An assessment of higher gradient theories from a continuum mechanics perspective

TL;DR

This paper critically examines higher gradient theories in continuum mechanics, revealing fundamental physical and mathematical inconsistencies, and argues they do not properly describe deformation or internal stresses.

Contribution

It provides a rigorous assessment showing that higher gradient theories are physically and mathematically flawed, challenging their validity in continuum mechanics.

Findings

Higher gradient measures are not proper deformation metrics.

Governing equations violate Newton's laws and angular momentum.

Higher gradient theories are inconsistent and physically invalid.

Abstract

In this paper, we investigate the inherent physical and mathematical character of higher gradient theories, in which the strain or distortion gradients are considered as the fundamental measures of deformation. Contrary to common belief, the first or higher strain or distortion gradients are not proper measures of deformation. Consequently, their corresponding energetically conjugate stresses are non-physical and cannot represent the state of internal stresses in the continuum. Furthermore, the governing equations in these theories do not describe the motion of infinitesimal elements of matter consistently. For example, in first strain gradient theory, there are nine governing equations of motion for infinitesimal elements of matter at each point; three force equations, and six unsubstantiated artificial moment equations that violate Newton's third law of action and reaction and the angular momentum theorem. This shows that the first strain gradient theory (F-SGT) is not an extension of rigid body mechanics, which then is not recovered in the absence of deformation. The inconsistencies of F-SGT and other higher gradient theories also manifest themselves in the appearance of strains, distortions or their gradients as boundary conditions and the requirement for many material coefficients in the constitutive relations.