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Hierarchies of plasticity models
Prof. Guy Houlsby
Department of Engineering Science, University of Oxford
Geomechanics: monotonic loading, large numbers of cycles and granular flows
Reggio Calabria, Italy
22 June 2016
Reggio Calabria Plasticity models 2
Development of hierarchies of plasticity models • 1-D models
• “Series” forms • (“Parallel” forms)
• 1-D models to continua
Hierarchies of plasticity models
Reggio Calabria Plasticity models 3
Elasticity
ijijij f
klijijijklij f ,
ij
ijij
f
Reggio Calabria Plasticity models 4
Energy
Elasticity
2
2
Ef
E
f
E
E
1
Reggio Calabria Plasticity models 5
?
Plasticity derived from potentials
ijff
ijij
f
Hyper-elasticity
Energy
Reggio Calabria Plasticity models 6
Plasticity derived from potentials
ijff
ijij
f
Hyper-elasticity
Energy
ijijijdd ,,
ijij
f
ijij
df
0
Hyper-plasticity
ijijff , Energy
Dissipation
Reggio Calabria Plasticity models 7
Plasticity derived from potentials
ijff
ijij
f
Hyper-elasticity
Energy
ijijijdd ,,
ijij
f
ijij
df
0
Hyper-plasticity
ijijff , Energy
Dissipation
ijij
f
ijij
d
ijij
Reggio Calabria Plasticity models 8
• Scalar functions
• Differentials
• Obeys thermodynamics
…. but does it describe plasticity?
The story so far …
ijijff , ijijijdd ,,
ijij
f
ijij
f
ijij
d
Reggio Calabria Plasticity models 9
Derivation of a plasticity model
E
f
22
E
f
kd
E
f
Sk
d
kE
E
1k
k
Reggio Calabria Plasticity models 10
Modified signum function
x
y = |x|
x
y = S(x)
1
x
y = sgn(x)
1
Reggio Calabria Plasticity models 11
Derivation of a plasticity model
E
f
22
E
f
kd
E
f
Sk
d
E
Sk
E
k
E
k
,0
,0
,0
kE
E
1k
k
Reggio Calabria Plasticity models 12
A plasticity model with kinematic hardening
E
f
22
22
HEf
kd
HE
f
Sk
d
E
Hk S
E
Hk
E
Hk
,0
,0
,0
E
H
k
k 2k
Reggio Calabria Plasticity models 13
Energy
Elasticity
2
2
Ef
E
f
E
E
1
Reggio Calabria Plasticity models 14
Energy
Dissipation
Perfect plasticity
22
E
f
kd
kE
E
1
k
Reggio Calabria Plasticity models 15
Energy
Dissipation
22
22
HEf
Hardening plasticity
kd
E
H
k
E
1
k
Reggio Calabria Plasticity models 16
Energy
Dissipation
Multisurface plasticity
N
nn
nN
nn
HEf
1
2
2
1 22
N
nnnkd
1
E
N
kN k1
H1
k2
H2
2 1
k1
k2
HN
Reggio Calabria Plasticity models 17
Take sum to logical conclusion?
N
nn
nN
nn
HEf
1
22
122
N
nnnkd
1
N
dkd
0
ˆˆ
NN
dH
dE
f
0
2
2
0
ˆ2
ˆˆ
2
n
n
1 2 3 4 5 6 7 8 9 10
100
^
ˆˆ
Reggio Calabria Plasticity models 18
Continuous curves
Reggio Calabria Plasticity models 19
Energy
Dissipation
NN
dH
dE
f0
2
2
0
ˆ2
ˆˆ
2
Continuous plasticity
N
dkd0
ˆˆ
Reggio Calabria Plasticity models 20
Hierarchy of series models
k1
k2
E
1
k
E
1
k
E
1
2
2
Ef
22
E
f kd
kd 22
22
HEf
N
nn
nN
nn
HEf
1
2
2
1 22
N
nnnkd
1
NN
dH
dE
f0
2
2
0
ˆ2
ˆˆ
2
N
dkd0
ˆˆ
Reggio Calabria Plasticity models 21
“Parallel” forms
Reggio Calabria Plasticity models 22
Energy
Dissipation
Multisurface plasticity (parallel form)
N
nn
nFJf
1
22
22
N
nnncd
1
1
J
F1c1
F2
2
c2
FN
N
cN
Reggio Calabria Plasticity models 23
From 1-D models to continua
Reggio Calabria Plasticity models 24
2
1
2
1
1-D elastic
2
2
Ef
22
21
2
Ef
2-D elastic
21
22
21 2
2112
Ef
2
1
2
1
1
1
211
E
03
E
2E
1
E
22
211
1 ,
E
fE
f
E
f
Reggio Calabria Plasticity models 25
… to continua
ijijjjii
ijijjjii
GK
GGKf
2
32
21
22
21
2213
221
22
21
221
2122
212
1
122112
211
GGK
EE
Ef
23
22
21
23213
2 GGKf
Reggio Calabria Plasticity models 26
222
21 ijijjjii and
2211 , ijij
21 , ij
22
21 ijij
S
22
21
11
iS
klkl
ijijij
S
Reggio Calabria Plasticity models 27
Example: kinematic hardening plasticity
kd
22
22
HEf
ijijijijijijjjjjiiii HGK
f 2
ijijkd 2
E
k
H
Reggio Calabria Plasticity models 28
1-D model Continuum
f d f d
ijijH ...
N
n
nij
nijnH
1
...
2
2E
22
E
22
22
HE
N
nn
nN
nn
HE
1
22
122
NN
dH
dE
0
2
2
0
ˆ2
ˆˆ
2
k
k
N
nnnk
1
N
dk
0
ˆˆ
ijijk 2
ijijk 2
N
n
nij
nijnk
1
2
N
ijij dk
0
ˆˆ2ˆ
ijijjjii GK
2
ijijijijG ...
N
ijij dH
0
ˆˆˆ...
Reggio Calabria Plasticity models 29
Continuous models: from kernel function to stress-strain curve
NN
dH
dE
f
0
2
2
0
ˆ2
ˆˆ
2
N
dkd
0
ˆˆ N
kk
ˆ
kN
dH
NkE
0ˆ
k
N
kNHd
d
ˆ1
2
2
1
2
2ˆ
d
d
k
NkNH
Reggio Calabria Plasticity models 30
From stress strain curve to kernel function
1
2
2ˆ
d
d
k
NkNH
kE
k
2
2
2
11
k
k
Ek
k
E
k
d
d
3
2
2
2 2
k
k
Ed
d
3
32
122
ˆ
k
ENk
k
E
k
NkNH
3
12
ˆ
N
ENH
NN
dN
ENd
Ef
0
23
2
0
ˆ14
ˆ2
N
dN
kd
0
k
1
E
Reggio Calabria Plasticity models 31
• Loading on “backbone curve” = b()
• On any unloading from 1,1 the backbone curve is doubled and reversed:
• On any reloading from 2,2 the backbone curve is doubled:
• If on reloading (or unloading) a previous loading (or unloading) curve, or the backbone curve, is encountered then that curve is followed
It can be shown that models exhibiting “pure kinematic hardening”, including the continuous hyperplasticity case, satisfy the Masing Rules for 1-D loading and unloading
Masing Rules
2211 b
2222 b
Reggio Calabria Plasticity models 32
Masing Rules
b
Reggio Calabria Plasticity models 33
From 1-D to Continuum
N
ijij
N
ijij
N
ijijjjii dN
GNddG
Kf
0
3
00
ˆˆ12
ˆˆ2
N
ijij dN
kd
0
ˆˆ2
NN
dN
ENd
Ef
0
23
2
0
ˆ14
ˆ2
N
dN
kd
0
Reggio Calabria Plasticity models 34
A consistent framework
2
2
Ef
Plasticity
Hardening
Multisurface
Continuum
Continuous
N
ijij
N
ijij
N
ijijjjii dN
GNddG
Kf
0
3
00
ˆˆ12
ˆˆ2
N
ijij dN
kd
0
ˆˆ2
Reggio Calabria Plasticity models 35
• Development of simple plasticity models in a consistent framework
Elasticity → Plasticity → Hardening → Multisurface → Continuous
• “Series” and “parallel” hierarchies
• 1-D → continuum
35
Summary
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