-
Computational Solid Mechanics Computational Plasticity
C. Agelet de Saracibar ETS Ingenieros de Caminos, Canales y
Puertos, Universidad Politcnica de Catalua (UPC), Barcelona,
Spain
International Center for Numerical Methods in Engineering
(CIMNE), Barcelona, Spain
Chapter 2. 1D Plasticity Algorithms
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Contents 1. Introduction 2. 1D Rate independent plasticity
models
1. Return mapping algorithm 2. Consistent elastoplastic tangent
modulus 3. Step by step algorithm 4. Nonlinear isotropic
hardening
3. 1D Rate dependent plasticity models 1. Return mapping
algorithm 2. Consistent elastoplastic tangent modulus 3. Step by
step algorithm 4. Nonlinear isotropic hardening
4. 1D Computational plasticity assignment
1D Plasticity Algorithms > Contents
Contents
April 1, 2015 Carlos Agelet de Saracibar 2
-
Contents 1. Introduction 2. 1D Rate independent plasticity
models
1. Return mapping algorithm 2. Consistent elastoplastic tangent
modulus 3. Step by step algorithm 4. Nonlinear isotropic
hardening
3. 1D Rate dependent plasticity models 1. Return mapping
algorithm 2. Consistent elastoplastic tangent modulus 3. Step by
step algorithm 4. Nonlinear isotropic hardening
4. 1D Computational plasticity assignment
1D Plasticity Algorithms > Contents
Contents
April 1, 2015 Carlos Agelet de Saracibar 3
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1D Plasticity Algorithms > Introduction
Time integration algorithm
April 1, 2015 Carlos Agelet de Saracibar 4
pnE 1
pn+E
1n+E
Time integration algorithm
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Contents 1. Introduction 2. 1D Rate independent plasticity
models
1. Return mapping algorithm 2. Consistent elastoplastic tangent
modulus 3. Step by step algorithm 4. Nonlinear isotropic
hardening
3. 1D Rate dependent plasticity models 1. Return mapping
algorithm 2. Consistent elastoplastic tangent modulus 3. Step by
step algorithm 4. Nonlinear isotropic hardening
4. 1D Computational plasticity assignment
1D Plasticity Algorithms > Contents
Contents
April 1, 2015 Carlos Agelet de Saracibar 5
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1. Additive split of strains
2. Constitutive equations
3. Associative plastic flow rule
4. Yield function
5. Kuhn-Tucker loading/unloading conditions
1D Plasticity Algorithms > Rate Independent Plasticity
Models
1D Rate independent plasticity model
April 1, 2015 Carlos Agelet de Saracibar 6
( ) { } { }, , , diag , ,e e e q q E K H = = =ES E CE S: , C
:=
{ } { } { }: : ,0,0 , : , , , : , ,e p p p e e = + = = = E E E ,
E E E
( )p f= SE S
( ) Yf q q = +S
( ) ( )0, 0, 0f f =S S
-
Associative plastic flow rule: plastic strains at time n+1
Using a Backward-Euler (BE) time integration scheme yields,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 7
( )p f= SE S
( )11 1 1n
p pn n n nf ++ + += + SE E S
( )
( )
1 1 1 1
1 1
1 1 1 1
sgn
sgn
p pn n n n n
n n n
n n n n n
q
q
+ + + +
+ +
+ + + +
= + = + =
-
Constitutive equations: stress state at time n+1
The time-discrete constituve equation at time n+1 takes the
form,
Substituting the plastic strain variables at time n+1
yields,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 8
( ) ( ) { }, diag , ,e e e p E K H= = = ES E CE C E E C :=
( )1 1 1 1e pn n n n+ + + += = S CE C E E
( )( )( ) ( )
1
1
1 1 1 1
1 1 1
n
n
pn n n n n
pn n n n
f
f
+
+
+ + + +
+ + +
=
=
S
S
S C E E S
C E E C S
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Trial state at time n+1 The trial state at time n+1 is defined
by freezing the plastic behaviour at the time step, yielding
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 9
( ) ( )( )
,1
, ,1 1 1
, ,1 1 1 1 1
1 1
:
:
:
:
p trial pn ne trial p trialn n n
trial e trial p trial pn n n n n n
trial trialn nf f
+
+ + +
+ + + + +
+ +
=
=
= = =
=
E E
E E E
S CE C E E C E E
S
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Return mapping algorithm The return mapping algorithm takes the
form,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 10
( )11 1 1 1n
trialn n n nf ++ + + += SS S C S
( )
( )
1 1 1 1 1
1 1 1
1 1 1 1 1
sgn
:
: sgn
trialn n n n n
trialn n n
trialn n n n n
E q
q q Kq q H q
+ + + + +
+ + +
+ + + + +
= = = +
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1. Additive split of strains at time n+1
2. Stresses at time n+1. Return mapping algorithm
3. Plastic internal variables at time n+1
4. Yield function at time n+1
5. Kuhn-Tucker loading/unloading conditions at time n+1
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 11
( )1 1 1 1 1:n n n n Y nf f q q + + + + += = +S
1 1 1 10, 0, 0n n n nf f + + + + =
( )11 1 1 1n
trialn n n nf ++ + + += SS S C S
1 1 1:e p
n n n+ + += +E E E
( )11 1 1n
p pn n n nf ++ + += + SE E S
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Theorem 1. Elastic step/plastic step If the yield function is
convex and the constitutive matrix is definite-positive, the
following condition holds,
and Kuhn-Tucker loading/unloading conditions can be decided just
in terms of the trial yield function according to,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 12
( ) ( )1 1trialn nf f+ +S S
( )( )
1 1
1 1
Elastic step
Plastic s
0 0
0 t0 ep
trialn n
trialn n
f
f
+ +
+ +
< =
> >
S
S
-
Proof 1. Convexity of the yield function yields,
Using the return mapping equation, yields,
Definite-positiveness of the constitutive matrix yields,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 13
( ) ( ) ( ) ( )11 1 1 1 1n
trial trialn n n n nf f f++ + + + + SS S S S S
( ) ( ) ( ) ( )1 11 1 1 1 1n n
trialn n n n nf f f f + ++ + + + + S SS S S C S
( )11 1 1 1n
trialn n n nf ++ + + += SS S C S
( ) ( ) ( ) ( )( ) ( )
1 11 1 1 1 1
1 1
0n n
trialn n n n n
trialn n
f f f f
f f
+ ++ + + + +
+ +
S SS S S C S
S S
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Proof 2. The trial yield function at time n+1 determines the
discrete plastic loading/elastic unloading conditions according
to,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 14
( )( ) ( )
( )( )
( )( ) ( )
1
1
1 1
1 1 1
1
1 1 1 1
1 1 1 1
if 0 then
0
0 Elastic step
Plastic
0
else if 0 then
0, 0 0 ste
en if
p
d
n
trialn
trialn n
n n n
trialn
trialn n n n
n n n n
f
f f
f
f
f
f f
+
+
+ +
+ + +
+
+ + + +
+ + + +
Rate Independent Plasticity Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 15
( )1 1arg min trialn n + += S S S S E
( )( ) ( )
1
211 12
1 11 12
trial trialn n
trial trialn n
+ +
+ +
=
= C
S S S S
S S C S S
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Proof 1. The stress state at time n+1, the
closest-point-projection of the trial stress state at n+1 onto the
space of admissible stresses, is the solution of the following
constrained minimization problem,
Using the Lagrange multipliers method, the constrained
minimization problem can be transformed into an unconstrained
minimization problem defined as,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 16
( )1 1arg min trialn n + += S S S S E
( ) ( ) ( )1 1, :trial trialn n f + += +S S; S S SL( )1 1 1arg
min ,trialn n n+ + += S S S; SL
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The optimality conditions of the unconstrained minimization
problem read, The return mapping algorithm arises as the solution
of the unconstrained minimization problem yielding,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 17
( ) ( ) ( )( ) ( )
1 1 1
1
1 1 1 1 1 1 1
11 1 1 1
, :
: 0
;n n n
n
trial trialn n n n n n n
trialn n n n
f
f
+ + +
+
+ + + + + + +
+ + + +
= +
= + =
S S S
S
S S S S S
C S S S
L
( ) ( )1 1 1 10, 0, 0n n n nf f + + + + =S S
( )11 1 1 1n
trialn n n nf ++ + + += SS S C S
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Return mapping algorithm The return mapping algorithm takes the
form,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 18
( )11 1 1 1n
trialn n n nf ++ + + += SS S C S
( )
( )
1 1 1 1 1
1 1 1
1 1 1 1 1
sgn
:
: sgn
trialn n n n n
trialn n n
trialn n n n n
E q
q q Kq q H q
+ + + + +
+ + +
+ + + + +
= = = +
-
The solution for the return mapping algorithm yields,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 19
( ) ( )1 1 1 1 1 1 1sgntrial trialn n n n n n nq q E H q + + + +
+ + + = +
( ) ( )( ) ( )
1 1 1 1 1 1 1 1
1 1 1
sgn sgn
sgn
trial trial trial trialn n n n n n n n
n n n
q q q q
E H q
+ + + + + + + +
+ + +
=
+
( )( ) ( )( )
1 1 1 1 1
1 1 1 1
sgn
sgnn n n n n
trial trial trial trialn n n n
q E H q
q q
+ + + + +
+ + + +
+ + =
=
( )( ) ( )
1 1 1 1 1
1 1 1 1sgn sgn
trial trialn n n n n
trial trialn n n n
q E H q
q q
+ + + + +
+ + + +
+ + =
=
-
For the non-trivial case (plastic loading), using the discrete
Kuhn-Tucker loading/unloading conditions, the discrete plastic
multiplier (or discrete plastic consistency parameter) reads,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 20
( )1 1 1 1if 0 then , , 0n n n nf q q + + + +> =( )
( )( ) ( )
1 1 1 1 1 1
1 1 1 1
1 1 1 1
, ,
, , 0
n n n n n Y n
trial trial trialn n n Y n
trial trial trialn n n n
f q q q q
q E K H q
f q q E K H
+ + + + + +
+ + + +
+ + + +
= +
= + + +
= + + =
( ) ( )11 1 1 1, , 0trial trial trialn n n nE K H f q q + + + +=
+ + >
( ) ( )1 1 1 1 1 1 1 10, , , 0, , , 0n n n n n n n nf q q f q q
+ + + + + + + + =
-
The return mapping algorithm takes the form,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 21
( )
( )
1 1 1 1 1
1 1 1
1 1 1 1 1
sgn
:
: sgn
trialn n n n n
trialn n n
trialn n n n n
E q
q q Kq q H q
+ + + + +
+ + +
+ + + + +
= = = +
( ) ( ) ( )( ) ( )( ) ( )
11 1 1 1 1 1 1
11 1 1 1 1
11 1 1 1 1 1 1
, , sgn
: , ,
: , , sgn
trial trial trial trial trial trialn n n n n n n
trial trial trial trialn n n n n
trial trial trial trial trialn n n n n n n
E K H f q q E q
q q E K H f q q K
q q E K H f q q H q
+ + + + + + +
+ + + + +
+ + + + + + +
= + +
= + +
= + + + ( )trial
-
The update of the plastic internal variables takes the form,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 22
( )
( )
1 1 1 1
1 1
1 1 1 1
sgn
sgn
p pn n n n n
n n n
n n n n n
q
q
+ + + +
+ +
+ + + +
= + = + =
( ) ( ) ( )( ) ( )( ) ( ) ( )
11 1 1 1 1 1
11 1 1 1
11 1 1 1 1 1
, , sgn
, ,
, , sgn
p p trial trial trial trial trialn n n n n n n
trial trial trialn n n n n
trial trial trial trial trialn n n n n n n
E K H f q q q
E K H f q q
E K H f q q q
+ + + + + +
+ + + +
+ + + + + +
= + + + = + + +
= + +
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Consistent elastoplastic tangent modulus The consistent
elastoplastic tangent modulus is computed taking the variation of
the stress at time n+1, yielding, where
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Consistent elastoplastic tangent modulus
April 1, 2015 Carlos Agelet de Saracibar 23
( ) ( )11 1 1 1 1sgntrial trial trial trialn n n n nE K H f E q
+ + + + += + + ( ) ( )11 1 1 1 1sgntrial trial trial trialn n n n
nd d E K H df E q + + + + += + +
( )( ) ( )
( ) ( )
,1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1
sgn
sgn sgn
trial e trial pn n n n n
trial trial trial trial trial trial trialn n n n n n n
trial trial trial trial trialn n n n n n
d Ed E d E d
df d q q d q
q d q E d
+ + + +
+ + + + + + +
+ + + + + +
= = =
= =
= =
-
The consistent elastoplastic tangent modulus takes the form,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Consistent elastoplastic tangent modulus
April 1, 2015 Carlos Agelet de Saracibar 24
( ) ( )
( ) ( ) ( )( )( )
11 1 1 1 1
1 11
1 1 1 1 1
11 1
sgn
sgn sgn
1
trial trial trial trialn n n n n
n n
trial trial trial trialn n n n n
n n
d d E K H df E q
d E d
E K H q E d E q
d E E E K H d
+ + + + +
+ +
+ + + + +
+ +
= + +
=
+ +
= + +
( )( )11 1, : 1ep epn nd E d E E E E K H + += = + +
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1D Plasticity algorithm Step 1. Given the strain at time n+1
(strain driven problem), and the stored plastic internal variables
at time n (plastic internal variables database) Step 2. Compute the
trial state at time n+1
1D Plasticity Algorithms > Rate Independent Plasticity
Models
1D Plasticity algorithm
April 1, 2015 Carlos Agelet de Saracibar 25
( ) ( )
,1
, ,1 1 1
, ,1 1 1 1 1
1 1 1 1
:
:
:
:
p trial pn ne trial p trialn n n
trial e trial p trial pn n n n n n
trial trial trial trialn n n Y nf q q
+
+ + +
+ + + + +
+ + + +
=
=
= = =
= +
E E
E E E
S CE C E E C E E
-
Step 3. Check the trial yield function at time n+1 Step 4.
Compute discrete plastic multiplier at time n+1
1D Plasticity Algorithms > Rate Independent Plasticity
Models
1D Plasticity algorithm
April 1, 2015 Carlos Agelet de Saracibar 26
( ) ( )1 1 1if 0 then set , and exittrialtrial ep
n n nf E E+ + + = =
( ) 11 1trialn nE K H f
+ += + +
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Step 5. Return mapping algorithm (closest-point-projection)
1D Plasticity Algorithms > Rate Independent Plasticity
Models
1D Plasticity algorithm
April 1, 2015 Carlos Agelet de Saracibar 27
( ) ( )( )( ) ( )
11 1 1 1 1
11 1 1
11 1 1 1 1
sgn
:
: sgn
trial trial trial trialn n n n n
trial trialn n n
trial trial trial trialn n n n n
E K H f E q
q q E K H f K
q q E K H f H q
+ + + + +
+ + +
+ + + + +
= + + = + +
= + + +
( )1
1 1 1 1trialn
trial trialn n n nf
++ + + += SS S C S
-
Step 6. Update plastic internal variables database at time n+1
Step 7. Compute the consistent elastoplastic tangent modulus
1D Plasticity Algorithms > Rate Independent Plasticity
Models
1D Plasticity algorithm
April 1, 2015 Carlos Agelet de Saracibar 28
( ) ( )( )( ) ( )
11 1 1 1
11 1
11 1 1 1
sgn
sgn
p p trial trial trialn n n n n
trialn n n
trial trial trialn n n n n
E K H f q
E K H f
E K H f q
+ + + +
+ +
+ + + +
= + + + = + + +
= + +
( )( )1: 1epE E E E K H = + +
( )1
1 1 1trialn
p p trialn n n nf
++ + += + SE E S
-
Nonlinear isotropic hardening Exponential saturation law +
linear hardening
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 29
( ) ( ):q = = =
( ) ( )( ): : 1 expYq K = = ( ) ( ) ( )( )1 expY K = +
-
Time discrete nonlinear isotropic hardening
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 30
( ) ( )( ) ( )
( ) ( )
1 1 1
1 1
1 1 1
:
:
:
n n n n
trial trialn n n n
trialn n n n n
q
q q
q q
+ + +
+ +
+ + +
= = +
= = =
= + +
-
Plastic loading: Yield function at time n+1 Nonlinear residual
scalar equation on the plastic multiplier at time n+1
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 31
( ) ( ) ( )( )1 1 1 1 0trialn n n n n nf f E H + + + + = + +
=
( )( ) ( ) ( )( )
1 1 1
1 1 1 1
: 0
: 0n n n
trialn n n n n n
g g f
g f E H
+ + +
+ + + +
= = =
= + + =
-
Newton-Raphson iterative solution algorithm Step 1. Initialize
iteration counter and plastic multiplier Step 2. Compute the
residual g at time n+1, iteration k Step 3. While the absolute
value of the current residual at time n+1, iteration k, is greater
than a tolerance Step 4. Solve the linarized equation Step 5.
Update the plastic multiplier at time n+1, iteration k+1 Step 6.
Compute the residual g at time n+1, iteration k+1 Step 7. Increment
iteration counter k=k+1 and go to Step 3
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 32
10, 0knk += =
1 1 1 0k k kn n ng Dg + + ++ =
11 1 1
k k kn n n ++ + += +
-
Newton-Raphson iterative solution algorithm
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 33
( ) ( ) ( )( )( ) ( )
( )( )
1 1 1 1
1 1 1 1 1
1 1
11 1 1
:
:
:
:
k trial k kn n n n n n
k k k k kn n n n n n
k kn n n
k k kn n n
g f E H
Dg E H
E H
+ + + +
+ + + + +
+ +
++ + +
= + +
= + +
= + + +
=
1 1 1 0k k kn n ng Dg + + ++ =
-
Consistent elastoplastic tangent modulus The consistent
elastoplastic tangent modulus is computed taking the variation of
the stress at time n+1, yielding, where the variations of the trial
stress tensor, and plastic multiplier, at time n+1, have to be
computed.
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 34
( )( )
1 1 1 1 1
1 1 1 1 1
sgn
sgn
trial trial trialn n n n n
trial trial trialn n n n n
E q
d d d E q
+ + + + +
+ + + + +
=
=
-
The variation of the plastic multiplier at time n+1 is computed
setting the variation of the residual at time n+1 equal to
zero,
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 35
( ) ( ) ( )( )( ) ( )
( )( )( )( )
1 1 1 1
1 1 1 1 1
1 1 1
1
1 1 1
: 0
:
: 0
trialn n n n n n
trialn n n n n n
trialn n n n
trialn n n n
g f E H
dg df d E H d
df d E H
d E H df
+ + + +
+ + + + +
+ + +
+ + +
= + + =
= + +
= + + + =
= + + +
-
Substituting the variations shown before, the following discrete
tangent constitutive equation is obtained, where the consistent
elastoplastic tangent modulus at time n+1 is given by
1D Plasticity Algorithms > Rate Independent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 36
1 1 1ep
n n nd E d + + +=
( )( )( )11 11epn n nE E E E H + += + + +
-
Contents 1. Introduction 2. 1D Rate independent plasticity
models
1. Return mapping algorithm 2. Consistent elastoplastic tangent
modulus 3. Step by step algorithm 4. Nonlinear isotropic
hardening
3. 1D Rate dependent plasticity models 1. Return mapping
algorithm 2. Consistent elastoplastic tangent modulus 3. Step by
step algorithm 4. Nonlinear isotropic hardening
4. 1D Computational plasticity assignment
1D Plasticity Algorithms > Contents
Contents
April 1, 2015 Carlos Agelet de Saracibar 37
-
1. Additive split of strains
2. Constitutive equations
3. Associative plastic flow rule
4. Yield function
5. Plastic multiplier
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
1D Rate dependent plasticity model
April 1, 2015 Carlos Agelet de Saracibar 38
( ) { } { }, , , diag , ,e e e q q E K H = = =ES E CE S: , C
:=
{ } { } { }: : ,0,0 , : , , , : , ,e p p p e e = + = = = E E E ,
E E E
( )p f= SE S
( ) Yf q q = +S
( )1 0f = S
-
Associative plastic flow rule: plastic strains at time n+1
Using a Backward-Euler (BE) time integration scheme yields,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 39
( )p f= SE S
( )11 1 1n
p pn n n nt f ++ + += + SE E S
-
Constitutive equations: stress state at time n+1
The time-discrete constituve equation at time n+1 takes the
form,
Substituting the plastic strains at time n+1 yields,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 40
( ) ( ) { }, diag , ,e e e p E K H= = = ES E CE C E E C :=
( )1 1 1 1e pn n n n+ + + += = S CE C E E
( )( )( ) ( )
1
1
1 1 1 1
1 1 1
n
n
pn n n n n
pn n n n
t f
t f
+
+
+ + + +
+ + +
=
=
S
S
S C E E S
C E E C S
-
Trial state at time n+1 The trial state at time n+1 is defined
by freezing the plastic behaviour at the time step, yielding
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 41
( ) ( )( )
,1
, ,1 1 1
, ,1 1 1 1 1
1 1
:
:
:
:
p trial pn ne trial p trialn n n
trial e trial p trial pn n n n n n
rial trialn nf f
+
+ + +
+ + + + +
+ +
=
=
= = =
=
E E
E E E
S CE C E E C E E
S
-
Return mapping algorithm The return mapping algorithm takes the
form,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 42
( )11 1 1 1n
trialn n n nt f ++ + + += SS S C S
( )
( )
1 1 1 1 1
1 1 1
1 1 1 1 1
sgn
:
: sgn
trialn n n n n
trialn n n
trialn n n n n
t E q
q q t Kq q t H q
+ + + + +
+ + +
+ + + + +
= = = +
-
1. Additive split of strains at time n+1
2. Stresses at time n+1. Return mapping algorithm
3. Plastic internal variables at time n+1
4. Yield function at time n+1
5. Plastic multiplier at time n+1
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 43
( )1 1 1 1 1:n n n n Y nf f q q + + + + += = +S
( )11 1 1 1n
trialn n n nt f ++ + + += SS S C S
1 1 1:e p
n n n+ + += +E E E
( )11 1 1n
p pn n n nt f ++ + += + SE E S
( )11 1 0n nf + += S
-
Return mapping algorithm The return mapping algorithm takes the
form,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 44
( )11 1 1 1n
trialn n n nt f ++ + + += SS S C S
( )
( )
1 1 1 1 1
1 1 1
1 1 1 1 1
sgn
:
: sgn
trialn n n n n
trialn n n
trialn n n n n
t E q
q q t Kq q t H q
+ + + + +
+ + +
+ + + + +
= = = +
-
The solution for the return mapping algorithm yields,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 45
( ) ( )1 1 1 1 1 1 1sgntrial trialn n n n n n nq q t E H q + + +
+ + + + = +
( ) ( )( ) ( )
1 1 1 1 1 1 1 1
1 1 1
sgn sgn
sgn
trial trial trial trialn n n n n n n n
n n n
q q q q
t E H q
+ + + + + + + +
+ + +
=
+
( )( ) ( )( )
1 1 1 1 1
1 1 1 1
sgn
sgnn n n n n
trial trial trial trialn n n n
q t E H q
q q
+ + + + +
+ + + +
+ + =
=
( )( ) ( )
1 1 1 1 1
1 1 1 1sgn sgn
trial trialn n n n n
trial trialn n n n
q t E H q
q q
+ + + + +
+ + + +
+ + =
=
-
For the non-trivial case (plastic loading), the yield function
and the discrete plastic multiplier are greater than zero, and the
following expressions hold,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 46
( )1 1 1 1 1if 0 then , , 0n n n n nf q q + + + + +> = >(
)
( )( ) ( )
1 1 1 1 1 1
1 1 1 1
1 1 1 1 1
, ,
, ,
n n n n n Y n
trial trial trialn n n Y n
trial trial trialn n n n n
f q q q q
q t E K H q
f q q t E K H
+ + + + + +
+ + + +
+ + + + +
= +
= + + +
= + + =
( )1
1 1 1 1, , 0trial trial trial
n n n nt E K H f q qt
+ + + + = + + + >
-
The return mapping algorithm takes the form,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 47
( )
( )
1 1 1 1 1
1 1 1
1 1 1 1 1
sgn
:
: sgn
trialn n n n n
trialn n n
trialn n n n n
t E q
q q t Kq q t H q
+ + + + +
+ + +
+ + + + +
= = = +
( ) ( )( )( ) ( )
11 1 1 1 1
11 1 1
11 1 1 1 1
sgn
:
: sgn
trial trial trial trialn n n n n
trial trialn n n
trial trial trial trialn n n n n
E K H t f E q
q q E K H t f K
q q E K H t f H q
+ + + + +
+ + +
+ + + + +
= + + + = + + +
= + + + +
-
The update of the plastic internal variables takes the form,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Return mapping algorithm
April 1, 2015 Carlos Agelet de Saracibar 48
( )
( )
1 1 1 1
1 1
1 1 1 1
sgn
:
: sgn
p pn n n n n
n n n
n n n n n
t q
tt q
+ + + +
+ +
+ + + +
= + = + =
( ) ( )( )( ) ( )
11 1 1 1
11 1
11 1 1 1
sgn
:
: sgn
p p trial trial trialn n n n n
trialn n n
trial trial trialn n n n n
E K H t f q
E K H t f
E K H t f q
+ + + +
+ +
+ + + +
= + + + + = + + + +
= + + +
-
Consistent elastoplastic tangent modulus The consistent
elastoplastic tangent modulus is computed taking the variation of
the stress at time n+1, yielding, where
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Consistent elastoplastic tangent modulus
April 1, 2015 Carlos Agelet de Saracibar 49
( ) ( )11 1 1 1 1sgntrial trial trial trialn n n n nE K H t f E
q + + + + += + + + ( ) ( )11 1 1 1 1sgntrial trial trial trialn n n
n nd d E K H t df E q + + + + += + + +
( )( ) ( )
( ) ( )
,1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1
sgn
sgn sgn
trial e trial pn n n n n
trial trial trial trial trial trial trialn n n n n n n
trial trial trial trial trialn n n n n n
d Ed E d E d
df d q q d q
q d q E d
+ + + +
+ + + + + + +
+ + + + + +
= = =
= =
= =
-
The consistent elastoplastic tangent modulus takes the form,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Consistent elastoplastic tangent modulus
April 1, 2015 Carlos Agelet de Saracibar 50
( ) ( )
( ) ( )( )( )
11 1 1 1 1
1 11 2
1 1 1
11 1
sgn
sgn
1
trial trial trial trialn n n n n
n n
trial trialn n n
n n
d d E K H t df E q
d E d
E K H t q E E d
d E E E K H t d
+ + + + +
+ +
+ + +
+ +
= + + +
=
+ + +
= + + +
( )( )11 1, : 1ep epn nd E d E E E E K H t + += = + + +
-
1D Plasticity algorithm Step 1. Given the strain at time n+1
(strain driven problem), and the stored plastic internal variables
at time n (plastic internal variables database) Step 2. Compute the
trial state at time n+1
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
1D Plasticity algorithm
April 1, 2015 Carlos Agelet de Saracibar 51
( ) ( )
,1
, ,1 1 1
, ,1 1 1 1 1
1 1 1 1
:
:
:
:
p trial pn ne trial p trialn n n
trial e trial p trial pn n n n n n
trial trial trial trialn n n Y nf q q
+
+ + +
+ + + + +
+ + + +
=
=
= = =
= +
E E
E E E
S CE C E E C E E
-
Step 3. Check the trial yield function at time n+1 Step 4.
Compute discrete plastic multipliet at time n+1
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
1D Plasticity algorithm
April 1, 2015 Carlos Agelet de Saracibar 52
( ) ( )1 1 1if 0 then set , and exittrialtrial ep
n n nf E E+ + + = =
( ) 11 1trialn nE K H t f
+ += + + +
-
Step 5. Return mapping algorithm (closest-point-projection)
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
1D Plasticity algorithm
April 1, 2015 Carlos Agelet de Saracibar 53
( ) ( )( )( ) ( )
11 1 1 1 1
11 1 1
11 1 1 1 1
sgn
:
: sgn
trial trial trial trialn n n n n
trial trialn n n
trial trial trial trialn n n n n
E K H t f E q
q q E K H t f K
q q E K H t f H q
+ + + + +
+ + +
+ + + + +
= + + + = + + +
= + + + +
( )1
1 1 1 1trialn
trial trialn n n nf
++ + + += SS S C S
-
Step 6. Update plastic internal variables database at time n+1
Step 7. Compute the consistent elastoplastic tangent modulus
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
1D Plasticity algorithm
April 1, 2015 Carlos Agelet de Saracibar 54
( ) ( )( )( ) ( )
11 1 1 1
11 1
11 1 1 1
sgn
sgn
p p trial trial trialn n n n n
trialn n n
trial trial trialn n n n n
E K H t f q
E K H t f
E K H t f q
+ + + +
+ +
+ + + +
= + + + + = + + + +
= + + +
( )( )1: 1epE E E E K H t = + + +
( )1
1 1 1trialn
p p trialn n n nf
++ + += + SE E S
-
Nonlinear isotropic hardening Exponential saturation law +
linear hardening
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 55
( ) ( ):q = = =
( ) ( )( ): : 1 expYq K = = ( ) ( ) ( )( )1 expY K = +
-
Time discrete nonlinear isotropic hardening
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 56
( ) ( )( ) ( )
( ) ( )
1 1 1
1 1
1 1 1
:
:
:
n n n n
trial trialn n n n
trialn n n n n
q t
q q
q q t
+ + +
+ +
+ + +
= = +
= = =
= + +
-
Plastic loading: Yield function at time n+1 Nonlinear residual
scalar equation on the plastic multiplier at time n+1
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 57
( ) ( ) ( )( )1 1 1 1 1 0trialn n n n n n nf f t E H t + + + + +
= + + = >
( )( ) ( )( )
1 1 1 1
1 1 1 1
: 0
: 0
n n n n
trialn n n n n n
g g f
g f t E H tt
+ + + +
+ + + +
= = =
= + + + =
-
Newton-Raphson iterative solution algorithm Step 1. Initialize
iteration counter and plastic multiplier Step 2. Compute the
residual g at time n+1, iteration k Step 3. While the absolute
value of the current residual at time n+1, iteration k, is greater
than a tolerance Step 4. Solve the linarized equation Step 5.
Update the plastic multiplier at time n+1, iteration k+1 Step 6.
Compute the residual g at time n+1, iteration k+1 Step 7. Increment
iteration counter k=k+1 and go to Step 3
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 58
10, 0knk += =
1 1 1 0k k kn n ng Dg + + ++ =
11 1 1
k k kn n n ++ + += +
-
Newton-Raphson iterative solution algorithm
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 59
( ) ( )( )
( )
( )
1 1 1 1
1 1 1 1 1
1 1
11 1 1
:
:
:
:
k trial k kn n n n n n
k k k k kn n n n n n
k kn n n
k k kn n n
g f t E H tt
Dg E H t t tt
E t H tt
+ + + +
+ + + + +
+ +
++ + +
= + + + = + + + = + + + +
=
1 1 1 0k k kn n ng Dg + + ++ =
-
Consistent elastoplastic tangent modulus The consistent
elastoplastic tangent modulus is computed taking the variation of
the stress at time n+1, yielding, where the variations of the trial
stress tensor, and plastic multiplier, at time n+1, have to be
computed.
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 60
( )( )
1 1 1 1 1
1 1 1 1 1
sgn
sgn
trial trial trialn n n n n
trial trial trialn n n n n
t E q
d d d t E q
+ + + + +
+ + + + +
=
=
-
The variation of the plastic multiplier at time n+1 is computed
setting the variation of the residual at time n+1 equal to
zero,
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 61
( ) ( )( )
( )
( )
( )
1 1 1 1
1 1 1 1 1
1 1 1
1
1 1
: 0
:
: 0
trialn n n n n n
trialn n n n n n
trialn n n n
n n n n
g f t E H tt
dg df d t E H t d tt
df d t E t Ht
d t E t H dft
+ + + +
+ + + + +
+ + +
+ + +
= + + + = = + + + = + + + + =
= + + + + 1
trial
-
Substituting the variations shown before, the following discrete
tangent constitutive equation is obtained, where the consistent
elastoplastic tangent modulus at time n+1 is given by
1D Plasticity Algorithms > Rate Dependent Plasticity
Models
Nonlinear isotropic hardening
April 1, 2015 Carlos Agelet de Saracibar 62
1 1 1ep
n n nd E d + + +=
( )1
1 11epn n nE E E E t H t
+ +
= + + + +
-
Contents 1. Introduction 2. 1D Rate independent plasticity
models
1. Return mapping algorithm 2. Consistent elastoplastic tangent
modulus 3. Step by step algorithm 4. Nonlinear isotropic
hardening
3. 1D Rate dependent plasticity models 1. Return mapping
algorithm 2. Consistent elastoplastic tangent modulus 3. Step by
step algorithm 4. Nonlinear isotropic hardening
4. 1D Computational plasticity assignment
1D Plasticity Algorithms > Contents
Contents
April 1, 2015 Carlos Agelet de Saracibar 63
-
Implement in MATLAB the BE time-stepping algorithm for 1D
rate-independent/rate-dependent hardening plasticity models,
including linear and nonlinear isotropic hardening, and linear
kinematic hardening
Perform the numerical simulation of uniaxial cyclic plastic
loading/elastic unloading examples for the following cases: o
Rate-independent/rate-dependent perfect plasticity o
Rate-independent/rate-dependent linear isotropic hardening
plasticity o Rate-independent/rate-dependent nonlinear isotropic
hardening
plasticity, considering an exponential saturation law o
Rate-independent/rate-dependent linear kinematic hardening
plasticity o Rate-independent/rate-dependent nonlinear isotropic
and linear
kinematic hardening plasticity
1D Plasticity Algorithms > 1D Computational Plasticity
Assignment
1D Computational plasticity assignment
April 1, 2015 Carlos Agelet de Saracibar 64
-
For the perfect plasticity models, plot the stress-strain curves
For the linear isotropic/linear kinematic hardening models,
plot the stress-strain curves showing the influence of the
isotropic/kinematic hardening parameters
For the nonlinear isotropic hardening model, plot the
stress-strain curves showing the influence of the exponential
coefficient of the exponential saturation law on the stress-strain
curves
For the rate-dependent plasticity models, plot the
stress-strain, and the stress-time curves showing the influence of
the viscosity parameter and the loading rate.
Show that the rate-independent response can be recovered from
the rate-dependent results using very small values for the
viscosity or the loading rate
1D Plasticity Algorithms > 1D Computational Plasticity
Assignment
1D Computational plasticity assignment
April 1, 2015 Carlos Agelet de Saracibar 65
-
Write a comprehensive deliverable report (10 pages) providing
the data of the cyclic loading and material properties considered,
the stress-strain curves, and the stress-time curves for the
rate-dependent plasticity examples. Add suitable comments on the
results, comparing the influence of the different material
parameters and loading conditions.
Add a printed copy of the subroutines as an Appendix
1D Plasticity Algorithms > 1D Computational Plasticity
Assignment
1D Computational plasticity assignment
April 1, 2015 Carlos Agelet de Saracibar 66
Computational Solid MechanicsComputational PlasticityNmero de
diapositiva 2Nmero de diapositiva 3Nmero de diapositiva 4Nmero de
diapositiva 5Nmero de diapositiva 6Nmero de diapositiva 7Nmero de
diapositiva 8Nmero de diapositiva 9Nmero de diapositiva 10Nmero de
diapositiva 11Nmero de diapositiva 12Nmero de diapositiva 13Nmero
de diapositiva 14Nmero de diapositiva 15Nmero de diapositiva
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diapositiva 19Nmero de diapositiva 20Nmero de diapositiva 21Nmero
de diapositiva 22Nmero de diapositiva 23Nmero de diapositiva
24Nmero de diapositiva 25Nmero de diapositiva 26Nmero de
diapositiva 27Nmero de diapositiva 28Nmero de diapositiva 29Nmero
de diapositiva 30Nmero de diapositiva 31Nmero de diapositiva
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diapositiva 35Nmero de diapositiva 36Nmero de diapositiva 37Nmero
de diapositiva 38Nmero de diapositiva 39Nmero de diapositiva
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diapositiva 43Nmero de diapositiva 44Nmero de diapositiva 45Nmero
de diapositiva 46Nmero de diapositiva 47Nmero de diapositiva
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diapositiva 51Nmero de diapositiva 52Nmero de diapositiva 53Nmero
de diapositiva 54Nmero de diapositiva 55Nmero de diapositiva
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diapositiva 59Nmero de diapositiva 60Nmero de diapositiva 61Nmero
de diapositiva 62Nmero de diapositiva 63Nmero de diapositiva
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