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, l 1, l 2, P P N1N1 N2N2

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

13. Displacements ASSUME NOW THAT APPLIED LOAD IS THEN UNLOAD IT

14. RESIDUAL STRESS

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

16. Shake Down Elastic solution: Limiting Load: Let us apply the Force P 1 to the system: Let us now unload the system: Let us apply the Force -P 2 to the system:

17. Limiting Cycle P1P1 P2P2 A BC D E F G H O OHGF – Elastic Regime ABGH and FGDE – system adjusts after first cycle; P 1 +P 2 <5N y BCD- cyclic plastic deformations Out of Big-square- Failure

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

27. More Examples

29. Punch and Its Force