Elastic-Plastic Deformation. Simple Constitutive Relations

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, l1, l2, Py

Pl

5

2

l5

3

l5

2

l5

3

N1

N2

PN

PN

5

25

3

2

1

Yield of Each Part

APN

ANA

P

NN

y

yy

y

2

1

21

elastic is still issection -cross second The

;53

AP y2 y

y

A

AP

Limiting or critical Force is:

Compare AP yy 35

Displacements

EA

Pl

EA

lP

25

652

53

Ey

yy

5

2then PP If

EA

lAPF y 5

3

:beam theofpart Second or the

ASSUME NOW THAT APPLIED LOAD IS

AP Y6

11*

E

lANAN y

YY 26

521

THEN UNLOAD IT

RESIDUAL STRESS

PN

PN

unload

unload

5

25

3

2

1

AN

AAAN

yunload

yyyresidual

10

110

11

2

101

1

Elements of Shake Down Method

ysteelyycoppery _32

_ ;

P

Ec=E; Es=2E;

AAAA steelcopper ;3

Shake DownPNPN cs 5

352 ;

yycyieldyysyield NANNAN 23; 32

cryycys PPPNPNNN 11 ;;

ycrcrcs NPPNN 3;

153

152 ; PNPN unloadcunloads

152

153

1

152

PNPNPN

PNN

yyresidualc

yresiduals

15

225

32

152

252

2

PNPN

PNPN

yc

ys

y

yy

NPP

NPNP

521

152

252

Elastic solution:Limiting Load:

Let us apply the Force P1 to the system:

Let us now unload the system:

Let us apply the Force -P2 to the system:

Limiting Cycle

yN25

yN5

yN3

P1

P2

yN3

A B C

D

EF

GH

O

OHGF – Elastic Regime

ABGH and FGDE – system adjusts after first cycle; P1+P2<5Ny

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

Examples

More Examples

More Examples

Punch and Its Force