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

TL;DR

This paper develops a metric-independent framework for higher-order continuum mechanics on manifolds, introducing hyper-stresses and hyper-tractions as advanced tools for analyzing boundary interactions.

Contribution

It introduces the concept of traction hyper-stresses in a metric-free setting, expanding the mathematical tools for higher-order continuum theories on manifolds.

Findings

Formulation of hyper-stresses acting on jets of velocities

Introduction of traction hyper-stresses inducing boundary hyper-tractions

Review of multilinear algebra relevant to these objects

Abstract

The paper considers the formulation of higher-order continuum mechanics on differentiable manifolds devoid of any metric or parallelism structure. For generalized velocities modeled as sections of some vector bundle, a variational kth order hyper-stress is an object that acts on jets of generalized velocities to produce power densities. The traction hyper-stress is introduced as an object that induces hyper-traction fields on the boundaries of subbodies. Additional aspects of multilinear algebra relevant to the analysis of these objects are reviewed.