# Stress theory for classical fields

## TL;DR

This paper develops a geometric framework for classical field theories and continuum mechanics, representing configurations as sections of fiber bundles and stresses as generalized force objects within this setting.

## Contribution

It introduces a unified geometric approach to classical fields and continuum mechanics, including a representation theorem for stresses and analysis of constitutive relations.

## Key findings

- Existence of a stress object representing forces, though non-unique
- Configuration space modeled as an infinite-dimensional manifold
- Role of constitutive relations analyzed within the geometric framework

## Abstract

Classical field theories together with the Lagrangian and Eulerian approaches to continuum mechanics are embraced under a geometric setting of a fiber bundle. The base manifold can be either the body manifold of continuum mechanics, space manifold, or space-time. Differentiable sections of the fiber bundle represent configurations of the system and the configuration space containing them is given the structure of an infinite dimensional manifold. Elements of the cotangent bundle of the configuration space are interpreted as generalized forces and a representation theorem implies that there exist a stress object representing forces, non-uniquely. The properties of stresses are studies as well as the role of constitutive relations in the present general setting.