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The Application of Tensor Algebra on Manifolds to Nonlinear Continuum Mechanics — Invited Survey Article
Author(s) -
Stumpf H.,
Hoppe U.
Publication year - 1997
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.19970770504
Subject(s) - tensor (intrinsic definition) , continuum mechanics , tensor calculus , tensor algebra , algebra over a field , mathematics , physics , pure mathematics , mathematical physics , theoretical physics , tensor field , classical mechanics , mathematical analysis , cellular algebra , algebra representation , exact solutions in general relativity
Some properties of the tensor algebra on manifolds are discussed with respect to the classical tensor algebra of continuum mechanics. Basic definitions and relations of linear maps are briefly recalled and applied to tensor spaces, where special attention is focused to the definition of dual and transposed maps. As application to nonlinear continuum mechanics, algebraic pull‐back and push‐forward maps, and Lie‐like time derivatives associated with linear maps are defined and used to build up commutative schemes for deformation measures, objective deformation rates, stresses, and frame‐invariant forms of the stress power. The transition to the classical formulation of tensor algebra by identifying vector spaces and their duals is examined. It is shown how this concept enables a straight‐forward derivation of the Doyle‐Ericksen formula and its possible variants.