Tensor series expansion of a spherical function for the use in constitutive theory of materials containing orientable particles [Sfäärilise funktsiooni tensorrea kasutamine orienteeritud osakesi sisaldava materjali olekuvõrrandi teoorias]

This paper presents a didactical introduction to a tensor series expansion of a spherical function for the use in constitutive theory of materials containing orientable particles. In several application areas a function of two angles, e.g. an orientation (density) distribution function, is expanded into a series of symmetric irreducible tensors. This paper will explain this series expansion, starting with reviewing the representation of a function defined on a unit sphere in terms of spherical harmonics, which are a possible choice for a basis. Then, the connection between spherical harmonics and symmetric traceless tensors is explained. This is the basis for introducing and understanding orientation and alignment tensors as well as their connection to the orientation distribution function. The style of presentation was chosen to be more on the didactical side, differently from the theorem–proof style found elsewhere, which directly starts from symmetric tensors.