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
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