Second‐neighbor interactions in classical field theories: invariance of the relative power and covariance

Abstract

We propose a model‐building framework unifying those continuum models of condensed matter accounting for second‐neighbor interactions. A notion of material isomorphism justifies restrictions that we impose to changes in observers on the material manifold. In the presence of dissipation due to evolution of inhomogeneities, we extend the notion of relative power including hyperstresses and derive pertinent balance equations by exploiting an invariance axiom. The scheme presented here permits an extension of the multi‐field model‐building framework for complex materials to account at a gross scale for second‐neighbor microstructural interactions. Copyright © 2016 John Wiley & Sons, Ltd.

Exact Controllability of a Faedo–Galërkin Scheme for the Dynamics of Polymer Fluids

“…A will play a role through its first variation. To evaluate it, we construct smooth test vector fields (for the origin of these relations see [23,Lemma 1]). Then, we consider a varied density L(u…”

“…the effect of a hyperstress proportional to ∇∇ν, i.e., an effect induced by second-neighborhood interactions due to the mutual entanglement of polymers, which probability to occur grows as the density of molecules increases in the ground fluid [23]. We refer to a Faedo-Galërkin scheme for the previous balances.…”

Strain Gradient Effects in a Thermo-Elastic Continuum with Nano-Pores

“…An interpretation which is in line with an old remark of Toupin [13] on the possibility of viewing hyper-elastic materials of second grade is as Cosserat's continua whose microrotations are constrained (see also Refs. [14][15][16][17][18]).…”

Crack occurrence in bodies with gradient polyconvex energies

“…In solids length scale effects appear to be non-negligible for sufficiently small test specimens in various geometries and loading programs; in particular, when plasticity occurs in poly-crystalline materials, such effects are associated with grain size and accumulation of both randomly stored and geometrically necessary dislocations [17], [16], [24]. These higher-order effects influence possible nucleation and growth of cracks because the corresponding hyperstresses enter the expression of Hamilton-Eshelby's configurational stress [34]. Here we refer to this kind of influence.…”

“…We look at energy minimization and consider a variational description of crack nucleation in a body with second-gradient energy dependence. We do not refer to higher order theories in abstract sense (see [13] for a general setting, [9] for a physical explanations in terms of microstructural effects, [34] for a generalization of [13] to higher-order complex bodies), rather we consider a specific energy, in which we account for the gradient of surface variations and confinement effects due to the spatial variation of volumetric strain. Specifically, the energy we consider reads as F (y, V ; B) := B Ŵ ∇y(x), ∇[cof∇y(x)], ∇[det ∇y(x)] dx…”