Elastic-Plastic Deformation

Simple Constitutive Relations

And Their Graphs

Flow Rule

Anisotropy

Yield Surfaces

Drucker postulate

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

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

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

Simple Beam Given: E, l 1, l 2, P P N1N1 N2N2

Yield of Each Part Limiting or critical Force is: Compare

Displacements ASSUME NOW THAT APPLIED LOAD IS THEN UNLOAD IT

RESIDUAL STRESS

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

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:

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

Slip Theory

Plasticity is Defined by Shear

Principal stress

Governing Equations

Slip Lines Equations

Hencky’s Equations

Hencky’s equations

Examples

More Examples

Punch and Its Force