Decomposition of third-order constitutive tensors

Abstract

Third-order tensors are widely used as a mathematical tool for modeling the physical properties of media in solid-state physics. In most cases, they arise as constitutive tensors of proportionality between basic physical quantities. The constitutive tensor can be considered the complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be used as a tool for proper identification of natural materials, such as crystals, and for designing artificial nanomaterials with prescribed properties. In this paper, we study the algebraic properties of a general asymmetric third-order tensor relative to its invariant decomposition. In correspondence with different groups acting on the basic vector space, we present the hierarchy of different types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a general asymmetric third-order tensor, these features are described explicitly. In the case of special tensors with a prescribed symmetry, the decomposition is demonstrated to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of pair symmetry and the Hall tensor as an example of pair skew-symmetry.

A review on octupolar tensors

“…These two latter do not enjoy the symmetry (76), whereas A (2) = A (2,1) + A (2,2) does. Moreover, as shown in [54],…”

“…While A (1) and A (3) are (uniquely identified) irreducible components of A, as pointed out in [54], the decomposition A (2) = A (2,1) + A (2,2) is irreducible, but not unique. It is also worth noting that by (12) and ( 13)…”

“…Following [99], Weyl [123] initiated a fully general theory of decomposition of tensors into irreducible symmetry parts, having especially in mind its application to quantum mechanics. An early description of the role of both Young's diagrams and tableaux can be retraced in [119]; here we follow a more recent approach [54].…”

“…where sgn(π) is the sign (or index) of the permutation. Finally, tensors under case (ii) are characterized by the following mixed symmetry relations (see also [54])…”

“…In classical elasticity, T ij = T ji , and so A enjoys the symmetry (76) and is represented by 18 independent parameters. The invariant decomposition of the piezoelectric tensor can help to classify piezoelectric crystals; its algebraic properties have recently received a renewed interest (see, for example, [90, chapter 7] and [46,54,61]). The decomposition of A as in (10) is affected by the extra symmetry requirement (76).…”