Geometric analysis of hyper-stresses

Abstract

A. A geometric analysis of high order stresses in continuum mechanics is presented. Virtual velocity fields take their values in a vector bundle W over the n-dimensional space manifold. A stress field of order k is represented mathematically by an n-form valued in the dual of the vector bundle of k-jets of W . While only limited analysis can be performed on high order stresses as such, they may be represented by non-holonomic hyper-stresses, n-forms valued in the duals of iterated jet bundles. For non-holonomic hyperstresses, the analysis that applies to first order stresses may be iterated. In order to determine a unique value for the tangent surface stress field on the boundary of a body and the corresponding edge interactions, additional geometric structure should be specified, that of a vector field transversal to the boundary.2000 Mathematics Subject Classification. 74A10; 53Z05; 58A32 .

Separately Global Solutions to Rate-Independent Processes in Large-Strain Inelasticity

“…Remark 1.1 (Hyper-loading). Second grade materials, whose stored energy density depends also on ∇ 2 y, may model various physical phenomena, for example the flow of Korteweg fluids (depending on the Eulerian gradient of the Eulerian density field), the deformation of woven fabrics [30,14], phase transitions [3,1,52], and multisymplectic field theory [31]; see also and [55,56,25,4,44,35,13,19,51,50] for further works on non-simple continua, the list definitely not being exhaustive. We point out that for the applications described in this paper it is not necessary to incorporate the hyper-loading, an additional terms representing conservative forces, typically in a form of an edge traction or the so-called couple-stress or double force acting on the boundary (see [12,11,22,23,45,41]), for no such physical phenomena are expected to arise in our intended applications.…”

On jets, almost symmetric tensors, and traction hyper-stresses

“…Variational hyper-stresses. In accordance with the variational approach to higher order continuum mechanics, we view variational hyper-stresses as fields that act on the derivatives of the virtual velocities to produce power densities (see [Seg17]). Thus, in the current setting, a variational hyper-stress object should act linearly on the k-jet of a field w to produce a density on X.…”

“…In particular, we have proposed the mathematical object that we believe should play the role of traction hyper-stress. While for the case k = 1, the traction stress has been defined in [Seg02,Seg13], no natural analogous definition has been presented in [Seg17]. In fact, in [Seg17] some of the difficulties have been indicated and subsequently avoided by using iterated jet bundles (the jet bundle of the jet bundle) and the corresponding dual objects rather than analyzing directly higher jet bundles and hyper-stresses.…”

“…As a generalization of the standard introduction of hyper-stresses in higherorder continuum mechanics, the kth order hyper-stress object, the variational hyper-stress, is dual to k-jets of sections of the vector bundle (see [Seg17]). Continuum mechanics on manifolds differs from standard formulations in Euclidean spaces in the following significant sense.…”

“…The variational stress acts on the jets of generalized velocity fields to produce power, while the traction stress induces the traction fields on the boundaries of subbodies. While the variational hyper-stress fields have been considered in [Seg86,Seg17], we propose here a suitable candidate for the role of traction hyper-stress.…”