Strain gradient elasticity theory for antiplane shear cracks. Part I: Oscillatory displacements

Solutions of crack problems for a strain gradient elastic theory whose crack opening displacements are monotonic rather than oscillatory in profile are examined in this (portion of a two-part) paper. A new analytical solution having a simple form is also obtained through the introduction of parabolic cylindrical coordinates and expansion of the imposed displacement along the crack surfaces in terms of orthogonal polynomials. Uniqueness of solution is also discussed, as well as finite-difference approximations of the governing equations for use in numerical solutions of boundary value problems.