Transversely Isotropic Tensor Closest in Euclidean Norm to a Given Anisotropic Elastic Modulus Tensor

Abstract

The problem of determining the transversely isotropic tensor closest in Euclidean norm to a given anisotropic elastic modulus tensor is considered. An orthonormal basis in the space of transversely isotropic tensors for any given axis of symmetry was obtained by decomposition of a transversely isotropic tensor in the general coordinate system into one isotropic part, two deviator parts, and one nonor part. The closest transversely isotropic tensor was obtained by projecting the general anisotropy tensor onto this basis. Equations for five coefficients of the transversely isotropic tensor were derived and solved. Three equations describing stationary conditions were obtained for the direction cosines of the axis of rotation (symmetry). Solving these equations yields the absolute minimum distance from the transversely isotropic tensor to a given anisotropic elastic modulus tensor. The transversely isotropic elastic modulus tensor closest to the cubic symmetry tensor was found.