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 inclement weather, and even human organs subjected to aging. Therefore, it would seem self-evident that cogent models capable of predicting such catastrophic 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 scientists 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 principle 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 conof 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 formulation 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 components 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 monumobstacles 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 compared 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 dissipation 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 contiguous 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 highspeed 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 completely closed. As mentioned above, there is still no completely agreed upon way of predicting when a crack will grow in a ductile medium. Furthermore, 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 sufficient 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 properties 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 homogeneous.” 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 heterogeneity is often not microscopic. Nevertheless, this approach has gained acceptance as a means of estimating the elastic properties of objects composed 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