Z. angew. Math. Phys. 48 (1997) 857–873 0044-2275/97/060857-17 $ 1.50+0.20/0 c 1997 Birkh¨auser Verlag, Basel Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Anisotropic polynomial constitutive equations for electroelastic crystals A. Danescu and A. M. Tarantino Abstract. This paper is concerned with polynomial constitutive relations for electroelastic crys- tals. We generalize previous results of [1], and we emphasize the role of disjoint union decom- positions in the representation problems. Our method, essentially a “group-average method” [2], provides the number of coefficients needed for the 32 crystallographic groups for polynomial relations up to degree 5. Mathematics Subject Classification (1991). 73B05, 73C30, 73R05. Keywords. Electroelasticity, constitutive equations, cristallographic point groups. 1. Introduction Constitutive equations in the theory of elasticity involve a dependence of the free energy density on a single second order tensor. For example, in the linear theory of elasticity, the measure of strains is the infinitesimal strain tensor [3], and in the nonlinear theory the Cauchy-Green strain tensor. For this reason, the symmetry groups of interest in that context are subgroups of the proper orthogonal group SO(3). In electroelastic theory, as in any other theory where constitutive functions depend on a first order tensor (vector) and a second order tensor, one must also consider groups that are not included in the proper orthogonal group. This paper generalizes a method presented in [1] to include the presence of a vector field in the form of the free energy and study some new qualitative aspects related to the symmetry properties of the coefficients for polynomial constitutive equations in this framework. The use of algebraic methods in the study of constitutive equations goes back to [4], [5] (see also [6]), [7]). The extensions provided by this work with respect to [1] are two fold: firstly, we treat the case of two independent variables (in the case of the electroelastic theory) and secondly, we extend the analysis to all 32 crystallographic point groups. Homogeneous polynomials of degree m ≥ 0 (in two variables) are obtained as a sum of terms, which are products of tensorial
Abstract. This paper is concerned with polynomial constitutive relations for electroelastic crys-tals. We generalize previous results of [1], and we emphasize the role of disjoint union decom-positions in the representation problems. Our method, essentially a “group-average method”[2], provides the number of coefficients needed for the 32 crystallographic groups for polynomialrelations up to degree 5.
Keywords. Electroelasticity, constitutive equations, cristallographic point groups.
1. Introduction
Constitutive equations in the theory of elasticity involve a dependence of the freeenergy density on a single second order tensor. For example, in the linear theoryof elasticity, the measure of strains is the infinitesimal strain tensor [3], and in thenonlinear theory the Cauchy-Green strain tensor. For this reason, the symmetrygroups of interest in that context are subgroups of the proper orthogonal groupSO(3). In electroelastic theory, as in any other theory where constitutive functionsdepend on a first order tensor (vector) and a second order tensor, one must alsoconsider groups that are not included in the proper orthogonal group. This papergeneralizes a method presented in [1] to include the presence of a vector field inthe form of the free energy and study some new qualitative aspects related to thesymmetry properties of the coefficients for polynomial constitutive equations inthis framework.
The use of algebraic methods in the study of constitutive equations goes backto [4], [5] (see also [6]), [7]). The extensions provided by this work with respectto [1] are two fold: firstly, we treat the case of two independent variables (in thecase of the electroelastic theory) and secondly, we extend the analysis to all 32crystallographic point groups. Homogeneous polynomials of degree m ≥ 0 (intwo variables) are obtained as a sum of terms, which are products of tensorial
858 A. Danescu and A. M. Tarantino ZAMP
and vectorial homogeneous quantities of degrees 0 ≤ k ≤ m and m − k. Withinthe framework of the elasticity theory, the symmetry properties of a homogeneouspolynomial of degree k were described by a single symmetry group Zk2 × Zk. Inthe present case, the symmetry properties are described by k + 2 groups eachacting on the real vector space of homogeneous polynomials of partial degrees(k,m−k). A particular aspect of the results presented in [1] was the use of disjointunion decomposition to reduce the general problem to cyclic groups. We devotea part of this paper to disjoint union decompositions for crystallographic groups,and we discuss their consequences on the number of independent coefficients forpolynomial constitutive equations.
The paper is organized as follows: the second section recall the general lines ofthe nonlinear electromagnetic deformable bodies (as presented in [8]). This formu-lation evidences that constitutive relations are completely specified by a free energyfunction Ψ, whose derivatives provide constitutive relations compatible with theClausius-Duhem inequality. As we mentioned above, in the context of electroe-lastic materials the free energy function depends only on two variables: one is asymmetric second order tensor (the right Cauchy-Green strain tensor), and theother is a vector (the electric field in the material description). The third sectionanalyses the structure of polynomial constitutive equations and emphasizes therelation between invariant polynomials and fixed point subspaces. This relationallows us to use the trace formula to compute the number of independent coeffi-cients in this context. The fourth section gives an explicit formula for the numberof independent coefficients and for arbitrary polynomial constitutive equations.
As in the context of nonlinear elasticity theory, we show how the groups de-scribing the symmetry properties of the coefficients involved in the descriptionof constitutive relations, determine completely the number of independent coeffi-cients. Based on the results of the fourth section, we explicitly compute in thefifth section the characters involved in the description of polynomial constitutiverelations for electroelastic materials up to degree 5. We evidence an interesting fea-ture of this framework: a purely algebraic argument shows that characters used tocompute coupling coefficients are expressed as products between characters arisingin elasticity theory and in dielectrics.
In the framework of nonlinear elasticity, in [1] the number of independent coeffi-cients for the dihedral groups, for the tetragonal group and for the cubic group hasbeen related to the number of coefficients for the cyclic groups. In the same spirit,in the sixth section we exploit this powerful result related to the trace formula.Specifically, we show that the number of independent coefficients for 8 crystallo-graphic groups allows the computation of the number of independent coefficientsfor all other 24 groups. The key of this result is the disjoint union decompositionformulas presented in section 6. Combining the general results, we find the num-ber of independent coefficients for polynomial constitutive relations up to degree5 for all 32 crystallographic groups.
For sake of completeness, the general lines of the nonlinear electromagnetic theoryare recalled following [8], [14]. Neglecting the jump conditions, the field equationsare:
1. Maxwell’s equations:
∇ ·D = qe, ∇×E +1c
∂B∂t
= 0, (2.1− 2)
∇ ·B = 0, ∇×H− 1c
∂D∂t
=1cJ, (2.3− 4)
supplemented by the relations
D = E + P, B = H + M. (2.5− 6)
2. Thermomechanical balance laws:
ρ+ ρ∇ · v = 0, (2.7)
tkl,k + ρ(fl − vl) + Mfl = 0, (2.8)
t[kl] = E[kPl] +B[kMl], (2.9)
ρε− tklvl,k −∇ · q− ρh− ρE · (P/ρ)· +M· B− I · E = 0, (2.10)
ρη −∇ · (q/θ) − ρh/θ ≥ 0. (2.11)
In the above equations the square brackets indicate the antisymmetry (e.g.,a[kl] = (akl − alk)/2), and the symbols employed denote: ρ - mass density, v -velocity field, t = (tkl) - Cauchy stress tensor, f = (fl) - mechanical body forcedensity, M f = (Mfl) - electromagnetic body force density, E - electric vector, P -polarization field, B - magnetic induction field, M - magnetization vector, ε - inter-nal energy density, q - heat flux, h - heat source density, θ - absolute temperature,η - entropy, D - dielectric displacement vector, qe - charge density, B - magneticinduction, H - magnetic induction, J - current vector. The electromagnetic fieldsin a moving frame are given by
E = E +1cv×B, H = H− 1
cv×D, (2.12− 13)
M = M− 1cv×P, I = J− qev. (2.14− 15)
The electromagnetic body force M f is defined by
M f = qeE +1cJ×B + (∇E) ·P + (∇B) ·M +
1c
[(P×B)vk],k. (2.16)
860 A. Danescu and A. M. Tarantino ZAMP
In the classical treatment [8], a canonical decomposition of the nonsymmetricalstress tensor tkl provides the symmetrical tensor Etkl, viz.
Etkl = tkl + PkEl +MkBl, (2.17)
and the Clausius-Duhem inequality imposes some restrictions on the constitutiverelations similar to those appearing in nonlinear elasticity. Let F denote the de-formation gradient from a fixed reference configuration and let Π and ET denotethe polarization vector corresponding to P in the material description and, re-spectively, the second order Piola-Kirchhoff stress tensor corresponding to Et, i.e.,Π = ρ0/ρF−1P and ET = ρ0/ρF−1(Et)F−T . If the free energy density Ψ, definedas
Ψ = ε− θη − 1ρ0E0 · Π, (2.18)
is supposed to depend on C (the right Cauchy-Green strain tensor), θ,∇θ, E0 (theelectric field in material description) and B, it can be shown that Ψ is a potentialfor the constitutive relations [8], namely,
ET = 2ρ0∂Ψ∂C
, Π = −ρ0∂Ψ∂E0
, (2.19− 20)
M = −ρ0∂Ψ∂B
, η = −∂Ψ∂θ
. (2.21− 22)
Finally, two constitutive relations (for the heat flux q and current I) are so re-stricted by the reduced Clausius-Duhem inequality
1θq · ∇θ + I · E ≥ 0. (2.23)
All above equations can be directly specialized to the nonlinear electroelastictheory by neglecting magnetic and temperature effects.
3. Polynomial constitutive relations
This section presents the general structure of polynomial constitutive relations andthe restrictions imposed by symmetry requirements. For the sake of simplicity weshall formally replace the tensorial variable by A and the vectorial one by v.
In the sequel, we denote by G a subgroup of the orthogonal group O(3) and byPn(Sym,R3) the real vector space of scalar valued polynomials of degree ≤ n intwo variables (A,v) ∈ Sym × R3. The actions of the symmetry group G on Symand R3, denoted by Φ1 and Φ2 respectively, are defined through
The natural action of the group G on Pn(Sym,R3), induced by the material sym-metry requirements Φ : G × Pn(Sym,R3) → Pn(Sym,R3) is defined, taking intoaccount (3.1) and (3.2), as:
Φ(Q, p(A,v)) = p(Φ1(Q,A),Φ2(Q,v)). (3.3)
The real vector subspace of Pn(Sym,R3) containing all polynomials p that satisfythe restriction
Φ(Q, p) = p, (3.4)
is designed by PnG(Sym,R3). In the next section a general result for the dimensionof PnG(Sym,R3) as a function of G and n, is provided. A first step of our analysisis the following reduction argument. A general polynomial form for the potentialΨ can always be written in the form
Ψ(A,v) =n∑
m=0
Hm(A,v), (3.5)
where Hm(A,v) are homogeneous polynomials of degree m. In turn, each homo-geneous polynomial Hm(A,v) can be expressed as
For 0 ≤ k ≤ m, we indicate with K(k,m−k) the real vector space of homo-geneous polynomials of total degree m (i.e., p(λA, λv) = λmp(A,v)) and partialhomogeneous degree k in the first variable (i.e., p(λA,v) = λkp(A,v)). Obviously,K(k,m−k) is invariant with respect to the action Φ, namely
p ∈ K(k,m−k) =⇒ Φ(Q, p) ∈ K(k,m−k), (3.7)
for every Q ∈ G and for every G subgroup of O(3). We denote by Φ(k,m−k)
the restriction to G × K(k,m−k) of the action Φ of G on PnG(Sym,R3). Thus,Φ(k,m−k) : G×K(k,m−k) → K(k,m−k) and
Φ(k,m−k)(Q, p(A,v)) = p(QAQT ,Qv). (3.8)
Let K(k,m−k)G be the real vector subspace of invariant polynomials of K(k,m−k);
thenK(k,m−k)G = K(k,m−k) ∩ PnG(Sym,R3). (3.9)
862 A. Danescu and A. M. Tarantino ZAMP
Obviously, we have
Pn(Sym,R3) =⊕
O≤k≤m≤nK(k,m−k) (3.10)
PnG(Sym,R3) =⊕
O≤k≤m≤nK(k,m−k)G . (3.11)
and to conclude our reduction argument we state the following proposition:
Proposition 3.1. The dimension of the vector space PnG(Sym,R3) equalsn∑
m=0
m∑k=0
dimK(k,m−k)G . (3.12)
Proof follows from relation (3.11). We are thus led to study the dimension ofthe real vector space of invariant homogeneous polynomials of the total degree ofhomogeneity m and the partial degree of homogeneity with respect to the firstvariable k, (k < m).
We recall that the action of the symmetry group on each vector space K(k,m−k)
provides a character, indicated by χ(k,m−k), which is expressed as
χ(k,m−k)(Q) = tr Φ(k,m−k)(Q, ·). (3.13)
Fix 0 ≤ k ≤ m ≤ n and denominate by M(k,m−k) the set of multilinear applica-tions from Symk ×R3(m−k) symmetric with respect to permutations of the first kvariables and symmetric with respect to permutations of the last m− k variables.Thus, the action of the symmetry group G on K(k,m−k) induces an action φk,m−k
= M [Φ1(A(1)), . . . ,Φ1(A(k)),Φ2(v(1)), . . . ,Φ2(v(m−k))]. (3.14)In this way the following proposition is obtained.
Proposition 3.2. The fixed-point subspace of M(k,m−k), with respect to the ac-tion φk,m−k defined in (3.14), is isomorphic to K(k,m−k)
G .
The exact symmetry requirements for elements inM(k,m−k) will be describedin the following section. We can now apply the trace formula to obtain a generalresult for the dimension of K(k,m−k)
G .
Theorem 3.1. If G is a subgroup of the orthogonal group O(3), then
In this section the computation of the characters χ(k,m−k) corresponding to repre-sentations of an arbitrary subgroup of the orthogonal group is performed. The firststep toward this end is a description of the symmetry properties of the multilinearforms in M(k,m−k). We once again recall that, for 0 ≤ k ≤ m ≤ n, M(k,m−k)
represents the set of multilinear applications from Symk×R3(m−k) symmetric withrespect to permutations of the first k variables and symmetric with respect to per-mutations of the last m− k variables. There is therefore a natural action Λ of thegroup S(k,m−k) = Zk2 ×Zk ×Zm−k on a set with k+m elements. If θ ∈ S(k,m−k),then θ = (σ1, . . . , σk, τ, ζ) and the action Λ is defined as
Each element M ∈ M(k,m−k) can be expressed, via coefficients Mi1i2...im+k whichsatisfy
Mi1i2...im+k = Mθ(i1i2...im+k), (4.2)
for every θ ∈ S(k,m−k). These symmetry properties have some direct implicationson the character formulas. For θ ∈ S(k,m−k), we denote by nc(θ) the number ofdisjoint cycles of θ and, for 1 ≤ j ≤ nc(θ), by l(cj) the length of the jth cycle of θ.
Theorem 4.1. If 0 ≤ k ≤ m, the following formula holds:
χ(k,m−k)(Q) =∑
θ∈S(k,m−k)
nc(θ)∏j=1
Ql(cj) (4.3)
Proof. The identity of L(M(k,m−k),M(k,m−k)) (i.e., the space of linear applica-tions fromM(k,m−k) into itself) is given by
for some θ and κ in S(k,m−k) and after interchanging the factors each of theseterms can be rewritten as
Qi1ζ(i1) . . . Qim+kζ(im+k), (4.9)
for some ζ ∈ S(k,m−k). There are 22k(m− k)!2k!2 terms like (4.8) providing2k(m− k)!k! different terms like (4.9) (each term (4.9) is found to occur exactly2k(m− k)!k! times). Thus we obtain
χ(k,m−k)(Q) =1
2k(m− k)!k!
∑κ∈S(k,m−k)
Qi1κ(i1) . . . Qim+kκ(im+k). (4.10)
Denoting by c1, . . . , cnc(κ) the disjoint cycles of κ, we finally note that
For practical reasons, to compute the character χ(k,m−k) one can decomposeall the permutations (that describe the symmetry properties of M) of the m + kindices of M in disjoint cycles, to count those for which the number and the lengthof disjoint cycles is the same and, finally, to write the character formula (4.3).This observation provides an interesting result which is peculiar to a frameworkinvolving two or more variables of different order, namely:
Theorem 4.2. For 0 ≤ k ≤ m, the following formula holds:
χ(k,m−k)(Q) = χ(k,0)(Q)χ(0,m−k)(Q). (4.11)
Proof. As its definition shows, the group S(k,m−k) has 2k(m − k)!k! elements. Infact S(k,m−k) can be regarded as a direct product S(k,0) ⊗ S(0,m−k). It followsthat: ∑
5. Formulas for characters for constitutive equations up to de-gree 5
We apply now the results obtained in the previous section to a polynomial con-situtive relation for electroelastic materials. Table I lists the characters χ(k,m−k)
for 0 ≤ k ≤ m ≤ 5. Note that the characters χ(m,0) recover polynomial relationsin elasticity, while the χ(0,m) refer to polynomial relations for rigid dielectrics.The theorem 4.2 gives the characters for the coupling coefficients with differentphysical meanings.
This section presents a major simplification of general results about dimensionof fixed spaces. The main ingredient is the disjoint union decomposition formulaand its role here is to exploit the algebraic structure of the symmetry group. It isinteresting to note that in the context of constitutive equations the disjoint uniondecomposition provides relations between the number of material parameters fordifferent symmetry groups. A new feature of this paper is to use the disjoint union
decomposition for all crystallographic groups in this context. We recall [10] that afinite group G admits a disjoint union decomposition if H1, . . . ,Hs subgroups ofG exist such that:
1. G = H1 ∪H2 ∪ . . . ∪Hs,2. for i 6= j, Hi ∩Hj = e (the identity in G).
We denote a disjoint union decomposition of G by G = H1∪H2∪ . . . ∪Hs ([10]). Adisjoint union decomposition of a finite group provides a powerful simplificationof the trace formula and leads to the following result (originally proved in [11]):
Proposition 6.1. Let G be a finite group acting on a vector space K and supposethat G = H1∪H2∪ . . . ∪Hn. Then
dimKG =1|G| (
n∑i=1
|Hi|dimKHi − (n− 1)dimK). (6.1)
In the following we state the disjoint union decomposition formulas valid forcrystallographic groups, and the consequences of the above proposition. Beforegoing into details, we recall for completeness the following algebraic result (for aproof see [10], [12]):
Lemma 6.1. Let G be a finite group of O(3). Then one of the following holds:(i) G is a finite group of SO(3) (these groups are also called of the first type),(ii) G = G+ ∪ (−1)G+ where G+ = G ∩ SO(3) (groups of the second type),(iii) G is neither of the first type nor of the second type, and in this case, G isa subgroup of index 2 of a group of the second type (these are groups of the thirdtype).
The above lemma divides the 32 crystallographic groups into three classes.There are 11 groups of the first type: isomorphic with the abstract groups: Z1,Z2, Z3, Z4, Z6 (the cyclic groups), D2, D3, D4, D6 (the dihedral groups), T (thetetraedral group) andO (the cubic group); 11 groups of the second type isomorphicto the abstract groups Z2 ⊕ G with G one of the 11 groups of the first type; 10groups of the third type isomorphic with the groups Z−2 , Z−4 , Z−6 , Dz2, Dz3, Dz4, Dz6,Dd4, Dd6, O−. For details concerning the algebraic features about crystallographicgroups we refer to [10] and [12]1.
The structure of groups of second type gives a simple result that we state inthe following.
Lemma 6.2. If G = G+∪G− is a crystallographic group of the second type, then:
dimK(k,m−k)G = 0, if m− k is odd,
1 Different authors use different notations in this context; for this reason we collect in theappendix notations from [8], [10] and [12].
868 A. Danescu and A. M. Tarantino ZAMP
dimK(k,m−k)G = dimK(k,m−k)
G+ , if m− k is even.
Proof. The proof follows noting simply that χ(k,m−k) is an homogeneous poly-nomial of degree m + k. Thus if m − k is odd so is m + k and in the traceformula for finite groups χ(k,m−k)(Q) cancel with χ(k,m−k)(−Q). If m− k is evenχ(k,m−k)(Q) = χ(k,m−k)(−Q) but the number of elements in G is two times thenumber of elements in G+ so the proof is complete. �
In the following lemma we collect together the disjoint union decompositionfor 13 groups (6 of the first type and 7 of the third type). The proofs are givenin [10] in the context of symmetry-breaking from O(3). In the following, ∪nZpmeans disjoint union of n copies of Zp.
Proposition 6.2. The following disjoint union decomposition formulas hold:
Dn = ∪nZ2∪Zn for n ≥ 2,
O = ∪3Z4∪4Z3∪6Z2,
T = ∪4Z3∪3Z2,
Dzn = Zn∪nZ−2 for n ≥ 2,
Dd2n = Z−2n∪nZ2∪nZ−2 for n ≥ 2,
O− = ∪3Z−4 ∪4Z3∪6Z−2 .
The disjoint union decomposition formulas listed above and the result of lemma6.2 show that in order to compute dimK(k,m−k)
G for all crystallographic groups itis enough to have it only for 8 groups, namely: Z1, Z2, Z3, Z4, Z6, Z−2 , Z−4 et Z−6 .
The exact formulas for dimK(k,m−k)G in the other cases are given by lemma 6.2 for
groups of the second type and by the following theorem for the other 13 groups.
Theorem 6.1. For 0 ≤ k ≤ m, the following formulas hold:
The next section presents numerical results for dimK(k,m−k)G for polynomial
constitutive equations and all crystallographic groups.
6.1. Numerical results
Tables II and III give the numerical results obtained by using the above characterformulas and the disjoint union decompositions for all 32 crystallographic groups.In a column corresponding to a pair (k,m− k) we give the number of invarianthomogeneous polynomials of total degree m and of partial degree k with respectto the tensorial variable.
The above results essentially capture the power of group representation techniques(and especially the “group-averaging method”) in the representation problems,with a special accent to the disjoint union decomposition in this context. The ex-plicit constitutive equations are not emphasized because one could easily computethe general forms for the polynomial case via a projection method. The simplestidea is to start with a basis in the space of coefficients and to average each ele-ment of the basis over the symmetry group. A classical Gram-Schmidt proceduretogether with the dimensions provided by our results are than sufficient to obtainthe general forms.
A detailed discussion for second order couplings (which involve third orderderivaties of the free energy) is provided in a previous work [13]. Results obtainedin this paper obviously cover that case, the elastic case (when k = m in termsof (3.5)) and the electric case (when k = 0). With respect to couplings a morepractical reference [15] shows that for fourth order couplings (this means m = 4 inour notations) the measures provide errors of more than 100%. Numerical valuesof different coefficients (second order (ı.e. the linear elastoelectric case), thirdorder and even fourth order) can be found for special materials that fall in thecubic class in [14].
Appendix
As references cited frequently in this paper use different notations for the 32crystallographic groups, we put together in the following table: the names, theHermann-Mauguin notation [8], the notations for abstract groups (used in [10]and in the present paper) and notations from [12].
872 A. Danescu and A. M. Tarantino ZAMP
Notations used for crystallographic groups.
Nr. System Hermann - Mauguin Class Notation from Notation fromnotation [8] name [10] [12]
[1] A. Danescu, The number of elastic moduli for anisotropic polynomial constitutive equations,Mech. Res. Comm. 24 (1997), 289-301.
[2] M. M. Smith, G. F. Smith, Group–averaging methods for generating constitutive equations,in Mechanics and Physics of Energy Density, eds. G.C. Sih and E.E. Gdoutos, KluwerAcademic Publishers, 1992, 167–178,
[3] M. E. Gurtin, The Linear Theory of Elasticity. Handbuch der Physik 6a2, Springer-Verlag,Berlin 1972.
[4] S. Bagavantham, T. Venkatarayudu, Theory of Groups and Its Applications to PhysicalProblems, Academic Press, New York 1969.
[5] G. E. Smith, Further results on the stored-energy function for anisotropic elastic materialsArch. Rational Mech. Anal. 10 (1962), 108–118.
[6] G.E. Smith, Constitutive Equations for Anisotropic and Isotropic Materials, North-Holland,Amsterdam 1994.
[7] A. J. M. Spencer, Theory of Invariants, in Continuum Physics (ed. K.A. Eringen), Aca-demic Press, New York 1971.
[8] E. Kiral, A. C. Eringen, Constitutive Equations of Nonlinear Electromagnetic-Elastic Crys-tals, Springer Verlag, Berlin 1990.
[9] G. Maugin, Continuum Mechanics of Electromagnetic Solids, North-Holland, Amsterdam1988.
[10] M. Golubitsky, I.N. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation The-ory, Vol. II, Applied Mathematical Sciences 69, Springer Verlag, Berlin 1988.
[11] E. Ihrig, M. Golubitsky, Pattern selection with O(3) symmetry, Physica 12D (1984), 1–33.[12] W. Miller, Symmetry Groups and Their Applications, Academic Press, New-York 1972.[13] A. Danescu, A. M. Tarantino Electro-elastic interactions and second order anisotropic
constitutive equations, Meccanica 31 (1996), 657-664.[14] G.A. Maugin, Nonlinear Electromechanical Effects and Applications, World Scientific, Sin-
gapore 1985.[15] G. Maugin, J. Pouget, R. Drouot, B. Collet, Nonlinear Electromechanical Couplings, John
Wiley & Sons, New York 1992.
A. DanescuLTDS - Ecole Centrale de LyonF 69131 EcullyFrance(Fax: +33.472.18.91.44)
A. M. TarantinoIstituto di Scienza e Tecnica delle CostruzioniUniversita di AnconaI 60131 Ancona, Italia(Fax: +39.71.2204576)
