Energy based methods applied in mechanics by using the extended Noether's formalism

TL;DR

This paper demonstrates how the extended Noether's formalism, combined with tensor algebra, can derive field equations in damage mechanics for generalized continua, specifically strain gradient elasticity.

Contribution

It introduces a practical application of extended Noether's formalism in damage mechanics using tensor algebra, bridging a gap between abstract mathematics and applied continuum mechanics.

Findings

Derived field equations for strain gradient elasticity using the extended Noether's formalism.

Showed the formalism's effectiveness in damage mechanics for generalized continua.

Connected mathematical formalism with practical continuum mechanics models.

Abstract

Physical systems are modeled by field equations; these are coupled, partial differential equations in space and time. Field equations are often given by balance equations and constitutive equations, where the former are axiomatically given and the latter are thermodynamically derived. This approach is useful in thermomechanics and electromagnetism, yet challenges arise once we apply it in damage mechanics for generalized continua. For deriving governing equations, an alternative method is based on a variational framework known as the extended Noether's formalism. Its formal introduction relies on mathematical concepts limiting its use in applied mechanics as a field theory. In this work, we demonstrate the power of extended Noether's formalism by using tensor algebra and usual continuum mechanics nomenclature. We demonstrate derivation of field equations in damage mechanics for generalized continua, specifically in the case of strain gradient elasticity.