On the axisymmetric nanoindentation of an exponentially graded coating–substrate structure with both surface and interface effects

This paper investigates the axisymmetric nanocontact of a gradient nanostructure, comprising an exponentially graded coating and a homogeneous half-space, under a rigid spherical indenter. The Steigmann–Ogden surface elastic theory is utilized to model the surface effects at the upper surface of the coating and the interface effects at the coating–substrate boundary. We derive nonclassical boundary conditions and, in conjunction with the displacement continuity across the interface, construct the integral equation describing the nanocontact using the Hankel integral transform. Along with the force equilibrium condition, this equation is discretized and collocated with Gauss–Chebyshev quadratures. An iterative algorithm is developed to solve the resulting algebraic system for contact pressure and radius of the contact circle. Validation against existing literature confirms the accuracy and reliability of the proposed solution method and numerical algorithm. Extensive parametric studies reveal the significant influence of surface and interface effects, the inhomogeneity index of the graded coating, and the indenter radius on nanocontact behavior. The surface effects, characterized by a reduction in contact radius, maximum stress, and subsidence, demonstrate a pronounced size dependency. Notably, soft coatings exhibit a more substantial impact, and the reduction of indenter radius or external load further amplifies these effects. The interface effects, though less pronounced than surface effects, also play a crucial role in affecting contact properties, particularly for hard graded coatings. These findings underscore the importance of considering both surface and interface effects in the design and analysis of nanostructured materials.