The nonlocal theory solution of a mode-i crack in functionally graded materials subjected to harmonic stress waves

In this paper, the dynamic behavior of a finite crack in functionally graded materials subjected to harmonic stress waves is investigated by means of nonlocal theory. The traditional concepts of nonlocal theory are extended to solve the dynamic fracture problem of functionally graded materials. To overcome mathematical difficulties, a one-dimensional nonlocal kernel is used instead of a twodimensional one for the dynamic problem to obtain the stress fields near the crack tips. To make the analysis tractable, it is assumed that the shear modulus and the material density vary exponentially and vertically with respect to the crack. Using the Fourier transform and defining the jumps of the displacements across the crack surfaces as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Unlike classical elasticity solutions, it is found that no stress singularities are present near crack tips. Numerical examples are provided to show the effects of the crack length, the parameter describing the functionally graded materials, the frequency of the incident waves, the lattice parameter of the materials and the material constants upon the dynamic stress fields near crack tips.