The Application of Tensor Algebra on Manifolds to Nonlinear Continuum Mechanics — Invited Survey Article

Abstract

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.

Mechanics of Shape Memory Alloy Materials – Constitutive Modeling and Numerical Implications

“…67 -68;Stumpf and Hoppe, 1997) to the basic equations. For instance, by performing a pushforward operation onto Eq.…”

“…93 -104;Schutz, 1999, pp. 73-79;Stumpf and Hoppe, 1997), defined as the convected derivative relative to the spatial configuration. Finally, i h stands for the expression of the scalar invariant functions i H in terms of the spatial variables (e, T, q) and the deformation gradient F, i l stands for the push -forward of the tensorial functions i L and i r denotes the (scalar invariant) loading rates which are written in the form…”

Anholonomic Configuration Spaces and Metric Tensors in Finite Elastoplasticity

“…Additionally in (20) the dual map operation corresponding to a horizontal exchange of indices is denoted by the superposed " * " [24,25]. From relations (20), we are in position to deÿne symmetric elastic and plastic strain tensors referred to the current and reference conÿgu-rations, respectively, e.g.…”

“…Multiplying (24) by through byg ÿ results iñ (23)) and its inverse (from (25)). We thus see that the plastic velocity gradient L p can only be decomposed into stretch rate and spin terms after a metric tensorg has been introduced, as noted by Maugin [18].…”

A geometric framework for the kinematics of crystals with defects

“…In continuum mechanics, both the reference and current configurations are typically viewed as three-dimensional Euclidean spaces, with the motion acting as a diffeomorphism (i.e. a differentiable homeomorphism, or a differentiable one-to-one, invertible mapping) between these two configurations [61]. Notice the distinction between Euclidean and Cartesian: the former (e.g.…”