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Analytical Modeling and Applications of Residual Stresses Induced by Shot Peening

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Analytical Modeling and Applications of Residual Stresses Induced by Shot Peening Analytical Modeling and Applications of Residual Stresses Induced by Shot Peening Julio Davis A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2012 Reading Committee: M. Ramulu, Chair A.S. Kobayashi D. Stuarti Program Authorized to Offer Degree: Mechanical Engineering University of Washington Abstract Analytical Modeling and Applications of Residual Stresses Induced by Shot Peening Julio Davis Chair of the Supervisory Committee: Professor M. Ramulu Mechanical Engineering 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 pre- diction 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, bend- ing, 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 ob- tained 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 resid- ual 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 ac- celeration term. Upon solving the ODE, expressions for the plastic strain and plastic strain rate as functions of time are immediately obtained. Comparisons with numer- ical 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 shake- down to calculate the infinite life fatigue limit of shot peened fatigue specimens under- going 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 cyclicly 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 shake- down 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. TABLE OF CONTENTS Page List of Figures . iv List of Tables . viii Chapter 1: Introduction . 5 1.1 A Brief History . 5 1.2 Practical Importance . 6 1.3 Most Influential Research . 7 1.4 Structure of the Thesis . 9 Chapter 2: Background and Literature Review . 11 2.1 Introduction . 11 2.2 Shot Peening Process and Parameters . 12 2.3 Analytical Modeling . 19 2.4 Numerical Simulations of Shot Peening Residual Stresses . 50 2.5 Experimental Findings . 56 2.6 Summary . 70 Chapter 3: Research Scope and Objectives . 72 3.1 Research Scope . 72 3.2 Goals and Objectives . 73 Chapter 4: Analytical Modeling of Shot Peening Residual Stresses by Evalu- ating the Elastic-Plastic Deviatoric Stresses Using J2-J3 Plasticity 75 4.1 Introduction . 75 4.2 Calculating the Elasto-Plastic Deviatoric Stress Tensor . 77 4.3 Iliushin's Plasticity Theory and the Elasto-Plastic Deviatoric Stresses 82 4.4 Elasto-Plastic Deviatoric Stresses From Incremental Plasticity . 84 i 4.5 Evaluation of the Elastic-Plastic Deviatoric Stresses Based on a Gen- eralized Isotropic Material . 86 4.6 Residual Stresses After Unloading . 88 4.7 Validation of Model . 91 4.8 Conclusions . 94 Chapter 5: A Semi-Analytical Model of Time Dependent Plastic Strains In- duced During Shot Peening . 96 5.1 Introduction . 96 5.2 Theoretical Development . 98 5.3 Numerical Simulations . 107 5.4 Validation of Model . 109 5.5 Conclusions . 110 Chapter 6: Strain Gradient Based Semi-Analytical Model of the Residual Stresses Induced by Shot Peening . 113 6.1 Introduction . 113 6.2 Theoretical Development . 115 6.3 Validation of Model . 120 6.4 Conclusions . 122 Chapter 7: Shakedown Prediction of Fatigue Life Extension After Residual Stress Relaxation via the Recovery Strain . 124 7.1 Introduction . 124 7.2 Shakedown, Creep and the Recovery Process . 125 7.3 Lower Bound Shakedown in the Presence of a Recovery Strain . 129 7.4 Lower Bound Shakedown . 130 7.5 Application of Shakedown at Room Temperature to Shot Peened Ti- 6Al-4V and Ti-5Al-5Mo-3Cr . 132 7.6 Application of Shakedown at Elevated Temperatures to Shot Peened 403 Stainless Steel . 134 7.7 Conclusion . 135 Chapter 8: Summary and Conclusion . 139 8.1 Summary . 139 ii 8.2 Conclusions . 142 8.3 Future Research Directions and Recommendations . 144 Appendix A: Equivalent Elasto-Plastic Stresses . 164 Appendix B: Mathematica Input and Output for Chapter 4 . 166 Appendix C: Mathematica Input and Output for Chapter 5 . 177 Appendix D: Derivation of the Plastic Strain as a Function of Depth . 180 Appendix E: Mathematica Input and Output for Chapter 6 . 181 iii LIST OF FIGURES Figure Number Page 2.1 Schematic diagram of shot peening air nozzle with relevant parameters 13 2.2 A typical intensity plot along with time to obtain saturation . 18 2.3 The arc height measurement process. (A) shows the peening of a test strip mounted in a fixture. (B) shows the bowing induced in the Almen strip as a result of the residual stresses in the metal. Test strips N, A, and C are also displayed along with dimensions. (C) depicts an Almen test strip mounted in an Almen gage . 19 2.4 Shot impacting a semi infinite surface with elastic plastic boundary separating the elastic and plastic zone . 21 2.5 a) Pressurized cavity model b) Radial and hoop stress in an elastic plastic sphere c) Residual hoop stress distribution d) Residual stress distribution with reversed yielding . 26 2.6 Elastic shakedown occurs with two intersecting elastic domains. 32 2.7 Plastic shakedown occurs when the elastic domains do not share a common intersection. 32 2.8 Residual stress in a finite structure is composed of three components. The residual stress in a semi infinite surface, normal stresses and bend- ing stresses. Note the residual stress in a semi infinite surface is not in equilibrium and completely lacks any balancing tensile stress. 34 2.9 Plot of the loading and unloading process [12] . 41 2.10 Graphical depiction of Nuebers theory relating the strain energy from the psuedo elastic stresses to the strain energy of the elastic plastic stresses . 45 2.11 Plastic strain, p(t), versus time obtained from [15]. 52 2.12 Influence of variable velocity on the equivalent plastic strain and resid- ual stress . 53 2.13 Plot of plastic strain rate versus time obtained from LS Dyna [16] . 54 2.14 Effect of shot peened and deep rolled treatments on fatigue life of austenitic steel AISI 304 [57] . 59 iv 2.15 SEM photographs of crack initiation below the surface [63] . 60 2.16 Compiled list of published research on cyclic relaxation of residual stresses for steel [5] . 63 2.17 Experimental residual stress versus depth measurements of shot peened Al 2024-T3 [75] . 67 2.18 Residual stress measurements for shot peening, laser shock peening and low plasticity burnishing of IN-718 are shown [5] . 70 4.1 Schematic of a single shot impacting a semi-infinite surface. Elastic- plastic boundary separates the confined plastic zone and the elastic domain. 77 4.2 Stress strain curve of the loading/unloading process for a single shot impact.
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