Strain gradient elasticity for antiplane shear cracks: A hypersingular integrodifferential equation approach

We consider Casal's strain gradient elasticity with two material lengths l, l′ associated with volumetric and surface energies, respectively. For a Mode III finite crack we formulate a hypersingular integrodifferential equation for the crack slope supplemented with the natural cracktip conditions. The full-field solution is then expressed in terms of the crack profile and the Green function, which is obtained explicitly. For l′ = 0, we obtain a closed form solution for the crack profile. The case of small l′ is shown to be a regular perturbation. The question of convergence, as l, l′ → 0, is studied in detail both analytically and numerically.