Hyper-Stresses in $k$-Jet Field Theories

TL;DR

This paper explores three distinct mathematical stress objects in high-order continuum mechanics and classical field theories, extending the traditional concept of stress tensors to more complex geometric settings.

Contribution

It introduces and analyzes variational, traction, and non-holonomic hyper-stresses, clarifying their properties and interrelations in $k$-jet field theories.

Findings

Identifies three types of hyper-stresses in high-order theories

Establishes relationships between variational, traction, and non-holonomic stresses

Provides a mathematical framework for stress analysis in advanced continuum mechanics

Abstract

For high-order continuum mechanics and classical field theories configurations are modeled as sections of general fiber bundles and generalized velocities are modeled as variations thereof. Smooth stress fields are considered and it is shown that three distinct mathematical stress objects play the roles of the traditional stress tensor of continuum mechanics in Euclidean spaces. These objects are referred to as the variational hyper-stress, the traction hyper-stress and the non-holonomic stress. The properties of these three stress objects and the relations between them are studied.