Symmetry types of the piezoelectric tensor and their identification

The third-order linear piezoelectricity tensor seems to be simpler than the fourth-order linear elasticity one, yet its total number of symmetry types is larger than the latter and the exact number is still inconclusive. In this paper, by means of the irreducible decomposition of the linear piezoelectricity tensor and the multipole representation of the corresponding four deviators, we conclude that there are 15 irreducible piezoelectric symmetry types, and thus further establish their characteristic web tree. By virtue of the notion of mirror symmetry and antisymmetry, we define three indicators with respect to two Euler angles and plot them on a unit disk in order to identify the symmetry type of a linear piezoelectricity tensor measured in an arbitrarily oriented coordinate system. Furthermore, an analytic procedure based on the solved axis-direction sets is also proposed to precisely determine the symmetry type of a linear piezoelectricity tensor and to trace the rotation transformation back to its natural coordinate system.