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 modeled by anisotropic strain-gradient elasticity theory is solved by the integral equation method. The gradient elasticity theory has two material characteristic lengths ℓand ℓ, 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)=G₀^eβx, where G₀ and βare material constants. A hypersingular integro-differential 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 ℓ, ℓ, and β. Formulas for the stress intensity factors, K _III, are derived and numerical results are provided.