《纺织复合材料》课程参考文献（Multiscale Modeling and Simulation of Composite Materials and Structures）Chapter 5 A Micromechanics-Based Notion of Stress for Use in the Determination of Continuum-Level Mechanical Properties via Molecular Dynamics

Chapter 5:A Micromechanics-Based Notion of Stress for Use in the Determination of Continuum-Level Mechanical Properties via Molecular Dynamics Francesco Costanzo and Gary L.Gray Department of Engineering Science and Mechanics,The Pennsylvania State University,University Park,PA 16802,USA 5.1 Introduction By formulating a continuum homogenization problem that includes inertia effects,a link is established between continuum homogenization and the estimation of effective mechanical properties for particle ensembles whose interactions are governed by potentials (e.g.,as is seen in molecular dynam- ics).The focus of this chapter is on showing that there is a fundamental consistency of ideas between continuum mechanics and the study of discrete particle systems,and that it is possible to define a notion of effective stress applicable to discrete systems that can be claimed to have the same meaning as it has in continuum mechanics. 5.2 Motivation,Objectives,and Organization The last 15 years have seen an astonishing growth in nanomechanics-related research.During this time,experimental and theoretical mechanicians alike have had to adapt to a fast-evolving research landscape.Like many others, the authors of this chapter found themselves delving into specialized fields of study such as molecular dynamics(MD)and struggling to learn new lan- guages and methodologies that were outside what they trained on during their graduate work.With this in mind,this chapter is in part the result of

Chapter 5: A Micromechanics-Based Notion of Stress for Use in the Determination of Continuum-Level Mechanical Properties via Molecular Dynamics Francesco Costanzo and Gary L. Gray Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802, USA 5.1 Introduction By formulating a continuum homogenization problem that includes inertia effects, a link is established between continuum homogenization and the estimation of effective mechanical properties for particle ensembles whose interactions are governed by potentials (e.g., as is seen in molecular dynam￾ics). The focus of this chapter is on showing that there is a fundamental consistency of ideas between continuum mechanics and the study of discrete particle systems, and that it is possible to define a notion of effective stress applicable to discrete systems that can be claimed to have the same meaning as it has in continuum mechanics. 5.2 Motivation, Objectives, and Organization The last 15 years have seen an astonishing growth in nanomechanics-related research. During this time, experimental and theoretical mechanicians alike have had to adapt to a fast-evolving research landscape. Like many others, the authors of this chapter found themselves delving into specialized fields of study such as molecular dynamics (MD) and struggling to learn new lan￾guages and methodologies that were outside what they trained on during their graduate work. With this in mind, this chapter is in part the result of

204 F.Costanzo and G.L.Gray the authors'learning experience in how to use MD to compute mechanical properties of solids.In going through this learning process,the authors had to confront the fundamental issue of what it means to compute the stress re- sponse of a particle system and how this measure of stress is related to the continuum mechanical notion of stress.Clearly,this question is not new, since it dates back to the pioneering work by Cauchy who formalized the very notion of stress.However,we feel that we have added something new to the discussion in that we have approached the problem from the view- point of continuum homogenization and,in so doing,not only were we able to extend the continuum homogenization notion of effective stress to MD, but we were also able to construct a practical Lagrangian MD scheme that is rigorously based on classical mechanics. From a conceptual viewpoint,the outcome of this work is that a good part of the MD that is used in nanomechanics can be comfortably under- stood with classical mechanics and homogenization ideas.In other words, it is possible to define an acceptable concept of stress for discrete systems without ever relying on ideas from statistical mechanics or a kinetic theory of matter.While this fact may be well understood by some researchers,we feel that it is not sufficiently known among classically trained engineers,and we hope that this chapter may reinforce the idea that there is a fundamental unity between the study of continuum and discrete systems. The organization of this chapter is based on the idea that classical ho- mogenization of heterogeneous systems is intimately related to MD,since both disciplines deal with the computation of effective properties of matter. Hence,we will start by reviewing some basic concepts of homogenization of linear elastic media.We will then discuss the extension of these concepts to the case of homogenization in the context of large deformation.Once this review is done,we will formulate a continuum homogenization problem that shares the basic properties of MD problems.We will show that the homog- enization scheme in question can be turned into an MD scheme in which stress is defined such that it can be said to have the same meaning that it has in continuum homogenization.Finally,we will compare the continuum homogenization-based stress concept with the virial stress,the latter being the stress concept typically used in MD. Before proceeding further,we wish to mention that some elements of this chapter have been presented in [1,2,8,9].The main contribution of this chapter lies in a presentation that is intended to give a coherent vision of how continuum homogenization and MD are related.With this said,this chap- ter does contain some new results consisting of more general proofs,with

the authors’ learning experience in how to use MD to compute mechanical properties of solids. In going through this learning process, the authors had to confront the fundamental issue of what it means to compute the stress re￾continuum mechanical notion of stress. since it dates back to the pioneering work by Cauchy who formalized the very notion of stress. However, we feel that we have added something new to the discussion in that we have approached the problem from the view￾point of continuum homogenization and, in so doing, not only were we able to extend the continuum homogenization notion of effective stress to MD, but we were also able to construct a practical Lagrangian MD scheme that is rigorously based on classical mechanics. From a conceptual viewpoint, the outcome of this work is that a good part of the MD that is used in nanomechanics can be comfortably under￾stood with classical mechanics and homogenization ideas. In other words, it is possible to define an acceptable concept of stress for discrete systems without ever relying on ideas from statistical mechanics or a kinetic theory of matter. While this fact may be well understood by some researchers, we feel that it is not sufficiently known among classically trained engineers, and we hope that this chapter may reinforce the idea that there is a fundamental unity between the study of continuum and discrete systems. The organization of this chapter is based on the idea that classical ho￾mogenization of heterogeneous systems is intimately related to MD, since both disciplines deal with the computation of effective properties of matter. Hence, we will start by reviewing some basic concepts of homogenization of linear elastic media. We will then discuss the extension of these concepts to the case of homogenization in the context of large deformation. Once this review is done, we will formulate a continuum homogenization problem that shares the basic properties of MD problems. We will show that the homog￾enization scheme in question can be turned into an MD scheme in which stress is defined such that it can be said to have the same meaning that it has in continuum homogenization. Finally, we will compare the continuum homogenization-based stress concept with the virial stress, the latter being the stress concept typically used in MD. Before proceeding further, we wish to mention that some elements of this chapter have been presented in [1, 2, 8, 9]. The main contribution of this chapter lies in a presentation that is intended to give a coherent vision of how continuum homogenization and MD are related. With this said, this chap￾ter does contain some new results consisting of more general proofs, with sponse of a particle system and how this measure of stress is related to the Clearly, this question is not new, 204 F. Costanzo and G.L. Gray

Chapter 5:A Micromechanics-Based Notion of Stress 205 respect to what had been previously published,on the equivalence between a continuum-based notion of effective stress and virial stress. 5.3 Notation The material system under consideration will be denoted by n in its de- formed configuration and will be denoted by Ss in its reference configura- tion.Both and are assumed to be regular subsets of a three-dimensional Euclidean point space.The boundaries of n and Ss will be denoted by on and on,respectively.The volumes of and will be denoted by Vol() and Vol(),respectively.The boundaries on and on are oriented by the outward unit normal vector fields n and n,respectively.The position of points in the reference configuration will be denoted by x and in the de- formed configuration by The operators“Div”and“div”indicate the divergence operators with respect to x and respectively.Similarly,the operators"Grad"and"grad" indicate the gradient operators with respect to x and respectively. We will use upper-case sans serif letters,such as A,to denote second- order tensors and lower-case bold italic letters,such as a,to denote vectors. The notation ab denotes the tensor product of the vectors a and b.The symbol will indicate a definition. 5.4 Homogenization of Linear Elastic Heterogeneous Media:A Brief Review To better illustrate how MD and continuum homogenization are related,it is useful to review some basic concepts from the theory of homogenization of linear elastic heterogeneous media.We will therefore review the essential objectives of homogenization theory and some basic definitions concerning effective mechanical properties.In subsequent sections,we will discuss how these definitions need to be adjusted to be useful in a fully nonlinear context in preparation for their application to discrete particle systems. 5.4.1 Homogenization Objectives Referring to Fig.5.1,consider a structural component made of a hetero- geneous material with overall dimensions that are much larger than the characteristic length over which the material's constitutive properties vary. Conceptually,under the assumption that the material is linear elastic,in

respect to what had been previously published, on the equivalence between a continuum-based notion of effective stress and virial stress. 5.3 Notation The material system under consideration will be denoted by Ω in its de￾formed configuration and will be denoted by Ωκ in its reference configura￾tion. Both Ω and Ωκ are assumed to be regular subsets of a three-dimensional Euclidean point space. The boundaries of Ω and Ωκ will be denoted by ∂Ω and ∂Ωκ, respectively. The volumes of Ω and Ωκ will be denoted by Vol(Ω) and Vol(Ωκ), respectively. The boundaries ∂Ω and ∂Ωκ are oriented by the outward unit normal vector fields n and nκ, respectively. The position of points in the reference configuration will be denoted by χ and in the de￾formed configuration by x. The operators “Div” and “div” indicate the divergence operators with respect to χ and x, respectively. Similarly, the operators “Grad” and “grad” indicate the gradient operators with respect to χ and x, respectively. We will use upper-case sans serif letters, such as A, to denote second￾order tensors and lower-case bold italic letters, such as a, to denote vectors. The notation a ⊗ b denotes the tensor product of the vectors a and b. The symbol , will indicate a definition. 5.4 Homogenization of Linear Elastic Heterogeneous Media: A Brief Review To better illustrate how MD and continuum homogenization are related, it is useful to review some basic concepts from the theory of homogenization of linear elastic heterogeneous media. We will therefore review the essential objectives of homogenization theory and some basic definitions concerning effective mechanical properties. In subsequent sections, we will discuss how these definitions need to be adjusted to be useful in a fully nonlinear context in preparation for their application to discrete particle systems. 5.4.1 Homogenization Objectives Referring to Fig. 5.1, consider a structural component made of a hetero￾geneous material with overall dimensions that are much larger than the characteristic length over which the material’s constitutive properties vary. Conceptually, under the assumption that the material is linear elastic, in Chapter 5: A Micromechanics-Based Notion of Stress 205

206 F.Costanzo and G.L.Gray 网。 effective strain Eeft AMMAAMMAY actual strain e Fig.5.1.A panel consisting of a heterogeneous material quasistatic conditions,and in the absence of body forces,the prediction of the component's stress/strain response requires the solution of a boundary value problem(BVP)of the following type BVPexact:Div(C(x)[e(x)])=0 along with BCs, (5.1) where x denotes position,C(x)is the (fourth-order)tensor of elastic mod- uli,e(x)is the small strain tensor field,and the expression "BCs"stands for"boundary conditions."For convenience,we denote by (x)the stress field corresponding to e(x),i.e.,(x)=C(x)[e(x)].Clearly,the struc- tural component's stress/strain response to some applied loading will reflect the spatial variability of the elastic moduli,as schematically represented by the solid line in Fig.5.1.Unfortunately,from a computational viewpoint, the spatial variability in question may make the solution of the problem in (5.1)difficult,if not impossible,to obtain.With this in mind,a practical way to approach the design of highly heterogeneous components is to construct

Fig. 5.1. A panel consisting of a heterogeneous material quasistatic conditions, and in the absence of body forces, the prediction of the component’s stress/strain response requires the solution of a boundary value problem (BVP) of the following type BVPexact : Div(C(χ)[ε(χ)]) = 0 along with BCs, (5.1) where χ denotes position, C(χ) is the (fourth-order) tensor of elastic mod￾uli, ε(χ) is the small strain tensor field, and the expression “BCs” stands for “boundary conditions.” For convenience, we denote by σ(χ) the stress field corresponding to ε(χ), i.e., σ(χ) = C(χ)[ε(χ)]. Clearly, the struc￾tural component’s stress/strain response to some applied loading will reflect the spatial variability of the elastic moduli, as schematically represented by the solid line in Fig. 5.1. Unfortunately, from a computational viewpoint, the spatial variability in question may make the solution of the problem in (5.1) difficult, if not impossible, to obtain. With this in mind, a practical way to approach the design of highly heterogeneous components is to construct 206 F. Costanzo and G.L. Gray

Chapter 5:A Micromechanics-Based Notion of Stress 207 a predictive capability that allows one to (1)model the material as homoge- neous so as to more easily determine the system's "average"response (see the dashed line in Fig.5.1)and (2)estimate the deviations from the "aver- age"behavior since this information is essential in assessing failure condi- tions.The purpose of homogenization is to have both types of predictive capability,though we will only explore the first type here.Before doing so, it is important to recognize that,at this stage,we do not know whether or not what we have called the "average"response will in fact be an average in a strict mathematical sense.Hence,we will refer to the "average"strain and stress response as the effective strain and stress response and we will denote these quantities as eefr and eff,respectively. As suggested above,a fundamental objective of continuum homogeniza- tion is to use the knowledge of the material's microstructure to formulate a BVP whose solution is the system's effective response,i.e.,homogenization theory delivers the possibility of predicting the effective system's response by solving the following BVP BVPerr:Div(Cef[eefr(x)])=0 along with BCs, (5.2) where it is essential to notice that,in the new BVP,the moduli Ceff,which are called the material's effective moduli,are not a function of position.There- fore,one way to interpret(5.2)is to say that homogenization theory takes information concerning the original heterogeneous material and maps it into the properties of an equivalent homogenous material.Finally,we will refer to the field Cerree(x)]as the effective stress field and we will denote it by oefr(x),i.e., ef Cef[Eeft(x)]. (5.3) So far,we have only sketched a conceptual map of what homogenization does without considering the important details needed to show that one can indeed go from the BVP in (5.1)to that in (5.2).Most of these "details"are outside the scope of this chapter and they can be easily found in the literature. For example,excellent references on the subject are the presentations in [18, 20,28,31].For discussions that are more technical from a mathematical viewpoint,one can see the presentations in [3,4,15].While we will stay away from the technical details of homogenization theory,a few important remarks are now needed for extending homogenization ideas to MD. Remark 1 (Representative Volume Element).To solve the BVP in(5.2),one must first determine the effective moduli Ceff and this can be done via several methods.Often,especially in engineering applications,the determination of

a predictive capability that allows one to (1) model the material as homoge￾neous so as to more easily determine the system’s “average” response (see the dashed line in Fig. 5.1) and (2) estimate the deviations from the “aver￾age” behavior since this information is essential in assessing failure condi￾tions. The purpose of homogenization is to have both types of predictive capability, though we will only explore the first type here. Before doing so, it is important to recognize that, at this stage, we do not know whether or not what we have called the “average” response will in fact be an average in a strict mathematical sense. Hence, we will refer to the “average” strain and stress response as the effective strain and stress response and we will denote these quantities as εeff and σeff, respectively. As suggested above, a fundamental objective of continuum homogeniza￾tion is to use the knowledge of the material’s microstructure to formulate a BVP whose solution is the system’s effective response, i.e., homogenization theory delivers the possibility of predicting the effective system’s response by solving the following BVP BVPeff : Div(Ceff[εeff(χ)]) = 0 along with BCs, (5.2) where it is essential to notice that, in the new BVP, the moduli Ceff, which are called the material’s effective moduli, are not a function of position. There￾fore, one way to interpret (5.2) is to say that homogenization theory takes information concerning the original heterogeneous material and maps it into the properties of an equivalent homogenous material. Finally, we will refer to the field Ceff[εeff(χ)] as the effective stress field and we will denote it by σeff(χ), i.e., σeff = Ceff[εeff(χ)]. (5.3) So far, we have only sketched a conceptual map of what homogenization does without considering the important details needed to show that one can indeed go from the BVP in (5.1) to that in (5.2). Most of these “details” are outside the scope of this chapter and they can be easily found in the literature. For example, excellent references on the subject are the presentations in [18, 20, 28, 31]. For discussions that are more technical from a mathematical viewpoint, one can see the presentations in [3, 4, 15]. While we will stay away from the technical details of homogenization theory, a few important remarks are now needed for extending homogenization ideas to MD. Remark 1 (Representative Volume Element). To solve the BVP in (5.2), one must first determine the effective moduli Ceff and this can be done via several methods. Often, especially in engineering applications, the determination of Chapter 5: A Micromechanics-Based Notion of Stress 207