Chapter 12:Multiscale Modeling for Damage Analysis Ramesh Talreja and Chandra Veer Singh Department of Aerospace Engineering,Texas A&M University, College Station,TX 77843-3141.USA 12.1 Introduction The increased computational power and programming capabilities in recent years have given impetus to the so-called multiscale modeling,which imple- ments the largely intuitive notion that physical phenomena occurring at a lower length or size scale determine the observed response at a higher scale.A logical outcome of this thought is an organization of differentiated scales-from the lowest,such as nanometer scale,to the highest scale typical of the part or structure in mind-giving a hierarchy of scales. Working up the scales produces a hierarchical multiscale modeling,in which the essential challenge consists of"bridging"the scales.The simulation tech- niques,such as molecular dynamics simulation (MDS),succeed mostly in revealing phenomena from one scale to the next;but proceeding to three or more scales often necessitates unrealistic computing power even with the most versatile facilities available.In addition,the limitation of independent physical validation of the simulated results questions the wisdom of total reliance on the multiscale hierarchical modeling strategy. When it comes to subcritical (prefailure)damage in composites,the multiscale modeling concept needs closer examination,firstly,because the length scales of constituents and heterogeneities are fixed while those of damage evolve progressively,and secondly,because the mechanisms of damage tend to segregate in modes with individual characteristic scales All this is the subject of this chapter,which will first describe and clarify the damage mechanisms in common types of composites followed by the induced response observed at the macroscale.The hierarchical modeling
Chapter 12: Multiscale Modeling for Damage Analysis Ramesh Talreja and Chandra Veer Singh Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA 12.1 Introduction The increased computational power and programming capabilities in recent years have given impetus to the so-called multiscale modeling, which implements the largely intuitive notion that physical phenomena occurring at a lower length or size scale determine the observed response at a higher scale. A logical outcome of this thought is an organization of differentiated scales – from the lowest, such as nanometer scale, to the highest scale typical of the part or structure in mind – giving a hierarchy of scales. Working up the scales produces a hierarchical multiscale modeling, in which the essential challenge consists of “bridging” the scales. The simulation techniques, such as molecular dynamics simulation (MDS), succeed mostly in revealing phenomena from one scale to the next; but proceeding to three or more scales often necessitates unrealistic computing power even with the most versatile facilities available. In addition, the limitation of independent physical validation of the simulated results questions the wisdom of total reliance on the multiscale hierarchical modeling strategy. When it comes to subcritical (prefailure) damage in composites, the multiscale modeling concept needs closer examination, firstly, because the length scales of constituents and heterogeneities are fixed while those of damage evolve progressively, and secondly, because the mechanisms of damage tend to segregate in modes with individual characteristic scales. All this is the subject of this chapter, which will first describe and clarify the damage mechanisms in common types of composites followed by the induced response observed at the macroscale. The hierarchical modeling
530 R.Talreja and C.V.Singh approach will be discussed against this knowledge;and a different approach, named synergistic multiscale modeling,will be advocated.Assessment will be offered of the current state of this modeling,and future activities aimed at accomplishing its objectives will be outlined. The following treatment of multiscale modeling will draw upon a recent paper by Talreja [61]as well as other previous works. 12.2 Phenomenon of Damage in Composite Materials Engineered structures must be capable of performing their functions through- out a specified lifetime while being exposed to a series of events that include loading,environment,and damage threats.These events,either individually or in combination,can cause structural degradation,which,in turn,can affect the ability of the structure to perform its function.The per- formance degradation in structures made of composites is quite different when compared to metallic components because the failure is not uniquely defined in composite materials.To understand how composites may lose the ability to perform satisfactorily,some basic definitions related to damage of composite materials must be reviewed.Section 12.2.1 contains a brief overview of significant mechanisms that can degrade a composite material. In a conventional sense,fracture is understood to be "breakage"of material,or at a more fundamental level,breakage of atomic bonds,which manifests itself in formation of internal surfaces.Examples of fractures in composites are fiber fragmentation,cracks in matrix,fiber/matrix debonding, and separation of bonded plies (delamination).The field of fracture mech- anics concerns itself with conditions for enlargement of the surfaces of material separation. Damage refers to a collection of all the irreversible changes brought about by energy dissipating mechanisms,of which atomic bond breakage is an example.Unless specified differently,damage is understood to refer to distributed changes.Examples of damage are multiple fiber-bridged matrix cracking in a unidirectional composite,multiple intralaminar cracking in a laminate,local delamination distributed in an interlaminar plane,and fiber/ matrix interfacial slip associated with multiple matrix cracking.These damage mechanisms are explained in some detail in Sect.12.2.1.The field of damage mechanics deals with conditions for initiation and progression of distributed changes as well as consequences of those changes on the response of a material (and by implication,a structure)to external loading. Failure is defined as the inability of a given material system(and con- sequently,a structure made from it)to perform its design function.Fracture
approach will be discussed against this knowledge; and a different approach, named synergistic multiscale modeling, will be advocated. Assessment will be offered of the current state of this modeling, and future activities aimed at accomplishing its objectives will be outlined. The following treatment of multiscale modeling will draw upon a recent paper by Talreja [61] as well as other previous works. 12.2 Phenomenon of Damage in Composite Materials Engineered structures must be capable of performing their functions throughout a specified lifetime while being exposed to a series of events that include loading, environment, and damage threats. These events, either individually or in combination, can cause structural degradation, which, in turn, can affect the ability of the structure to perform its function. The performance degradation in structures made of composites is quite different when compared to metallic components because the failure is not uniquely defined in composite materials. To understand how composites may lose the ability to perform satisfactorily, some basic definitions related to damage of composite materials must be reviewed. Section 12.2.1 contains a brief overview of significant mechanisms that can degrade a composite material. In a conventional sense, fracture is understood to be “breakage” of material, or at a more fundamental level, breakage of atomic bonds, which manifests itself in formation of internal surfaces. Examples of fractures in composites are fiber fragmentation, cracks in matrix, fiber/matrix debonding, and separation of bonded plies (delamination). The field of fracture mechanics concerns itself with conditions for enlargement of the surfaces of material separation. Damage refers to a collection of all the irreversible changes brought about by energy dissipating mechanisms, of which atomic bond breakage is an example. Unless specified differently, damage is understood to refer to distributed changes. Examples of damage are multiple fiber-bridged matrix cracking in a unidirectional composite, multiple intralaminar cracking in a laminate, local delamination distributed in an interlaminar plane, and fiber/ matrix interfacial slip associated with multiple matrix cracking. These damage mechanisms are explained in some detail in Sect. 12.2.1. The field of damage mechanics deals with conditions for initiation and progression of distributed changes as well as consequences of those changes on the response of a material (and by implication, a structure) to external loading. Failure is defined as the inability of a given material system (and consequently, a structure made from it) to perform its design function. Fracture 530 R. Talreja and C.V. Singh
Chapter 12:Multiscale Modeling for Damage Analysis 531 is one example of a possible failure;but,generally,a material could fracture (locally)and still perform its design function.Upon suffering damage,e.g.,in the form of multiple cracking,a composite material may still continue to carry loads and,thereby,meet its load-bearing requirement but fail to deform in a manner needed for its other design requirements, such as vibration characteristics and deflection limits. Structural integriry is defined as the ability of a load-bearing structure to remain intact or functional upon the application of loads.In contrast to metals,remaining intact(not breaking up in pieces)for composites is not necessarily the same as remaining functional.Composites can lose their functionality by suffering degradation in their stiffness properties while still carrying significant loads. 12.2.1 Mechanisms of Damage Due to extreme levels of anisotropy and inhomogeneity of composites,a variety of damage mechanisms cause degradation in the material behavior. These can occur separately or in combination.A short description of each damage mechanism follows. Multiple matrix cracking Matrix cracks are usually the first observed form of damage in composite laminates [45].These are intralaminar or ply cracks,transverse to loading direction,traversing the thickness of the ply and running parallel to the fibers in that ply.The terms matrix microcracks,transverse cracks,intra- laminar cracks,and ply cracks are invariably used to refer to this very same phenomenon.Matrix cracks are observed during tensile loading, fatigue loading,changes in temperature,and thermocycling.Figure 12.1 a{.0°/90°/0°..} b{..+45/90°-45°.} Fig.12.1.Examples of matrix cracks observed on the free edges induced due to fatigue loading in composite laminates [37]
is one example of a possible failure; but, generally, a material could fracture (locally) and still perform its design function. Upon suffering damage, e.g., in the form of multiple cracking, a composite material may still continue to carry loads and, thereby, meet its load-bearing requirement but fail to deform in a manner needed for its other design requirements, such as vibration characteristics and deflection limits. Structural integrity is defined as the ability of a load-bearing structure to remain intact or functional upon the application of loads. In contrast to metals, remaining intact (not breaking up in pieces) for composites is not necessarily the same as remaining functional. Composites can lose their functionality by suffering degradation in their stiffness properties while still carrying significant loads. 12.2.1 Mechanisms of Damage Due to extreme levels of anisotropy and inhomogeneity of composites, a variety of damage mechanisms cause degradation in the material behavior. These can occur separately or in combination. A short description of each damage mechanism follows. Multiple matrix cracking Matrix cracks are usually the first observed form of damage in composite laminates [45]. These are intralaminar or ply cracks, transverse to loading direction, traversing the thickness of the ply and running parallel to the fibers in that ply. The terms matrix microcracks, transverse cracks, intralaminar cracks, and ply cracks are invariably used to refer to this very same phenomenon. Matrix cracks are observed during tensile loading, fatigue loading, changes in temperature, and thermocycling. Figure 12.1 Fig. 12.1. Examples of matrix cracks observed on the free edges induced due to fatigue loading in composite laminates [37] Chapter 12: Multiscale Modeling for Damage Analysis 531
532 R.Talreja and C.V.Singh illustrates matrix cracks observed on the free edges induced due to fatigue loading in composite laminates [37].Although matrix cracking does not cause structural failure by itself,it can result in significant degradation in material stiffness and also can induce more severe forms of damage,such as delamination and fiber breakage [44].Numerous studies of micro- cracking initiation were performed in the 1970s and early 1980s [4,13-15, 29,48,49].It was observed that the strain to initiate microcracking increases as the thickness of 90 plies decreases.Also,these microcracks form almost instantaneously across the width of the specimen. The first attempt to predict the strain to first microcrack used the first ply failure theory [18]where it is assumed that the first crack develops when the strain in the plies reaches the strain to failure in the plies.The predictions were not in agreement with the experimental observations since the first ply failure theory predicts that the strain to initiate micro- cracking will be independent of the ply thickness.The experimental obser- vations on laminates with a 90 layer on the surface [90/0s show that the strain to initiate microcracking is lower for laminates with cracks in surface plies than for laminates with cracks in central plies [52,54]. The simplest way to model transverse matrix cracks in composite laminates is to completely neglect the transverse stiffness of cracked plies, called the ply discount method.This method underestimates the stiffness of cracked laminates,since cracked plies,in reality,can take some loading Another simple way is shear lag analysis,wherein the load transfer between plies is assumed to take place in shear layers between neighboring plies. The normal stress in the external load direction is assumed to be constant over the ply thickness.The thicknesses and stiffness of these shear layers are generally unknown,and the variations in the thickness direction of local ply stresses and strains are also neglected in the shear lag theory.The shear lag theory has limited success for crossply laminates [19,25,39,62]. For crossply laminates,the most successful approach is the variational method.By application of the principle of minimum complementary potential energy,Hashin [21,22]derived estimates for thermomechanical properties and local ply stresses,which were in good agreement with experimental data.Varna and Berglund [65]later made improvements to the Hashin model by use of more accurate trial stress functions.A disadvantage of the variational method is that it is extremely difficult to use for laminate lay- ups other than crossplies.McCartney [43]used Reissner's energy function to derive governing equations similar to Hashin's model.He applied this approach to doubly cracked crossply laminates assuming that the in-plane normal stress dependence on the two in-plane coordinates is given by two independent functions.Gudmundson and coworkers [16,17]considered
illustrates matrix cracks observed on the free edges induced due to fatigue loading in composite laminates [37]. Although matrix cracking does not cause structural failure by itself, it can result in significant degradation in material stiffness and also can induce more severe forms of damage, such as delamination and fiber breakage [44]. Numerous studies of microcracking initiation were performed in the 1970s and early 1980s [4, 13–15, 29, 48, 49]. It was observed that the strain to initiate microcracking increases as the thickness of 90° plies decreases. Also, these microcracks form almost instantaneously across the width of the specimen. The first attempt to predict the strain to first microcrack used the first ply failure theory [18] where it is assumed that the first crack develops when the strain in the plies reaches the strain to failure in the plies. The predictions were not in agreement with the experimental observations since the first ply failure theory predicts that the strain to initiate microcracking will be independent of the ply thickness. The experimental observations on laminates with a 90° layer on the surface [90n/0m]s show that the strain to initiate microcracking is lower for laminates with cracks in surface plies than for laminates with cracks in central plies [52, 54]. The simplest way to model transverse matrix cracks in composite laminates is to completely neglect the transverse stiffness of cracked plies, called the ply discount method. This method underestimates the stiffness of cracked laminates, since cracked plies, in reality, can take some loading. Another simple way is shear lag analysis, wherein the load transfer between plies is assumed to take place in shear layers between neighboring plies. The normal stress in the external load direction is assumed to be constant over the ply thickness. The thicknesses and stiffness of these shear layers are generally unknown, and the variations in the thickness direction of local ply stresses and strains are also neglected in the shear lag theory. The shear lag theory has limited success for crossply laminates [19, 25, 39, 62]. For crossply laminates, the most successful approach is the variational method. By application of the principle of minimum complementary potential energy, Hashin [21, 22] derived estimates for thermomechanical properties and local ply stresses, which were in good agreement with experimental data. Varna and Berglund [65] later made improvements to the Hashin model by use of more accurate trial stress functions. A disadvantage of the variational method is that it is extremely difficult to use for laminate layups other than crossplies. McCartney [43] used Reissner’s energy function to derive governing equations similar to Hashin’s model. He applied this approach to doubly cracked crossply laminates assuming that the in-plane normal stress dependence on the two in-plane coordinates is given by two independent functions. Gudmundson and coworkers [16, 17] considered 532 R. Talreja and C.V. Singh
Chapter 12:Multiscale Modeling for Damage Analysis 533 laminates with general layup and used the homogenization technique to derive expressions for stiffness and thermal expansion coefficient of laminates with cracks in layers of three-dimensional(3D)laminates.These expressions correlate damaged laminate thermoelastic properties with parameters characterizing crack behavior:the average crack opening displacement (COD)and the average crack face sliding.These parameters follow from the solution of the local boundary value problem,and their determination is a very complex task.Also,the effect of neighboring layers on crack face displacements was neglected;and the displacements were determined assuming a periodic system of cracks in an infinite homogeneous, transversely isotropic medium(90 layer).The application of their methodo- logy by other researchers has been rather limited due to the fairly complex form of the presented solutions. An alternative way to describe the mechanical behavior of matrix- cracked laminates is to apply concepts of damage mechanics.Generally speaking,the continuum damage mechanics(CDM)approaches [1,2,56, 57]may be used to describe the stiffness of laminates with intralaminar cracks in off-axis plies of any orientation.The damage is represented by internal state variables (ISVs),and the laminate constitutive equations are expressed in general forms containing ISV and a certain number of material constants.These constants must be determined for each laminate configuration considered either experimentally,measuring stiffness for a laminate with a certain crack density,or using finite element(FE)analysis. This limitation is partially removed in synergistic damage mechanics (SDM)suggested by Talreja [60],which incorporates micromechanics information in determining the material constants.The SDM approach has proved to be quite efficient for a variety of laminate layups and material systems.The present chapter builds on this methodology,and relevant details will be discussed later. Interfacial debonding The performance of a composite is markedly influenced by the properties of the interface between the fiber and matrix resin.The adhesion bond at the interfacial surface affects the macroscopic mechanical properties of the composite.The interface plays a significant role in stress transfer between fiber and matrix.Controlling interfacial properties thus leads to the control of composite performance.In unidirectional composites,debonding occurs at the interface between fiber and matrix when the interface is weak.The longitudinal interfacial debonding behavior of single-fiber composites has been studied in detail by the use of the pullout [26,38,73]and frag- mentation [10,12,24,72]tests.The mechanics of interfacial debonding in
laminates with general layup and used the homogenization technique to derive expressions for stiffness and thermal expansion coefficient of laminates with cracks in layers of three-dimensional (3D) laminates. These expressions correlate damaged laminate thermoelastic properties with parameters characterizing crack behavior: the average crack opening displacement (COD) and the average crack face sliding. These parameters follow from the solution of the local boundary value problem, and their determination is a very complex task. Also, the effect of neighboring layers on crack face displacements was neglected; and the displacements were determined assuming a periodic system of cracks in an infinite homogeneous, transversely isotropic medium (90° layer). The application of their methodology by other researchers has been rather limited due to the fairly complex form of the presented solutions. An alternative way to describe the mechanical behavior of matrixcracked laminates is to apply concepts of damage mechanics. Generally speaking, the continuum damage mechanics (CDM) approaches [1, 2, 56, 57] may be used to describe the stiffness of laminates with intralaminar cracks in off-axis plies of any orientation. The damage is represented by internal state variables (ISVs), and the laminate constitutive equations are expressed in general forms containing ISV and a certain number of material constants. These constants must be determined for each laminate configuration considered either experimentally, measuring stiffness for a laminate with a certain crack density, or using finite element (FE) analysis. This limitation is partially removed in synergistic damage mechanics (SDM) suggested by Talreja [60], which incorporates micromechanics information in determining the material constants. The SDM approach has proved to be quite efficient for a variety of laminate layups and material systems. The present chapter builds on this methodology, and relevant details will be discussed later. Interfacial debonding The performance of a composite is markedly influenced by the properties of the interface between the fiber and matrix resin. The adhesion bond at the interfacial surface affects the macroscopic mechanical properties of the composite. The interface plays a significant role in stress transfer between fiber and matrix. Controlling interfacial properties thus leads to the control of composite performance. In unidirectional composites, debonding occurs at the interface between fiber and matrix when the interface is weak. The longitudinal interfacial debonding behavior of single-fiber composites has been studied in detail by the use of the pullout [26, 38, 73] and fragmentation [10, 12, 24, 72] tests. The mechanics of interfacial debonding in Chapter 12: Multiscale Modeling for Damage Analysis 533
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