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Application of a coordinate‐free tensor formalism to the numerical implementation of a material model
Author(s) -
Shutov A.V.,
Kreißig R.
Publication year - 2008
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200800017
Subject(s) - cartesian tensor , mathematics , tensor contraction , symmetric tensor , tensor (intrinsic definition) , tensor calculus , tensor field , tensor density , uniqueness , tensor product of hilbert spaces , algebra over a field , mathematical analysis , exact solutions in general relativity , pure mathematics
We analyze a coordinate‐free tensor setting in ℝ 3 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 ℝ 3 , 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.