A periodic array of cracks in a functionally graded nonhomogeneous medium loaded under in-plane normal and shear

In this paper, the problem of a periodic array of parallel cracks in a functionally graded medium is investigated based on the theory of plane elasticity for a nonhomogeneous continuum. Both the in-plane normal (mode I) and shear (mode II) loading conditions are considered. It is assumed that the material nonhomogeneity is represented as the spatial variation of the shear modulus in the form of an exponential function along the direction of cracks, and the Poisson's ratio is constant. For each of the individual loading modes, a hypersingular integral equation is derived, in a separate but parallel manner in which the crack surface displacements are the unknown functions. As the basic parameters in applying the linear elastic fracture mechanics criteria, the mode I and mode II stress intensity factors are defined from the stress fields with the square-root singularity ahead of the crack tips. Numerical results are obtained to illustrate the variations of the stress intensity factors as a function of the crack periodicity for different values of the material nonhomogeneity. The crack surface displacements are also presented for the prescribed loading, material, and geometric combinations.