# Geometric Analysis of Hyper-Stresses

## TL;DR

This paper presents a geometric framework for analyzing high order stresses in continuum mechanics using jet bundles and differential forms, enabling a deeper understanding of boundary interactions and hyper-stresses.

## Contribution

It introduces a geometric approach to high order stresses using non-holonomic hyper-stresses and jet bundle theory, extending classical stress analysis.

## Key findings

- Representation of high order stresses as n-forms in dual jet bundles
- Iterative analysis of non-holonomic hyper-stresses
- Requirement of additional geometric structure for boundary stress determination

## Abstract

A geometric analysis of high order stresses in continuum mechanics is presented. Virtual velocity fields take their values in a vector bundle \vbts 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 \vbts. 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 hyper-stresses, 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.