Homogenization of Polycrystalline Aggregates

Roberto Paroni

University of Kentucky, Lexington, KY 40506-0027

Submitted to Archive for Rational Mechanics and Analysis

Abstract

The effective elastic properties of a polycrystalline material depend on the single crystal elastic constants of the crystallites comprising the polycrystal and on the manner in which the crystallites are arranged. In this paper we apply the techniques of homogenization to put the problem of determining effective elastic constants in a precise mathematical framework. Under our theory, we shall derive an expression for the effective elasticity tensor which will take into account not only the orientations of the crystallites but also the shapes and arrangements of the grains. We also study how the homogenized elasticity tensor changes as the probability characterizing the ensemble changes. We show, under the assumption that the field of orientations of the crystallographic axes of the crystallites is an independent random field, that our theory is compatible with the formulation used in texture analysis. In particular, under this assumption we are able to prove that the physical assumption adopted by Man (Arch. Rational Mech. Anal., in press) in his study of weaklytextured polycrystals holds true. We also establish some elementary bounds on the material constants that characterize the effective elasticity tensor of weakly-textured orthorhombic aggregates of cubic crystallites.

1 Introduction

A polycrystalline material is an aggregate of numerous crystallites or grains, each having its own shape and particular orientation of its crystallographic axes. We will consider here only materials whose crystallites have the same composition. The elastic constants of an unstressed polycrystalline material at a given point are determined by the orientation and the elastic constants of the crystallite covering the point in consideration; hence, the elasticity tensor is a piecewise constant tensor (it is constant over each grain). For most structural materials the dimensions of the grains are very small as compared with the dimensions of the body; thus the elasticity tensor is highly discontinuous, with jumps of discontinuity that repeat at a very small scale. For such materials it is convenient to define an effective elasticity tensor, which in a precise sense averages out the discontinuities and describes the macroscopic behavior of the polycrystalline material. Usually the effective elasticity tensor for a given body is defined through the following procedure: impose a boundary displacement which would produce a uniform strain if the body were homogeneous; the effective elasticity tensor is then defined as the constant tensor which puts the average strain in a linear relation with the average stress. It turns out that the effective elastic properties of a polycrystalline material depend on the single crystal elastic constants of the crystallites comprising the polycrystal and on the manner in which the crystallites are arranged. Theoretically, the effective elastic constants can be evaluated if a detailed description of the microstructure (i.e., the orientation and arrangement of the crystallites comprising the aggregate) is known; but in practice the interlocking aggregate of grains is so complex that it is hardly feasible to define the complicated microstructure throughout the aggregate. It is more likely, instead, that only some statistical information is available.

Unless the orientation of the crystallites are completely random, we say that the polycrystalline aggregate has preferred crystallographic orientations and is textured. In the materials science literature a polycrystalline material is considered to be isotropic if the crystallographic axes of the grains are randomly arranged, and hence it is tacitly assumed that only the orientation of the grains have influence on the effective elastic constants: the shapes and the arrangements of the grains are neglected. Along this line a huge amount of research has been done [1, 2, 3], including studies in acoustoelasticity [4, 5, 6, 7, 8, 9]. In fact, against a random background, certain percentages of the grains of most structural metals tend to cluster around some orientations (and hence the material has texture) during manufacturing processes such as cold-working and annealing. The resulting slight anisotropy, while in most structural applications is negligible, has significant consequences in acoustoelasticity [7, 8, 9]. Despite the strong assumption made, this line of research has proved to be very fruitful. In fact nowadays texture analysis is applied not only in acoustoelasticity but even directly in production processes. In most of the above references the effective elastic constants of a textured polycrystalline material are estimated by averaging the elastic constants of the single crystallite in a generic orientation over all orientations, where the orientational average is weighted by a crystallite orientation distribution function (ODF). Very recently, Man [10] and Paroni and Man [11] obtained, under a physical hypothesis, representations of constitutive equations depending on the ODF. In particular Man [10] showed, again under a physical hypothesis, that the elasticity tensor ℂ of a weakly-textured orthorhombic aggregate of cubic crystallites with orientation distribution function w can be written as:

ℂ(w)[e] = λ(tr e)I + 2μe + c 𝕎(w)[e];  (1)

here e is the infinitesimal strain, a symmetric second-order tensor; I is the identity tensor; λ, μ and c are material constants; 𝕎(w) is an explicitly determined orthorhombic fourth-order tensor which, under a Cartesian coordinate system where the coordinate axes agree with the axes of orthorhombic symmetry of the texture and with the axes of cubic symmetry of the reference crystallite, can be written as:

\begin{aligned}(𝕎_{11}, 𝕎_{22}, 𝕎_{33}) & = & −(a_{2} + a_{3}, a_{3} + a_{1}, a_{1} + a_{2}), \\ (𝕎_{23}, 𝕎_{13}, 𝕎_{12}) & = & (a_{1}, a_{2}, a_{3}), \\ (𝕎_{44}, 𝕎_{55}, 𝕎_{66}) & = & (a_{1}, a_{2}, a_{3}), \end{aligned}  (2)

where

\begin{aligned}a_{1} & = & −4(W_{400} + \sqrt{5/2} W_{420}), \\ a_{2} & = & −4(W_{400} − \sqrt{5/2} W_{420}), \\ a_{3} & = & (W_{400} − \sqrt{70} W_{440}), \end{aligned}  (3)

and W_{lmn} are the expansion coefficients when we express w as an infinite series of the generalized spherical functions. In writing the components of the tensor 𝕎(w) we have used the Voigt index notation (cf. Auld [12]).

In the last thirty years the mathematical community has developed another method of deriving effective elasticities, namely, the homogenization method (see for instance [13, 14, 15, 16]). In this paper we apply homogenization techniques to recast the problem of determining effective elastic constants in a precise mathematical framework. Under our theory, we shall derive an expression for the effective elasticity tensor which will take into account not only the orientations of the crystallites but also the shapes and arrangements of the grains. From our formulation the ODF will acquire a precise mathematical meaning. We shall also prove the physical hypothesis made by Man [10] in his derivation of the representation written above.

As said previously, it is very difficult (if not impossible) to ascertain the microstructural information pointwise; for this reason we shall follow a statistical approach. In Section 2 we describe a probabilistic framework appropriate for studying the homogenization of a polycrystalline material. Under this model the orientation of a crystallite is viewed as a random variable. In constructing the probability space we try to emphasize the relationship of the model with practical experiments; in particular, we shall devote some attention to the probability measure ℘ which describes the orientation of the crystallite at a single point in space. In Section 3 the homogenization problem is considered. The homogenization of equations with random coefficients was considered by several authors, including Papanicolaou and Varadhan [17], Kozlov [18] and Dal Maso and Modica [19, 20]. All these studies, however, are concerned with the conductivity of random media. Papanicolaou and Varadhan [17] developed a very elegant abstract framework and proved the convergence theorem, classical in the theory of homogenization, by applying the energy method of Tartar (see for instance [13]). Kozlov [18] used instead the theory of G-convergence together with some of the ideas of De Giorgi and Spagnolo [21] as well as several properties of the Green’s function of the conductivity problem. Nonlinear problems were studied by Dal Maso and Modica [19, 20], using De Giorgi’s Γ-convergence. The stochastic homogenization of the elasticity system does not seem to have been dealt with in detail in the literature. By using a framework similar to that of Papanicolaou and Varadhan and using also some of Kozlov’s ideas, we give a relatively short and simple proof of the convergence problem based on the Div-Curl lemma of Murat [22] and Tartar [23]. In Section 3 we also establish our notation. In Section 4 we study how the homogenized elasticity tensor changes as the probability characterizing the ensemble changes. We show, under the assumption that the orientations of the crystallographic axes of the crystallites constitute an independent random field, that our theory is compatible with the formulation used in texture analysis. In particular, under this assumption we are able to prove that the physical assumption adopted by Man [10] in his derivation of Eq. (1) holds true. In Section 5 we study the symmetries of the homogenized elasticity tensor. After motivating the definition of “group of texture symmetry” and analyzing the restrictions imposed on the orientation measure ℘ by the symmetries of the crystallites, we conclude Section 5 with a lemma relating the “material symmetry” of the homogenized elasticity tensor with the texture symmetry of the polycrystalline aggregate. Finally, in Section 6 we establish some elementary bounds on the material constants λ, μ and c (cf. Eq. (1)).

Notation

Throughout this paper, Einstein’s summation convention is used. The Euclidean space considered is three-dimensional. We now introduce some definitions that we shall use in what follows:

e_{i} the i-th element of an orthonormal basis · the Frobenius inner product which is defined as u · v = u_{i}v_{i} for any

two vectors u, v, and as a · b = a_{ij}b_{ij} for any two second-order
tensors a, b

⊗ the tensor product which is defined by (a ⊗ b)u = a(b · u) for any

tensors a, b and u of the same order

SO(3) the group of all rotations L^{n} the Lebesgue measure on ℝ^{n} L^{2}(X, Y) the Lebesgue space of square-integrable functions on X ⊂ ℝ^{n} with

range in Y

L_{loc}^{2}(X, Y) the Lebesgue space of locally square-integrable functions on X ⊂ ℝ^{n}

with range in Y

H^{1}(X, Y) the Sobolev space of functions in L^{2}(X, Y) whose first derivatives

(in the sense of distributions) are also in L^{2}(X, Y)

H_{0}^{1}(X, Y) the Sobolev space of functions in H^{1}(X, Y) with null trace on the

boundary of X

H^{−1}(X, Y) the dual space of H_{0}^{1}(X, Y) Dv the matrix whose (i, j)th entry is the jth first derivative of v_{i} M^{n×n} the space of n × n matrices

2 Probability space for a polycrystalline material

Let us consider a single crystallite, which has an unstressed natural configuration λ, and a polycrystalline material C in a configuration κ. Throughout this section we shall denote κ(C) by X, and we shall suppose the polycrystalline aggregate C occupies the entire Euclidean space so that X = ℝ^{3}. For almost every x ∈ X (i.e., those points not at grain boundaries) there is a crystallite covering the point x; let ω(x) ∈ SO(3) be the orientation of that crystallite with respect to the configuration λ. The function ω : X → SO(3) defined by x ↦ ω(x) not only specifies the orientation of the grains but also the shape and the arrangement of them in the configuration κ (the region where the function ω is constant denotes a grain). Hence, we may think of the map ω : X → SO(3) as describing the configuration of a polycrystalline material. Let Ω be the set of all configurations of a polycrystalline aggregate. Having identified a polycrystalline material with a map from X into SO(3), we can take Ω = SO(3)^{X}, i.e., the set of all functions from X into SO(3). Henceforth, we endow the space Ω with the σ-algebra Σ = B^{X}, where B is the Borel σ-algebra on SO(3). For the sequel it will be convenient to define the map R_{x} : Ω → SO(3) by R_{x}(ω) = ω(x). We note that R_{x} is nothing but the natural projection of Ω into one of its components. Physically, we may interpret R_{x}(ω) as the orientation of the crystallite at x in the polycrystalline material ω. Let F be the class of nonempty finite subsets of X; for instance β = {x_{1}, …, x_{n}} ∈ F. We define R_{β} : Ω → SO(3)^{β} by R_{β}(ω) = {R_{x}(ω) : x ∈ β} = {ω(x) : x ∈ β} = ω_{β}. Since, by definition, Σ = B^{X} is equal to the smallest σ-algebra over ∪_{β∈F}R_{β}^{−1}(B^{β}), we have that each R_{β} is a measurable map from (Ω, Σ) to (SO(3)^{β}, B^{β}). In what follows we shall also assume that the mapping (x, ω) ↦ R_{x}(ω) is jointly measurable when the space X × Ω is endowed with the product σ-algebra L(ℝ^{3}) ⊗ Σ, where L(ℝ^{3}) is the σ-algebra of Lebesgue measurable sets in ℝ^{3}. We now define a measure on the measurable space (Ω, Σ).

Definition 2.1 Let A ∈ B, and x ∈ X. We define ℘_{x}(A) to be the probability to find the crystallite at x with orientation represented by an element of A.

In the above definition and unless otherwise stated the contrary, by orientation we will always mean the orientation of a crystallite in the configuration κ with respect the configuration λ.

Similarly, we make the following

Definition 2.2 Let β = {x_{1}, …, x_{n}} ∈ F, and let A = {A_{x_{i}} : i = 1, …, n, A_{x_{i}} ∈ B} ∈ B^{β}. We define ℘_{β}(A) = ℘_{{x_{1},...,x_{n}}}({A_{x_{1}}, …, A_{x_{n}}}) = probability to find the crystal at x_{i} represented by an element of A_{x_{i}}, for every i = 1, …, n, simultaneously.

Hereafter we suppose that all the set functions ℘_{β} are given for each β ∈ F; moreover we suppose that ℘_{β} is countably additive on the set of all finite measurable rectangles, for every β ∈ F. Since the set of all finite measurable rectangles forms a semialgebra, we can extend the set functions ℘_{β}, for every β ∈ F, to the smallest algebra over the set of all finite measurable rectangles, and by the Caratheodory-Hahn extension theorem (see Gariepy & Ziemer [24] p.105), such a measure can be extended so that it is on B^{β}, the Borel σ-algebra on SO(3)^{β}. We still denote this extension by ℘_{β}. From the definition (2.2) it is clear that if α, β ∈ F with α ⊂ β and if R_{αβ} is the natural projection of SO(3)^{β} onto SO(3)^{α}, we have ℘_{β}(R_{αβ}^{−1}(A)) = ℘_{α}(A) for every A ∈ B^{α}, i.e., the measures ℘_{β} are compatible. Now, by the Kolmogorov-Bochner extension theorem, (see Rao [25] p.365), there is a unique measure P on (Ω, Σ) such that

P (R_{β}^{−1}(A)) = ℘_{β}(A), for every A ∈ B^{β} and β ∈ F.  (4)

Let W ∈ Ω. Then P (W) is the probability to have a polycrystalline aggregate ω represented by an element in W. In the next section we will use the measure space (Ω, Σ, P) in the setting of homogenization.

Hereafter in order to simplify the problem we make some assumptions.

We suppose the random field {R_{x} : x ∈ X} to be strictly stationary. We recall that a random field {ϕ_{t} : t ∈ Z} from (Ω, Σ, P) to (Λ, Γ) is said to be strictly stationary if for every finite set {t_{1}, …, t_{n}} of indices of Z, every h ∈ Z, and all sets A_{1}, …, A_{n} ∈ Γ we have

P {ϕ_{t_{1}} ∈ A_{1}, …, ϕ_{t_{n}} ∈ A_{n}} = P {ϕ_{h+t_{1}} ∈ A_{1}, …, ϕ_{h+t_{n}} ∈ A_{n}}.

Let us analyze the consequences of this assumption. Let x, h ∈ X and A ∈ B, then by stationarity we have

P (R_{x}^{−1}(A)) = P (R_{x} ∈ A) = P (R_{x+h} ∈ A) = P (R_{x+h}^{−1}(A)),

and hence from Eq. (4) we obtain

℘(A) = ℘_{x}(A), for every x ∈ X and A ∈ B,  (5)

i.e., all points have the same “one-point correlation” measure.

Definition 2.3 Let T (h) : Ω → Ω be defined by

(T (h)ω)(x) = ω(x + h)  (6)

for every h ∈ X.

It is easily seen that T (0) = I, T (x)T (y) = T (x + y), and that the mapping (x, ω) ↦ T (x)ω is jointly measurable (recall that we have assumed the map (x, ω) ↦ R_{x}(ω) to be jointly measurable). Moreover, T (x) is P -measurable, i.e., P (T (x)A) = P (A) for every x ∈ X and A ∈ Σ. To see this, let Σ̃_{x} = {A ∈ Σ : P (T (x)A) = P (A)}. From the assumed strictly stationarity of R_{x} we have that the semialgebra of finite rectangles is contained in Σ̃_{x}, and hence the algebra generated by such a semialgebra is also contained in Σ̃_{x}. But, Σ̃_{x} is a monotone class, as it can be easily checked, and hence we have that Σ ⊂ Σ̃_{x}.

Hence, T (x) is a dynamical system (see Tempel’man [26]). By means of the dynamical system T (x) we can write

R_{x}(ω) = ω(x) = ω(0 + x) = (T (x)ω)(0) = R_{0}(T (x)ω).

Hence setting

R(ω) = R_{0}(ω),  (7)

we have

R_{x}(ω) = R(T (x)ω).  (8)

Let us denote by ℂ(x, ω) the elasticity tensor of the crystallite at x in the material point ω, and by ℂ^{∘} the elasticity tensor of the crystallite in the configuration λ. Recall that R_{x}(ω) is the orientation of the crystallite covering the point x in the configuration κ with respect to the configuration λ. It is well known from the theory of linear elasticity (see Gurtin [27]) that

ℂ̃(x, ω)[e] = R_{x}(ω)ℂ^{∘}[R_{x}^{T} (ω)eR_{x}(ω)]R_{x}^{T} (ω).  (9)

By defining

ℂ̃(ω)[e] = R(ω)ℂ^{∘}[R(ω)^{T} eR(ω)]R(ω)^{T}  (10)

for every symmetric matrix e, we find

ℂ(x, ω) = ℂ̃(T (x)ω).  (11)

This last equation implies that the random field {ℂ(x, ω) : x ∈ X} is strictly stationary.

Finally, we assume that the dynamical system T (x) is ergodic, so that Birkhoff’s ergodic theorem (see Tempel’man [26] or Cornfeld et al. [28]) applies. This assumption leads to the following characterization of ℘.

Lemma 2.1 Let ω be any point in Ω, and A ∈ B. Then if we define

f_{A}(x) = \{ \begin{matrix}1 & ω(x) ∈ A \\ 0 & otherwise \end{matrix}  (12)

we have

℘(A) = lim_{r→∞} \frac{1}{L^{3}(B_{r})} ∫_{B_{r}} f_{A}(x)dL^{3}(x),  (13)

where B_{r} denotes the ball centered at the origin with radius r.

Proof. Note that ω(x) ∈ A ⇔ R_{x}(ω) ∈ A ⇔ R(T (x)ω) ∈ A ⇔ T (x)ω ∈ R^{−1}(A). Hence if we define

g̃_{A}(ω) = χ_{R^{−1}(A)}(ω) = \{ \begin{matrix}1 & ω ∈ A \\ 0 & otherwise \end{matrix}

we have that g̃_{A}(T (x)ω) = f_{A}(x). Hence applying Birkhoff’s ergodic theorem, we obtain

lim_{r→∞} \frac{1}{L^{3}(B_{r})} ∫_{B_{r}} f_{A}(x)dL^{3}(x) = lim_{r→∞} \frac{1}{L^{3}(B_{r})} ∫_{B_{r}} g̃_{A}(T (x)ω)dL^{3}(x)  (14)
= ∫_{Ω} g̃_{A}(ω)dP (ω) = P (R^{−1}(A)) = ℘(A).
■

The right hand-side of Eq. (13) represents what material scientists do in practice to determine ℘ for a given sample. Take the method Randle [29] of electron back-scatter diffraction (EBSD) for instance. There the orientations of hundreds or thousands operator-selected grains are measured, and the data is fitted with a smooth distribution function as if all the grains are located at the same “material point”. Hence, this Lemma tells us that the orientation measure used in practice is equivalent to our definition.

3 Homogenization problem

Let (Ω, Σ, P) be the probability measure space and T (x) the dynamical system defined in the previous section.

Let H = L^{2}(Ω, Σ, P) be the Lebesgue space of functions which are square integrable with respect to the measure P on Ω. The space H, when endowed with the inner product

(f̃, h̃) = ∫_{Ω} f̃(ω)h̃(ω) dP (ω),  (15)

with f̃, h̃ ∈ H, becomes a Hilbert space.

For f̃ ∈ H we define the operators U(x) : H → H by

(U(x)f̃)(ω) = f̃(T (x)ω),  (16)

i.e., U(x)f̃ = f̃ ∘ T (x). The operators U(x) form a unitary group of operators (cf. Jikov, Kozlov and Oleinik [15]), and therefore they have closed densely defined infinitesimal generators, which we are now going to define (cf. Dunford and Schwartz [30]). For s ∈ ℝ we define the operator ∂_{s}^{(i)} : H → H, for i = 1, 2, 3, by

∂_{s}^{(i)}f̃ = \frac{U(se_{i})f̃− f̃}{s}.  (17)

Let D(∂_{i}) be the set of all f̃ ∈ H for which the H-limit (i.e., the limit in the L^{2}(Ω, Σ, P) sense) exists. We define the operator ∂_{i} : D(∂_{i}) → H by the formula

∂_{i}f̃ = H − lim_{s→0} ∂_{s}^{(i)}f̃.  (18)

The operator ∂_{i} with domain D(∂_{i}) is called the i-infinitesimal generator of the unitary group U(x).

The linear subspace

H^{1} = ⋂_{i=1}^{3} D(∂_{i})  (19)

is a dense subset of H and, when equipped with the inner product

(f̃, g̃)_{H^{1}} = (f̃, g̃)_{H} + (∂_{i}f̃, ∂_{i}g̃)_{H},  (20)
= ∫_{Ω} f̃(ω)g̃(ω)dP (ω) + ∫_{Ω} ∂_{i}f̃(ω)∂_{i}g̃(ω)dP (ω),

turns out to be a Hilbert space, since the ∂_{i} are closed. Moreover, for f̃ ∈ H^{1} we have

\frac{∂}{∂x_{i}} U(x)f̃(ω) = ∂_{i}f̃(T (x)ω),  (21)

where the differentiation is in the H sense (cf. Dunford and Schwartz [30]). For f̃ ∈ (H^{1})^{3}, i.e., f̃_{i} ∈ H^{1} for i = 1, 2, 3, define ∂ : (H^{1})^{3} → (H)^{3×3} by

(∂f̃)_{ij} = ∂_{j}f̃_{i},  (22)

for i, j = 1, 2, 3, and define also ẽ : (H^{1})^{3} → (H)^{3×3} by

ẽ(f̃) = \frac{1}{2}(∂f̃+ ∂f̃^{T}).  (23)

Let A be the space of symmetric-matrix-valued functions in H such that they satisfy the “generalized” Saint-Venant equations of compatibility (cf. Nečas and Hlavacek [31]) and with null average over Ω. More precisely, we introduce

Definition 3.1

A = {f̃_{ij} ∈ H : f̃_{ij} = f̃_{ji}, δ_{ihr}δ_{jks}∂_{r}∂_{s}f̃_{ij} = 0 weakly,  (24)
for h, k = 1, 2, 3, and ∫_{Ω} f̃_{ij}(ω)dP (ω) = 0}.

In the above definition δ_{ijk} is the so-called Levi-Civita skew-symmetric tensor. Roughly speaking, the “generalized” Saint-Venant equations of compatibility imply that if f̃ ∈ A, then there exists a vector-valued function u(x, ω) such that f̃_{ij}(T (x)ω) = \frac{1}{2} (\frac{∂u_{i}(x,ω)}{∂x_{j}} + \frac{∂u_{j}(x,ω)}{∂x_{i}}); for a precise statement see Lemma (7.2). It is easily seen that the space A is a Hilbert space.

Hereafter, we suppose that the elasticity tensor ℂ^{∘} in the reference λ be symmetric and positive definite, i.e., there exists a Λ_{1} such that ℂ^{∘}[ξ] · ξ ≥ Λ_{1}|ξ|^{2} for every second order symmetric tensor ξ, and that R(ω) be a bounded function for P -a.e. ω ∈ Ω. Hence, from Eq. (10), we deduce that ℂ̃(ω) = ℂ̃^{T} (ω) for P -a.e. ω ∈ Ω, and there exist Λ_{1}, Λ_{2} > 0 such that

ℂ̃(ω)[ξ] · ξ ≥ Λ_{1}|ξ|^{2},  (25)

for every second order symmetric tensor ξ and for P -a.e. ω ∈ Ω, and

max ess sup |ℂ̃_{ijkl}(ω)| = Λ_{2} < ∞,  (26)

where the max is taken over all i, j, k, l = 1, 2, 3 and the ess sup over all ω ∈ Ω. Henceforth we denote the symmetric tensor product by ⊙, i.e.,

α ⊙ β = \frac{1}{2}(α ⊗ β + β ⊗ α),  (27)

for any two vectors α and β.

Lemma 3.1 For any k, l = 1, 2, 3, there exists a unique ẽ^{kl} ∈ A such that

∫_{Ω} ℂ̃(ω)[e_{k} ⊙ e_{l} + ẽ^{kl}(ω)] · ψ̃(ω)dP (ω) = 0,  (28)

for all ψ̃ ∈ A.

The proof follows easily by an application of the Lax-Milgram lemma.

Definition 3.2 If ẽ^{kl} ∈ A is the unique solution of Eq. (28), define

ℂ^{h}[e_{k} ⊙ e_{l}] = ∫_{Ω} ℂ̃(ω)[e_{k} ⊙ e_{l} + ẽ^{kl}(ω)] dP (ω).  (29)

It is not too difficult to show that the homogenized elasticity tensor ℂ^{h} can be evaluated by means of the following formula (cf. Paroni [32])

ℂ^{h}[η] · η = inf_{ψ̃∈A} ∫_{Ω} ℂ̃(ω)[η + ψ̃(ω)] · (η + ψ̃(ω)) dP (ω),  (30)

which holds for every constant symmetric tensor η. We note that for every ψ̃ ∈ A we have, using Eq. (25),

∫_{Ω} ℂ̃(ω)[η + ψ̃(ω)] · (η + ψ̃(ω)) dP (ω) ≥ Λ_{1} ∫_{Ω} |η + ψ̃(ω)|^{2} dP (ω)
≥ Λ_{1}(η · η + 2η · ∫_{Ω} ψ̃(ω) dP (ω)) = Λ_{1}|η|^{2},

and therefore from Eq. (30) we find

ℂ^{h}[η] · η ≥ Λ_{1}|η|^{2}  (31)

for all symmetric tensors η.

We now report some lemmas that shall be very useful in proving the main theorem of this section.

Let V ⊂ ℝ^{3} be a simply connected domain with smooth boundary. Henceforth, if u is a differentiable function from V to ℝ^{3} we denote by e(u) the symmetric part of the gradient of u, i.e., e(u) = \frac{1}{2}(Du + Du^{T}).

Theorem 3.1 (Korn’s first inequality) The inequality

∥Du∥_{L^{2}(V,M^{3×3})} ≤ \sqrt{2}∥e(u)∥_{L^{2}(V,M^{3×3})}

holds for all u ∈ H_{0}^{1}(V, ℝ^{n}).

For a proof see for instance Nečas and Hlavacek [31]. The next Lemma can be found in Kozlov [18].

Lemma 3.2 Let f̃ ∈ H, and let’s denote f(x, ω) = f̃(T (x)ω). Then as ε → 0

f(\frac{·}{ε}, ω) ⇀< f̃ >= ∫_{Ω} f̃(ω)dP (ω),

in L_{loc}^{2}(ℝ^{3}) for P -a.e. ω ∈ Ω.

In the proof of the next theorem we will use several times the Div-Curl lemma of Murat [22] and Tartar [23], in the form for the elasticity system.

Lemma 3.3 Let {v_{k}}_{k=1}^{∞}, {σ_{k}}_{k=1}^{∞} be two sequences in H^{1}(V, ℝ^{3}) and L^{2}(V, M^{3×3}), respectively, such that

1. {div σ_{k}}_{k=1}^{∞} lies in a compact subset of H^{−1}(V, ℝ^{3}), and σ_{k} = σ_{k}^{T} for every k;

2. e_{k} = e(v_{k}) is a bounded sequence in L^{2}(V, M^{3×3}).

Suppose further e_{k} ⇀ e and σ_{k} ⇀ σ in L^{2}(V, M^{3×3}), then

e_{k} · σ_{k} ⇀ e · σ

in the sense of distributions.

For a proof see Murat [22], Francfort and Murat [33] (cf. also Evans [34]). We now state and prove the main theorem of this section. In the proof we will use two lemmas which are proved in the Appendix.

Theorem 3.2 Let f ∈ L^{2}(V, ℝ^{3}). For every ϕ(·, ω) ∈ H_{0}^{1}(V; ℝ^{3}) for P -a.e ω ∈ Ω there exists a unique u^{ε}(·, ω) ∈ H_{0}^{1}(V; ℝ^{3}) for P -a.e ω ∈ Ω such that the identity

∫_{V} ℂ (\frac{x}{ε}, ω) [e(u^{ε}(x, ω))] · e(ϕ(x, ω))dL^{3}(x) = ∫_{V} f(x) · ϕ(x, ω)dL^{3}(x)  (32)

holds for P -a.e ω ∈ Ω. Moreover, the solutions u^{ε} converge weakly in H_{0}^{1}(V; ℝ^{3}), for P -a.e, to the solution u ∈ H_{0}^{1}(V; ℝ^{3}) of the deterministic variational problem

∫_{V} ℂ^{h}[e(u(x))] · e(ϕ(x))dL^{3}(x) = ∫_{V} f(x) · ϕ(x)dL^{3}(x)  (33)

which holds for every ϕ ∈ H_{0}^{1}(V; ℝ^{3}).

Proof. Set S^{ε}(x, ω) = ℂ(\frac{x}{ε}, ω)e(u^{ε}(x, ω)). Taking ϕ = u^{ε} in Eq. (32) and using the coercivity of ℂ, (see Eq. (25)), we find

∫_{V} |e(u^{ε}(x, ω))|^{2}dL^{3}(x) ≤ C (∫_{V} |u^{ε}(x, ω))|^{2}dL^{3}(x))^{\frac{1}{2}} ,

and from Poincare’s and first Korn’s inequalities (cf. Theorem (3.1)) we get

∫_{V} |e(u^{ε}(x, ω))|^{2}dL^{3}(x) ≤ C (∫_{V} |e(u^{ε}(x, ω))|^{2}dL^{3}(x))^{\frac{1}{2}} ,

for P -a.e ω ∈ Ω. By using again Poincare’s and first Korn’s inequalities and the boundeness of ℂ, (see Eq. (26)), we obtain

∫_{V} (|u^{ε}(x, ω)|^{2} + |Du^{ε}(x, ω)|^{2})dL^{3}(x) ≤ C

and

∫_{V} |S^{ε}(x, ω)|^{2}dL^{3}(x) ≤ C,

where the constant C is independent of ε. By passing to a subsequence, still denoted by ε, we have

u^{ε}(·, ω) ⇀ u^{∗}(·, ω), in H_{0}^{1}(V), P − a.e ω ∈ Ω,  (34)
S^{ε}(·, ω) ⇀ S^{∗}(·, ω), in L^{2}(V, M^{3×3}), P − a.e ω ∈ Ω.  (35)

Set

𝔹̃(ω) = ℂ̃(ω)[e_{i} ⊙ e_{j} + ẽ^{ij}(ω)] ⊗ (e_{i} ⊙ e_{j}),

where ẽ^{ij} is the solution of Eq. (28). As always we set 𝔹(x, ω) = 𝔹̃(T (x)ω) and e^{ij}(x, ω) = ẽ^{ij}(T (x)ω). An easy computation shows that 𝔹(x, ω)[e_{p} ⊙ e_{q}] = ℂ(x, ω)[e_{p} ⊙ e_{q} + e^{pq}(x, ω)] so that by Eq. (29) we have

< 𝔹̃(ω)[e_{p} ⊙ e_{q}] > = ∫_{Ω} 𝔹̃(ω)[e_{p} ⊙ e_{q}]dP (ω) = ℂ^{h}e_{p} ⊙ e_{q},

and hence from Lemma 3.2 we obtain

𝔹 (\frac{·}{ε}, ω) [e_{p} ⊙ e_{q}] ⇀ ℂ^{h}[e_{p} ⊙ e_{q}] in L^{2}(V, M^{3×3}), P − a.e ω ∈ Ω. (36)

From Lemma 3.1 and using the definition of 𝔹̃, we have

∫_{Ω} 𝔹̃(ω)[e_{k} ⊙ e_{l}] · ψ̃(ω)dP (ω) = 0

for all ψ̃ ∈ A, and by Lemma 7.1 this implies that div(𝔹(\frac{x}{ε}, ω)[e_{k} ⊙ e_{l}]) is compact in H^{−1}(V), for P -a.e. ω ∈ Ω. Thus, from Eq. (34), Eq. (36), and Lemma 3.3, we obtain

𝔹 (\frac{·}{ε}, ω) [e_{p} ⊙ e_{q}] · e(u^{ε}(·, ω))) ⇀ ℂ^{h}[e_{p} ⊙ e_{q}] · e(u^{∗}(·, ω)),  (37)

in the sense of distributions, for P -a.e. ω ∈ Ω. By Lemma 3.2, and using the fact that ẽ^{pq} ∈ A, we find

e^{pq} (\frac{·}{ε}, ω) ⇀< ẽ^{pq} >= 0  (38)

in L^{2}(V, M^{3×3}) for P -a.e. ω ∈ Ω. From the Div-Curl Lemma 3.3 and Lemma 7.2, we find

S^{ε}(·, ω) · e^{pq} (\frac{·}{ε}, ω) ⇀ 0  (39)

in the sense of distributions, for P -a.e. ω ∈ Ω. Now, a simple computation yields

S^{ε}(·, ω) · e^{pq} (\frac{·}{ε}, ω) = 𝔹 (\frac{·}{ε}, ω) [e_{p} ⊙ e_{q}] · e(u^{ε}(·, ω)) − S^{ε}(·, ω) · (e_{p} ⊙ e_{q}), (40)

and therefore using Eqs. (35), (37) and (39) we obtain

ℂ^{h}[e_{p} ⊙ e_{q}] · e(u^{∗}(·, ω)) = S^{∗}(·, ω) · e_{p} ⊙ e_{q},

for P -a.e. ω ∈ Ω, which is equivalent to

S^{∗}(·, ω) = ℂ^{h}e(u^{∗}(·, ω)),  (41)

for P -a.e. ω ∈ Ω. Taking the limit of Eq. (32), with Eqs. (35) and (41) in mind, we find

∫_{V} ℂ^{h}[e(u^{∗}(x, ω))] · e(ϕ(x, ω))dL^{3}(x) = ∫_{V} f(x) · ϕ(x, ω)dL^{3}(x),  (42)

which should hold for every ϕ(·, ω) ∈ H_{0}^{1}(V) and for P -a.e. ω ∈ Ω. Since ℂ^{h} does not depend on ω, we have that the solution u^{∗} ∈ H_{0}^{1}(V), and the uniqueness of the problem yields u^{∗} = u, where u is the solution of Eq. (33). ■

4 Representation of the homogenized elasticity tensor

In this section we study how the homogenized elasticity tensor changes if we change the probability measure P, and we prove the physical assumption adopted by Man [10] in his derivation of Eq. (1).

Fix Q ∈ SO(3), and consider an ensemble of polycrystalline aggregates characterized by a probability measure P̂ defined by

P̂(A) = P (π_{Q}^{−1}A),  (43)

for every A ∈ Σ, where π_{Q} : Ω → Ω is defined by

(π_{Q}ω)(x) = Qω(Q^{T} x).  (44)

Note that for every ω ∈ Ω and h ∈ X we have π_{Q}(T (Q^{T} h)ω)(x) = Qω(Q^{T} x + Q^{T} h) = (π_{Q}ω)(x + h) = (T (h)π_{Q}ω)(x), from which we deduce that

π_{Q}(T (Q^{T} h)ω) = T (h)π_{Q}ω.  (45)

By using Eq. (45) it is easy to see that T (x) is an ergodic dynamical system with respect to the measure P̂.

Let Ĥ = L^{2}(Ω, Σ, P̂) and let Ĥ^{1} be the set of functions f̃ ∈ Ĥ for which the Ĥ-limit of ∂_{s}^{(i)}f̃ exists for every i = 1, 2, 3. For f̃ ∈ Ĥ^{1}, set ˆ∂_{i}f̃ = Ĥ − lim_{s→0} ˆ∂_{s}^{(i)}f̃.

Analogous to what we did in Section 2, we define

Â = {f̃_{ij} ∈ Ĥ : f̃_{ij} = f̃_{ji}, δ_{ihr}δ_{jks} ˆ∂_{r} ˆ∂_{s}f̃_{ij} = 0 weakly,  (46)
for h, k = 1, 2, 3, and ∫_{Ω} f̃_{ij}(ω)dP̂(ω) = 0},

and

ℂ^{h}(P̂)[η] · η = inf_{ψ̃∈Â}∫_{Ω} ℂ̃(ω)[η + ψ̃(ω)] · (η + ψ̃(ω))dP̂(ω).  (47)

It is easy to see that for every f̃ ∈ L^{1}(Ω, Σ, P̂) the following identity holds:

∫_{Ω} f̃(ω)dP̂(ω) = ∫_{Ω} f̃(π_{Q}ω)dP (ω),  (48)

or equivalently

∫_{Ω} f̃(π_{Q}^{−1}ω)dP̂(ω) = ∫_{Ω} f̃(ω)dP (ω).  (49)

Lemma 4.1 If f̃ is any function defined on Ω, we set (ζ_{Q}f̃)(ω) = f̃(π_{Q}ω). For any f̃ ∈ (H^{1})^{3}, the following identity holds:

ζ_{Q}(ˆ∂(Qf̃)) = Q∂(ζ_{Q}f̃)Q^{T} ,  (50)

i.e., ζ_{Q}(ˆ∂_{i}(Q_{lp}f̃_{p})) = Q_{lp}∂_{i}(ζ_{Q}f̃_{p})Q_{ij}.

Proof. Let Qf̃ ∈ (Ĥ^{1})^{3}. Then, from Eq. (45) and Eq. (49), we obtain

0 = lim_{s→0} ∫_{Ω} |\frac{Q_{lp}f̃_{p}(T (se_{i})ω) − Q_{lp}f̃_{p}(ω)}{s} − ˆ∂_{i}(Q_{lp}f̃_{p})(ω)|^{2} dP̂(ω),  (51)

= lim_{s→0} ∫_{Ω} |Q_{lp} \frac{f̃_{p}(T (se_{i})π_{Q}ω) − f̃_{p}(π_{Q}ω)}{s} − ˆ∂_{i}(Q_{lp}f̃_{p})(π_{Q}ω)|^{2} dP (ω),

= lim_{s→0} ∫_{Ω} |Q_{lp} \frac{ζ_{Q}f̃_{p}(T (sQ_{ir}e_{r})ω) − ζ_{Q}f̃_{p}(ω)}{s} − ζ_{Q} ˆ∂_{i}(Q_{lp}f̃_{p})(ω)|^{2} dP (ω),
= ∫_{Ω} |Q_{lp}∂_{r}(ζ_{Q}f̃_{p})(ω)Q_{ir} − ζ_{Q} ˆ∂_{i}(Q_{lp}f̃_{p})(ω)|^{2} dP (ω),

which implies

Q∂(ζ_{Q}f̃)Q^{T} = ζ_{Q} ˆ∂(Qf̃).  (52)
■

The previous Lemma suggests that if f̃ ∈ Â, then Q^{T} ζ_{Q}f̃Q ∈ A. We study if this is the case in the next Lemma.

Lemma 4.2 If g̃ is any function defined on Ω, we set (ζ_{Q}g̃)(ω) = g̃(π_{Q}ω). The map from A to Â defined by

g̃ ↦ Qζ_{Q^{T}} g̃Q^{T}  (53)

is a bijection with inverse defined by

g̃ ↦ Q^{T} ζ_{Q}g̃Q.  (54)

Proof. From Eq. (48) and Eq. (49), we notice that ψ̃ ∈ Ĥ if and only if ζ_{Q}ψ̃ ∈ H; moreover, by repeating the proof of Lemma (4.1), we see that ψ̃ ∈ Ĥ^{1} if and only if ζ_{Q}ψ̃ ∈ H^{1} and

ζ_{Q} ˆ∂_{i}ψ̃ = Q_{ir}∂_{r}(ζ_{Q}ψ̃),  (55)

for i = 1, 2, 3. From the definition of π_{Q} in Eq. (44), it is easy to see that π_{Q}^{−1} = π_{Q^{T}}, and from this equality it immediately follows that ζ_{Q}^{−1} = ζ_{Q^{T}}. We can therefore rewrite Eq. (55) as

ˆ∂_{i}ψ̃ = Q_{ir}ζ_{Q^{T}} ∂_{r}(ζ_{Q}ψ̃),  (56)

for i = 1, 2, 3. Let H^{2} = {f̃ ∈ H^{1} : ∂_{i}f̃exists for every i = 1, 2, 3}, and let Ĥ^{2} = {f̃ ∈ Ĥ^{1} : ˆ∂_{i}f̃exists for every i = 1, 2, 3}. Then by repeating the argument above we have that ψ̃ ∈ Ĥ^{2} if and only if ζ_{Q}ψ̃ ∈ H^{2} and

ˆ∂_{l} ˆ∂_{i}ψ̃ = Q_{lp}ζ_{Q^{T}} (∂_{p}(ζ_{Q}ζ_{Q^{T}} Q_{ir}∂_{r}(ζ_{Q}ψ̃))) = Q_{lp}Q_{ir}ζ_{Q^{T}} (∂_{p}∂_{r}(ζ_{Q}ψ̃)),  (57)

for l, i = 1, 2, 3. We are now ready to check that the map defined by Eq. (53) is well defined. Let g̃ ∈ A. We want to show that f̃ = Qζ_{Q^{T}} g̃Q^{T} ∈ Â. That f̃ is symmetric follows immediately from the definition of f̃ itself and from the fact that g̃ is symmetric. From Eq. (49) we find

∫_{Ω} f̃(ω) dP̂(ω) = ∫_{Ω} Qg̃(π_{Q^{T}} ω)Q^{T} dP̂(ω) = Q ∫_{Ω} g̃(ω) dP (ω)Q^{T} = 0.

We now check that f̃ satisfies the “generalized” Saint-Venant condition of compatibility. Let ψ̃ ∈ Ĥ^{2}. Then by using Eq. (57) and Eq. (49) we have

δ_{ikl}δ_{jmn} ∫_{Ω} f̃_{km}(ω)ˆ∂_{l} ˆ∂_{n}ψ̃(ω) dP̂(ω)
= δ_{ikl}δ_{jmn} ∫_{Ω} Q_{kp}Q_{mr}g̃_{pr}(π_{Q^{T}} ω)ˆ∂_{l} ˆ∂_{n}ψ̃(ω) dP̂(ω)
= δ_{ikl}δ_{jmn}Q_{kp}Q_{mr}Q_{lb}Q_{na} ∫_{Ω} g̃_{pr}(π_{Q^{T}} ω)∂_{b}∂_{a}(ζ_{Q}ψ̃)(π_{Q^{T}} ω) dP̂(ω)
= δ_{ikl}δ_{jmn}Q_{kp}Q_{mr}Q_{lb}Q_{na} ∫_{Ω} g̃_{pr}(ω)∂_{b}∂_{a}(ζ_{Q}ψ̃)(ω) dP (ω).

Note that

δ_{ikl}Q_{kp}Q_{lb} = δ_{ikl}Qe_{p} · e_{k} Qe_{b} · e_{l} = Qe_{p} × Qe_{b} · e_{i}
= Q(e_{p} × e_{b}) · e_{i} = e_{p} × e_{b} · Q^{T} e_{i} = δ_{dpb}Q_{id},

where × denotes the cross product, and hence from the previous two equations we obtain

δ_{ikl}δ_{jmn} ∫_{Ω} f̃_{km}(ω)ˆ∂_{l} ˆ∂_{n}ψ̃(ω) dP̂(ω)
= Q_{id}Q_{js}δ_{dpb}δ_{sra} ∫_{Ω} g̃_{pr}(ω)∂_{b}∂_{a}(ζ_{Q}ψ̃)(ω) dP (ω) = 0,

for g̃ ∈ A and ζ_{Q}ψ̃ ∈ H^{2}. That the map defined by Eq. (53) is a bijection is obvious, and that the map defined by Eq. (54) is its inverse follows from the fact that ζ_{Q}^{−1} = ζ_{Q^{T}}. ■

For the remaining part of this section we write ℂ^{h}(P) in place of ℂ^{h}, i.e.,

ℂ^{h}(P)[η] · η = inf_{ψ̃∈A} ∫_{Ω} ℂ̃(ω)[η + ψ̃(ω)] · (η + ψ̃(ω))dP (ω).  (58)

Theorem 4.1 For every symmetric matrix e, the following identity holds:

ℂ^{h}(P̂)[e] = Qℂ^{h}(P)[Q^{T} eQ]Q^{T} .  (59)

Proof. Note that R(π_{Q}ω) = (π_{Q}ω)(0) = Qω(0) = QR(ω). Hence for every symmetric matrix e, we have

ℂ̃(π_{Q}ω)[e] = R(π_{Q}ω)ℂ^{∘}[R(π_{Q}ω)^{T} eR(π_{Q}ω)]R(π_{Q}ω)^{T}
= QR(ω)ℂ^{∘}[R(ω)^{T} Q^{T} eQR(ω)]R(ω)^{T} Q^{T}
= Qℂ̃(ω)[Q^{T} eQ]Q^{T} .

By using this identity and Eq. (49), we obtain

ℂ^{h}(P)[η] · η =
= inf_{ψ̃∈A} ∫_{Ω} Q^{T} ℂ̃(π_{Q}ω)[QηQ^{T} + Qψ̃(ω)Q^{T}]Q · (η + ψ̃(ω)) dP (ω)
= inf_{ψ̃∈A} ∫_{Ω} ℂ̃(π_{Q}ω)[QηQ^{T} + Qψ̃(ω)Q^{T}] · (QηQ^{T} + Qψ̃(ω)Q^{T}) dP (ω)
= inf_{ψ̃∈A} ∫_{Ω} ℂ̃(ω)[QηQ^{T} + Qψ̃(π_{Q}^{−1}ω)Q^{T}] · (QηQ^{T} + Qψ̃(π_{Q}^{−1}ω)Q^{T}) dP̂(ω).

Now, from Lemma 4.2 we can rewrite the previous equation as

ℂ^{h}(P)[η] · η = inf_{ϕ̃∈Â}∫_{Ω} ℂ̃(ω)[QηQ^{T} + ϕ̃(ω)] · (QηQ^{T} + ϕ̃(ω)) dP̂(ω)
= ℂ^{h}(P̂)[QηQ^{T}] · QηQ^{T} .

Finally from the polarization identity, we obtain

ℂ^{h}(P)[α] · η = \frac{1}{2}(ℂ^{h}(P)[α + η] · (α + η) −
ℂ^{h}(P)[α] · α − ℂ^{h}(P)[η] · η)
=  \frac{1}{2}(ℂ^{h}(P̂)[Q(α + η)Q^{T}] · Q(α + η)Q^{T} −
ℂ^{h}(P̂)[QαQ^{T}] · QαQ^{T} − ℂ^{h}(P̂)[QηQ^{T}] · QηQ^{T})
=  ℂ^{h}(P̂)[QαQ^{T}] · QηQ^{T} ,

and hence the theorem is proved. ■

We now suppose that {R_{x} : x ∈ X} is a family of independent random variables. Thus, for every β = {x_{1}, …, x_{n}} ∈ F and any A = {A_{x} : x ∈ β} ∈ B^{β}, we have

P (R_{β}^{−1}(A)) = P (R_{x_{1}} ∈ A_{x_{1}}, …, R_{x_{n}} ∈ A_{x_{n}})  (60)
= P (R_{x_{1}} ∈ A_{x_{1}}) ⋯ P (R_{x_{n}} ∈ A_{x_{n}})
= P (R_{x_{1}}^{−1}(A_{x_{1}})) ⋯ P (R_{x_{n}}^{−1}(A_{x_{n}})),

and Eq. (4) yields

℘_{β}(A) = ℘_{{x_{1},...,x_{n}}}({A_{x_{1}}, ⋯, A_{x_{n}}}) = ℘(A_{x_{1}}) ⋯ ℘(A_{x_{n}}).  (61)

Hence ℘_{β} is uniquely determined by the “one-point correlation” measure ℘; in this case P is nothing but the product measure P = ⊗_{x∈X}℘. This leads us to conclude that there is a one-to-one correspondence between the measure ℘ and the measure P.

Hereafter, if ω ∈ Ω we denote by Qω ∈ Ω the function defined by (Qω)(x) = Q(ω(x)). Hence if A ∈ Ω we have

QA = {ω ∈ Ω : ∃ ω̃ ∈ A s.t. Qω̃ = ω} = {Qω : ω ∈ A}.

Lemma 4.3 Define

T_{Q}℘(A) = ℘(Q^{T} A)  for every A ∈ B.  (62)

Then

P̂ = ⊗_{x∈X}T_{Q}℘.  (63)

Proof. Let β = {x_{1}, …, x_{n}} and A = {A_{1}, …, A_{n}}, with each A_{i} ∈ B. Define Q^{T} β = {Q^{T} x_{1}, …, Q^{T} x_{n}} and Q^{T} A = {Q^{T} A_{1}, …, Q^{T} A_{n}}. Then R_{β}^{−1}(A) = {ω ∈ Ω : ω(x_{i}) ∈ A_{i}, for i = 1, …, n} = {ω ∈ Ω : Q^{T} ω(x_{i}) ∈ Q^{T} A_{i}, for i = 1, …, n} = {ω ∈ Ω : (π_{Q}^{−1}ω)(Q^{T} x_{i}) ∈ Q^{T} A_{i}, for i = 1, …, n} = π_{Q}{ω ∈ Ω : ω(Q^{T} x_{i}) ∈ Q^{T} A_{i}, for i = 1, …, n} = π_{Q}R_{Q^{T} β}^{−1}(Q^{T} A). Hence

R_{β}^{−1}(A) = π_{Q}R_{Q^{T} β}^{−1}(Q^{T} A).  (64)

From Eqs. (61), (62) and (64), we deduce that

T_{Q}℘(A_{i}) ⋯ T_{Q}℘(A_{n}) = ℘_{Q^{T} β}(Q^{T} A) = P (R_{Q^{T} β}^{−1}(Q^{T} A))
= P̂(π_{Q}R_{Q^{T} β}^{−1}(Q^{T} A)) = P̂(R_{β}^{−1}(A)),

and hence the lemma is proved. ■

Thus if the orientations form an independent random field, we have that there is a one-to-one correspondence between the probability measures P and ℘, and between P̂ and T_{Q}℘; we can therefore write ℂ^{h}(℘) for ℂ^{h}(P) and ℂ^{h}(T_{Q}℘) for ℂ^{h}(P̂). In this case it follows from the Theorem 4.1 the following corollary.

Corollary 4.1 For every symmetric matrix e, the following identity holds:

ℂ^{h}(T_{Q}℘)[e] = Qℂ^{h}(℘)[Q^{T} eQ]Q^{T} .  (65)

Eq. (65) is the physical assumption adopted by Man [10] in his derivation of Eqs. (1), (2) and (3) for an aggregate of cubic crystallites with weak orthorhombic texture. We refer to [10] for a proof of that representation.

5 Symmetries of the homogenized material

In continuum mechanics the symmetries of a given body are defined by observing the macroscopic behavior of the body; no knowledge of the microstructure of the body is needed. One might therefore think, as far as the number of independent elastic constants is concerned, that there is no difference between an orthorhombic material and an orthorhombic aggregate of cubic crystallites, for instance. We now show that there is a fundamental difference.

Lemma 5.1 If I is an eigenvector of ℂ^{∘} then ℂ^{∘}[I] = ℂ^{h}[I], i.e., if ℂ^{∘}[I] = αI for some α ≠ 0, then ℂ^{∘}[I] = ℂ^{h}[I] = αI.

Proof. Let η be any symmetric second order tensor and ψ̃ ∈ A. From Eq. (10), we have

ℂ̃(ω)[I] = R(ω)ℂ^{∘}[R(ω)^{T} IR(ω)]R(ω)^{T} = R(ω)αIR(ω)^{T} = ℂ^{∘}[I],

and using this equality we find

∫_{Ω} ℂ̃(ω)[I + η + ψ̃(ω)] · (I + η + ψ̃(ω)) dP (ω) =
= ∫_{Ω} ℂ̃(ω)[I] · I + 2ℂ̃(ω)[I] · (η + ψ̃(ω))
+ ℂ̃(ω)[η + ψ̃(ω)] · (η + ψ̃(ω)) dP (ω),
= ℂ^{∘}[I] · I + 2ℂ^{∘}[I] · η + 2ℂ^{∘}[I] · ∫_{Ω} ψ̃(ω) dP (ω)
+ ∫_{Ω} ℂ̃(ω)[η + ψ̃(ω)] · (η + ψ̃(ω)) dP (ω),
= ℂ^{∘}[I] · I + 2ℂ^{∘}[I] · η + ∫_{Ω} ℂ̃(ω)[η + ψ̃(ω)] · (η + ψ̃(ω)) dP (ω).

Taking the infimum over all ψ̃ ∈ A on both sides of the preceding equality, we obtain (cf. Eq. (30))

ℂ^{h}[I + η] · (I + η) = ℂ^{∘}[I] · I + 2ℂ^{∘}[I] · η + ℂ^{h}[η] · η.

Multiplying out the left hand side and simplyfing, we deduce the identity

ℂ^{h}[I] · I + 2ℂ^{h}[I] · η = ℂ^{∘}[I] · I + 2ℂ^{∘}[I] · η,

which holds for every symmetric tensor η. By taking η = I we find ℂ^{h}[I]·I = ℂ^{∘}[I]·I; thus the preceding equation reduces to

ℂ^{h}[I] · η = ℂ^{∘}[I] · η.

The proof of the lemma is finished, for this last equation holds true for every symmetric second order tensor η. ■

The elasticity tensor ℂ^{∘} of a cubic crystal has an eigenvector equal to the identity I. If we consider an orthorhombic aggregate of cubic crystallites, the homogenized elasticity tensor ℂ^{h} will therefore enjoy not only the symmetries of an orthorhombic material but also the property to have ℂ^{h}[I] parallel to I. This extra condition reduces to seven the number of independent elastic constants needed to describe an orthorhombic aggregate of cubic crystallites. We recall that a generic orthorhombic material has nine independent elastic constants.

We now try to characterize the symmetries of the aggregate. We start by studying the symmetries of the measure ℘; to do so we find it convenient to change the reference configurations λ and κ. Let λ^{∗} be a reference configuration which can be obtained by a rotation Q of the reference λ; we write λ^{∗} = Qλ. Let ℘^{λ^{∗},κ} be the probability measure associated with the references λ^{∗} and κ. Then, if R and R^{∗} are the orientations of a crystallite in the configuration κ with respect to the configurations λ and λ^{∗}, respectively, we have that R^{∗} = RQ^{T}. Thus if A ⊂ SO(3) we have R^{∗} ∈ A ⇔ R ∈ AQ, and from this equivalence it follows that ℘(A) = ℘^{λ^{∗},κ}(AQ). We write this last result in the following most suggestive way:

℘(A) = ℘^{λ,κ}(A) = ℘^{Qλ,κ}(AQ),  (66)

for every Q ∈ SO(3). Similarly, by changing the reference κ and by keeping fixed the reference λ, we obtain

℘(A) = ℘^{λ,κ}(A) = ℘^{λ,Qκ}(QA),  (67)

for every Q ∈ SO(3).

Let us denote by G_{cr} the symmetry group of the crystallite in the reference configuration λ. Then, by definition we have

ℂ^{∘}[e] = Qℂ^{∘}[Q^{T} eQ]Q^{T} ,
λ^{∗} = Qλ = λ,

for every Q ∈ G_{cr} and symmetric tensor e. From Eq. (66) and the previous equation, we conclude that the measure ℘ has to satisfy the following identity

℘(A) = ℘(AQ),  (68)

for every Q ∈ G_{cr} and every Borel A ⊂ SO(3). Analogously, we define G_{tex} to be the set of Q ∈ SO(3) for which ℘^{λ,κ} = ℘^{λ,Qκ}; the group G_{tex} measures the symmetries of the reference configuration κ in a probabilistic sense. From Eq. (67) we find that

℘(A) = ℘(QA),  (69)

for every Q ∈ G_{tex} and every Borel A ⊂ SO(3). Following Man & Paroni [7], we call G_{tex} the group of texture symmetry.

From Corollary (4.1) we immediately find:

Lemma 5.2 Suppose the orientations of the crystallites in an aggregate form an independent random field. If Q ∈ G_{tex}, then Q belongs to the symmetry group of the material.

6 Bounds on material constants

In this section we restrict our attention to the case of an aggregate of cubic crystallites with weak orthorhombic texture symmetry. We find it convenient to choose a coordinate system whose coordinate axes are the axes of orthorhombic symmetry of the texture and we pick for the crystallites a reference orientation whose axes of cubic symmetry fall on the axes of the coordinate system. We further suppose the orientation measure ℘ to be absolutely continuous with respect to the unique probability Haar measure ℘_{H} on SO(3), ℘ ≪ ℘_{H} (cf. Rudin [35, 36]). We can then define the orientation distribution function w in terms of ℘ and ℘_{H} by

w = \frac{1}{8π^{2}} \frac{d℘}{d℘_{H}} ,  (70)

where d℘/d℘_{H} denotes the Radon-Nikodym derivative of ℘ with respect to ℘_{H}.

Henceforth we assume that w is square-integrable on SO(3) with respect to the Haar measure ℘_{H}. After parametrizing the rotation group SO(3) with the Euler angles (ψ, θ, ϕ), the definition of which we follow the convention adopted by Roe [1], we may expand w as an infinite series of the generalized spherical functions:

w(ψ, θ, ϕ) = ∑_{l=0}^{∞} ∑_{m=−l}^{l} ∑_{n=−l}^{l} W_{lmn}Z_{lmn}(cos θ)e^{−imψ}e^{−inϕ} ,  (71)

where Z_{lmn} are the normalized Jacobi polynomials. We call the expansion coefficients W_{lmn} the texture coefficients.

Now, under the assumption that the orientations of the crystallites form an independent random field, we can write the homogenized elasticity tensor ℂ^{h} as

ℂ^{h}[e] = λ(tr e)I + 2μe + c 𝕎[e];  (72)

here e is a second-order symmetric tensor, I is the second-order identity tensor and the fourth-order tensor 𝕎 is defined through Eqs. (2) and (3); for brevity here we have written 𝕎 in place of 𝕎(w). We would like to point out once more that the representation given by Eq. (72) was proved by Man in [10] under the assumption that our Corollary (4.1) would hold and under the assumption of weak texture, i.e., terms quadratic or of higher degree in the texture coefficients W_{lmn} are negligible. If we let 𝕀 to be the fourth-order identity tensor and we define the fourth-order tensors 𝕁 and 𝕂 by

𝕁[a] = \frac{tr a}{3}I,  𝕂 = 𝕀 − 𝕁,  (73)

where Eq. (73)_{1} holds true for every symmetric second-order tensor a, we can rewrite Eq. (72) as

ℂ^{h} = (3λ + 2μ)𝕁 + 2μ𝕂 + c𝕎.  (74)

By using these definitions it is easily seen that

\begin{matrix}𝕁𝕁 = 𝕁, 𝕂𝕂 = 𝕂, 𝕁𝕂 = 𝕂𝕁 = 0, \\ 𝕁𝕎 = 𝕎𝕁 = 0, 𝕂𝕎 = 𝕎𝕂 = 𝕎, \end{matrix}  (75)

and hence using Eq. (75) and the assumption of weak texture we find

ℂ^{h^{−1}} = \frac{1}{3λ + 2μ}𝕁 + \frac{1}{2μ}𝕂 − \frac{c}{4μ^{2}} 𝕎.  (76)

Since the crystallites are cubic, we can write ℂ^{∘}, under the chosen coordinate system, as (cf. Walpole [37])

ℂ^{∘} = (2c_{12} + c_{11})𝕁 + 2c_{44}(𝕀 − 𝔾) + 3(c_{11} − c_{12})(𝔾 − 𝕁)  (77)

where 𝔾 is the fourth-order tensor defined by

𝔾 = ∑_{i=1}^{3}(e_{i} ⊗ e_{i}) ⊗ (e_{i} ⊗ e_{i}).  (78)

We note that c_{11}, c_{12} and c_{44} are the three independent components of ℂ^{∘} written in the abbreviated subscript notation, and also that

ℂ^{∘^{−1}} = \frac{1}{2c_{12} + c_{11}} 𝕁 + \frac{1}{2c_{44}} (𝕀 − 𝔾) + \frac{1}{3(c_{11} − c_{12})}(𝔾 − 𝕁).  (79)

In Hirao et al. [4] it is shown by straight integration (cf. also Man and Paroni [7]) that

ℂ^{V} [e] = ∫_{SO(3)} rℂ^{∘}[r^{t}er]r^{t} d℘(r) = (3λ^{V} + 2μ^{V})𝕁[e] + 2μ^{V} 𝕂[e] + c^{V} 𝕎[e], (80)

where

μ^{V} = \frac{c_{11} − c_{12} + 3c_{44}}{5} λ^{V} = \frac{c_{11} + 4c_{12} − 2c_{44}}{5} ,

c^{V} = c_{11} − c_{12} − 2c_{44}.

By using Eq. (75) and Eq. (80) we find

ℂ^{R}[e] = (∫_{SO(3)} rℂ^{∘^{−1}}[r^{t}er]r^{t} d℘(r))^{−1} = (3λ^{R}+2μ^{R})𝕁[e]+2μ^{R}𝕂[e]+c^{R}𝕎[e], (81)

where

μ^{R} = 5\frac{(c_{11} − c_{12})c_{44}}{3c_{11} − 3c_{12} + 4c_{44}} λ^{R} = \frac{2c_{11}c_{44} − 6c_{12}c_{44} − c_{11}^{2} − c_{11}c_{12} + 2c_{12}^{2}}{3c_{11} − 3c_{12} + 4c_{44}} ,

c^{R} = 50\frac{(c_{11} − c_{12})c_{44}(c_{11} − c_{12} − 2c_{44})}{(3c_{11} − 3c_{12} + 4c_{44})^{2}}.

Before we state and prove the theorem on the bounds on λ, μ and c, we point out that the variational principle dual (cf. Ekeland and Temam [38] and also Kohn and Milton [39]) to Eq. (30) is (cf. Paroni [32]):

ℂ^{h^{−1}}[η] · η = inf_{\begin{matrix}ψ̃ ∈ A^{⊥} \\ < ψ̃ >= η \end{matrix}} ∫_{Ω} ℂ̃(ω)^{−1}[ψ̃(ω)] · ψ̃(ω) dP (ω),  (82)

where A^{⊥} denotes the orthogonal complement of A as a subset of H^{3×3}. We recall that < ψ̃ >= ∫_{Ω} ψ̃(ω) dP (ω).

Theorem 6.1 With the notation above and

Γ = (a_{1}^{2} + a_{2}^{2} + a_{3}^{2} − a_{1}a_{2} − a_{1}a_{3} − a_{2}a_{3})^{\frac{1}{2}} ,

where the a_{i}’s are defined in Eq. (3), we have

μ^{R} + c^{R}a_{i} ≤ μ + ca_{i} ≤ μ^{V} + c^{V} a_{i} for i = 1, 2, 3,
2μ^{R} − c^{R}(a_{1} + a_{2} + a_{3} ± Γ) ≤ 2μ − c(a_{1} + a_{2} + a_{3} ± Γ)
≤ 2μ^{V} − c^{V} (a_{1} + a_{2} + a_{3} ± Γ),
2μ + 3λ = 2μ^{R} + 3λ^{R} = 2μ^{V} + 3λ^{V} .

Proof. With the help of Eq. (10), a change of variables and by taking ψ̃(ω) = 0 in Eq. (30), we find

ℂ^{h}[η] · η ≤ ∫_{Ω} ℂ̃(ω)[η] · η dP (ω) = ∫_{Ω} R(ω)ℂ^{∘}[R(ω)^{T} ηR(ω)]R(ω)^{T} · η dP (ω)
= ∫_{SO(3)} rℂ^{∘}[r^{t}ηr]r^{t} d℘(r) · η = ℂ^{V} [η] · η.

Similarly, by taking ψ̃(ω) = η ∈ A^{⊥}, in Eq. (82) we find ℂ^{h^{−1}}[η] · η ≤ ℂ^{R^{−1}}[η] · η, and hence we have

ℂ^{R}[η] · η ≤ ℂ^{h}[η] · η ≤ ℂ^{V} [η] · η,

because ℂ^{h} is symmetric and positive definite (cf. Eq. (31)). This last equation implies that the eigenvalues of ℂ^{V} − ℂ^{h} and of ℂ^{h} − ℂ^{R} must be nonnegative. It is easily seen that these conditions on the eigenvalues imply the bounds on λ, μ and c claimed in the statement of the theorem. ■

From the previous theorem we see that the quantity 2μ + 3λ is uniquely determined, while we have a total of 10 inequalities which give us restrictions on μ and c. We conclude this section by applying the previous theorem to the following particular case:

Aluminum  Texture coefficients
data after Sarma & Reddy [40]  data after Hirsch and Lücke [41]
c_{11} = 106.75 GPa  W_{400} = −0.0026
c_{12} = 60.41 GPa  W_{420} = +0.0030
c_{44} = 28.34 GPa  W_{400} = −0.0078

In this case the pair (μ, c) has to fall inside the darker region in Figure 1.

Figure 1: Bounds on μ and c. 26

7 Appendix

The scope of this appendix is to prove the two lemmas that we used in the proof of Theorem 3.2. Before stating and proving the lemmas we make some definitions.

Let η be the standard mollifier on ℝ^{3} (cf. Evans and Gariepy [42]) and set η^{δ}(x) = \frac{1}{δ^{3}} η(\frac{x}{δ}). For f̃ ∈ H, we define

f̃^{δ}(ω) = ∫ℝ^{3} η^{δ}(x)f̃(T (x)ω)dL^{3}(x).  (83)

Note that if f̃ ∈ H, then f̃(T (·)ω) ∈ L_{loc}^{2}(ℝ^{3}) for P -a.e. ω ∈ Ω (cf. Jikov et. al. [15]) and hence the above definition makes sense. It is easy to prove that if f̃ ∈ H, then f̃^{δ} ∈ H^{1} and

∂_{i}f̃^{δ}(ω) = − ∫ℝ^{3} \frac{∂}{∂x_{i}} η^{δ}(x)f̃(T (x)ω)dL^{3}(x).  (84)

Moreover, by applying Fubini’s theorem we obtain

∫_{Ω} f̃^{δ}(ω)g̃(ω) dP (ω) = ∫_{Ω} f̃(ω)g̃^{δ}(ω) dP (ω), for every f̃, g̃ ∈ H,  (85)

and

∫_{Ω} ∂f̃(ω)g̃(ω) dP (ω) = − ∫_{Ω} f̃(ω)∂g̃(ω) dP (ω), for every f̃, g̃ ∈ H^{1}.  (86)

We shall also use the identity

∂(f̃^{δ}) = (∂f̃)^{δ},  (87)

which, as it can be easily checked, holds true for every f̃ ∈ H^{1}.

Lemma 7.1 Let L̃ ∈ H^{3×3} such that L̃ = L̃^{T} and

∫_{Ω} L̃(ω) · ψ̃(ω) dP (ω) = 0

for every ψ̃ ∈ A. Then for every ξ ∈ C_{0}^{∞}(ℝ^{3}, ℝ^{3}) we have

∫ℝ^{3} L̃(T (x)ω) · Dξ(x)dL^{3}(x) = 0,

for P -a.e. ω ∈ Ω.

Proof. Let ϕ̃_{i} ∈ H^{1}, for i = 1, 2, 3. Then ψ̃_{ij} = \frac{1}{2}(∂_{j}ϕ̃_{i}^{δ}+∂_{i}ϕ̃_{j}^{δ}) belongs to A, for every i, j = 1, 2, 3. In fact, by definition ψ̃_{ij} = ψ̃_{ji}. Also ψ̃_{ij} satisfies the “generalized” Saint-Venant compatibility conditions because the operators ∂_{i} commute, while the average of the ψ̃_{ij} over Ω is equal to zero as a result of Eq. (86) because the ∂_{i} are unitary operators. Using the symmetry of L̃ and Eqs. (85)–(87) we obtain

0 = ∫_{Ω} L̃(ω) · ψ̃(ω) dP (ω) = ∫_{Ω} L̃(ω) · ∂ϕ̃^{δ}(ω) dP (ω) =  (88)
= ∫_{Ω} L̃^{δ}(ω) · ∂ϕ̃(ω) dP (ω) = − ∫_{Ω} ∂_{i}L̃_{ij}^{δ}(ω)ϕ̃_{j}(ω) dP (ω);

which implies

∂_{i}L̃_{ij}^{δ} = 0,  (89)

because H^{1} is dense in H, for every j = 1, 2, 3. This last equation and Eq. (21) imply

\frac{∂}{∂x_{i}} L̃_{ij}^{δ}(T (x)ω) = 0  (90)

for every x and ω where the differentiation is meant in the H sense. But convergence in the H sense implies convergence in the L_{loc}^{2}(ℝ^{3}) sense for P -a.e.; hence Eq. (90) holds for P -a.e. ω ∈ Ω with differentiation in the L^{2} sense. Choose ξ_{j} ∈ C_{0}^{∞}(ℝ^{3}, ℝ^{3}) for j = 1, 2, 3, we have

0 = ∫ℝ^{3} \frac{∂}{∂x_{i}} L̃_{ij}^{δ}(T (x)ω)ξ_{j}(x)dL^{3}(x) = − ∫ℝ^{3} L̃_{ij}(T (x)ω)\frac{∂}{∂x_{i}} ξ_{j}^{δ}(x)dL^{3}(x). (91)

Now letting δ → 0 (cf. Evans and Gariepy [42]) we obtain the statement of the lemma. ■

Lemma 7.2 For any f̃ ∈ A, there exists a ξ(·, ω) ∈ H^{1}(V; ℝ^{3}) for P -a.e. ω ∈ Ω such that

f̃(T (x)ω) = \frac{1}{2} (Dξ(x, ω) + Dξ(x, ω)^{T} )

for P -a.e. ω ∈ Ω.

Proof. Let H^{2} = {f̃ ∈ H^{1} : ∂_{i}f̃exists for every i = 1, 2, 3}. Then H^{2} is dense in H^{1}, and hence in H. If f̃ ∈ A we have that

δ_{ihr}δ_{jks} ∫_{Ω} f̃_{ij}(ω)∂_{r}∂_{s}ϕ̃(ω) dP (ω) = 0,

for every ϕ̃ ∈ H^{2} and for every h, k = 1, 2, 3. By repeating the proof of Lemma 7.1 we find

δ_{ihr}δ_{jks} ∫ℝ^{3} f̃_{ij}(T (x)ω)\frac{∂^{2}}{∂x_{r}∂x_{s}} ζ(x)dL^{3}(x) = 0,

for every ζ ∈ C_{o}^{∞}(ℝ^{3}), and for every h, k = 1, 2, 3,. From this last equation it follows the statement of the lemma (cf. Paroni [32] or Nečas and Hlavacek [31]). ■

Acknowledgment

The present paper is an edited version of a part of my dissertation, which was accepted by the University of Kentucky in May 1998. First of all, I wish to thank my advisor Chi-Sing Man for his continuous meticulous teaching, for the many interesting discussions I have had with him, for sharing with me some of his acute ideas, for correcting my mistakes, for his financial support and for being not only a teacher but also a friend. I would also like to thank Cesare Davini; it was he who, after my undergraduate degree, strongly advised me to pursue postgraduate studies, and it was he who introduced me to Dr. Man. He has always shown much interest in my work, and I have learned a lot from my interactions with him. He also gave me financial support and continuous encouragement. I also wish to thank G. Dal Maso and P. Hislop for their suggestions and comments pertaining to the problem treated in this paper. The research reported here was supported in part by a grant (No. CMS-9522829) from the National Science Foundation.

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