Decomposition of third-order constitutive tensors

Third-order tensors are widely used as a mathematical tool for modeling physical properties of media in solid state physics. In most cases, they arise constitutive proportionality between basic physics quantities. The tensor can be considered complete set parameters medium. algebraic features the seen proper identification natural material, crystals, and design artificial nano-materials with prescribed properties. this paper, we study generic 3-rd order relative to its invariant decomposition. correspondence different groups acted on vector space, present hierarchy types decomposition into subtensors. particular, discuss problem non-uniqueness reducibility high-order For tensor, these described explicitly. case special symmetry, turns out irreducible unique. We explicit results two physically interesting models: piezoelectric an example pair symmetry Hall skew-symmetry.