Form invariants and orthonormal basis elements

The decomposition methods of tensors have many applications in different subjects of engineering. In the mechanics of continuous media i. e. in elasticity studies; so far, the stress and strain tensors are decomposed into spherical (hydrostatic) and deviatoric parts, the hydrostatic pressure is connected to the change of volume without change of shape, whereas the change of shape is connected to the deviatoric part of the stress.

The anisotropic elastic properties represented by the fourth rank tensor of elastic coefficients is designated as the elasticity tensor. The constitutive relation for linear anisotropic elasticity is the generalized Hooke’s law

7..= C.., є, (1)

ij ijkm km v ‘

which is the most general linear relation between the stress tensor whose components are

<7.. and the strain tensor whose components are є. The coefficients of linearity, ij r km }’

namely C.., , are the components of the fourth rank elastic stiffness tensor. The elastic

J ijkm r

properties of crystals appear to be well described in terms of symmetry planes. Symmetry planes (i. e. planes of mirror symmetry) were defined, for example, by Spencer (Spencer, 1983). Cowin et al. (Cowin & Mehrabadi, 1987) classified the known elastic symmetries of materials and ordered materials on the basis of symmetry planes. Cowin et al. (Cowin & Mehrabadi, 1987), and Hue and Del Piero(Hue & Del Piero, 1991) listed ten symmetry classes. There are three important symmetry restrictions on C^m that are independent of

Form invariants and orthonormal basis elements Подпись: : Form invariants and orthonormal basis elements Подпись: Cijkm Ckmij Подпись: (2)

those imposed by material symmetry:

which follow from the symmetry of the stress tensor, the symmetry of the strain tensor, and

the thermodynamic requirement that no work be produced by the elastic material in a closed loading cycle, respectively(Srinivasan, 1998; Blinowski, A. & Rychlewski, 1998) . The number of independent components of a fourth rank tensor in three dimensions is 81, but the restrictions in (2) reduce the number of independents of Cijkm to 21, which

corresponds to the most asymmetric elastic solid, namely triclinic media. Since it has 21 independent components, there is considerable information on the material properties apparent a decomposition of Cijkm into orthonormal tensor basis would be of interest.

The determination of the class system of an elastic medium from its elastic constants in an arbitrary coordinate is not a trivial matter. The problem has been studied thoroughly by several authors (Srinivasan, 1969; Srinivasan, 1985; Spencer, 1983; Cowin & Mehrabadi, 1987; Hue & Del Piero, 1991; Srinivasan, 1998; Blinowski & Rychlewski, 1998; Tu, 1968). Another interesting material property in anisotropic solids is the direct piezoelectric effect that comprises a group of phenomena in which the mechanical stresses or strains induce in crystals an electric polarization (electric field) proportional to those factors. Besides, the mechanical and electrical quantities are found to be linearly related as following (Srinivasan, 1998).

: d.., o., ijk jk

Подпись: (3)where Pi and a jk denote the components of the electric polarization vector and the

components of the mechanical stress tensor, respectively, and d^ are the piezoelectric

coefficients forming a third rank tensor. The piezoelectric tensor is a third rank tensor symmetric with respect to the last two indices

Подпись: (4)dijk=dikj

with 18 coefficients for the noncentrosymmetric triclinic case. Considerable information on the material properties apparent a decomposition of d^ into orthonormal tensor basis would be of interest, as well.

In writing out tensors which represent physical properties of solid materials, it is customary to choose a Cartesian frame reference which has a specific orientation with respect to the material coordinate axes. A physical property is characterized by n rank tensor that has two kinds of symmetry properties. The first kind is due to an intrinsic symmetry derives from the nature of the physical property itself, and this can be established by the thermodynamical arguments or from the indispensability of some of the quantities involved. The second kind of symmetry is due to the geometric or crystallographic symmetry of the system described.

The symmetry properties of the material may be defined by the group of orthonormal transformations which transform any of these triads into its equivalent positions. For each of the symmetry classes, we will choose as reference system a rectangular Cartesian coordinate system Oxyz, so related to the material directions v1,v2,v3 in the material under consideration that the symmetry of the material may be described by one or more of the transformations. Transformations in which the coefficients satisfy the orthogonality relations are called linear orthogonal transformations. In this formulation, the number of elastic constants and their values do not depend on the choice of the coordinate system.

Form invariants and orthonormal basis elements Подпись: = v .V, .A , ai bj ab Подпись: (5)

The form-invariant expressions for the electrical susceptibility components, the piezoelectric coefficients and the elastic stiffness coefficients are, respectively

Подпись: (6)d.- = v. v – v і A і
ijk ai bj ck abc

C.., = v. v, .v, v, A, , (7)

ijkm ai bj ck dm abcd v ‘

Where summation is implied by repeated indices, vai are the components of the unit vectors va ( a=1,2,3) along the material directions axes. The quantities Aah, Aahc, Aahcd are invariants in the sense that when the Cartesian system is rotated around Ox’y’z’, where v1,v2,v3 form a linearly independent basis in three dimensions but are not necessarily always orthogonal. Their relative orientations in the seven crystal systems are well known (Ikeda, 1990). The corresponding reciprocal triads satisfy the relations

vaiv. = 5.. a] Ч