The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity

TL;DR

This paper investigates the asymptotic behavior of sharp notches in microstructured solids using dipolar gradient elasticity, revealing significant differences from classical fracture mechanics predictions.

Contribution

It introduces an asymptotic analysis of notches within dipolar gradient elasticity, incorporating microstructural length scales and boundary-layer techniques.

Findings

Significant deviation from classical fracture mechanics predictions.

Asymptotic fields characterized by an eigenvalue problem.

Analysis of crack and half-space as special cases.

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.