Application of a coordinate-free tensor formalism to the numerical implementation of a material model

Abstract

We analyze a coordinate-free tensor setting in ℝ3 within the context of the classical tensor analysis. To this end, we formulate in a basis-free manner the notions of second- and fourth-rank tensors in ℝ3, and corresponding operations on tensors. Among the large number of different approaches to the tensor setting, we give the preference to the convenient ones, concerning the specific needs of computational solid mechanics. We use the well-known Fréchet derivative to define the derivative of a function with respect to its tensor argument in a natural way. Furthermore, such aspects as the derivative with respect to a symmetric tensor argument and its uniqueness are covered in this paper. For the sake of completeness we present the coordinate representation of tensors and tensor operations. This representation is obtained in a straight-forward manner from the coordinate-free one. In particular, we elaborate the computation of the inverse of a fourth-rank tensor and the inverse of a linear transformation on the space of symmetric second-rank tensors. The tensor formalism is applied to the analysis of a nonlinear system of differential and algebraic equations governing visoplastic material response. An implicit time-stepping algorithm is formulated and the numerical treatment of the algorithm is discussed. © 2008 WILEY-VCH Verlag GmbH & Co. KGaA.