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Elastic-Plastic Deformation

Elastic-Plastic Deformation

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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

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

  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