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 micromechanical 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 unidirectional fibers with a fiber volume fraction of Vf. Making use of symmetry, 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 interfaces can be classified into weak and strong bond interfaces according to interfacial bonding strength. If the fiber/matrix interface is strong, no interfacial 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 interface 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 propagation 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 tensile 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 farfield 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 interfacial 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 constitutive 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 understand the size scaling of the composite strength and, thus, predict strengths of very large samples, requires hundreds of simulations of failure in composites 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 capturing 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