THE APPLICATION OF TENSOR ALGEBRA ON MANIFOLDS TO NONLINEAR CONTINUUM-MECHANICS - INVITED SURVEY ARTICLE

Some properties of the tensor algebra on manifolds are discussed with respect to the classical tensor algebra of continuum um mechanics, Bas ic definitions and relations of linear maps are briefly recalled and a pplied to tensor spaces, where special attention is focused to the def inition of dual and transposed maps. As application to nonlinear conti nuum mechanics, algebraic pull-back and push-forward maps, and Lie-lik e time derivatives associated with linear maps are defined and used to build up commutative schemes for deformation measures. objective defo rmation rates, stresses, and frame-invariant forms of the stress power . The transition to the classical formulation of tensor algebra by ide ntifying vector spaces and their duals is examined. It is shown how th is concept enables a straightforward derivation of the Doyle-Ericksen formula and its possible variants.