Random dynamic response of crack in functionally graded materials layer for plane problem

In order to dynamically analyze a crack in a functionally graded materials layer for plane problem under dynamic loadings, a stochastic model is established for plane problem in which the material properties of the functionally graded materials layer vary randomly in the thickness direction, and the crack is parallel to the materials faces. A pair of dynamic loadings applied on the crack faces are treated as stationary stochastic processes of time. By dividing the functionally graded materials layer into several sub-layers, this problem is reduced to the analysis of laminated composites containing a crack, the material properties of each layer being random variables. A fundamental problem is constructed for the solution. Based on the use of Laplace and Fourier transforms, the boundary conditions are reduced to a set of singular integral equations, which can be solved by the Chebyshev polynomial expansions. The stress intensity factor history with its statistics is analytically derived. Numerical calculations are provided to show the effects of the related parameters. The results show that the increase of crack length, random field parameter β and crack location ratio h ₂/h leads to the increase of the mathematical expectation and standard deviation of normalized stress intensity factor history.