Gradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials - Part I: Crack Perpendicular to the Material Gradation

Anisotropic strain gradient elasticity theory is applied to the solution of a mode III crack in a functionally graded material. The theory possesses two material characteristic lengths, ℓ and ℓ′, which describe the size scale effect resulting from the underlining microstructure, and are associated to volumetric and surface strain energy, respectively. The governing differential equation of the problem is derived assuming that the shear modulus is a function of the Cartesian coordinate y, i.e., G = G(y) = G ₀e^γγ, where G⁰ and γ are material constants. The crack boundary value problem is solved by means of Fourier transforms and the hypersingular integrodifferential equation method. The integral equation is discretized using the collocation method and a Chebyshev polynomial expansion. Formulas for stress intensity factors, K _III, are derived, and numerical results of K_III for various combinations of ℓ, ℓ′, and γ are provided. Finally, conclusions are inferred and potential extensions of this work are discussed.