Plasticity Models 1

Phase2 Theory

Plasticity Models In this document, the expressions for yield surface and plastic potential are based on a compression positive sign convention.

Mohr-Coulomb

Yield (failure) surface:

1 21sin cos sin sin cos

3 3sIf J c

Plastic potential flow surface:

1 21sin cos sin sin cos

3 3s dilIg J c

For associated flow use dil

Where:

3211 I

2222222 21

zxyzxyzyx sssJ

2223 2 xyzxzyyzxzxyzxyzyx ssssssJ

113i i

s I

1 33

22

1 3 3sin3 2

J

J

Plasticity Models 2

Phase2 Theory

Hoek-Brown

Yield (failure) surface:

0.5

1 222cos sin 3 cos3 3s c c

I Jmf J s

Plastic potential flow surface:

0.5

1 222cos sin 3 cos3 3

dils c

c

m I Jg J s

For associated flow use mmdil , recommended 4mmdil

Generalized Hoek-Brown

Yield (failure) surface:

1 222cos sin 3 cos3 3

a

s cc

I Jmf J s

Plastic potential flow surface:

1 222cos sin 3 cos3 3

a

dils c

c

m I Jg J s

For associated flow use bbdil mm , recommended: 4b

bdilmm

Plasticity Models 3

Phase2 Theory

Drucker-Prager

Yield (failure) surface:

12 3s

If J q k

Plastic potential flow surface:

12 3s dil

Ig J q k

For associated flow use qq dil

Cam Clay

Material constants:

M, , , (or N), constant G or (Poissons ratio)

vpK

If G is provided then 3 22 6K GG K

If is provided then (3 6 )2(1 )

KG

User supplied input parameters:

op , initp , initq

( )( )

NN

Determine the initial specific volume, vinit, and the equation for the initial Swelling Line (Unloading-Reloading Line):

lno ov N p

Plasticity Models 4

Phase2 Theory

lninit

inito

o

pv vp

lno ov v p

Yield Function:

0

ln 0pF q Mpp

1 2 313

p

2 2 21 2 2 3 3 112

q

Plastic Potential:

0

ln 0pP q Mpp

Critical State Line (CSL):

q Mp

Intermediate parameters:

qp

Hardening/softening rule (increment/decrement in op once strength envelope is exceeded):

p

o vo

p vp

Elastic strains:

Plasticity Models 5

Phase2 Theory

13

ev

es

pvp

qG

Set of elastoplastic constitutive equations considering an explicit scheme over a finite time increment (step n to 1n ),:

0

ln 0pF q Mpp

Yield function and plastic potential

0dF The consistency condition

1e e

n n n D Hookes law

e p The additivity postulate

p

n

Fd

The flow rule

0 01 expp

n vn n

p p

Hardening/softening rule

The derivative of the yield function with respect to the stress tensor/vector:

F F p F qp q

; 1nnn n

qF FMp p q

The set of constitutive equations presented above can be simplified in one nonlinear equation with one independent variable, i.e. d , which can be calculated as

0

0

e

epv

F

dpF F F F

p p

D

D

Plasticity Models 6

Phase2 Theory

Modified Cam Clay

Material constants:

M, , , (or N), constant G or (Poissons ratio)

vpK

If G is provided then 3 22 6K GG K

If is provided then (3 6 )2(1 )

KG

User supplied input parameters:

op , initp , initq

( ) ln 2( ) ln 2

NN

Determine the initial specific volume, vinit, and the equation for the initial Swelling Line (Unloading-Reloading Line):

lno ov N p

lninit

inito

o

pv vp

lno ov v p

Yield Function:

22

2 1 0opqF M

p p

, where

1 2 313

p

2 2 21 2 2 3 3 112

q

Plasticity Models 7

Phase2 Theory

Plastic Potential Function:

22

2 1 0opqP M

p p

Critical State Line (CSL):

q Mp

Hardening/softening rule (increment/decrement in op once strength envelope is exceeded):

p

o vo

p vp

Nonlinear elastic behavior:

vpK

e ev vvpp K

Integrating the above equation over a finite time increment (step n to 1n ), assuming that the change in specific volume is insignificant, results in the following incremental equation

1 expen

n n vv

p p

Assuming an average bulk modulus from step n to 1n :

1e

n n vp p p K

exp 1en n vev

p vK

In case the shear modulus is not constant, i.e. the poisons ration is constant, the shear modulus can be calculated as

(3 6 )2(1 )

G rK K

Plasticity Models 8

Phase2 Theory

Set of elastoplastic constitutive equations, in terms of stress invariants, considering an implicit scheme over a finite time increment (step n to 1n ),:

2

11 1 1 02 1

0nn n n nqF p p pM

Yield function and plastic potential

1

1 3

en n v

en n q

p p Kq q G

Hookes law

e pv v v

e pq q q

The additivity postulate

1 0 11 1

12

1 1

; 2

2;

pv n n

n n

p nq

n n

F Fd p pp p

qF Fdq q M

The flow rule

0 01 expp

n vn np p

Hardening/softening rule

The set of constitutive equations presented above can be simplified in a nonlinear set of equations with 4 equations and 4 unknowns

( 1 1 0 1, , ,n n np q p d ) 4 independent unknowns

1 1 1 0 1

12 1 2

21

3 1 1 02 1

4 0 0 1 01 1

exp 2 0

23 0

0

exp 2 0

nn n v n n

nn n q

nn n n

nnn n n

g p p d p p

qg q q G dM

qg p p p

M

g p p d p p

4 independent nonlinear equations

Plasticity Models 9

Phase2 Theory

The integration/solution algorithm for a material point starts from an initial state of stress and hardening parameters ( 0, , ,n n nnp q p ) with the introduction of the increment of strains ( v , q ).

The solution algorithm is based on a Newton iterative technique and follows these steps, sequentially:

1- Initializing the unknown variables:

1n np p , 1n nq q , 0 01n np p , 0d

2- Calculate the ig functions, and check if they are all close enough to zero. If yes terminate the process, if not go to step 3 to modify the current values of unknowns.

3- Update the unknowns by solving the linear system of equations presented below, and then go to step2. The term ,i jg represents the partial derivative of the function

ig with respect to the j th variable.

11,1 1,2 1,3 1,4 1

12,1 2,2 2,3 2,4 2

03,1 3,2 3,3 3,4 31

4,1 4,2 4,3 4,4 41

n

n

n

n

pg g g g gqg g g g gpg g g g g

g g g g gdp

After finding the updated state of stress and state variables, i.e. 1 1 0 1, , ,n n np q p , the transformation to general state of stress, , is straightforward.