Elastic solids with strain-gradient elastic boundary surfaces

TL;DR

This paper develops a comprehensive strain-gradient elastic boundary surface model for brittle solids, ensuring bounded stresses and strains at crack tips across all fracture modes, improving upon previous theories.

Contribution

It formulates an exact general theory for 3D solids with strain-gradient surface elasticity, unifying and extending Steigmann-Ogden models, and applies it to mode-III fracture with proven bounded stress predictions.

Findings

Bounded stresses and strains at crack tips for all fracture modes.

Existence and uniqueness of classical solutions in the linearized theory.

Model consistency with linearization assumptions and resistance to geodesic distortion.

Abstract

Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle Green-elastic solid with Steigmann-Ogden surface elasticity yields a model that predicts bounded stresses and strains at the crack tips for plane-strain problems. However, singularities persist for anti-plane shear (mode-III fracture) under far-field loading, even when Steigmann-Ogden surface elasticity is incorporated. This work is motivated by obtaining a model of brittle fracture capable of predicting bounded stresses and strains for all modes of loading. We formulate an exact general theory of a three-dimensional solid containing a boundary surface with strain-gradient surface elasticity. For planar reference surfaces parameterized by flat coordinates, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface covariant derivative of the stretching, the theory reduces to that of Steigmann and Ogden. We discuss material symmetry using Murdoch and Cohen's extension of Noll's theory. We present a model small-strain surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory. Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate under far-field loading. We prove that there always exists a unique classical solution to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the linearization assumption in predicting finite stresses and strains at the crack tips.