated) Load Terms Fi F1 F2 F3 F4 F5 F6 1.00 -1.00 1.00 4.00E-08 4.00E-08 2.00E-08 DMM 1.49E+03 -1.49E+03 1.49E+03 5.96E-05 5.96E-05 2.98E-05 Quadratic Theory 1.28E+03 -1.28E+03 1.28E+03 5.13E-05 5.13E-05 2.56E-05 Ratio 0.86 1.00 1.00 0.52 1.33E-08 1.33E-08 1.33E-08 DMM 3.00E+03 3.00E+03 1.55E+03 4.00E-05 4.00E-05 4.00E-05 Quadratic Theory 2.61E+03 2.61E+03 1.35E+03 3.48E-05 3.48E-05 3.48E-05 Ratio 0.87 1.00 -0.73 -0.33 -1.33E-08 1.33E-08 1.33E-08 DMM -3.00E+03 2.20E+03 1.00E+03 4.00E-05 -4.00E-05 -4.00E-05 Quadratic Theory -2.85E+03 2.09E+03 9.50E+02 3.80E-05 -3.80E-05 -3.80E-05 Ratio 0.95 1.00 -0.67 0.67 -2.67E-08 5.33E-08 -3.33E-08 DMM -1.50E+03 1.00E+03 -1.00E+03 4.00E-05 -8.00E-05 5.00E-05 Quadratic Theory -1.44E+03 9.60E+02 -9.60E+02 3.84E-05 -7.68E-05 4.80E-05 Ratio 0.96 1.00 0.59 0.88 3.53E-08 3.53E-08 4.71E-08 DMM -1.70E+03 -1.00E+03 -1.50E+03 -6.00E-05 -6.00E-05 -8.00E-05 Quadratic Theory -1.45E+03 -8.50E+02 -1.28E+03 -5.10E-05 -5.10E-05 -6.80E-05 Ratio 0.85 Optimization of Failure Coefficients The failure coefficients of Table 4-2 and 4-3 have been obtained from procedures designed to be compatible with physical expe riments. However, further accuracy may be obtained through analytical methods to optim ize the coefficients. The DMM has shown that for this particular weave pattern, a failu re theory of the form of Equation 4.4 is an effective predictor of strength in multiple fail ure spaces, and that its inclusion of moment

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54 resultants at the micro level represents a cr itical consideration. After this groundwork, the failure coefficients may be found from the above procedure either by using the DMM or physical testing, or by employing a leastsquares routine which calculates the failure coefficients that maximize the accur acy of the obtained solution. To this end, the Matlab subroutine lsqnonlin has been employed. The coefficients of Table 4-2 and 4-3 are supplied to the routine, which was then limited to its perturbation of each coefficient by a factor of 10%. This represents an optimization which is constrained to maintain some proximity to the physically interpretable constants of Table 4-2 and 4-3. Further perturbation co uld be allowed, with a danger of overfitting, but this initial study is presen ted to elucidate the opportunity for further optimization. As fitting data, the 15 above test cases were provided, in addition to the data from the 27 tests used to fit failure coefficients by the pr evious procedure, for a total of 42 fit-points. Resulting coefficients as determined by the optimization algorithm are shown in Table 47 and Table 4-8. Significant reduction was show n in the minimization of the error of the objective function of Equation 4.4. Over th e 15 test cases of Ta ble 4-4 through 4-6, which previously exhibited an average error of 9.3%, the optimized failure coefficients reduce error to 7.7% over these cases. Note that average error is calculated as the absolute value of deviation from a perfect prediction. Table 4-7: Optimized Failure Coefficients Cij for Quadratic Failure Equation. (Coefficient Cmn is in mth Row and nth Column) n = 1 2 3 4 5 6 m = 1 1.115 -0.891 2.205 0.135 0.135 -0.099 2 1.029 2.205 0.135 0.135 -0.099 3 8.361 0.135 0.135 -1.408 4 0.9 -0.715 0.261 5 0.9 0.261 6 2.745

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55 Table 4-8: Optimized Failure Coefficients Di for Quadratic Failure Equation D1 -0.0121 D2 -0.099 D3 0 D4 0 D5 0 D6 0 Table 4-9: Data to Indicate Re sults of Coefficient Optimization Q-1 Q-1 (Q-1)2 (Q-1)2 index Normalized Force or Moment Resultants Fistandardoptim standard optim 1 0.1875 0 0.171880.432430 0.432430.11964 -0.0063 0.01431373 3.96887E-05 2 -0.35938 0.3125 0 -0.486490.174590 -0.03526 -0.0351 0.00124341 0.00123208 3 0.1875 0.34375 0 0.432430 0.43243-0.07631 -0.01636 0.00582322 0.000267584 4 -0.78125 -0.3125 0 -0.486490.174590 -0.09002 -0.07669 0.00810414 0.005881816 5 -0.625 0.3125 0 0.348650.174590 -0.25237 -0.2142 0.06369062 0.04588164 6 0.28125 0.15625 0.257810.378380.378380.432430.73672 0.52306 0.54275636 0.273591764 7 -0.23438 0.15625 -0.156250.21622-0.432430.27027-0.06791 -0.01178 0.00461177 0.000138721 8 0.46875 0.46875 0.242190.216220.216220.216220.58519 0.43468 0.34244734 0.188946702 9 -0.26563 -0.15625 -0.23438-0.32432-0.32432-0.432430.53185 0.34624 0.28286442 0.119882138 10 -0.46875 0.34375 0.156250.21622-0.21622-0.21622-0.03722 -0.03568 0.0013854 0.001272706 11 0.1875 0.1875 0.1875 0.405410 0.432430.13935 0.096473 0.01941842 0.00930704 12 -0.23438 -0.3125 -0.234380.27027-0.216220 0.078115-0.01175 0.00610195 0.000138039 13 0.23438 0.39063 0.281250.378380.432430 0.0630190.045837 0.00397139 0.002101031 14 -0.34375 0.15625 -0.15625-0.270270 -0.37838-0.03917 -0.01162 0.00153437 0.000135048 15 -0.23438 0.3125 0.25 0.37838-0.270270 0.13059 0.064909 0.01705375 0.004213178 16 1 0 0 0 0 0 0.009 0.1029 0.000081 0.01058841 17 0 1 0 0 0 0 0.009 0.0195 0.000081 0.00038025 18 0 0 0 1 0 0 1.00E-07-0.1 1E-14 0.01 19 0 0 0 0 1 0 1.00E-07-0.1 1E-14 0.01 20 0 0 0.328130 0 0 0.000217-0.0998 4.7228E-08 0.009960838 21 0 0 0 0 0 0.572970.001309-0.09882 1.7135E-06 0.009765788 22 0 0 0 1.1892 1.1892 0 0.00913 0.05343 8.3357E-05 0.002854765 23 0.25312 0 0.253120 0 0 -0.01852 -0.02546 0.00034306 0.000648364 24 0 0.25312 0.253120 0 0 -0.01852 -0.02596 0.00034306 0.000673714 25 0 0 0 0.464860 0.46486-0.00621 -0.01559 3.86E-05 0.00024311 26 0 0 0 0 0.464860.46486-0.00621 -0.01559 3.86E-05 0.00024311 27 0.51 0 0 0 0 0.51 0.029588-0.02794 0.00087545 0.000780364 28 0 0.51 0 0 0 0.51 0.029588-0.04908 0.00087545 0.002408552 29 0.66 0 0 0.66 0 0 -0.0062 -0.07145 3.845E-05 0.005104531 30 0 0.66 0 0.66 0 0 -0.0062 -0.10728 3.845E-05 0.011508998 31 0 0 0.32 0 0 0.32 0.013254-0.00692 0.00017567 4.79515E-05 32 0.66 0 0 0 0.66 0 -0.0062 -0.07145 3.845E-05 0.005104531

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56 Table 4-9: Continued Q-1 Q-1 (Q-1)2 (Q-1)2 index Normalized Force or Moment Resultants Fistandardoptim standard optim 33 0 0.66 0 0 0.66 0 -0.0062 -0.10728 3.845E-05 0.011508998 34 0 0 0.66 0.66 0 0 -0.00092 0.15173 8.4839E-07 0.023021993 35 0 0 0.66 0 0.66 0 -0.00092 0.15173 8.4839E-07 0.023021993 36 1.58 1.58 0 0 0 0 9.80E-070.013 9.604E-13 0.000169 37 -0.91 0 0 0 0 0 8.80E-070.0072 7.744E-13 0.00005184 38 0 -0.91 0 0 0 0 8.80E-070.001404 7.744E-13 1.97122E-06 39 0 0 -0.328 0 0 0 0.000217-0.0998 4.7228E-08 0.009960838 40 0 0 0 -1 0 0 1.00E-07-0.1 1E-14 0.01 41 0 0 0 0 -1 0 1.00E-07-0.1 1E-14 0.01 42 0 0 0 0 0 -0.573 0.001309-0.09882 1.7135E-06 0.009765788 Sum 1.31841456 0.830844875 rms error 0.17717455 0.140648572 Table 4-9 indicates the output of the coeffici ent optimization routine. The first column simply indexes each of the 42 load cases used for fitting. This is followed by columns indicating the normalized force and moment resultants in eac h case. As before, force resultants are normalized with the critical Nx in tension, and mo ment resultants are normalized with respect to critical Mx. The next two columns display the evaluation of the quantity (Q-1) for each load case (where Q is the calculation of Equation 4.4, which should ideally produce unity at failure) as eval uated with standard or optimized failure coefficients. These columns would then display zero for a completely accurate evaluation. The final two columns display the square of this error, which is then summed and divided by 42, the square root of which then yields the indicated rms error. The optimization produces a significant improvement wh ich, as has been mentioned, yields to an improvement in accuracy of predicted test cases.

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57 Chapter Summary Based upon failure envelopes constructe d by analysis of the microstresses developed in a representative volume el ement (RVE), two alternate methods for predicting failure envelopes of a plain-weave textile compos ite have been developed. The previously developed Direct Microm echanics Method (DMM) has been used to construct failure envelopes for a plain w eave carbon/epoxy textile composite in plane stress. To allow for the accommodation of stress gradients, micromechanical analysis had been performed in terms of classical laminate theory force and moment resultants [ N ] [ M ] and constitutive matrices [ A ] [ B ] [ D ]. A parametric ellipse-fitting scheme which accurately predicts trends in failure envelopes for a given failure space has been developed by analysis of failure ellipse pa rameters. This method for predicting failure envelopes was found to agree with DMM resu lts to within a few percent. A second method involves development of a 27-term quadr atic failure criteri on to predict failure under general loading conditions. The quadratic failure criterion was found to agree with DMM results within an average deviation of 9.3%, but the method is more robust in terms of its ability to predict failure from more complex loading cases.

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58 CHAPTER 5 MULTI-LAYER ANALYSIS The methods of the previous chapters invol ve characterization of the stiffness and strength of a single RVE, or a single-layer plain weave textile com posite. When multiple layers are present, the layer properties will re main the same, but the stiffness and strength of the overall composite will change. To develop a fully general failure theory, these methods must be adapted to accommodate a te xtile composite of an arbitrary number of layers. This can be predicte d through direct simulation of a layup to directly determine stiffness properties. This will yield accura te results, but creating a new FEM model for each new layup is highly impractical. Thus the single layer char acterization methods must be adapted to predict the properties of a layup of an arbitrary number of layers. This allows material characterization simulati ons of a single RVE to be applicable to a layup of an arbitrary number of layers, el iminating the need for further material characterization. Thus a practical tool for failure analysis and design of a plain weave textile composite has been developed. These previous methods are adapted via three multi-layer an alysis techniques: direct simulation of a multi-layer composite (which provides an accurate basis for comparison), an adjustment of the data out put from single-layer FEM simulation, and implementation of the quadrati c failure theory (without the requirement of determining a new set of failure coefficients). Failure point s have been predicted for a variety of load cases.

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59 The entire body of work is then applied to several practical ex amples of strength prediction to illustrate their implementation. The results are compared to conventional methods utilizing common failure theories not specifically developed for textile composites. Design of a dual-la yer textile plate under uniform pressure is considered, for several geometries and width-to-t hickness ratios. Also shown is a test case of a pressure vessel in which stress-gradients are less prev alent. Results are shown in terms of the maximum allowable pressure, as well as comp arison of point-by-point factor of safety values. Stiffness Prediction of Mult i-Layer Textile Composites A direct simulation of the behavior of a dual layer composite has been performed. The single RVE is replaced with two stacked RV Es which simulate the dual-layer textile composite. Using the procedures of Chapter 2, 6 unit strain and curvature cases have again been carried out to dire ctly determine the mechanical response. Then macro level force and moment resultants can be com puted, and the constitutive matrices are determined. Referring back to Chapter 3, a single layer of the pl ain weave textile under consideration will exhibit the following the following constitutive matrices ) ( 10 18 0 0 0 0 14 4 52 0 0 52 0 14 4 ] [6m Pa A 0 ] [ B (5.1) ) ( 10 35 1 0 0 0 70 7 53 2 0 53 2 70 7 ] [3 3m Pa D Now for a two-layer textile, the constitutive matrices, as determined from a two-layer model utilizing the DMM, are seen to be

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60 ) ( 10 36 0 0 0 0 28 8 04 1 0 04 1 28 8 ] [6m Pa A 0 ] [ B (5.2) ) ( 10 51 8 0 0 0 8 148 4 22 0 4 22 8 148 ] [3 3m Pa D The in-plane properties of the [ A ] stiffness matrix remain effectively unchanged; the doubling in value of this matrix is a bookk eeping artifact commensurate with the fact that there is twice as much material presen t with two layers, as compared to one. The same is true in part for the bending stiffness [ D ] as well. However, bending properties are affected by the moment of in ertia of the material that is present. For the two-layer plain-weave under investigation, this represents an approxi mately 10-fold increase in bending stiffness per layer, or an overall 20-fold increase in bending stiffness for both layers collectively. The same amount of material present will have higher bending stiffness the further it is from the neut ral axis of bending. Consequently, bending stiffness follows a relationship analogous to the Parallel Axis Theorem, which governs the increased moment of inertia of an area of material that is moved away from the bending axis. These expressions that govern th e overall stiffness of a multi-layer textile can be represented as SL ij DL ijA A 2 (5.3) ) ( 22d A D DSL ij SL ij DL ij (5.4) where the superscripts DL and SL represent double layer or single layer properties, and d represents the distance from the center of a layer to the bending axis. Similar results have been shown in [35] for tex tile composites under flexure. From this

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61 expression, the double layer stiffness matri ces (Equation 5.2) can effectively be calculated with knowledge only of the single layer stiffness properties (Equation 5.1). This can be extrapolated to a material of an arbitrary number of layers ( N ), once the DMM has been used to characterize a layer ( n ) or one RVE. Equation 5.5 and Equation 5.6 may be used to evaluate the stiffness matrices and no further analysis is needed. n ij N n ijA A1 (5.5) ) (2 1 n n ij n ij N n ijd A D D (5.6) Strength Prediction of Mult i-Layer Textile Composites Once stiffness has been determined by the me thods of the previous section, strength of a multi-layer textile may be determined. The following three sub-sections describe the various modeling approaches used to analyze laminated plain weave composite structures and comparison of various methodologies in m odeling the strength of textile composites. For clarity, the various approaches are summarized in Table 5-1. Table 5-1: Summary of the Various Methods Employed in Multi-Layer Strength Analysis Method Acronym Summary Direct Micromechanics Method DMM Method of chapter 2, used to characterize strength and stiffness of an RVE Dual Layer Direct Micromechanics Method DDMM Direct simulation of two stacked RVEs, used to characterize multiple layers Adapted Direct Micromechanics Method for Dual Layer Analysis ADMM Method to adapt data output from the DMM to obtain multiple layer strength prediction Quadratic Failure Theory QFT Met hod for implementing the quadratic failure theory introduced in chapter 4

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62 Direct FEM Simulation of Multi-Layer Textile Composites Using the DMM for Failure Analysis (DDMM) The same methods described above for singl e layer strength prediction can be used for direct simulation of a two layer RVE. This direct simulation paints an accurate picture of the load capacity of a multilayer textile, at the expense of model preparation and calculation time. As the methods for the DDMM approach are essentially the same as the DMM described in Chapter 2, with a dual-layer RVE in place of the single RVE, the details are not repeated. The two sections to follow then describe two methods based upon these results (ADMM and QFT) which can be used to predict strength of a textile composite of an arbitrary number of la yers without having to employ direct FEM simulation. Through the DDMM, it is seen that, as also seen for stiffness properties, the inplane strength properties do not change. The cr itical force resultant doubles as a result of the doubling of material presen t, but otherwise th e load capacity is unchanged. SL crit ij DL crit ijN N, ,2 (5.7) Here the subscript crit denotes the critical load that is the maximum allowable when all other loads are zero (or the streng th for each individual load type Nij). As previously seen, the superscripts DL and SL represent associations with single-layer or dual-layer properties, respectively. The strength under bending will change signi ficantly for the two-layer textile. As a direct consequence of increased bending stiffnes s, the critical applied moment is seen to increase tenfold. SL crit ij DL crit ijM M, ,10 (5.8)

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63 Note that this relationship will depend on th e thickness of each layer, and thus is an observation specific to this RVE micro geomet ry (conversely, Equation 5.7 will be true for comparing single and double laye r properties for any thickness). The DDMM serves as a check upon which a more generalized approach may be developed, which can predict for an arbitr ary number of layers under arbitrary load conditions with mixed load types (not simply mono-loading cases of one critical force or moment resultant). Adaptation of the Single Layer DMM Resu lts to Predict Stren gth for Multi-Layer Textile Composites (ADMM) This method for taking the results of si ngle layer material characterization and analysis and using them for multi-layer stre ngth prediction involves adapting the results of single layer FEM analysis directly fr om the DMM. Thus only one material characterization is needed (single layer) to predict strength for any number of layers. In general, using the DMM, the stress fiel d from an applied load is calculated by determining the resultant strains and curvatur es, then superposing scaled multiples of the single-layer stress field resulting from each un it strain or curvature. In the case of a single layer under bending, the so le stress source is the re sulting curvature, thus the resulting stress field can be calculated by scal ing the stress field from a single layer under unit curvature. Now in the case of two la yers under bending, this stress source of curvature is still present, but there is an additional stress source that must be accounted for. Normal strains will result from the la yer offset from the axis of bending. Thus scaled multiples of these unit strain cases must also be applied to find the total stress field for a multi-layer textile under bending.

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64 Stress source from curvature Single stress source from curvature Additional stress source from strains that result from the layer offset from the axis of bending Single Layer Bending Multi-Layer Bending Stress source from curvature Single stress source from curvature Additional stress source from strains that result from the layer offset from the axis of bending Single Layer Bending Multi-Layer Bending Figure 5-1: Schematic Illustration of the Si ngle-Layer Strain and Curvature Stress Fields (As Found via the DMM) That Must Be Superposed in Calculation of the Total Stresses Resulting from Multi-Layer Bending As detailed in classical laminate theory [34], the magnitude of the normal strain in offset layers will be directly proportional to the curvature ( ) that is present and the distance from the layer midplane to the bending axis ( d ). ij ijd (5.9) This strain will be tensile or compressive (p ositive or negative) depending on the position of the layer with respect to the bending axis. The failure envelope for any force and mo ment resultants can be determined using the above procedure. To briefly illustrate the procedure, consider a two-layer textile under bending, with an x -curvature of 0.5 and a layer thic kness of 0.2. Then the total stress field is found by superposing scaled mu ltiples of the stress fields resulting from unit strain ( ij =1) and curvature ( ij =1) cases 1 2 2 0 1 1 15 0 5 0 x x x xij ij ij ij x ij x ijd (5.10)

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65 where the indicates that the stress field in the tensile layer will be calculated as the sum of the two terms in each equation, and conversely, stress in the compressive layer will be calculated as the difference of the two stress fields. As detailed earlier, failure from the total stress field in each la yer is then checked on an element-by-element basis to determine ove rall failure of the composite. Proveout and comparisons of this method will be shown after the next section. Implementation of the Qu adratic Failure Theory to Predict Strength for MultiLayer Textile Composites (QFT) The previously developed 27-term quadratic failure theory for textile composites, as determined from the single-layer DMM, can be implemented to predict failure for a multi-layer specimen. Once the original failure coefficients have been determined, no further FEM or experimental analysis will be needed. Implementation of this procedure is accomplished by adjusting the force and moment resultants applied to the multi-layer specimen to reflect the true stre ss state in each layer, on a laye r-by-layer basis. First, the mid-plane multi-layer strain and curvature ar e calculated from the applied macro-level force and moment resultant, along with the constitutive matrix representing the doublelayer ( DL ) material properties. ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [1M N D B B ADL o (5.11) The dual-layer midplane strain must now be modi fied to represent the actual strain state in each layer. ij o ijd (5.12)

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66 Note that in plate analysis, mid-plane curvat ure and layer-level curvature will always be the same, as curvature is always constant through-thickness for a given layup, regardless of thickness or number of layers (alth ough strains may show variation). The layer-level force and moment resultant are then calculated using this modified strain from Equation 5. 12, along with the single layer ( SL ) constitutive matrix. ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ SL SLD B B A M N (5.13) The adjusted force and moment resultants capture what is seen in each layer offset from the bending axis. These are then directly input to the quadrat ic failure theory of Equation 4.4 with coefficients as per Table 4-2 and 4-3 (as developed from one layer or RVE). Failure analyses are performed independently in each layer. This is to say that, the single layer force and moment resultants for each la yer must be independently calculated input to the quadratic failure theory (computations which can still be automated). Note that, via the above procedure, a pure moment result ant applied to the two-layer composite will correspond to both a force and moment resultant in each layer. Comparison of the Results of Multi-Layer Failure Analysis Methods Several cases are now presented which illust rate the relative effectiveness of the multi-layer analysis methods shown in the prece ding sections of this chapter. Direct FEM simulation provides the most accurate pr ediction of the stress field and failure envelope of the multi-layer textile. Thus the two techniques for predicting failure of a multi-layer composite without additional materi al characterization tests can be compared to this in order to estimate their accuracy. One method of comparison is to look dire ctly at the predicted stress field under several loading conditions. However, th ese point-by-point (or element-by-element)

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67 comparisons prove to be not the most eff ective summarization of accuracy. Comparison of a few points can yield an inaccurate sample of results that appear to differ greatly, even if the majority of the stress field compares quite closely. However, comparison of the calculated stresses at many (or all) poi nts and taking an average accuracy is not necessarily the best metric eith er, as standard deviation could potentially be large. Thus, certain portions of the stress field might be predicted very well, whereas certain portions might not be accurately predicted. In this case, an average of point-by-point stress deviation might appear small, but in fact such situation should not be considered as an accurate prediction. Furthermore, some pr edicted stress components at a given point might compare well to direct simulation, whereas other components do not, which opens the door for further ambiguity. This having b een said, in general, in a point-by-point comparison, it has been observe d that predicted stress fields generally show an average accuracy on the order of 90% w ith roughly 10% standard deviation for the two prediction methods (ADMM and QFT) as compared to direct simulation (DDMM). Comparison of predicted failure points (t he maximum allowable force and moment resultants under combined loading) proves to be the best and most germane method of comparing the multiple prediction methods. To this end, failure has been predicted for a variety of load cases based on the data from the direct simulation of the DDMM. The results of the ADMM and the QFT are then comp ared to this. Both methods are shown to compare well to the DDMM results, though use of the QFT is computationally faster and more practical once failure coefficients have been determined. Table 5-2 and Table 5-3 below show a comparison of the various met hods to calculate failu re for a multi-layer textile. As in Chapter 4, load cases are shown in terms of a Load Ratio ( ) defined as

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68 1F Fi i (5.14) As introduced in Chapter 4, by maintaining th e same load ratios, all predicted failure loads will maintain a single ratio with resp ect to DDMM failure points. Thus one ratio can characterize the congruence of these solutions. A ratio of one will imply complete agreement. A ratio less than one indicates a conservative failure prediction, and a ratio greater than one implies a nonconservative prediction. The last two failure prediction comparisons of Table 5-3 are shown in different terms in order to employ a more effec tive normalization which accommodates the incongruity of the magnitude of force result ants and moment result ants. In these two cases, failure points are found for which A M A Ncrit i crit i, (5.15) for an unknown value A at failure. The first case, indicated by a (+) in Table 5-2, is the solution for failure in the tensile layer, in which the applied moment generates tensile forces that accelerate failure. The second case, indicated by a (-) is the solution for failure in the compressive layer, in which case the applied moment offsets applied tensile force resultants. This approach also esse ntially solves for the factor by which load capacity changes when the additional load is applied. In other words, an A of 0.48 in the compressive layer implies that twice as mu ch force resultant may be applied to the compressive layer before failure, when comp ared to the failure load under no applied moment. Conversely, in the tensile layer, the A of 2.01 indicates that the additional load Mx accelerates failure such that the allowable Nx is halved (this becomes the limiting case for ultimate failure).

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69 Table 5-2: Example Load Cases to Determ ine the Accuracy of Multi-Layer Analysis Methods. Accuracy is Indicated by a Ratio as Compared to DDMM Results. DDMM ADMM Accuracy QFT Accuracy 1 1.35E+04 1.31E+04 0.97 1.30E+04 0.96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1.91E+04 2.01E+04 1.05 2.02E+04 1.06 1 1.91E+04 2.01E+04 2.02E+04 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4.23E+03 4.14E+03 0.98 4.18E+03 0.99 0 0 0 0 0 0 0 0 0 0 0 0 1 3.73E+03 3.52E+03 0.94 3.42E+03 0.92 1 3.73E+03 3.52E+03 3.42E+03 1 3.73E+03 3.52E+03 3.42E+03 0 0 0 0 0 0 0 0 0 0 0 0 The Adjusted DMM and the Adjusted QFT show 5.2% and 5.5% error, respectively over these load cases. Most often, this error is conservative in comparison to the direct FEM simulation.

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70 Table 5-3: Further Example Load Cases (In cluding Moment Resultants) to Determine the Accuracy of Multi-Layer Analysis Me thods. Accuracy is Indicated by a Ratio as Compared to DDMM Results. DDMM ADMM Accuracy QFT Accuracy 0 0 0 1 1.68E-03 1.60E-03 0.95 1.54E-03 0.92 0 0 0 0 0 1 2.59E-03 2.67E-03 1.03 2.72E-03 1.05 1 2.59E-03 2.67E-03 2.72E-03 0 N/A (+) 2.01 2.08 1.03 1.9 0.95 0 0 M/A 2.01 2.08 1.9 0.95 0 0 N/A (-) 0.48 0.44 0.92 0.42 0.88 0 0 M/A 0.48 0.44 0.42 0.88 0 0 Practical Examples to Illustrate Stren gth Prediction of a Two-Layer Textile Composite Plate In order to show the application of th e preceding failure prediction approaches to practical examples, design of a two-layer plai n weave-textile plate is considered. Also shown is an example of a closed-end thin-w alled pressure vessel. Classical analysis

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71 procedures are employed to determine the loads that are then input to the various failure analysis techniques. As shown in Figure 5-2, a uniform pressure is applied to a si mply-supported plate. Three different plate sizes, as shown in Table 5-4, are consid ered to explore the different mechanical regimes of varying width-to-thickne ss ratios and to cons ider a square versus rectangular geometry. Plate Th eory [36] is employed to dete rmine the loads at each point in the plate, which are then checked for failure. Once moments and curvatures (per unit pressure) have been determined from SDPT, failure in the plate is analyzed vi a several methods. From this, the maximum allowable pressure can be determined, and results for each method are compared. The most reliable method is direct simulation through the DDMM. This again provides a basis of comparison for the remaining methods. Figure 5-2: Schematic of th e Simply Supported Textile Pl ate under Uniform Pressure Table 5-4: Geometry of the Simply S upported Textile Plate under Uniform Pressure a b t Case 1 0.102 m 0.102 m 0.508 mm Case 2 0.0102 m 0.0102 m 0.508 mm Case 3 0.102 m 0.051 m 0.508 mm

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72 The second method represents a conventi onal approach, which employs failure analysis methods not developed for textile co mposites, and for which stress gradients are not considered. Classical anal ysis techniques are used to fi nd strains and stresses, which are compared to a conventional failure theory. The third method is the Quadratic Failure Theory developed in Chapter 4. The four th method is the aforementioned ADMM. For each case, the plate is discretized into a 21 by 21 point grid of points that are each checked for failure (441 total points). The conventional method is accomplished by determining the curvature per unit pressure at each point via the SDPT. Strain is then determined by Equation 5.12 as per classical laminated plate theory. Stress is calculated by approximating a stiffness matrix [ Q ] from the [ A ] matrix, that was determined via the DMM as indicated by Equation 5.16, and multiplying by the corresponding strain. It should be noted that this in itself an represent an improvement over conventiona l methods, as a stiffness matrix would generally be calculated from homogenized mate rial properties or estimations rather than from direct simulation or experiment. (However, unlike bendi ng properties, these methods can often be acceptable fo r in-plane stiffness properties). At Q (5.16) in which t is the thickness of a layer and A represents the in-plane stiffness matrix of a layer. This stress can then be compared to a maximum allowable stress via the Tsai-Wu Failure Theory (for which failure coefficien ts can be found via the DMM or experimental methods). The procedures for the direct simulation (DDMM) as well as the QFT and ADMM methods have been detailed above, which is not repeated. In these cases, the maximum

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73 allowable moment per unit pressure is found, and the maximum allowable pressure can then be compared. Note that in the dire ct two-layer DDMM, the stress field in both layers is treated as a whole, thus fa ilure is not calculated per layer. Tables 5-5 through 5-7 below tabulate th e maximum allowable pressure for each of the three geometries under consideration, for each of the four prediction methods. The critical pressure for the compressive as well as tensile side (limiting case) of the plate in bending are shown. The relativ e accuracy of prediction is indicated as a ratio with respect to the DDMM direct simulation. A ratio greater than one indicates a nonconservative prediction. Table 5-5: Maximum Pressure for the Text ile Plate of Figure 5-2 with the Case 1 Geometry of Table 5-14 as Predicted from Various Multi-Layer Analysis Methods Case 1 (a = b a/t = 200) pmax (+) (kPa) pmax (-) (kPa) Ratio/DDMM DDMM 19.5 n/a Conventional 24.0 27.0 1.23 QFT 20.5 22.7 1.05 ADMM 20.1 22.3 1.03 Table 5-6: Maximum Pressure for the Text ile Plate of Figure 5-2 with the Case 2 Geometry of Table 5-14 as Predicted from Various Multi-Layer Analysis Methods Case 2 (a = b a/t = 20) pmax (+) (kPa) pmax (-) (kPa) Ratio/DDMM DDMM 135.3 n/a Conventional 169.1 189.2 1.25 QFT 142.1 154.8 1.05 ADMM 139.4 153.5 1.03

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74 Table 5-7: Maximum Pressure for the Text ile Plate of Figure 5-2 with the Case 2 Geometry of Table 5-14 as Predicted from Various Multi-Layer Analysis Methods Case 3 (a/b = 2, a/t = 200) pmax (+) (kPa) pmax (-) (kPa) Ratio/DDMM DDMM 34.7 n/a Conventional 40.6 45.5 1.17 QFT 32.6 36.2 0.94 ADMM 33.7 37.1 0.97 For all three cases, the QFT and ADMM represent a significant improvement over the conventional approach. This is due to the presence of significant stress gradients across the thickness dimension of the RVE (as accounted for with the moment resultant matrix), contrary to common isostrain assumptions used in textile micromechanics or failure theory development. The relative accuracy of conventional methods increases somewhat for Case 3. The disparity between the conventio nal and DMM-based approaches, which include consideration of stre ss gradients, will diminish as the relative presence of stress gradients diminish w ith respect to other loads present. In general, the ADMM will be more accurate than the QFT, as the QFT is an approximation method which is slightly furthe r removed from the developmental data. Though both methods involve multi-layer a pproximations, the QFT must also approximate the DMM stress field data. For Ca se 1 versus Case 2, the agreement of the two mutli-layer analysis methods (ADMM or QFT) with the direct FEM simulation is similar, for both mechanical regimes. Alt hough the low aspect ratio plate is naturally able to withstand a much higher pressure, prediction accuracies are similar. These predictions are both non-conserva tive, though Tables 5-2 and 5-3 have shown that this is not generally true. For Case 3, these methods are conservative compared to the DDMM.

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75 In this case, at the failure point, there is a different load ratio (Mx = 2My, rather than the Mx = My of Case 1 and 2), thus a different portion of the failure space is being predicted. As most often seen in the results of Chap ter 3, the initial failure mode for these design cases is transverse failure of the fibe r tow, beginning in the tensile layer. This represents a fiber pull-apart initiation, or an intra-tow matrix cracking. For the unbiased (Mx = My) biaxial bending of Cases 1 and 2, failur e initiates in both fiber tows, as the transverse stresses will be equal for both tows. Tables 5-8 through 5-11 belo w further elucidate the failu re initiation of the design cases by illustrating the Factor of Safety (ratio of allowable pressure with respect to the failure point) at several point s on the plate. Only one quart er-section of the plates is considered, due to the symmetr y of the plates and of the resulting stress field. The bottom-rightmost cell represents the plate center. Position is defined as a fraction of the total width dimension, such that a position of (0 .5,0.5) represents the center of the plate. Although the maximum allowable pressure will differ, the di fference in th e distribution of factor of safety in Case 1 and 2 will be negligible and is thus co-tabulated. Case by case, both the QFT and conventiona l methods yield a similar distribution of factor of safety, although the QFT shows a greater variation in the center-to-edge Table 5-8: Case 1 and 2 F actor of Safety Across the Plate as Determined via the Conventional Approach Position 0 0.09 0.18 0.27 0.36 0.43 0.5 0 0.09 4.81 3.03 2.50 2.28 2.20 2.24 0.18 3.03 2.31 1.86 1.68 1.30 1.62 0.27 2.50 1.86 1.58 1.40 1.33 1.32 0.36 2.28 1.68 1.40 1.25 1.18 1.16 0.43 2.20 1.60 1.33 1.18 1.10 1.07 0.5 2.24 1.62 1.32 1.16 1.07 1.00

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76 Table 5-9: Case 1 and 2 Fact or of Safety Across the Plate as Determined via the QFT Position 0 0.09 0.18 0.27 0.36 0.43 0.5 0 0.09 7.99 4.81 3.81 3.36 3.04 3.07 0.18 4.81 3.03 2.31 1.99 1.71 1.75 0.27 3.81 2.31 1.86 1.58 1.35 1.33 0.36 3.36 1.99 1.58 1.40 1.18 1.14 0.43 3.04 1.71 1.35 1.18 1.10 1.06 0.5 3.07 1.75 1.33 1.14 1.06 1.00 Table 5-10: Case 3 Factor of Safety Across the Plate as Determined via the Conventional Approach Position 0 0.09 0.18 0.27 0.36 0.43 0.5 0 0.09 6.74 4.04 3.16 2.77 2.48 2.49 0.18 4.80 2.72 2.05 1.75 1.51 1.49 0.27 4.12 2.30 1.71 1.44 1.21 1.17 0.36 3.94 2.17 1.59 1.33 1.10 1.05 0.43 3.88 2.13 1.56 1.29 1.06 1.00 0.5 3.99 2.18 1.58 1.30 1.06 1.00 Table 5-11: Case 3 Factor of Safety Ac ross the Plate as Determined via the QFT Position 0 0.09 0.18 0.27 0.36 0.43 0.5 0 0.09 6.90 4.15 3.25 2.85 2.55 2.57 0.18 4.86 2.77 2.09 1.79 1.55 1.53 0.27 4.14 2.32 1.73 1.46 1.24 1.20 0.36 3.92 2.17 1.60 1.34 1.12 1.07 0.43 3.83 2.11 1.55 1.29 1.06 1.01 0.5 3.90 2.14 1.56 1.29 1.06 1.00 magnitude of factor of safety. Note that fact or of safety rises to infinity in the topmost row and leftmost column which represent the simply-supported edge of the plate, as the bending moment and curvature are theoretically zero here. Also note that the maximum pressure to which each factor of safety is normalized will be different for the different prediction methods. Along any radial path fr om center to edge of the plates, factor of

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77 safety is seen to be nearly inversely proportional to grid posit ion. This trend remains for all calculation methods, over all geometry cases. In contrast to the design of a textile plate under uniform pressure, there are other common design cases for which there would be no improvement in accuracy in employing the QFT or ADMM rather than c onventional methods. For example, in designing a thin-walled pressure vessel (see Figure 5-3 and Table 512), a biaxial stress state with negligible stress gr adients will exist. In this case, conventional methods will predict similar maximum allowable pressure when compared to the QFT or ADMM. Stresses and stress distributions are found fr om basic pressure ve ssel theory [37]. Force resultants and moment resultants (whi ch result from a slight radial stress distribution and will be quite small) can then be easily calculated from this result. Note that since there is no curvature, the multilayer analysis ADMM and adjustments to the QFT inputs (as detailed in previous sections ) will not be needed. Thus, this example serves as a comparison of the DDMM simulation and the QFT, contrasted to conventional methods, for a test case in which stress gradients are small. Results for this design case are shown in Table 5-13 below. Figure 5-3: Schematic of the Du al-Layer Textile Pressure Vessel

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78 Table 5-12: Geometry of the Textile Pressure Vessel radius (r) thickness (t) Case 4 10.2 mm 0.508 mm Table 5-13: Maximum Pressure for the Textil e Pressure Vessel of Figure 5-3 with the Geometry of Table 5-12 as Predicted from Various Multi-Layer Analysis Methods Case 4 (r/t = 20) pmax (MPa) Ratio/DDMM DDMM 1.89 Conventional 1.93 1.02 QFT 1.86 0.98 As expected, it is seen that predicted ma ximum allowable pressures are similar for all methods. Although there are stress gradients along the radial directio n, the variation is relatively small, thus the moment resultant that is present will be nearly negligible. These results will hold true for any th in-walled pressure vessel. Chapter Summary For analysis of textile com posites of an arbitrary number of layers, the methods of previous chapters have been adapted. Through the results direct FEM simulation, stiffness of a multi-layer text ile has been shown to be governed through an expression similar in form to that of the Parallel Axis Theorem. Fo r the plain-weave textile under investigation, this represents a roughly 20-fold increase in stiffness when thickness is increased from one to two layers. For streng th prediction, two analysis methods have been presented, again based upon and presente d in comparison to the results of direct FEM simulation. The adapted single layer Direct Micromechanics Method (ADMM) is based upon a correction to the stress field as determined by the DMM (see Chapter 3) commensurate with the layer-by-layer strains wh ich result from the offset from the axis

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79 of bending. A method for employing the quadrat ic failure theory (QFT, see Chapter 4) has also been developed. Without the need to modify the single-layer failure coefficients, the inputs to the QFT are adapted to represen t the true strain and curvature in each layer that is offset from the bending axis. Both the ADMM and QFT methods have been shown to predict multi-layer failure within roughly 5% accuracy. Several simple but significant design cases have been presented as a practical application of the methods presented in this dissertation. Multi-layer failure prediction methods have been shown to be sufficiently accurate, and the importance of the consideration of stress gradients in a common design situation is shown.

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80 CHAPTER 6 CONCLUSIONS AND FUTURE WORK In this paper, robust methods have been developed for predicting stiffness and strength of multi-layer textile composites, with techniques designed to address the difficulties that arise when considering a textile microstructure. Currently existing failure criteria for composite materials are genera lly developed for and based upon usage with unidirectional composite laminates. Though these theories may to some extent be applied in an adapted form to the analysis of textile composites, as has been shown herein, many inherent simplifying assumptions no longer apply. Given the increased complexity of analysis of textile composites, there are several outstanding issues with regards to textile composites that have been a ddressed in this research. One of the most important issues addressed here is a robus t model for prediction of strength. Though much attention has been given to the predic tion of stiffness, little work has focused upon strength prediction for textile composites. Additionally, conventional micromechanical models for textile composites assume that the state of stress is uniform over a di stance comparable to the dimensions of the representative volume element (RVE). Ho wever, due to complexity of the weave geometry, the size of the RVE in textile com posites can be large compared to structural dimensions. In such cases, severe non-uniformities in the stress state will exist, which conventional models do not account for. Me thods for including the consideration of stress gradients have been developed, and th e importance of such considerations has been demonstrated.

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81 Stiffness properties have been predicted for a plain-weave textile composite. The results indicate good agreement with expected values from literature and material supplier data. The constitutive matrices have been calculated directly from the micromechanics model without any assumptions as in traditional plate theories. The results are quite different from commonly em ployed approximations. By comparison to the direct micromechanics results of the D MM, conventional methods will misrepresent flexural stiffness values D11, D12, and D66 by as much as factors of 2.9, 1.1, and 0.7 respectively. The DMM results imply that there is no consistent relation between inplane and flexural properties, alt hough the two properties are related. Failure envelopes have been presented and the comparisons to and improvements over conventional methods have been shown. Under relatively simple loading conditions in which no stress gradients are present acr oss the RVE, the DMM failure envelope was shown to compare most closely with the Ts ai-Wu Failure Theory. The Maximum Stress Failure Theory and Maximum Strain Failure Theory were less close in comparison. Fiber pull-apart or failure of the transverse fiber tows was shown to be the dominant mode of failure. In limited instances of large biaxial st resses, failure of the interstitial matrix was seen to be the mode of initial failure. Further failure envelopes have been pres ented which illustrate the importance of consideration of stress gradients, and the in ability of conventiona l failure models to account for this load type. In these cases, traditional failure prediction methods can greatly overpredict the failure envelope. The presence of applied moment resultants [ M ], as would exist in cases of non-uniform load across the RVE, was shown to have a

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82 significant effect on the failure envelope. Thus its consideration, not covered in conventional failure models, can be critical. Based upon failure envelopes constructe d by analysis of the microstresses developed in a representative volume elemen t (RVE), alternate methods for predicting failure envelopes of a plain-w eave textile composite have been developed. A parametric ellipse-fitting scheme which accurately predicts trends in failure envelopes for a given failure space has been developed by analysis of failure ellipse parameters. This method for predicting failure envelopes was found to agree with DMM results to within a few percent. However, it is impractical in its implementation, and is limited to consideration of one particular failure space in which only th ree concurrent force or moment resultants may be considered at once. The method is us eful for a solid visual ization and lends itself to a firm understanding of simpler load cases. A second method involves development of a 27-term quadratic failure cr iterion to predict failure under general loading conditions. The quadratic failure criterion was found to ag ree with DMM results within an average deviation of 9.3%, but the method is more robus t in terms of its abil ity to predict failure from more complex loading cases. The methods thusfar have been further modi fied to accommodate analysis of textile composites of an arbitrary number of layers. Stiffness of a multi-layer textile has been shown to be governed through an expression simila r in form to that of the Parallel Axis Theorem. For strength prediction, two anal ysis methods have been presented. The adapted single layer Direct Micromechanic s Method (ADMM) is ba sed upon a correction to the stress field as determined by the DMM. A method for employing the quadratic failure theory (QFT) has also been developed. Both the ADMM and QFT methods have

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83 been shown to predict multi-layer failure wi thin roughly 5% accuracy This adaptation thus allows for methods for the accurate pr ediction of strength of a multi-layer textile with a minimum of characterization requireme nts based upon micromechanics of a single RVE. Finally, several design cases have been pres ented as a practical application of the methods presented in this disse rtation. Multi-layer failure prediction methods have been shown to be sufficiently accurate, and the importance of the consideration of stress gradients in a common design situation is shown. For a simple design case of a two layer textile pressure vessel, a state of biaxial fo rce resultant with negligible stress gradients exists. In this case, the OFT and conven tional methods predict a similar allowable pressure. In consideration of the design of several textile plates of varying geometries under uniform pressure, significant stress grad ients exist. Conventional methods will overpredict allowable pressures by 17% to 25%, whereas the DMM based QFT method will predict strength within 5% to 6% accuracy. Several suggestions are offered here fo r potential future work that may be completed to extend the current body of work, both in terms of further development and in terms of useful application. Incorporating a model of progressive failu re represents one potential issue for future consideration. After initial failure, a component may still retain some stiffness and load bearing capacity. Continued loading l eads to a progressive property loss as more and more of the constituent material becomes degraded. This can be simulated within the finite element micromechanical model by redefinition of the stiffness matrix (or

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84 redefinition of material propert ies) after single element failures. The simulation is then rerun, and additional element stiffness matr ices are appropriately recalculated as additional element failures occur. Incorporation of thermal stresses and inve stigation of the coefficient of thermal expansion is another potential avenue for fu rther development of the failure modeling. Due to mismatches between the coefficient of thermal expansion of constituent materials, thermal stresses can build up during manufacture or during operation. This makes inclusion of such effects critical to th e accuracy of a strength prediction model. Furthermore, textile composites have been shown to perform well compared to other composites at cryogenic temperatures. Thus i nvestigation of such thermal effects should be of considerable interest. In the current work, the plain-weave architecture has been used to develop and demonstrate an effective micromechanical methodology and failure theory for textile composites. The same methods can easily be us ed to investigate the mechanical behavior of other textile weave or braid patterns. In these cases, the RVE will be larger and more complicated, but the DMM approach remains the same. Further, given that other architectures are geometrically larger, it sta nds to reason that the importance of stress gradient effects as presented herein will be of even greater importance. The DMM could also be employed to perfor m a parametric study of the effect of weave architecture and geometry on mechanical behavior. For example, successive characterizations of the same weave type w ith varying two spacing, or tow undulation, etc. could provide a useful insight into the effect these parameters have on the stiffness

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85 and strength properties of a te xtile composite. This could also potentially lead to the ability to optimize the microarchit ecture to a specific application.

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86 APPENDIX PERIODIC BOUNDARY CONDITIONS The derivation of periodic boundary conditions for unit extension and unit shear is presented here. Unit curvature has been presented in the text in Chapter 2. Unit Extension From the definition of stra in, and utilizing integration c x z y x u x ux ) , ( 1 c z y u c a z y a u ) , 0 ( ) , ( a z y u z y a u ) , 0 ( ) , ( Unit Shear 1 2 1 21 Assuming, (,,)xyuv yx uvu yxy uxyzyc c z x u c b z b x u ) 0 ( ) , (2 1 b z x u z b x u2 1) 0 ( ) , ( Similarly, a z y v z y a v2 1) , 0 ( ) , (

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87 LIST OF REFERENCES 1. Cox, B.N. and Flanagan, G. Handbook of Analytical Methods for Textile Composites, NASA CR4750, Celveland, OH, 1997. 2. Tsai, S.W. and H.T. Hahn. Introduction to Composite Materials, Technomic Publishing Co, Lancaster, PA. 1980. 3. Dasgupta, A.S., S. Bhandarkar, M. Pecht, and D. Barkar. Thermoelastic Properties of Woven-Fabric Composites using Homoge nization Techniques, Proceedings of the American Society for Composites, Fi fth Technical Conference, Lansing, MI, 1990. p1001-1010. 4. Foye, R.I. Approximating the Stress Fi eld within the Unit Cell of a Fabric Reinforced Composite Using Repl acement Elements, NASA CR-191422, Cleveland, OH, 1993. 5. Naik, R.A. Analysis of Woven and Br aided Fabric Reinforced Composites, NASA CR-194930, Cleveland, OH, 1994. 6. Whitcomb, J.D. Three-Dimensional Stress Analysis of Plain Weave Composites, Composite Materials Fatigue and Fract ure (Third Volume), ASTM STP 1110, 1991. p417-438. 7. Ishikawa, T., and T.W. Chou, One-dimen sional Micromechanical Analysis of Woven Fabric Composites, AIAA Journal Vol 21, 1983. p1714-1721. 8. Ma, C.L., J.M. Yang, and T.W. Chou, Ela stic Stiffness of Three-dimensional Braided Textile Structural Composites, Composite Materials: Testing and Design (7th Conference), ASTM STP 893, 1986. p404-421. 9. Raju, I.S., R.L. Foye, and V.S. Avva, A Review of Analytical Methods for Fabric and Textile Composites, Proceedings of the Indo-U.S. Workshop on Composite Materials for Aerospace Applications, Bangalore, India, Part I, 1990. p129-159. 10. Whitcomb, J.K. and K. Srirengan, Effect of Various Approximations on Predicted Progressive Failure in Plain Weave Composites, Composite Structures 34, 13-20, 1996. 11. Lomov, S.E. et al, Textile Com posites: Modelling Strategies, Composites: Part A 32, 1379-1394, 2001.

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88 12. Hochard, C., Aubourg, P.A., and Charles, J.P., Modelling of the Mechanical Behavior of Woven-Fabric CFRP Laminates up to Failure, Composites Science and Technology 61, 221-230, 2001. 13. Huang, Z.M. and Ramakrishna, S., Modeling Inelastic and Streng th Properties of Textile Laminates: A Unified Approach, Composites Science and Technology 63, 445-466, 2002. 14. Bigaud, D. and Hamelin, P., Stiffness and Fa ilure Modelling of 2D and 3D Textile Reinforced Composites by Means of Imbr icate Type Elements Approaches, Computers and Structures 80, 2253-2264, 2002. 15. Woo, K. and Whitcomb, J.D., A Post Proce ssor Approach for Stress Analysis of Woven Textile Composites, Composites Science and Technology 60, 693-704, 2000. 16. Tang, X. and Whitcomb, J.D., General Te chniques for Exploiting Periodicity and Symmetries in Micromechanics An alysis of Textile Composites, Journal of Composite Materials vol 37, 13, 2003. 17. Cox, B.N., and M.S. Dadkhah, A Binary Model of Textile Composites: IFormulation, Acta Metallurgica et Materiala 42, 10, 3463, 1994. 18. Yang, Q. and Cox, B.N., Predicting Local Strains in Textile Composites Using the Binary Model Formulation, Proceedings of the ICCM 2003, San Diego, CA, July, CD-ROM, 2003. 19. Bogdanovich, A.E. and Pastore, C.M., M aterial-Smart Analysis of TextileReinforced Structures, Composites Science and Technology 56, 291-309, 1996. 20. Bogdanovich, A.E., Multiscale Predictive Analysis of 3-D Woven Composites, SAMPE 35th International Technical Conf erence, Dayton, OH, September 2003, CD-ROM. 21. Quek, S.C., Waas, A., et al. Compre ssive Response and Failure of Braided Textile Composites: Part 2Computations, International Journal of Nonlinear Mechanics 39, 649-663, 2004. 22. Quek, S.C., Waas, A., et al. Compre ssive Response and Failure of Braided Textile Composites: Part 1Experiments, International Journal of Nonlinear Mechanics 39, 635-648, 2004. 23. Cox, B.N., M.S. Dadkhah, and W.L. Morris, Failure Mechanis ms of 3D Woven Composites in Tension, Compression, and Bending, Acta Metallurgica et Materiala 42, 12, 3967-84, 1994.

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89 24. Pochiraju, K., T.W. Chou, and B.M. Shah, Modeling Stiffness and Strength of 3D Textile Structural Composite s, Proceedings of the 37th Joint AIAA/ ASME/ ASCE/ AMS/ ASC Structures, Structural Dynamics, and Materials Conference, Salt Lake City, UT, p91-105, 1996. 25. Dadhkah, M.S., W.L. Morris, T. Kniveton, and B.N. Cox, Simple Models for Triaxially Braided Composites, Composites 26, 1995. p91-102. 26. Fleck, N.A. and P.M. Jelf, Deformation and Failure of a Carbon Fiber Composite under Combined Shear and Transverse Loading, Acta Metallurgica et Materiala, 43, 8, 3001-7. 1995. 27. Swanson, S.R. and L.V. Smith, Multiaxial Stiffness and Strength Characterization of 2D Braid Carbon/Epoxy Fiber Compos ites, Mechanics of Textile Composites Conference, Hampton, VA, C.C. Poe, ed. NASA Conference Publication 3311. p126-138, 1995. 28. Karkkainen, R.L. and B.V. Sankar, "A Direct Micromechanics Method for Failure Initiation of Plain Weave Textile Composites," Composites Science and Technology 66, p137-150, 2006. 29. Zhu, H., B.V. Sankar, and R.V. Marrey, Eva luation of Failure Criteria for Fiber Composites Using Finite Element Micromechanics, Journal of Composite Materials 32, 8, 1998. p766-782. 30. Marrey, R.V. and B.V. Sankar, Micromech anical Models for Textile Structural Composites, NASA CR198229. Cleveland, OH, 1995. 31. Sankar, B.V. and R.V. Marrey, "Analytical Method for Micromechanics of Textile Composites," Composites Scienc e and Technology 57, 6, p703-713. 1997. 32. Marrey R.V. and B.V. Sankar, "A Microm echanical Model for Textile Composite Plates," Journal of Composite Materials 31, 12, p1187-1213. 1997. 33. Carvelli V. and C. Poggi, A Homoge nization Procedure for the Numerical Analysis of Woven Fabric Composites, Composites Part A: Applied Science and Manufacturing 32, 1425-1432, 2001. 34. Gibson, R.F. Principles of Compos ite Material Mechanics McGraw-Hill, Inc, New York, 1994. 35. Whitcomb, J.D., C.D. Chapman, and K. Sr irengan, Analysis of Plain-Weave Composites Subjected to Flexure, Mechanics of Composite Materials and Structures 5:41-53, 1998.

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90 36. Whitney, J.M. Structural Analysis of La minated Anisotropic Plates Technomic Publishing Co, Lancaster, PA, 1987. 37. Norton, R.L. Machine Design: An Integrated Approach Prentice-Hall Inc, Upper Saddle River, NJ, 1996.

PAGE 103

91 BIOGRAPHICAL SKETCH Ryan Karkkainen was born in Pensacola, Florida, in 1975. He grew up predominantly in the subtropical climes of Baton Rouge, Louisiana, and received his Bachelor of Science in mechanical engineer ing from Louisiana State University in May 1998. From there he proceeded to Knoxvill e, Tennessee, in successful accomplishment of a Master of Science in mechanical engine ering in May 2000, with research in the area of processing and mechanics of polymer com posite materials. Karkkainen then moved on to Detroit, Michigan, where he was empl oyed for several years as a crash safety research engineer at Ford Mot or Company. In this capacity his job duties involved work both in finite element method crash simulations as well as in experime ntal investigations of the applicability of composite materials to advanced lightweight crash structures. After completion of the current body of work at the University of Florida in the area of micromechanics of textile composite structures, Karkkainen will begin work at the Army Research Laboratory in Aberdeen, Maryland, to contribute his efforts to the development of national defense projects.


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STRESS GRADIENT FAILURE THEORY FOR TEXTILE STRUCTURAL
COMPOSITES


















By

RYAN KARKKAINEN


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


2006

































Copyright 2006

by

Ryan Karkkainen

































This document is dedicated to everyone curious enough to ask the questions, persistent
enough to find the answers, and honest enough to respect the truth always.
















ACKNOWLEDGMENTS

I offer my sincere thanks to my committee and colleagues for their kind and

invaluable assistance. Further thanks are due to the Army Research Office and Army

Research Laboratory for their gracious funding of this project, and for their feedback and

assistance.
















TABLE OF CONTENTS

page

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

LIST OF TABLES ....................................................... ............ .............. .. vii

LIST OF FIGURES ......... ......................... ...... ........ ............ ix

ABSTRACT ........ .............. ............. ...... ...................... xi

CHAPTER

1 INTRODUCTION ............... ................. ........... ................. ... ..... 1

2 APPROACH ..................................... ................................ .......... 10

Finite Element M icromechanical M ethod ...................................... ............... 10
Direct Micromechanics Method for Failure Analysis ..............................................19
Phenom ecological Failure Criteria................................... ............................. ....... 21

3 STIFFNESS AND STRENGTH DETERMINATION ...........................................25

Stiffn ess P rop erties ........................................................................... .................... 2 5
Strength Properties.............................................................. .. ....28
M icrostress Field Contour Plots ................... ... ................... ............... ... 34
Inspection of Failure Envelopes in Additional Stress Spaces ..................................35
Design of Experimental Verification ................................................... .......... 38
Chapter Sum m ary ........................ .................... .. .. .... ........ ......... 40

4 PREDICTION OF FAILURE ENVELOPES..........................................................41

A Parametric Approach to Predicting Failure Envelopes for a Given Stress Space ..41
Development of a Quadratic Failure Criterion to Predict Failure for General
L o a d in g .............. .......... ................................................................ ............... 4 6
Optim ization of Failure Coefficients ...................................................................... 53
C chapter Sum m ary ............................................ .. .. .... ........ .... .. ... 57

5 M ULTI-LAYER ANALYSIS .............................................................................58

Stiffness Prediction of Multi-Layer Textile Composites................ .............. ....59
Strength Prediction of Multi-Layer Textile Composites ................. ............. .....61









Direct FEM Simulation of Multi-Layer Textile Composites Using the DMM
for Failure Analysis ("DDMM") ............................................................ 62
Adaptation of the Single Layer DMM Results to Predict Strength for Multi-
Layer Textile Composites ("ADM M ")........................................................ 63
Implementation of the Quadratic Failure Theory to Predict Strength for
M ulti-Layer Textile Composites ("QFT") ......................................................65
Comparison of the Results of Multi-Layer Failure Analysis Methods ..............66
Practical Examples to Illustrate Strength Prediction of a Two-Layer Textile
C om posite Plate ....................... .............. ............... .... ....... 70
C chapter Sum m ary .............................................. ... .... ........ ......... 78

6 CONCLUSIONS AND FUTURE WORK..............................................................80

APPENDIX: PERIODIC BOUNDARY CONDITIONS ...............................................86

L IST O F R E F E R E N C E S ........................................................................ .....................87

BIO GRAPH ICAL SK ETCH .................................................. ............................... 91
















LIST OF TABLES


Table p

2-1 RVE D im tensions ......... ..... .. ........................ ................. ...... 12

2-2 Periodic Displacement Boundary Conditions .................................. ............... 17

3-1 Fiber Tow and M atrix M material Properties............................ .. .......... ........ 27

3-2 Stiffness Properties for Plain W eave Textile Plate ........................................ ......27

3-3 Fiber Tow and Matrix Failure Strength Properties (MPa) .......................................28

4-1 Strength Values for Independent Load Conditions ................................................48

4-2 Normalized Failure Coefficients C, for Quadratic Failure Equation. (Coefficient
Cmn is in mth Row and nth Colum n)..................................... .......................... 49

4-3 Normalized Failure Coefficients D, for Quadratic Failure Equation ....................... 49

4-4 Comparison of Quadratic Failure Equation Predictions with DMM Results. Test
Cases Include 4 Populated Load Terms F..................... ...............51

4-5 Comparison of Quadratic Failure Equation Predictions with DMM Results. Test
Cases Include 5 Populated Load Terms F..................... ...............52

4-6 Comparison of Quadratic Failure Equation Predictions with DMM Results. Test
Cases Include 6 (Fully Populated) Load Terms F............... ..................53

4-7 Optimized Failure Coefficients C, for Quadratic Failure Equation. (Coefficient
Cmn is in mth Row and nh Colum n)..................................... .......................... 54

4-8 Optimized Failure Coefficients D, for Quadratic Failure Equation .......................55

4-9 Data to Indicate Results of Coefficient Optimization ...........................................55

5-1 Summary of the Various Methods Employed in Multi-Layer Strength Analysis....61

5-2 Example Load Cases to Determine the Accuracy of Multi-Layer Analysis
Methods. Accuracy is Indicated by a Ratio as Compared to DDMM Results........69









5-3 Further Example Load Cases (Including Moment Resultants) to Determine the
Accuracy of Multi-Layer Analysis Methods. Accuracy is Indicated by a Ratio
as Compared to DDM M Results. ........................................................................... 70

5-4 Geometry of the Simply Supported Textile Plate under Uniform Pressure............71

5-5 Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 1 Geometry
of Table 5-14 as Predicted from Various Multi-Layer Analysis Methods...............73

5-6 Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 2 Geometry
of Table 5-14 as Predicted from Various Multi-Layer Analysis Methods ..............73

5-7 Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 2 Geometry
of Table 5-14 as Predicted from Various Multi-Layer Analysis Methods ..............74

5-8 Case 1 and 2 Factor of Safety Across the Plate as Determined via the
C onv entional A approach ......................................... .............................................75

5-9 Case 1 and 2 Factor of Safety Across the Plate as Determined via the QFT.........76

5-10 Case 3 Factor of Safety Across the Plate as Determined via the Conventional
A p p ro ach ............................................................................. 7 6

5-11 Case 3 Factor of Safety Across the Plate as Determined via the QFT...................76

5-12 Geometry of the Textile Pressure Vessel ............. .............................................. 78

5-13 Maximum Pressure for the Textile Pressure Vessel of Figure 5-3 with the
Geometry of Table 5-12 as Predicted from Various Multi-Layer Analysis
M methods ............................................................... ..... ..... ......... 78
















LIST OF FIGURES


Figure pge

1-1 A Schematic Illustration of Several Common Weave Patterns Employed with
Textile Composites. A Box Indicating the Unit Cell Borders the Smallest
Repeatable Geometry Element for Each Pattern ............... .............. .....................3

1-2 A Schematic Illustration of Several Common Braid Patterns Employed with
Textile Composites. A Box Indicating the Unit Cell Borders the Smallest
Repeatable Geometry Element for Each Pattern ............... .............. .....................3

1-3 A Schematic Illustration of Several Common 3D Weave Patterns Employed
with Textile Composites ......................................... .... ........... .. .. ......... 4

2-1 Example Load Cases to Illustrate the Importance of Including Moment Terms in
the A analysis of a T extile R V E ....................................................... .....................12

2-2 RVE Geometry of a Plain Weave Textile Composite........................... ...........13

2-3 Flowchart for Failure Analysis Using the Direct Micromechanics Method ...........20

3-1 Comparison of DMM Failure Envelopes with Common Failure Theories..............29

3-2 Effect of Bending Moment on the Failure Envelope.. .......................................... 31

3-3 Effect of Micro-Level Tow Failure Theory on the DMM Failure Envelope ...........32

3-4 Effect of Changing the Definition of Micro-Level Failure on the Failure
E n v e lo p e ...................................................................... 3 3

3-5 Stress Contours for Plain-Weave Fiber Tows in Uniaxial Extension......................34

3-6 DMM Failure Envelopes for Biaxial Loading with Multiple Constant Moment
Resultants. For Illustration, an Ellipse is Fit to Each Data Set Using a Least-
S qu areas M eth o d .................................................... ................ 3 6

3-7 DMM Failure Envelopes for Biaxial Bending with Constant Applied Twisting
M om ent. .......................................................... ................ 3 7

3-8 DMM Failure Envelopes with Force and Moment Resultants for Constant
A applied Shear. ..................................................... ................. 37









3-9 Schematic of the Specimen for Off-Axis Uniaxial Testing. ............................39

3-10 Critical Force Resultant under Uniaxial Loading as Calculated via Various
F ailu re T h eories.............................................................................. ............... 39

4-1 Effect of Applied Moment Resultant on the Major Axis of Elliptical Failure
E envelopes. ...........................................................................42

4-2 Effect of Applied Moment Resultant on the Minor Axis of Elliptical Failure
E envelopes. ...........................................................................43

4-3 Effect of Applied Moment Resultant on the Center Point Coordinates (uo, Vo) of
Elliptical Failure Envelopes. ............................................ ............................ 43

4-4 Failure Envelopes Predicted with the Parametric Approach as Compared to
DMM Results (Applied Moment Resultant of 0.65 Critical Moment)....................45

4-5 Failure Envelopes Predicted with the Parametric Approach as Compared to
DMM Results (Applied Moment Resultant of 0.9 Critical Moment)..................... 46

4-6 DMM Failure Envelopes for Biaxial Bending with Constant Applied Twisting
Moment as Compared to the Quadratic Failure Theory Predictions......................49

4-7 DMM Failure Envelopes with Force and Moment Resultants for Constant
Applied Shear as Compared to the Quadratic Failure Theory Predictions .............50

5-1 Schematic Illustration of the Single-Layer Strain and Curvature Stress Fields
(As Found via the DMM) That Must Be Superposed in Calculation of the Total
Stresses Resulting from M ulti-Layer Bending..................... .. .. .................. 64

5-2 Schematic of the Simply Supported Textile Plate under Uniform Pressure ............71

5-3 Schematic of the Dual-Layer Textile Pressure Vessel .................. ..............77















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

STRESS GRADIENT FAILURE THEORY FOR TEXTILE STRUCTURAL
COMPOSITES

By

Ryan Karkkainen

May 2006

Chair: Bhavani Sankar
Major Department: Mechanical and Aerospace Engineering

Micromechanical methods for stiffness and strength prediction are presented, the

results of which have led to an effective failure theory for prediction of strength.

Methods to account for analysis of multi-layer textile composites are also developed.

This allows simulation of a single representative volume element (RVE) to be applicable

to a layup of an arbitrary number of layers, eliminating the need for further material

characterization. Thus a practical tool for failure analysis and design of a plain weave

textile composite has been developed. These methods are then readily adaptable to any

textile microarchitecture of interest.

A micromechanical analysis of the RVE of a plain-weave textile composite has

been performed using the finite element method. Stress gradient effects are investigated,

and it is assumed that the stress state is not uniform across the RVE. This is unlike most

models, which start with the premise that an RVE is subjected to a uniform stress or

strain. For textile geometries, non-uniform stress considerations are important, as the









size of a textile RVE will typically be several orders of magnitude larger than that of a

unidirectional RVE. The stress state is defined in terms of the well-known laminate

theory force and moment resultants [N] and [M]. Structural stiffness coefficients

analogous to the [A], [B], [D] matrices are defined, and these are computed directly using

the Direct Micromechanics Method (DMM), rather than making estimations based upon

homogenized properties.

Based upon these results, a robust 27-term quadratic failure criterion has been

developed to predict failure under general loading conditions. For multi-layer analysis,

the methods are adapted via three techniques: direct simulation of a multi-layer

composite, an adjustment of the data output from single-layer FEM simulation, and an

adjustment of the quadratic failure theory (without the requirement of determining a new

set of failure coefficients). The adjusted single-layer data analysis and the adjusted

quadratic failure criterion show 5.2% and 5.5% error over a variety of test cases.

The entire body of work is then applied to several practical examples of strength

prediction to illustrate their implementation. In many cases, comparisons to conventional

methods show marked improvements.














CHAPTER 1
INTRODUCTION

Though composites in general are not a new material, continuing advancements in

constituent materials, manufacturing techniques, and microstructures require that

considerable amounts of research be devoted to the study of composite mechanics. The

associated knowledge base is much smaller when compared to more conventional

materials, such as metals or ceramics. Composites have yet to be absorbed into

widespread use across multiple industries, limiting their economy and familiarity. Much

remains to be developed in the way of design methodologies and effective employment in

optimized structural applications.

By far the two most commonly employed and studied composite materials are

randomly oriented chopped-fiber composites, as well as laminated polymer composites

with embedded unidirectional carbon, glass, or kevlar fibers. Such composites have

received a great amount of treatment in the literature, and much exists in the way of stress

analysis techniques, and effective prediction of stiffness, fatigue life, strength, and other

such mechanical analyses.

Textile composites are a subgroup of composite materials that are formed by the

weaving or braiding of bundles of fibers (called tows or yams), which are resin-

impregnated and cured into a finished component. Though heavily adapted, in some

sense they draw upon traditional textile weaving processes and fabrication machines such

as mandrels and looms, akin to those of the textile clothing industries. Much of the

terminology of structural textile composites draws upon this classical sense as well.









The value of textile composites stems from many advantages, such as speed and

ease of manufacture of even complex components, consequent economy compared to

other composite materials, and out-of-plane reinforcement that is not seen in traditional

laminated composites. Further, textile composites do not lose the classically valued

advantage that composite materials possess over their metal or traditional counterparts, in

that textile composites have an inherent capacity for the material itself to be adapted to

the mechanical needs of the design. This is to say that the strength and stiffness of the

material can be oriented in needed directions, and no material weight is wasted in

providing reinforcement in unnecessary directions. For a conventional laminated

composite, this is accomplished by oriented stacking of layers of unidirectional resin-

impregnated fibers, such that fibers are aligned with any preferred loading axes. A textile

composite may also be so adapted by several methods, such as unbalanced weaves. The

woven fiber tows in a preferred direction may be larger (containing more constituent

fibers per tow) than in other directions. Also, an extremely diverse set of woven or

braided patterns may be employed, from a simple 2D plain weave to an eight-harness

satin weave or a 3D orthogonal weave pattern, any of which may exhibit a useful bias in

orientation of material properties. Figure 1-1 through Figure 1-3 [1] illustrate some of

the more common of these patterns.

The economy of textile composites arises mainly from the fact that manufacturing

processes can be highly automated and rapidly accomplished on loom and mandrel type

machinery. This can lead to easier and quicker manufacture of a finished product, though

curing times may still represent a weak link in the potential speed of manufacture.

























a) Plain Weave


c) FiveH-alame Satin Weave


d) Eight-Harness Satin Weave


Figure 1-1. A Schematic Illustration of Several Common Weave Patterns Employed with
Textile Composites. A Box Indicating the Unit Cell Borders the Smallest
Repeatable Geometry Element for Each Pattern.


a] Ix Bias Braid


b) lx1 Triaxial Braid. Axial Yarns at c) Ixi Triaxial Braid. Axial Yarns at
Altemate Crossovers Each Crossover


Figure 1-2. A Schematic Illustration of Several Common Braid Patterns Employed with
Textile Composites. A Box Indicating the Unit Cell Borders the Smallest
Repeatable Geometry Element for Each Pattern.


Unit
Cell


b) Crows-Foot Satin Weave










4 Filler (Weft)
StuflEr (Straight Warp)

/'\ Warp Weaver


a) Through-Thickness Angle Inltrlock

Surface Warp Weaver Body Warp







b) Layer-Layer Angle Intedrlck
Straight-Interlacing Structure


dl Orthgonal Interlock









c) Layer-t-Layer Angle Interlock
Wavy-hmrlacing Structre


Figure 1-3. A Schematic Illustration of Several Common 3D Weave Patterns Employed
with Textile Composites.

The out-of-plane reinforcement provided by textile composites comes by virtue of

the fact that the constituent weave patterns lead inherently to undulation and interlacing

of the woven fiber tows, which become oriented out of plane. A 2D weave will typically

exhibit tow undulation varying from 0 to 15 degrees out of plane. A 3D weave such as

an orthogonal interlock may actually have fiber tows directly aligned in the out of plane

direction.

Naturally, there are also several tradeoffs and disadvantages associated with

laminated and textile composites. The most significant added disadvantage that textile

composites possess may be the increased complexity of mechanical analysis. This is

directly owing to the undulation inherent to the weave or braid patterns, the complexity

of microstructure, as well as the multiscale nature of the textile microstructure. The fiber

tows woven together and embedded in a bulk polymer matrix are the most representative









microstructure, but the tows themselves have an inherent microstructure of aligned fibers

and intra-tow polymer matrix.

Given this increased complexity of analysis, there are several outstanding issues

with regards to textile composites. One of the most important issues, and the issue which

is to be addressed in the current body of research, is a robust model for prediction of

strength.

Current failure theories are generally developed for unidirectional composites and

do not capture the unique characteristics of textile composites. Though these theories

may to some extent be applied in an adapted form to the analysis of textile composites,

many inherent simplifying assumptions will no longer apply, and in general such

techniques will not be suitable to the increased complexity intrinsic to textile geometry.

Even at the micro scale, textile composites maintain a relatively complicated

microstructure. Even under simple loading conditions, a textile microstress state will be

shown to be quite complex, and elastic constants are non-uniform due to the waviness of

a woven fiber tow. Laminate analysis, property homogenization, and other common

approaches will no longer apply. Thus, current designs of textile structures will not be

optimized for maximum damage resistance and light-weight.

Conventional micromechanical models for textile composites assume that the state

of stress is uniform over a distance comparable to the dimensions of the representative

volume element (RVE). However, due to complexity of the weave geometry, the size of

the RVE in textile composites can be large compared to structural dimensions. In such

cases, severe non-uniformities in the stress state will exist, and conventional models may

fail. Such stress gradients also exist when the load is applied over a very small region, as









in static contact or foreign object impact loading, and when there are stress concentration

effects such as open holes in a structure.

Although micromechanical models have been successfully employed in predicting

thermo-elastic constants of fiber-reinforced composite materials, their use for strength

prediction in multiaxial loading conditions is not practical, as computational analysis

must be performed in each loading case. Thus phenomenological failure criteria are still

the predominant choice for design in industry. There are three major types of

engineering failure criteria for unidirectional composite materials: maximum stress

criterion, maximum strain criterion, and quadratic interaction criterion, such as the Tsai-

Hill and Tsai-Wu failure theories [2].

Most of the micromechanical modeling work done thus far has focused on

predicting thermo-mechanical properties [3-6]. To facilitate the use of textile composites

in lightweight structures, it is required to have a lucid understanding of failure

mechanisms, and design engineers must have an accurate and practical model for

prediction of failure stress. Most of the current analytical and numerical methodologies

developed to characterize textile composites [7-14] assume that the textile is a

homogeneous material at the macroscopic scale.

Finite element analysis of initial failure of a plain weave composite [10] has shown

that failure due to inter-tow normal stresses are the predominant mode of failure, and

there is generally little or no damage volume of the bulk matrix between tows. This work

is extended to a thorough investigation of progressive failure analysis under axial

extension using several different property knockdown schemes. This has shown stiffness

losses on the order of 40% after initial failure. More recently this work has been









extended to include the capability for more detailed stress fields in the RVE under

investigation [15], and techniques have been developed to minimize required

computation times by employing boundary conditions based on thorough exploitation of

symmetry and periodicity of RVE geometry [16].

The Binary Model [17, 18] allows for quick and efficient analysis of any textile

weave of interest. It has been shown to provide for accurate prediction of stiffness

properties. Further, it is robust and readily adaptable to provide insight into effects of

alteration of parameters such as tow waviness, tow misalignment, varying weave

architectures, etc. This technique does not yield a detailed map of RVE stress fields or

allow for cross-sectional variation of tow geometry, as the fiber tow is simulated as an

embedded 1-D line element with representative material properties. Thus some micro-

level detail is lost to provide for computational efficiency and macro-level representation.

The Mosaic Model and its adaptations [7, 19, 20] represent a textile composite

RVE as a collection of homogenized blocks, each with unidirectional composite or

matrix properties. These blocks are then assembled to represent the weave geometry

under consideration. In this way, classical laminate plate theory can be used to determine

the global stiffness matrix of the RVE. For macroscopically homogeneous load cases,

good agreement has been shown with experimental data, including three-dimensional

weave geometries.

Effective prediction of compressive strength of braided textile composites using a

detailed FEM micromechanical model has been performed [21], which shows good

comparison to experimental results in a parallel study [22]. A detailed 3D solid model

was formed to exactly model a 2D triaxially braided composite RVE. Biaxial loading is









considered in both the experimental and computational analyses. Buckling analysis has

been performed, and the effects of tow waviness and microarchitecture on the

compressive strength are shown.

For uniaxially loaded textile composites, consistent but optimistic strength

estimates have been made by comparing the strength of fiber tows with the predicted

stresses in the fiber yarns that are aligned with the loading axis [23-25]. The off-axis

tows are given little consideration, without much effect on the outcome, as they play little

part in such uniaxial loading cases. Multi-axial loading presents an obvious escalation in

modeling complexity. The failure envelope for combined transverse tension and in-plane

shear has been presented as an ellipse [26], according to quadratic strength rules

developed for unidirectional composites. Further, proposals for multiaxial loadings

submit that axial strain in the textile geometry should be compared to a critical value of

tow strain, analogous to a first-ply failure criterion for unidirectional composites [27].

A previous study [28] extended a method, known as the Direct Micromechanics

Method [29] (DMM), to develop failure envelopes for a plain-weave textile composite

under plane stress in terms of applied macroscopic stresses. In this study, it was assumed

that the state of stress is uniform across the RVE. The micro scale stresses within the

RVE were computed using finite element methods. The relation between the average

macrostress and macrostrains provides the constitutive relations for the idealized

homogeneous material. The microstresses are used to predict the failure of the yam or

matrix, which in turn translates to failure of the textile composite.

In the current research, micromechanical finite element analysis is performed to

determine the constitutive relations and failure envelope for a plain-weave graphite/epoxy









textile composite. The model is based upon the analysis of an RVE, which is subjected to

force and moment resultants of classical laminate theory. Thus there is no assumption

about the uniformity of an applied load or strain, as any load can be represented by a

combination of force and moment resultants. The micro-scale stresses within the RVE

are computed using the finite element method. The relation between the average

macrostress and macrostrains provides the constitutive relations for the material. Thus

constitutive characterization matrices [A], [B], [D] are found directly from

micromechanics. The microstresses are also used to predict the failure of the yar or

matrix, which in turn translates to failure of the textile composite. Using the DMM, the

failure envelope is developed for in-plane force resultants, with and without applied

moment resultants. No currently accepted failure criteria exist that may be used

explicitly for the analysis of textile composites. Thus the methods and results employed

herein are used to develop phenomenological failure criteria for textile composites. The

results are compared to conventional methods that are not specifically developed for the

analysis of textile composites, as a basis for evaluation.














CHAPTER 2
APPROACH

Micromechanical finite element analysis is performed to determine the constitutive

relations and failure envelope for a plain-weave graphite/epoxy textile composite. The

model is based upon the analysis of an RVE, which is subjected to force and moment

resultants of classical laminate theory. Thus there is no assumption about the uniformity

of an applied load or strain, as any load can be represented by a combination of force and

moment resultants. The micro-scale stresses within the RVE are computed using the

finite element method. The relation between the average macrostress and macrostrains

provides the constitutive relations for the material. Thus constitutive characterization

matrices [A], [B], [D] are found directly from micromechanics. The microstresses are

also used to predict the failure of the yarn or matrix, which in turn translates to failure of

the textile composite.

Finite Element Micromechanical Method

In the current study, stress gradient effects are investigated, and it is assumed that

the stress state is not uniform across the RVE. This represents an extension of the

micromechanical models used to predict the strength of textile composites [28-32]. The

stress state is defined in terms of the well-known laminate theory load matrices [N] and

[M] which describe force and moment resultants. Furthermore, structural stiffness

coefficients analogous to the [A], [B], [D] matrices are defined. In this approach, these

structural stiffness coefficients are computed directly from the micromechanical models,

rather than making estimations based upon the homogeneous Young's modulus and plate









thickness. Accordingly, individual unit strains and unit curvatures can be applied to the

micromechanical finite element model, and the resulting deformations are used to define

the stiffness coefficient matrices. Conventional models essentially neglect the presence

of [M] terms that result from non-uniformity or gradients in applied force resultants, thus

assuming a uniform stress state for which only the [N] matrix is populated. The

additional analysis of the [M] term includes information about the distribution, or

gradient, of a non-uniform load. This can greatly increase the ability of a failure model to

accurately predict failure for load cases in which such effects may well be predominant,

such as in thin plates, concentrated loading, or impact loading.

The significance of including the analysis of moment terms is further illustrated in

Figure 2-1. The moment term describes the distribution, not only the magnitude, of

applied loading. Depending on the stress state of an RVE, an analysis incorporating

stress gradient effects and inclusion of the consideration of applied moment could be of

critical importance. In Figure 2-la, the force resultant (N) is non-zero, but the uniform

loading results in zero moment resultant (M). However, in load cases such as Figure 2-1b

and 2-1c, the non-uniformity of applied loading leads directly to an appreciable moment

term, which must be included in the analysis. In fact, in some cases it is possible that the

net force resultant is zero while the effective moment resultant is non-zero, in which case

conventional analysis techniques cannot be employed.

The present micromechanical analysis of a plain-weave textile composite is

performed by analyzing the representative volume element (RVE) using the finite

element method. A typical weave-architecture has been selected and this RVE is detailed

in Table 2-1 and also in Figure 2-2. This architecture was chosen from a literature source











N O
M=O




N O
M*O



N=0
M*O


Figure 2-1. Example Load Cases to Illustrate the Importance of Including Moment Terms
in the Analysis of a Textile RVE. Distributed Loads across the Large
Dimensions of a Textile RVE Will Produce Bending Moments That Must Be
Incorporated into Stiffness and Strength Prediction.

[33] that provided a complete and detailed description of the needed geometrical

parameters, such as those shown in Table 2-1. Given parameters are representative of

microarchitectures as experimentally observed via SEM or standard microscope.

Total fiber volume fraction, given these dimensions, will be 25%, incorporating

the fact that the resin-impregnated tow itself has a fiber volume fraction of 65% (this is

calculated directly from ABAQUS software, which yields element volumes as outputs,

thus the volume of all matrix elements can be compared to the volume of all tow

elements). Though this volume fraction may seem low for structural uses, it can be

representative of many significant low-load, impact-resistant applications, such as

automotive lightweight body panels.

Table 2-1: RVE Dimensions
Dimension a, b c p t w
Length (mm) 1.68 0.254 0.84 0.066 0.70






















Figure 2-2: RVE Geometry of a Plain Weave Textile Composite

It should be noted that a number of parameters are required to exactly specify the

textile geometry. These specifications will have a significant effect on micromechanical

modeling. Consequently, care should be taken when comparing the results of various

studies that the textile geometries under comparison are truly equivalent.

In order to evaluate the stiffness and strength properties of the textile weave under

consideration, the DMM is essentially employed as an analytical "laboratory" that

quickly and effectively replaces physical testing and experimental procedures. Though

experimental verification always provides a baseline of veracity to FEM analysis, this

procedure effectively overcomes the limitations of physical apparatus. Furthermore, as

will be shown later, this allows for a dense population of analysis points; thus a failure

envelope may be quickly and fully constructed with a large number of data points, and

there is no need for interpolation of limited discrete experimental data points. Also,

results achieved from the DMM are completely three-dimensional stress or strain fields.

Thus, the results can be visualized throughout the thickness of the specimen. This

overcomes the limitations inherent to physical application of experimental stress analysis

techniques, which are labor-intensive and generally limited to surface visualizations.









Specification of relative displacements on opposite faces of the RVE can represent

any general macro deformation under investigation. Displacements are applied using

periodic boundary conditions. The periodic displacement boundary conditions isolate the

mechanical effects of application of unit strains or curvatures, and ensure the

repeatability of deformations. Thus, the RVE is not only repeatable as a representative

geometry, but is also mechanically repeatable in that each RVE has an identical response

to strains and curvatures regardless of the location of that RVE in a textile plate or

component, for example

u, a Ou, 02w 02w
u =w -=w (2.1)
Sx = x+a x x+a

To ensure continuity of microstresses and compatibility of displacements across an

RVE, periodic traction and displacement boundary conditions must be employed. A

macroscopically homogeneous deformation can be represented as

um, = H x ij = 1,2,3 (2.2)

e.g. u, (a, x, x3) u (0, x, x,) = H,,a

e.g. u, (x,,b, x) -u, (x,O, x3) = H, b

The derivation of the periodic boundary condition for unit curvature is presented

below, and further examples are presented in the appendix.

The periodic displacement boundary condition corresponding to unit curvature

along the x-axis (Ki) will be derived. All other curvatures will be zero. Curvatures are

defined as follows


K= 1 Ky i= Ky0 = 0 (2.3)
ax2 x Q2 xy 9x8y









The periodic displacement boundary condition will be derived from the definition

of curvature along the x-axis (Kx). Integrating once with respect to x yields

Ow
= x + f(y) (2.4)
ox

wheref(y) is an arbitrary function ofy. Differentiation of this expression with respect to

y, together with the requirement that K is set as zero, indicates that

2w
K =- -f'(y)=0 (2.5)


Due to the above expression,f(y) must therefore be a constant, since Kxy = 0.

Furthermore, this constant must be zero due to the specification that slope at the origin is

Ow
zero, such that = 0 Next, Equation 2.4 is integrated with respect to x, giving rise to


the following expression and an arbitrary function h(y)

X2
x-
w = + h(y) (2.6)
2

Now coordinate values for opposite faces of the RVE (see Figure 2-2) may be

substituted into Equation 2.6

w(a,y,z) = -a + h(y) (2.7)
w(O, y, z)= h(y)

These are then subtracted from each other, eliminating the unknown h(y), and

effectively prescribing the relative displacement on opposite faces that can be used to

apply unit curvature.

w(a,y,z)- w(0,y,z) = -a2 (2.8)









However, a further boundary condition is required to remove the transverse shear

forces that will be present due to the application of this displacement. In this way, the

mechanical effect of curvature is isolated. Thus the requirement is that transverse shear

strain is zero, which is defined as

au Ow
Y + 0 (2.9)
dz ox

aw.
This can be rearranged as below, and the value of slope is known from
Ox

Equation 2.4 above (where the arbitrary function has been shown to be zero).

du aw
x (2.10)
dz ox

Similar to the above procedure, this expression can then be integrated with respect

to z, evaluated with coordinate values of opposite faces, which are then subtracted from

each other to specify the relative displacement that must be proscribed.

u = zx+c (2.11)

u(a,y,z) u(,y,z)= za (2.12)

The periodic boundary conditions as shown in Equations 2.8 and 2.12 must be

simultaneously applied to isolate the effects of an applied unit-curvature.

In order to satisfy equilibrium, traction boundary conditions are applied to ensure

equal and opposite forces on opposite faces of the RVE. The traction boundary

conditions for traction forces on the lateral faces of the RVE are

F,(a, y,z)= -F (0, y,z)
F (x,b, z)= -F (x,0,z) (2.13)
F(x, y, c)= -F (x, y,O)









In the Direct Micromechanics Method (DMM), the RVE is subjected to

macroscopic force and moment resultants, which are related to macroscopic strain and

curvature according to

S[N]1 I[A] [B]~ [E]} (2.14)
[M]J L[B] [D]_ [r]j

Thus the constitutive matrices must be evaluated to determine this correlation.

Once this has been determined, a macroscopic deformation can be applied using an FEM

code. In this way, the FEM results for stress in each element yield the microstresses

resulting from an applied force or moment resultant.

The RVE is subjected to independent macroscopic unit deformations in order to

evaluate the stiffness matrices of Equation 2.14. In each of the six cases shown in Table

2-2 below, a single unit strain or unit curvature is applied, and all other deformation

terms are set to zero, and the appropriate periodic boundary conditions are applied.

The four-node linear tetragonal elements in the commercial ABAQUSTM (Standard)

FEM software package were used to model the yam and matrix for all cases. An h-

refinement convergence study was performed in which analysis was performed for a

progressively finer mesh of four-node linear tetragonal elements. For several reasons, the

final mesh chosen employs 68,730 such elements.

Table 2-2: Periodic Displacement Boundary Conditions
u(a,y)- v(a,y)- w(a,y)- u(x,b)- v(x,b)- w(x,b)-
u(O,y) v(0,y) w(0,y) u(x,0) v(x,0) w(x,0)
1 M =1 a 0 0 0 0 0
2 ,M= 1 0 0 0 0 b 0
3 xM 1 0 a/2 0 b/2 0 0
4 Kx= 1 az 0 -a2/2 0 0 0
5 M=1 0 0 0 0 bz -b2/2
6 VM = 1 0 az/2 -ay/2 bz/2 0 -bx/2









This mesh refinement is significantly beyond the point of numerical convergence

for which output element stresses can be assured to be accurate. This also allows for a

mesh that accurately covers the "corners" of an ellipsoidal tow cross-section without

sacrificing element quality in such regions. Furthermore, a refined mesh can capture the

intricacies of stress contours and stress gradients expected to be seen through the RVE.

Note that the shared nodes are employed between each tow and its surrounding interstitial

matrix. There are no tow-tow shared nodes, thus tows are not bound to each other, but

only to the interstitial matrix. As a final note on the character and quality of mesh

employed in this analysis, the following quality assurance metrics are indicated: fewer

than 0.1% of elements have an interior angle less than 20 degrees, fewer than 0.3% have

an interior angle greater than 120 degrees, and less than 0.2% have an aspect ratio greater

than 3 (the average aspect ratio is 1.66).

The FEM results for each element yield the microstresses resulting from an applied

macro-level strain and curvature. The corresponding macro-level force and moment

resultants in each case can be computed by averaging the microstresses over the entire

volume of the RVE

N, = (,b)ZV (2.15)

M,= (b)-za ejV (2.16)

where e denotes summation over all elements in the FE model of the RVE, yV is the

volume of the eth element, and a and b are the dimensions of the RVE as shown in Figure

2-2.

Thus the constitutive matrices of Equation 2.14 can be found by independently

evaluating the six cases shown in Table 2-2, in tandem with Equations 2.15 and 2.16. By









applying the appropriate displacements according to Table 2-2 which correspond to a

given unit strain or curvature case, the stiffness coefficients in a column corresponding to

the non-zero strain can be evaluated directly from the force and moment resultant values

as calculated from the finite element micro stresses via Equations 2.15 and 2.16. Thus

the six load cases completely describe the six columns of the [A], [B], [D] matrix.

This information having been determined, one is then able to evaluate the

microstress field resulting from general loading cases via the following steps: Step 1)

Relate applied force and moment resultants to applied macro strain and curvature via the

[A], [B], [D] matrices and Equation 2.14, Step 2) Apply this macro strain and curvature

to the RVE using an FEM code, and Step 3) The element stresses from FEM results yield

the microstress field in the yarn and matrix. The present study assumes there are no

residual stresses or pre-stresses in the composite. The significance of residual stresses

would depend on the particular cure cycle employed in manufacturing the composite, as

well as upon the weave pattern under investigation. For a plain-weave textile, it would

be expected that residual pre-stresses would affect the "center point" of the failure

envelope, given that 1) symmetry of microarchitecture would lead to a level of symmetry

of residual stresses, and 2) pre-stresses shift, rather than shrink, an existing failure

envelope as the applied loads can either add to or be offset by the residual stresses. The

magnitude of residual stresses could conceivably reach on the order of 10% of the failure

strength.

Direct Micromechanics Method for Failure Analysis

The method described above can be used to predict strength by comparing the

computed microstresses in each element against failure criteria for the constituent yarn

and matrix of the textile composite. Interface failure is not considered in the current










study, but will be incorporated into future work. The microstresses in each element can

be extrapolated from the preliminary RVE analysis (described above) of each of the six

linearly independent macrostrain components. The microstress state for a general applied

force or moment resultant is obtained by superposing multiples of the results from the

unit macrostrain analysis


{ }= [Fe] } (2.17)


Where the 6x6 matrix [Y] contains the microstress in each element resulting from

the unit strain and curvature analysis. For example, the microstress oy in the RVE for e0o

=0.05 and Ky= 0.003 m-1 is calculated as cy =0.05F21+0.003F25.

Failure is checked on an element-by-element basis, and the failure criterion of each

element can be selected appropriately based upon whether it is a yam or matrix element.

It is assumed that the entire textile composite has failed, even if only one of the yarn or

matrix elements has failed. Although this may be considered conservative, it is

realistically representative of the initial failure of the composite.

Select the state of macrostress [NJ, [M]

Compute macrostrain and curvature via Eq 14

Compute microstress {cL)} in ele meant e
(} = [F' }()+ [PF] J(M}
I Check element failure
No s Composite Stop
=+les fails
C eh + eI N o If e = e f I
Cl -ck -xt-* ____t __ Yes _
Composite does not fail Stop


Figure 2-3: Flowchart for Failure Analysis Using the Direct Micromechanics Method









For the isotropic matrix elements, the Maximum Principal Stress criterion is used to

evaluate element failure. For fiber tow elements, the Tsai-Wu failure criterion is used.

This criterion is more suitable to the orthotropic nature of the fiber tow, which is

essentially a unidirectional composite at the micro level. Microstresses in the yarn are

transformed to local coordinates tangent to the path of the yam and compared to strength

coefficients for a unidirectional composite, using the Tsai-Wu criterion.

A flow chart that describes the DMM procedure is shown in Figure 2-3. Failure

envelopes are generated by first selecting a macrostress state to investigate. Then the

macrostrains and curvatures resulting from this applied loading are calculated from

Equation 2.14. The resulting stress field for the entire RVE is then calculated by

Equation 2.17, based on the scaled superposition of the results from FEM analysis of the

unit load cases shown in Table 2-2. Failure is then checked in each element against a

given failure criterion. This cycle is then repeated while progressively increasing a

selected force or moment resultant and holding all others constant until an element level

failure criterion is exceeded. If a particular failure criterion is exceeded, the element and

the RVE are considered to have failed, which then defines the threshold of the failure

envelope at a given point. Thus failure envelopes for the textile composite can be

generated in various force and moment resultant spaces. The scope of the current study

considers analysis of in-plane force and moment resultants [N] and [M], though the

methods are applicable to any general loading conditions.

Phenomenological Failure Criteria

In addition to being used to determine failure of the fiber tow or the matrix phase at

the element or micro level, the Tsai-Wu phenomenological failure criterion is used as a

basis for comparison to the DMM for the macroscopic failure of the textile composite.









Since the Tsai-Wu criterion was formulated in terms of stresses, an adapted form of this

criterion in terms of applied force resultants is used.
2 2 2
FINX +F22N +F 66N +2FNN +FN. +FN, = 1 (2.18)

Employment of this criterion essentially represents fitting an ellipse of Tsai-Wu

form to the DMM failure data. The failure coefficients (F,j) that appear in Equation 2.18

are based on failure data from the DMM. These parameters are based on the failure

strength of the material under various loading conditions, and are typically determined by

conducting physical tests on the specimen. For example, in order to obtain F11 and F1, a

load of Nx is applied, and all other stresses are set to zero. Then Equation 2.18 reduces to

F,11N +FN =1 (2.19)

Since the DMM has been used to determine the maximum uniaxial tensile and

compressive force resultant, each of which satisfy Equation 2.19, these two independent

equations can be solved for F11 and F,

F, = (/xT) (c) (2.20)

F 1 = xTxc (2.21)

where XT and Xc are the failure values of Nx for tension and compression, respectively.

Using similar procedures, strength coefficients F22, F2, and F66 are evaluated. The

resulting values are as follows

F2 = (Y) -(C) (2.22)

F22 = r(,) (2.23)

F66 = /2 (2.24)









In the above equations Y and S are the strengths in terms of the force resultants Ny

and Nxy, and C and T denote compression and tension, respectively. In the literature, there

exist many proposed methods for determining an appropriate F12. In this study, the

coefficient F12 is determined by subjecting the unit-cell to a state of biaxial stress such

that Nx = Ny while Nxy=0, and then determining the maximum value of applied Nx = Ny =

Nmax. The resulting coefficient takes the following form

F12 = [- FI + F22)Nm-F, + F2ma (2.25)
2Nmax

Please note that for Equations 2.18 through 2.25, strength values must be in terms

of force resultants (N), as noted in the nomenclature. This is different from the textbook

definition of the Tsai-Wu failure theory, which is in terms of stresses. Also note that, as

will be discussed later, the Tsai-Wu failure theory includes no provision for the

incorporation of applied moment resultants into the prediction of failure.

The Maximum Stress Theory is another failure theory to which the results of the

DMM may be compared. Simply stated, failure occurs when any single stress component

exceeds an allowable level.

Xc < N, < X
Yc < Ny < Y, (2.26)
N^
And similarly, the Maximum Strain failure theory states that failure occurs when

any single mid-plane strain component exceeds an allowable level.

ec < x0o < er
ec < yo < eT (2.27)

Yy < eLT









For example, the limiting mid-plane strain in a given direction is

Nma


(2.28)


The bounds of the Maximum Strain Theory failure envelope take the shape of a

parallelogram whose sides are defined by lines of the form

N, = vXN + S, (2.29)

As was mentioned regarding the Tsai-Wu failure theory, the Maximum Stress and

Maximum Strain theories were developed in terms of applied stresses, thus adapted forms

of these criteria in terms of applied force resultants are used. Again, currently there is no

provision for the incorporation of applied moments into the prediction of failure.














CHAPTER 3
STIFFNESS AND STRENGTH DETERMINATION

Using the DMM procedure as detailed in Chapter 2, the failure envelope for a plain

weave textile composite is developed for in-plane force resultants, with and without

applied moment resultants. No currently accepted failure criteria exist that may be used

explicitly for the analysis of textile composites. Thus the methods and results employed

herein are used to develop phenomenological failure criteria for textile composites. The

results are compared to conventional failure envelopes that are not specifically developed

for the analysis of textile composites, as a basis for evaluation.

Stiffness Properties

The fiber tow was assumed to have material properties of a unidirectional

composite (weave geometry is taken into account in the finite element model), in this

case AS/3501 graphite-epoxy.

The constitutive matrices relating macroscopic force and moment resultants to

strains and curvatures were found using the aforementioned procedures and are found to

be

4.14 0.52 0
[A]= 0.52 4.14 0 x106(Pa-m) [B] 0 (3.1)
0 0 0.18

7.70 2.53 0
[D]= 2.53 7.70 0 x10 3(Pa-m3)
0 0 1.35









The character of the constitutive matrices is analogous to an orthotropic stiffness

matrix with identical elastic constants in the material principal directions (also referred to

as a tetragonal stiffness matrix). Flexural stiffness values of the [D] matrix may seem

slightly low, but it should be noted that the RVE under consideration is relatively thin at

0.254 mm. Also note that although the zero terms in the above matrices were not

identically zero, they were several orders of magnitude below comparable matrix terms,

and thus have been neglected with little or no effect on end results.

The above constitutive matrices have been calculated directly from the

micromechanics model without any assumptions on the deformation of the composite

such as plane sections remain plane etc. as in traditional plate theories. The results are

quite different from commonly employed approximations. From classical laminate

theory, a plane stiffness matrix is calculated from homogenized continuum stiffness

properties (the familiar E, G, and v). The flexural stiffness matrix [D] is then calculated

from this homogenized stiffness and the thickness of the textile RVE. By comparison to

the direct micromechanics results of the DMM, these methods will misrepresent flexural

stiffness values Dll, D12, and D66 by as much as factors of 2.9, 1.1, and 0.7 respectively.

The DMM results imply that there is no consistent relation between in-plane and flexural

properties, although the two properties are related.

In-plane axial and shear stiffness values (E, G) are calculated directly from the [A]

matrix. Compared to the bare properties [34] of the constituent fiber tows (Table 3-1),

stiffness is lower by an order of magnitude. But it must be noted that the textile

composite under consideration here has on overall volume fraction of 25%, whereas the

tow properties of Table 3-1 reflect the 65% volume fraction as seen within the tow itself.









As a basis for comparison, the in-plane homogenized stiffness properties are

presented in Table 3-2, as calculated directly from the [A] and [D] matrices above.

Flexural moduli, which represent the bending stiffness of the textile composite, are also

presented in Table 3-2 and are calculated from the relations shown in Equation 3.2

through Equation 3.5.

-Mx 12
Ex = (3.2)
I Kx t'3D,


S 12
E =
t tD22

S 12
t3D66


(3.3)



(3.4)



(3.5)


Flexural moduli are material-dependent properties, but are strongly geometry-

dependent as well. Similarly to what is seen for the in-plane stiffness, the flexural

stiffness is shown to be much lower than the constituent tow properties, for the relatively

thin plate under consideration here. The bending properties are strongly influenced by

the plate thickness and the weave architecture.

Table 3-1: Fiber Tow and Matrix Material Properties
Material E1 (GPa) E2 (GPa) G12 (GPa) V12
AS/3501 Graphite/Epoxy 138 9.0 6.9 0.30
(65% Fiber Volume)
3501 Matrix 3.5 3.5 1.3 0.35

Table 3-2: Stiffness Properties for Plain Weave Textile Plate
In-Plane Ex= E (GPa) Gxy (GPa) vxY
Properties 16.0 0.71 0.13
Flexural Ef = Ef, (GPa) GfAy (GPa) VfX
Properties 5.0 0.88 0.33


D12
Vfx = D=









Strength Properties

Table 3-3 [34] shows the textbook values for failure strengths of the yarn and

matrix materials. The subscripts "L" and "T" refer to the longitudinal and transverse

directions, respectively. The superscripts "+" and "-" refer to tensile and compressive

strength. These were used with the Tsai-Wu failure criterion to determine failure of the

yarn at the micro level within an element.

Figure 3-1 shows a comparison of the DMM failure envelope for the plain weave

graphite/3501 textile composite with several common failure theories: the Tsai-Wu,

Maximum Stress, and Maximum Strain failure theories. Failure envelopes are shown in

the plane of biaxial force resultants with no applied moment present. Since the DMM is

used to define the macro level failure strength, all theories share the same uniaxial

strengths and are fit to these points.

For the most part, the Maximum Stress Theory is much more conservative than all

other theories. However, it is less conservative in Quadrants II and IV, since this failure

theory does not account for the interaction of biaxial stresses. The Maximum Strain and

Tsai-Wu failure theories compare more closely to the DMM failure envelope, especially

the latter.

For zero applied moment (Figure 3-1), the DMM failure envelope follows closely

with the form of a Tsai-Wu failure envelope. For the most part, the initial failure mode is

transverse failure of the fiber tows. However, at the extremes of the major axis of the

Table 3-3: Fiber Tow and Matrix Failure Strength Properties (MPa)











Failure Envelopes, Mx = 0

---------------1 505 104--------------------




-. -' - I/

DMM
z -Tsai-Wu
| M-- -Max Strain
Max Stress
-1 50-+84 -1 00EE4 C1ODEt 3 00 50 0E5 1 OOEt4 1 501 -04
I I 1 0EI-+-4 I 4E 3 0 -














Adapted Forms of the Tsai-Wu, Maximum Strain, and Maximum Stress
Failure Theories are Shown. Since the DMM is Used to Define the Macro
Level Failure Strength, all Theories Share the Same Uniaxial Strengths and
are Fit to these Points.

failure envelope (the outer corners of quadrants I and III), the initial failure mode

transitions to failure of the matrix material. Thus the DMM failure envelope is cut short
Force Resultant, Nx (Pa-m)









at tFigure ends3-1: Comparisoned to the failure envelope thatEnvelopes would exist if matrix failure Theories. not

considered) anpted Fois squared-off in the Tsai-Wu, Maximum Strain, and Maximum failure stress

(force resultant) criteries arThe DMM envelope is more consed to Define the Macroi-Wu
Level Failure Strength, all Theories Share the Same Uniaxial Strengths and
are Fit to these Points.

failure envelope (the outer corners of quadrants I and III), the initial failure mode

transitions to failure of the matrix material. Thus the DMM failure envelope is cut short

at the ends (compared to the failure envelope that would exist if matrix failure were not

considered) and is squared-off in these regions and resembles the maximum failure stress

(force resultant) criterion. The DMM envelope is more conservative than the Tsai-Wu

type criterion in quadrant I, and slightly less conservative in quadrant III.

Though failure loads are not generally reported in terms of force and moment

resultants, a translation of the failure envelope force resultant values to traditional stress

values shows that strength magnitudes are reasonable based upon comparison to literature









and supplier-reported values for this material, geometry, and the relatively low fiber

volume fraction (25%). Since the yarn takes much of the load, the matrix does not begin

to fail until a much higher load than its bare tensile strength. Although the strength value

is a fraction of the pure tow strength, the woven tow is not completely aligned in the

loading directions. Some of the tow is curved into the thickness direction, thus providing

through-thickness reinforcement. Furthermore, after initial transverse failure of the fiber

tow (indicative of the introduction of intra-tow micro-cracking), the structure will still

maintain load-bearing capacity, though stress concentrations will begin to build up and

part integrity will be degraded. Also note that, due to symmetry of the textile RVE about

the x- and y- axes, the failure envelope exhibits this symmetry as well.

The effect of an added moment M on the failure envelope is shown in Figure 3-2

with the applied moment M, equal to half the critical value that would cause failure if it

were the only applied load. The figure also includes results from Figure 3-1, the DMM

results and quadratic failure envelope for the case of zero applied bending moment.

There is no Tsai-Wu, Maximum Stress, or Maximum Strain failure envelope to include

applied moment resultants, as these theories are not developed to include such load types.

As has been mentioned, strength estimates will be somewhat conservative given that

failure is defined as failure of a single element that surpasses the maximum allowable

microstress, but this presents a realistic definition of initial failure.

Continuing to inspect Figure 3-2, an applied moment in the x-direction has the

expectable effect of shrinking the failure envelope in regions where tensile applied loads

dominate. However, when only compressive loads are applied, an applied moment can

actually increase the in-plane load capacity by offsetting some of the compressive stress










with bending-induced tension. The magnitude of this load-capacity increase is limited,

however, by the eventual failure of the matrix. As with the case of pure in-plane loading,

the failure envelope at the outer corner of quadrant III is dominated by matrix failure.

The effects of applied moment on the failure envelope of the plain weave textile

represents the importance of the consideration of stress gradients, or load non-

uniformities. The appreciable difference that arises suggests that such consideration

could be critical to the successful design or optimization of a textile structural

component.


Failure Envelopes

1 50E+04






S* Mx=0
X Tsal-Wu
SX (1/2)Mx, crnt
S-1 50+04 -1 OOE+04 -500E+03 0 00 +00 50 1 00E+04 1 50 ;+04

o






1 50E104
Applied Force Resultant, Nx (Pa-m)


Figure 3-2: Effect of Bending Moment on the Failure Envelope. The Overall Envelope
Decreases Significantly in Size. The Tsai-Wu Failure Ellipse is Unaltered
when Moment is Applied, as the Theory Includes No Provision for this Load
Type.

The effect of changing the failure theory used to define failure of a tow element at

the micro (elemental) level is shown in Figure 3-3. The DMM can easily be modified to

employ any appropriate failure criterion for the constituent phases at the










micromechanical level. Here, the Maximum Stress Failure Theory (MSFT) is used to

replace the Tsai-Wu failure theory to determine first-element failure for the fiber tows.

Though the overall character of the failure envelope is unchanged, the effect is quite

significant in that the failure envelope becomes roughly half as conservative. It should be

noted that, even in the case of simple uniaxial macro applied loads, the micro stress field

that results is fully three-dimensional and non-homogeneous across the RVE. Thus,

especially for the orthotropic fiber tows, a failure theory that includes multi-dimensional

stress interaction effects (such as Tsai-Wu) should be more appropriate.


Failure Envelopes








I DMM
S-- Tsal-Wu
5 DMMMS
i -1 75 +04 -1 25E+04 -7 50E+ -250E+03 2 50E+03 7 50E+03 1 25E+04 1 75 +04


C-




-1 75E+04 -
Applied Force Resultant, Nx (Pa-m)



Figure 3-3: Effect of Micro-Level Tow Failure Theory on the DMM Failure Envelope.
A Comparison is Shown between the Original DMM Using the Tsai-Wu
Theory to Analyze Failure of a Tow Element and an Altered DMM in which
the MSFT is Used

As mentioned earlier, we have assumed that the entire textile composite has failed,

even if only one of the yarn or matrix elements has failed. It is possible to change the











definition of failure by stating that failure is considered to have occurred when 1% of the


total number of elements have failed. Figure 3-4 shows the effect of changing the


definition of failure and consequently the macro failure envelope. Note that failure points


are shown in the plane of uniaxial applied force and moment resultants, in order to


illustrate these effects under different loading types. About a 30% increase in maximum


allowable force or moment resultant results from changing the definition of initial failure.


This is shown to present the possibility of a more stochastic or a less conservative


approach to determining the point at which initial failure occurs, thus only a few points


are presented.


Failure Points


4-
E



1C




E
0
V
CL
C.
4


*1% Elems
-+04 1 Elem


Applied Force Resultant, Nx (Pa-m)





Figure 3-4: Effect of Changing the Definition of Micro-Level Failure on the Failure
Envelope. The Allowable Force or Moment is Shown to Increase by 30% if
Failure is Defined as the Point at which 1% of Elements Have Exceeded
Failure Criterion, as Compared to the Point at which One Single Element Has
Failed.


1 OE 04-

----------------------------4 OO E ---------------------





























V": 70"_W. Urg 3 ^N*W." n9: 0 %::) tsF 9"M- 1tp" UP

I 1 [rKEfhn' SI .B.U r"-- L.OCSe
.xlnmcy VT : 5, ;J]
.f.hnr-rj Wr: Y- D.tlqr-l .n 5,0.L. 7l-c ,r: I. .-DT


Figure 3-5: Stress Contours for Plain-Weave Fiber Tows in Uniaxial Extension.
Detailed Microstress Field Results from the DMM Provide Valuable Insight
into the Mechanical Response of an RVE under Any Loading Condition. Tow
Elements or Matrix Elements Can Be Isolated to Provide Further Detail.

Microstress Field Contour Plots

Detailed stress-field contour plots are one output of the DMM that provide great

insight into the failure modes and points of maximum stress in the RVE. Tow or matrix

portions of the RVE can be isolated for individual scrutiny. Figure 3-5 shows the stress

field for the plain-weave fiber tows in uniaxial tension. Most of the load in this case is

taken by the fiber tows aligned in the loading direction. However, the failure mode is

transverse failure of the cross-axis tows, as the strength is much lower in this direction.

Tows aligned in the direction of loading tend to be pulled straight as load is applied. This

has the secondary effect of applying bending to the cross-axis tows, creating significant

bending stresses. The maximum stress levels in this micro stress field tend to occur

around the matrix pockets between tows, which tend to act as a micromechanical stress

concentration. Similar inspection of the stress field in the matrix surrounding the fiber


A ti. CUrb.: -Ii 1


-] .23.+.O
- S .1 t
U-NI3. f,+3









tows (not shown here) shows that the maximum stress tends to occur along the relatively

sharp edges of the lenticular fiber tows, which again tend to act as a micromechanical

geometrical stress concentration.

Further stress contour plots illustrating results and capabilities of the DMM show

the RVE behavior in shear and bending load cases. For unit bending load cases, it is

shown that the majority of the load is again taken by the fiber tows along the direction of

curvature. Furthermore, just as there is a transition from tension to compression across

the RVE in bending, a rapid stress gradient from tension to compression can be seen in an

individual fiber tow. This again suggests the importance of the consideration of stress

gradients in the micromechanical characterization of a textile RVE.

Inspection of Failure Envelopes in Additional Stress Spaces

Based on an extension of the results presented also in [28] and shown earlier in

Figure 3-2, failure envelopes in the space ofNx Ny Mx (a practical and useful failure

space which illustrates the limits of biaxial loading and the importance of consideration

of stress gradients across an RVE) are seen to be characteristically elliptical in nature.

This is due to the prevalence of stress interaction effects, as well as the symmetry of the

plain weave geometry under analysis.

Figure 3-6 shows the discrete failure points as determined from the DMM for cases

of biaxial loading with constant applied moments of 0, 0.3, 0.5, and 0.8 times the critical

moment that would cause failure if it were the only load present. Phenomenologically, it

can be seen that the failure envelopes become smaller as a larger moment (stress

gradient) is applied. An applied moment has the expectable effect of shrinking the failure

envelope, though in limited regions it has been seen that the complexity of the

superposed stress fields may have an offsetting or beneficial effect.



















0 I i / K OS Cnl
Sx 0 CN:rll









Force Reultant Nx (Pa-m)

Figure 3-6: DMM Failure Envelopes for Biaxial Loading with Multiple Constant
Moment Resultants. For Illustration, an Ellipse is Fit to Each Data Set Using
a Least-Squares Method.

Also shown in this figure are elliptical fits to each failure envelope, which were

computed by a MatlabTM based routine that was used to determine a least squares fit to

the DMM data points. Regular analytical trends in these failure envelopes lead directly

to the development of failure prediction methods which will be detailed in Chapter 4.

In general, failure envelopes in spaces other than those shown in Figure 3-6 will not

necessarily be elliptical in nature, as has been observed for cases including shear, twist,

and multiple moment loading terms. Figures 3-7 and 3-8 show failure envelopes in

spaces of Mx My I and Nx Mx Ny respectively. Discrete points represent the

failure envelopes as determined by the DMM.

Figure 3-7 isolates the relative effect of moment resultants, or stress gradients, of

varying types on the failure envelope, as well as the interaction of multiple moments.

Each envelope is mapped out with a constant applied t\\ ist f,.









37





Failure Envelopes
with constant twist


*M Xy-0
S0 5 MW crit
-04 AMx 0 OFT
x 0 5 Mxy crt OFT


Moment Resultant, Mx (Pa-m2)


Figure 3-7: DMM Failure Envelopes for Biaxial Bending with Constant Applied

Twisting Moment.




Failure Envelopes
with constant in-plane shear


Nxy= -0
S0 5 Nx crlt
+03 ANxy 0 OFT
x 0 5 Ny crt OFT


Force Resultant, Nx (Pa-m)


Figure 3-8: DMM Failure Envelopes with Force and Moment Resultants for Constant

Applied Shear.


E



r
S-35(
(U


v
ce
5St

E
0
2


------------------------------ &UL-U4-------------------------




*

1 50E-04 -

*
5 00E 05-


E-04 -250E-04 -1 50E-04 -5 00E-05 5 00E-05 1 50E-04 2 50E-04 350








-2 50E-04--
*11


4^





C



E
0
2


_+03 -6 00E+03 -4 00E+03 -2 00E+03 000 +00 2 00E+03 4 00E+03 6 00E+03 800





-1 50E- 04
---------------------------' 1 5 0 E 0 4-------------------

-----------------------------------------------^ i ^ -------------------------------------------------------









The overall character of Figure 3-7 is in some ways similar to that of Figure 3-6, in

that the failure envelope symmetry reflects the symmetry of geometry and loading.

Further, the strength limits in quadrants II and IV are lower due to stress interactions.

Whereas for biaxial loading, strength is decreased when stresses act in the direction of the

natural tendency of Poisson effects, in bending the strength is decreased when biaxial

bending acts in the direction of the natural tendency of anticlastic curvature. As is

generally seen in all failure analyses that have been performed in this study, the dominant

mode of failure is transverse failure of the fiber tow.

The N Mx Ny failure envelope (Figure 3-8) provides a visualization of the effect

of the magnitude of stress gradient, or loading non-uniformity, for a given force resultant.

Each envelope is drawn for constant in-plane shear to further incorporate the effects of

multiple loading types. The envelope is symmetric about the x-axis due to the

mechanical equality of positive or negative bending moment. This symmetry is not seen

about the y-axis since the carbon-epoxy plain weave responds differently when in tension

or compression.

Design of Experimental Verification

A procedure is suggested here which may be used to illustrate the differences

between DMM and conventional failure points (a point which will also be visited in

Chapter 5). Figure 3-9 shows a schematic representation of an off-axis test specimen.

This specimen can be used under uniaxial tension to investigate different stress states.

The orientation angle (0) represents the angle of the principal material axes of the plain-

weave with respect to specimen bounds.




















Nx -X





Figure 3-9: Schematic of the Specimen for Off-Axis Uniaxial Testing.


8.00E+03

7.00E+03

6.00E+03

5.00E+03

4.00E+03

3.00E+03

2.00E+03

1.00E+03

0.00E+00


- DMM
- Tsai-Wu
-A- MaxStress


0 10 20 30 40 50
Orientation Angle (degrees)


Figure 3-10: Critical Force Resultant under Uniaxial Loading as Calculated via Various
Failure Theories.

Figure 3-10 shows a comparison of the maximum allowable force resultant Nx that

may be applied, as calculated via the DMM, Tsai-Wu Failure Theory, and the Maximum

Stress Theory, for various orientation angles. Data collected for tests performed at 0.08

radians (4.5 degrees) will produce data that should illustrate the greatest disparity


IIU

- d









between DMM and the Tsai-Wu Failure Theory. At this point, the DMM predicts a

failure load 17% less conservative than Tsai-Wu. Thus this can serve as an effective

illustration to compare and contrast the two theories. Similarly, tests performed for a

0.31 radians (17.8 degrees) specimen will provide a comparison point most useful to

investigate differences between predictions from DMM vs. Max Stress failure theories.

At this point, the DMM predicts failure levels 32% more conservative.

Chapter Summary

By analysis of the microstresses developed in a representative volume element

(RVE), the Direct Micromechanics Method (DMM) has been used to construct failure

envelopes for a plain weave carbon/epoxy textile composite in plane stress. To allow for

the accommodation of stress gradients, or non-uniform applied loads, micromechanical

analysis had been performed in terms of classical laminate theory force and moment

resultants [N], [M] and constitutive matrices [A], [B], [D]. The predicted values of the

stiffness matrices and ultimate strength values compare well to expectable magnitudes.

The DMM failure envelope was shown to be largely elliptical of the form of a Tsai-Wu

failure criterion and dominated by transverse fiber tow failure. But in cases of large

biaxial tension or biaxial compression loads, the DMM failure envelope compared to the

form of the Maximum Stress Criterion, and matrix failure was the mode of initial failure.

The presence of applied moment resultants [M], as would exist in cases of non-uniform

load across the RVE, was shown to have a significant effect on the failure envelope.

Thus its consideration, not covered in conventional failure models, can be critical.

The diversity of the failure spaces seen here is a harbinger of the further need to

develop analytical methods to predict failure without pre-knowledge of the nature of the

failure envelope. This will be discussed in Chapter 4.














CHAPTER 4
PREDICTION OF FAILURE ENVELOPES

The current chapter presents methods for utilizing the results presented in Chapter

3 to develop a failure criterion for textile composites. Using the DMM, the failure

envelope is developed for in-plane force resultants, with and without moment resultants.

No currently accepted failure criteria exist that may be used explicitly for the analysis of

textile composites, or which include this ability to account for stress gradients at the

micromechanical level. Thus the methods and results employed herein are used to

develop phenomenological failure criteria for textile composites. Based on the DMM

results, two methods are presented which may be used to predict failure of a textile

composite. The first is a parametric method based on prediction of regular trends in the

failure envelopes of a given 3D stress space. The second method represents the

formulation of a 27-term quadratic failure equation that can be evaluated to determine

failure of the textile under any general force and moment resultants.

A Parametric Approach to Predicting Failure Envelopes for a Given Stress Space

Referring again to the results of Figure 3-6, the observation is repeated that,

phenomenologically, the failure envelopes become smaller as a larger moment (stress

gradient) is applied. An applied moment has the expectable effect of shrinking the failure

envelope, though in limited regions it has been seen that the complexity of the

superposed stress fields may have an offsetting or beneficial effect.

Analytically, it can be seen that there are definite trends in the axes and placement

of the failure ellipses. Failure ellipses were then characterized with parameters such as







42


major axis length, minor axis length, ellipse axis orientation angle, and the ellipse center

point. Any general ellipse in (x, y) space can be represented by the expression


x cosO+ ysin- u, -xsinO +ycos -v (4.1)
a b


where Ois the orientation angle of the major axis of the ellipse with respect to the x-axis,

uo and vo are the coordinates of the ellipse center point, a is half the major axis length,

and b is half the minor axis length. For each failure ellipse shown in Figure 3-6, these

parameters were then plotted as functions of the moment applied for each case, in order

to inspect parametric trends.

For example, Figures 4-1 and 4-2 show the major and minor axis length of several

Nx Ny Mx failure envelopes plotted against the moment resultant M, applied in each

case. A limited number of fitting cases were used, in order to reserve an adequate

number of test cases and to prevent over-fitting of the trends.




16


12






,_--77777X'4 1144l -5821X'+ 731x* 1458ii

0.


0 02 04 00 0 1 1
X-Dlrectlon Moment Resultant as a Fraction of Critical Moment


Figure 4-1: Effect of Applied Moment Resultant on the Major Axis of Elliptical Failure
Envelopes.


















45




x 4
S3





c 2

( 15
a. vw-$a692Y' 52113?- 0953~ 4eo89





C ol? 5 o0 02 1 1?
X-Direction Moment Resultant as a Fraction of Critical Moment



Figure 4-2: Effect of Applied Moment Resultant on the Minor Axis of Elliptical Failure
Envelopes.


--I'rlv [C)
--Pollj (u)


X-Direction Moment Resultant as a Fraction of Critical Moment


Figure 4-3: Effect of Applied Moment Resultant on the Center Point Coordinates (uo, vo)
of Elliptical Failure Envelopes.


As mentioned earlier and as seen in these above figures, a larger applied moment


with a given failure envelope has the effect of shrinking the envelope's major and minor









axes. These trends were regular enough to be closely approximated with a polynomial

trend line. At the critical moment, the ellipse axes lengths become zero, as a load

sufficient for failure is already applied, and no additional force resultants may be applied.

In addition to shrinking the failure envelope, larger applied moments also tend to

cause a small but significant shift in the center point of the failure ellipse. This is caused

by the fact that an applied moment is still a directional loading, and thus produces a

directional bias in the location of the failure envelope. These trends were also plotted

(Figure 4-3) and approximated with polynomial trend lines. Ellipse orientation angle (0)

was found to be nearly constant, thus no trend plot is shown.

Based on the above, an elliptical failure envelope of the form of Equation 4.1 can

be predicted by evaluating the expressions for ellipse parameters in terms of applied

moment resultant, as shown below

Nx, cos 0 + N, sin0 u N sin0 +N, cos 0 v 1 (4.2)
+ = 1 (4.2)
a b


hM Mapphed
Mcnhcal


a = -7.77M4 +11.44M3 5.85M2 + 0.73M +1.4586


b = -9.47M3 + 5.81M2 0.95M + 4.6 (4.3)


u = -3.48M2 -0.51M


v = 3.26M2 5.92M

Thus any general Nx Ny Mx failure envelope can be predicted with the above

procedure and expressions. For several test cases, a failure ellipse is predicted and











compared to discrete failure points as calculated directly from the DMM. These


comparisons are shown in Figure 4-4 and Figure 4-5. The average deviation between the


failure envelope predicted from the parametric curve fitting as compared with the direct


results of micromechanical modeling was less than 2%.


The greatest value of this parametric approach to predicting failure is that it


provides good insight into the exact nature of the failure space under consideration.


Further, the results are quite accurate and could be useful for design purposes. However,


the load cases are limited, and the methods presented here would have to be extended if


more than three simultaneous loads were to be applied and analyzed.


Figure 4-4: Failure Envelopes Predicted with the Parametric Approach as Compared to
DMM Results (Applied Moment Resultant of 0.65 Critical Moment)


Conparison of Predicted Failure Envelope with DMM Results
6000-
40Or *
4000





z -2000 -
4000
/















-12000
/*








-10000 -



-12000 -DD -8000 -6000 -40D -2000 0 2O- 4000 6000
Force Resultant, Nx(Pa-m)







46


Comparison of Predicted Failure Envelope with DMM Results
4000








-200 '
I A "
ir


-6000 /







Figure 4-5: Failure Envelopes Predicted with the Parametric Approach as Compared to
DMM Results (Applied Moment Resultant of 0.9 Critical Moment)

Development of a Quadratic Failure Criterion to Predict Failure for General
Loading

In order to bind together the failure spaces, which can be quite different in nature,

as can be seen in Figure 3-6 through Figure 3-8, methods of the previous section will not

be readily applicable. Therefore, development of an additional analytical method

becomes necessary. Further, an analytical approach to binding the results of multiple

failure envelopes as calculated via the DMM, along with the capacity to accommodate

any general plate loading condition has a practical value. Given the quadratic interactive

nature of the stress state in determination of failure, an expression of the below form has

been developed to predict failure.

C,F,F + D,F, = 1 (4.4)


where F, represent general load terms (NxNyN ,1 fI, -f ,), and C, or D, represent 27

failure coefficients such that Equation 4.4 defines failure when its magnitude exceeds 1.









Failure coefficients C, and D, can be solved given a sufficient amount of known

failure points or failure envelopes. However, Equation 4.4 is numerically ill-conditioned

given the great disparity in the magnitude ofN,j and M,, loads which will cause failure.

These values will differ by many orders of magnitude when typical units are utilized (Pa-

m and Pa-m2). This makes accurate solution of failure coefficients impossible when both

such load types are present. Thus, Equation 4.4 must be defined in terms ofF, terms that

are normalized with respect to a critical Nmax or Mmax that would cause failure if it were

the only load present.

Coefficients C,, and D, can be solved by evaluating Equation 4.4 with a load ofF,

and setting all other loads at zero. For example, in order to obtain C11 and D1 a load of F

= Nx is applied, all others are set to zero, and Equation 4.4 reduces to

CN + DINx = 1 (4.5)

Since the DMM has been used to determine the maximum tensile and compressive

allowable Nx, each of which must satisfy Equation 4.5, two independent equations are

generated which can be simultaneously solved to yield C1 and D1. Note that, as

mentioned earlier, loads have been normalized for numerical robustness (in this case, Nx /

Nx,ctical ). The Nx,crntca for the plain weave carbon-epoxy in this study was found to be

6.40 x 103 Pa-m ; this was used to normalize all force resultant terms. The Mxcntcal used

to normalize all moment resultant terms was 1.85 x 10-4 Pa-m2. A complete table of

strength values for the carbon-epoxy plain weave textile composite is shown in Table 4-1.

Remaining coefficients C, can be solved by evaluating Equation 4.4 with the

maximum possible F, = F, as determined by the DMM results. All other loads are set to

zero. This reduction in terms, along with knowledge of previously determined









Table 4-1: Strength Values for Independent Load Conditions
Strength (+) Strength (-)
N, 6.40e3 Pa-m 5.86e3 Pa-m
N, 6.40e3 Pa-m 5.86e3 Pa-m
N,, 2.11e3 Pa-m 2.11e3 Pa-m
M, 1.85e-4 Pa-m2 1.85e-4 Pa-m2
M, 1.85e-4 Pa-m2 1.85e-4 Pa-m2
MxV 1.06e-4 Pa-m2 1.06e-4 Pa-m2


coefficients C,, and D, allows for the solution of all remaining coefficients. For example,

C14 can be determined by applying the maximum possible Fi and F4 such that

N M
F, = F4 (4.6)
Scntlcal Mcrhcal

The failure coefficient is then solved from Equation 4.4 as

4 1 (I- iF12 -C44F42 -D F -D4F4) (4.7)
2FF4

The full results of the above procedures are shown in Table 4-2 and Table 4-3 below.

Note that coefficients D3 through D6 are equal to zero since positive and negative failure

values are the same for any shear or moment loads.

For complete evaluation of the 27 failure coefficients, it will generally not be

necessary to complete 27 separate physical or simulated experimental evaluations.

Exploitation of geometry can lead to a significant reduction. The plain weave under

investigation requires 13 evaluations to determine all coefficients, and more complicated

weaves will still often exhibit symmetry such that only this amount is required. For the

most general case, 27 evaluations may be required. Twelve involve a single load F, both

in tension and compression. Fifteen evaluations will be needed that involve every

combination of two loads applied equally (as normalized) until failure. These 27

evaluations may be performed either by physical experiment or simulated via the DMM.










Normalized Failure Coefficients C, for Quadratic Failure Equation.


Table 4-3:


(Coefficient Cmn is in m Row and n' Column)
n=l 2 3 4 5 6
m = 1 1.02 -0.81 2.45 0.15 0.15 -0.09
2 1.02 2.45 0.15 0.15 -0.09
3 9.29 0.15 0.15 -1.28
4 1.00 -0.65 0.29
5 1.00 0.29
6 3.05


Normalized Failure Coefficients D, for Quadratic Failure Equation
D, -0.011
D2 -0.011
D3 0.000
D4 0.000
Ds 0.000
D6 0.000


M-,OFT
-- 1 yQrl OFT


Momenr Resultant. Mx (Pa-m')


Figure 4-6: DMM Failure Envelopes for Biaxial Bending with Constant Applied
Twisting Moment as Compared to the Quadratic Failure Theory Predictions.

Referring to Figures 4-6 and 4-7, the failure spaces calculated via the DMM and

previously shown in Figures 3-7 and 3-8 are compared to the elliptical curves labeled


Table 4-2:







50


QFT, which represent the failure envelopes as predicted by the quadratic failure theory of

Equation 4.4. The overall agreement in such test cases is seen to be quite suitable, but as

seen in Figure 4-6, there will be "corners" or portions of the 6D failure space that will be

missed with the essentially 6D ellipse space of the quadratic failure theory. In general,

the QFT predictions tend to be conservative in areas of disagreement.


FCZ.-04 ,








Force Resultant. Nx (Pa-m)

Figure 4-7: DMM Failure Envelopes with Force and Moment Resultants for Constant
Applied Shear as Compared to the Quadratic Failure Theory Predictions.

For further comparison and to incorporate more complex loading conditions,

several test cases were computed to determine the difference in solutions computed from

Equation 4.4 as compared to the results of the DMM. In general, F, may be populated by

as many as 6 terms from among (NxNyN, 3f -1f, 3if ,) For cases in which 1, 2, or 3 terms

are populated, the solution is accurate to within a few percent. For test cases in which 4,

5, or 6 terms are populated, the average error was seen to be 9.3%, and several example

test cases are tabulated below. Care was taken to select a broad spectrum of cases such
E *\ _-O:Oi.**.


























test cases are tabulated below. Care was taken to select a broad spectrum of cases such









that both positive and negative values are employed, and multiple load ratios and load

types are employed. Load ratios (c) are shown to characterize the test cases, defined as


(4.8)


By maintaining the same load ratios, all predicted failure loads will maintain a single

ratio with respect to DMM failure points. Thus one ratio can characterize the congruence

of these two solutions. A ratio of one will imply complete agreement between the two

solutions. A ratio less than one indicates a conservative failure prediction, and a ratio

greater than one implies the converse. Tables 4-4 through 4-6 summarize the results.

Table 4-4: Comparison of Quadratic Failure Equation Predictions with DMM Results.
Test Cases Include 4 Populated Load Terms F,
F1 F2 F3 F4 F5 F6
o 1.00 0.00 0.92 6.67E-08 0.00 6.67E-08
DMM 1.20E+03 0.00E+00 1.10E+03 8.00E-05 0.00E+00 8.00E-05
Quadratic
Theory 1.12E+03 0.00E+00 1.02E+03 7.44E-05 0.00E+00 7.44E-05
Ratio 0.93

a 1.00 1.83 0.00 6.67E-08 0.00 6.67E-08
DMM 1.20E+03 2.20E+03 0.00E+00 8.00E-05 0.00E+00 8.00E-05
Quadratic
Theory 1.24E+03 2.27E+03 0.00E+00 8.24E-05 0.00E+00 8.24E-05
Ratio 1.03

ao 1.00E+00 -0.87 0.00 3.91E-08 -1.40E-08 0.00
-9.00E-
DMM -2.30E+03 2.00E+03 0.00E+00 05 3.23E-05 0.00E+00
Quadratic -9.99E-
Theory -2.55E+03 2.22E+03 0.00E+00 05 3.59E-05 0.00E+00
Ratio 1.11

oa 1.00 0.40 0.00 1.80E-08 -6.46E-09 0.00
-9.00E-
DMM -5.00E+03 -2.00E+03 0.00E+00 05 3.23E-05 0.00E+00
Quadratic -9.90E-
Theory -5.50E+03 -2.20E+03 0.00E+00 05 3.55E-05 0.00E+00
Ratio 1.1









Table 4-4: Continued
F, F2 F3 F4 F5 F6
-1.61E-
a 1.00 -0.50 0.00 08 -8.08E-09 0.00
DMM -4.00E+03 2.00E+03 0.00E+00 6.45E-05 3.23E-05 0.00E+00
Quadratic
Theory -3.72E+03 9.30E+02 1.53E+03 6.51E-05 6.51E-05 7.44E-05
Ratio 0.93


Table 4-5: Comparison of Quadratic Failure Equation Predictions with DMM Results.
Test Cases Include 5 Populated Load Terms F,
F1 F2 F3 F4 F5 F6
o 1.00 1.00 1.00 6.25E-08 0.00 6.67E-08
DMM 1.20E+03 1.20E+03 1.20E+03 7.50E-05 0.00E+00 8.00E-05
Quadratic
Theory 1.04E+03 1.04E+03 1.04E+03 6.53E-05 0.00E+00 6.96E-05
Ratio 0.87

a 1.00 1.67 1.20 4.67E-08 5.33E-08 0.00
DMM 1.50E+03 2.50E+03 1.80E+03 7.00E-05 8.00E-05 0.00E+00
Quadratic
Theory 1.32E+03 2.20E+03 1.58E+03 6.16E-05 7.04E-05 0.00E+00
Ratio 0.88

-4.67E-
a 1.00 -1.33 -1.07 08 3.33E-08 0.00
DMM -1.50E+03 2.00E+03 1.60E+03 7.00E-05 -5.00E-05 0.00E+00
Quadratic
Theory -1.32E+03 1.76E+03 1.41E+03 6.16E-05 -4.40E-05 0.00E+00
Ratio 0.88

-3.33E-
a 1.00 1.33 1.00 08 2.67E-08 0.00
DMM -1.50E+03 -2.00E+03 -1.50E+03 5.00E-05 -4.00E-05 0.00E+00
Quadratic
Theory -1.32E+03 -1.76E+03 -1.32E+03 4.40E-05 -3.52E-05 0.00E+00
Ratio 0.88

a 1.00 -0.45 0.45 2.27E-08 0.00 3.18E-08
-5.00E-
DMM -2.20E+03 1.00E+03 -1.00E+03 05 0.00E+00 -7.00E-05
Quadratic -4.90E-
Theory -2.16E+03 9.80E+02 -9.80E+02 05 0.00E+00 -6.86E-05
Ratio 0.98









Table 4-6: Comparison of Quadratic Failure Equation Predictions with DMM Results.
Test Cases Include 6 (Fully Populated) Load Terms F,
F1 F2 F3 F4 F5 F6
a 1.00 -1.00 1.00 4.00E-08 4.00E-08 2.00E-08
DMM 1.49E+03 -1.49E+03 1.49E+03 5.96E-05 5.96E-05 2.98E-05
Quadratic
Theory 1.28E+03 -1.28E+03 1.28E+03 5.13E-05 5.13E-05 2.56E-05
Ratio 0.86

a 1.00 1.00 0.52 1.33E-08 1.33E-08 1.33E-08
DMM 3.00E+03 3.00E+03 1.55E+03 4.00E-05 4.00E-05 4.00E-05
Quadratic
Theory 2.61E+03 2.61E+03 1.35E+03 3.48E-05 3.48E-05 3.48E-05
Ratio 0.87

a 1.00 -0.73 -0.33 -1.33E-08 1.33E-08 1.33E-08
DMM -3.00E+03 2.20E+03 1.00E+03 4.00E-05 -4.00E-05 -4.00E-05
Quadratic
Theory -2.85E+03 2.09E+03 9.50E+02 3.80E-05 -3.80E-05 -3.80E-05
Ratio 0.95

a 1.00 -0.67 0.67 -2.67E-08 5.33E-08 -3.33E-08
DMM -1.50E+03 1.00E+03 -1.00E+03 4.00E-05 -8.00E-05 5.00E-05
Quadratic
Theory -1.44E+03 9.60E+02 -9.60E+02 3.84E-05 -7.68E-05 4.80E-05
Ratio 0.96

a 1.00 0.59 0.88 3.53E-08 3.53E-08 4.71E-08
DMM -1.70E+03 -1.00E+03 -1.50E+03 -6.00E-05 -6.00E-05 -8.00E-05
Quadratic
Theory -1.45E+03 -8.50E+02 -1.28E+03 -5.10E-05 -5.10E-05 -6.80E-05
Ratio 0.85


Optimization of Failure Coefficients

The failure coefficients of Table 4-2 and 4-3 have been obtained from procedures

designed to be compatible with physical experiments. However, further accuracy may be

obtained through analytical methods to optimize the coefficients. The DMM has shown

that for this particular weave pattern, a failure theory of the form of Equation 4.4 is an

effective predictor of strength in multiple failure spaces, and that its inclusion of moment









resultants at the micro level represents a critical consideration. After this groundwork,

the failure coefficients may be found from the above procedure either by using the DMM

or physical testing, or by employing a least-squares routine which calculates the failure

coefficients that maximize the accuracy of the obtained solution.

To this end, the MatlabTM subroutine "lsqnonlin" has been employed. The

coefficients of Table 4-2 and 4-3 are supplied to the routine, which was then limited to its

perturbation of each coefficient by a factor of + 10%. This represents an optimization

which is constrained to maintain some proximity to the physically interpretable constants

of Table 4-2 and 4-3. Further perturbation could be allowed, with a danger of overfitting,

but this initial study is presented to elucidate the opportunity for further optimization. As

fitting data, the 15 above test cases were provided, in addition to the data from the 27

tests used to fit failure coefficients by the previous procedure, for a total of 42 fit-points.

Resulting coefficients as determined by the optimization algorithm are shown in Table 4-

7 and Table 4-8. Significant reduction was shown in the minimization of the error of the

objective function of Equation 4.4. Over the 15 test cases of Table 4-4 through 4-6,

which previously exhibited an average error of 9.3%, the optimized failure coefficients

reduce error to 7.7% over these cases. Note that average error is calculated as the

absolute value of deviation from a perfect prediction.

Table 4-7: Optimized Failure Coefficients C, for Quadratic Failure Equation.
(Coefficient Cmn is in mth Row and th Column)
n=l 2 3 4 5 6
m=1 1.115 -0.891 2.205 0.135 0.135 -0.099
2 1.029 2.205 0.135 0.135 -0.099
3 8.361 0.135 0.135 -1.408
4 0.9 -0.715 0.261
5 0.9 0.261
6 _2.745











Table 4-8: Optimized Failure Coefficients D, for Quadratic Failure Equation
D, -0.0121
D2 -0.099
D3 0
D4 0
Dg 0
D6 0



Table 4-9: Data to Indicate Results of Coefficient Optimization

Q-1 Q-1 (Q-1)2 (Q-l)2
index Normalized Force or Moment Resultants F standard optim standard optim
1 0.1875 0 0.17188 0.43243 0 0.43243 0.11964 -0.0063 0.01431373 3.96887E-05
2 -0.35938 0.3125 0 -0.48649 0.17459 0 -0.03526 -0.0351 0.00124341 0.00123208
3 0.1875 0.34375 0 0.43243 0 0.43243 -0.07631 -0.01636 0.00582322 0.000267584
4 -0.78125 -0.3125 0 -0.48649 0.17459 0 -0.09002 -0.07669 0.00810414 0.005881816
5 -0.625 0.3125 0 0.34865 0.17459 0 -0.25237 -0.2142 0.06369062 0.04588164
6 0.28125 0.15625 0.25781 0.37838 0.37838 0.43243 0.73672 0.52306 0.54275636 0.273591764
7 -0.23438 0.15625 -0.15625 0.21622 -0.43243 0.27027 -0.06791 -0.01178 0.00461177 0.000138721
8 0.46875 0.46875 0.24219 0.21622 0.21622 0.21622 0.58519 0.43468 0.34244734 0.188946702
9 -0.26563 -0.15625 -0.23438 -0.32432 -0.32432 -0.43243 0.53185 0.34624 0.28286442 0.119882138
10 -0.46875 0.34375 0.15625 0.21622 -0.21622 -0.21622 -0.03722 -0.03568 0.0013854 0.001272706
11 0.1875 0.1875 0.1875 0.40541 0 0.43243 0.13935 0.096473 0.01941842 0.00930704
12 -0.23438 -0.3125 -0.23438 0.27027 -0.21622 0 0.078115 -0.01175 0.00610195 0.000138039
13 0.23438 0.39063 0.28125 0.37838 0.43243 0 0.063019 0.045837 0.00397139 0.002101031
14 -0.34375 0.15625 -0.15625 -0.27027 0 -0.37838 -0.03917 -0.01162 0.00153437 0.000135048
15 -0.23438 0.3125 0.25 0.37838 -0.27027 0 0.13059 0.064909 0.01705375 0.004213178
16 1 0 0 0 0 0 0.009 0.1029 0.000081 0.01058841
17 0 1 0 0 0 0 0.009 0.0195 0.000081 0.00038025
18 0 0 0 1 0 0 1.00E-07 -0.1 1E-14 0.01
19 0 0 0 0 1 0 1.00E-07 -0.1 1E-14 0.01
20 0 0 0.32813 0 0 0 0.000217 -0.0998 4.7228E-08 0.009960838
21 0 0 0 0 0 0.57297 0.001309 -0.09882 1.7135E-06 0.009765788
22 0 0 0 1.1892 1.1892 0 0.00913 0.05343 8.3357E-05 0.002854765
23 0.25312 0 0.25312 0 0 0 -0.01852 -0.02546 0.00034306 0.000648364
24 0 0.25312 0.25312 0 0 0 -0.01852 -0.02596 0.00034306 0.000673714
25 0 0 0 0.46486 0 0.46486 -0.00621 -0.01559 3.86E-05 0.00024311
26 0 0 0 0 0.46486 0.46486 -0.00621 -0.01559 3.86E-05 0.00024311
27 0.51 0 0 0 0 0.51 0.029588 -0.02794 0.00087545 0.000780364
28 0 0.51 0 0 0 0.51 0.029588 -0.04908 0.00087545 0.002408552
29 0.66 0 0 0.66 0 0 -0.0062 -0.07145 3.845E-05 0.005104531
30 0 0.66 0 0.66 0 0 -0.0062 -0.10728 3.845E-05 0.011508998
31 0 0 0.32 0 0 0.32 0.013254 -0.00692 0.00017567 4.79515E-05
32 0.66 0 0 0 0.66 0 -0.0062 -0.07145 3.845E-05 0.005104531










Table 4-9: Continued
Q-1 Q-1 (Q-l1)2 (Q-l)2
index Normalized Force or Moment Resultants Fistandard optim standard optim
33 0 0.66 0 0 0.66 0 -0.0062 -0.10728 3.845E-05 0.011508998
34 0 0 0.66 0.66 0 0 -0.00092 0.15173 8.4839E-07 0.023021993
35 0 0 0.66 0 0.66 0 -0.00092 0.15173 8.4839E-07 0.023021993
36 1.58 1.58 0 0 0 0 9.80E-07 0.013 9.604E-13 0.000169
37 -0.91 0 0 0 0 0 8.80E-07 0.0072 7.744E-13 0.00005184
38 0 -0.91 0 0 0 0 8.80E-07 0.001404 7.744E-13 1.97122E-06
39 0 0 -0.328 0 0 0 0.000217 -0.0998 4.7228E-08 0.009960838
40 0 0 0 -1 0 0 1.00E-07 -0.1 1E-14 0.01
41 0 0 0 0 -1 0 1.00E-07 -0.1 1E-14 0.01
42 0 0 0 0 0 -0.573 0.001309 -0.09882 1.7135E-06 0.009765788
Sum 1.31841456 0.830844875

rms error 0.17717455 0.140648572


Table 4-9 indicates the output of the coefficient optimization routine. The first column

simply indexes each of the 42 load cases used for fitting. This is followed by columns

indicating the normalized force and moment resultants in each case. As before, force

resultants are normalized with the critical Nx in tension, and moment resultants are

normalized with respect to critical Mx. The next two columns display the evaluation of

the quantity (Q-1) for each load case (where "Q" is the calculation of Equation 4.4, which

should ideally produce unity at failure) as evaluated with standard or optimized failure

coefficients. These columns would then display zero for a completely accurate

evaluation. The final two columns display the square of this error, which is then summed

and divided by 42, the square root of which then yields the indicated rms error. The

optimization produces a significant improvement which, as has been mentioned, yields to

an improvement in accuracy of predicted test cases.









Chapter Summary

Based upon failure envelopes constructed by analysis of the microstresses

developed in a representative volume element (RVE), two alternate methods for

predicting failure envelopes of a plain-weave textile composite have been developed.

The previously developed Direct Micromechanics Method (DMM) has been used to

construct failure envelopes for a plain weave carbon/epoxy textile composite in plane

stress. To allow for the accommodation of stress gradients, micromechanical analysis

had been performed in terms of classical laminate theory force and moment resultants

[N], [M] and constitutive matrices [A], [B], [D]. A parametric ellipse-fitting scheme

which accurately predicts trends in failure envelopes for a given failure space has been

developed by analysis of failure ellipse parameters. This method for predicting failure

envelopes was found to agree with DMM results to within a few percent. A second

method involves development of a 27-term quadratic failure criterion to predict failure

under general loading conditions. The quadratic failure criterion was found to agree with

DMM results within an average deviation of 9.3%, but the method is more robust in

terms of its ability to predict failure from more complex loading cases.














CHAPTER 5
MULTI-LAYER ANALYSIS

The methods of the previous chapters involve characterization of the stiffness and

strength of a single RVE, or a single-layer plain weave textile composite. When multiple

layers are present, the layer properties will remain the same, but the stiffness and strength

of the overall composite will change. To develop a fully general failure theory, these

methods must be adapted to accommodate a textile composite of an arbitrary number of

layers. This can be predicted through direct simulation of a layup to directly determine

stiffness properties. This will yield accurate results, but creating a new FEM model for

each new layup is highly impractical. Thus, the single layer characterization methods

must be adapted to predict the properties of a layup of an arbitrary number of layers.

This allows material characterization simulations of a single RVE to be applicable to a

layup of an arbitrary number of layers, eliminating the need for further material

characterization. Thus a practical tool for failure analysis and design of a plain weave

textile composite has been developed.

These previous methods are adapted via three multi-layer analysis techniques:

direct simulation of a multi-layer composite (which provides an accurate basis for

comparison), an adjustment of the data output from single-layer FEM simulation, and

implementation of the quadratic failure theory (without the requirement of determining a

new set of failure coefficients). Failure points have been predicted for a variety of load

cases.









The entire body of work is then applied to several practical examples of strength

prediction to illustrate their implementation. The results are compared to conventional

methods utilizing common failure theories not specifically developed for textile

composites. Design of a dual-layer textile plate under uniform pressure is considered, for

several geometries and width-to-thickness ratios. Also shown is a test case of a pressure

vessel in which stress-gradients are less prevalent. Results are shown in terms of the

maximum allowable pressure, as well as comparison of point-by-point factor of safety

values.

Stiffness Prediction of Multi-Layer Textile Composites

A direct simulation of the behavior of a dual layer composite has been performed.

The single RVE is replaced with two stacked RVE's which simulate the dual-layer textile

composite. Using the procedures of Chapter 2, 6 unit strain and curvature cases have

again been carried out to directly determine the mechanical response. Then macro level

force and moment resultants can be computed, and the constitutive matrices are

determined. Referring back to Chapter 3, a single layer of the plain weave textile under

consideration will exhibit the following the following constitutive matrices

4.14 0.52 0
[A]= 0.52 4.14 0 x106(Pa-m) [B] &0 (5.1)
0 0 0.18

7.70 2.53 0
[D]= 2.53 7.70 0 x103(Pa -m3)
0 0 1.35

Now for a two-layer textile, the constitutive matrices, as determined from a two-layer

model utilizing the DMM, are seen to be









8.28 1.04 0
[A] = 1.04 8.28 0 x106(Pa- m) [B] 0 (5.2)
0 0 0.36

148.8 22.4 0
[D]= 22.4 148.8 0 x103(Pa-m3)
0 0 8.51

The in-plane properties of the [A] stiffness matrix remain effectively unchanged;

the doubling in value of this matrix is a bookkeeping artifact commensurate with the fact

that there is twice as much material present with two layers, as compared to one. The

same is true in part for the bending stiffness [D] as well. However, bending properties

are affected by the moment of inertia of the material that is present. For the two-layer

plain-weave under investigation, this represents an approximately 10-fold increase in

bending stiffness per layer, or an overall 20-fold increase in bending stiffness for both

layers collectively. The same amount of material present will have higher bending

stiffness the further it is from the neutral axis of bending. Consequently, bending

stiffness follows a relationship analogous to the Parallel Axis Theorem, which governs

the increased moment of inertia of an area of material that is moved away from the

bending axis. These expressions that govern the overall stiffness of a multi-layer textile

can be represented as

AL = 2As (5.3)

DL =2(DAL + A d2) (5.4)

where the superscripts DL and SL represent "double layer" or "single layer" properties,

and d represents the distance from the center of a layer to the bending axis. Similar

results have been shown in [35] for textile composites under flexure. From this










expression, the double layer stiffness matrices (Equation 5.2) can effectively be

calculated with knowledge only of the single layer stiffness properties (Equation 5.1).

This can be extrapolated to a material of an arbitrary number of layers (N), once the

DMM has been used to characterize a layer (n) or one RVE. Equation 5.5 and Equation

5.6 may be used to evaluate the stiffness matrices and no further analysis is needed.

N
A,= A, (5.5)
n=1

N
D = (D; + And2~ (5.6)
n=1

Strength Prediction of Multi-Layer Textile Composites

Once stiffness has been determined by the methods of the previous section, strength

of a multi-layer textile may be determined. The following three sub-sections describe the

various modeling approaches used to analyze laminated plain weave composite structures

and comparison of various methodologies in modeling the strength of textile composites.

For clarity, the various approaches are summarized in Table 5-1.

Table 5-1: Summary of the Various Methods Employed in Multi-Layer Strength
Analysis
Method Acronym Summary
Direct Micromechanics DMM Method of chapter 2, used to characterize
Method strength and stiffness of an RVE
Dual Layer Direct DDMM Direct simulation of two stacked RVE's,
Micromechanics Method used to characterize multiple layers
Adapted Direct ADMM Method to adapt data output from the DMM
Micromechanics Method to obtain multiple layer strength prediction
for Dual Layer Analysis
Quadratic Failure Theory QFT Method for implementing the quadratic
failure theory introduced in chapter 4









Direct FEM Simulation of Multi-Layer Textile Composites Using the DMM for
Failure Analysis ("DDMM")

The same methods described above for single layer strength prediction can be used

for direct simulation of a two layer RVE. This direct simulation paints an accurate

picture of the load capacity of a multilayer textile, at the expense of model preparation

and calculation time. As the methods for the DDMM approach are essentially the same

as the DMM described in Chapter 2, with a dual-layer RVE in place of the single RVE,

the details are not repeated. The two sections to follow then describe two methods based

upon these results (ADMM and QFT) which can be used to predict strength of a textile

composite of an arbitrary number of layers without having to employ direct FEM

simulation.

Through the DDMM, it is seen that, as also seen for stiffness properties, the in-

plane strength properties do not change. The critical force resultant doubles as a result of

the doubling of material present, but otherwise the load capacity is unchanged.

NrDnL = 2NsL (5.7)
N9j,cnt = 2N j,crnt

Here the subscript crit denotes the critical load that is the maximum allowable when all

other loads are zero (or the strength for each individual load type Nij). As previously

seen, the superscripts DL and SL represent associations with single-layer or dual-layer

properties, respectively.

The strength under bending will change significantly for the two-layer textile. As

a direct consequence of increased bending stiffness, the critical applied moment is seen to

increase tenfold.

MDL& I OMSL (5.8)
9,cnt ,cnit









Note that this relationship will depend on the thickness of each layer, and thus is an

observation specific to this RVE micro geometry (conversely, Equation 5.7 will be true

for comparing single and double layer properties for any thickness).

The DDMM serves as a check upon which a more generalized approach may be

developed, which can predict for an arbitrary number of layers under arbitrary load

conditions with mixed load types (not simply mono-loading cases of one critical force or

moment resultant).

Adaptation of the Single Layer DMM Results to Predict Strength for Multi-Layer
Textile Composites ("ADMM")

This method for taking the results of single layer material characterization and

analysis and using them for multi-layer strength prediction involves adapting the results

of single layer FEM analysis directly from the DMM. Thus only one material

characterization is needed (single layer) to predict strength for any number of layers.

In general, using the DMM, the stress field from an applied load is calculated by

determining the resultant strains and curvatures, then superposing scaled multiples of the

single-layer stress field resulting from each unit strain or curvature. In the case of a

single layer under bending, the sole "stress source" is the resulting curvature, thus the

resulting stress field can be calculated by scaling the stress field from a single layer under

unit curvature. Now in the case of two layers under bending, this stress source of

curvature is still present, but there is an additional stress source that must be accounted

for. Normal strains will result from the layer offset from the axis of bending. Thus

scaled multiples of these unit strain cases must also be applied to find the total stress field

for a multi-layer textile under bending.










Single stress source
from curvature

Single Layer Bending




L I + I I


SB Additional stress source from
Multi-Layer Bending strains that result from the
Stress source from layer offset from the axis of
curvature bending

Figure 5-1: Schematic Illustration of the Single-Layer Strain and Curvature Stress Fields
(As Found via the DMM) That Must Be Superposed in Calculation of the
Total Stresses Resulting from Multi-Layer Bending

As detailed in classical laminate theory [34], the magnitude of the normal strain in offset

layers will be directly proportional to the curvature (K) that is present and the distance

from the layer midplane to the bending axis (d).

E, = dK- (5.9)

This strain will be tensile or compressive (positive or negative) depending on the position

of the layer with respect to the bending axis.

The failure envelope for any force and moment resultants can be determined using

the above procedure. To briefly illustrate the procedure, consider a two-layer textile

under bending, with an x-curvature of 0.5 and a layer thickness of 0.2. Then the total

stress field is found by superposing scaled multiples of the stress fields resulting from

unit strain (,ij"1) and curvature (aij' 1) cases

o,= +1 dK (5.10)
o- = 0.50-c 0 (2Xo.s)









where the "" indicates that the stress field in the tensile layer will be calculated as the

sum of the two terms in each equation, and conversely, stress in the compressive layer

will be calculated as the difference of the two stress fields.

As detailed earlier, failure from the total stress field in each layer is then checked

on an element-by-element basis to determine overall failure of the composite. Proveout

and comparisons of this method will be shown after the next section.

Implementation of the Quadratic Failure Theory to Predict Strength for Multi-
Layer Textile Composites ("QFT")

The previously developed 27-term quadratic failure theory for textile composites,

as determined from the single-layer DMM, can be implemented to predict failure for a

multi-layer specimen. Once the original failure coefficients have been determined, no

further FEM or experimental analysis will be needed. Implementation of this procedure

is accomplished by adjusting the force and moment resultants applied to the multi-layer

specimen to reflect the true stress state in each layer, on a layer-by-layer basis. First, the

mid-plane multi-layer strain and curvature are calculated from the applied macro-level

force and moment resultant, along with the constitutive matrix representing the double-

layer (DL) material properties.

{E, [] [A] [B]- 1 [N]
I[] [B] [D] DL [M]

The dual-layer midplane strain must now be modified to represent the actual strain state

in each layer.


E = cE, dK


(5.12)









Note that in plate analysis, mid-plane curvature and layer-level curvature will always be

the same, as curvature is always constant through-thickness for a given layup, regardless

of thickness or number of layers (although strains may show variation).

The layer-level force and moment resultant are then calculated using this

modified strain from Equation 5.12, along with the single layer (SL) constitutive matrix.

S[N] F[A] [B] [] (5.13)[]
M]L [B] [D]L [ic]

The adjusted force and moment resultants capture what is seen in each layer offset from

the bending axis. These are then directly input to the quadratic failure theory of Equation

4.4 with coefficients as per Table 4-2 and 4-3 (as developed from one layer or RVE).

Failure analyses are performed independently in each layer. This is to say that, the single

layer force and moment resultants for each layer must be independently calculated input

to the quadratic failure theory (computations which can still be automated). Note that,

via the above procedure, a pure moment resultant applied to the two-layer composite will

correspond to both a force and moment resultant in each layer.

Comparison of the Results of Multi-Layer Failure Analysis Methods

Several cases are now presented which illustrate the relative effectiveness of the

multi-layer analysis methods shown in the preceding sections of this chapter. Direct

FEM simulation provides the most accurate prediction of the stress field and failure

envelope of the multi-layer textile. Thus the two techniques for predicting failure of a

multi-layer composite without additional material characterization tests can be compared

to this in order to estimate their accuracy.

One method of comparison is to look directly at the predicted stress field under

several loading conditions. However, these point-by-point (or element-by-element)









comparisons prove to be not the most effective summarization of accuracy. Comparison

of a few points can yield an inaccurate sample of results that appear to differ greatly,

even if the majority of the stress field compares quite closely. However, comparison of

the calculated stresses at many (or all) points and taking an average accuracy is not

necessarily the best metric either, as standard deviation could potentially be large. Thus,

certain portions of the stress field might be predicted very well, whereas certain portions

might not be accurately predicted. In this case, an average of point-by-point stress

deviation might appear small, but in fact such situation should not be considered as an

accurate prediction. Furthermore, some predicted stress components at a given point

might compare well to direct simulation, whereas other components do not, which opens

the door for further ambiguity. This having been said, in general, in a point-by-point

comparison, it has been observed that predicted stress fields generally show an average

accuracy on the order of 90% with roughly 10% standard deviation for the two prediction

methods (ADMM and QFT) as compared to direct simulation (DDMM).

Comparison of predicted failure points (the maximum allowable force and moment

resultants under combined loading) proves to be the best and most germane method of

comparing the multiple prediction methods. To this end, failure has been predicted for a

variety of load cases based on the data from the direct simulation of the DDMM. The

results of the ADMM and the QFT are then compared to this. Both methods are shown to

compare well to the DDMM results, though use of the QFT is computationally faster and

more practical once failure coefficients have been determined. Table 5-2 and Table 5-3

below show a comparison of the various methods to calculate failure for a multi-layer

textile. As in Chapter 4, load cases are shown in terms of a Load Ratio (ca) defined as










a, = ,- (5.14)


As introduced in Chapter 4, by maintaining the same load ratios, all predicted failure

loads will maintain a single ratio with respect to DDMM failure points. Thus one ratio

can characterize the congruence of these solutions. A ratio of one will imply complete

agreement. A ratio less than one indicates a conservative failure prediction, and a ratio

greater than one implies a non-conservative prediction.

The last two failure prediction comparisons of Table 5-3 are shown in different

terms in order to employ a more effective normalization which accommodates the

incongruity of the magnitude of force resultants and moment resultants. In these two

cases, failure points are found for which


Ncnt Mc (5.15)
A A

for an unknown value A at failure. The first case, indicated by a (+) in Table 5-2, is the

solution for failure in the tensile layer, in which the applied moment generates tensile

forces that accelerate failure. The second case, indicated by a (-), is the solution for

failure in the compressive layer, in which case the applied moment offsets applied tensile

force resultants. This approach also essentially solves for the factor by which load

capacity changes when the additional load is applied. In other words, an A of 0.48 in the

compressive layer implies that twice as much force resultant may be applied to the

compressive layer before failure, when compared to the failure load under no applied

moment. Conversely, in the tensile layer, the A of 2.01 indicates that the additional load

Mx accelerates failure such that the allowable Nx is halved (this becomes the limiting case

for ultimate failure).









Table 5-2:


Example Load Cases to Determine the Accuracy of Multi-Layer Analysis
Methods Accuracy is Indicated by a Rati s


DDMM ADMM Accuracy QFT Accuracy



1 1.35E+04 1.31E+04 0.97 1.30E+04 0.96
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

1 1.91E+04 2.01E+04 1.05 2.02E+04 1.06
1 1.91E+04 2.01E+04 2.02E+04
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0

0 0 0 0
0 0 0 0
1 4.23E+03 4.14E+03 0.98 4.18E+03 0.99
0 0 0 0
0 0 0 0
0 0 0 0

1 3.73E+03 3.52E+03 0.94 3.42E+03 0.92
1 3.73E+03 3.52E+03 3.42E+03
1 3.73E+03 3.52E+03 3.42E+03
0 0 0 0
0 0 0 0
0 0 0 0


The Adjusted DMM and the Adjusted QFT show 5.2% and 5.5% error, respectively over

these load cases. Most often, this error is conservative in comparison to the direct FEM


simulation.









Table 5-3: Further Example Load Cases (Including Moment Resultants) to Determine
the Accuracy of Multi-Layer Analysis Methods. Accuracy is Indicated by a
Ratio as Compared to DDMM Results.
DDMM ADMM Accuracy QFT Accuracy



0
0
0
1 1.68E-03 1.60E-03 0.95 1.54E-03 0.92
0
0

0
0
0
1 2.59E-03 2.67E-03 1.03 2.72E-03 1.05
1 2.59E-03 2.67E-03 2.72E-03
0

N/A(+) 2.01 2.08 1.03 1.9 0.95
0
0
M/A 2.01 2.08 1.9 0.95
0
0

N/A (-) 0.48 0.44 0.92 0.42 0.88
0
0
M/A 0.48 0.44 0.42 0.88
0
0


Practical Examples to Illustrate Strength Prediction of a Two-Layer Textile
Composite Plate

In order to show the application of the preceding failure prediction approaches to

practical examples, design of a two-layer plain weave-textile plate is considered. Also

shown is an example of a closed-end thin-walled pressure vessel. Classical analysis









procedures are employed to determine the loads that are then input to the various failure

analysis techniques.

As shown in Figure 5-2, a uniform pressure is applied to a simply-supported plate.

Three different plate sizes, as shown in Table 5-4, are considered to explore the different

mechanical regimes of varying width-to-thickness ratios and to consider a square versus

rectangular geometry. Plate Theory [36] is employed to determine the loads at each point

in the plate, which are then checked for failure.

Once moments and curvatures (per unit pressure) have been determined from

SDPT, failure in the plate is analyzed via several methods. From this, the maximum

allowable pressure can be determined, and results for each method are compared. The

most reliable method is direct simulation through the DDMM. This again provides a

basis of comparison for the remaining methods.










t JA
a


Figure 5-2: Schematic of the Simply Supported Textile Plate under Uniform Pressure

Table 5-4: Geometry of the Simply Supported Textile Plate under Uniform Pressure
a b t
Case 1 0.102 m 0.102 m 0.508 mm
Case 2 0.0102 m 0.0102 m 0.508 mm
Case 3 0.102 m 0.051 m 0.508 mm









The second method represents a conventional approach, which employs failure

analysis methods not developed for textile composites, and for which stress gradients are

not considered. Classical analysis techniques are used to find strains and stresses, which

are compared to a conventional failure theory. The third method is the Quadratic Failure

Theory developed in Chapter 4. The fourth method is the aforementioned ADMM. For

each case, the plate is discretized into a 21 by 21 point grid of points that are each

checked for failure (441 total points).

The conventional method is accomplished by determining the curvature per unit

pressure at each point via the SDPT. Strain is then determined by Equation 5.12 as per

classical laminated plate theory. Stress is calculated by approximating a stiffness matrix

[Q] from the [A] matrix, that was determined via the DMM as indicated by Equation

5.16, and multiplying by the corresponding strain. It should be noted that this in itself an

represent an improvement over conventional methods, as a stiffness matrix would

generally be calculated from homogenized material properties or estimations rather than

from direct simulation or experiment. (However, unlike bending properties, these

methods can often be acceptable for in-plane stiffness properties).

c = QE AtE (5.16)

in which t is the thickness of a layer and A represents the in-plane stiffness matrix of a

layer. This stress can then be compared to a maximum allowable stress via the Tsai-Wu

Failure Theory (for which failure coefficients can be found via the DMM or experimental

methods).

The procedures for the direct simulation (DDMM) as well as the QFT and ADMM

methods have been detailed above, which is not repeated. In these cases, the maximum









allowable moment per unit pressure is found, and the maximum allowable pressure can

then be compared. Note that in the direct two-layer DDMM, the stress field in both

layers is treated as a whole, thus failure is not calculated per layer.

Tables 5-5 through 5-7 below tabulate the maximum allowable pressure for each

of the three geometries under consideration, for each of the four prediction methods. The

critical pressure for the compressive as well as tensile side (limiting case) of the plate in

bending are shown. The relative accuracy of prediction is indicated as a ratio with

respect to the DDMM direct simulation. A ratio greater than one indicates a non-

conservative prediction.

Table 5-5: Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 1
Geometry of Table 5-14 as Predicted from Various Multi-Layer Analysis
Methods
Case 1
(a = b a/t = 200)
Pmax (+) max (-) Ratio/DDMM
(kPa) (kPa)
DDMM 19.5 n/a-
Conventional 24.0 27.0 1.23
QFT 20.5 22.7 1.05
ADMM 20.1 22.3 1.03


Table 5-6: Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 2
Geometry of Table 5-14 as Predicted from Various Multi-Layer Analysis
Methods
Case 2
(a = b a/t = 20)
Pmax (+) max (-) Ratio/DDMM
(kPa) (kPa)
DDMM 135.3 n/a-
Conventional 169.1 189.2 1.25
QFT 142.1 154.8 1.05
ADMM 139.4 153.5 1.03









Table 5-7: Maximum Pressure for the Textile Plate of Figure 5-2 with the Case 2
Geometry of Table 5-14 as Predicted from Various Multi-Layer Analysis
Methods
Case 3
(a/b = 2, a/t = 200)
Pmax (+) Pmax (-) Ratio/DDMM
(kPa) (kPa)
DDMM 34.7 n/a-
Conventional 40.6 45.5 1.17
QFT 32.6 36.2 0.94
ADMM 33.7 37.1 0.97


For all three cases, the QFT and ADMM represent a significant improvement over

the conventional approach. This is due to the presence of significant stress gradients

across the thickness dimension of the RVE (as accounted for with the moment resultant

matrix), contrary to common isostrain assumptions used in textile micromechanics or

failure theory development. The relative accuracy of conventional methods increases

somewhat for Case 3. The disparity between the conventional and DMM-based

approaches, which include consideration of stress gradients, will diminish as the relative

presence of stress gradients diminish with respect to other loads present.

In general, the ADMM will be more accurate than the QFT, as the QFT is an

approximation method which is slightly further removed from the developmental data.

Though both methods involve multi-layer approximations, the QFT must also

approximate the DMM stress field data. For Case 1 versus Case 2, the agreement of the

two mutli-layer analysis methods (ADMM or QFT) with the direct FEM simulation is

similar, for both mechanical regimes. Although the low aspect ratio plate is naturally

able to withstand a much higher pressure, prediction accuracies are similar. These

predictions are both non-conservative, though Tables 5-2 and 5-3 have shown that this is

not generally true. For Case 3, these methods are conservative compared to the DDMM.









In this case, at the failure point, there is a different load ratio (Mx = 2My, rather than the

Mx = My of Case 1 and 2), thus a different portion of the failure space is being predicted.

As most often seen in the results of Chapter 3, the initial failure mode for these

design cases is transverse failure of the fiber tow, beginning in the tensile layer. This

represents a fiber pull-apart initiation, or an intra-tow matrix cracking. For the unbiased

(Mx = My) biaxial bending of Cases 1 and 2, failure initiates in both fiber tows, as the

transverse stresses will be equal for both tows.

Tables 5-8 through 5-11 below further elucidate the failure initiation of the design

cases by illustrating the Factor of Safety (ratio of allowable pressure with respect to the

failure point) at several points on the plate. Only one quarter-section of the plates is

considered, due to the symmetry of the plates and of the resulting stress field. The

bottom-rightmost cell represents the plate center. Position is defined as a fraction of the

total width dimension, such that a position of (0.5,0.5) represents the center of the plate.

Although the maximum allowable pressure will differ, the difference in the distribution

of factor of safety in Case 1 and 2 will be negligible and is thus co-tabulated.

Case by case, both the QFT and conventional methods yield a similar distribution

of factor of safety, although the QFT shows a greater variation in the center-to-edge

Table 5-8: Case 1 and 2 Factor of Safety Across the Plate as Determined via the
Conventional Approach
Position 0 0.09 0.18 0.27 0.36 0.43 0.5
0 O0 O0 O0 O0 O0 O0 0O
0.09 oo 4.81 3.03 2.50 2.28 2.20 2.24
0.18 oo 3.03 2.31 1.86 1.68 1.30 1.62
0.27 _o 2.50 1.86 1.58 1.40 1.33 1.32
0.36 oo 2.28 1.68 1.40 1.25 1.18 1.16
0.43 oo 2.20 1.60 1.33 1.18 1.10 1.07
0.5 0o 2.24 1.62 1.32 1.16 1.07 1.00









Table 5-9: Case 1 and 2 Factor of Safety Across the Plate as Determined via the QFT
Position 0 0.09 0.18 0.27 0.36 0.43 0.5
0 O0 O0 O0 O0 O0 O0 0O
0.09 oo 7.99 4.81 3.81 3.36 3.04 3.07
0.18 o 4.81 3.03 2.31 1.99 1.71 1.75
0.27 oo 3.81 2.31 1.86 1.58 1.35 1.33
0.36 oo 3.36 1.99 1.58 1.40 1.18 1.14
0.43 oo 3.04 1.71 1.35 1.18 1.10 1.06
0.5 oo 3.07 1.75 1.33 1.14 1.06 1.00


5-10: Case 3 Factor
Annrn rch


of Safety


Across


the Plate


as Determined via the


Conventional


Position 0 0.09 0.18 0.27 0.36 0.43 0.5
0 co co o o o o o
0.09 oo 6.74 4.04 3.16 2.77 2.48 2.49
0.18 0o 4.80 2.72 2.05 1.75 1.51 1.49
0.27 oo 4.12 2.30 1.71 1.44 1.21 1.17
0.36 o 3.94 2.17 1.59 1.33 1.10 1.05
0.43 oo 3.88 2.13 1.56 1.29 1.06 1.00
0.5 0o 3.99 2.18 1.58 1.30 1.06 1.00


Table 5-11: Case 3 Factor


of Safety


Across


the Plate


as Determined via the


QFT


Position 0 0.09 0.18 0.27 0.36 0.43 0.5
0 co co co co co co co
0.09 oo 6.90 4.15 3.25 2.85 2.55 2.57
0.18 0o 4.86 2.77 2.09 1.79 1.55 1.53
0.27 oo 4.14 2.32 1.73 1.46 1.24 1.20
0.36 o 3.92 2.17 1.60 1.34 1.12 1.07
0.43 oo 3.83 2.11 1.55 1.29 1.06 1.01
0.5 0o 3.90 2.14 1.56 1.29 1.06 1.00


magnitude of factor of safety. Note that factor of safety rises to infinity in the topmost

row and leftmost column which represent the simply-supported edge of the plate, as the

bending moment and curvature are theoretically zero here. Also note that the maximum

pressure to which each factor of safety is normalized will be different for the different

prediction methods. Along any "radial" path from center to edge of the plates, factor of


Table









safety is seen to be nearly inversely proportional to grid position. This trend remains for

all calculation methods, over all geometry cases.

In contrast to the design of a textile plate under uniform pressure, there are other

common design cases for which there would be no improvement in accuracy in

employing the QFT or ADMM rather than conventional methods. For example, in

designing a thin-walled pressure vessel (see Figure 5-3 and Table 5-12), a biaxial stress

state with negligible stress gradients will exist. In this case, conventional methods will

predict similar maximum allowable pressure when compared to the QFT or ADMM.

Stresses and stress distributions are found from basic pressure vessel theory [37].

Force resultants and moment resultants (which result from a slight radial stress

distribution and will be quite small) can then be easily calculated from this result. Note

that since there is no curvature, the multi-layer analysis ADMM and adjustments to the

QFT inputs (as detailed in previous sections) will not be needed. Thus, this example

serves as a comparison of the DDMM simulation and the QFT, contrasted to

conventional methods, for a test case in which stress gradients are small. Results for this

design case are shown in Table 5-13 below.


Figure 5-3: Schematic of the Dual-Layer Textile Pressure Vessel









Table 5-12: Geometry of the Textile Pressure Vessel
radius (r) thickness (t)
Case 4 10.2 mm 0.508 mm


Table 5-13: Maximum Pressure for the Textile Pressure Vessel of Figure 5-3 with the
Geometry of Table 5-12 as Predicted from Various Multi-Layer Analysis
Methods
Case 4
(r/t = 20)
pmax Ratio/DDMM
(MPa)
DDMM 1.89-
Conventional 1.93 1.02
QFT 1.86 0.98


As expected, it is seen that predicted maximum allowable pressures are similar for

all methods. Although there are stress gradients along the radial direction, the variation is

relatively small, thus the moment resultant that is present will be nearly negligible. These

results will hold true for any thin-walled pressure vessel.

Chapter Summary

For analysis of textile composites of an arbitrary number of layers, the methods of

previous chapters have been adapted. Through the results direct FEM simulation,

stiffness of a multi-layer textile has been shown to be governed through an expression

similar in form to that of the Parallel Axis Theorem. For the plain-weave textile under

investigation, this represents a roughly 20-fold increase in stiffness when thickness is

increased from one to two layers. For strength prediction, two analysis methods have

been presented, again based upon and presented in comparison to the results of direct

FEM simulation. The adapted single layer Direct Micromechanics Method (ADMM) is

based upon a correction to the stress field as determined by the DMM (see Chapter 3)

commensurate with the layer-by-layer strains which result from the offset from the axis









of bending. A method for employing the quadratic failure theory (QFT, see Chapter 4)

has also been developed. Without the need to modify the single-layer failure coefficients,

the inputs to the QFT are adapted to represent the true strain and curvature in each layer

that is offset from the bending axis. Both the ADMM and QFT methods have been

shown to predict multi-layer failure within roughly 5% accuracy.

Several simple but significant design cases have been presented as a practical

application of the methods presented in this dissertation. Multi-layer failure prediction

methods have been shown to be sufficiently accurate, and the importance of the

consideration of stress gradients in a common design situation is shown.














CHAPTER 6
CONCLUSIONS AND FUTURE WORK

In this paper, robust methods have been developed for predicting stiffness and

strength of multi-layer textile composites, with techniques designed to address the

difficulties that arise when considering a textile microstructure. Currently existing failure

criteria for composite materials are generally developed for and based upon usage with

unidirectional composite laminates. Though these theories may to some extent be

applied in an adapted form to the analysis of textile composites, as has been shown

herein, many inherent simplifying assumptions no longer apply. Given the increased

complexity of analysis of textile composites, there are several outstanding issues with

regards to textile composites that have been addressed in this research. One of the most

important issues addressed here is a robust model for prediction of strength. Though

much attention has been given to the prediction of stiffness, little work has focused upon

strength prediction for textile composites.

Additionally, conventional micromechanical models for textile composites assume

that the state of stress is uniform over a distance comparable to the dimensions of the

representative volume element (RVE). However, due to complexity of the weave

geometry, the size of the RVE in textile composites can be large compared to structural

dimensions. In such cases, severe non-uniformities in the stress state will exist, which

conventional models do not account for. Methods for including the consideration of

stress gradients have been developed, and the importance of such considerations has been

demonstrated.









Stiffness properties have been predicted for a plain-weave textile composite. The

results indicate good agreement with expected values from literature and material

supplier data. The constitutive matrices have been calculated directly from the

micromechanics model without any assumptions as in traditional plate theories. The

results are quite different from commonly employed approximations. By comparison to

the direct micromechanics results of the DMM, conventional methods will misrepresent

flexural stiffness values Dll, D12, and D66 by as much as factors of 2.9, 1.1, and 0.7

respectively. The DMM results imply that there is no consistent relation between in-

plane and flexural properties, although the two properties are related.

Failure envelopes have been presented and the comparisons to and improvements

over conventional methods have been shown. Under relatively simple loading conditions

in which no stress gradients are present across the RVE, the DMM failure envelope was

shown to compare most closely with the Tsai-Wu Failure Theory. The Maximum Stress

Failure Theory and Maximum Strain Failure Theory were less close in comparison. Fiber

pull-apart or failure of the transverse fiber tows was shown to be the dominant mode of

failure. In limited instances of large biaxial stresses, failure of the interstitial matrix was

seen to be the mode of initial failure.

Further failure envelopes have been presented which illustrate the importance of

consideration of stress gradients, and the inability of conventional failure models to

account for this load type. In these cases, traditional failure prediction methods can

greatly overpredict the failure envelope. The presence of applied moment resultants [M],

as would exist in cases of non-uniform load across the RVE, was shown to have a









significant effect on the failure envelope. Thus its consideration, not covered in

conventional failure models, can be critical.

Based upon failure envelopes constructed by analysis of the microstresses

developed in a representative volume element (RVE), alternate methods for predicting

failure envelopes of a plain-weave textile composite have been developed. A parametric

ellipse-fitting scheme which accurately predicts trends in failure envelopes for a given

failure space has been developed by analysis of failure ellipse parameters. This method

for predicting failure envelopes was found to agree with DMM results to within a few

percent. However, it is impractical in its implementation, and is limited to consideration

of one particular failure space in which only three concurrent force or moment resultants

may be considered at once. The method is useful for a solid visualization and lends itself

to a firm understanding of simpler load cases. A second method involves development of

a 27-term quadratic failure criterion to predict failure under general loading conditions.

The quadratic failure criterion was found to agree with DMM results within an average

deviation of 9.3%, but the method is more robust in terms of its ability to predict failure

from more complex loading cases.

The methods thusfar have been further modified to accommodate analysis of textile

composites of an arbitrary number of layers. Stiffness of a multi-layer textile has been

shown to be governed through an expression similar in form to that of the Parallel Axis

Theorem. For strength prediction, two analysis methods have been presented. The

adapted single layer Direct Micromechanics Method (ADMM) is based upon a correction

to the stress field as determined by the DMM. A method for employing the quadratic

failure theory (QFT) has also been developed. Both the ADMM and QFT methods have









been shown to predict multi-layer failure within roughly 5% accuracy. This adaptation

thus allows for methods for the accurate prediction of strength of a multi-layer textile

with a minimum of characterization requirements based upon micromechanics of a single

RVE.

Finally, several design cases have been presented as a practical application of the

methods presented in this dissertation. Multi-layer failure prediction methods have been

shown to be sufficiently accurate, and the importance of the consideration of stress

gradients in a common design situation is shown. For a simple design case of a two layer

textile pressure vessel, a state of biaxial force resultant with negligible stress gradients

exists. In this case, the OFT and conventional methods predict a similar allowable

pressure. In consideration of the design of several textile plates of varying geometries

under uniform pressure, significant stress gradients exist. Conventional methods will

overpredict allowable pressures by 17% to 25%, whereas the DMM based QFT method

will predict strength within 5% to 6% accuracy.



Several suggestions are offered here for potential future work that may be

completed to extend the current body of work, both in terms of further development and

in terms of useful application.

Incorporating a model of progressive failure represents one potential issue for

future consideration. After initial failure, a component may still retain some stiffness and

load bearing capacity. Continued loading leads to a progressive property loss as more

and more of the constituent material becomes degraded. This can be simulated within the

finite element micromechanical model by redefinition of the stiffness matrix (or









redefinition of material properties) after single element failures. The simulation is then

rerun, and additional element stiffness matrices are appropriately recalculated as

additional element failures occur.

Incorporation of thermal stresses and investigation of the coefficient of thermal

expansion is another potential avenue for further development of the failure modeling.

Due to mismatches between the coefficient of thermal expansion of constituent materials,

thermal stresses can build up during manufacture or during operation. This makes

inclusion of such effects critical to the accuracy of a strength prediction model.

Furthermore, textile composites have been shown to perform well compared to other

composites at cryogenic temperatures. Thus investigation of such thermal effects should

be of considerable interest.

In the current work, the plain-weave architecture has been used to develop and

demonstrate an effective micromechanical methodology and failure theory for textile

composites. The same methods can easily be used to investigate the mechanical behavior

of other textile weave or braid patterns. In these cases, the RVE will be larger and more

complicated, but the DMM approach remains the same. Further, given that other

architectures are geometrically larger, it stands to reason that the importance of stress

gradient effects as presented herein will be of even greater importance.

The DMM could also be employed to perform a parametric study of the effect of

weave architecture and geometry on mechanical behavior. For example, successive

characterizations of the same weave type with varying two spacing, or tow undulation,

etc. could provide a useful insight into the effect these parameters have on the stiffness






85


and strength properties of a textile composite. This could also potentially lead to the

ability to optimize the microarchitecture to a specific application.















APPENDIX
PERIODIC BOUNDARY CONDITIONS

The derivation of periodic boundary conditions for unit extension and unit shear is

presented here. Unit curvature has been presented in the text in Chapter 2.

Unit Extension

From the definition of strain, and utilizing integration

au
g 1 ]

u(x,y,z)= x+c

u(a,y,z) =a+c
u(, y, z) = c

u(a,y,z) u(O,y,z) = a

Unit .\he,'1r

_u av
xy =-+--=1

au av au
Assuming- -, 1
8y 8x 8y
u(x,y,z)= y+c

u(x,b,z)= 1b+c
u(x,, z) = c

u(x, b, z) u(x,O, z) = b

Similarly,

v(a, y,z)- v(, y,z) = a
2a
















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