《纺织复合材料》课程参考文献（Multiscale Modeling and Simulation of Composite Materials and Structures）Chapter 2 Multiscale Modeling of Tensile Failure in Fiber-Reinforced Composites

Chapter 2:Multiscale Modeling of Tensile Failure in Fiber-Reinforced Composites Zhenhai Xia'and W.A.Curtin2 'Department of Mechanical Engineering,The University of Akron,Akron, OH 44325,USA PDivision of Engineering,Brown University,Providence,RI 02912,USA 2.1 Multiscale Damage and Failure of Fiber-Reinforced Composites Fiber-reinforced composites can be engineered to exhibit high strength high stiffness,and high toughness,and are,thus,attractive alternatives to monolithic polymer,metals,and ceramics in structural applications.To engineer the material for high performance,the relationship between material microstructure and its properties must be established to accurately predict the deformation and failure.Such a relationship between under- lying constituent material properties and composite performance can also aid selection and/or optimization of new composite systems.Successful models can yield predictive insight into the origins of damage tolerance, size scaling,and reliability of existing composite systems and can be extended to investigate damage and failure under more complex loading and environmental conditions,such as fatigue and stress rupture. Damage relevant to macroscopic failure of fiber-reinforced composite occurs at many length scales and by a variety of physical mechanisms.At the smallest scale,preexisting defects in the fibers propagate and form fiber cracks that impinge on the matrix and the interface.Debonding, sliding,and/or matrix yielding at the crack perimeter inhibit crack pro- pagation into the matrix;but the ensuing deformations are complex.The load carried by the broken fiber is then redistributed among the remaining

Chapter 2: Multiscale Modeling of Tensile Failure in Fiber-Reinforced Composites Zhenhai Xia1 and W.A. Curtin2 1 Department of Mechanical Engineering, The University of Akron, Akron, OH 44325, USA 2 Division of Engineering, Brown University, Providence, RI 02912, USA 2.1 Multiscale Damage and Failure of Fiber-Reinforced Composites Fiber-reinforced composites can be engineered to exhibit high strength, high stiffness, and high toughness, and are, thus, attractive alternatives to monolithic polymer, metals, and ceramics in structural applications. To engineer the material for high performance, the relationship between material microstructure and its properties must be established to accurately predict the deformation and failure. Such a relationship between under￾lying constituent material properties and composite performance can also aid selection and/or optimization of new composite systems. Successful models can yield predictive insight into the origins of damage tolerance, size scaling, and reliability of existing composite systems and can be extended to investigate damage and failure under more complex loading and environmental conditions, such as fatigue and stress rupture. Damage relevant to macroscopic failure of fiber-reinforced composite occurs at many length scales and by a variety of physical mechanisms. At the smallest scale, preexisting defects in the fibers propagate and form fiber cracks that impinge on the matrix and the interface. Debonding, sliding, and/or matrix yielding at the crack perimeter inhibit crack pro￾pagation into the matrix; but the ensuing deformations are complex. The load carried by the broken fiber is then redistributed among the remaining

38 Z.Xia and W.A.Curtin unbroken fibers and matrix as determined by the detailed conditions at the debonded fiber/matrix interface and in the matrix.Subsequent damage occurs in and around other fibers according to the statistical distribution of flaws in the fibers and the stresses acting on those flaws due to the applied stress and the stress redistribution.Eventually,macrocracks will form and grow,leading to failure of the composites.Figure 2.1 illustrates the damage evolution of fiber-reinforced composites at each length scale under different loading conditions. Fatigue Life Size-dependent Strength Rupture Life 鹗 Macro-scale Number of Cycles Sample size Times Critical Damage State:Failure Meso-scale Multiple Matrix Multiple Fiber Interacting cracking breaking Damage Evolution Micro-scale Fiber,matrix and interface crack growth Nano-scale Atomic bond breaking Fig.2.1.Multiscale damage and failure in fiber-reinforced composites Although the modeling path is conceptually clear,direct simulation of composite materials is still not a viable option despite advances in com- putational techniques and computing power.Finite element models that can capture micromechanical effects of cracks at the fiber/matrix/interface scale generally must employ mesh sizes of the order of the size of the microstructure and can result in an algebraic system with many millions of unknowns.It is insufficient,however,to focus only on one scale,i.e.,a fiber break and the myriad details associated with it.On the other hand, homogenization and averaging techniques for analyzing heterogeneous materials,while possibly leading to manageable problem sizes,do not

unbroken fibers and matrix as determined by the detailed conditions at the debonded fiber/matrix interface and in the matrix. Subsequent damage occurs in and around other fibers according to the statistical distribution of flaws in the fibers and the stresses acting on those flaws due to the applied stress and the stress redistribution. Eventually, macrocracks will form and grow, leading to failure of the composites. Figure 2.1 illustrates the damage evolution of fiber-reinforced composites at each length scale under different loading conditions. Size-dependent Strength Micro-scale Nano-scale Macro-scale Fatigue Life Rupture Life Fiber, matrix and interface crack growth Multiple Fiber breaking Atomic bond breaking Multiple Matrix cracking Critical Damage State: Failure Interacting Damage Evolution Meso-scale Number of Cycles Sample size Times Stress Stress Stress Fig. 2.1. Multiscale damage and failure in fiber-reinforced composites Although the modeling path is conceptually clear, direct simulation of composite materials is still not a viable option despite advances in com￾putational techniques and computing power. Finite element models that can capture micromechanical effects of cracks at the fiber/matrix/interface scale generally must employ mesh sizes of the order of the size of the microstructure and can result in an algebraic system with many millions of unknowns. It is insufficient, however, to focus only on one scale, i.e., a fiber break and the myriad details associated with it. On the other hand, homogenization and averaging techniques for analyzing heterogeneous materials, while possibly leading to manageable problem sizes, do not 38 Z. Xia and W.A. Curtin

Chapter 2:Multiscale Modeling of Tensile Failure 39 provide information about the microscopic fields needed,for example,to predict failure.Thus,there is a need for accurate and computationally effi- cient techniques that take into account the most important scales involved in the goal of the simulation while permitting the analyst to choose the level of accuracy and detail of description desired.Therefore,a multiscale modeling strategy is needed to accurately handle the evolution of damage at the larger scales while retaining important small-scale details and,thus, to accurately predict mechanical properties and performance of fiber- reinforced composites. There are two main multiscale modeling techniques for materials: seamless coupling of methods in a single computational framework and hierarchical information transfer.Direct coupling methods are not viable for fiber composite problems because the damage spans a range of scales, and it is not possible to focus on one microscopic region in detail sur- rounded by a less-detailed description.Thus,the hierarchical multiscale modeling approach,in which the information of simulations at small length scales is processed and fed into larger-scale models,is preferable.The need for multiscale analyses has been well recognized;but until recently there has not been a direct connection made between the detailed structures at the fiber/matrix/interface scale,the multifiber damage problem,and large- scale component performance.Most work has assumed some approximate representation of the behavior at the smallest scale and pursued the larger- scale damage evolution.Such approaches are certainly warranted for under- standing broad trends,identifying characteristic length scales associated with the damage,and for guiding the development of analytic models [5,21,311.Other work has investigated the detailed stress states around damaged fibers,matrix,and/or interfaces but then employed only very simple models of overall composite behavior to indicate the important role of the microscale damage [12].Specific system design and optimization requires attention to the detailed micromechanics of damage and load trans- fer around individual fiber breaks and the inclusion of such information directly into accurate larger-scale models. In this chapter,one multiscale modeling approach for predicting tensile strengths of unidirectional fiber composites,including metal,poly- mer,and ceramic matrix composites will be reviewed.The quantitative success of this approach in predicting the tensile strength and its size dependence in a carbon fiber-reinforced plastic (CFRP),silicon carbide fiber/titanium matrix composites (TMCs),and alumina fiber/aluminum matrix composite(AMC)will be demonstrated.Finally,the approach will be extended to the prediction of strength and low-cycle fatigue life of TMCs

provide information about the microscopic fields needed, for example, to predict failure. Thus, there is a need for accurate and computationally effi￾cient techniques that take into account the most important scales involved in the goal of the simulation while permitting the analyst to choose the level of accuracy and detail of description desired. Therefore, a multiscale modeling strategy is needed to accurately handle the evolution of damage at the larger scales while retaining important small-scale details and, thus, to accurately predict mechanical properties and performance of fiber￾reinforced composites. There are two main multiscale modeling techniques for materials: seamless coupling of methods in a single computational framework and hierarchical information transfer. Direct coupling methods are not viable for fiber composite problems because the damage spans a range of scales, and it is not possible to focus on one microscopic region in detail sur￾rounded by a less-detailed description. Thus, the hierarchical multiscale modeling approach, in which the information of simulations at small length scales is processed and fed into larger-scale models, is preferable. The need for multiscale analyses has been well recognized; but until recently there has not been a direct connection made between the detailed structures at the fiber/matrix/interface scale, the multifiber damage problem, and large￾scale component performance. Most work has assumed some approximate representation of the behavior at the smallest scale and pursued the larger￾scale damage evolution. Such approaches are certainly warranted for under￾standing broad trends, identifying characteristic length scales associated with the damage, and for guiding the development of analytic models [5, 21, 31]. Other work has investigated the detailed stress states around damaged fibers, matrix, and/or interfaces but then employed only very simple models of overall composite behavior to indicate the important role of the microscale damage [12]. Specific system design and optimization requires attention to the detailed micromechanics of damage and load trans￾fer around individual fiber breaks and the inclusion of such information directly into accurate larger-scale models. In this chapter, one multiscale modeling approach for predicting tensile strengths of unidirectional fiber composites, including metal, poly￾mer, and ceramic matrix composites will be reviewed. The quantitative success of this approach in predicting the tensile strength and its size dependence in a carbon fiber-reinforced plastic (CFRP), silicon carbide fiber/titanium matrix composites (TMCs), and alumina fiber/aluminum matrix composite (AMC) will be demonstrated. Finally, the approach will be extended to the prediction of strength and low-cycle fatigue life of TMCs. Chapter 2: Multiscale Modeling of Tensile Failure 39

40 Z.Xia and W.A.Curtin This review emphasizes the published work of the present authors on multiscale modeling and simulation cast into a single overall framework Progress in the field at one or several coupled scales has been made by many workers,with important insights and advances.In addition,analytic models for many problems in composite failure have been devised,but those works are not discussed here.Hence,the work presented here is not a comprehensive review of the literature.Interested readers can refer to several previous significant review articles [7,22,27]as well as other papers [19]. 2.2 Multiscale Modeling via Information Transfer 2.2.1 Model Description and General Strategy The fiber-reinforced composites considered here consist of continuous cylindrical fibers embedded in a matrix material in a unidirectional (aligned)arrangement.Such a composite can also be considered as a ply,a basic unit of a laminated composite structure.To develop a relationship between macroscopic properties and microstructure of the composite,a hierarchical set of models addressing physical phenomena at successive larger lengths scale,with coupling through information transfer,is intro- duced,as illustrated in Fig.2.2.Figure 2.2 shows the full possible range of studies relevant to the problem.At the smallest scales,an atomistic or quantum analysis can assess features such as interface fracture energy and crack deflection at the bimaterial fiber/matrix interface.Key information on interfacial debonding and sliding is then passed into a continuum interface model,e.g.,a cohesive zone,used in a micromechanical unit cell model consisting of matrix and a number of fibers to compute the stress redistribution around a fiber break for a particular material system.The stress redistribution is condensed into stress concentration factors on un- broken fibers,and,perhaps,stress intensity factors on matrix cracks,and this information is transferred to a larger-scale Monte Carlo model that tracks the evolution of fiber and/or matrix damage with increasing applied load.Details of the deformation around each fiber break are not retained at this scale,only their effects on stress concentrations.The Monte Carlo model is used to simulate damage up to the point of tensile failure,leading to a predicted average strength and statistical distribution for a composite sample that is small on the scale of practical samples but large compared to the critical damage size that drives failure.The tensile strength distribu- tion calculated from the Monte Carlo model is then employed in analytic

This review emphasizes the published work of the present authors on multiscale modeling and simulation cast into a single overall framework. Progress in the field at one or several coupled scales has been made by many workers, with important insights and advances. In addition, analytic models for many problems in composite failure have been devised, but those works are not discussed here. Hence, the work presented here is not a comprehensive review of the literature. Interested readers can refer to 2.2 Multiscale Modeling via Information Transfer 2.2.1 Model Description and General Strategy The fiber-reinforced composites considered here consist of continuous cylindrical fibers embedded in a matrix material in a unidirectional basic unit of a laminated composite structure. To develop a relationship between macroscopic properties and microstructure of the composite, a hierarchical set of models addressing physical phenomena at successive larger lengths scale, with coupling through information transfer, is intro￾duced, as illustrated in Fig. 2.2. Figure 2.2 shows the full possible range of studies relevant to the problem. At the smallest scales, an atomistic or quantum analysis can assess features such as interface fracture energy and crack deflection at the bimaterial fiber/matrix interface. Key information on interfacial debonding and sliding is then passed into a continuum interface model, e.g., a cohesive zone, used in a micromechanical unit cell model consisting of matrix and a number of fibers to compute the stress redistribution around a fiber break for a particular material system. The stress redistribution is condensed into stress concentration factors on un￾broken fibers, and, perhaps, stress intensity factors on matrix cracks, and this information is transferred to a larger-scale Monte Carlo model that tracks the evolution of fiber and/or matrix damage with increasing applied load. Details of the deformation around each fiber break are not retained at this scale, only their effects on stress concentrations. The Monte Carlo model is used to simulate damage up to the point of tensile failure, leading to a predicted average strength and statistical distribution for a composite sample that is small on the scale of practical samples but large compared to the critical damage size that drives failure. The tensile strength distribu￾tion calculated from the Monte Carlo model is then employed in analytic 40 Z. Xia and W.A. Curtin several previous significant review articles [7, 22, 27] as well as other (aligned) arrangement. Such a composite can also be considered as a ply, a papers [19]

Chapter 2:Multiscale Modeling of Tensile Failure 41 weak-link size-scaling models to predict ply strength and its statistical distribution as a function of physical size.Finally,the ply strength is used in standard laminated composite models to predict the strength and reliabi- lity of the composite component.In the last stage,other damage phenomena such as interply delamination can occur and change the local stresses in the plies themselves.In such cases,the ply strength vs.size can be used at smaller scales to assess the onset of local ply damage due to these other damage modes. Component Engineering design Strength,Life, m Reliability... Multiple Plies Laminate Analyses Ply mm Multiple Strengths Fibers Size-scaling Strength Monte Carlo Distribution Fiber/ Interface Damage Evolution Stress matrix m 3D Finite Element Concentration Micromechanics Factors Atoms Molecular Interface nm Dynamics Behavior Electrons Inter-atomic Quantum Mechanics Potential Fig.2.2.Approach to multiscale modeling:scale coupling via information transfer It is not necessary to always start from the quantum mechanical scale and progress upward.In fact,the goal of composite design is to shift the critical scale of damage from the nanoscale,e.g.,the crack tip,to the much larger,observable,and detectable scale of collective fiber damage.Since a single fiber break does not initiate macroscopic failure,the details of the behavior at the smaller scales,while important,are not sufficient to predict failure.Therefore,one strategy is to envision possible modes of interface debonding and fiber/matrix constitutive behavior,as motivated by experi- ments or other theoretical models,and then use the multiscale modeling approach starting at the micromechanical scale.Parametric studies of the effect of interface and matrix behavior on the macroscopic fracture can then point to issues at smaller scales that would merit more detailed treat- ment.The work presented in this chapter focuses on the multiscale modeling

weak-link size-scaling models to predict ply strength and its statistical distribution as a function of physical size. Finally, the ply strength is used in standard laminated composite models to predict the strength and reliabi￾lity of the composite component. In the last stage, other damage phenomena such as interply delamination can occur and change the local stresses in the plies themselves. In such cases, the ply strength vs. size can be used at smaller scales to assess the onset of local ply damage due to these other damage modes. Fig. 2.2. Approach to multiscale modeling: scale coupling via information transfer It is not necessary to always start from the quantum mechanical scale and progress upward. In fact, the goal of composite design is to shift the critical scale of damage from the nanoscale, e.g., the crack tip, to the much larger, observable, and detectable scale of collective fiber damage. Since a single fiber break does not initiate macroscopic failure, the details of the behavior at the smaller scales, while important, are not sufficient to predict failure. Therefore, one strategy is to envision possible modes of interface debonding and fiber/matrix constitutive behavior, as motivated by experi￾ments or other theoretical models, and then use the multiscale modeling approach starting at the micromechanical scale. Parametric studies of the effect of interface and matrix behavior on the macroscopic fracture can then point to issues at smaller scales that would merit more detailed treat￾ment. The work presented in this chapter focuses on the multiscale modeling Chapter 2: Multiscale Modeling of Tensile Failure 41

42 Z.Xia and W.A.Curtin of unidirectional fiber-reinforced composites starting from the micro- mechanical scale taking the interface behavior as a parametric input with quantities such as the interfacial coefficient of friction and interfacial strength obtained from experiments when applications to a particular material system are made. 2.2.2 Micromechanics at the Fiber/Matrix/Interface Scale The goal of modeling at the micromechanics scale is to compute the detailed stress redistribution around broken fibers with various interfacial deformation models and extract from such studies the average stress concentrations induced in the surrounding unbroken fibers and the stress recovery along the broken fiber due to interface shear resistance.Since introduction of a fiber break or a matrix crack causes large stress changes only in the vicinity of the crack,a small-scale model with high spatial refinement is used.The model used consists of a hexagonal array of uni- directional fibers with a fiber volume fraction of Vr.Making use of sym- metry,the model can be restricted to a 30 wedge,as shown in Fig.2.3a. Each fiber in this wedge section represents a distinct set of neighbors relative to the central fiber.A 3D finite element representation of this model is then constructed to calculate the stress distributions around broken fibers (Fig.2.3b,c).The axial length of the model depends on the interface and matrix behavior and is generally chosen such that the stress distribution at the end of the model is not affected by the stress redistributions caused by the introduction of fiber or matrix damage at the midplane.The size of the model in the radial direction(perpendicular to the fibers)is chosen so that the deformation of fibers at the outer perimeter is not affected by the imposed fiber damage.For example,with a single central broken fiber,we use the nearest eight sets of neighbors(43 fibers total).With seven broken fibers (fibers 1 and 2 broken in the 30 wedge section),a larger model extending out to tenth neighbors and containing a total of 91 fibers is used The mesh sizes are selected to obtain converged results,for which there is no a priori guidance except that there should be at least several elements in the matrix region between the fibers and within the fibers themselves.The model is subjected to tensile loading along the axis of the fibers,and the appropriate boundary conditions are shown in Fig.2.3b.The nodes of uncracked material at the crack plane (=0)have fixed displacements in the z-direction while the outer surface of the model is traction free

of unidirectional fiber-reinforced composites starting from the micro￾mechanical scale taking the interface behavior as a parametric input with quantities such as the interfacial coefficient of friction and interfacial strength obtained from experiments when applications to a particular material system are made. 2.2.2 Micromechanics at the Fiber/Matrix/Interface Scale The goal of modeling at the micromechanics scale is to compute the detailed stress redistribution around broken fibers with various interfacial deformation models and extract from such studies the average stress concentrations induced in the surrounding unbroken fibers and the stress recovery along the broken fiber due to interface shear resistance. Since introduction of a fiber break or a matrix crack causes large stress changes only in the vicinity of the crack, a small-scale model with high spatial refinement is used. The model used consists of a hexagonal array of uni￾directional fibers with a fiber volume fraction of Vf. Making use of sym￾metry, the model can be restricted to a 30° wedge, as shown in Fig. 2.3a. Each fiber in this wedge section represents a distinct set of neighbors relative to the central fiber. A 3D finite element representation of this model is then constructed to calculate the stress distributions around broken fibers (Fig. 2.3b,c). The axial length of the model depends on the interface and matrix behavior and is generally chosen such that the stress distribution at the end of the model is not affected by the stress redistributions caused by the introduction of fiber or matrix damage at the midplane. The size of the model in the radial direction (perpendicular to the fibers) is chosen so that the deformation of fibers at the outer perimeter is not affected by the imposed fiber damage. For example, with a single central broken fiber, we use the nearest eight sets of neighbors (43 fibers total). With seven broken fibers (fibers 1 and 2 broken in the 30° wedge section), a larger model extending out to tenth neighbors and containing a total of 91 fibers is used. The mesh sizes are selected to obtain converged results, for which there is no a priori guidance except that there should be at least several elements in the matrix region between the fibers and within the fibers themselves. The model is subjected to tensile loading along the axis of the fibers, and the appropriate boundary conditions are shown in Fig. 2.3b. The nodes of uncracked material at the crack plane (z = 0) have fixed displacements in the z-direction while the outer surface of the model is traction free. 42 Z. Xia and W.A. Curtin

Chapter 2:Multiscale Modeling of Tensile Failure 43 Mid-plane 309 .6 (a) (b) 369+03 344+03 3t9+03 296+03 70+3 16+0 1+0 97+0 2+ 48+0 2340 986+0 41+00 96402 251+0 583400 (c) Fig.2.3.(a)Optical image of Ti/SiC composite microstructure,(b)wedge section of model hexagonal distribution of fiber composite with boundary conditions for finite element analysis,and (c)a 30 wedge of finite element model showing the axial stress distribution in the fibers and matrix around a central broken fiber (reprinted with permission from [38])

(c) Fig. 2.3. (a) Optical image of Ti/SiC composite microstructure, (b) wedge section of model hexagonal distribution of fiber composite with boundary conditions for finite element analysis, and (c) a 30° wedge of finite element model showing the axial stress distribution in the fibers and matrix around a central broken fiber 43 (a) (b) 30o z r 2 5 3 4 8 6 9 10 11 1 7 12 16 15 14 13 Mid-plane z r 2 5 3 4 8 6 9 10 11 1 7 12 16 15 14 13 2 5 3 4 8 6 9 10 11 1 7 12 16 15 14 13 2 5 3 4 8 6 9 10 11 1 7 12 16 15 14 13 Mid-plane Chapter 2: Multiscale Modeling of Tensile Failure (reprinted with permission from [38])

44 Z.Xia and W.A.Curtin Modeling of loading transfer through a fiber/matrix interface is a key step to properly simulate the stress distributions in the fibers.The inter- faces can be classified into weak and strong bond interfaces according to interfacial bonding strength.If the fiber/matrix interface is strong,no inter- facial debonding occurs.Modeling of such an interface is simple.Since there is no sliding between the fiber and matrix,the matrix and fiber elements are compatible and shear the same nodes at the interface in the finite element model.However,if the interfacial bonding is weak,the inter- face will debond,leading to sliding during loading.In this case,contact elements can be used to simulate stress transfer across the fiber/matrix interface.If the residual thermal stresses (axial tension in the matrix,axial compression in the fibers,and radial compression o,at the interface)are high,the fiber/matrix bond strength is usually assumed to be zero for simplification.Interfacial stress transfer is then realized by Coulomb friction at the interface so that the friction shear stress r along the interface in the slip zone is simply =-uo,where u is the coefficient of friction. The introduction of a fiber break in the central fiber at the midplane of the model induces significant changes in the local stresses around the break (e.g.,Fig.2.3c).The stress distribution around a broken fiber is very complex.Multiscale modeling progresses by assuming that all of these details are not relevant to the desired macroscopic behavior.For the pro- pagation of damage among fibers,the tensile stresses in the unbroken fibers drive the growth of preexisting flaws in those fibers if the tensile stress is large enough.It is assumed that it is sufficient to consider the average ten- sile stress through the cross-section of any fiber,rather than maintain the full spatial variation.While it is certainly true that any particular fiber can have a flaw that experiences a stress higher or lower than the average [20, 33],the influence of such an effect has not been considered.Condensing the detailed information from studies such as that shown in Fig.2.3c, consider the stress in the broken fiber and the stresses in the surrounding fibers.The stress in the broken fiber is zero at the break point and recovers along the broken fiber,as shown in Fig.2.4a.Shear deformation along the interface,by either shear yielding of a well-bonded plastically deforming matrix or frictional sliding along a debonded interface,leads to a nearly linear recovery of axial stress in the fiber.Figure 2.4b shows the average axial stress concentration factor (SCF=actual stress normalized by far- field applied fiber stress)in the plane of the fiber break on the successive sets of neighbors around the broken fiber.The stresses in the neighboring fibers are increased to compensate for the loss of load-carrying capacity in the broken fiber,with the SCF decreasing with increasing distance from

Modeling of loading transfer through a fiber/matrix interface is a key step to properly simulate the stress distributions in the fibers. The inter￾faces can be classified into weak and strong bond interfaces according to interfacial bonding strength. If the fiber/matrix interface is strong, no inter￾facial debonding occurs. Modeling of such an interface is simple. Since there is no sliding between the fiber and matrix, the matrix and fiber elements are compatible and shear the same nodes at the interface in the finite element model. However, if the interfacial bonding is weak, the inter￾face will debond, leading to sliding during loading. In this case, contact elements can be used to simulate stress transfer across the fiber/matrix interface. If the residual thermal stresses (axial tension in the matrix, axial compression in the fibers, and radial compression σr at the interface) are high, the fiber/matrix bond strength is usually assumed to be zero for simplification. Interfacial stress transfer is then realized by Coulomb friction at the interface so that the friction shear stress τ along the interface in the slip zone is simply r τ = −µσ , where µ is the coefficient of friction. The introduction of a fiber break in the central fiber at the midplane of the model induces significant changes in the local stresses around the complex. Multiscale modeling progresses by assuming that all of these details are not relevant to the desired macroscopic behavior. For the pro￾pagation of damage among fibers, the tensile stresses in the unbroken fibers drive the growth of preexisting flaws in those fibers if the tensile stress is large enough. It is assumed that it is sufficient to consider the average ten￾sile stress through the cross-section of any fiber, rather than maintain the full spatial variation. While it is certainly true that any particular fiber can the detailed information from studies such as that shown in Fig. 2.3c, consider the stress in the broken fiber and the stresses in the surrounding fibers. The stress in the broken fiber is zero at the break point and recovers along the broken fiber, as shown in Fig. 2.4a. Shear deformation along the interface, by either shear yielding of a well-bonded plastically deforming matrix or frictional sliding along a debonded interface, leads to a nearly linear recovery of axial stress in the fiber. Figure 2.4b shows the average axial stress concentration factor (SCF = actual stress normalized by far￾field applied fiber stress) in the plane of the fiber break on the successive sets of neighbors around the broken fiber. The stresses in the neighboring fibers are increased to compensate for the loss of load-carrying capacity in the broken fiber, with the SCF decreasing with increasing distance from 44 Z. Xia and W.A. Curtin have a flaw that experiences a stress higher or lower than the average [20, break (e.g., Fig. 2.3c). The stress distribution around a broken fiber is very 33], the influence of such an effect has not been considered. Condensing

Chapter 2:Multiscale Modeling of Tensile Failure 45 1.08 1 1.06 0.8 Qe, 1.04 1.02 0 14 21 28 0 Normalized distance from fiber break,z/R, Distance from broken fibre/fiber spacing,d/s (a) (b) 1.08 1.06 1.04 1.02 0.98 0.96 14 21 28 35 Distance from the fiber break,z/R (c) Fig.2.4.(a)Axial stress distribution on the central broken fiber along the fiber direction z/R(R=fiber radius),normalized by the far-field fiber stress,(b)axial stress concentration factor (SCF)on the fibers as a function of the distance away from the broken fiber,normalized by fiber spacing s,and (c)average axial stress concentrations on the near-neighbor fibers along the fiber direction =Dashed lines in (a)and (c)show the approximated stress concentrations using a constant inter- facial shear stress r model that is employed in one of the larger-scale models (Green's function model) the broken fiber.The average stress concentration on the near-neighbor fibers vs.the distance z away from the crack plane is shown in Fig.2.4c. Near the plane of the break,the neighboring fiber stresses are larger than

0 0.2 0.4 0.6 0.8 1 1.2 0 7 14 21 28 Normalized distance from fiber break, z/R, Normalized axial stress, SCF 1 1.02 1.04 1.06 1.08 012345 Distance from broken fibre/fiber spacing, d/s SCF 1 1.02 1.04 1.06 1.08 012345 Distance from broken fibre/fiber spacing, d/s SCF (a) (b) 0.96 0.98 1 1.02 1.04 1.06 1.08 0 7 14 21 28 35 Distance from the fiber break, z/R Stress concentration factor, SCF 0.96 0.98 1 1.02 1.04 1.06 1.08 0 7 14 21 28 35 Distance from the fiber break, z/R Stress concentration factor, SCF ( c) Fig. 2.4. (a) Axial stress distribution on the central broken fiber along the fiber direction z/R (R = fiber radius), normalized by the far-field fiber stress, (b) axial stress concentration factor (SCF) on the fibers as a function of the distance away from the broken fiber, normalized by fiber spacing s, and (c) average axial stress concentrations on the near-neighbor fibers along the fiber direction z. Dashed lines in (a) and (c) show the approximated stress concentrations using a constant inter￾facial shear stress τ model that is employed in one of the larger-scale models (Green’s function model) the broken fiber. The average stress concentration on the near-neighbor fibers vs. the distance z away from the crack plane is shown in Fig. 2.4c. Near the plane of the break, the neighboring fiber stresses are larger than Chapter 2: Multiscale Modeling of Tensile Failure 45

46 Z.Xia and W.A.Curtin in the far-field.Within increasing distance z,the broken fiber recovers its load-carrying capacity and the SCFs of the surrounding fibers thus decrease over a similar length scale.The SCF on the neighboring fibers can actually fall below unity before recovering to unity at larger distances, which is due to bending that arises from the need to satisfy compatibility. The details,such as those shown in Fig.2.4,depend on the input con- stitutive properties:the fiber elastic modulus,the matrix elastic modulus and plastic flow behavior,if any,and the interface constitutive model. However,the results are generically those shown in Fig.2.4,and the SCFs and length scales of stress recovery are the information derived from the detailed micromechanical model that is passed to a larger-scale damage accumulation model. 2.2.3 Mesoscale Modeling of Fiber Damage Evolution The finite element(FE)models provide the detailed stress state around a single broken fiber.Larger clusters of broken fibers can be investigated, but such a direct numerical approach is limited to symmetric clusters of breaks due to the symmetry of the unit cell.Decreasing the symmetry of the unit cell is possible but computationally difficult.Furthermore,to under- stand the size scaling of the composite strength and,thus,predict strengths of very large samples,requires hundreds of simulations of failure in com- posites having several hundred fibers.Here,two alternative approaches to obtaining reasonably accurate but computationally more feasible results: the 3D shear-lag and Green's function methods are discussed.The goal of these methods is to reliably calculate the stress states in any surviving fibers given an arbitrary spatial distribution of fiber breaks,while cap- turing the proper SCFs and length scales computed from the detailed finite element method (FEM)models. Shear-lag method The shear-lag model(SLM)for fiber SCFs has a long history,dating back to the work of Hedgepeth and Hedgepeth and Van Dyke [4,14,15,30].In this model,the fibers are treated as one-dimensional extensional elements of modulus Er while the matrix is treated as a material with modulus Gm that transfers tensile loads among fibers via shear deformation only and carries no tensile loads.Here we discuss a 3D SLM developed by Okabe and Takeda [25]that incorporates interface sliding due to friction and/or

in the far-field. Within increasing distance z, the broken fiber recovers its load-carrying capacity and the SCFs of the surrounding fibers thus decrease over a similar length scale. The SCF on the neighboring fibers can actually fall below unity before recovering to unity at larger distances, which is due to bending that arises from the need to satisfy compatibility. The details, such as those shown in Fig. 2.4, depend on the input con￾stitutive properties: the fiber elastic modulus, the matrix elastic modulus and plastic flow behavior, if any, and the interface constitutive model. and length scales of stress recovery are the information derived from the detailed micromechanical model that is passed to a larger-scale damage accumulation model. 2.2.3 Mesoscale Modeling of Fiber Damage Evolution The finite element (FE) models provide the detailed stress state around a single broken fiber. Larger clusters of broken fibers can be investigated, but such a direct numerical approach is limited to symmetric clusters of breaks due to the symmetry of the unit cell. Decreasing the symmetry of the unit cell is possible but computationally difficult. Furthermore, to under￾stand the size scaling of the composite strength and, thus, predict strengths of very large samples, requires hundreds of simulations of failure in com￾posites having several hundred fibers. Here, two alternative approaches to the 3D shear-lag and Green’s function methods are discussed. The goal of these methods is to reliably calculate the stress states in any surviving fibers given an arbitrary spatial distribution of fiber breaks, while cap￾turing the proper SCFs and length scales computed from the detailed finite element method (FEM) models. Shear-lag method The shear-lag model (SLM) for fiber SCFs has a long history, dating back this model, the fibers are treated as one-dimensional extensional elements of modulus Ef while the matrix is treated as a material with modulus Gm that transfers tensile loads among fibers via shear deformation only and carries no tensile loads. Here we discuss a 3D SLM developed by Okabe 46 Z. Xia and W.A. Curtin to the work of Hedgepeth and Hedgepeth and Van Dyke [4, 14, 15, 30]. In obtaining reasonably accurate but computationally more feasible results: and Takeda [25] that incorporates interface sliding due to friction and/or However, the results are generically those shown in Fig. 2.4, and the SCFs