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 dynamics). 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 languages 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 recontinuum 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 viewpoint 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 understood 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 homogenization 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 homogenization 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 chapter 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 deformed configuration and will be denoted by Ωκ in its reference configuration. 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 deformed 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 secondorder 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 heterogeneous 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 moduli, 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 structural 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 homogeneous 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 “average” behavior since this information is essential in assessing failure conditions. 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 homogenization 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. Therefore, 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
208 F.Costanzo and G.L.Gray the effective moduli is carried out by solving a special BVP defined over a portion of the material such that both the composition and the geometry of this portion are able to represent the material as a whole.This subset of material is called a representative volume element (RVE),which is schemat- ically shown in Fig.5.2.In general the determination of the RVE may not be o< o< o RVE 0 △ 0 0- △ 0 0 0△ 0△ 0△ -80K Fig.5.2.A representative volume element for the special case of a periodic medium straightforward,however,for periodic media,the RVE is readily identified with the periodic cell of the material.Furthermore,in the case of periodic media,there are rigorous proofs showing that the determination of Cefr by asymptotic expansion methods (see,e.g.,[3,4])delivers the same result as the solution of the RVE BVP so long as the periodicity of the material is properly accounted for. Remark 2 (Definition of Effective Quantities).Roughly speaking,in formal homogenization theory,eefr(x)and efr(x)are defined as the leading terms of an asymptotic expansion of the fields e(x)and (x)with respect to a scaling parameter,say A,defined as the ratio between the length over which the moduli vary and the overall (large)dimension of the component (for example,referring to Fig.5.1,one can set A =h/L).With this in mind, and referring to Fig.5.2,one can show that,when using the RVE as a way to determine the effective moduli,these definitions can be given the following form 1 Ceff- 2Vol(S)Jag (u⑧nx+nx⑧u)dA, (5.4) 1 Ceff= (5.5) Vol(S)Jas (anx⑧X)dA, where u denotes the displacement field.The essential feature of these defi- nitions is that eeff(x)and oefr(x)are determined by gathering information on the boundary of the RVE rather than its interior.If the RVE is a simply
the effective moduli is carried out by solving a special BVP defined over a portion of the material such that both the composition and the geometry of this portion are able to represent the material as a whole. This subset of material is called a representative volume element (RVE), which is schematically shown in Fig. 5.2. In general the determination of the RVE may not be Fig. 5.2. A representative volume element for the special case of a periodic medium straightforward, however, for periodic media, the RVE is readily identified with the periodic cell of the material. Furthermore, in the case of periodic media, there are rigorous proofs showing that the determination of Ceff by asymptotic expansion methods (see, e.g., [3, 4]) delivers the same result as the solution of the RVE BVP so long as the periodicity of the material is properly accounted for. Remark 2 (Definition of Effective Quantities). Roughly speaking, in formal homogenization theory, εeff(χ) and σeff(χ) are defined as the leading terms of an asymptotic expansion of the fields ε(χ) and σ(χ) with respect to a scaling parameter, say λ, defined as the ratio between the length over which the moduli vary and the overall (large) dimension of the component (for example, referring to Fig. 5.1, one can set λ = h/L). With this in mind, and referring to Fig. 5.2, one can show that, when using the RVE as a way to determine the effective moduli, these definitions can be given the following form εeff = 1 2 Vol(Ωκ) Z ∂Ωκ (u ⊗ nκ + nκ ⊗ u)dA, (5.4) σeff = 1 Vol(Ωκ) Z ∂Ωκ (σnκ ⊗ χ)dA, (5.5) where u denotes the displacement field. The essential feature of these defi- nitions is that εeff(χ) and σeff(χ) are determined by gathering information on the boundary of the RVE rather than its interior. If the RVE is a simply 208 F. Costanzo and G.L. Gray
Chapter 5:A Micromechanics-Based Notion of Stress 209 connected regular domain,a straightforward application of the divergence theorem tells us that 1 Ceff- Vol(n) and Ceff= odv, (5.6) Vol(S)J where the second of (5.6)requires that the pointwise!balance law is Div(C(x)[e(x)])=0.Equation(5.6)implies that there are cases in which the word“effective”'does mean“volume average,”but,in general,.effective strain and stress must be understood as given in (5.4)and(5.5)to be use- ful mathematical constructs.In addition,there is a strong physically based reason for defining effective quantities via boundary integrals,as eloquently remarked by Hill ([13];see also [18,27,28]): Experimental determinations of mechanical behaviour rest ultimately on measured loads or mean displacements over pairs of opposite faces of a representative cube.Macro-variables in- tended for constitutive laws should thus be capable of definition in terms of surface data alone,either directly or indirectly.It is not necessary,by any means,that macro-variables so defined should be unweighed volume averages of their microscopic counterparts. Remark 3 (Basic Properties of eeff and oeff).In a small strain theory,one expects the strain and stress measures to be symmetric tensors.Referring to (5.4),it is easy to see that the effective strain is,by definition,a symmetric tensor.Furthermore,one can easily show that under most conditions oeff is symmetric.What needs to be observed here is that,at least at first glance, no special steps are needed to make sure that the above-defined effective quantities have the properties that one usually expects of the corresponding pointwise quantities.As we will see,this is certainly not the case when dealing with the definitions of the effective stress and deformation concepts in nonlinear homogenization. 5.4.2 Boundary Conditions for the RVE Problem When relying on an RVE for the determination of the effective moduli,one must pose and solve a BVP over the RVE in question.This BVP is usually The "point"in pointwise refers to a continuum material point,by which we mean a point in a regular subset of R3.This is not to be confused with a material particle,by which we mean an abstract physical entity endowed with a given fixed mass
connected regular domain, a straightforward application of the divergence theorem tells us that εeff = 1 Vol(Ωκ) Z Ωκ ε dV and σeff = 1 Vol(Ωκ) Z Ωκ σ dV, (5.6) where the second of (5.6) requires that the pointwise1 balance law is Div(C(χ)[ε(χ)]) = 0. Equation (5.6) implies that there are cases in which the word “effective” does mean “volume average,” but, in general, effective strain and stress must be understood as given in (5.4) and (5.5) to be useful mathematical constructs. In addition, there is a strong physically based reason for defining effective quantities via boundary integrals, as eloquently remarked by Hill ([13]; see also [18, 27, 28]): Experimental determinations of mechanical behaviour rest ultimately on measured loads or mean displacements over pairs of opposite faces of a representative cube. Macro-variables intended for constitutive laws should thus be capable of definition in terms of surface data alone, either directly or indirectly. It is not necessary, by any means, that macro-variables so defined should be unweighed volume averages of their microscopic counterparts. Remark 3 (Basic Properties of εeff and σeff). In a small strain theory, one expects the strain and stress measures to be symmetric tensors. Referring to (5.4), it is easy to see that the effective strain is, by definition, a symmetric tensor. Furthermore, one can easily show that under most conditions σeff is symmetric. What needs to be observed here is that, at least at first glance, no special steps are needed to make sure that the above-defined effective quantities have the properties that one usually expects of the corresponding pointwise quantities. As we will see, this is certainly not the case when dealing with the definitions of the effective stress and deformation concepts in nonlinear homogenization. 5.4.2 Boundary Conditions for the RVE Problem When relying on an RVE for the determination of the effective moduli, one must pose and solve a BVP over the RVE in question. This BVP is usually 1 The “point” in pointwise refers to a continuum material point, by which we mean a point in a regular subset of R3. This is not to be confused with a material particle, by which we mean an abstract physical entity endowed with a given fixed mass. Chapter 5: A Micromechanics-Based Notion of Stress 209
210 F.Costanzo and G.L.Gray called a localization problem and its governing partial differential equations are those in (5.1).As far as the BCs are concerned,these need to be carefully stated to match the particular nature of the problem.In fact,in a localization problem one is not interested in computing the solution's pointwise behavior. Rather,one needs to control the solution's effective behavior in such a way that the effective moduli can be calculated.With this in mind,it turns out that it is indeed possible to control the value of the effective strain or stress by specifying some specific sets of BCs.These BCs are as follows: 1.Uniform strain:u=ex on onx,with e a given symmetric second- order tensor. 2.Uniform stress:onk=onk on onk,with aa given symmetric second-order tensor. 3.Periodic:If the RVE is a periodic cell,then the displacement field is decomposed such that u =ex+u*everywhere in the RVE,with e a given symmetric second-order tensor and with u*being an unknown vector field whose boundary values are constrained to be periodic,i.e., u*is constrained to take on identical values on homologous points of the boundary.Furthermore,in addition to constraining the boundary values of the field u*,one must also constrain the behavior of the field ong to be antiperiodic. If one chooses BCs of type 1 or 3,it is relatively straightforward to prove (see,e.g.,[18,28])that the controlled parameter determines the value of the effective strain,i.e.,sefr =e.If one chooses condition 2,then it is not difficult to show that the controlled parameter o determines the value of the effective stress,i.e.,oefr=o. From a conceptual viewpoint,the determination of the elastic moduli in RVE-based linear homogenization is carried out by the following procedure. Choosing uniform strain BCs for the sake of discussion,one can set=1 and all other components of e equal to zero.Then,one solves the RVE BVP and thus determines the o component of the solution.Next,one uses the o field in question,along with (5.5),to determine oeff.Finally,due to the linearity of problem and referring to(5.3),the efr just computed coincides with the "ij11"components of Ce (ij =1,2,3).This process is then re- peated by selecting setting all components of e equal to zero,except say pg, which is set to unity so that the"ijpg"components of the elastic moduli can be found
called a localization problem and its governing partial differential equations are those in (5.1). As far as the BCs are concerned, these need to be carefully stated to match the particular nature of the problem. In fact, in a localization problem one is not interested in computing the solution’s pointwise behavior. Rather, one needs to control the solution’s effective behavior in such a way that the effective moduli can be calculated. With this in mind, it turns out that it is indeed possible to control the value of the effective strain or stress by specifying some specific sets of BCs. These BCs are as follows: 1. Uniform strain: u = εχˆ on ∂Ωκ, with εˆ a given symmetric secondorder tensor. 2. Uniform stress: σnκ = σnˆ κ on ∂Ωκ, with σˆ a given symmetric second-order tensor. 3. Periodic: If the RVE is a periodic cell, then the displacement field is decomposed such that u = εχˆ + u∗ everywhere in the RVE, with εˆ a given symmetric second-order tensor and with u∗ being an unknown vector field whose boundary values are constrained to be periodic, i.e., u∗ is constrained to take on identical values on homologous points of the boundary. Furthermore, in addition to constraining the boundary values of the field u∗, one must also constrain the behavior of the field σnκ to be antiperiodic. If one chooses BCs of type 1 or 3, it is relatively straightforward to prove (see, e.g., [18, 28]) that the controlled parameter εˆ determines the value of the effective strain, i.e., εeff = εˆ. If one chooses condition 2, then it is not difficult to show that the controlled parameter σˆ determines the value of the effective stress, i.e., σeff = σˆ. From a conceptual viewpoint, the determination of the elastic moduli in RVE-based linear homogenization is carried out by the following procedure. Choosing uniform strain BCs for the sake of discussion, one can set εˆ11 = 1 and all other components of εˆ equal to zero. Then, one solves the RVE BVP and thus determines the σ component of the solution. Next, one uses the σ field in question, along with (5.5), to determine σeff. Finally, due to the linearity of problem and referring to (5.3), the σeff just computed coincides with the “ij11” components of Ceff (ij = 1, 2, 3). This process is then repeated by selecting setting all components of εˆ equal to zero, except say εˆpq, which is set to unity so that the “ijpq” components of the elastic moduli can be found. 210 F. Costanzo and G.L. Gray
Chapter 5:A Micromechanics-Based Notion of Stress 211 5.5 The RVE Problem and Large Deformations In this section,we discuss the concepts of effective strain and stress in a con- text of large deformations.This discussion is again meant to properly setup a stage for the extension of continuum homogenization ideas to MD prob- lems.Choosing to work in a large deformation context is motivated by the fact that we want the discussion to be as general as possible.2 Before pro- ceeding to the presentation of effective measures of deformation and stress, it is important to remark that the field of nonlinear homogenization is not as well developed as the corresponding linear theory.In particular,there are fewer theoretical results linking an asymptotic approach to homogenization to the RVE-based averaging procedures.With this in mind,as has been done by other authors (see,e.g.,[12-14,25]),we will simply assume that the RVE problem is a valid way to compute effective properties.This assumption allows us to focus our attention on the RVE approach to homogenization, as opposed to considering the (more technically difficult)formal asymptotic approach. 5.5.1 Definition of Effective Deformation and Stress In general,in a context of large deformation,one must take into consider- ation two measures of stress,namely the Cauchy stress and the first Piola- Kirchhoff stress,depending on whether one chooses the deformed or reference configurations,respectively,to write the system's equation of motion.The two notions of stress are related by the well-known relation (see,e.g.,[11]) S=det(F)T(F-1)T, (5.7) where S denotes the first Piola-Kirchhoff stress tensor,F denotes the defor- mation gradient,T denotes the Cauchy stress tensor,and the superscript T denotes transposition.The relationship in (5.7)reminds us that,when we define effective deformation and stress in a context of large deformation, we need to(1)provide definitions that are based both on the reference and the deformed configurations and(2)discuss how these definitions relate to one another.With this in mind,we introduce two independent measures of effective deformation:the effective deformation gradient tensor,denoted by 2In choosing to work in a regime of large deformations,we assume that the kinematics at both the micro-and macroscales is fully nonlinear.Correspondingly,we do not assume that any aspect of the constitutive theory is linear
5.5 The RVE Problem and Large Deformations In this section, we discuss the concepts of effective strain and stress in a context of large deformations. This discussion is again meant to properly setup a stage for the extension of continuum homogenization ideas to MD problems. Choosing to work in a large deformation context is motivated by the fact that we want the discussion to be as general as possible.2 Before proceeding to the presentation of effective measures of deformation and stress, it is important to remark that the field of nonlinear homogenization is not as well developed as the corresponding linear theory. In particular, there are fewer theoretical results linking an asymptotic approach to homogenization to the RVE-based averaging procedures. With this in mind, as has been done by other authors (see, e.g., [12–14,25]), we will simply assume that the RVE problem is a valid way to compute effective properties. This assumption allows us to focus our attention on the RVE approach to homogenization, as opposed to considering the (more technically difficult) formal asymptotic approach. 5.5.1 Definition of Effective Deformation and Stress In general, in a context of large deformation, one must take into consideration two measures of stress, namely the Cauchy stress and the first Piola– Kirchhoff stress, depending on whether one chooses the deformed or reference configurations, respectively, to write the system’s equation of motion. The two notions of stress are related by the well-known relation (see, e.g., [11]) S = det(F)T(F−1) T, (5.7) where S denotes the first Piola–Kirchhoff stress tensor, F denotes the deformation gradient, T denotes the Cauchy stress tensor, and the superscript T denotes transposition. The relationship in (5.7) reminds us that, when we define effective deformation and stress in a context of large deformation, we need to (1) provide definitions that are based both on the reference and the deformed configurations and (2) discuss how these definitions relate to one another. With this in mind, we introduce two independent measures of effective deformation: the effective deformation gradient tensor, denoted by 2 In choosing to work in a regime of large deformations, we assume that the kinematics at both the micro- and macroscales is fully nonlinear. Correspondingly, we do not assume that any aspect of the constitutive theory is linear. Chapter 5: A Micromechanics-Based Notion of Stress 211
212 F.Costanzo and G.L.Gray RVE Deformation Fig.5.3.The RVE in its reference(left)and deformed(right)configurations [F],and the effective inverse deformation tensor,denoted by.These quantities are defined as follows IF]≌ 1 Vol()Jag x⑧nxdA,[F-1]e X8nda,(5.8) where the RVE is subject to a motion =(x,t),and the symbols are defined in Fig.5.3.As far as stress is concerned,we will define the effec- tive first Piola-Kirchhoff stress tensor and the effective Cauchy stress tensor, denoted by [S]and [T],respectively,as follows: IS] 1 (Sn)xdA.[V Vol(S)Jas Vol()Jan (Tn)⑧xda. (5.9) To the best of the authors'knowledge,the definitions of effective deforma- tion and effective stress in a regime of large deformation were first systemati- cally discussed in [13](see also [14]).Important contributions to this subject also include the works in [12,26].More recent discussions have been given in [16,25].It should be pointed out that Hill [13,14]does not include in his discussions the definition ofF.However,we feel that the definition of [F-1]is important because it has a bearing on the type of phenomena that we will choose as being physically meaningful when extending the above notions of effective deformation and stress to discrete systems. Going back to (5.8)and(5.9),it is important to notice that we have once more defined effective quantities via boundary integrals rather than via vol- ume averages.However,as was done in Sect.5.4,under standard regularity and smoothness assumptions,a straightforward application of the divergence theorem yields(cf.[13]) 1 [F]= Fv nd oi F-1du.(5.10)
Fig. 5.3. The RVE in its reference (left) and deformed (right) configurations JFK, and the effective inverse deformation tensor, denoted by q F−1 y . These quantities are defined as follows JFK , 1 Vol(Ωκ) Z ∂Ωκ x⊗nκ dA, JF−1K , 1 Vol(Ω) Z ∂Ω χ⊗n da, (5.8) where the RVE is subject to a motion x = x(χ, t), and the symbols are defined in Fig. 5.3. As far as stress is concerned, we will define the effective first Piola–Kirchhoff stress tensor and the effective Cauchy stress tensor, denoted by JSK and JTK, respectively, as follows: JSK , 1 Vol(Ωκ) Z ∂Ωκ (Snκ) ⊗ χ dA, JTK , 1 Vol(Ω) Z ∂Ω (Tn) ⊗ x da. (5.9) To the best of the authors’ knowledge, the definitions of effective deformation and effective stress in a regime of large deformation were first systematically discussed in [13] (see also [14]). Important contributions to this subject also include the works in [12, 26]. More recent discussions have been given in [16, 25]. It should be pointed out that Hill [13, 14] does not include in his discussions the definition of JF−1K. However, we feel that the definition of JF−1K is important because it has a bearing on the type of phenomena that we will choose as being physically meaningful when extending the above notions of effective deformation and stress to discrete systems. Going back to (5.8) and (5.9), it is important to notice that we have once more defined effective quantities via boundary integrals rather than via volume averages. However, as was done in Sect. 5.4, under standard regularity and smoothness assumptions, a straightforward application of the divergence theorem yields (cf. [13]) JFK = 1 Vol(Ωκ) Z Ωκ F dV and q F−1y = 1 Vol(Ω) Z Ω F−1 dv. (5.10) 212 F. Costanzo and G.L. Gray
