Fourth-order tensors – tensor differentiation with applications to continuum mechanics. Part II: Tensor analysis on manifolds

Abstract

Key words fourth-order tensors, tensor differentiation, tangent operators, anisotropic hyperelasticity MSC (2000) 03E25In Part II of this contribution the theory of fourth-order tensors and tensor differentiation introduced in Part I will be applied to finite deformation problems using a mathematical framework which is mainly borrowed from tensor analysis on manifolds. The main aspects are the explicit consideration of component variance, the introduction of different configurations and its associated metric tensors. Some relevant problems of continuum mechanics like the construction of tangent operators are discussed. Special attention is given to the conjugated formulation of the corresponding tangent operator in terms of the left Cauchy-Green-tensor.In the first part of this contribution a tensor formalism for fourth-order tensors has been introduced which has been used in the process of tensor differentiation with respect to a second-order tensor according to a recent approach of Itskov [8]. The proposed rules allow an easy application of the tensor differentiation law in absolute tensor notation which is in particular useful for the linearization of nonlinear functionals, the differentation of isotropic tensor functions e.g. in pow er series or the construction of tangent operators. In the first part the laws have been applied in a mathematical framework referred to as classical tensor analysis [1,16,11,12]. However, as far as finite deformation or large strain problems are concerned, the mathematical framework of tensor analysis on manifolds [2,13,24] is more appropriate since it delivers a rigorous and profound mathematical theory for the study of such kinds of problems. This mathematical framework has attained popularity especially through the work of Marsden & Hughes [13]. In a number of overview articles the topic of tensor algebra on manifolds has been addressed in a comprehensive way (see e.g. [4,5,22]).Due to the relevance of this mathematical field the present contribution will focus on the algebra of tensor analysis on manifolds and aims at a unified presentation considering the theory of fourth-order tensors and the rules of tensor differentiation. The consideration of large strain problems rests upon the introduction of different configurations of a body and its associated metric tensors. Following the typical approach of tensor algebra in dual spaces these metric tensors, which are used to construct invariants of tensors, are introduced as independent argument tensors in the corresponding tensor functions since the push-forward or pull-back of a metric tensor is not trivially the identity. According to this approach the derivative with respect to a metric tensor is a well-established quantity. We introduce the well-known Doyle-Ericksenformula [17,19], which is given as derivative of the strain energy function with respect to the spatial metric tensor g, and discuss its conjugated formulation in terms of the left Cauchy-Green-tensor b. The latter problem has been addressed in earlier contribution...

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