Representation of continuum equations in physical components for arbitrary curved surfaces

TL;DR

This paper reformulates continuum equations in terms of physical components on arbitrary curved surfaces, making them more accessible for practical physical problem modeling on complex geometries.

Contribution

It introduces a general formulation of continuum equations in physical components on differentiable manifolds, facilitating their application to curved surface problems.

Findings

The formulation simplifies solving continuum equations on curved surfaces.

Examples demonstrate the practical use of the new formulation.

The approach enhances modeling flexibility for complex geometries.

Abstract

Continuum equations are ubiquitous in physical modelling of elastic, viscous, and viscoelastic systems. The equations of continuum mechanics take nontrivial forms on curved surfaces. Although the curved surface formulation of the continuum equations are derived in many excellent references available in the literature, they are not readily usable for solving physical problems due to the covariant, contravariant or mixed nature of the stress and strain tensors in the equations. We present the continuum equations in terms of physical components in a general differentiable manifold. This general formulation of the continuum equations can be used readily for modelling physical problems on arbitrary curved surfaces. We demonstrate this with the help of some examples.