534 R.Talreja and C.V.Singh a unidirectional fiber-reinforced composite are depicted in Fig.12.2.When fracture strain of the fiber is greater than that of the matrix,i.e.,r>&m,a crack originating at a point of stress concentration,e.g.,voids,air bubbles, or inclusions,in the matrix is either halted by the fiber,if the stress is not high enough,or it may pass around the fiber without destroying the interfacial bond(Fig.12.2a).As the applied load increases,the fiber and matrix deform differentially,resulting in a buildup of large local stresses in the fiber.This causes local Poisson contraction;and eventually shear force developed at the interface exceeds the interfacial shear strength,resulting in interfacial debonding at the crack plane that extends some distance along the fiber at the interface (Fig.12.2c). (a) (b) (c) Fig.12.2.Mechanics of interfacial debonding in a simple composite [20] Interfacial sliding Interfacial sliding between constituents in a composite can take place by differential displacement of the constituents.One example of this is when fibers and matrix in a composite are not bonded together adhesively but by a"shrink-fit"mechanism,due to difference in thermal expansion properties of the constituents.On thermomechanical loading,the shrink-fit (residual) stresses can be removed,leading to a relative displacement(sliding)at the interface.The relief of interfacial normal stress can also occur when a matrix crack tip approaches or hits the interface. When the two constituents are bonded together adhesively,interfacial sliding can occur subsequent to debonding if a compressive normal stress on the interface is present.The debonding can be induced by a matrix crack,or it can result from growth of interfacial defects.Thus,interfacial sliding that follows debonding can be a separate damage mode or it can be a damage mode coupled with matrix damage
a unidirectional fiber-reinforced composite are depicted in Fig. 12.2. When fracture strain of the fiber is greater than that of the matrix, i.e., εf > εm, a crack originating at a point of stress concentration, e.g., voids, air bubbles, or inclusions, in the matrix is either halted by the fiber, if the stress is not high enough, or it may pass around the fiber without destroying the interfacial bond (Fig. 12.2a). As the applied load increases, the fiber and matrix deform differentially, resulting in a buildup of large local stresses in the fiber. This causes local Poisson contraction; and eventually shear force developed at the interface exceeds the interfacial shear strength, resulting in interfacial debonding at the crack plane that extends some distance along the fiber at the interface (Fig. 12.2c). Fig. 12.2. Mechanics of interfacial debonding in a simple composite [20] Interfacial sliding Interfacial sliding between constituents in a composite can take place by differential displacement of the constituents. One example of this is when fibers and matrix in a composite are not bonded together adhesively but by a “shrink-fit” mechanism, due to difference in thermal expansion properties of the constituents. On thermomechanical loading, the shrink-fit (residual) stresses can be removed, leading to a relative displacement (sliding) at the interface. The relief of interfacial normal stress can also occur when a matrix crack tip approaches or hits the interface. When the two constituents are bonded together adhesively, interfacial sliding can occur subsequent to debonding if a compressive normal stress on the interface is present. The debonding can be induced by a matrix crack, or it can result from growth of interfacial defects. Thus, interfacial sliding that follows debonding can be a separate damage mode or it can be a damage mode coupled with matrix damage. 534 R. Talreja and C.V. Singh
Chapter 12:Multiscale Modeling for Damage Analysis 535 When the interface between the matrix and the fiber debonds,this relieves the tensile residual stresses in the matrix.Due to different stresses in the matrix and the fiber at the interface,the fibers slide on the interfacial surface.Subsequently,the sliding surfaces cause degradation of material due to frictional wear at the interface.Pullout and pushback tests are useful in determining the stress required to cause interfacial sliding.This mostly depends upon the strength of the adhesive bond between the matrix and the fiber at the interface. Fiber microbuckling When a unidirectional composite is loaded in compression,the failure is governed by the matrix and occurs through a mechanism known as micro- buckling of fibers.There are two basic modes of microbuckling deformation: extensional”and“shear'”modes[5l],as shown in Fig.l2.3,depending upon whether the fibers deform“out of phase”or“in phase.”The com- pressive strength corresponds to the onset of instability and is given as Extensional Mode Shear Mode Fig.12.3.Extensional and shear modes of microbuckling [51]
When the interface between the matrix and the fiber debonds, this relieves the tensile residual stresses in the matrix. Due to different stresses in the matrix and the fiber at the interface, the fibers slide on the interfacial surface. Subsequently, the sliding surfaces cause degradation of material due to frictional wear at the interface. Pullout and pushback tests are useful in determining the stress required to cause interfacial sliding. This mostly depends upon the strength of the adhesive bond between the matrix and the fiber at the interface. Fiber microbuckling When a unidirectional composite is loaded in compression, the failure is governed by the matrix and occurs through a mechanism known as microbuckling of fibers. There are two basic modes of microbuckling deformation: “extensional” and “shear” modes [51], as shown in Fig. 12.3, depending upon whether the fibers deform “out of phase” or “in phase.” The compressive strength corresponds to the onset of instability and is given as Fig. 12.3. Extensional and shear modes of microbuckling [51] Chapter 12: Multiscale Modeling for Damage Analysis 535
536 R.Talreja and C.V.Singh V:E Em (12.1) 0。=2 31-') for the extension mode and G (12.2) o.1- for the shear mode,where E and G denote Young's modulus and shear modulus,respectively,and subscripts“f”and“m”designate fiber and matrix,respectively. Although these expressions are based on energy balance,they do not agree with experimental observations.As an alternative,it has been argued that manufacturing of composites tends to cause misalignment of fibers, which can induce localized kinking of fiber bundles.The kinking process is driven by local shear,which depends on the initial misalignment angle o [3].The critical compressive stress corresponding to instability is given by (12.3) 4 where zy represents the interlaminar shear strength.Budiansky [6] considered the kink band geometry and derived the following estimate for the kink band angle B in terms of the transverse modulus Er and shear modulus G of a two-dimensional(2D)composite layer: -0 (-1G-<tanB< (12.4) E E To account for shear deformation effects,Niu and Talreja [46] modeled the fiber as a generalized Timoshenko beam with the matrix as an elastic foundation.It was observed that not only an initial fiber mis- alignment but also any misalignment in the loading system can affect the critical stress for kinking. Delamination Delamination as a result of low-velocity impact loading is a major cause of failure in fiber-reinforced composites [7,9,40].Delamination can occur below the surface of a composite structure with a relatively light impact, such as that from a dropped tool,while the surface remains undamaged to visual inspection [9,28,50].The growth of delamination cracks under the subsequent application of external loads leads to a rapid deterioration of the mechanical properties and may cause catastrophic failure of the com-
ffm c f f 2 3(1 ) VEE V V σ = − (12.1) for the extension mode and m c f 1 G V σ = − (12.2) for the shear mode, where E and G denote Young’s modulus and shear modulus, respectively, and subscripts “f ” and “m” designate fiber and matrix, respectively. Although these expressions are based on energy balance, they do not agree with experimental observations. As an alternative, it has been argued that manufacturing of composites tends to cause misalignment of fibers, which can induce localized kinking of fiber bundles. The kinking process is driven by local shear, which depends on the initial misalignment angle φ 0 [3]. The critical compressive stress corresponding to instability is given by y c 0 , τ σ φ = (12.3) where τy represents the interlaminar shear strength. Budiansky [6] considered the kink band geometry and derived the following estimate for the kink band angle β in terms of the transverse modulus ET and shear modulus G of a two-dimensional (2D) composite layer: 2 2 c c T T ( 2 1) tan . G G E E σ σ β − − − << (12.4) To account for shear deformation effects, Niu and Talreja [46] modeled the fiber as a generalized Timoshenko beam with the matrix as an elastic foundation. It was observed that not only an initial fiber misalignment but also any misalignment in the loading system can affect the critical stress for kinking. Delamination Delamination as a result of low-velocity impact loading is a major cause of failure in fiber-reinforced composites [7, 9, 40]. Delamination can occur below the surface of a composite structure with a relatively light impact, such as that from a dropped tool, while the surface remains undamaged to visual inspection [9, 28, 50]. The growth of delamination cracks under the subsequent application of external loads leads to a rapid deterioration of the mechanical properties and may cause catastrophic failure of the com- 536 R. Talreja and C.V. Singh
Chapter 12:Multiscale Modeling for Damage Analysis 537 posite structure [55].Delamination is a substantial problem because the composite laminates,although having strength in the fiber direction,lack strength in the through-thickness direction.This essentially limits the strength of a traditional 2D composite to the properties of the brittle matrix alone [71].The development of interlaminar stresses is the primary cause of delamination in laminated fibrous composites.Delamination occurs when the interlaminar stress level exceeds the interlaminar strength.The inter- laminar stress level is associated with the specimen geometry and loading parameters,while the interlaminar strength is related to the material pro- perties [40,71].From an energy point of view,delamination cracks will grow when the energy required to overcome the cohesive force of the atoms is equal to the dissipation of the strain energy that is released by the crack [11].The delamination can be reduced by either improving the frac- ture toughness of the material or modifying the fiber architecture [8,421. Typically,a low-speed impact overstresses the matrix material,pro- ducing local subcritical cracking (microcracking).This does not necessarily produce fracture;however,it will result in stress redistribution and the concentration of energy and stress at the interply regions where large differences in material stiffness exist.The onset and rapid propagation of a crack results in sudden variations in both section properties and load paths within the composite local to the impactor.This requires an adaptive method to track the progression of damage and fracture growth. Fiber fracture As the applied load is increased,progressive matrix cracks lead to fiber/ matrix interfacial debonding and delamination;and the stress state inside laminate material becomes quite complex.Ultimately,when the laminate strain reaches fiber failure strain,the fibers start to fail;and multiple cracks develop in the fibers.The multiple fiber cracks also develop due to stress transfer in the regions where the matrix is not able to take any more load. Since at this load level other damage modes are also present,the real reason for ultimate failure is often not clear.At ultimate failure load,the matrix is shattered;and,evidently,the fibers carry the full failure load.The composites usually support large load and deformation at failure,although the measured ultimate strength clearly may not be reliable in actual applications [47].All fibers are not of the same strength,and a statistical variation of strength between fibers and along fiber lengths is used.In addition to strength and modulus,another important property of a fiber- reinforced composite is its resistance to fracture.The fracture toughness of a composite depends not only on the properties of the constituents but also significantly on the efficiency of bonding across the interface [33]
posite structure [55]. Delamination is a substantial problem because the composite laminates, although having strength in the fiber direction, lack strength in the through-thickness direction. This essentially limits the strength of a traditional 2D composite to the properties of the brittle matrix alone [71]. The development of interlaminar stresses is the primary cause of delamination in laminated fibrous composites. Delamination occurs when the interlaminar stress level exceeds the interlaminar strength. The interlaminar stress level is associated with the specimen geometry and loading parameters, while the interlaminar strength is related to the material properties [40, 71]. From an energy point of view, delamination cracks will grow when the energy required to overcome the cohesive force of the atoms is equal to the dissipation of the strain energy that is released by the crack [11]. The delamination can be reduced by either improving the fracture toughness of the material or modifying the fiber architecture [8, 42]. Typically, a low-speed impact overstresses the matrix material, producing local subcritical cracking (microcracking). This does not necessarily produce fracture; however, it will result in stress redistribution and the concentration of energy and stress at the interply regions where large differences in material stiffness exist. The onset and rapid propagation of a crack results in sudden variations in both section properties and load paths within the composite local to the impactor. This requires an adaptive method to track the progression of damage and fracture growth. Fiber fracture As the applied load is increased, progressive matrix cracks lead to fiber/ matrix interfacial debonding and delamination; and the stress state inside laminate material becomes quite complex. Ultimately, when the laminate strain reaches fiber failure strain, the fibers start to fail; and multiple cracks develop in the fibers. The multiple fiber cracks also develop due to stress transfer in the regions where the matrix is not able to take any more load. Since at this load level other damage modes are also present, the real reason for ultimate failure is often not clear. At ultimate failure load, the matrix is shattered; and, evidently, the fibers carry the full failure load. The composites usually support large load and deformation at failure, although the measured ultimate strength clearly may not be reliable in actual applications [47]. All fibers are not of the same strength, and a statistical variation of strength between fibers and along fiber lengths is used. In addition to strength and modulus, another important property of a fiberreinforced composite is its resistance to fracture. The fracture toughness of a composite depends not only on the properties of the constituents but also significantly on the efficiency of bonding across the interface [33]. Chapter 12: Multiscale Modeling for Damage Analysis 537
538 R.Talreja and C.V.Singh The damage mechanisms described above have different characteristics depending on a variety of geometric and material parameters.Each mech- anism has different governing length scales and evolves differently when the applied load is increased.Interactions between individual mechanisms further complicate the damage picture.As the loading increases,stress transfer takes place from a region of high damage to that of low damage, and the composite failure results from the criticality of the last load-bearing element or region.For clarity of treatment,the full range of damage can be separated into damage modes,treating them individually followed by examining their interactions.This approach will be discussed in detail in later sections with respect to ceramic matrix composites (CMCs)and polymer matrix composites(PMCs). 12.2.2 Damage-Induced Response of Composites The presence of damage in a composite induces permanent changes in the response with respect to the virgin state.One objective of multiscale model- ing is to relate these changes to the damage,specifically taking into account the scale(s)at which damage mechanisms operate.In this section,a simple case of unidirectional continuous fiber composites,which respond linear elastically in the virgin state,will be examined to illustrate how the response can be varied when multiple matrix cracking damage exists.Two cases will be considered (1)a constrained PMC loaded in tension trans- verse to fibers and (2)an unconstrained CMC loaded in tension along fibers. Constrained PMC loaded in tension transverse to fibers When a unidirectional PMC is loaded in uniform tension normal to fibers. it responds linear elastically until failure initiates from matrix or interfacial cracking.However,if this composite is bonded to stiff elastic elements and then loaded,still transverse to fibers,its failure changes from single fracture to multiple matrix cracking as described above.The response of the combined composite and the stiff elements changes as the multiple cracking progresses,i.e.,its intensity,measured by,e.g.,crack number density,increases.The changes in response induced by cracking depend on the constraining effect of the stiff elements.This phenomenon is con- veniently illustrated in Fig.12.4 by an axially loaded crossply composite [O/90ls in which the degree of constraint to transverse ply cracking can be varied by selecting the m/n ratio.Considering the strain apr at which first cracking occurs in the constrained transverse ply,Talreja [56] classified the constraint in four categories (Fig.12.5)(A)no constraint,(B)
The damage mechanisms described above have different characteristics depending on a variety of geometric and material parameters. Each mechanism has different governing length scales and evolves differently when the applied load is increased. Interactions between individual mechanisms further complicate the damage picture. As the loading increases, stress transfer takes place from a region of high damage to that of low damage, and the composite failure results from the criticality of the last load-bearing element or region. For clarity of treatment, the full range of damage can be separated into damage modes, treating them individually followed by examining their interactions. This approach will be discussed in detail in later sections with respect to ceramic matrix composites (CMCs) and polymer matrix composites (PMCs). 12.2.2 Damage-Induced Response of Composites The presence of damage in a composite induces permanent changes in the response with respect to the virgin state. One objective of multiscale modeling is to relate these changes to the damage, specifically taking into account the scale(s) at which damage mechanisms operate. In this section, a simple case of unidirectional continuous fiber composites, which respond linear elastically in the virgin state, will be examined to illustrate how the response can be varied when multiple matrix cracking damage exists. Two cases will be considered (1) a constrained PMC loaded in tension transverse to fibers and (2) an unconstrained CMC loaded in tension along fibers. Constrained PMC loaded in tension transverse to fibers When a unidirectional PMC is loaded in uniform tension normal to fibers, it responds linear elastically until failure initiates from matrix or interfacial cracking. However, if this composite is bonded to stiff elastic elements and then loaded, still transverse to fibers, its failure changes from single fracture to multiple matrix cracking as described above. The response of the combined composite and the stiff elements changes as the multiple cracking progresses, i.e., its intensity, measured by, e.g., crack number density, increases. The changes in response induced by cracking depend on the constraining effect of the stiff elements. This phenomenon is conveniently illustrated in Fig. 12.4 by an axially loaded crossply composite [0m/90n]s in which the degree of constraint to transverse ply cracking can be varied by selecting the m/n ratio. Considering the strain εFPF at which first cracking occurs in the constrained transverse ply, Talreja [56] classified the constraint in four categories (Fig. 12.5) (A) no constraint, (B) 538 R. Talreja and C.V. Singh