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