An elastic–plastic analysis of spherical indentation: Constitutive equations for single-indentation unloading and development of plasticity due to repeated indentation

The unloading response of elastic–plastic materials indented by a rigid spherical indenter and the development of plasticity due to repeated indentation are examined in the light of finite element simulations performed for a wide range of effective elastic modulus-to-yield strength ratio E∗/Y and strain hardening exponent. Equations of the dimensionless residual indentation depth and plastic work vs maximum indentation depth are extracted from finite element solutions. A relation of the residual indentation depth is derived for isotropic strain hardening materials using the concept of effective strain. Constitutive contact equations are given for elastic–perfectly plastic and isotropic strain hardening materials subjected to indentation unloading. The evolution of plasticity is tracked in four consecutive indentation cycles. For elastic–perfectly plastic materials with high E∗/Y, only the first indentation cycle is inelastic, implying elastic shakedown behavior characterizes steady-state deformation, whereas for materials with low E∗/Y, plastic deformation continues to occur with increasing indentation cycles, indicating that plastic work is dissipated in all subsequent indentation cycles. Repeated indentation of elastic–perfectly plastic materials with low E∗/Y shows the existence of elastic deformation, shakedown, and ratcheting regions, with plastic strain accumulation occurring near the center of the contact region without the growth of the plastic zone, as opposed to continuous plastic deformation in the vicinity of the contact edge and plastic zone growth with increasing indentation cycles for kinematic strain hardening behavior, which is attributed to the shift of the yield surface.