Analysis of nanoindentation of functionally graded layered bodies with surface elasticity

• An iterative-incremental FEM is developed for analyzing FG nanoindentation process.
• Surface/interface energy effect is simulated using Gurtin–Murdoch elasticity theory.
• Graded FE technique is used to incorporate the material property gradient.
• Three different nanoindentation problems are fully analyzed and investigated.
• Both size and grading index have a significant effect on nanoindentation response.

Due to the high surface to volume ratio of the nanoscale domain, the surface stress effect is a major concern in the analysis of mechanical response of the nanomaterials and nanostructures. This paper presents a nonlinear variational numerical model for the analysis of frictionless indentation of a functionally graded layered homogeneous elastic bodies considering surface/interface stress effect in the context of contact mechanics. To accommodate the influence of surface/interface energy, the complete constitutive relation of the Gurtin–Murdoch elasticity theory is adopted, while the bulk material follows the classical theory of linear elasticity. The graded, nonhomogeneous property of the nanolayer follows a power law variation of the shear modulus, while a constant Poisson׳s ratio is considered. Utilizing the generalized isoparametric formulation, the graded finite element method is utilized to incorporate the material property gradient at the element scale. Throughout the indentation interface, the Lagrange multiplier approach is adopted to exactly satisfy the inequality contact constraints, where the indentation forces and the displacement field are treated as independent variables. Three nanoindentation problems of different natures are analyzed using the developed variational-based finite element computational model. The achieved results show the significance of the surface/interface energy, size-dependency, and the material distribution of functionally graded layer on the mechanical response of indentation processes.