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