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 $ℓ$ and $ℓ′$⁠, 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., $G=G(x)=G0eβx$⁠, 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 $ℓ$⁠, $ℓ′$⁠, and $β$⁠. Formulas for the stress intensity factors, $KIII$⁠, are derived and numerical results are provided.