Higher-order indentation model based on mixture unified gradient with surface elasticity: A theoretical study

This study proposes a higher-order framework for half-space indentation based on mixture unified gradient theory (MUGT) with surface elasticity (SE). MUGT, a well-posed theory that captures both nonlocal and strain gradient properties, is essential for understanding size effects in nano/micro-scale materials and structures. However, indentation problems considering MUGT remain unexplored. We develop efficient analytical and numerical methods to address the problem. In the 3D context, the stress components are analytically determined using 2D Fourier transform applied to constitutive relations that incorporate stress gradient elasticity. Regarding the contact pressure, the problem results in integral equations whose kernel is challenging to obtain explicitly. These are numerically solved using the sum of independent functions, rather than relying on discrete point values as done in previous studies on singular integral equations. Our findings demonstrate that stress gradient elasticity leads to greater surface vertical displacement, whereas strain gradient and surface elasticity result in smaller surface vertical displacement, highlighting the softening and hardening behaviors respectively. Drastically different contact pressure distributions and surface vertical displacements can be obtained compared to existing theories. Particularly, both hardening and softening of size-dependent indentation hardness are intrinsically captured, aligning with available experimental observations. These behaviors, however, are challenging to simultaneously reflect in existing indentation theories due to the exclusion of stress gradient elasticity. The study enhances the understanding of contact mechanics and is of practically significance for nano/micro-scale materials and structures.