The Mathematical and Physical Inconsistencies of Strain-Gradient Theories

In this paper, we examine the inherent mathematical and physical inconsistencies of strain-gradient theories. It is shown that strain gradients are not proper measures of deformation, because 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 equilibrium or motion of infinitesimal elements of matter properly. In the first strain-gradient theory (F-SGT), there are nine explicit 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. This shows that F-SGT is not an extension of rigid-body mechanics, which is, therefore, recovered in the absence of deformation. Moreover, F-SGT would require the existence of six additional fictitious symmetries of space-time according to Noether’s theorem, and a complete revision of the well-established concept of static indeterminacy in introductory mechanics. The inconsistencies of F-SGT also manifest themselves in the appearance of strains as boundary conditions.