International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 24-25
Section 1.1.4.10.3. Reduction of the number of independent components of third-rank polar tensors due to the symmetry of the strain and stress tensors^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
1.1.4.10.3. Reduction of the number of independent components of third-rank polar tensors due to the symmetry of the strain and stress tensors
Equation (1.1.4.5) can be written
The sums for have a definite physical meaning, but it is impossible to devise an experiment that permits and to be measured separately. It is therefore usual to set them equal:
It was seen in Section 1.1.4.8.1 that the components of a third-rank tensor can be represented as a matrix which can be subdivided into three submatrices:
Relation (1.1.4.7) shows that submatrices 1 and 2 are identical.
One puts, introducing a two-index notation, Relation (1.1.4.7) becomes
The coefficients may be written as a matrix: This matrix is constituted by two submatrices. The left-hand one is identical to the submatrix 1, and the right-hand one is equal to the sum of the two submatrices 2 and 3:
The inverse piezoelectric effect expresses the strain in a crystal submitted to an applied electric field: where the matrix associated with the coefficients is a matrix which is the transpose of that of the coefficients used in equation (1.1.4.5), as shown in Section 1.1.1.4.
The components of the Voigt strain matrix are then given by This relation can be written simply as where the matrix of the coefficients is a matrix which is the transpose of the matrix.
There is another set of piezoelectric constants (see Section 1.1.5) which relates the stress, , and the electric field, , which are both intensive parameters: where a new piezoelectric tensor is introduced, . Its components can be represented as a matrix:
Both sides of relation (1.1.4.8) remain unchanged if the indices i and j are interchanged, on account of the symmetry of the stress tensor. This shows that
Submatrices 2 and 3 are equal. One introduces here a two-index notation through the relation , and the matrix can be written
The relation between the full and the reduced matrix is therefore different for the and the tensors. This is due to the particular property of the strain Voigt matrix (1.1.4.6), and as a consequence the relations between nonzero components of the reduced matrices are different for certain point groups (3, 32, , , ).