The Explicit Determination of the Logarithm of a Tensor and Its Derivatives (original) (raw)

An objective of this paper is to reconcile the "symmetry" approach with the "symmetry groups" approach as these two different points of view presently coexist in the literature. Here we will be concerned exclusively with linearly elastic materials. The starting point for an analysis of the inherent symmetry of elastic materials is the notion of a symmetry transformation. Particularly, we paid attention to the compliance tensor for cubic and hexagonal crystals.

In this paper, a uni ed explicit tensorial relation is sought between two stress tensors conjugate to arbitrary and general Hill strains. The approach used for deriving the tensorial relation is based on the eigenprojection method. The result is, indeed, a generalization of the relations that were derived by Farahani and Naghadabadi [1] in 2003 from a component to intrinsic form. The result is uni ed in the sense that it is valid for all cases of distinct and coalescent principal stretches. Also, in the case of three dimensional Euclidean inner product space, using the derived uni ed relation, some expressions for the conjugate stress tensors are presented.

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor, hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor that is endowed with a particular symmetry and is closest to the given elasticity tensor.