Gradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials – Part II: Crack Parallel to the Material Gradation

A mode III crack problem in a functionally graded material (FGM) modeled by anisotropic strain gradient elasticity theory is solved by the integral equation method. The gradient elasticity theory has two material characteristic lengths ` and `′, which are responsible for volumetric and surface strain-gradient terms, respectively. The governing differential equation of the problem is derived assuming that the shear modulus G is a function of x, i.e. G = G(x) = G0e, where G0 and β are material constants. A hypersingular integrodifferential equation is derived and discretized by means of the collocation method and a Chebyshev polynomial expansion. Numerical results are given in terms of the crack opening displacements, strains, and stresses with various combinations of the parameters `, `′, and β. Formulas for the stress intensity factors (SIFs), KIII , are derived and numerical results are provided. ∗ Department of Computer and Mathematical Sciences, University of Houston–Downtown, One Main Street, Houston, Texas 77002, U.S.A. †Department of Civil and Environmental Engineering, University of Illinois, 2209 Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801, U.S.A. ‡Department of Mathematics, University of California, Davis, CA 95616, U.S.A. 1