The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity
Abstract
In this paper, we deal with the asymptotic problem of a body of infinite extent with a notch (re-entrant corner) under remotely applied plane-strain or anti-plane shear loadings. The problem is formulated within the framework of the Toupin-Mindlin theory of dipolar gradient elasticity. This generalized continuum theory is appropriate to model the response of materials with microstructure. A linear version of the theory results by considering a linear isotropic expression for the strain-energy density that depends on strain-gradient terms, in addition to the standard strain terms appearing in classical elasticity. Through this formulation, a microstructural material length is introduced, in addition to the standard Lam\'e constants . The faces of the notch are considered to be traction-free and a boundary-layer approach is followed. The boundary value problem is attacked with the asymptotic Knein-Williams technique. Our analysis leads to an eigenvalue problem, which, along with the restriction of a bounded strain energy, provides the asymptotic fields. The cases of a crack and a half-space are analyzed in detail as limit cases of the general notch (infinite wedge) problem. The results show significant departure from the predictions of the standard fracture mechanics.
Cite
@article{arxiv.2508.03595,
title = {The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity},
author = {P. A. Gourgiotis and M. D. Sifnaiou and H. G. Georgiadis},
journal= {arXiv preprint arXiv:2508.03595},
year = {2025}
}