Balance laws versus the Principle of Virtual Work and the limited scope of Noll's theorem

TL;DR

This paper explores the fundamental relationship between balance laws and the Principle of Virtual Work in continuum mechanics, revealing limitations of Noll's theorem in higher-gradient continua and clarifying the role of contact interactions.

Contribution

It demonstrates that balance laws alone do not guarantee equilibrium in higher-gradient continua and clarifies the conditions under which Noll's theorem applies or fails.

Findings

Balance laws are insufficient for equilibrium in n≥2 gradient continua.

Noll's assumptions exclude curvature-dependent contact forces in general.

Curvature-dependent contact forces can exist without contradicting Noll's theorem.

Abstract

The relationship between balance laws and the Principle of Virtual Work as well as the structure of contact interactions in continua remain foundational issues in Mechanics. In this work, we revisit these issues within the distributional framework emphasized by Paul Germain. We show that while the Principle of Virtual Work implies balance of forces and moments for $n$th-gradient continua, balance laws alone do not suffice to characterize equilibrium for $n \geq 2$. We then reexamine Noll's classical theorem asserting that surface contact forces depend solely on the unit normal to the surface and identify the precise role of his additional assumptions, namely the absence of edge and wedge contact interactions and the boundedness of the surface contact density on the space of oriented surfaces. We demonstrate that these hypotheses fail for general higher-gradient continua. Consequently, the presence of curvature-dependent surface contact forces in such materials does not conflict with Noll's theorem and refutes his claim of their nonexistence.