When a semi-infinite surface is shot peened the in-plane inelastic strains are zero because of compatibility. As a result, the in-plane residual stress and plastic strain are linearly related. These boundary conditions motivate a second order differential equation similar in mathematical form to a modified Kelvin solid model with a strain acceleration term. Solving the resulting equation gives a closed form expression for the plastic strain as a function of time. The plastic strain rate is solved by taking the derivative with respect to time. Comparisons with published finite element data show good agreement and are within 10% for most of the loading and unloading period.

Prager developed a yield criterion for a generalized isotropic material that is more capable of predicting material behavior that occurs from combined loading. A less studied topic is the effect of combined loading on the residual stress state. The purpose of the present work is to develop a theoretical model for a residual stress based on Prager's relation and establish how the predictions are influenced. Prager's yield criterion which includes the third deviatoric invariant, J3, provides a way to estimate the influence J3 has on the residual stress. The analyzed residual stress originates from shot peening because much experimental data exists to compare and validate the theoretical results.