A random field formulation of Hooke's law in all elasticity classes

For each of the $8$ isotropy classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set $\mathsf{V}$ of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders $1$ and $2$ of such a field, and the field's spectral expansion.


Introduction
Microstructural randomness is present in just about all solid materials. When dominant (macroscopic) length scales are large relative to microscales, one can safely work with deterministic homogeneous continuum models. However, when the separation of scales does not hold and spatial randomness needs to be accounted for, various concepts of continuum mechanics need to be re-examined and new methods developed. This involves: (1) being able to theoretically model and simulate any such randomness, and (2) using such results as input into stochastic field equations. In this paper, we work in the setting of linear elastic random media that are statistically wide-sense homogeneous and isotropic.
Regarding the modelling motivation (1), two basic issues are considered in this study: (i) type of anisotropy, and (ii) type of correlation structure. Now, with reference to Fig. 1 showing a planar Voronoi tessellation of E 2 which serves as a planar geometric model of a polycrystal (although the same arguments apply in E 3 ), each cell may be occupied by a differently oriented crystal, with all the crystals belonging to any specific crystal class. The latter include: • transverse isotropy modelling, say, sedimentary rocks at long wavelengths; • tetragonal modelling, say, wulfenite (PbMoO 4 ); • trigonal modelling, say, dolomite (CaMg(CO 3 ) 2 ); • orthotropic modelling, say wood or orthoclase feldspar; • triclinic modelling, say, microcline feldspar.
Thus, we need to be able to model rank 4 tensor random fields, point-wise taking values in any crystal class. While the crystal orientations from grain to grain are random, in general they are not spatially independent of each other -the assignment of crystal properties over the tessellation is not white noise. This is precisely where the two-point characterisation of the random field of elasticity tensor is needed, so as to account for any mathematically admissible correlation structures as dictated by the statistically wide-sense homogeneous and isotropic assumption. A specific correlation can then be fitted to physical measurements.
Regarding the modelling motivation (1), it may also be of interest to work with a mesoscale random continuum approximation defined by placing a mesoscale window at any spatial position as shown in Fig. 1(b). Clearly, the larger is the mesoscale window, the weaker are the random fluctuations in the mesoscale elasticity tensor: this is the trend to homogenise the material when upscaling from a statistical volume element (SVE) to a representative volume element (RVE), e.g. [30,32]. A simple paradigm of this upscaling, albeit only in terms of a scalar random field, is the opacity of a sheet of paper held against light: the further away is the sheet from our eyes, the more homogeneous it appears. Similarly, in the case of upscaling of elastic properties, on any finite scale there is (almost surely) an anisotropy and this anisotropy, with mesoscale increasing, tends to zero hand-in-hand with the fluctuations and it is in the infinite mesoscale limit (i.e. RVE) that material isotropy is obtained as a consequence of the statistical isotropy.
Regarding the motivation (2) of this study, i.e. input of elasticity random fields into stochastic field equations, there are two principal routes: stochastic partial differential equations (SPDE) and stochastic finite elements (SFE). The classical paradigm of SPDE [22] can be written in terms of the anti-plane elastostatics (with u ≡ u 3 ): ∇ · (C (x, ω) ∇u) = 0, x ∈ E 2 , ω ∈ Ω with C (·, ω) being spatial realisations of a scalar RF. In view of the foregoing discussion, (1) is well justified for a piecewise-constant description of realisations of a random medium such as a multiphase composite made of locally isotropic grains. However, in the case of a boundary value problem set up on coarser (i.e. mesoscales) scales, a rank 2 tensor random field (TRF) of material properties would be much more appropriate, Fig. 1(b). The field equation should then read where C is the rank 2 tensor random field. Indeed, this type of upscaling is sorely needed in the stochastic finite element (SFE) method, where, instead of assuming the local isotropy of the elasticity tensor for each and every material volume (and, hence, finite element), full triclinic-type anisotropy is needed [31].
Here u is the displacement field, λ and µ are two Lamé constants, and ρ is the mass density. This equation is often (e.g. in stochastic wave propagation) used as an Ansatz, typically with the pair (λ, µ) taken ad hoc as a "vector" random field with some simple correlation structure for both components. However, in order to properly introduce the smooth randomness in λ and µ, one has to go one step back in derivation of (3) and write µ∇ 2 u + (λ + µ) ∇ (∇ · u) + ∇µ ∇u + (∇u) + ∇λ∇ · u = ρü (4) or µu i , jj + (λ + µ) u j , ji +µ, j (u j , i +u i , j ) + λ, i u j , j = ρü i .
While two extra terms are now correctly present on the left-hand side, this equation still suffers from the drawback (just as (1)) of local isotropy so that, again by micromechanics upscaling arguments, it should be replaced by ∇ · (C∇ · u) = ρü or C ijkl u (k , l) , j = ρü i .
Here C (= C ijkl e i ⊗ e j ⊗ e k ⊗ e l ), which, at any scale finitely larger than the microstructural scale, is almost surely (a.s.) anisotropic. Clearly, instead of (4) one should work with the SPDE (5) for u. While the mathematical theory of SPDEs with anisotropic realisations is not developed, one powerful way to numerically solve such equations is through stochastic finite elements (SFE). However, the SFE, just like the SPDE, require a general representation of the random field C [31], so it can be fitted to micromechanics upscaling studies, as well as its spectral expansion. Observe that each and every material volume (and, hence, the finite element) is an SVE of Fig. 1(b), so that a full triclinic-type anisotropy is needed: all the entries of the rank 4 stiffness tensor C are non-zero with probability one. While a micromechanically consistent procedure for upscaling has been discussed in [35] and references cited there, general forms of the correlation tensors are sorely needed.
In this paper we develop second-order TRF models of linear hyperelastic media in each of the eight elasticity classes. That is, for each class, the fourthrank elasticity tensor is taken as an isotropic and homogeneous random field in a three-dimensional Euclidean space, for which the one-point (mean) and two-point correlation functions need to be explicitly specified. The simplest case is that of an isotropic class, which implies that two Lamé constants are random fields. Next, we develop representations of seven higher crystal classes: cubic, transversely isotropic, trigonal, tetragonal, orthotropic, monoclinic, and triclinic. We also find the general form of field's spectral expansion for each of the eight isotropy classes.

The formulation of the problem
Let E = E 3 be a three-dimensional Euclidean point space, and let V be the translation space of E with an inner product (·, ·). Following [37], the elements A of E are called the places in E. The symbol B − A is the vector in V that translates A into B.
Let B ⊂ E be a deformable body. The strain tensor ε(A), A ∈ B, is a configuration variable taking values in the symmetric tensor square S 2 (V ) of dimension 6. Following [28], we call this space a state tensor space.
The stress tensor σ(A) also takes values in S 2 (V ). This is a source variable, it describes the source of a field [36].
We work with materials obeying Hooke's law linking the configuration variable ε(A) with the source variable σ(A) by Here the elastic modulus C is a linear map C(A) : S 2 (V ) → S 2 (V ). In linearised hyperelasticity, the map C(A) is symmetric, i.e., an element of a constitutive tensor space V = S 2 (S 2 (V )) of dimension 21.
We assume that C(A) is a single realisation of a random field. In other words, denote by B(V) the σ-field of Borel subsets of V. There is a probability space (Ω, F, P) and a mapping C : B × Ω → V such that for any A 0 ∈ B the mapping C(A 0 , ω) : Ω → V is (F, B(V))-measurable.
Translate the whole body B by a vector x ∈ V . The random fields C(A + x) and C(A) have the same finite-dimensional distributions. It is therefore convenient to assume that there is a random field defined on all of E such that its restriction to B is equal to C(A). For brevity, denote the new field by the same symbol C(A) (but this time A ∈ E). The random field C(A) is strictly homogeneous, that is, the random fields C(A + x) and C(A) have the same finite-dimensional distributions. In other words, for each positive integer n, for each x ∈ V , and for all distinct places A 1 , . . . , A n ∈ E the random elements C(A 1 ) ⊕ · · · ⊕ C(A n ) and C(A 1 + x) ⊕ · · · ⊕ C(A n + x) of the direct sum on n copies of the space V have the same probability distribution.
Let K be the material symmetry group of the body B acting in V . The group K is a subgroup of the orthogonal group O(V ). Fix a place O ∈ B and identify E with V by the map f that maps A ∈ E to A − O ∈ V . Then K acts in E and rotates the body B by g · A = f −1 gf A, g ∈ K, A ∈ B.
Let A 0 ∈ B. Under the above action of K the point A 0 becomes g · A 0 . The random tensor C(A 0 ) becomes S 2 (S 2 (g))C(A 0 ). The random fields C(g·A) and S 2 (S 2 (g))C(A) must have the same finite-dimensional distributions, because g · A 0 is the same material point in a different place. Note that this property does not depend on a particular choice of the place O, because the field is strictly homogeneous.
To formalise the non-formal considerations of the above paragraph, note that the map g → S 2 (S 2 (g)) is an orthogonal representation of the group K, that is, a continuous map from K to the orthogonal group O(V) that respects the group operations: S 2 (S 2 (g 1 g 2 )) = S 2 (S 2 (g 1 ))S 2 (S 2 (g 2 )), g 1 , g 2 ∈ K.
of the group K = O(3) in the space V = S 2 (S 2 (R 3 )). The symmetry group of an elasticity tensor C ∈ V is Note that the symmetry group K(g · C) is conjugate through g to K(C): Whenever two bodies can be rotated so that their symmetry groups coincide, they share the same symmetry class. Mathematically, two elasticity tensors C 1 and C 2 are equivalent if and only if there is g ∈ O(3) such that K(C 1 ) = K(g·C 2 ). In view of (6), C 1 and C 2 are equivalent if and only if their symmetry groups are conjugate. The equivalence classes of the above relation are called the elasticity classes. The first column of Table 1 adapted from [2], contains the name of an elasticity class. The second column represents a collection of subgroups H of O(3) such that H is conjugate to a symmetry group of any elasticity tensor of the given class. In other words, the above symmetry group lies in the conjugacy class [H] of the group H. The third column contains the notation for the normaliser N (H): Here Z c 2 = {I, −I}, where I is the 3×3 identity matrix, Z n is generated by the rotation about the z-axis with angle 2π/n, O (2) is generated by rotations about the z-axis with angle θ, 0 ≤ θ < 2π and the rotation about the x-axis with angle π, D n is the dihedral group generated by Z n and the rotation about the x-axis with angle π, and O is the octahedral group which fixes an octahedron. See also [29,Appendix B] for the correspondence between the above notation and notation of Hermann-Mauguin [19,25] and Schönfließ [34].
The importance of the group N (H) can be clarified as follows. Consider the fixed point set of H: By [2,Lemma 3.1], if H is the symmetry group of some tensor C ∈ V, then N (H) is the maximal subgroup of O(3) which leaves V H invariant. In the language of the representation theory, V H is an invariant subspace of the representation g → S 2 (S 2 (g)) of any group K that lies between H and N (H), that is, S 2 (S 2 (g))C ∈ V H for all g ∈ K and for all C ∈ V H . Denote by U (g) the restriction of the above representation to V H .
The problem is formulated as follows. For each elasticity class [H] and for each group K that lies between H and N (H), consider an V H -valued homogeneous random field C(x) on R 3 . Assume that C(x) is isotropic with respect to U : C(gx) = U (g) C(x) , C(gx), C(gy) = (U ⊗ U )(g) C(x), C(y) .
We would like to find the general form of the one-and two-point correlation tensors of such a field and the spectral expansion of the field itself in terms of stochastic integrals.
To explain what we mean consider the simplest case when the answer is known. Put K = H = O(3), V H = R 1 , and U (g) = 1, the trivial representation of K. Recall that a measure Φ on the σ-field of Borel sets of a Hausdorff topological space X is called tight if for any Borel set B, Φ(B) is the supremum of Φ(K) over all compact subsets K of B. A measure Φ is called locally finite if every point of X has a neighbourhood U for which Φ(U ) is finite. A measure Φ is called a Radon measure if it is tight and locally finite. In what follows we consider only Radon measures and call them just measures.
Schoenberg [33] proved that the equation establishes a one-to-one correspondence between the class of two-point correlation tensors of homogeneous and isotropic random fields τ (x) and the class of finite measures on [0, ∞). Let L 2 0 (Ω) be the Hilbert space of centred complex-valued random variables with finite variance. Let Z be a L 2 0 (Ω)-valued measure on the σ-field of Borel sets of a Hausdorff topological space X. A measure Φ is called the control measure for Z, if for any Borel sets B 1 and B 2 we have Yaglom [39] and independently M.Ȋ. Yadrenko in his unpublished PhD thesis proved that the field τ (x) has the form are real-valued spherical harmonics, J +1/2 (λρ) are the Bessel functions of the first kind of order + 1/2, and Z m is a sequence of centred uncorrelated real-valued orthogonal random measures on [0, ∞) with the measure Φ as their common control measure.
Other known results include the case of V H = R 3 , and U (g) = g. Yaglom [38] found the general form of the two-point correlation tensor. Malyarenko and Ostoja-Starzewski [24] found the spectral expansion of the field. In the same paper, they found both the general form of the two-point correlation tensor and the spectral expansion of the field for the case of V H = S 2 (R 3 ), and U (g) = S 2 (g). In [23] they solved one of the cases for two-dimensional elasticity, when V = R 2 , K = O(2), V H = S 2 (S 2 (R 2 )), and U (g) = S 2 (S 2 (g)). Remark 1. The set of possible values of elasticity tensors is a proper subset of V H , namely, the intersection of V H with the cone K of symmetric nonnegative operators in S 2 (V ). The complete description of homogeneous and isotropic random fields taking values in V H ∩K is not known even in the simplest case, when V H = R 1 and K = [0, ∞). It is possible to construct various particular classes of such random fields using the ideas of Guilleminot and Soize [13,14,16,15,17]. The advantage of their approach is that the random field depends on a few real parameters and may be easily simulated and calibrated. Our approach is based on general spectral expansions, whereby the above questions become more complicated and will be considered in forthcoming publications.

A general result
The idea of this Section is as follows. Let V be a finite-dimensional real linear space, let K be a closed subgroup of the group O(3), and let U be an orthogonal representation of the group K in the space V. Consider a homogeneous and isotropic random field C(x), x ∈ R 3 , and solve the problem formulated in Section 2. In Section 5, apply general formulae to our cases. The resulting Theorems 1-16 are particular cases of general Theorem 0.
To obtain general formulae, we describe all homogeneous random fields taking values in V and throw away non-isotropic ones. The first obstacle here is as follows. The complete description of such fields is unknown. We use the following result instead.
Let V C be a complex finite-dimensional linear space with an inner product (·, ·) that is linear in the second argument, as is usual in physics. Let J be a real structure on V C , that is, a map J : V C → V C satisfying the following conditions: for all α, β ∈ C and for all C 1 , C 2 ∈ V C . In other words, J is a multidimensional and coordinate-free generalisation of complex conjugation. The set of all eigenvectors of J that correspond to eigenvalue 1, constitute a real linear space, denote it by V. Let H be the real linear space of Hermitian linear operators in V C . The real structure J induces a linear operator J in H. For any A ∈ H, the operator JA acts by In coordinates, the operator J is just the transposition of a matrix. The result by Cramér [5] in coordinate-free form is formulated as follows. Equation establishes a one-to-one correspondence between the class of two-point correlation tensors of homogeneous mean-square continuous V C -valued random fields C(x) and the class of Radon measures on the σ-field of Borel sets of the wavenumber domainR 3 tasking values in the set of nonnegative-definite Hermitian linear operators in V C . For V-valued random fields, there is only a necessary condition: if C(x) is V-valued, then the measure F satisfies Introduce the trace measure µ by µ(B) = tr F (B), B ∈ B(R 3 ) and note that F is absolutely continuous with respect to µ. This means that Equation (8) may be written as where f (p) is a measurable function on the wavenumber domain taking values in the set of all nonnegative-definite Hermitian linear operators in V C with unit trace, that satisfies the following condition Using representation theory, it is possible to prove the following. Let C 1 , By linearity, this action may be extended to an isomorphism L between V ⊗ V and H. The orthogonal operators LU ⊗ U (g)L −1 , g ∈ K, constitute an orthogonal representation of the group K in the space H, equivalent to the tensor square U ⊗ U of the representation U . The operator L is an intertwining operator between the spaces V ⊗ V and H where equivalent representations U ⊗ U and LU ⊗ U L −1 act. In what follows, we are working only with the latter representation, for simplicity denote it again by U ⊗ U and note that it acts in the space H by Denote H + = LS 2 (V). In coordinates, it is the subspace of Hermitian matrices with real-valued matrix entries. If −I ∈ K, then the second equation in (7) and Equation (9) together are equivalent to the following conditions: and The description of all measures µ satisfying Equation (10) is well known, see [3]. There are finitely many, say M , orbit types for the action of K inR 3 by (gp, x) = (p, g −1 x).
Denote by (R 3 /K) m , 0 ≤ m ≤ M − 1 the set of all orbits of the mth type. It is known, see [2], that all the above sets are manifolds. Assume for simplicity of notation that there are charts λ m such that the domain of λ m is dense in (R 3 /K) m . The orbit of the mth type is the manifold K/H m , where H m is a stationary subgroup of a point on the orbit. Assume that the domain of a chart ϕ m is a dense set in K/H m , and let dϕ m be the unique probabilistic K-invariant measure on the σ-field of Borel sets of K/H m . There are the unique measures Φ m on the σ-fields of Borel sets in ( To find all functions f satisfying Equation (11), proceed as follows. Fix an orbit λ m and denote by ϕ 0 m the coordinates of the intersection of the orbit λ m with the set (R 3 /K) m . Let U m be the restriction of the representation S 2 (U ) to the group H m . We have g(λ m , ϕ 0 m ) = (λ m , ϕ 0 m ) for all g ∈ H m , because H m is the stationary subgroup of the point (λ m , ϕ 0 m ). For g ∈ H m , Equation (11) becomes Any orthogonal representation of a compact topological group in a space H has at least two invariant subspaces: {0} and H. The representation is called irreducible if no other invariant subspaces exist. The space of any finite-dimensional orthogonal representation of a compact topological group can be uniquely decomposed into a direct sum of isotypic subspaces. Each isotypic subspace is the direct sum of finitely many subspaces where the copies of the same irreducible representation act. Equation (12) means that the operator f (λ m , ϕ 0 m ) lies in the isotypic subspace H m which corresponds to the trivial representation of the group H m . The intersection of this subspace with the convex compact set of all nonnegative-definite operators in H + with unit trace is again a convex compact set, call it C m . As λ m runs over (R 3 /K) m , f (λ m , ϕ 0 m ) becomes an arbitrary measurable function taking values in C m .
An irreducible orthogonal representation of the group K is called a representation of class 1 with respect to the group H m if the restriction of this representation to H m contains at least one copy of the trivial representation of H m . Let S 2 (U ) m be the restriction of the representation S 2 (U ) to the direct sum of the isotypic subspaces of the irreducible representation of class 1 with respect to H m . Let g ϕm be an arbitrary element of K such that g ϕm (ϕ 0 m ) = ϕ m . Two such elements differ by an element of H m , therefore the second equation in (11) becomes The two-point correlation tensor of the field takes the form Choose an orthonormal basis T 1 , . . . , T dim V in the space V. The tensor square V ⊗ V has several orthonormal bases. The coupled basis consists of tensor products The mth uncoupled basis is build as follows. Let U m,1 , . . . , U m,km be all non-equivalent irreducible orthogonal representations of the group K of class 1 with respect to H m such that the representation S 2 (U ) contains isotypic subspaces where c mk copies of the representation U m,k act, and let the restriction of the representation U m,k to H m contains d mk copies of the trivial representation of H m . Let T mkln , 1 ≤ l ≤ d mk , 1 ≤ n ≤ c mk be an orthonormal basis in the space where the nth copy act. Complete the above basis to the basis T mkln , 1 ≤ l ≤ dim U m,k and call this basis the mth uncoupled basis. The vectors of the coupled basis are linear combinations of the vectors of the mth uncoupled basis: where dots denote the terms that include the tensors in the basis of the space S 2 (V) S 2 (V) m . In the introduced coordinates, Equation (13) takes the form The choice of bases inside the isotypic subspaces is not unique. One has to choose them in such a way that calculation of the transition coefficients c mkln ij is as easy as possible.
To calculate the inner integrals, we proceed as follows. Consider the action of K on R 3 by matrix-vector multiplication. Let (R 3 /K) m , 0 ≤ m ≤ M − 1 be the set of all orbits of the mth type. Let ρ m be such a chart that its domain is dense in (R 3 /K) m . Let ψ m be a chart in K/H m with a dense domain, and let dψ m be the unique probabilistic K-invariant measure on the σ-field of Borel sets of K/H m . It is known that the sets of orbits of one of the types, say ( and consider the plane wave as a function of two variables ϕ M −1 and ψ M −1 with domain (K/H M −1 ) 2 . This function is K-invariant: Denote byK H M −1 the set of all equivalence classes of irreducible representations of K of class 1 with respect to H M −1 , and let the restriction of the representation U q ∈K H M −1 to H M −1 contains d q copies of the trivial representation of H M −1 . By the Fine Structure Theorem [20], there are some numbers d q ≤ d q such that the set be the corresponding Fourier coefficients. The uniformly convergent Fourier expansion takes the form This expansion is defined on the dense set and may be extended to all ofR 3 × R 3 by continuity. Substituting the extended expansion to Equation (14), we obtain the expansion Theorem 0. Let −I ∈ K. The one-point correlation tensor of a homogeneous and (K, U )-isotropic random field lies in the space of the isotypic component of the representation U that corresponds to the trivial representation of K and is equal to 0 if no such isotypic component exists. Its two-point correlation tensor is given by Equation (17).
The results by [23,24,33,38,39] as well as Theorems 1-16 below are particular cases of Theorem 0. The expansion (17) is the first necessary step in studying random fields connected to Hooke's law.
Later we will see that it is easy to write the spectral expansion of the field directly if the group K is finite. Otherwise, we write the Fourier expansion (16) for plane waves e i(p,y) and e −i(p,x) separately and substitute both expansions to Equation (14). As a result, we obtain the expansion of the two-point correlation tensor of the field in the form where Λ is a set, and where F is a measure on a σ-field L of subsets of Λ taking values in the set of Hermitian nonnegative-definite operators on V C . Moreover, the set { u(x, λ) : x ∈ R 3 } is total in the Hilbert space L 2 (Λ, Φ) of the measurable complex-valued functions on Λ that are square-integrable with respect to the measure Φ, that is, the set of finite linear combinations c n u(x n , λ) is dense in the above space. By Karhunen's theorem [21], the field C(x) has the following spectral expansion: where Z is a measure on the measurable space (Λ, L) taking values in the Hilbert space of random tensors Z : The measure F is the control measure of the measure Z, i.e., The components of the random tensor Z(A) are correlated, which creates difficulties when one tries to use Equation (18) for computer simulation. It is possible to use Cholesky decomposition and to write the expansion of the field using uncorrelated random measures, see details in [24].

Preliminary calculations
The possibilities for the group K are as follows. In the triclinic class, there exist infinitely many groups between Z c 2 and O(3), we put K 1 = Z c 2 and K 2 = O(3). Similarly, for the monoclinic class put K 3 = Z 2 × Z c 2 and The possibilities for the orthotropic class are Here T is the tetrahedral group which fixes a tetrahedron. In the trigonal class, we have

The structure of the representation U
The notation for irreducible orthogonal representation is as follows. If K i is a finite group, we use the Mulliken notation [26], see also [1,Chapter 14] to denote the irreducible unitary representation of K i . For an irreducible orthogonal representation, consider its complexification. A standard result of representation theory, see, for example, [6,Proposition 4.8.4], states that there are three possibilities: • The complexification is irreducible, say U . Then, it is a sum of two equivalent orthogonal representations, and we denote each of them by U .
• The complexification is a direct sum of two mutually conjugate representation U 1 and U 2 , that is, U 2 (g) = U 1 (g). We denote the orthogonal representation by U 1 ⊕ U 2 .
• The complexification is a direct sum of two copies of an irreducible representation U . We denote the orthogonal representation by U ⊕ U .
For infinite groups, the notation is as follows. For K 2 = O(3), we denote the representations by U g (the tensor product of the representation U of the group SO(3) and the trivial representation A g of Z c 2 ) and U u (that of U and the nontrivial representation Fist, we determine the structure of the representation g → g of the group K i . For finite groups, the above structure is given in Table n.10 in [1], where n in the number given in the second column of Table 2. For K 2 and K 16 , this representation is U 1u , for K 4 and K 14 it is U 1u ⊕ U 0uu . Then we determine the structure of the representations S 2 (g) and S 2 (S 2 (g)). For finite groups, we use Table n.8. For infinite groups, we use the following multiplication rules. The product of two isomorphic irreducible representations of Z c 2 is A g , that of two different representations is A u . For SO(3), we have If K i = H, then the space V is spanned by the spaces of the copies of all trivial representations of K i that belong to S 2 (S 2 (g)). This gives us a method for calculation of the dimension dim V alternative to that in [2]. Otherwise, it is spanned by the spaces of all irreducible representations of K i that contain at least one copy of the trivial representation of H. To determine such representations, we use Table n.9.
We call them the Godunov-Gordienko coefficients. Malyarenko and Ostoja-Starzewski [24] calculated the tensors of the basis of the 21-dimensional space S 2 (S 2 (R 3 )) for the group K 2 in terms of the above coefficients. Using MAT-LAB Symbolic Math Toolbox, we calculate the elements of the bases for the groups K 1 , K 3 -K 16 as linear combinations of the tensors of the basis for the group K 2 , see Table 3.
Continued at next page Continued at next page Continued at next page The isotropy subgroups for the groups K i Table 4 shows the isotropy subgroups of the groups K i . In this table, Z − 2 is the order 2 group generated by the reflection through the yz-plane, andZ 2 is the group generated by a reflection leavind an edge of a cube invariant [11]. The group H 0 is always equal to K i and therefore is omitted.

The orbit type stratification
The following formulae describe the orbit type stratification of the orbit spacê R 3 /K i . The zeroth stratum is always equal to {0} and therefore is omitted.

The results
In Theorem m below we denote by Km T ijkl the tensors of the basis given in Table 3 in the lines marked by K m , 1 ≤ m ≤ 16. We say "a triclinic (orthotropic, etc) random field" instead of more rigourous "a random field with triclinic (orthotropic, etc) symmetry".

The triclinic class
Theorem 1 (A triclinic random field in the triclinic class). The one-point correlation tensor of a homogeneous and (Z c where C m ∈ R. Its two-point correlation tensor has the form where f (p) is a Φ-equivalence class of measurable functions acting from R 3 /Z c 2 to the set of nonnegative-definite symmetric linear operators on V Z c 2 with unit trace, and Φ is a finite measure onR 3 /Z c 2 . The field has the form To formulate the next theorem, we need to introduce some notation. Let f (λ), λ ≥ 0 be a measurable function on [0, ∞) taking values in the set of real symmetric nonnegative-definite matrices of size 21 × 21 with unit trace. Assume that and 18,19 (λ), Assume also that all the entries of the matrix f (λ) that lie over its main diagonal and were not mentioned previously, are equal to 0. Put and The set C of the possible values of the function f (λ) is a convex compact. The set of extreme points of C consists of three connected components. The are coordinates in the closed convex hull of the first (resp. second, resp. third) component. The possible values for coordinates are determined by the following conditions: the principal minors of the matrix f (λ) are nonnegative.
Let < be the lexicographic order on the sequences tuijkl, where ijkl are indices that numerate the 21 component of the elasticity tensor, t ≥ 0, and −t ≤ u ≤ t. Consider the infinite symmetric positive definite matrices given with 1 ≤ m ≤ 13. Let L(m) be the infinite lower triangular matrices of the Cholesky factorisation of the matrices b t u i j k l tuijkl (m) constructed in [18]. Let Z mtuijkl be the sequence of centred scattered random measures with Φ m as their control measures. Define Theorem 2 (An isotropic random field in the triclinic class). The one-point correlation tensor of a homogeneous and where C 1 , C 2 ∈ R. Its two-point correlation tensor has the spectral expansion where the functions N nq (λ, ρ) are given in Table 5 and the functions L q iikli j k l are given in Table 6. The measures Φ n (λ) satisfy the condition The spectral expansion of the field has the form where S u t (θ, ϕ) are real-valued spherical harmonics.
Continued at next page Continued at next page Continued at next page Continued at next page                  20 33 v 26 (λ)]j 6 (λρ) Continued at next page    Let ν be a nonnegative integer. The Ogden tensor [27] I ν of rank 2ν + 2 is determined inductively as where there is a summation over p. In what follows we will omit the upper index.
i···l 4(I iji j I klk l + I ijk l I kli j ) L 7 i···l 4(I iji k I klj l + I iji l I klj k + I ijj k I kli l + I ijj l I kli k ) Continued at next page

The monoclinic class
Theorem 3 (A monoclinic random field in the monoclinic class). The onepoint correlation tensor of a homogeneous and ( where C m ∈ R. Its two-point correlation tensor has the form where f (p) is a Φ-equivalence class of measurable functions acting from R 3 /Z 2 × Z c 2 to the set of nonnegative-definite symmetric linear operators on V Z 2 ×Z c 2 with unit trace, and Φ is a finite Radon measure onR 3 /Z 2 × Z c 2 . The field has the form where (Z n 1 (p), . . . , Z n 13 (p)) are four centred uncorrelated V Z 2 ×Z c 2 -valued random measures onR 3 /Z 2 × Z c 2 with control measure f (p) dΦ(p).
Theorem 4 (A transverse isotropic random field in the monoclinic class).
The one-point correlation tensor of a homogeneous and where C m ∈ R. Its two-point correlation tensor has the form where Φ is a measure onR 3 /O(2) × Z c 2 , and f (p) is a Φ-equivalence class of measurable functions onR 3 /O(2) × Z c 2 with values in the compact set of all nonnegative-definite linear operators in the space V Z 2 ×Z c 2 with unit trace of the form  where A is a nonnegative-definite 5 × 5 matrix, and B m , 1 ≤ m ≤ 7 are 2 × 2 matrices proportional to the identity matrix. The field has the form where (Z i 1 (p), . . . , Z i 13 (p)) are centred uncorrelated V Z 2 ×Z c 2 -valued random measures onR 3 /O(2)×Z c 2 with control measure f (p) dΦ(p), J are the Bessel functions, and

The orthotropic class
Theorem 5 (An orthotropic random field in the orthotropic class). The one-point correlation tensor of a homogeneous and (D 2 × Z c 2 , 9A g )-isotropic where C m ∈ R. Its two-point correlation tensor has the form where f (p) is a Φ-equivalence class of measurable functions acting from R 3 /D 2 × Z c 2 to the set of nonnegative-definite symmetric linear operators on V D 2 ×Z c 2 with unit trace, and Φ is a finite measure onR 3 /D 2 × Z c 2 . The field has the form where (Z n 1 (p), . . . , Z n 9 (p)) are eight centred uncorrelated V D 2 ×Z c 2 -valued random measures onR 3 /D 2 × Z c 2 with control measure f (p) dΦ(p), and where u n (p, x) are eight different product of sines and cosines of p r x r .
Consider a 9 × 9 symmetric nonnegative-definite matrix with the unit trace of the following structure: where A is a 6 × 6 matrix. Introduce the following notation: Let Φ be a finite measure onR 3 /D 4 × Z c 2 . Let f 0 (p) be a Φ-equivalence class of measurable functions acting from (R 3 /D 4 × Z c 2 ) m , 0 ≤ m ≤ 1 to the set of nonnegative-definite symmetric matrices with unit trace satisfying B = 0. Let f + (p) be a Φ-equivalence class of measurable functions acting from (R 3 /D 4 × Z c 2 ) m , 2 ≤ m ≤ 4 to the set of nonnegative-definite symmetric linear operators on V D 2 ×Z c 2 with unit trace, and let f − (p) is obtained from f + (p) by multiplying B and B by −1.
Theorem 6 (A tetragonal random field in the orthotropic class). The onepoint correlation tensor of a homogeneous and (D 4 ×Z c 2 , 6A 1g ⊕3B 1g )-isotropic random field C(x) is where C m ∈ R. Its two-point correlation tensor has the form (27) The field has the form where (Z n0 1 (p), . . . , Z n0 9 (p)) (resp. (Z n+ 1 (p), . . . , Z n+ 9 (p)) , resp. (Z n− 1 (p), . . . , Z n− 9 (p)) ) are centred uncorrelated V D 2 ×Z c 2 -valued random measures on the spaces ( with control measure f 0 (p) dΦ(p) (resp. f + (p) dΦ(p), resp. f − (p) dΦ(p)), u n (p, x) are different product of sines and cosines of p r x r for 1 ≤ n ≤ 8 and eight different product of sines and cosines of p 1 x 2 , p 2 x 1 , and p 3 x 3 for 9 ≤ n ≤ 16, and Consider a 9 × 9 symmetric nonnegative-definite matrix with unit trace of the following structure where stars are arbitrary numbers, c i are vectors with two components, and A i are 2 × 2 matrices of the form Let Φ be a finite measure onR 3 /D 6 × Z c 2 . Let f 0 (p) be a Φ-equivalence class of measurable functions acting from (R 3 /D 6 × Z c 2 ) m , 0 ≤ m ≤ 1 to the set of nonnegative-definite symmetric matrices with unit trace such that c i = 0 and A i are proportional to the identity matrix. Let f − (p) be a Φequivalence class of measurable functions acting from (R 3 /D 6 × Z c 2 ) 2 to the set of nonnegative-definite symmetric matrices with unit trace such that A i are symmetric. Let f + (p) be a Φ-equivalence class of measurable functions acting from (R 3 /D 6 × Z c 2 ) m , 3 ≤ m ≤ 4 to the set of nonnegative-definite symmetric matrices with unit trace. Consider matrices and functions of Table 7.
Let f −i (p) is obtained from f − (p) by replacing all c j with g i c j and the vectors (b, c) in all A j with g i (b, c) . Let f +i (p) is obtained from f + (p) by replacing all c j with g i c j and all A j with g i A j g −1 i .
Theorem 7 (A hexagonal random field in the orthotropic class). The onepoint correlation tensor of a homogeneous and ( Table 7: The matrices g n and the functions j n (p, z) for the group D 6 × Z c 2 n g n j n (p, z) where C m ∈ R. Its two-point correlation tensor has the form The field has the form where (Z 0n 1 (p), . . . , Z 0n 9 (p)) (resp. (Z −ns 1 (p), . . . , Z −ns 9 (p)) , resp. (Z +ns 1 (p), . . . , Z +ns with control measure f 0 (p) dΦ(p) (resp. f −s (p) dΦ(p), resp. f +s (p) dΦ(p)), u n (p, x), 1 ≤ n ≤ 8 are different product of sines and cosines of angles in Table 7, and where Consider a 9 × 9 symmetric nonnegative-definite matrix with unit trace of the following structure where stars are arbitrary numbers, c i are vectors with two components, and A i are 2 × 2 matrices. Let Φ be a finite measure onR 3 /T × Z c 2 . Let f 0 (p) be a Φ-equivalence class of measurable functions acting from (R 3 /T × Z c 2 ) m , 0 ≤ m ≤ 1 to the set of nonnegative-definite symmetric linear operators on V D 2 ×Z c 2 with unit trace such that c i = 0 and A i are proportional to the identity matrix. Let f 1 (p) be a Φ-equivalence class of measurable functions acting from (R 3 /T × Z c 2 ) m , 2 ≤ m ≤ 4 to the set of nonnegative-definite symmetric linear operators on V D 2 ×Z c 2 with unit trace. Denote Let f + (p) (resp. f − (p)) is obtained from f 1 (p) by replacing all c i with gc 1 (resp. with g −1 c i ) and all A i with gA i g −1 (resp. g −1 A i g). Finally, let j m (p, z) be functions from Table 8.
Consider a 9 × 9 symmetric nonnegative-definite matrix with unit trace of the following structure where stars are arbitrary numbers, c i are vectors with two components, and A i are 2 × 2 matrices of the form (28). Let Φ be a finite measure to the set of nonnegative-definite symmetric matrices with unit trace such that c i = 0 and A i are proportional to the identity matrix. Let f − (p) be a Φ-equivalence class of measurable functions acting from (R 3 /O × Z c 2 ) 2 to the set of nonnegative-definite symmetric matrices with unit trace such that A i are symmetric. Let f + (p) be a Φ-equivalence class of measurable functions acting from (R 3 /O × Z c 2 ) m , 3 ≤ m ≤ 6 to the set of nonnegative-definite symmetric matrices with unit trace. Consider matrices and functions of Table 9.
Let f −i (p) is obtained from f − (p) by replacing all c j with g i c i and the vectors (b, c) in all A j with g i (b, c) . Let f +i (p) is obtained from f + (p) by replacing all c j with g i c i and all A j with g i A j g −1 i .

The trigonal class
Introduce the following notation: Theorem 10 (A trigonal random field in the trigonal class). The one-point correlation tensor of a homogeneous and (D 3 × Z c 2 , 6A 1g )-isotropic random field C(x) is where C m ∈ R. Its two-point correlation tensor has the form where f (p) is the Φ-equivalence class of measurable functions acting from R 3 /D 3 × Z c 2 to the set of nonnegative-definite symmetric linear operators on V D 3 ×Z c 2 with unit trace, and Φ is a finite measure onR 3 /D 3 × Z c 2 . The field has the form with control measure f (p) dΦ(p), and where u n (p, x), 1 ≤ n ≤ 4 are four different products of sines and cosines of p 1 x 1 + p 3 x 3 and p 2 x 2 , u n (p, x), 5 ≤ n ≤ 8 are four different product of sines and cosines of 1 , 9 ≤ n ≤ 12 are four different product of sines and cosines of 1 Consider a 6 × 6 symmetric nonnegative-definite matrix with unit trace of the following structure where stars and c i are arbitrary numbers. Let Φ be a finite measure on R 3 /D 6 × Z c 2 . Let f 0 (p) be a Φ-equivalence class of measurable functions acting from (R 3 /D 6 × Z c 2 ) m , 0 ≤ m ≤ 2 to the set of nonnegative-definite symmetric matrices with unit trace such that c i = 0. Let f + (p) be a Φequivalence class of measurable functions acting from (R 3 /D 6 × Z c 2 ) m , 3 ≤ m ≤ 4 to the set of nonnegative-definite symmetric matrices with unit trace, and let f − (p) be a Φ-equivalence class of measurable functions acting from to the set of nonnegative-definite symmetric matrices with unit trace such that all c i s are multiplied by −1.

The tetragonal class
Theorem 12 (A tetragonal random field in the tetragonal class). The onepoint correlation tensor of a homogeneous and (D 4 × Z c 2 , 6A 1g )-isotropic random field C(x) is where C m ∈ R. Its two-point correlation tensor has the form where f (p) is a Φ-equivalence class of measurable functions acting from R 3 /D 4 × Z c 2 to the set of nonnegative-definite symmetric linear operators on V D 4 ×Z c 2 with unit trace, and Φ is a finite measure onR 3 /D 4 × Z c 2 . The field has the form 2 with control measure f (p) dΦ(p), and where u n (p, x) are eight different product of sines and cosines of p r x r for 1 ≤ n ≤ 8 and eight different product of sines and cosines of p 1 x 2 , p 2 x 1 , and p 3 x 3 for 9 ≤ n ≤ 16.
Consider a 6 × 6 symmetric nonnegative-definite matrix with unit trace of the structure (30). Let Φ be a finite measure onR 3 /D 8 × Z c 2 . Let f 0 (p) be a Φ-equivalence class of measurable functions acting from (R 3 /D 8 × Z c 2 ) m , 0 ≤ m ≤ 1 to the set of nonnegative-definite symmetric matrices with unit trace such that c i = 0. Let f + (p) be a Φ-equivalence class of measurable functions acting from (R 3 /D 8 × Z c 2 ) m , 2 ≤ m ≤ 4 to the set of nonnegativedefinite symmetric matrices with unit trace, and let f − (p) be a Φ-equivalence class of measurable functions acting from (R 3 /D 8 × Z c 2 ) m , 2 ≤ m ≤ 4 to the set of nonnegative-definite symmetric matrices with unit trace such that all c i s are multiplied by −1.
Introduce the following notation.

The transverse isotropic class
Theorem 14 (A transverse isotropic random field in the transverse isotropic class). The one-point correlation tensor of the homogeneous and (O(2) × Z c 2 , 5U 0gg )-isotropic mean-square continuous random field C(x) has the form where C m ∈ R. Its two-point correlation tensor has the form where Φ is a measure onR 3 /O(2) × Z c 2 , and f (p) is a Φ-equivalence class of measurable functions onR 3 /O(2) × Z c 2 with values in the compact set of all nonnegative-definite linear operators in the space V O(2)×Z c 2 with unit trace. The field has the form where (Z i1 (p), . . . , Z i5 (p)) are centred uncorrelated V O(2)×Z c 2 -valued random measures onR 3 /O(2) × Z c 2 with control measure f (p) dΦ(p), and where J are the Bessel functions.

The cubic class
Theorem 15 (A cubic random field in the cubic class). The one-point correlation tensor of the homogeneous and (O × Z c 2 , 3A 1g )-isotropic mean-square continuous random field C(x) has the form where C m ∈ R. Its two-point correlation tensor has the form where the functions j m (z, p) are shown in Table 9, Φ is a measure onR 3 /O× Z c 2 , and f (p) is a Φ-equivalence class of measurable functions onR 3 with values in the compact set of all nonnegative-definite linear operators in the space V O×Z c 2 with unit trace. The field has the form where (Z 1n (p), . . . , Z 3n (p)) are 48 centred uncorrelated V O×Z c 2 -valued random measures onR 3 /O × Z c 2 with control measure f (p) dµ(p), and where u n (x, p) are different products of sines and cosines of angles from Table 9.

The isotropic class
Theorem 16 (An isotropic random field in the isotropic class). The onepoint correlation tensor of the homogeneous and (O(3), 2U 0g )-isotropic meansquare continuous random field C(x) has the form Its two-point correlation tensor has the form where Φ(λ) is a finite measure on [0, ∞), is the set of mutually uncorrelated V O(3) -valued random measures with f (λ) dΦ(λ) as their common control measure.

A sketch of proofs of Theorems 1-16
The first display formulae in Theorems 1-16 follow directly from Theorem 0. Now we need to prove that (17) is equivalent to the second display formulae in each theorem. The easiest cases arise when K = H, i.e., in Theorems 1, 3, 5, 10, 12, 14-16. Then the representation U is the direct sum of the dim V copies of the trivial representation of the group K, the matrix f (p) is nonnegative-definite with unit trace and no further restrictions appear. In Theorems 14 and 16, the group K is infinite, and the integral in (15) is calculated directly. Otherwise, the group K is discrete. The sets (R 3 /K) M −1 ⊂ R 3 and (R 3 /K) M −1 ⊂R 3 have nonempty interior. The coordinate ρ M −1 ∈ (R 3 /K) M −1 may be identified with the coordinate x ∈ R 3 , similarly for λ M −1 ∈ (R 3 /K) M −1 and p ∈R 3 . The representation U is trivial. Equation (15) takes the form where |G| is the number of elements in G. The matrix entries g ij of the matrix g ∈ K in the Wigner basis may be found in [1,Table N.7]. To calculate the entries g ij in the Gordienko basis, use the following result obtained in [10]: where u ik are the matrix entries of the unitary matrix In Theorem 2 we proceed as follows. By Theorem 0, The basis of the 21-dimensional space V is formed by the basis tensors of the group K 2 shown in Table 7. We are interested in the tensors of the uncoupled basis of the space S 2 (V) that lie in the spaces of the irreducible components U 2t,g . They are shown in Table 10. Table 10: The tensors of the uncoupled basis of the space S 2 (V) that lie in the spaces of the irreducible components U 2t,g .

Function Value
To prove the last part of each theorem, we first observe that any homogeneous random field C(x) may be written as The first term in the right hand side is the same as that in the spectral expansions in Theorems 1-16. The second term is centred and has the same two-point correlation tensor as C(x) has. Assume that the above tensor has the form C(x), C(y) = Λ u(x, λ)u(y, λ) dF (λ), where Λ is a set, and where F is a measure on a σ-field L of subsets of Λ taking values in the set of Hermitian nonnegative-definite operators on V C . Let Φ be the following measure: Assume that the set { u(x, λ) : x ∈ R 3 } is total in the Hilbert space L 2 (Λ, Φ) of the measurable complex-valued functions on Λ that are square-integrable with respect to the measure Φ, that is, the set of finite linear combinations c n u(x n , λ) is dense in the above space. By Karhunen's theorem [21], the field C(x) has the following spectral expansion: where Z is a measure on the measurable space (Λ, L) taking values in the Hilbert space of random tensors Z : Ω → V C with E[Z] = 0 and E[ Z 2 ] < ∞. The measure F is the control measure of the measure Z, i.e., where J is the real structure in the space V C : J(v + iw) = v − iw.
We illustrate the use of Kahrunen's theorem in Theorem 1. The two-point correlation tensor of the random field C(x) has the form (35), where Λ is the union Z c 2 \Ê 1 ∪ Z c 2 \Ê 2 of two copies of the space Z c 2 \Ê and u(x, p) = cos(p, x), if p ∈ Z c 2 \Ê 1 , sin(p, x), if p ∈ Z c 2 \Ê 2 .
This follows from the elementary formula cos(p, y−x) = cos(p, x) cos(p, y)+ sin(p, x) sin(p, y). Similar considerations are applicable in Theorems 3, 5-13, and 15, where the group K is discrete. In Theorems 2 and 16, where K = O(3), we represent the plane wave e i(p,y−x) in the form u(x, λ)u(y, λ), using the real version of the Rayleigh expansion (33). We see that Λ is the union of countably many copies of the half-line [0, ∞) enumerated by the pairs of integers ( , m) with ≥ 0 and − ≤ m ≤ , and the function u(y, λ) has the form u(y, λ m ) = 2 √ πi j (λ m ρ)S m (θ y , ϕ y ).
Note that the random fields C(x) and e iϕ C(x) have the same two-point correlation tensor. Using this freedom, we can always force the random measure Z to become V-valued rather than V C -valued.

Conclusions
Hooke's law describes the physical phenomenon of elasticity and belongs to the family of linear constitutive laws, see [28]. A physical quantity is a tensor of rank p over V , that is, an element of the space V ⊗p . Usually, physical quantities have symmetries. To describe symmetries mathematically, consider a subgroup Σ of the symmetric group Σ p on p symbols. Let τ be linear operator acting from V p to V ⊗p by τ (x 1 , . . . , x p ) = x 1 ⊗ · · · ⊗ x p .
This action can be extended by linearity to V ⊗p . Define the linear operator P Σ : V ⊗p → V ⊗p by where |Σ| is the number of elements in Σ. The range of the operator P Σ is called the state tensor space. A linear constitutive law C is a linear map between two state tensor spaces, say V 1 and V 2 . It may be identified with an element of the tensor product V 1 ⊗ V 2 , because the state tensor spaces inherit the Euclidean metric from V . A linear constitutive law C describes proper physics or a single physical phenomenon if V 1 = V 2 and C is symmetric. Otherwise, C describes coupled physics, or a coupling between two different physics.
For example, Hooke's law corresponds to the case when V 1 = V 2 = P Σ 2 V ⊗2 and C is symmetric. It describes the single physical phenomenon, elasticity. On the other hand, the photoelasticity tensor is a general linear map C : P Σ 2 V ⊗2 → P Σ 2 V ⊗2 . It couples two different physics and maps the space of strain tensors to the space of the increments of dielectric tensors, see [8]. The piezoelectricity tensor maps the space P Σ 2 V ⊗2 of strain tensors to the space V of electric displacement vectors and couples two different physics, see [9].
In general, a linear constitutive law is an element of a subspace of the tensor product V ⊗(p+q) , where p (resp. q) is the rank of tensors in the first (resp. second) state tensor space. Denote by U the restriction of the representation g → g ⊗(p+q) to the above subspace. Consider U as a group action. The orbit types of this action are called the classes of the phenomenon under consideration (e.g., photoelasticity classes, piezoelectricity classes and so on). All symmetry classes of all possible linear constitutive laws were described in [28,29].
For each class, one can consider its fixed point set V H ⊂ V ⊗(p+q) , a group K with H ⊆ K ⊆ N (H), and the restriction U of the representation g → g ⊗(p+q) of the group K to V H . Calculating the general form of the onepoint and two-point correlation tensors of the corresponding homogeneous and (K, U )-isotropic random field and the spectral expansion of the field in terms of stochastic integrals with respect to orthogonal scattered random measures is an interesting research question.
The part of the above question concerning the one-point correlation tensor is almost trivial: it is any tensor lying in the direct sum of all one-dimensional • for a given microstructure, determine the one-and two-point statistics using some experimental and/or image-based computational methods; • calibrate the entire correlation structure of the elasticity TRF; • simulate the realisations of this TRF.
The second application of our results is their use as input of a random mesoscale continuum (Fig. 1(c)) into stochastic field equations such as SPDEs and SFEs.