Section 1.1.4.10.1. Introduction

Many tensors representing physical properties or physical quantities appear in relations involving symmetric tensors. Consider, for instance, the strain resulting from the application of an electric field E (the piezoelectric effect): where the first-order terms represent the components of the third-rank converse piezoelectric tensor and the second-order terms represent the components of the fourth-rank electrostriction tensor. In a similar way, the direct piezoelectric effect corresponds to the appearance of an electric polarization P when a stress is applied to a crystal:

Owing to the symmetry properties of the strain and stress tensors (see Sections 1.3.1 and 1.3.2 ) and of the tensor product , there occurs a further reduction of the number of independent components of the tensors which are engaged in a contracted product with them, as is shown in Section 1.1.4.10.3 for third-rank tensors and in Section 1.1.4.10.5 for fourth-rank tensors.