In this paper, the initial fracture in graded coatings under sliding rigid flat stamp contact loading is considered. This problem could be represented in the form of weakly singular Fredholm of the second kind. Shoukralla et al [1], made progress towards the application of some advanced single and double barycentric Lagrange interpolants and how to adapt them to be applicable to completely isolating the kernel singularity and find accurate solutions of the weakly singular Fredholm equations of the second kind which we will use in this paper for studying the effect of many factors such as the coefficient of friction, material inhomogeneity constants, and other length parameters on the critical stresses that may affect the break of the coating