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# Elastic-Plastic Deformation

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1. Elastic-Plastic Deformation

2. Simple Constitutive Relations

3. And Their Graphs

4. Flow Rule

5. Anisotropy

6. Yield Surfaces

7. Drucker postulate

8. Kinematic hardening Kinematic hardening is a monotonically growing & saturating function of strain and is a complex function of temperature

9. Isotropic Hardening Latent hardening is a monotonically growing and saturating function of strain and is a complex function of temperature

10. Example on the simple Beams • Let us consider the simple problem or two, which should give us general feeling what is the plasticity is about • We look at 1D problem • We look at non-hardening problem • We look at isothermal problem • Nothing is more illustrative as beam examples

11. Simple Beam • Given: E, l1, l2, P N1 P N2

12. Yield of Each Part Limiting or critical Force is: Compare

14. RESIDUAL STRESS

15. Elements of Shake Down Method Ec=E; Es=2E; P

16. Shake Down Limiting Load: Elastic solution: Let us apply the Force P1 to the system: Let us now unload the system: Let us apply the Force -P2 to the system:

17. Limiting Cycle OHGF – Elastic Regime ABGH and FGDE – system adjusts after first cycle; P1+P2<5Ny BCD- cyclic plastic deformations P2 B C A Out of Big-square- Failure G H D O F E P1

18. Slip Theory

19. Plasticity is Defined by Shear

20. Principal stress

21. Governing Equations

22. Slip Lines Equations

23. Hencky’s Equations

24. Hencky’s equations

25. Examples

26. Examples

27. More Examples

28. More Examples

29. Punch and Its Force