INTERPOLATION METHODS FOR TENSOR FUNCTIONS

Abstract

The classical interpolation methods due to Lagrange and, alternatively, to Newton are extended to symmetric tensor-valued functions of one and two symmetric argument tensors. The restriction on symmetric tensors is made in view of the application to continuum mechanics. Based upon the Hamilton-Cayley matrix theorem, which can be used for second rank tensors and extended for tensors of rank four, it can be shown that the tensor-valued remainders vanish and that both interpolation methods (necessarily) yield to the same isotropic tensor function. The scalar coeffiecients in the polynomial representations are functions of the integrity basis the elements of which are the irreducible invariants of the argument tensors. These scalar functions are determined and expressed by the principal values of the argument tensors. Thus, we have an interpolation at the principal values, which are considered as interpolating points or mesh-points. Some special applications are discussed, e.g., the third root of a tensor or the tensorial representation of transcendental functions. Especially, a tensorial logarithmic strain measure can be represented as an isotropic tensor function. Furthermore, the extension of empirically uniaxial stress-strain relations to constitutive equations for materials in a state of multi-axial stress is discussed.