## Abstract

Under consideration is the problem of size and response of the Representative Volume Element (RVE) in the setting of finite elasticity of statistically homogeneous and ergodic random microstructures. Through the application of variational principles, a scale dependent hierarchy of strain energy functions (i.e. mesoscale bounds) is derived for the effective strain energy function of the RVE. In order to account for thermoelastic effects, the variational principles are first generalized, and then analogous bounds on the effective thermoelastic response are derived. It is also shown that, in contradiction to results obtained for random linear composites, the hierarchy on the effective strain energy function in nonlinear elasticity cannot be split into volumetric and isochoric terms, while the hierarchy on the effective free energy function cannot be divided into purely mechanical and thermal contributions.

## 1. Introduction

The prediction of macroscopic properties of heterogeneous materials is of great importance in many engineering applications. Evidently, any material displays heterogeneity on a microscale and has different properties depending on its inherent physical and chemical composition. The issue which commonly arises when dealing with such materials—especially now, in the age of bio- and nanotechnology—is the validity of the *separation of scales* of continuum mechanics:(1.1)where *L* is the size of the *representative volume element* (RVE). As is well known, (1.1) allows one to smooth the heterogeneous microstructure, characterized by the microscale *d*, and work with an effective continuum over the range of scales from *L* through *L*_{macro}. The RVE is clearly set-up in two basic cases (Ostoja-Starzewski 1999, 2001): (i) a unit cell in a periodic microstructure and (ii) a domain containing infinitely many microscale elements (e.g. crystals) in a random medium. Overwhelmingly, in the case of random media, the prediction of effective properties of heterogeneous materials is usually based on an *a priori* assumption of existence of the RVE, without clearly specifying its size *L* relative to *d*.

In this paper, we develop scale-dependent bounds on the effective response of random composites for *physically nonlinear* materials subjected to *large isothermal* and *non-isothermal* deformations. We follow the approach that originated some 15 years ago (Huet 1990; Sab 1992), also driven by the need to derive continuum random field models from microstructures (Ostoja-Starzewski & Wang 1989; see also the review Ostoja-Starzewski 2001). We shall deal with ‘apparent’ material properties at finite strain, that is, when the body is smaller than the RVE and when deformations are large. The idea is to consider properties of a composite material of the size smaller than the representative volume and apply variational principles along with different types of boundary conditions to investigate scale-dependent relationships for various types of material microstructure. These results lay the ground for a quantitative estimation of the real size of the RVE, currently under study.

## 2. General theory

Consider a thermoelastic body undergoing equilibrium deformation in a three-dimensional Euclidian space. Let * x* and

*denote the position vectors of a particle inside the body in reference and current configurations, respectively; then the deformation of the body can be described as . The deformation gradient tensor is given by(2.1)where Grad*

**X***is the displacement gradient in material description,*

**u***δ*

_{ij}is the Kronecker delta, and the indices,

*i*,

*j*,

*k*, take on the values 1, 2, 3. Equation of equilibrium in the absence of body forces has the form(2.2)Here,

*denotes a generally non-symmetric first Piola–Kirchhoff stress tensor and Div is the divergence calculated with respect to the reference configuration.*

**P**The key assumption of the finite hyperelasticity theory is the existence of a strain energy function *ψ*=*ψ*(*F*_{ij}) per unit volume of an undeformed body, which depends on the deformation of the object and its material properties. Restricting ourselves to the reference configuration, the equation of state of the material can be written as(2.3)If temperature changes are to be considered, the strain energy density should be replaced by the free energy as a function of both deformation gradient and temperature.

It is worth mentioning that the first Piola–Kirchhoff stress tensor and deformation gradient tensor are the most convenient tensors to be used since the averaging theorems and Hill condition can be formulated for these quantities in the way analogous to the infinitesimal deformation theory (for a discussion of these theorems, the reader is referred to §4). Combining (2.3) and (2.1), equation (2.2) can be rewritten as(2.4)Here and elsewhere, *u*_{i,j} denotes the deformation gradient tensor, where the unit tensor is omitted for simplicity of notation.

## 3. Minimum theorems in finite elasticity and thermoelasticity

Variational principles for a material undergoing finite strains have been studied extensively by many writers (e.g. Zubov 1971; de Veubeke 1972; Lee & Shield 1980; Ogden 1984; Gao 1999). If the principle of stationary potentional energy in finite elasticity can be formulated similarly to that in linear elasticity, the complementary energy principle cannot be established without certain restrictions placed on the strain-energy function. In nonlinear elasticity, the strain-energy function may be non-convex and, therefore, non-invertible, i.e. the deformation gradient cannot be expressed in terms of stress. Even the inversion of convex strain-energy functions is extremely difficult in general, which places a limitation on the use of the complementary energy theorem.

To overcome the difficulty of inversion, different ‘complementary-type’ variational principles have been introduced in the literature. From the point of view of the bounding problem, the variational principles proposed by Lee & Shield (1980) seem more advantageous to use. It allows one to avoid a difficult and often impossible task of inverting the strain-energy density function. In §3*a* we outline this theory and in §3*b* extend it to the case of nonlinear thermoelastostatics.

### (a) Variational principles in finite elastostatics

Consider the functional(3.1)where *V*_{0} is material volume in reference state, *U*_{i} is an admissible displacement field such that on the portion of the boundary, where displacement is prescribed, *S*_{U}, and *t*_{i}^{0} is prescribed nominal traction on the remaining part of the boundary *S*_{T}. Equation (3.1) represents the finite elasticity counterpart of the principle of stationary potential energy in linear elasticity theory. The functional *P*{*U*_{i}} is stationary for the actual solution *u*_{i}, which satisfies the equilibrium equation (2.4) in *V*_{0} and natural boundary conditions(3.2)Here, *n*_{j} denotes an outward normal vector to the boundary surface in the reference configuration. Expanding the quantity *P*{*u*_{i}+δ*u*_{i}, *u*_{i,j}+δ*u*_{i,j}} in Taylor's series and ignoring terms of the third and higher order in δ*u*_{i}, one can obtain the condition under which the functional *P*{*U*_{i}} assumes a local minimum for the actual solution *u*_{i} for all non-zero δ*u*_{i} such that δ*u*_{i}=0 on *S*_{U} (Lee & Shield 1980):(3.3)

If the condition (3.3) holds, the principle of stationary potential energy becomes the minimum principle. Assuming interface continuity of the displacement and traction field inside the body, the above variational principle becomes valid for multiphase material as well as for materials with continuously varying properties and can be used to estimate bounds on the effective material response.

We now turn to the investigation of the principle of minimum complementary energy. To overcome the difficulties of inversion in formulating the complementary energy functional, Lee & Shield (1980) proposed to consider the complementary strain-energy function *ψ*_{c} to be a function of the deformation gradient rather than stresses:(3.4)Then, the complementary energy functional can be written as(3.5)where *U*_{ij} is an admissible displacement gradient field satisfying(3.6)

Using the Lagrange multiplier method, it can be shown that the functional (3.5) is stationary for *U*_{ij}=*u*_{i,j}, where *u*_{i} is the actual solution of a given elasticity problem. In order for *Q* to assume a local minimum, the following condition should be fulfilled:(3.7)for all non-zero δ*u*_{ij} satisfying(3.8)

Equation (3.7) was obtained by applying Taylor series expansion up to the second order in δ*u*_{ij} to the quantity *Q*{*u*_{i,j}+δ*u*_{ij}}, taking into account that equations (3.8) need to be satisfied by both actual and admissible solution.

An application of the divergence theorem leads to(3.9)which is a well-known identity from the calculus of variations between total potential and complementary energies.

Equations (3.1)–(3.9) define variational principles in finite elastostatics for elastic bodies not subjected to constraints. For the case when internal constraints such as incompressibility are present, the components of the deformation gradient tensor are not arbitrary and additional conditions need to be considered in formulating minimum potential and complementary energy theorems (Lee & Shield 1980).

### (b) Variational principles in finite thermoelastostatics

Here, we consider an uncoupled thermoelastic problem when the material is at rest and the heat flow is steady. In this case, the equation of heat conduction can be solved separately from the remaining field equations and the temperature field *T* is determined uniquely by thermal boundary conditions. Minimum energy principles in this case can be derived using the Helmholtz free energy function *ψ*=*ψ*(* F*,

*T*) and the Gibbs thermodynamic potential

*G*=

*G*(

*,*

**P***T*) in place of the isothermal strain energy and the complementary strain energy, respectively.

*G*is associated with

*ψ*by the transformation(3.10)The potential energy functional (3.1) can be rewritten as(3.11)where

*θ*

_{0}is the prescribed temperature field. For simplicity, we will assume temperature to be uniform throughout the volume. The Helmholtz free energy in nonlinear elasticity has the form (Holzapfel 2000)(3.12)where

*T*

_{0}is the temperature at reference state, Δ

*T*=

*T*−

*T*

_{0}is the temperature change,

*ψ*

_{0}is the free energy at reference temperature,

*e*

_{0}is the internal energy at reference temperature and is a purely thermal contribution defined as(3.13)Here,

*c*

_{F}(

*T*) is specific heat capacity at constant deformation. For infinitesimal elastic deformations (3.12) can be easily reduced to a well-known Helmholtz free energy form by taking , and assuming

*c*

_{F}to be a constant over a given small temperature change.

Varying independently the quantities *u*_{i} and *u*_{i,j}, while keeping the temperature *θ*_{0} constant, it can be proved by the same procedure as used in §3*a* that the functional (3.11) is stationary for the actual solution *U*_{i}=*u*_{i} and provides a local minimum if(3.14)for all non-zero δ*u*_{i} vanishing over the surface upon which the displacement is prescribed.

Applying the transformation (3.10), the complementary-type functional becomes(3.15)where *G* can be written as(3.16)

The functional (3.15) assumes a local minimum provided(3.17)for all non-zero δ*u*_{ij} satisfying (3.8), in which *ψ* is replaced by the free energy function (3.12). The energetic contribution *e*_{0} to (3.12) appears only in the modified entropic theory of rubber thermoelasticity, and is equal to zero in the case of a purely entropic theory, for which conditions (3.14) and (3.17) simply reduce to the isothermal inequalities (3.3) and (3.7).

Note that (3.3) and (3.7) are the convexity conditions on *ψ*. They play an important role in hyperelasticity by ensuring the stability of the material; the strain-energy models fitted to available experimental data are commonly designed so as to avoid the lack of convexity (Holzapfel *et al*. 2000).

## 4. Averaging theorems in finite deformation theory

Averaging theorems play a key role in the estimation of overall material properties of composites. It is known that, in a small deformation theory, volume averages of both infinitesimal strain and stress fields can be fully determined from the surface data (e.g. Nemat-Nasser & Hori 1999). In finite elasticity theory, the relation between strain and displacement gradient is nonlinear, which, as a result, leads to difficulties in evaluation of the volume averages. Accordingly, different deformation measures should be considered when dealing with a problem involving finite strain. In this section, we will formulate averaging theorems along with the Hill condition for two conjugate quantities, namely, the deformation gradient tensor and the first Piola–Kirchhoff stress tensor.

### (a) Average deformation gradient and stress theorems

Deformation gradient tensor is the simplest kinematic variable and the only deformation measure for which the average theorem can be derived in finite elasticity theory. Assuming no jumps in displacement and traction across the boundary between different phases of the composite, the averaging theorems for the deformation gradient tensor and the first Piola–Kirchhoff stress tensor take the form(4.1)(4.2)where is the prescribed deformation gradient and is a prescribed traction field acting on the boundary *S*_{0} in the reference configuration, accordingly. The superposed bar here and elsewhere denotes the volume average of the overall macroscopic quantities. Equations (4.1) and (4.2) can be readily proved by transforming the volume integral via the Green–Gauss theorem. One of the main consequences of these theorems is that they allow one to control the value of the effective deformation gradient tensor and the first Piola–Kirchhoff stress tensor of the composite under consideration.

Equations (4.1) and (4.2) were first considered by Hill (1972) and then re-examined by many authors (Nemat-Nasser 1999; Löhnert & Wriggers 2003; Costanzo *et al*. 2005). In fact, in finite elasticity, the average stress theorem holds for both spatial and material descriptions (Hill 1972), i.e. for both Cauchy and first Piola–Kirchhoff stress tensors. This is not surprising since both tensors represent the same resultant force acting on a surface element of the deformed body.

It is worth mentioning, that (4.1) holds only for a static process in the absence of body forces. If this assumption fails, one has to consider the effective value of a target quantity, but not the ‘true’ average over the volume (Costanzo *et al*. 2005).

### (b) Hill condition for finite deformations

A well-known criterion in a homogenization problem is the so-called *Hill condition*. It represents a condition under which the mechanical and energetic definitions of the effective properties are compatible. For the deformation gradient tensor and the first Piola–Kirchhoff stress tensor, it can be stated as follows (Hill 1972; Nemat-Nasser 1999):(4.3)This identity is trivially satisfied by a homogeneous body, but imposes certain restrictions on the boundary conditions if the body is heterogeneous.

Applying the Green–Gauss theorem and noting that from the equilibrium equation in the absence of body forces , we get(4.4)

When the boundary conditions are such that the right-hand side integral is vanishing, the average of the product of the deformation gradient tensor and the stress tensor is equal to the product of averages. This implies three types of boundary conditions, which allow the use of the Hill condition when dealing with inhomogeneous bodies:

uniform kinematic boundary condition (prescribing a volume average deformation gradient):(4.5)

uniform static boundary condition (prescribing a volume average nominal stress):(4.6)

uniform orthogonal-mixed boundary condition:(4.7)where the averaging theorems §4

*a*have been used.

In the following, we shall use boundary conditions (4.5) and (4.6) along with variational principles introduced in §3 for estimation of bounds on the effective response of nonlinear composites.

## 5. Hierarchies of mesoscale bounds

### (a) Energy bounds in finite elasticity and thermoelasticity of random composites

When dealing with a random microstructure one needs to impose certain restriction on the internal geometry of the composite in order to be able to estimate average properties of the material and the RVE size itself. It turns out that sufficient conditions for the bounds on the effective properties to converge with the increasing sample size are the assumptions of statistical homogeneity and mean-ergodicity of the material. In the present case, spatial homogeneity implies that all points in a given realization *B*(*ω*) of the random composite * B*={

*B*(

*ω*);

*ω*∈

*Ω*}, where

*Ω*is a sample space, are equivalent, i.e. there is no preferred location in the structure. The ergodicity assumption states that any one realization of the composite is a representative of the entire ensemble,(5.1)where denotes ensemble average and

*V*

_{0}is the material volume in reference state.

Next, we introduce a non-dimensional scale parameter *δ*=*L*/*d*, where *d* is the heterogeneity size (e.g. single grain) and *L* is the size of a *mesoscale* domain. Consider a partition of a body of size *δ* into *n* smaller square elements of size (*n*=2 in figure 1). Defining two types of boundary conditions (e.g. Ostoja-Starzewski 2001)—*restricted* (the boundary condition specified on the boundary of each element) and *unrestricted* (the boundary condition specified on the boundary of the whole body)—one observes that the deformation of the material under the restricted boundary condition represents an admissible field for the unrestricted boundary condition, but not conversely.

Consider now the uniform displacement boundary condition (4.5) (i.e. when the volume average deformation gradient is being prescribed). In this case, *S*_{T} is identically equal to zero and the functional (3.1) reduces to(5.2)where the strain-energy function is now a function of both deformation gradient and coordinates. Under the assumption that the minimum theorem holds the energy stored in the body under the restricted boundary conditions is related to the unrestricted one as(5.3)where , is the elastic energy of any given realization *B*(*ω*) of the composite, and the superscript ‘r’ denotes the effective properties obtained under restricted boundary conditions. Henceforth, in order to simplify the notation we shall write *ψ* instead of *ψ*(*ω*, * X*,

*). With*

**F***Δ*and 1 denoting the RVE size and inhomogeneity size, respectively, upon ensemble averaging we obtain for the upper bound(5.4)

The hierarchy (5.4) can be generalized for the case of thermoelastostatics to be(5.5)

We now turn to the investigation of an analogous reciprocal expression for the lower bounds. Under the uniform traction boundary condition (4.6), the functional *Q* reduces to(5.6)Thus, the following inequality between responses under restricted and unrestricted boundary conditions holds:(5.7)where From this, upon ensemble averaging, we can derive a scale dependent hierarchy of a lower bound on the effective properties(5.8)which in thermoelastostatics takes the form(5.9)

Noting that can be equivalently expressed as * P* :

*we can writewhere averaging theorems §4*

**F***a*have been employed. At the macroscale, we will always recover the regular Legendre transformation(5.11)since for the RVE size composite material and . Equation (5.11) allows one to estimate effective strain energy function as .

Note that, strictly speaking, *Δ* (denoting the RVE size *L* relative to the microheterogeneity *d*) is infinite for a random medium lacking any spatial periodicity, but the convergence of upper and lower bounds is achieved within an accuracy of a few percent at finite scales; see e.g. Jiang *et al*. (2001) and Ostoja-Starzewski & Castro (2003) for case studies of two physically nonlinear composites.

Now, suppose that we have a composite with two different phases (one stiffer than the other), it follows from our theory that, for the lower bound,(5.12)Let us denote the right-hand side of the inequality (5.12) by *ψ*_{L}. Under the assumption of constant stress in the body, the application of restricted boundary conditions does not alter the displacement field, so that , which, as , leads to(5.13)

Equation (5.13) is a generalization of the well-known Reuss bound of linear elasticity. It agrees with the derivation via a different route by Ponte Castañeda (1989); see also Ogden (1978) for previous work on bounds. Generalization of the upper Voigt bound can be done in a similar way.

Introducing a power-law type energy function into the inequalities (5.4) and (5.8), a hierarchy of bounds on elastic tensors can be obtained in the same manner as in the infinitesimal strain theory (Hazanov 1999). In the case of more complex energy functions (such as those described by Ogden (1972)), it is more advantageous to work directly with the energy functions.

Here, it is worth mentioning that the main assumption involved in the derivation of (5.4), (5.5), (5.8) and (5.9) is that minimum principles described in §3 hold for any kinematically admissible displacement. In reality, inequalities (3.3), (3.7), (3.14) and (3.17) provide a *global* minimizer criteria only if the strain energy function is convex, which is just a mathematical assumption. For materials described by non-convex strain-energy functions, bounds (5.4)–(5.9) hold only locally in the range of validity of inequalities (3.3)–(3.17), which places a limitation on their use. If the inversion of the constitutive relation (2.3) is not unique, a different complementary variational principle such as the one proposed by Gao (1999) might be of better use.

### (b) Linear versus nonlinear bounding problems

Effective properties of a random composite in linear thermoelasticity were recently considered by Du & Ostoja-Starzewski (2005); a scale-dependent hierarchy of bounds on the thermal expansion coefficient was derived in the way analogous to the one considered above. Now, contrary to the linear elasticity case, the behaviour of random *nonlinear* composites subjected to temperature changes is not governed by a thermal expansion coefficient *α*_{ij} alone but by a number of temperature dependant constants. Moreover, the only materials undergoing finite strains and temperature changes relative to equilibrium state are biological soft tissues and rubber-like materials (Holzapfel 2000), whose thermomechanical behaviour is almost entirely based on an entropy concept. Total stress in these materials is caused by a change in entropy with deformation, while internal energy does not change with deformation at all. Therefore, the behaviour of composites in finite thermoelasticity is expected to be very different from the linear elasticity one. The central aim of this section is to compare the above-derived hierarchies in nonlinear elasticity with the results in the linear elasticity theory.

Consider an isothermal elasticity problem. In linear elasticity, the volume average of strain energy of the body can be fully determined from surface data if one of the ‘canonical’ boundary conditions (4.5) or (4.6) are applied. One of the main consequences is that, for a macroscopically isotropic composite with isotropic phases, the hierarchy on effective infinitesimal strain-energy function can be separated into volumetric and deviatoric parts, which cannot be done in nonlinear elasticity. Indeed, the isotropic strain-energy function for small deformation elasticity can be put in the following form:(5.14)where *κ* is the bulk modulus, *μ* is the shear modulus, *e*=*ϵ*_{kk} is volume strain and is deviatoric part of the strain field. The strain-energy form (5.14) plays an important role in proving the existence and uniqueness of solution in linear elastostatics. The first and second terms of (5.14) are independently variable and represent volumetric and deviatoric parts of *ψ*, respectively. Equation (5.14) can be rewritten as(5.15)

Here, is the mean pressure and is the deviatoric part of the stress field. Since the deviatoric and volumetric stress and strain tensors are mutually orthogonal, it follows from the Hill condition that(5.16)Hence, the average strain energy under uniform volumetric or isochoric boundary conditions is fully determined by the corresponding volumetric or isochoric stress and strain field on the boundary. It is important to notice that, defining an apparent bulk modulus as and an apparent shear modulus as , one can estimate bounds on effective volumetric and isochoric responses. Thus, applying uniform volumetric strain on the boundary after ensemble averaging we get(5.17)whereas uniform isochoric kinematic boundary condition leads to(5.18)The reciprocal expression for the lower bound on the effective response can be obtained from the complementary energy function, which gives(5.19)

(5.20)

In the nonlinear elasticity of compressible rubber-like materials, the strain-energy function *ψ* can be also split into a volumetric part *ψ*_{vol} and isochoric part *ψ*_{iso} (Holzapfel 2000). However, the application of purely volumetric boundary condition does not give a zero isochoric contribution to the average strain energy and vice versa. The reason for this is the complex nature of the strain-energy function, which in general cannot be expressed as a product of the nominal stress and deformation gradient just as it was done in (5.19).

In the potential (3.12), the only term independent of deformation is the purely thermal contribution . This term does not depend on the type of mechanical loading and, therefore, is the same under restricted and unrestricted boundary conditions:(5.21)

Hence, the purely thermal contribution can be ignored in hierarchies (5.5) and (5.9).

As pointed out in §3*b*, the energetic contribution *e*_{0} to the free energy function (3.12) is equal to zero in the case of a purely entropic theory, in which the thermoelastic bounding problem becomes identical to the purely elastic one. In the most general situation, *e*_{0} is assumed to be a function of the volume ratio (Chadwick & Creasy 1984). Thus, in contrast to the linear elasticity theory, the hierarchy (5.5) cannot be separated into purely mechanical and thermal parts and one has to consider the first two terms of *ψ* in (3.12) jointly.

## 6. Conclusions

In this paper, the variational principles of Lee & Shield (1980) have been extended to the case of finite thermoelastostatics. It was shown that the uniform displacement and traction boundary conditions can be used to obtain, respectively, the upper and lower bounds on the effective energy functions of a composite at finite deformations under isothermal and non-isothermal loading. The approach developed can be used to estimate the scale dependence (i.e. mesoscale bounds) on effective response of random microstructures in finite elasticity and thermoelasticity and, hence, the size of the RVE without unnecessary and unwieldy inversion of constitutive relations. Such results will be reported in the subsequent publication.

## Acknowledgements

Anonymous referees' comments are gratefully appreciated. The work reported herein has been made possible through support by the Werner Graupe International Fellowship, the NSERC, and the Canada Research Chairs program.

## Footnotes

- Received July 6, 2005.
- Accepted November 14, 2005.

- © 2006 The Royal Society