Modeling and analysis of the nonlinear indentation problems of functionally graded elastic layered solids

In tribological and thermo-electro-mechanical applications with an appropriate gradation of properties, functionally graded materials provide superior performance for damage and wear resistance of contact systems and thermal and mechanical responses for other mechanical devices. This paper presents a nonlinear mixed variational-based computational model for the analysis of nonlinear plane indentation problems of elastic functionally graded layered elastic solids. The functionally graded elastic layer is perfectly bonded to the elastic substrate and is normally loaded by a cylindrical rigid punch. In addition to nonlinearity inherent to the contact problem, the developed model accounts for the geometric nonlinearity in the sense of larger displacements and rotations, but smaller strains. The modulus of elasticity as well as Poisson's ratio of the graded layer are varying along the thickness of the layer according to both exponential and power laws. On the contrary, in the conventional homogeneous finite element formulation the isoparametric graded finite element formulation is adopted on the level of Gaussian integration points to realize the gradation in material properties. The friction at the contact interface is modeled using Coulomb's friction law. Moreover, the inequality contact constraints are exactly satisfied throughout the contact interface by employing the Lagrange multiplier method, where the indentation force and the displacement fields are treated as independent variables. The obtained results of contact pressure, tangential contact stress, stress distribution, and indentation force show a significant influence of the material distribution, gradation law, gradient index, and thickness of the functionally graded layer.