metal plasticity - 东南大学土木工程学院...isotropic strain hardening 14 •plastic...
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Metal Plasticity
Outline
• Introduction(引言) • 1D plasticity theory(一维塑性) • Rough values of yield stress(常用工程材料屈服应力值) • 3D plasticity theory(三维塑性理论) • Decomposition of strain(应变分解) • Yield criterion(屈服判据) • Yield surface(屈服面) • Isotropic strain hardening(各向同性强化) • Kinematic strain hardening(运动强化) • Principal of maximum plastic resistance(最大塑阻原理) • Law of plastic flow(塑性流动定律) • Elastic unloading conditions(弹性卸载的条件)
2
• Linear elastic first; Plastic (permanent deformation); Unloading
follows linear curve; Relaxation; Creep; Bauschinger effect; Cyclic
hardening/softening; Rate, loading history and temperature
dependent
Introduction
3
• In 1930’s, Taylor and scientists experimentally measured the response of thin-
walled tubes under combined torsion, axial loading, and hydrostatic pressure.
• Hydrostatic stress has no effects on plastic deformation.
• Plastic behavior doesn’t induce volume change of a material.
• Plastic deformation is caused by shearing of atomic planes via propagation of a
type of lattice defects called dislocations.
• During plastic loading, the principal components of the plastic strain rate tensor
are parallel to the components of stress acting on the solid.
• Levy–Mises flow rule relates the principal plastic strain increment to the
principal stresses.
Introduction
4
p p p p p p
I II II III III I
I II II III III I
d d d d d d
• Decomposition of strain, yield criteria, strain hardening
rules, plastic flow rule, elastic unloading criterion
• We restrict attention to small deformations (≤10%).
Introduction
5
• Decomposition of strain:
1D Plasticity
6
, e p ed d d d Ed
• Yield criterion: p
Y
• Strain hardening rules govern the
functional dependence of yield stress
on plastic strain.
Rigid-perfectly plastic model
const.Y
1D Plasticity
7
Elastic-perfectly plastic model
const.Y
Linear hardening model
0 0
10 0
1
,
,
Y Y
pY
Y Y Y
EE
E E
1D Plasticity
8
Power law hardening model
0 0
0 0
0
,
1 ,
Y Y
Np
Y
Y Y
Y
E
• Here σY0, E, and N can be treated as fitting parameters to
experimental data. 0≤N<1 is called the hardening index.
• Tangent modulus of the stress-plastic strain curve
1
0
1
N
P
P Y
Edh EN
d
1D Plasticity
9
• Law of plastic flow:
, , 0e p
Y
d dd d d d
E h
0h consth 0
0
1
N
PY Y
Y
E
• Elastic unloading condition:
0d
Rough Values of Yield Stress
10
3D Plasticity Theory
11
• von Mises yield criterion: take the distortional part of
elastic strain energy as a criterion for the onset of plastic
deformation
2 2 21 3, + 0.
2 2 ij ij
p p p
ij I II II III III I Y Yf
• Decomposition of strain: , e p e
ij ij ij ij ijkl kld d d d C d
2
2 2 2
2
2 2 2
2
2 2 2
8
3 3 3 13
2 2 2 3
1 +
2
3 31 , 0 :
2 2
1 2 1+
3 3 3
ij ij
ij ij ij ij
ij
I II III I I II III
I II II III III I
I Y II III Y e Y
I II II III III I e
J
D
2
2
3ijJ
• von Mises yield function:
• Tresca yield criterion:
Yield Criterion
12
3
2 3 3
1
1 1: 2 2
YY
• Tresca yield function:
, max , , 0.p p
ij I II II III I III Yf
• Given the current stress σij applied to the material, we need to
determine current yield stress σY based on an effective plastic strain:
11 11 22 22 33 33
11 11 11 11 11 11 11
2 2,
3 3
21 , 0 :
3
2 1 12
3 2 2
p p p p p p p
ij ij ij ij
p p p p p p p
I Y II III
p p p p p p p
d d d d d d
D d d d d d d d
d d d d d d d
Yield Surface
13
• Geometric representation of von Mises
yield condition in stress space.
• If the state of stress falls within the
cylinder, the material is below yield and
responds elastically.
• If the state of stress lies on the surface of
the cylinder, the material yields and
deforms plastically.
• The stress state cannot lie outside the
cylinder; this would lead to an infinite
plastic strain.
2 2 2
i.e. : , , 02 2
1+
2
1 1 1 3 22 +
2 2 2 2 3
I II III
e I II II III III I
Y Y
R R
R R R
Isotropic Strain Hardening
14
• Plastic deformation typically causes the metal to strain
harden, as obviously seen in 1D.
• Strain hardening can be modeled by relating the size and
shape of the yield surface to plastic strain in some
appropriate way.
• The easiest way to model strain hardening is to make the
yield surface increase in size but remain the same shape.
2 2, ,
3 3
p p p p p p p p
ij ij ij ij Yd d d d d d
• Determine the updated radius via the
effective plastic strain, 1D hardening
functions, and new yield stress.
• An isotropic hardening law does not
account for the Bauschinger effect.
• Kinematic hardening drag the yield
surface in the direction of increasing
stress as you deform the material in
tension.
• This softens the material in compression,
however.
• So, this law can model cyclic plastic
deformation.
Kinematic Strain Hardening
0
3, 0.
2 ij ij
p p p
ij ij ij ij ij Yf
2
3
p
ij ijd cd • Linear kinematic hardening law:
15
• The von Mises yield surface is convex:
• The plastic strain increment is normal to the yield surface
for plastically stable solids.
Principal of Maximum Plastic Resistance
16
* 0p
ij ij ijd p p
ij ijd d f
• Experimental results (Levy-Mises theory) suggest that
plastic strains can be derived from the yield criterion.
Law of Plastic Flow for Isotropic Hardening
• Law of plastic flow for isotropic hardening:
3 3 1 3
2 2 2
ij ijp p p klij kl
ij Y Y Y
fd d d d
h
3, 0 0 , ,
2 ij ij
p p p p p
ij Y ij ij ijf f d d f
1 1 3
2
1 1 3 3 1 32
1
32 3
3
2
1
2
2 2
pq i
k
pq
p
j
pq p
l
k
q
q k
l k
p
ij p
ij
ijp p pYij ij ij ijp
ij ij ij Y
ip jq ij pq
ij ij
ij ij YY
f fd d
f f fd d d hd d d d
h h
f
1 1
3 3pq ip jq ik jk pq ip jq ij pq
• Some intermediate results
17
• Law of plastic flow for kinematic hardening:
Law of Plastic Flow for Kinematic Hardening
18
0
3, 0 0 , ,
2 ij ijij ij ij ij Y ij ij ij ij ij ijf f d d f
2
0 0
1
0
000
3 3
2 2
3,
3 3
2
2
2
ij ij ij ij
ij
jij i
ij
ij ij ij p
ij ij
Y Y
ijp
ij
Y
ij
ij
ij
ij
ijp
Y Y
ij
Y
d
d d cd
f
dc
fd
cd
d
0 0
0 0
1 1 3 32 .
2 2 23
2 2
2
3;
3 3
3
;2
2
ij ij
pq
kl
ij ijij ij
ij
ip jq pq
ij Y
kl
ijp
p p p
ij ij
Y
i
l
j
Y
Y
kl k
d cd c d c
f f
d
d
0 0 0
3 3 3
2 2 2
ij ijklij ijklp p p
ij kl
ij Y Y Y
fd d d d
c
• Some intermediate results
Elastic Unloading Conditions
19
p p p p p p
I II II III III I
I II II III III I
d d d d d d
• Both plastic flow laws are consistent with the Levy-Mises
theory, which is based on experimental observations.
• Elastic unloading conditions
0, 0.ij ij ij ij ijd d
• In both cases, the solid deforms
elastically (no plastic strain) if the
condition is satisfied.
• Summary of isotropically hardening elastic-plastic model
• Given:
Summary of 3D Plasticity Theory
20
, , ,p p
Y YE h d d
1 1 1
2 9
0, 0
3 1 3, 0
2 2
e p
ij
kl kl
ij ij
e
ij ij kk ij ij ij kk
p
e Y
p
ij ij p
e Y
Y Y
d d d
d d d d d
d
E E G K
d
h
• It will correctly predict the conditions necessary to initiate
yield under multiaxial loading.
• It will correctly predict the plastic strain rate under an
arbitrary multiaxial stress state.
• It can model accurately any uniaxial stress-strain curve.
• Summary of linear kinematically hardening model
• Given:
Summary of 3D Plasticity Theory
21
0, , ,YE c
0
0
0 0
1 1 1
2 9
30, 0
2
3 3 3, 0
2 2 2
ij
k
ij
ij
ij ij
l kl kl
e p
ij ij ij
e
ij ij kk ij ij ij kk
ij ij Y
p
ij
ij
ij ij Y
Y Y
d d
d
d
d d d d dE E G K
d
c
• This constitutive equation is used primarily to model
cyclic plastic deformation or plastic flow under
nonproportional loading (in which principal axes of stress
rotate significantly during plastic flow).