《纺织复合材料》课程参考文献（Multiscale Modeling and Simulation of Composite Materials and Structures）Chapter 11 Multiscale Modeling of the Evolution of Damage in Heterogeneous Viscoelastic Solids

Chapter 11:Multiscale Modeling of the Evolution of Damage in Heterogeneous Viscoelastic Solids David H.Allen and Roberto F.Soares University of Nebraska-Lincoln,Lincoln,NE 68520,USA 11.1 Introduction It has long been recognized that one of the primary failure modes in solids is due to crack growth,whether it be a single or multiple cracks.It is known, for instance,that Da Vinci [14]proposed experiments of this type in the late fifteenth century.Indeed,modern history is replete with accounts of events wherein fracture-induced failure of structural components has caused the loss of significant life.Such events are common in buildings subjected to acts of nature,such as earthquakes,aircraft subjected to incle- ment weather,and even human organs subjected to aging.Therefore,it would seem self-evident that cogent models capable of predicting such cata strophic events could be utilized to avoid much loss of life.However, despite the fact that such events occur regularly,the ability to predict the evolution of cracks,especially in inelastic media,continues to elude scien- tists and engineers.This appears to be at least due,in part,to two as yet unresolved issues (1)there is still no agreed upon model for predicting crack extension in inelastic media and(2)the prediction of the extension of multiple cracks simultaneously in the same object is as yet untenable. While it would be presumptuous to say that the authors have resolved these two outstanding issues,there is at least a glimmer of hope that these two issues may be resolved by using an approach not unlike that proposed herein.This chapter outlines an approach for predicting the evolution of multiple cracks in heterogeneous viscoelastic media that ultimately leads to failure of the component to perform its intended task.Examples of such components would include geologic formations,cementitious roadways

Chapter 11: Multiscale Modeling of the Evolution of Damage in Heterogeneous Viscoelastic Solids David H. Allen and Roberto F. Soares University of Nebraska-Lincoln, Lincoln, NE 68520, USA 11.1 Introduction It has long been recognized that one of the primary failure modes in solids is due to crack growth, whether it be a single or multiple cracks. It is known, for instance, that Da Vinci [14] proposed experiments of this type in the late fifteenth century. Indeed, modern history is replete with accounts of events wherein fracture-induced failure of structural components has caused the loss of significant life. Such events are common in buildings subjected to acts of nature, such as earthquakes, aircraft subjected to incle￾ment weather, and even human organs subjected to aging. Therefore, it would seem self-evident that cogent models capable of predicting such cata￾strophic events could be utilized to avoid much loss of life. However, despite the fact that such events occur regularly, the ability to predict the evolution of cracks, especially in inelastic media, continues to elude scien￾tists and engineers. This appears to be at least due, in part, to two as yet unresolved issues (1) there is still no agreed upon model for predicting crack extension in inelastic media and (2) the prediction of the extension of multiple cracks simultaneously in the same object is as yet untenable. While it would be presumptuous to say that the authors have resolved these two outstanding issues, there is at least a glimmer of hope that these two issues may be resolved by using an approach not unlike that proposed herein. This chapter outlines an approach for predicting the evolution of multiple cracks in heterogeneous viscoelastic media that ultimately leads to failure of the component to perform its intended task. Examples of such components would include geologic formations, cementitious roadways

496 D.H.Allen and R.F.Soares human organs,and advanced structures,including composite aircraft com- ponents and defensive armor such as that used on tanks.Implicit in the need for deploying such a model as that proposed herein are the following two requirements (1)at least some subdomain of the medium must be inelastic and (2)cracks must grow on at least two significantly different length scales prior to failure of the component. The model that is proposed herein for addressing this problem is posed entirely within the confines of the fundamental assumption embodied in continuum mechanics,i.e.,that the mass density of a body is continuously differentiable in spatial coordinates on all length scales of interest so that cracks that initiate on the scale of single atoms or molecules cannot be modeled by this approach,implying that the smallest scale that can be considered is of the order of tens to hundreds of nanometers. This chapter opens with a short historical review of developments that have led up to the current state of knowledge on this subject,followed by a detailed description of the methodology proposed by the authors for addressing this problem.This will be followed by a few example problems that are meant to illustrate how the approach described herein can be utilized to make predictions of practical significance. 11.2 Historical Review The discipline of mechanics,the study of the motion of bodies,dates to the ancients.Chief among these is Archimedes [32],who enunciated the prin- ciple of the lever among other achievements.However,the first systematic study of the mechanics of bodies is attributed to Galileo [19]in the early seventeenth century.These accomplishments were not withstanding,it was not until the early nineteenth century that concerted efforts were made to study the motions of deformable bodies within the context of continuum mechanics.These efforts appear to have been initiated with the study of plates by Germain [20]and were followed shortly thereafter by the seminal papers by Navier [26]and Cauchy [10]on the prediction of deformations in elastic bodies.These formulations utilized Newton's laws of motion [27],together with definitions of strain and the necessary idea of the con- stitution of an elastic body,first enunciated by Hooke [8],a contemporary of Newton.These initial formulations did not encompass the notion of dissipation of energy,so the prediction of failure was not a component of these models.However,over the course of the succeeding century,the for- mulation of fundamental concepts of thermodynamics led to the first cogent theory of fracture by Griffith [21]in 1920

need for deploying such a model as that proposed herein are the following two requirements (1) at least some subdomain of the medium must be inelastic and (2) cracks must grow on at least two significantly different length scales prior to failure of the component. The model that is proposed herein for addressing this problem is posed entirely within the confines of the fundamental assumption embodied in continuum mechanics, i.e., that the mass density of a body is continuously differentiable in spatial coordinates on all length scales of interest so that cracks that initiate on the scale of single atoms or molecules cannot be modeled by this approach, implying that the smallest scale that can be considered is of the order of tens to hundreds of nanometers. This chapter opens with a short historical review of developments that have led up to the current state of knowledge on this subject, followed by a detailed description of the methodology proposed by the authors for addressing this problem. This will be followed by a few example problems that are meant to illustrate how the approach described herein can be utilized to make predictions of practical significance. 11.2 Historical Review The discipline of mechanics, the study of the motion of bodies, dates to the ancients. Chief among these is Archimedes [32], who enunciated the prin￾ciple of the lever among other achievements. However, the first systematic study of the mechanics of bodies is attributed to Galileo [19] in the early seventeenth century. These accomplishments were not withstanding, it was not until the early nineteenth century that concerted efforts were made to study the motions of deformable bodies within the context of continuum mechanics. These efforts appear to have been initiated with the study of plates by Germain [20] and were followed shortly thereafter by the seminal in elastic bodies. These formulations utilized Newton’s laws of motion [27], together with definitions of strain and the necessary idea of the con￾of Newton. These initial formulations did not encompass the notion of dissipation of energy, so the prediction of failure was not a component of these models. However, over the course of the succeeding century, the for￾mulation of fundamental concepts of thermodynamics led to the first cogent theory of fracture by Griffith [21] in 1920. human organs, and advanced structures, including composite aircraft com￾ponents and defensive armor such as that used on tanks. Implicit in the D.H. Allen and R.F. Soares stitution of an elastic body, first enunciated by Hooke [8], a contemporary papers by Navier [26] and Cauchy [10] on the prediction of deformations 496

Chapter 11:Multiscale Modeling of the Evolution of Damage 497 Griffith proposed that a crack would extend in an elastic body whenever GZGc, (11.1) where G is the energy released per unit area of crack produced and Gc is assumed to be a material constant called the critical energy release rate. Though succeeding progress has been slow to develop,this monum- ental proposition seems to have been the key step that was necessary to begin to make somewhat accurate predictions of crack growth.Two obstacles lay in the way before the usefulness of Griffith's proposition could be ascertained.The first obstacle was centered around the right-hand side of inequality (11.1):how to measure the material property required to make cogent predictions.The answer to this question was suggested in a paper by Rice [29]and proven mathematically a decade later by Gurtin [22].Subsequently,techniques have been developed for quite accurately measuring the critical energy release rate for a broad range of materials. The other obstacle arose due to the left-hand side of inequality (11.1):how to accurately calculate the available energy in a body necessary to produce new crack surface area.This issue is complicated by the fact that,in an imaginary elastic body,it is necessary for the stresses at a crack tip to be singular in order for there to be a nonzero energy available for crack extension.This problem has been studied in significant detail over the past half-century with some success.However,it would be presumptive to say that the subject is resolved;because in reality,it is not possible for the stresses at a crack tip to be singular. Initial experimental results for brittle materials indicated that Griffith's proposition was accurate.However,when experimental results were obtained for ductile materials,such as crystalline metals,experimental results com- pared less favorably to predictions.For some time,efforts were made to improve upon the calculations of the available energy for crack growth in ductile materials;and to make these calculations,researchers turned to the more advanced constitutive theory,such as that embodied in plasticity theory [24].However,it is now widely understood that Griffith's proposition is not accurate for some ductile materials due to the fact that energy dissipa- tion occurs in a variety of ways other than crack extension,and in ways that depend on the history of loading of the body.In these circumstances,it may be more appropriate to envision the critical energy release rate Ge as a history-dependent material property rather than a material constant.In the meantime,other approaches have been developed,such as cohesive zone models [7,16],that do not require the concept of a critical energy release

Griffith proposed that a crack would extend in an elastic body whenever C G G≥ , (11.1) where G is the energy released per unit area of crack produced and GC is assumed to be a material constant called the critical energy release rate. Though succeeding progress has been slow to develop, this monum￾obstacles lay in the way before the usefulness of Griffith’s proposition could be ascertained. The first obstacle was centered around the right-hand side of inequality (11.1): how to measure the material property required to make cogent predictions. The answer to this question was suggested in a paper by Rice [29] and proven mathematically a decade later by Gurtin [22]. Subsequently, techniques have been developed for quite accurately measuring the critical energy release rate for a broad range of materials. The other obstacle arose due to the left-hand side of inequality (11.1): how to accurately calculate the available energy in a body necessary to produce new crack surface area. This issue is complicated by the fact that, in an imaginary elastic body, it is necessary for the stresses at a crack tip to be singular in order for there to be a nonzero energy available for crack extension. This problem has been studied in significant detail over the past half-century with some success. However, it would be presumptive to say that the subject is resolved; because in reality, it is not possible for the stresses at a crack tip to be singular. Initial experimental results for brittle materials indicated that Griffith’s proposition was accurate. However, when experimental results were obtained for ductile materials, such as crystalline metals, experimental results com￾pared less favorably to predictions. For some time, efforts were made to improve upon the calculations of the available energy for crack growth in ductile materials; and to make these calculations, researchers turned to the more advanced constitutive theory, such as that embodied in plasticity theory [24]. However, it is now widely understood that Griffith’s proposition is not accurate for some ductile materials due to the fact that energy dissipa￾tion occurs in a variety of ways other than crack extension, and in ways that depend on the history of loading of the body. In these circumstances, it may be more appropriate to envision the critical energy release rate Gc as a history-dependent material property rather than a material constant. In the meantime, other approaches have been developed, such as cohesive zone models [7, 16], that do not require the concept of a critical energy release Chapter 11: Multiscale Modeling of the Evolution of Damage ental proposition seems to have been the key step that was necessary to begin to make somewhat accurate predictions of crack growth. Two 497

498 D.H.Allen and R.F.Soares rate to predict crack extension (although energy release rates can be calculated by this approach);and these have met some success in modeling crack growth in ductile media. Simultaneously,over the past half-century,two more or less con- tiguous developments have led to significant improvements in calculating the available energy for crack extension in both elastic and a variety of inelastic (including elastoplastic,viscoplastic,and both linear and nonlinear viscoelastic)media.One of these developments was the rise of the high- speed computer,whose power has made it possible to make billions of calculations of the type needed to estimate the energy required for crack extension,even in bodies of quite complicated geometry and material makeup.The other development is the finite element method,which grew out of the so-called flexibility method used in the aerospace and civil engineering communities in the first half of the twentieth century.This methodology came under scrutiny by the applied math community after World War II and was subsequently identified as a member of the method of weighted residuals for solving sets of coupled partial differential equations. Today,quite a few finite element codes are available for calculating stresses in both elastic and inelastic bodies. 11.3 The Current State of the Art While significant progress has been made in the ability to predict when a crack will grow and where it will go,the subject has not yet been com- pletely closed.As mentioned above,there is still no completely agreed upon way of predicting when a crack will grow in a ductile medium.Further- more,when there are multiple cracks,the computational requirements needed to utilize the finite element method go up significantly.Even with today's high-speed computers,it is not yet possible to predict,with suf- ficient accuracy,the available energy for crack extension for the physical circumstance wherein a few cracks are simultaneously imbedded in a body.And yet,it is known from experimental observation that many,many cracks can occur simultaneously in all manner of structural components and that these cracks can coalesce into a single crack that leads to structural failure.It can be said here without reservation that the state of the art of fracture mechanics is not to the point where the evolution of large numbers of cracks of evenly distributed sizes in a single inelastic body can be predicted.However,there is one case involving multiple cracks that may be a tenable problem at this time.That is the case wherein the cracks in the body are distributed by size into widely separated length

rate to predict crack extension (although energy release rates can be calculated by this approach); and these have met some success in modeling crack growth in ductile media. Simultaneously, over the past half-century, two more or less con￾tiguous developments have led to significant improvements in calculating the available energy for crack extension in both elastic and a variety of inelastic (including elastoplastic, viscoplastic, and both linear and nonlinear viscoelastic) media. One of these developments was the rise of the high￾speed computer, whose power has made it possible to make billions of calculations of the type needed to estimate the energy required for crack extension, even in bodies of quite complicated geometry and material makeup. The other development is the finite element method, which grew out of the so-called flexibility method used in the aerospace and civil engineering communities in the first half of the twentieth century. This methodology came under scrutiny by the applied math community after World War II and was subsequently identified as a member of the method of weighted residuals for solving sets of coupled partial differential equations. Today, quite a few finite element codes are available for calculating stresses in both elastic and inelastic bodies. 11.3 The Current State of the Art While significant progress has been made in the ability to predict when a crack will grow and where it will go, the subject has not yet been com￾pletely closed. As mentioned above, there is still no completely agreed upon way of predicting when a crack will grow in a ductile medium. Further￾more, when there are multiple cracks, the computational requirements needed to utilize the finite element method go up significantly. Even with today’s high-speed computers, it is not yet possible to predict, with suf￾ficient accuracy, the available energy for crack extension for the physical circumstance wherein a few cracks are simultaneously imbedded in a body. And yet, it is known from experimental observation that many, many cracks can occur simultaneously in all manner of structural components and that these cracks can coalesce into a single crack that leads to structural failure. It can be said here without reservation that the state of the art of fracture mechanics is not to the point where the evolution of large numbers of cracks of evenly distributed sizes in a single inelastic body can be predicted. However, there is one case involving multiple cracks that may be a tenable problem at this time. That is the case wherein the cracks in the body are distributed by size into widely separated length 498 D.H. Allen and R.F. Soares

Chapter 11:Multiscale Modeling of the Evolution of Damage 499 scales,with a small number of cracks observed at the largest scale,termed the global or macroscale,upon which failure ultimately occurs.This,then, is the subject of this chapter:to develop a modeling approach for predicting the evolution of multiple cracks on widely separated length scales in heterogeneous viscoelastic bodies.To affect a solution technique, the problem will be solved by using the concept of multiscaling,as described below. The concept of multiscaling in continuous media is an old one that is based on classical elasticity theory.In this approach,constitutive pro- perties of the elastic object are required to predict deformations,stresses, and strains in a structural part.To obtain these properties,a constitutive test is performed on a specimen made of the material of interest.For the test to be valid,not only should the state of stress and strain in the body be measurable by observing boundary displacements of the object when it is loaded,but also it is necessary that the object be "statistically homo- geneous."This is a sometimes ill-defined term;but what is meant by the term is that any asperities in the test specimen are several orders of magnitude smaller than the specimen itself,so that the spatial variations in the magnitudes of the observed stresses and strains in the test specimen are small compared to the mean stresses and strains observed during the test to obtain the constitutive properties.This type of experiment essentially embodies the concept of multiscaling.By assuming that the response of the test specimen is statistically homogeneous,the smaller length scale on which asperities might be observed is separated from the larger scale of the structural component. This separation of length scales has long been understood,having been considered in some detail by nineteenth-century scientists such as Maxwell and Boltzmann,as well as in the early twentieth century by Einstein,to explain macroscale observations(visible to the naked eye)of molecular phenomena in liquids and gases.Capitalizing on this approach,a number of researchers developed rigorous mathematical techniques in the 1960s for bounding the elastic properties of multiphase elastic continua [17,23, 251.Such methods earned the descriptor"micromechanics,"although this designator is perhaps not the best terminology,since the observed hetero- geneity is often not microscopic.Nevertheless,this approach has gained acceptance as a means of estimating the elastic properties of objects com- posed of multiple elastic phases which are small compared to the size of the body of interest.The advantage of such models (over the experimental approach described above)for measuring elastic properties is that the volume fractions (as well as shape,orientations,etc.)of the constituents can be changed without the necessity of redoing sometimes costly constitutive experiments.Thus,this approach,that inherently involves multiscaling,has

scales, with a small number of cracks observed at the largest scale, termed the global or macroscale, upon which failure ultimately occurs. This, then, is the subject of this chapter: to develop a modeling approach for predicting the evolution of multiple cracks on widely separated length scales in heterogeneous viscoelastic bodies. To affect a solution technique, the problem will be solved by using the concept of multiscaling, as described below. The concept of multiscaling in continuous media is an old one that is based on classical elasticity theory. In this approach, constitutive pro￾perties of the elastic object are required to predict deformations, stresses, and strains in a structural part. To obtain these properties, a constitutive test is performed on a specimen made of the material of interest. For the test to be valid, not only should the state of stress and strain in the body be measurable by observing boundary displacements of the object when it is loaded, but also it is necessary that the object be “statistically homo￾geneous.” This is a sometimes ill-defined term; but what is meant by the term is that any asperities in the test specimen are several orders of magnitude smaller than the specimen itself, so that the spatial variations in the magnitudes of the observed stresses and strains in the test specimen are small compared to the mean stresses and strains observed during the test to obtain the constitutive properties. This type of experiment essentially embodies the concept of multiscaling. By assuming that the response of the test specimen is statistically homogeneous, the smaller length scale on which asperities might be observed is separated from the larger scale of the structural component. This separation of length scales has long been understood, having been considered in some detail by nineteenth-century scientists such as Maxwell and Boltzmann, as well as in the early twentieth century by Einstein, to explain macroscale observations (visible to the naked eye) of molecular phenomena in liquids and gases. Capitalizing on this approach, a number of researchers developed rigorous mathematical techniques in the 1960s for bounding the elastic properties of multiphase elastic continua [17, 23, 25]. Such methods earned the descriptor “micromechanics,” although this designator is perhaps not the best terminology, since the observed hetero￾geneity is often not microscopic. Nevertheless, this approach has gained acceptance as a means of estimating the elastic properties of objects com￾posed of multiple elastic phases which are small compared to the size of the body of interest. The advantage of such models (over the experimental approach described above) for measuring elastic properties is that the volume fractions (as well as shape, orientations, etc.) of the constituents can be changed without the necessity of redoing sometimes costly constitutive experiments. Thus, this approach, that inherently involves multiscaling, has Chapter 11: Multiscale Modeling of the Evolution of Damage 499

500 D.H.Allen and R.F.Soares become quite popular in the engineering field.Furthermore,because the resulting body is elastic,the analyses on the smaller and larger scales can be performed independently of one another,so that no coupling between the two-length scales is necessary. In the case of inelastic media,this,unfortunately,is not the case.When materials undergo load-induced energy dissipation,such as that occurs in elastoplastic or viscoelastic media,the micromechanical description does not decouple from the analysis to be performed on the larger scale.In other words,the material properties become spatially variable and dependent on the load history,so that coupling between the macro-and microscale is unavoidable.Therefore,it becomes essential to develop modeling approaches that account for this fundamental increase in the level of complexity of the problem if there is to be any hope of achieving accuracy of prediction. For the better part of the last half of the twentieth century,efforts to account for this complexity in inelastic media centered on development of ever more complicated constitutive theories for the microscale,similar to that used successfully to model heterogeneous elastic media,as described above.This had the pragmatic basis that one could perform a finite element analysis on a single length scale,which was just about the limit that computers of that time could handle.However,as it became apparent that microscale cracking would have to be included in constitutive models of heterogeneous media at the macroscale,efforts began to bog down and become very complicated indeed.To account for observed behavior in test specimens with time-dependent microcracking,more and more (often un- explained)phenomenological parameters had to be introduced into models. This approach developed the name"continuum damage mechanics."It also inherited the unpalatable complication that sometimes many experi- mentally measured material parameters were required,especially when it became necessary to model evolving microcracks. Enter the twenty-first century and more and more powerful computers. What required a supercomputer 10 years ago now requires only a desktop computer.Therefore,it is now possible to conceive of algorithms that obviate the necessity to perform many complicated experiments at the microscale.Furthermore,these new algorithms have the added advantage that,by performing simultaneous computations on both the micro-and global scales,they possess the flexibility to include heretofore unmanage- able design variables at the microscale in the global design process,and without recourse to expensive constitutive testing

become quite popular in the engineering field. Furthermore, because the resulting body is elastic, the analyses on the smaller and larger scales can be performed independently of one another, so that no coupling between the two-length scales is necessary. In the case of inelastic media, this, unfortunately, is not the case. When materials undergo load-induced energy dissipation, such as that occurs in elastoplastic or viscoelastic media, the micromechanical description does not decouple from the analysis to be performed on the larger scale. In other words, the material properties become spatially variable and dependent on the load history, so that coupling between the macro- and microscale is unavoidable. Therefore, it becomes essential to develop modeling approaches that account for this fundamental increase in the level of complexity of the problem if there is to be any hope of achieving accuracy of prediction. For the better part of the last half of the twentieth century, efforts to account for this complexity in inelastic media centered on development of ever more complicated constitutive theories for the microscale, similar to that used successfully to model heterogeneous elastic media, as described above. This had the pragmatic basis that one could perform a finite element analysis on a single length scale, which was just about the limit that computers of that time could handle. However, as it became apparent that microscale cracking would have to be included in constitutive models of heterogeneous media at the macroscale, efforts began to bog down and become very complicated indeed. To account for observed behavior in test specimens with time-dependent microcracking, more and more (often un￾explained) phenomenological parameters had to be introduced into models. This approach developed the name “continuum damage mechanics.” It also inherited the unpalatable complication that sometimes many experi￾mentally measured material parameters were required, especially when it became necessary to model evolving microcracks. Enter the twenty-first century and more and more powerful computers. What required a supercomputer 10 years ago now requires only a desktop computer. Therefore, it is now possible to conceive of algorithms that obviate the necessity to perform many complicated experiments at the microscale. Furthermore, these new algorithms have the added advantage that, by performing simultaneous computations on both the micro- and global scales, they possess the flexibility to include heretofore unmanage￾able design variables at the microscale in the global design process, and without recourse to expensive constitutive testing. 500 D.H. Allen and R.F. Soares

Chapter 11:Multiscale Modeling of the Evolution of Damage 501 11.4 Multiscale Modeling in Inelastic Media with Damage In this section,a multiscale model is proposed for predicting the evolution of damage on multiple scales in inelastic media.The formulation is taken from [5]. 11.4.1 Microscale Model Consider an approach proposed herein that can be used on any number of length scales observed in a solid object.The number of scales n utilized is determined by the physics of the problem on the one hand and the amount of computational speed and size available on the other hand.To that end,consider a solid object with a region wherein microcracks are evolving on the smallest length scale considered /as shown in Fig.11.1. Macrocrack Macroscale RVE for Microscale Microcracks Fig.11.1.Scale problem with cracks on both length scales

11.4 Multiscale Modeling in Inelastic Media with Damage In this section, a multiscale model is proposed for predicting the evolution of damage on multiple scales in inelastic media. The formulation is taken from [5]. 11.4.1 Microscale Model Consider an approach proposed herein that can be used on any number of length scales lµ observed in a solid object. The number of scales n utilized is determined by the physics of the problem on the one hand and the amount of computational speed and size available on the other hand. To that end, consider a solid object with a region wherein microcracks are evolving on the smallest length scale considered l1, as shown in Fig. 11.1. Fig. 11.1. Scale problem with cracks on both length scales Chapter 11: Multiscale Modeling of the Evolution of Damage 501 Macroscale RVE for Microscale Macrocrack Microcracks l µ+1 l µ x2 µ+1 x2 µ x1 µ x3 µ x1 µ+1 x3 µ+1

502 D.H.Allen and R.F.Soares While it is not necessary (or even always correct)that a representative volume of the object on this length scale be accurately modeled by con- tinuum mechanics,it is assumed that this is the case in this chapter to simplify the discussion.Suppose that the object can be treated as linear viscoelastic,again for simplicity,so that the following initial-boundary value problem(IBVP)may be posed. Conservation of linear momentum i.6。+pf=0,re', (11.2) where is the Cauchy stress tensor defined on length scale u,p is the mass density,and f is the body force vector per unit mass.Note that inertial effects have been neglected,implying that the length scale of interest is small compared to the next larger length scale,thus neglecting the effects of waves at this scale on the next scale up.Ultimately,it will be convenient within this context to model waves only on the largest,or global,scale Strain-displacement equations ,=i。+a,)'门 (11.3) where is the strain tensor on the length scale u and i is the displacement vector on the length scale u.Note that the linearized form of the strain tensor has been taken for simplicity,although a nonlinear form may be employed without loss of generality. Constitutive equations G(4,)=2{(4,t)}, (11.4) where is the coordinate location in the object on the length scale 4, which has interior V and boundary oV The above description implies that the entire history of strain at any point in the body is mapped into the current stress,which is termed a viscoelastic material model.Because only the value of strain (the sym- metric part of the deformation gradient is used in this model)is required at

While it is not necessary (or even always correct) that a representative volume of the object on this length scale be accurately modeled by con￾tinuum mechanics, it is assumed that this is the case in this chapter to simplify the discussion. Suppose that the object can be treated as linear viscoelastic, again for simplicity, so that the following initial–boundary value problem (IBVP) may be posed. Conservation of linear momentum f xV 0, , ∇⋅ + = ∀ ∈ σ ρ µ µ µ G G G  (11.2) where σ µ  is the Cauchy stress tensor defined on length scale µ, ρ is the mass density, and f G is the body force vector per unit mass. Note that inertial effects have been neglected, implying that the length scale of interest is small compared to the next larger length scale, thus neglecting the effects of waves at this scale on the next scale up. Ultimately, it will be convenient within this context to model waves only on the largest, or global, scale. Strain–displacement equations 1 T [ ( ) ], 2 u u µ µµ ε ≡ ∇ +∇ G G G G  (11.3) where µ ε is the strain tensor on the length scale µ and uµ G is the displacement vector on the length scale µ. Note that the linearized form of the strain tensor has been taken for simplicity, although a nonlinear form may be employed without loss of generality. Constitutive equations ( , ) { ( , )}, t xt x τ σ µµ τ µµ ε τ = = Ω =−∞ G G   (11.4) where xµ G is the coordinate location in the object on the length scale µ, which has interior Vµ and boundary Vµ ∂ . The above description implies that the entire history of strain at any point in the body is mapped into the current stress, which is termed a viscoelastic material model. Because only the value of strain (the sym￾metric part of the deformation gradient is used in this model) is required at 502 D.H. Allen and R.F. Soares

Chapter 11:Multiscale Modeling of the Evolution of Damage 503 the point of interest,it is sometimes called a simple (or local)model [15]. Note that a local elastic material model,such as Hooke's law [321,is a special case of(11.4). Equations(11.2)(11.4)must apply in the body,together with appro- priate initial and boundary conditions.These are then adjoined with a frac- ture criterion that is capable of predicting the growth of new or existing cracks anywhere in the object.There are multiple possibilities but,for example,the Griffith criterion given by inequality (11.1)can be taken.The above then constitutes a well-posed boundary value problem,albeit non- linear due to the crack growth criterion(perhaps as well as the constitutive model (11.4)). Obtaining solutions for this problem,even for simple geometries,is in itself a difficult challenge,as anyone who has every attempted to do so will attest.Nevertheless,assume that by some means(most likely computational) a solution can be obtained for the boundary conditions,geometry,and precise form of the constitutive (11.4)at hand.Assume,furthermore,that the cracks that are predicted within the model dissipate so much energy locally that they may have further deleterious effects on the response at the next larger length scale.As an example,the so-called microcracks may in some way influence the development or extension of one or more macrocracks on the next larger length scale /2.It will be assumed that the cracks on the next larger length scale are much larger than those on the current scale and that this restriction applies to all length scales for cracks in the object of interest 141之14，l=1,n, (11.5) where n is the number of different length scales observed in the solid. Note that the above restriction is a necessary condition (but not sufficient)for the multiscale methodology proposed herein to produce reasonably accurate predictions on the larger length scale(s).If this con- dition is not satisfied,as in the case of a so-called localization problem, then there may indeed be no alternative to performing an exhaustive analysis at a single scale that takes into account all of the asperities simultaneously. 11.4.2 Homogenization Principle Connecting the Microscale to the Macroscale To perform an analysis of the solid on the next length scale up from the local scale (termed the macroscale herein for simplicity),it is necessary to find a means of linking the state variables predicted on the microscale to

the point of interest, it is sometimes called a simple (or local) model [15]. Note that a local elastic material model, such as Hooke’s law [32], is a special case of (11.4). Equations (11.2)–(11.4) must apply in the body, together with appro￾priate initial and boundary conditions. These are then adjoined with a frac￾ture criterion that is capable of predicting the growth of new or existing cracks anywhere in the object. There are multiple possibilities but, for example, the Griffith criterion given by inequality (11.1) can be taken. The above then constitutes a well-posed boundary value problem, albeit non￾linear due to the crack growth criterion (perhaps as well as the constitutive model (11.4)). Obtaining solutions for this problem, even for simple geometries, is in itself a difficult challenge, as anyone who has every attempted to do so will attest. Nevertheless, assume that by some means (most likely computational) a solution can be obtained for the boundary conditions, geometry, and precise form of the constitutive (11.4) at hand. Assume, furthermore, that the cracks that are predicted within the model dissipate so much energy locally that they may have further deleterious effects on the response at the next larger length scale. As an example, the so-called microcracks may in some way influence the development or extension of one or more macrocracks on the next larger length scale l2. It will be assumed that the cracks on the next larger length scale are much larger than those on the current scale and that this restriction applies to all length scales for cracks in the object of interest 1 ll n , 1, , , µ µ +  µ = … (11.5) where n is the number of different length scales observed in the solid. Note that the above restriction is a necessary condition (but not sufficient) for the multiscale methodology proposed herein to produce reasonably accurate predictions on the larger length scale(s). If this con￾dition is not satisfied, as in the case of a so-called localization problem, then there may indeed be no alternative to performing an exhaustive analysis at a single scale that takes into account all of the asperities simultaneously. 11.4.2 Homogenization Principle Connecting the Microscale to the Macroscale To perform an analysis of the solid on the next length scale up from the local scale (termed the macroscale herein for simplicity), it is necessary to find a means of linking the state variables predicted on the microscale to Chapter 11: Multiscale Modeling of the Evolution of Damage 503

504 D.H.Allen and R.F.Soares those on the macroscale.Of course,the state variables at the microscale are predicted at an infinite collection of material points in the local domain V+V,so that there is plenty of information available to supply to the next larger length scale.However,the objective herein is to find an effi- cient means of constructing this link without sacrificing too much accuracy. In other words,it is propitious to utilize the minimum data obtained at the local scale necessary to make a sufficiently accurate prediction at the macro- scale.One way is to link the microscale to the macroscale via the use of mean fields.To see how this might work,consider the following mathe- matical expansion for the macroscale stress in terms of the microscale stress (11.6) where G.dv (11.7) is the volume averaged (or mean)stress at the microscale,and it is assumed that the local coordinate system is set at the geometric centroid of the microscale volume. Note that,since the microscale domain V+ov can be placed arbitrarily within the domain on the next larger length scale V+ the mean stress is a continuously varying function of coordinates on the next larger length scale u+1,as shown in Fig.11.1.Note also that the terms within the summation in(11.6)represent higher area moments of the stress tensor. Now,it may be said without loss of generality that microscale con- servation of momentum (11.2)also applies to the macroscale (assuming that quasistatic conditions still hold at this length scale) .61+pf=0,∈'n (11.8) By using (11.6),it can be shown that lim () (11.9) 1al→0 and (11.8)reduces to the following: .6n+pf=0,441∈V+ (11.10)

those on the macroscale. Of course, the state variables at the microscale are predicted at an infinite collection of material points in the local domain V V µ + ∂ µ , so that there is plenty of information available to supply to the next larger length scale. However, the objective herein is to find an effi￾cient means of constructing this link without sacrificing too much accuracy. In other words, it is propitious to utilize the minimum data obtained at the local scale necessary to make a sufficiently accurate prediction at the macro￾scale. One way is to link the microscale to the macroscale via the use of mean fields. To see how this might work, consider the following mathe￾matical expansion for the macroscale stress in terms of the microscale stress 1 ( ) 1 1 d , j j V j x V V x µ µ µ µ µ σ σ σσ ∞ + = = +∑ − ∫ G    G (11.6) where 1 d V V V µ µ µ µ σ σ ≡ ∫   (11.7) is the volume averaged (or mean) stress at the microscale, and it is assumed that the local coordinate system is set at the geometric centroid of the microscale volume. Note that, since the microscale domain V V µ + µ ∂ can be placed arbitrarily within the domain on the next larger length scale V V µ+1 1 + ∂ µ+ , the mean stress σ µ  is a continuously varying function of coordinates 1 xµ+ G on the next larger length scale µ + 1, as shown in Fig. 11.1. Note also that the terms within the summation in (11.6) represent higher area moments of the stress tensor. Now, it may be said without loss of generality that microscale con￾servation of momentum (11.2) also applies to the macroscale (assuming that quasistatic conditions still hold at this length scale) 1 1 1 f xV 0, . ∇⋅ + = ∀ ∈ σ ρ µ µ + + + µ G G G  (11.8) By using (11.6), it can be shown that 1 1 / 0 lim ( ) l l µ µ σ µ σ + + →   = (11.9) and (11.8) reduces to the following: D.H. Allen and R.F. Soares 1 1 f xV 0, . ∇⋅ + = ∀ ∈ σ ρ µ µ+ µ+ G G G  (11.10) 504