The complex response of metals to the shot peening process is described by many fields of study including elasticity, plasticity, contact mechanics, and fatigue. This dissertation consists of four unique contributions to the field of shot peening. All are based on the aforementioned subjects.;The first contribution is an analytical model of the residual stresses based on J2-J3 incremental plasticity. Utilizing plasticity requires a properly chosen yield criteria so that yielding at a given stress state of a particular material type can be predicted. Yielding of ductile metals can be accurately predicted with the Tresca and Von Mises yield criteria if loading is simple, but if a material undergoes combined loading prediction of yielding requires an alternate criteria. Edelman and Drucker formulated alternate yield criteria for materials undergoing combined loading using the second and third deviatoric stress invariants, J2 and J3. The residual stress is determined from the yield state and is thus influenced by the third invariant. From Hertzian theory a triaxial stress state forms directly below a single shot and is defined in terms of three principal stresses. The stress state becomes substantially more complex when a surface is repeatedly bombarded with shots. The material experiences shear, bending, and axial stresses simultaneously along with the induced residual stress. The state of stress easily falls under the category of combined loading. The residual stress is calculated from both the elastic and elastic-plastic deviatoric stress. Incremental plasticity is used to calculate the elastic-plastic deviatoric stress that depends on both invariants J2 and J3. Better predictions of experimental residual stress data are obtained by incorporating the new form of the elastic-plastic deviatoric stress into Li's theoretical framework of the residual stress.;The second contribution is a time dependent model of the plastic strain and residual stress. A general dynamic equation of the residual displacements in the workpiece is introduced. The equation is then expressed in terms of the inelastic strain. The imposed boundary conditions lead to an elegant second order differential equation in which the plastic strain acceleration is a natural result. The time dependent model is similar in mathematical form to the Kelvin Solid model, aside from the strain acceleration term. Upon solving the ODE, expressions for the plastic strain and plastic strain rate as functions of time are immediately obtained. Comparisons with numerical results are within 10%. To the author's knowledge this approach has never been published.;The third contribution is an extension of the second. Parameterizing the plastic strain leads to a simple transformation of variables so that the temporal derivatives can be written in terms of spatial gradients. Solving the second order ODE gives a solution for the plastic strain and hence residual stress (via Hooke's law) as a function of depth, z. Comparisons made with two aluminum alloys, 7050-T7452 and 7075-T7351, are in good agreement and within 10%.;The fourth and final contribution of the dissertation applies the theory of shakedown to calculate the infinite life fatigue limit of shot peened fatigue specimens undergoing high temperature fatigue. The structure is said to shakedown when the material will respond either as perfectly elastic or with closed cycles of plastic strain (elastic shakedown and plastic shakedown respectively). Tirosh uses shakedown to predict the infinite life fatigue limit for shot peened fatigue specimens being cyclically loaded at room temperature. The main complication that occurs during high temperature fatigue is residual stress relaxation. At high temperatures the magnitude of the shot peening residual stress will decrease which leads to diminishing fatigue benefits. A strain quantity known as the recovery strain is directly responsible for the relaxation of the shot peened residual stress. We incorporate the recovery strain into the shakedown model and prove that shakedown is still valid even when the residual stress is time dependent because of relaxation. The reduction of the infinite life fatigue limit is calculated for shot peened Ti-6-4, Ti-5-5-3, and 403 stainless steel.
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