Chapter 5:A Micromechanics-Based Notion of Stress 213 Furthermore,if to these assumptions one adds that the underlying deforma- tion process is governed by div T=0(or DivS =0),i.e.,governed by a guasistatic form of the local balance of linear momentum without body forces,an application of the divergence theorem allows one to show that [S]=Vol①x)J. 1 sdv and I可=vol@Jh Tdv. (5.11) Before proceeding further,we should keep in mind that one of the objec- tives of this chapter is to extend continuum homogenization notions of effec- tive stress and strain to the discrete systems analyzed via MD.Although often disregarded in continuum homogenization of elastic systems,time evolution and time averaging are central to MD calculations.Hence,we introduce here a time averaging operation that will be employed later in the chapter.This operation will be denoted by the use of angle brackets and defined as follows 1 rt0十T (f）≌1im f(t)dt, (5.12) r→oT Jto where t denotes time,f(t)is a generic function of time,and to is the initial time.Without loss of generality,we will assume that to =0.Adopting concepts from statistical mechanics,we will view the time average operation defined in (5.12)as a way of translating the effects of fast dynamics into corresponding thermal effects. 5.5.2 Meaningful Deformation Processes Now that we have introduced the definition of effective deformation and stress in a regime of large deformation,we need to address the problem of clearly identifying those RVE motions for which the definitions in question are useful in some sense.Specifically,it should be observed that each of the definitions we have given is independent of the others.Therefore,one cannot expect that,for example,[F]=[F]for all possible RVE motions.By the same reasoning,in general,we cannot expect that the effective Cauchy and the first Piola-Kirchhoff stresses are related by a relationship such as (5.7),i.e.,the relation satisfied the corresponding pointwise stress measures. These observations indicate that there is a need for the establishment of conditions that guarantee the ability to attach physical meaning to the definitions given above.In fact,it can be argued that Hill's macrohomo- geneity conditions [13,14],i.e.,those conditions under which certain prod- ucts of effective quantities are equal to the volume average of the product
Furthermore, if to these assumptions one adds that the underlying deformation process is governed by div T = 0 (or Div S = 0), i.e., governed by a quasistatic form of the local balance of linear momentum without body forces, an application of the divergence theorem allows one to show that JSK = 1 Vol(Ωκ) Z Ωκ S dV and JTK = 1 Vol(Ω) Z Ω Tdv. (5.11) Before proceeding further, we should keep in mind that one of the objectives of this chapter is to extend continuum homogenization notions of effective stress and strain to the discrete systems analyzed via MD. Although often disregarded in continuum homogenization of elastic systems, time evolution and time averaging are central to MD calculations. Hence, we introduce here a time averaging operation that will be employed later in the chapter. This operation will be denoted by the use of angle brackets and defined as follows hfi , limτ→∞ 1 τ Z t0+τ t0 f(t)dt, (5.12) where t denotes time, f(t) is a generic function of time, and t0 is the initial time. Without loss of generality, we will assume that t0 = 0. Adopting concepts from statistical mechanics, we will view the time average operation defined in (5.12) as a way of translating the effects of fast dynamics into corresponding thermal effects. 5.5.2 Meaningful Deformation Processes Now that we have introduced the definition of effective deformation and stress in a regime of large deformation, we need to address the problem of clearly identifying those RVE motions for which the definitions in question are useful in some sense. Specifically, it should be observed that each of the definitions we have given is independent of the others. Therefore, one cannot expect that, for example, JF−1K = JFK−1 for all possible RVE motions. By the same reasoning, in general, we cannot expect that the effective Cauchy and the first Piola–Kirchhoff stresses are related by a relationship such as (5.7), i.e., the relation satisfied the corresponding pointwise stress measures. These observations indicate that there is a need for the establishment of conditions that guarantee the ability to attach physical meaning to the definitions given above. In fact, it can be argued that Hill’s macrohomogeneity conditions [13, 14], i.e., those conditions under which certain products of effective quantities are equal to the volume average of the product Chapter 5: A Micromechanics-Based Notion of Stress 213
214 F.Costanzo and G.L.Gray of the corresponding local quantities,play the role of defining the set of physically meaningful averaging processes.Here,since we are interested in a rigorous extension of the continuum concepts of Cauchy stress and first Piola-Kirchhoff stress to discrete systems,we propose slightly more strin- gent requirements(with respect to Hill's macrohomogeneity conditions).We therefore introduce the following definition. Definition 1 (Meaningful Deformation Processes).By a large deformation process with meaningful space averages,we mean a deformation process possessing all of the following properties: 1.[IF]-1=[F-1],with det(IF])>0. 2.Vol()=det(IF])Vol(x). 3.[S]det([F])[T]([F]-1)T. This definition is motivated by a desire to have effective quantities that formally behave just like their local counterparts.Now,similarly to Hill's approach(cf.[14)),instead of attempting to derive necessary and sufficient conditions for satisfying Definition 1,we will only provide a list of sufficient conditions.These conditions are found by demanding that the RVE motions satisfy specific BCs.As observed in Sect.5.4,the"right"choice of BCs is crucial for successfully solving the RVE problem that delivers the effective elastic moduli.In this section,we see that the right choice of BCs is crucial for establishing the very meaning of the definitions of effective quantities. In determining the type of BCs in question,one can start with analyzing the three "canonical"BCs we have discussed in Sect.5.4.2.With this in mind,the first step is to properly redefine these BCs in a regime of large deformations.We do this next. In the present context,we define uniform strain BCs as follows x(x,t)=F(t)x for xEok, (5.13) where,for all t of interest,F(t)is a prescribed second-order tensor with positive determinant.The definition given here matches the definition given byHi[13,14].3 Uniform stress BCs are now defined as follows T(x,t)n(r,t)=∑(t)n(c,t)forc∈a2, (5.14) Equation(5.13)is presented under the assumption that the origin of the coordinate sys- tem is at the mass center and that the total linear momentum of the system is zero
of the corresponding local quantities, play the role of defining the set of physically meaningful averaging processes. Here, since we are interested in a rigorous extension of the continuum concepts of Cauchy stress and first Piola–Kirchhoff stress to discrete systems, we propose slightly more stringent requirements (with respect to Hill’s macrohomogeneity conditions). We therefore introduce the following definition. Definition 1 (Meaningful Deformation Processes). By a large deformation process with meaningful space averages, we mean a deformation process possessing all of the following properties: 1. JFK −1 = JF−1K, with det(JFK) > 0. 2. Vol(Ω) = det(JFK) Vol(Ωκ). 3. JSK = det(JFK)JTK(JFK −1) T. This definition is motivated by a desire to have effective quantities that formally behave just like their local counterparts. Now, similarly to Hill’s approach (cf. [14]), instead of attempting to derive necessary and sufficient conditions for satisfying Definition 1, we will only provide a list of sufficient conditions. These conditions are found by demanding that the RVE motions satisfy specific BCs. As observed in Sect. 5.4, the “right” choice of BCs is crucial for successfully solving the RVE problem that delivers the effective elastic moduli. In this section, we see that the right choice of BCs is crucial for establishing the very meaning of the definitions of effective quantities. In determining the type of BCs in question, one can start with analyzing the three “canonical” BCs we have discussed in Sect. 5.4.2. With this in mind, the first step is to properly redefine these BCs in a regime of large deformations. We do this next. In the present context, we define uniform strain BCs as follows x(χ, t) = Fˆ(t)χ for χ ∈ ∂Ωκ, (5.13) where, for all t of interest, Fˆ(t) is a prescribed second-order tensor with positive determinant. The definition given here matches the definition given by Hill [13, 14].3 Uniform stress BCs are now defined as follows T(x, t)n(x, t) = Σˆ(t)n(x, t) for x ∈ ∂Ω, (5.14) 3 Equation (5.13) is presented under the assumption that the origin of the coordinate system is at the mass center and that the total linear momentum of the system is zero. 214 F. Costanzo and G.L. Gray