The spectral expansion of the elasticity

The spectral expansion of the elasticity random fieldAnatoliy Malyarenko and Martin Ostoja-Starzewski Citation: AIP Conference Proceedings 1637, 647 2014; doi: 10.1063/1.4904635 View onl

The spectral expansion of the elasticity random field

Anatoliy Malyarenko and Martin Ostoja-Starzewski

Citation: AIP Conference Proceedings 1637, 647 (2014); doi: 10.1063/1.4904635

View online: http://dx.doi.org/10.1063/1.4904635

View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1637?ver=pdfcov

Published by the AIP Publishing

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The spectral expansion of the elasticity random field

∗Mälardalen University, Sweden

†University of Illinois at Urbana-Champaign, USA

Abstract We consider a deformable body that occupies a region D in the plane In our model, the body’s elasticity tensor

H(x) is the restriction to D of a second-order mean-square continuous random field Under translation, the expected value and

the correlation tensor of the field H(x) do not change Under action of an arbitrary element k of the orthogonal group O(2),

expansion of the correlation tensor R(x) of the elasticity field as well as the expansion of the field itself in terms of stochastic

integrals with respect to a family of orthogonal scattered random measures

Keywords: elasticity, random field, spectral expansion

PACS: 02.50.Ey, 46.25.Cc

1 INTRODUCTION Tensor random fields are a necessity when one is interested in studying field problems in random media Overall,

while the literature on scalar random fields is vast (e.g Christakos [1], Cressie [2], Porcu et al [3]), that on the vector

random fields is very limited, and the case of higher tensor ranks (second, fourth) is, effectively, at an early stage of

development We make a distinction between random fields governed by some field equations and those representing

some spatially inhomogeneous constitutive properties In the first case, the correlation function (itself a tensor of twice

higher rank when a two-point information is involved) is subject to constraints such as the equilibrium equation or

strain-displacement relation (Ostoja-Starzewski et al [4]) In the second case, the correlation function of, say, the

elasticity tensor must be positive-definite In this paper, we consider the second case

Let d be a positive integer, and let E = Ed be a Euclidean space of dimension d with an inner product (·, ·) (the

space domain) Hooke’s law in the theory of elasticity says that σ (x) = H(x)ε(x), where σ (x) is the stress tensor of

a deformable body, ε(x) be its strain tensor, and where H(x) is a symmetric linear operator on the space S2(E) of the

symmetric rank 2 tensors over E called the elasticity (or stiffness) tensor In what follows we consider the case of d = 2

(plane elasticity problems)

Assume that H(x) is a second-order mean-square continuous random field taking values in the space S2(S2(E))

This means the following

• Random field: there is a probability space (Ω,F ,P), and H is a function of two variables

H(x, ω): E × Ω → S2(S2(E)) such that for any fixed x0∈ E the function H(x0, ω): Ω → S2(S2(E)) is measurable

• Second-order:

E[kH(x)k2] < ∞, x ∈ E

• Mean-square continuous:

lim kx−x 0 k→0E[kH(x) − H(x0)k2] = 0, x0∈ E

If one shifts the origin of a coordinate system, the tensor H(x) does not change value It follows that the random

field H(x) is wide-sense homogeneous, i.e.,

R(x, y) := E[(H(x) − E(x)) ⊗ (H(y) − E(y))] = R(x − y)

Let O(E) be the group of the orthogonal transformations of the space E Apply an arbitrary transformation k ∈ O(E)

to the tensor field H(x) After the transformation k:

• the point k−1x becomes the point x;

• the tensor H(k−1x) is transformed by the rotation into S2(S2(k))H(k−1x), where S2(S2(k)) is the symmetric tensor square of the orthogonal representation k 7→ S2(k) of the group O(E)

It is easy to prove that

E(kx) = S2(S2(k))E(x), (2) R(k(x − y)) = (S2(S2(k)) ⊗ S2(S2(k)))R(x − y) (3) for all k ∈ O(E) Such a field is called wide-sense isotropic In what follows we omit the words “wide-sense”

We would like to find the general form of the expected value and correlation tensor of a homogeneous and isotropic

random field H(x) as well as the spectral expansion of the field itself

To formulate the answer, we need to introduce some notation in Section 2 We present our results in Section 3 A

sketch of proofs is given in Section 4 We conclude in Section 5

2 NOTATION The group O(E) consists of two connected components In an orthonormal basis {e−1, e1} of the space E, the

connected component of the identity element consists of the elements

kϕ=

 cos ϕ sin ϕ

− sin ϕ cos ϕ

 , 0 ≤ ϕ < 2π, (4) while the second component consists of the elements

kϕ=



− cos ϕ − sin ϕ

− sin ϕ cos ϕ

 , 2π ≤ ϕ < 4π (5)

Note that the map k 7→ kϕis an isomorphism of the group O(E) onto the group O(2) of the 2 × 2 orthogonal matrices

The map kϕ7→ kϕ is an irreducible orthogonal representation of the group O(2), which will be denoted by U1 The

complete list of representatives of the equivalence classes of the irreducible orthogonal representations of the group

O(2) is as follows: U0,+(kϕ) = 1 (the trivial representation), U0,−(kϕ) = det kϕ, and Un, n ≥ 1, which maps the matrix

(4) to the matrix

Un(kϕ) =

 cos(nϕ) sin(nϕ)

− sin(nϕ) cos(nϕ)



and the matrix (5) to the matrix

Un(kϕ) =



− cos(nϕ) − sin(nϕ)

− sin(nϕ) cos(nϕ)



In particular, we have

U1(k0) =



1 0

0 1

 , U1(k2π) =



−1 0

0 1



In other words, the one-dimensional subspace of the space E generated by e−1 carries the irreducible orthogonal

representation U0,− of the subgroup O(1) = {k0, k2π} of the group O(2), while that generated by e1 carries the

representation U0,+of O(1), hence the notation Similarly, we denote by {en

−1, en} the orthonormal basis of the space

of the representation Un, in which the matrices of the representation have the form (6)–(7) We recommend Adams [5]

as a standard reference on representation theory of compact topological groups

The following tensor products of the two irreducible orthogonal representations of the group O(2) are irreducible:

U0,+⊗U = U, U0,−⊗U0,−= U0,+, U0,−⊗Un= Un The tensor product Un⊗ Un is equivalent to the direct sum U0,+⊕ U0,−⊕ U2n, while the tensor product Un⊗ Uq,

q6= n, is equivalent to the direct sum U|n−q|⊕Un+q

TABLE 1 The uncoupled basis of the space S (S (S (E))).

Ti···m0+,1,00 1

δi jδ`mδi 0 j 0δ` 0 m 0

Ti···m0+,2,00 1

4[δi jδ`m∑t∈{−1,1}gt[i

0 , j 0 ] 2[1,1]gt[`

0 ,m 0 ] 2[1,1] + δi 0 j 0δ` 0 m 0∑t∈{−1,1}gt[i, j]2[1,1]gt[`,m]2[1,1]]

Ti···m0+,3,00 1

∑t,t 0 ∈{−1,1}gt[i, j]2[1,1]gt[`,m]2[1,1]gt

0 [i 0 , j 0 ] 2[1,1]gt

0 [` 0 ,m 0 ] 2[1,1]

Ti···m0+,4,00 1

4 √

2∑t∈{−1,1}(δi jgt[`,m]2[1,1]+ δ`mgt[i, j]2[1,1])(δi 0 j 0gt[`

0 ,m0] 2[1,1] + δ` 0 m 0gt[i

0 , j0] 2[1,1])

Ti···m0+,5,00 √ 1

2∑t,u,v,u 0 ,v 0 ∈{−1,1}gt[u,v]4[2,2]gt[u4[2,2]0,v0]gu[i, j]2[1,1]gu2[1,1]0[i0, j0]gv[`,m]2[1,1]gv2[1,1]0[`0,m0]

Ti···m2,1,s0 1

4 √

2[δi jδ`m(δi 0 j 0gs[`

0 ,m 0 ] 2[1,1] + δ` 0 m 0gs[i

0 , j 0 ] 2[1,1]) + δi 0 j 0δ` 0 m 0(δi jgs[`,m]2[1,1]+ δ`mgs[i, j]2[1,1])]

Ti···m2,2,s0 1

4[(δi 0 j 0gs[`2[1,1]0,m0]+ δ` 0 m 0gs[i2[1,1]0, j0]) ∑t∈{−1,1}gt[i, j]2[1,1]gt[`,m]2[1,1]+ (δi jgs[`,m]2[1,1]+ δ`mgs[i, j]2[1,1]) ∑t∈{−1,1}gt[i2[1,1]0, j0]gt[`2[1,1]0,m0]]

Ti···m2,3,s0 1

2 √

2[∑t,t 0 ,u 0 ,v 0 ∈{−1,1}gs[t,t2[2,4]0]gt4[2,2]0[u0,v0]gu2[1,1]0[i0, j0]gv2[1,1]0[`0,m0](δi jgt[`,m]2[1,1]+ δ`mgt[i, j]2[1,1]) + ∑t,t 0 ,u,v∈{−1,1}gs[t

0 ,t]

2[4,2]gt[u,v]4[2,2]gu[i, j]2[1,1]gv[`,m]2[1,1](δi 0 j 0gt

0 [` 0 ,m 0 ] 2[1,1] + δ` 0 m 0gt

0 [i 0 , j 0 ] 2[1,1])]

Ti···m4,1,s0 1

2 √

2[δi jδ`m∑u 0 ,v 0 ∈{−1,1}gs[u

0 ,v 0 ] 4[2,2]gu

0 [i 0 , j 0 ] 2[1,1] gv

0 [` 0 ,m 0 ] 2[1,1] + δi 0 j 0δ` 0 m 0∑u,v∈{−1,1}gs[u,v]4[2,2]gu[i, j]2[1,1]gv[`,m]2[1,1]]

Ti···m4,2,s0 1

2[∑t,u 0 ,v 0 ∈{−1,1}gt[i, j]2[1,1]gt[`,m]2[1,1]gs[u4[2,2]0,v0]gu2[1,1]0[i0, j0]gv2[1,1]0[`0,m0]+ ∑t0 ,u,v∈{−1,1}gt2[1,1]0[i0, j0]gt2[1,1]0[`0,m0]gs[u,v]4[2,2]gu[i, j]2[1,1]gv[`,m]2[1,1]]

Ti···m4,3,s0 1

4∑t,t 0 ∈{−1,1}gs[t,t

0 ] 4[2,2](δi jgt[`,m]2[1,1]+ δ`mgt[i, j]2[1,1])(δi 0 j 0gt

0 [` 0 ,m 0 ] 2[1,1] + δ` 0 m 0gt

0 [i 0 , j 0 ] 2[1,1])

Ti···m6,1,s0 1

2 √

2[∑t,t 0 ,u 0 ,v 0 ∈{−1,1}gs[t,t

0 ] 6[2,4]gt

0 [u 0 ,v 0 ] 4[2,2] gu

0 [i 0 , j 0 ] 2[1,1] gv

0 [` 0 ,m 0 ] 2[1,1] (δi jgt[`,m]2[1,1]+ δ`mgt[i, j]2[1,1]) + ∑t 0 ,t,u,v∈{−1,1}gs[t

0 ,t]

6[2,4]gt[u,v]4[2,2]gu[i, j]2[1,1]gv[`,m]2[1,1](δi 0 j 0gt

0 [`0,m0] 2[1,1] + δ` 0 m 0gt

0 [i0, j0] 2[1,1])]

Ti···m8,1,s0 ∑t,t 0 ,u,v,u 0 ,v 0 ∈{−1,1}gs[t,t8[4,4]0]gt[u,v]4[2,2]gt4[2,2]0[u0,v0]gu[i, j]2[1,1]gu2[1,1]0[i0, j0]gv[`,m]2[1,1]gv2[1,1]0[`0,m0]

The rank 2 tensors en

i⊗ eqj, i, j ∈ {−1, 1}, form the orthonormal basis in the space of the representation Un⊗ Uq, which is called the coupled basis If n 6= q, then the basis vectors en+q±1 and e|n−q|±1 form the uncoupled basis of the above

space Otherwise, if n = q, choose a unit vector e0,+0 in the space of the representation U0,+and a unit vector e0,−0 in

that of the representation U0,− The vectors e0,+0 , e0,−0 , and e2n±1form the uncoupled basis

In the coupled basis, the vectors of the uncoupled basis become 2 × 2 matrices We denote them as follows:

en+q±1 = g±1n+q[n,q], e|n−q|±1 = g±1|n−q|[n,q], e0,+0 = g00+[n,n], e0,−0 = g00−[n,n]

The entries of the above matrices, say g±1[i, j]n+q[n,q], i, j ∈ {−1, 1}, are the Clebsch–Gordan coefficients of the group O(2)

For any two positive integers n and q with n 6= q we have

g−1|n−q|[n,q] = g00−[n,n]=



0 1/√

2

−1/√2 0

 , g1|n−q|[n,q]= g00+[n,n]=

 1/√

2 0

0 1/√

2

 ,

g−1n+q[n,q] =



0 1/√

2 1/√

2 0

 , g1n+q[n,q]



−1/√2 0

0 1/√

2



In particular, the matrices g00+[1,1]and g±12[1,1]are symmetric, while the matrix g00−[1,1] is skew-symmetric It follows

that the symmetric tensor square S2(U1) is equivalent to the direct sum U0,+⊕ U2 What is more, the representation

U0,+ acts in the space generated by the identity matrix, while the representation U2acts in the space of symmetric

traceless 2 × 2 matrices

Similarly, one can prove that the representation S2(S2(S2(U1))) is equivalent to the direct sum of m0= 5 copies of

the representation U0,+, m2= 3 copies of the representation U2, m4= 3 copies of the representation U4, m6= 1 copy

of the representation U6, and m8= 1 copy of the representation U8 The uncoupled basis of the space of the above

representation consists of the rank 8 tensors Ti···m0+,q,00 , 1 ≤ q ≤ m0, and Ti···m2n,q,s0 , 1 ≤ n ≤ 4, 1 ≤ q ≤ m2n, s ∈ {−1, 1}

shown in Table 1 We use a shortcut i · · · m0:= i j`mi0j0`0m0

Let f (λ ) be a measurable function acting from [0, ∞) to the set of 6 × 6 symmetric nonnegative-definite matrices

with unit trace as follows:

λ 7→









u1(λ ) u7(λ ) u8(λ ) u10(λ ) 0 0

u7(λ ) u2(λ ) u9(λ ) u11(λ ) 0 0

u8(λ ) u9(λ ) u3(λ ) u12(λ ) 0 0

u10(λ ) u11(λ ) u12(λ ) u4(λ ) 0 0

0 0 0 0 u5(λ ) u13(λ )

0 0 0 0 u13(λ ) u6(λ )









The set of all possible values of the function f (λ ) is a convex compact subsetC of the real 12-dimensional space The

set of extreme points of the setC has two connected components: the topological boundary of the set C4of all 4 × 4

symmetric nonnegative-definite matrices with unit trace and that of the setC2 of all 2 × 2 symmetric

nonnegative-definite matrices with unit trace

Introduce the coordinates v1(λ ), v2(λ ), , v9(λ ) inC4as follows:

vi(λ ) =

( ui(λ )

u1(λ )+···+u4(λ ), 1 ≤ i ≤ 3,

ui+3(λ )

u1(λ )+···+u4(λ ), 4 ≤ i ≤ 9, (9) and the coordinates v10(λ ), v11(λ ) inC2as follows:

v10(λ ) = u5(λ )

u5(λ ) + u6(λ ), v11(λ ) =

u13(λ )

u5(λ ) + u6(λ ). (10) Let N2n,q,t(λ ), 0 ≤ n ≤ 4, 1 ≤ q ≤ m2n, t = 1, 2, be the functions shown in Table 2

To simplify the subsequent formulae, put Ti···m0,q,−10 = 0 and Ti···m0,q,10 = Ti···m0+,q,00 , 1 ≤ q ≤ 5 Let (ρ, ϕr) be the polar

coordinates in the space domain, and let (λ , ϕp) be those in the wavenumber domain Introduce the following notation:

Mi···m2n,q0(ϕr) = Ti···m2n,q,−10 sin(2nϕr) + Ti···m2n,q,10 cos(2nϕr) (11) Let i be the imaginary unit with i2= −1, and let Judenote the Bessel function of the first kind of order u

3 THE RESULTS Theorem 1 The expected value of the elasticity random field H(x) is

Ei j`m(x) = C1δi jδ`m+C2(δi`δjm+ δimδj`), C1,C2∈ R (12) The correlation tensor of the above field has the form

Ri···m 0(ρ, ϕr) =

2

∑

t=1

0

4

∑

n=0

i2nJ2n(λ ρ)

m2n

∑

q=1

N2n,q,t(λ )Mi···m2n,q0(ϕr) dΦt(λ ), (13)

where Φ1and Φ2are two finite measures on[0, ∞) satisfying the condition

Φ1({0}) ≥ 2Φ2({0}) (14) Introduce the following notation:

Iss−+0 n = δs−s 0 ,2n+ δs0 −s,2n− δs+s 0 ,2n− δ−s−s 0 ,2n,

Iss+−0 n = δs−2n,s0+ δ2n−s,s 0− δs+2n,s 0− δ−s−2n,s0,

Iss−−0 n = δs0 −2n,s+ δ2n−s 0 ,s− δs0 +2n,s− δ−s 0 −2n,s,

Is0n++ = δs,2n,

I0s++0 n = δs0 ,2n,

Iss++0 n = δs−s 0 ,2n+ δs0 −s,2n+ δs+s0 ,2n+ δ−s−s 0 ,2n,

TABLE 2 The functions N2n,q,t(λ ).

N0,1,1(λ ) −24

31v1(λ ) −2431v2(λ ) −2831v3(λ ) +1631v4(λ ) −312v5(λ ) +3031v6(λ ) −312v7(λ ) +3031v8(λ ) +314v9(λ ) +3231

N0,2,1(λ ) 7962v1(λ ) +7962v2(λ ) +2831v3(λ ) +1531v4(λ ) +3331v5(λ ) +311v6(λ ) +3331v7(λ ) +311v8(λ ) +5831v9(λ ) −3231

N0,3,1(λ ) −1231v1(λ ) −1231v2(λ ) +4831v3(λ ) +318v4(λ ) +3031v5(λ ) −1631v6(λ ) +3031v7(λ ) −1631v8(λ ) −6031v9(λ ) +1631

N0,4,1(λ ) 1

2 √

2 √

2v4(λ )

N0,5,1(λ ) −6 √

2

√ 2

√ 2

√ 2

√ 2

√ 2

√ 2

√ 2

31 v8(λ )

√ 2

√ 2 31

N2,1,1(λ ) 12v1(λ ) −12v2(λ ) + v6(λ ) − v8(λ )

N2,2,1(λ ) 2√1

2v8(λ )

N2,3,1(λ ) 14v1(λ ) −14v2(λ ) − v5(λ ) −12v6(λ ) + v7(λ ) +12v8(λ )

N4,1,1(λ ) 7962v1(λ ) +7962v2(λ ) +2831v3(λ ) +1531v4(λ ) −2931v5(λ ) +311v6(λ ) −2931v7(λ ) +311v8(λ ) −6631v9(λ ) −3231

N4,2,1(λ ) −12 √

2

√ 2

√ 2

√ 2

√ 2

√ 2

√ 2

√ 2

31 v8(λ )

√ 2

√ 2 31

N4,3,1(λ ) 2√1

2v4(λ )

N6,1,1(λ ) 14v1(λ ) −14v2(λ ) − v5(λ ) −12v6(λ ) + v7(λ ) +12v8(λ )

N8,1,1(λ ) −6

√ 2

√ 2

√ 2

√ 2

√ 2

√ 2

√ 2

√ 2

31 v8(λ )

√ 2

√ 2 31

N0,1,2(λ ) 318v11(λ ) −314

N0,2,2(λ ) 318v11(λ ) +314

N0,3,2(λ ) −318v11(λ ) −312

N0,4,2(λ ) 2√

N0,5,2(λ ) 60

√ 2

√ 2 31

N2,1,2(λ ) 0

N2,2,2(λ ) 0

N2,3,2(λ ) −4v10(λ ) + 2

N4,1,2(λ ) −8

N4,2,2(λ ) 4

√ 2

√ 2 31

N4,3,2(λ ) −2√2v11(λ ) −√2

N6,1,2(λ ) 4v10(λ ) − 2

N8,1,2(λ ) 64

√ 2

√ 2 31

where s > 0, s0> 0 in the last equality Let Zi j`mnqst±be the set of centred scattered random measures on [0, ∞) such that

for any Borel sets A and B we have

E[Zi j`mnqst−(A)Zin00jq0 `0s00mt00−(B)] = is−s0δnn0δqq0δtt0Iss−−0 nTi···m2n,q,10 Φt(A ∩ B), E[Zi j`mnqst−(A)Zin00jq0 `0s00mt00+(B)] = is−s0δnn0δqq0δtt0Iss−+0 nTi···m2n,q,−10 Φt(A ∩ B), (15) E[Zi j`mnqst+(A)Zin00jq0 `0s00mt00−(B)] = is−s0δnn0δqq0δtt0Iss+−0 nTi···m2n,q−10 Φt(A ∩ B),

E[Zi j`mnqst+(A)Zin00jq0 `0s00mt00+(B)] = is−s0δnn0δqq0δtt0Iss++0 nTi···m2n,q,10 Φt(A ∩ B)

The introduced random measures are correlated To produce uncorrelated measures, we proceed as follows Introduce

the lexicographic order ≤ on the set of indices i j`mnqs±, and let Mt be two infinite matrices indexed by the above

ordered set whose elements are equal to the right hand sides of (15) divided by Φt(A ∩ B) Apply to Mt the algorithm

of Cholesky decomposition of infinite positive-definite matrices described by Flinta [6] Let Lt be the infinite lower

triangular matrices satisfying LtL>t = Mt Then we have

Zi j`mnqst±(A) = ∑

(i 0 j 0 ` 0 m 0 n 0 q 0 s 0 ± 0 )≤(i j`mnqs±)

(Lt)ii j`mnqs±0j0`0m0n0q0s0±0Wi j`mnqst±(A), (16)

where Wi j`m is the sequence of uncorrelated scattered random measures with Φtas their control measures, i.e.,

E[Wi j`mnqst±(A)Wi j`mnqst±(B)] = Φt(A ∩ B)

Theorem 2 The elasticity random field H(x) has the spectral expansion

Hi j`m(ρ, ϕr) = C1δi jδ`m+C2(δi`δjm+ δimδj`) +

4

∑

n=0

m2n

∑

q=1

∞

∑

s=0

2

∑

t=1

Z ∞

0

q

N2n,q,t(λ )Js(λ ρ) sin(2sϕr) dZnqst−i j`m (λ )

+

0

q

N2n,q,t(λ )Js(λ ρ) cos(2sϕr) dZi j`mnqst+(λ )

 ,

where the centred scattered random measures Znqst±i j`m are defined by (16)

4 SKETCH OF PROOFS

It follows from (1) and (2) that the tensor E lies in the direct sum of the two one-dimensional subspaces of the space

S2(S2(E)) where two copies of the trivial representation act The basis rank 4 tensors of the above subspaces are:

Ti j`m0,1,1=1

2δi jδ`m, T

0,2,1

i j`m =√1

2(g

−1[i, j]

2[1,1] g−1[`,m]2[1,1] + g1[i, j]

2[1,1]g1[`,m]2[1,1])

It is possible to prove that

Ti j`m0,2,1= 1

2√

2(−δi jδ`m+ δi`δjm+ δimδj`), (17) hence (12) holds true, whereby C1and C2are recognised as two Lamé constants

The idea of proof of equation (13) is simple First, we describe all homogeneous random fields Second, we choose

those of them that are isotropic

Let HCbe a finite-dimensional complex vector space It is known (see Yaglom [7]) that formula

R(x, y) =

Z

ˆ E

establishes a one-to-one correspondence between the set of correlation tensors of mean-square continuous

homoge-neous HC-valued random fields on E and the set of all measures F on the Borel sets of the wavenumber domain ˆE

taking values in the cone of Hermitian nonnegative-definite operators on HC Moreover, let J be a real structure on HC,

i.e., an operator J: HC→ HCsatisfying the following conditions:

J(x + y) = J(x) + J(y), J(αx) = αJ(x), J2(x) = −x for all x, y ∈ HC, α ∈ C The set H of all eigenvectors of J with eigenvalue 1 is a real vector space If the random

field takes values in H, then the measure F satisfies the condition F(−A) = JF(A) for all Borel sets of ˆE, where

−A = { −p: p ∈ A }

Let µ be the trace measure: µ(A) = tr F(A) It is possible to prove that the measure F is absolutely continuous

with respect to µ, and the corresponding density, say f (p), is a measurable function on ˆEtaking values in the set of

Hermitian nonnegative-definite operators with unit trace on HC Equation (18) takes the form

R(x, y) =

Z

ˆ E

ei(p,x−y)f(p) dµ(p)

Using (3), it is possible to prove that the field is isotropic if and only if

µ (kA) = µ (A), f(kp) = (S2(S2(k)) ⊗ S2(S2(k))) f (p), k∈ O(E) (19) Moreover, if the field takes values in H, then

f(−p) = J f (p), p ∈ ˆE (20)

TABLE 3 The functions ui(λ ).

2 √

8 √

2 √

4 √

2f8,1(λ )

4 √

2 √

8 √

2f8,1(λ )

2f8,1(λ )

8 √

4 √

8 √

2f8,1(λ )

2f8,1(λ )

2f8,1(λ )

2 √

8 √

4 √

2 √

2f4,3(λ )

8 √

2f8,1(λ )

8 √

4 √

8 √

2f8,1(λ )

2f8,1(λ )

2f8,1(λ )

8 √

4 √

4 √

8 √

2f8,1(λ )

2f8,1(λ )

8 √

8 √

8 √

8 √

2f8,1(λ )

In polar coordinates (λ , ϕp) the measure µ takes the form

dµ(λ , ϕp) = 1

2πdϕpdν(λ ), where ν is a finite measure on [0, ∞) The correlation tensor takes the form

R(x, y) = 1

2π

0

0

ei(p,x−y)f(λ , ϕp) dϕpdν(λ ) (21) Put f (λ ) := f (λ , 0) Using the second equation in (19) and (20), it is possible to prove the following The linear

form f (0): S2(S2(E)) ⊗ S2(S2(E)) → C may have nonzero values only on the basis tensors Ti···m0,q,10, 1 ≤ q ≤ 5 The

linear form f (λ ), λ > 0, takes nonzero values only on the basis tensors Ti···m2n,q,10 , 0 ≤ n ≤ 4, 1 ≤ q ≤ m2n Denote

fn,q(λ ) := f (λ )(Ti···m2n,q,10 ) By linearity, the tensor entry fi···m 0(λ ) is

fi···m0(λ ) =

4

∑

n=0

m 2n

∑

q=1

Ti···m2n,q,10 fn,q(λ ) (22)

By (11) and the second equation in (19) we have

fi···m0(λ , ϕp) =

4

∑

n=0

m 2n

∑

q=1

Mi···m2n,q0(ϕp) fn,q(λ ) (23)

Enumerate the indices i j`m in the following order: −1 − 1 − 1 − 1, 1111, −11 − 11, −1 − 111, 11 − 11, −1 − 1 − 11

Using (22), calculate the entries ui(λ ) of the matrix (8) as linear combinations of the functions fn,q(λ ) The results of

calculations are shown in Table 3 They were obtained using MATLAB® Symbolic Math Toolbox

Using Table 3, it is easy to prove that the matrix f (0) is nonnegative-definite with unit trace if and only if f0,q(0)

are nonnegative real numbers with

3

4f

0,1 (0) +1

4f

0,2 (0) +1

2f

0,3 (0) + 5

4√

2f

0,4 (0) + 3

4√

2f 0,5 (0) = 1

Introduce the measures Φ1and Φ2by

dΦ1(λ ) = (u1(λ ) + · · · + u4(λ )) dν(λ ), dΦ2(λ ) = (u5(λ ) + u6(λ )) dν(λ )

We have Φ1({0}) = (u1(0) + · · · + u4(0))ν({0}) and Φ2({0}) = (u5(0) + u6(0))ν({0}) On the other hand,

u1(0) + · · · + u4(0) = 3

4f

0,1(0) +1

4f

0,2(0) +1

2f

0,3(0) +√1

2f

0,4(0) + 1

2√

2f 0,5(0),

u5(0) + u6(0) = 1

4√

2( f 0,4(0) + f0,5(0))

Assume that ν({0}) > 0 It is easy to see that the system

3

4f

0,1 (0) +1

4f

0,2 (0) +1

2f

0,3 (0) +√1

2f

0,4 (0) + 1

2√

2f

0,5 (0) = Φ1({0})/ν({0}), 1

4√

2( f

0,4 (0) + f0,5(0)) = Φ2({0})/ν({0}) has a nonnegative solution if and only if (14) holds true

Express the functions fn,q(λ ) in terms of ui(λ ), substitute the result to (23) and apply (9), (10), and (21) We obtain

Ri···m 0(x − y) = 1

2π

2

∑

t=1

4

∑

n=0

m2n

∑

q=1

0

0

ei(p,x−y)N2n,q,t(λ )Mi···m2n,q0(ϕp) dϕpdΦt(λ ) (24)

The Jacobi–Anger formula states that

ei(p,r)= J0(kpkkrk) + 2

∞

∑

s=1

isJs(kpkkrk)(cos(nϕp) cos(nϕr) + sin(nϕp) sin(nϕr))

Substitute this formula to (24) and integrate over the unit circle 0 ≤ ϕp≤ 2π We obtain (13),

To prove Theorem 2, write down the Jacobi–Anger formula twice: the first time with ei(p,x)in the left hand side, the

second time with e−i(p,y) Substitute both formulae to (24) and use theorem by Karhunen [8]

5 CONCLUDING REMARKS The problem considered in the paper can be generalised as follows Let r be a positive integer, let k 7→ k⊗r be the

tensor product of r copies of the orthogonal representation k 7→ k of the group O(Ed), let L be an invariant subspace

of the above representation, and let U be the restriction of the above representation to L Let T(x) be an L-valued

mean-square continuous homogeneous random field on E = Ed, isotropic in the following sense:

E(kx) = U (k)E(x), R(k(x − y)) = (U ⊗U )(k)R(x − y)

Our case corresponds to d = 2, r = 4, L = S2(S2(E)) Other interesting cases include the tensor-valued random fields of

electric polarisation (r = 1, L = E), piezoelectricity (r = 3, L = S2(E) ⊗ E), photoelasticity (r = 4, L = S2(E) ⊗ S2(E)),

and so on The general form of the expected value and correlation tensor of the field T(x) as well as the spectral

expansion of the field itself may be found by adding necessary details to the above sketch

In equation (17) we expressed the tensor Ti j`m0,2,1in terms of more simple tensors δi jδ`m and δi`δjm+ δimδj` The

general problem, i.e., how to express the tensor-valued functions Mi2n,q

1 ···i2r(p) in terms of more simple tensor-valued functions, the so called polynomial invariants of the group O(E), is currently solved only in particular cases

Another interesting problem is to study random fields that are isotropic with respect to a proper subgroup of the

group O(E)

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2 N Cressie, Statistics for Spatial Data, Wiley–Interscience, 1993

3 E Porcu, J.-M Montero, and M Schlather, Advances and Challenges in Space-time Modelling of Natural Events, vol 207 of

Lect Notes in Statist., Springer, 2012

4 M Ostoja-Starzewski, L Shen, and A Malyarenko, Math Mech Solids (2013), URL http://mms.sagepub.com/

content/early/2013/08/19/1081286513498524.full.pdf+html

5 J Adams, Lectures on Lie Groups, W.A Benjamin, 1965

T Simos, and G Maroulis, Amer Inst Phys., 2009, vol 2, pp 778–780

7 A Yaglom, Teor Veroyatnost i Primenen 2, 292–338 (1957)

8 K Karhunen, Ann Acad Sci Fennicae Ser A I Math.-Phys 1, 3–79 (1947)