Riemannian and Euclidean material structures in anelasticity

Abstract

In this paper, we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to non-linear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold. This allows one to define, in addition to the two geometric structures, a Weitzenböck connection on the material manifold. We use this connection to express natural uniformity in a geometrically meaningful way. The concept of uniformity is then extended to the Riemannian and Euclidean structures. Finally, we discuss the role of non-uniformity in the form of material forces that appear in the configurational form of the balance of linear momentum with respect to the two structures.

The mathematical foundations of anelasticity: existence of smooth global intermediate configurations

“…φ maps from the Riemannian manifold false(B,Mfalse), and φfalse~ maps from false(B,Hfalse), but both of these morphisms become φ:Bfalse→C under the forgetful functor that forgets metric tensor fields. Additionally, by construction, the map idscriptB defined above induces the same strain as G , since H is the pullback of h under G , as can be explicitly checked in components δABHACδCD=GαBhαβGβD. This last expression is written in terms of the disjoint union of the frames falsefalse{boldeαfalsefalse} [42]; the interpretation of these as a moving frame depends on the smoothness of G . Specifically, G can be (highly) discontinuous in which case we just have a collection of frames, one frame for each tangent space, with no general relationship between the frames at different points.…”

“…This last expression is written in terms of the disjoint union of the frames {e α } [42]; the interpretation of these as a moving frame depends on the smoothness of G. Specifically, G can be (highly) discontinuous in which case we just have a collection of frames, one frame for each tangent space, with no general relationship between the frames at different points. As a pathological example, G could be square and invertible on points with rational coordinates, and non-square (but still full rank) on the other points, in which case {e α } at each point would not even have a consistent number of elements, let alone be interpretable as a moving frame.…”

Accretion–ablation mechanics

“…1 This leads to the notion of the so-called ‘intermediate configuration’, which is usually defined only locally. 2 A more natural approach would be to follow Eckart [30] and Kondo [52,53] and use a Riemannian material manifold, which is a fixed manifold with a possibly time-dependent metric if the source of anealsticity is time dependent [54]. 3 For example, in the case of bodies undergoing bulk growth the reference configuration is a Riemannian manifold false(B,boldGtfalse), where B is a fixed 3-manifold with a time-dependent Riemannian metric boldGt [48].…”

Universality in Anisotropic Linear Anelasticity

“…In nonlinear anelasticity, in the notion first defined in [7], strain has an elastic and an anelastic part. In terms of deformation gradient it is written as F = F e F a , where F e and F a are the elastic and anelastic deformation tensors, respectively [17,37,39]. The hybrid German-English portmanteau term eigenstrain has its origin in the pioneering paper of Hans Reissner [31] (Eigenspannung means proper or self stress) and was further popularized by Mura [25,30].…”