Formulation of Deformation Stress Fields and Constitutive Equations in Rational Mechanics

TL;DR

This paper develops a comprehensive framework for deformation stress fields and constitutive equations in continuum mechanics, clarifying stress concepts and their geometric and energetic foundations for complex materials.

Contribution

It introduces a geometric and energetic formulation of deformation stresses and derives general constitutive equations using exterior differentials, enhancing understanding of material behavior.

Findings

Deformation stress concept is rigorously formulated.

Additive energy decomposition leads to defining static continuum.

General constitutive equations are derived using exterior calculus.

Abstract

In continuum mechanics, stress concept plays an essential role. For complicated materials, different stress concepts are used with ambiguity or different understanding. Geometrically, a material element is expressed by a closed region with arbitral shape. The internal region is acted by distance dependent force (internal body force), while the surface is acted by surface force. Further more, the element as a whole is in a physical background (exterior region) which is determined by the continuum where the element is embedded (external body force). Physically, the total energy can be additively decomposed as three parts: internal region energy, surface energy, and the background energy. However, as forces, they cannot be added directly. After formulating the general forms of physical fields, the deformation tensor is introduced to formulate the force variations caused by deformation. As the force variation is expressed by the deformation tensor, the deformation stress concept is well formulated. Furthermore, as a natural result, the additive decomposition gives out the definition of static continuum, which determines the material parameters in constitutive equations. Through using the exterior differentials, the constitutive equations are formulated in general form. Throughout the paper, when it is suitable, the related results are simplified to classical results for easier understanding.