A Note on the Objectivity (Rotational Invariance) of the Stored Energy Density in Continuum Physics

TL;DR

This paper examines the rotational invariance of stored energy density in continuum physics, deriving its general form in 2D and discussing reductions and extensions to 3D cases.

Contribution

It provides a simple derivation of the general form of energy density based on rotational invariance in 2D and discusses its extension to 3D.

Findings

Derived the general form of energy density in 2D based on rotational invariance.

Showed that the energy density form can be further simplified.

Discussed the extension of the form to three-dimensional space.

Abstract

This short note is concerned with the rotational invariance of the stored energy density in continuum physics as a scalar function of a few vectors. A simple derivation is presented for the determination of the general form of the energy density in the case of a two-dimensional space. It is also shown that the general form of the energy density so determined may be further reduced. The three-dimensional case is also discussed.