computational solid mechanicsagelet.rmee.upc.edu/master/chapter 4. j2 plasticity algorithms...
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Computational Solid Mechanics Computational Plasticity
C. Agelet de Saracibar ETS Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Cataluña (UPC), Barcelona, Spain
International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain
Chapter 4. J2 Plasticity Algorithms
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Contents 1. Introduction 2. J2 Rate independent plasticity algorithms
1. Return mapping algorithm 2. Consistent elastoplastic tangent operator 3. Step by step algorithm 4. Nonlinear isotropic hardening
3. J2 Rate dependent plasticity algorithms 1. Return mapping algorithm 2. Consistent elastoplastic tangent operator 3. Step by step algorithm 4. Nonlinear isotropic hardening
4. J2 Computational plasticity assignment
J2 Plasticity Algorithms > Contents
Contents
April 20, 2015 Carlos Agelet de Saracibar 2
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Contents 1. Introduction 2. J2 Rate independent plasticity algorithms
1. Return mapping algorithm 2. Consistent elastoplastic tangent operator 3. Step by step algorithm 4. Nonlinear isotropic hardening
3. J2 Rate dependent plasticity algorithms 1. Return mapping algorithm 2. Consistent elastoplastic tangent operator 3. Step by step algorithm 4. Nonlinear isotropic hardening
4. J2 Computational plasticity assignment
J2 Plasticity Algorithms > Contents
Contents
April 20, 2015 Carlos Agelet de Saracibar 3
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J2 Plasticity Algorithms > Introduction
Time integration algorithm
April 20, 2015 Carlos Agelet de Saracibar 4
pnE 1
pn+E
1n+E
Time integration algorithm
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Contents 1. Introduction 2. J2 Rate independent plasticity algorithms
1. Return mapping algorithm 2. Consistent elastoplastic tangent operator 3. Step by step algorithm 4. Nonlinear isotropic hardening
3. J2 Rate dependent plasticity algorithms 1. Return mapping algorithm 2. Consistent elastoplastic tangent operator 3. Step by step algorithm 4. Nonlinear isotropic hardening
4. J2 Computational plasticity assignment
J2 Plasticity Algorithms > Contents
Contents
April 20, 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
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
J2 Rate independent plasticity model
April 20, 2015 Carlos Agelet de Saracibar 6
( ) { } { }23, , , diag , ,e
e e q K Hψ= ∂ = = σ q 1E
S E CE S: , C :=
{ } { } { }: : ,0,0 , : , , , : , ,e p p p e eξ ξ= + = = = − −ε ε ξ ε ξE E E , E E E
( )p fγ= ∂
SE S
( ) ( ) ( )23
: , , : dev Yf f q qσ= = − − −σ q σ qS
( ) ( )0, 0, 0f fγ γ≥ ≤ =S S
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Associative plastic flow rule: plastic strains at time n+1
Using a Backward-Euler (BE) time integration scheme yields,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 7
( )p fγ= ∂
SE S
( )11 1 1n
p pn n n nfγ
++ + += + ∂SE E S
1 1 1
1 1
1 1 1
2 3
p pn n n n
n n n
n n n n
γ
ξ ξ γ
γ
+ + +
+ +
+ + +
= + = + = −
ε ε n
ξ ξ n
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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,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 8
( ) ( ) { }23, diag , ,e
e e p K Hψ= ∂ = = − 1E
S E CE C E E C :=
( )1 1 1 1e p
n 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
γ
γ+
+
+ + + +
+ + +
= − − ∂
= − − ∂
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
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 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,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 10
( )11 1 1 1n
trialn n n nfγ
++ + + += − ∂SS S C S
1 1 1 1 1 1 1
1 1 1
21 1 1 13
: 2
: 2 3
:
trial trialn n n n n n n
trialn n n
trialn n n n
q q K
H
γ γ µ
γ
γ
+ + + + + + +
+ + +
+ + + +
= − = − = − = +
σ σ n σ n
q q n
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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
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 11
( ) ( )21 1 1 1 13: : devn n n n Y nf f qσ+ + + + += = − − −σ qS
1 1 1 10, 0, 0n n n nf fγ γ+ + + +≥ ≤ =
( )11 1 1 1n
trialn n n nfγ
++ + + += − ∂SS S C S
1 1 1: e pn n n+ + += +E E E
( )11 1 1n
p pn n n nfγ
++ + += + ∂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 state according to,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 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
γ
γ
+ +
+ +
< ⇒ =
> ⇒ >
S
S
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Theorem 2. Closest-point-projection The stress state at time n+1 is the closest-point-projection of the trial stress state at n+1 onto the space of admissible stresses, measured in the complementary energy norm, where the complementary energy is given by,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 15
( )1 1arg min trialn n σ+ += Ξ − ∀ ∈S S S S E
( )( ) ( )
1
211 12
1 11 12
trial trialn n
trial trialn n
−+ +
−+ +
Ξ − = −
= − −C
S S S S
S S C S S
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Return mapping algorithm The return mapping algorithm takes the form,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 18
( )11 1 1 1n
trialn n n nfγ
++ + + += − ∂SS S C S
1 1 1 1 1 1 1
1 1 1
21 1 1 13
: 2
: 2 3
:
trial trialn n n n n n n
trialn n n
trialn n n n
q q K
H
γ γ µ
γ
γ
+ + + + + + +
+ + +
+ + + +
= − = − = − = +
σ σ n σ n
q q n
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Return mapping algorithm Taking into account that the return mapping algorithm takes place on the deviatoric plane, it can be written as,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 19
1 1 1 1 1 1 1
1 1 1
21 1 1 13
dev dev : dev 2
: 2 3
:
trial trialn n n n n n n
trialn n n
trialn n n n
q q K
H
γ γ µ
γ
γ
+ + + + + + +
+ + +
+ + + +
= − = − = − = +
σ σ n σ n
q q n
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The solution for the return mapping algorithm yields,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 20
( )21 1 1 1 1 13dev dev 2trial trial
n n n n n nHγ µ+ + + + + +− = − − +σ q σ q n
( )21 1 1 1 1 1 1 13dev dev 2trial trial trial
n n n n n n n nHγ µ+ + + + + + + +− = − − +σ q n σ q n n
( )( )21 1 1 1 1 1 13dev 2 dev trial trial trial
n n n n n n nHγ µ+ + + + + + +− + + = −σ q n σ q n
( )21 1 1 1 13
1 1
dev 2 dev trial trialn n n n n
trialn n
Hγ µ+ + + + +
+ +
− + + = −
=
σ q σ q
n n
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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,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 21
( )1 1 1 1if 0 then , , 0n n n nf qγ + + + +> =σ q
( ) ( )
( ) ( )( ) ( )
21 1 1 1 1 13
2 2 21 1 1 13 3 3
2 21 1 1 1 3 3
, , dev
dev 2
, , 2 0
n n n n n Y n
trial trial trialn n n Y n
trial trial trialn n n n
f q q
K H q
f q K H
σ
γ µ σ
γ µ
+ + + + + +
+ + + +
+ + + +
= − − −
= − − + + − −
= − + + =
σ q σ q
σ q
σ q
( ) 12 21 13 32 trial
n nK H fγ µ−
+ += + +
( ) ( )1 1 1 1 1 1 1 10, , , 0, , , 0n n n n n n n nf q f qγ γ+ + + + + + + +≥ ≤ =σ q σ q
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The return mapping algorithm takes the form,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 22
( )( )( )
12 21 1 1 13 3
12 21 1 13 3
12 21 1 1 13 3
2 2
: 2 2 3
2: 23
trial trial trialn n n n
trial trialn n n
trial trial trialn n n n
K H f
q q K H f K
K H f H
µ µ
µ
µ
−
+ + + +
−
+ + +
−
+ + + +
= − + +
= − + + = + + +
σ σ n
q q n
1 1 1 1 1 1 1
1 1 1
21 1 1 13
: 2
: 2 3
:
trial trialn n n n n n n
trialn n n
trialn n n n
q q K
H
γ γ µ
γ
γ
+ + + + + + +
+ + +
+ + + +
= − = − = − = +
σ σ n σ n
q q n
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Return mapping algorithm: Geometric interpretation
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 23
, 1nσ +
1dev n+σ
1n+q
( )1 1
23n nYR qσ+ += −
1n+n
1dev trialn+σ
dev nσ
,nσnq
( )23n nYR qσ= −
1trialn+n
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The update of the plastic internal variables takes the form,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 24
1 1 1
1 1
1 1 1
2 3
p pn n n n
n n n
n n n n
γ
ξ ξ γ
γ
+ + +
+ +
+ + +
= + = + = −
ε ε n
ξ ξ n
( )( )( )
12 21 1 13 3
12 21 13 3
12 21 1 13 3
2
2 2 3
2
p p trial trialn n n n
trialn n n
trial trialn n n n
K H f
K H f
K H f
µ
ξ ξ µ
µ
−
+ + +
−
+ +
−
+ + +
= + + + = + + + = − + +
ε ε n
ξ ξ n
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Consistent elastoplastic tangent operator The consistent elastoplastic tangent operator is computed taking the variation of the stress tensor at time n+1, yielding, where the variations of the trial stress tensor, trial yield function, and trial unit normal to the yield surface at time n+1, have to be computed.
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 25
( )( )
( )
12 21 1 1 13 3
12 21 1 1 13 3
12 21 13 3
2 2
2 2
2 2
trial trial trialn n n n
trial trial trialn n n n
trial trialn n
K H f
d d K H df
K H f d
µ µ
µ µ
µ µ
−
+ + + +
−
+ + + +
−
+ +
= − + +
= − + +
− + +
σ σ n
σ σ n
n
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The variation of the trial stress tensor at time n+1 takes the form,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 26
( ) ( )( ) ( ) ( ) ( )( ) ( )( )
, , , ,1 1 1 1 1
1 1 1 1
, , , ,1 1 1 1 1
1
tr 2 tr 2 dev
tr 2 tr 2 dev
tr 2 tr 2 dev
tr
trial e trial e trial e trial e trialn n n n n
p pn n n n n n
trial e trial e trial e trial e trialn n n n n
n
d d d
d
λ µ κ µ
λ µ κ µ
λ µ κ µ
λ
+ + + + +
+ + + +
+ + + + +
+
= + = +
= + − = + −
= + = +
=
σ ε 1 ε ε 1 ε
ε 1 ε ε ε 1 ε ε
σ ε 1 ε ε 1 ε
ε 1 ( )( ) ( ) ( )( )
1 1 1
11 1 13
2 tr 2 dev
: 2 2 :
n n n
n n n
d d d
d d d
µ κ µ
λ µ κ µ
+ + +
+ + +
+ = +
= ⊗ + = ⊗ + − ⊗
ε ε 1 ε
1 1 ε ε 1 1 1 1 ε
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The variation of the trial yield function at time n+1 takes the form,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 27
( )( )
21 1 1 13
21 3
1 1 1 1 1 1
1 1
dev
dev
dev : dev : 2 dev
: 2
trial trial trial trialn n n Y n
trialn n Y n
trial trial trial trial trialn n n n n n n
trialn n
f q
q
df d d d
d
σ
σ
µ
µ
+ + + +
+
+ + + + + +
+ +
= − − −
= − − −
= − = =
=
σ q
σ q
σ q n σ n ε
n ε
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The variation of the unit normal to the yield surface at time n+1 takes the form,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 28
( )( )
1
1
1 1 11
1 1 1
11 1 1 1dev
11 1 1dev
dev devdev dev
dev dev
: dev
trialn n
trialn n
trial trial trialtrial n n n nn trial trial trial
n n n n
trial trial trial trialn n n n n
trial trial trn n n
d d d
d+
+
+ + ++
+ + +
+ + + +−
+ + +−
− −= =
− −
= − −
= − ⊗
σ q
σ q
σ q σ qnσ q σ q
n σ n σ q
n n σ
( )1
1
11 1 1dev
11 1 1dev
: 2 dev
1 : 23
trialn n
trialn n
ial
trial trialn n n
trial trialn n n
d
d
µ
µ
+
+
+ + +−
+ + +−
= − ⊗
= − ⊗ − ⊗
σ q
σ q
n n ε
1 1 n n ε
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Substituting the variations shown before, the following discrete tangent constitutive equation can be obtained, where the consistent elastoplastic tangent operator at time n+1 is given by
and the following parameters have been introduced
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 29
1 1 1:epn n nd d+ + +=σ ε
1 1 1 1 112 23
ep trial trialn n n n nκ µδ µδ+ + + + +
= ⊗ + − ⊗ − ⊗
1 1 I 1 1 n n
( )11 1 12 2
1 3 3
2 2: 1 , : 12dev
nn n ntrial
n n K Hµ γ µδ δ δ
µ+
+ + ++
= − = − −+ +−σ q
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J2 Plasticity algorithm Step 1. Given the strain tensor 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
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 30
( ) ( )( )
,1
, ,1 1 1
, ,1 1 1 1 1
21 1 1 13
:
:
:
: dev
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 qσ
+
+ + +
+ + + + +
+ + + +
=
= −
= = − = −
= − − −σ q
E E
E E E
S CE C E E C E E
![Page 27: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/27.jpg)
Step 3. Check the trial yield function at time n+1 Step 4. Compute the discrete plastic multiplier at time n+1
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 31
( ) ( )1 11 1if 0 then set , and exittrialtrial ep
n nn nf + ++ +
≤ • = • =
( ) 12 21 13 32 trial
n nK H fγ µ−
+ += + +
![Page 28: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/28.jpg)
Step 5. Return mapping algorithm (closest-point-projection)
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 32
( )1
1 1 1 1trialn
trial trialn n n nfγ
++ + + += − ∂
SS S C S
( )( )( )
12 21 1 1 13 3
12 21 1 13 3
12 21 1 1 13 3
2 2
: 2 2 3
2: 23
trial trial trialn n n n
trial trialn n n
trial trial trialn n n n
K H f
q q K H f K
K H f H
µ µ
µ
µ
−
+ + + +
−
+ + +
−
+ + + +
= − + +
= − + + = + + +
σ σ n
q q n
![Page 29: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/29.jpg)
Step 6. Update plastic internal variables database at time n+1
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 33
( )1
1 1 1trialn
p p trialn n n nfγ
++ + += + ∂
SE E S
( )( )( )
12 21 1 13 3
12 21 13 3
12 21 1 13 3
2
2 2 3
2
p p trial trialn n n n
trialn n n
trial trialn n n n
K H f
K H f
K H f
µ
ξ ξ µ
µ
−
+ + +
−
+ +
−
+ + +
= + + + = + + + = − + +
ε ε n
ξ ξ n
![Page 30: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/30.jpg)
Step 7. Compute the consistent elastoplastic tangent operator
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 34
1 1 1 1 112 23
ep trial trialn n n n nκ µδ µδ+ + + + +
= ⊗ + − ⊗ − ⊗
1 1 I 1 1 n n
( )11 1 12 2
1 3 3
2 2: 1 , : 12dev
nn n ntrial
n n K Hµ γ µδ δ δ
µ+
+ + ++
= − = − −+ +−σ q
![Page 31: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/31.jpg)
Nonlinear isotropic hardening Exponential saturation law + linear isotropic hardening
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 35
( ) ( ):q ξ ξψ ξ ξ′= −∂ = −∂ Π = −Π
( ) ( )( ): : 1 expYq Kξψ σ σ δξ ξ∞= −∂ = − − − − −
( ) ( ) ( )( )1 expY Kξ σ σ δξ ξ∞′Π = − − − +
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Time discrete nonlinear isotropic hardening
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 36
( ) ( )( ) ( )
( ) ( )
1 1 1
1 1
1 1 1
: 2 3
:
: 2 3
n n n n
trial trialn n n n
trialn n n n n
q
q q
q q
ξ ξ γ
ξ ξ
ξ γ ξ
+ + +
+ +
+ + +
′ ′= −Π = −Π +
′ ′= −Π = −Π =
′ ′= −Π + +Π
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Plastic loading: Yield function at time n+1 Nonlinear residual scalar equation on the plastic multiplier at time n+1
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 37
( ) ( ) ( )( )2 2 21 1 1 13 3 3
2 0trialn n n n n nf f Hγ µ ξ γ ξ+ + + +′ ′= − + − Π + −Π =
( )( ) ( ) ( )( )
1 1 1
2 2 21 1 1 13 3 3
: 0
: 2 0
n n n
trialn n n n n n
g g f
g f H
γ
γ µ ξ γ ξ
+ + +
+ + + +
= = =
′ ′= − + − Π + −Π =
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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
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 38
10, 0knk γ += =
1 1 1 0k k kn n ng Dg γ+ + ++ ∆ =
11 1 1
k k kn n nγ γ γ++ + += + ∆
![Page 35: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/35.jpg)
Newton-Raphson iterative solution algorithm
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 39
( )( ) ( )( )
( ) ( )( )( )
21 1 1 3
2 213 3
2 2 21 1 1 1 13 3 3
2 221 13 33
11 1 1
: 2
: 2
: 2
:
k trial kn n n
kn n n
k k k k kn n n n n n
k kn n n
k k kn n n
g f H
Dg H
H
γ µ
ξ γ ξ
γ µ γ ξ γ γ
µ ξ γ γ
γ γ γ
+ + +
+
+ + + + +
+ +
++ + +
= − +
′ ′− Π + −Π
′′∆ = − + ∆ − Π + ∆
′′= − + Π + + ∆
∆ = −
1 1 1 0k k kn n ng Dg γ+ + ++ ∆ =
![Page 36: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/36.jpg)
Consistent elastoplastic tangent operator The consistent elastoplastic tangent operator is computed taking the variation of the return mapping equation, yielding, where the variations of the trial stress tensor, plastic multiplier, and trial unit normal to the yield surface at time n+1, have to be computed.
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 40
1 1 1 1
1 1 1 1 1 1
2
2 2
trial trialn n n n
trial trial trialn n n n n nd d d d
γ µ
γ µ γ µ+ + + +
+ + + + + +
= −
= − −
σ σ nσ σ n n
![Page 37: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/37.jpg)
The variation of the plastic multiplier at time n+1 is computed setting the variation of the residual equal to zero,
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 41
( ) ( ) ( )( )( ) ( )
( )( )( )( )
2 2 21 1 1 13 3 3
2 2 21 1 1 1 13 3 3
2 221 1 13 33
12 22
1 1 13 33
: 2
: 2
: 2 0
2
trialn n n n n n
trialn n n n n n
trialn n n n
trialn n n n
g f H
dg df d H d
df d H
d H df
γ µ ξ γ ξ
γ µ ξ γ γ
γ µ ξ γ
γ µ ξ γ
+ + + +
+ + + + +
+ + +
−
+ + +
′ ′= − + − Π + −Π
′′= − + − Π +
′′= − + Π + + =
′′= + Π + +
![Page 38: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/38.jpg)
Substituting the variations shown before, the following discrete tangent constitutive equation can be obtained, where the consistent elastoplastic tangent operator at time n+1 is given by
and the following parameters have been introduced
J2 Plasticity Algorithms > J2 Rate Independent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 42
1 1 1:epn n nd d+ + +=σ ε
1 1 1 1 112 23
ep trial trialn n n n nκ µδ µδ+ + + + +
= ⊗ + − ⊗ − ⊗
1 1 I 1 1 n n
( ) ( )11 1 12 22
1 13 33
2 2: 1 , : 1dev 2
nn n ntrial
n n n n Hµ γ µδ δ δ
µ ξ γ+
+ + ++ +
= − = − −− ′′+ Π + +σ q
![Page 39: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/39.jpg)
Contents 1. Introduction 2. J2 Rate independent plasticity algorithms
1. Return mapping algorithm 2. Consistent elastoplastic tangent modulus 3. Step by step algorithm 4. Nonlinear isotropic hardening
3. J2 Rate dependent plasticity algorithms 1. Return mapping algorithm 2. Consistent elastoplastic tangent modulus 3. Step by step algorithm 4. Nonlinear isotropic hardening
4. J2 Computational plasticity assignment
J2 Plasticity Algorithms > Contents
Contents
April 20, 2015 Carlos Agelet de Saracibar 43
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1. Additive split of strains
2. Constitutive equations
3. Associative plastic flow rule
4. Yield function
5. Plastic multiplier
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
J2 Rate dependent plasticity model
April 20, 2015 Carlos Agelet de Saracibar 44
( ) { } { }23, , , diag , ,e
e e q K Hψ= ∂ = = σ q 1E
S E CE S: , C :=
{ } { } { }: : ,0,0 , : , , , : , ,e p p p e eξ ξ= + = = = − −ε ε ξ ε ξE E E , E E E
( )p fγ= ∂
SE S
( ) ( ) ( )23
: , , : dev Yf f q qσ= = − − −σ q σ qS
( )1 0fηγ = ≥S
![Page 41: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/41.jpg)
Associative plastic flow rule: plastic strains at time n+1
Using a Backward-Euler (BE) time integration scheme yields,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 45
( )p fγ= ∂
SE S
( )11 1 1n
p pn n n nt fγ
++ + += + ∆ ∂SE E S
![Page 42: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/42.jpg)
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,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 46
( ) ( ) { }23, diag , ,e
e e p K Hψ= ∂ = = − 1E
S E CE C E E C :=
( )1 1 1 1e p
n 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
γ
γ+
+
+ + + +
+ + +
= − − ∆ ∂
= − − ∆ ∂
S
S
S C E E S
C E E C S
![Page 43: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/43.jpg)
Trial state at time n+1 The trial state at time n+1 is defined by freezing the plastic behaviour
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 47
( ) ( )( )
,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
![Page 44: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/44.jpg)
Return mapping algorithm The return mapping algorithm takes the form,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 48
( )11 1 1 1n
trialn n n nt fγ
++ + + += − ∆ ∂SS S C S
1 1 1 1 1 1 1
1 1 1
21 1 1 13
: 2
: 2 3
:
trial trialn n n n n n n
trialn n n
trialn n n n
t t
q q t K
t H
γ γ µ
γ
γ
+ + + + + + +
+ + +
+ + + +
= − ∆ = − ∆ = − ∆ = + ∆
σ σ n σ n
q q n
![Page 45: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/45.jpg)
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
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 49
( ) ( )21 1 1 1 13: : devn n n n Y nf f qσ+ + + + += = − − −σ qS
( )11 1 1 1n
trialn n n nt fγ
++ + + += − ∆ ∂SS S C S
1 1 1: e pn n n+ + += +E E E
( )11 1 1n
p pn n n nt fγ
++ + += + ∆ ∂SE E S
( )11 1 0n nfηγ + += ≥S
![Page 46: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/46.jpg)
Return mapping algorithm The return mapping algorithm takes the form,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 50
( )11 1 1 1n
trialn n n nt fγ
++ + + += − ∆ ∂SS S C S
1 1 1 1 1 1 1
1 1 1
21 1 1 13
: 2
: 2 3
:
trial trialn n n n n n n
trialn n n
trialn n n n
t t
q q t K
t H
γ γ µ
γ
γ
+ + + + + + +
+ + +
+ + + +
= − ∆ = − ∆ = − ∆ = + ∆
σ σ n σ n
q q n
![Page 47: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/47.jpg)
Return mapping algorithm Taking into account that the return mapping algorithm takes place on the deviatoric plane, it can be written as,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 51
1 1 1 1 1 1 1
1 1 1
21 1 1 13
dev dev : dev 2
: 2 3
:
trial trialn n n n n n n
trialn n n
trialn n n n
t t
q q t K
t H
γ γ µ
γ
γ
+ + + + + + +
+ + +
+ + + +
= − ∆ = − ∆ = − ∆ = + ∆
σ σ n σ n
q q n
![Page 48: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/48.jpg)
The solution for the return mapping algorithm yields,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 52
( )21 1 1 1 1 13dev dev 2trial trial
n n n n n nt Hγ µ+ + + + + +− = − − ∆ +σ q σ q n
( )21 1 1 1 1 1 1 13dev dev 2trial trial trial
n n n n n n n nt Hγ µ+ + + + + + + +− = − − ∆ +σ q n σ q n n
( )( )21 1 1 1 1 1 13dev 2 dev trial trial trial
n n n n n n nt Hγ µ+ + + + + + +− + ∆ + = −σ q n σ q n
( )21 1 1 1 13
1 1
dev 2 dev trial trialn n n n n
trialn n
t Hγ µ+ + + + +
+ +
− + ∆ + = −
=
σ q σ q
n n
![Page 49: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/49.jpg)
For the non-trivial case (plastic loading), the discrete plastic multiplier reads,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 53
( )1 1 1 1 1if 0 then , ,n n n n nf qγ γ η+ + + + +> =σ q
( ) ( )
( ) ( )( ) ( )
21 1 1 1 1 13
2 2 21 1 1 13 3 3
2 21 1 1 1 13 3
, , dev
dev 2
, , 2
n n n n n Y n
trial trial trialn n n Y n
trial trial trialn n n n n
f q q
t K H q
f q t K H
σ
γ µ σ
γ µ γ η
+ + + + + +
+ + + +
+ + + + +
= − − −
= − − ∆ + + − −
= − ∆ + + =
σ q σ q
σ q
σ q
12 2
1 13 32 trialn nt K H f
tηγ µ
−
+ + ∆ = + + + ∆
![Page 50: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/50.jpg)
The return mapping algorithm takes the form,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 54
12 2
1 1 1 13 3
12 2
1 1 13 3
12 2
1 1 1 13 3
2 2
: 2 2 3
2: 23
trial trial trialn n n n
trial trialn n n
trial trial trialn n n n
K H ft
q q K H f Kt
K H f Ht
ηµ µ
ηµ
ηµ
−
+ + + +
−
+ + +
−
+ + + +
= − + + + ∆ = − + + + ∆ = + + + + ∆
σ σ n
q q n
1 1 1 1 1 1 1
1 1 1
21 1 1 13
: 2
: 2 3
:
trial trialn n n n n n n
trialn n n
trialn n n n
t t
q q t K
t H
γ γ µ
γ
γ
+ + + + + + +
+ + +
+ + + +
= − ∆ = − ∆ = − ∆ = + ∆
σ σ n σ n
q q n
![Page 51: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/51.jpg)
The update of the plastic internal variables takes the form,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Return mapping algorithm
April 20, 2015 Carlos Agelet de Saracibar 55
1 1 1
1 1
1 1 1
2 3
p pn n n n
n n n
n n n n
t
t
t
γ
ξ ξ γ
γ
+ + +
+ +
+ + +
= + ∆ = + ∆ = − ∆
ε ε n
ξ ξ n1
2 21 1 13 3
12 2
1 13 3
12 2
1 1 13 3
2
2 2 3
2
p p trial trialn n n n
trialn n n
trial trialn n n n
K H ft
K H ft
K H ft
ηµ
ηξ ξ µ
ηµ
−
+ + +
−
+ +
−
+ + +
= + + + + ∆ = + + + + ∆ = − + + + ∆
ε ε n
ξ ξ n
![Page 52: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/52.jpg)
Consistent elastoplastic tangent operator The consistent elastoplastic tangent operator is computed taking the variation of the stress tensor at time n+1, yielding,
where the variations of the trial stress tensor, trial yield function, and trial unit normal to the yield surface at time n+1, have to be computed.
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 56
12 2
1 1 1 13 3
12 2
1 1 1 13 3
12 2
1 13 3
2 2
2 2
2 2
trial trial trialn n n n
trial trial trialn n n n
trial trialn n
K H ft
d d K H dft
K H f dt
ηµ µ
ηµ µ
ηµ µ
−
+ + + +
−
+ + + +
−
+ +
= − + + + ∆
= − + + + ∆
− + + + ∆
σ σ n
σ σ n
n
![Page 53: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/53.jpg)
The variation of the trial stress tensor at time n+1 takes the form,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 57
( ) ( )( ) ( ) ( ) ( )( ) ( )( )
, , , ,1 1 1 1 1
1 1 1 1
, , , ,1 1 1 1 1
1
tr 2 tr 2 dev
tr 2 tr 2 dev
tr 2 tr 2 dev
tr
trial e trial e trial e trial e trialn n n n n
p pn n n n n n
trial e trial e trial e trial e trialn n n n n
n
d d d
d
λ µ κ µ
λ µ κ µ
λ µ κ µ
λ
+ + + + +
+ + + +
+ + + + +
+
= + = +
= + − = + −
= + = +
=
σ ε 1 ε ε 1 ε
ε 1 ε ε ε 1 ε ε
σ ε 1 ε ε 1 ε
ε 1 ( )( ) ( ) ( )( )
1 1 1
11 1 13
2 tr 2 dev
: 2 2 :
n n n
n n n
d d d
d d d
µ κ µ
λ µ κ µ
+ + +
+ + +
+ = +
= ⊗ + = ⊗ + − ⊗
ε ε 1 ε
1 1 ε ε 1 1 1 1 ε
![Page 54: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/54.jpg)
The variation of the trial yield function at time n+1 takes the form,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 58
( )( )
21 1 1 13
21 3
1 1 1 1 1 1
1 1
dev
dev
dev : dev : 2 dev
: 2
trial trial trial trialn n n Y n
trialn n Y n
trial trial trial trial trialn n n n n n n
trialn n
f q
q
df d d d
d
σ
σ
µ
µ
+ + + +
+
+ + + + + +
+ +
= − − −
= − − −
= − = =
=
σ q
σ q
σ q n σ n ε
n ε
![Page 55: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/55.jpg)
The variation of the unit normal to the yield surface at time n+1 takes the form,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 59
( )( )
1
1
1 1 11
1 1 1
11 1 1 1dev
11 1 1dev
dev devdev dev
dev dev
: dev
trialn n
trialn n
trial trial trialtrial n n n nn trial trial trial
n n n n
trial trial trial trialn n n n n
trial trial trn n n
d d d
d+
+
+ + ++
+ + +
+ + + +−
+ + +−
− −= =
− −
= − −
= − ⊗
σ q
σ q
σ q σ qnσ q σ q
n σ n σ q
n n σ
( )1
1
11 1 1dev
11 1 1dev
: 2 dev
1 : 23
trialn n
trialn n
ial
trial trialn n n
trial trialn n n
d
d
µ
µ
+
+
+ + +−
+ + +−
= − ⊗
= − ⊗ − ⊗
σ q
σ q
n n ε
1 1 n n ε
![Page 56: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/56.jpg)
Substituting the variations shown before, the following discrete tangent constitutive equation can be obtained, where the consistent elastoplastic tangent operator at time n+1 is given by
and the following parameters have been introduced
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Consistent elastoplastic tangent operator
April 20, 2015 Carlos Agelet de Saracibar 60
1 1 1:epn n nd d+ + +=σ ε
1 1 1 1 112 23
ep trial trialn n n n nκ µδ µδ+ + + + +
= ⊗ + − ⊗ − ⊗
1 1 I 1 1 n n
( )11 1 1
2 213 3
2 2: 1 , : 1dev 2
nn n ntrial
n n
t
K Ht
µ γ µδ δ δηµ
++ + +
+
∆= − = − −
− + + +∆
σ q
![Page 57: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/57.jpg)
J2 Plasticity algorithm Step 1. Given the strain tensor 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
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 61
( ) ( )( )
,1
, ,1 1 1
, ,1 1 1 1 1
21 1 1 13
:
:
:
: dev
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 qσ
+
+ + +
+ + + + +
+ + + +
=
= −
= = − = −
= − − −σ q
E E
E E E
S CE C E E C E E
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Step 3. Check the trial yield function at time n+1 Step 4. Compute the discrete plastic multiplier at time n+1
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 62
( ) ( )1 11 1if 0 then set , and exittrialtrial ep
n nn nf + ++ +
≤ • = • =
12 2
1 13 32 trialn nt K H f
tηγ µ
−
+ + ∆ = + + + ∆
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Step 5. Return mapping algorithm (closest-point-projection)
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 63
( )1
1 1 1 1trialn
trial trialn n n nt fγ
++ + + += − ∆ ∂
SS S C S
12 2
1 1 1 13 3
12 2
1 1 13 3
12 2
1 1 1 13 3
2 2
: 2 2 3
2: 23
trial trial trialn n n n
trial trialn n n
trial trial trialn n n n
K H ft
q q K H f Kt
K H f Ht
ηµ µ
ηµ
ηµ
−
+ + + +
−
+ + +
−
+ + + +
= − + + + ∆ = − + + + ∆ = + + + + ∆
σ σ n
q q n
![Page 60: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/60.jpg)
Step 6. Update plastic internal variables database at time n+1
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 64
( )1
1 1 1trialn
p p trialn n n nt fγ
++ + += + ∆ ∂
SE E S
12 2
1 1 13 3
12 2
1 13 3
12 2
1 1 13 3
2
2 2 3
2
p p trial trialn n n n
trialn n n
trial trialn n n n
K H ft
K H ft
K H ft
ηµ
ηξ ξ µ
ηµ
−
+ + +
−
+ +
−
+ + +
= + + + + ∆ = + + + + ∆ = − + + + ∆
ε ε n
ξ ξ n
![Page 61: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/61.jpg)
Step 7. Compute the consistent elastoplastic tangent operator
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
J2 Plasticity algorithm
April 20, 2015 Carlos Agelet de Saracibar 65
1 1 1 1 112 23
ep trial trialn n n n nκ µδ µδ+ + + + +
= ⊗ + − ⊗ − ⊗
1 1 I 1 1 n n
( )11 1 1
2 213 3
2 2: 1 , : 1dev 2
nn n ntrial
n n
t
K Ht
µ γ µδ δ δηµ
++ + +
+
∆= − = − −
− + + +∆
σ q
![Page 62: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/62.jpg)
Nonlinear isotropic hardening Exponential saturation law + linear isotropic hardening
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 66
( ) ( ):q ξ ξψ ξ ξ′= −∂ = −∂ Π = −Π
( ) ( )( ): : 1 expYq Kξψ σ σ δξ ξ∞= −∂ = − − − − −
( ) ( ) ( )( )1 expY Kξ σ σ δξ ξ∞′Π = − − − +
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Time discrete nonlinear isotropic hardening
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 67
( ) ( )( ) ( )
( ) ( )
1 1 1
1 1
1 1 1
: 2 3
:
: 2 3
n n n n
trial trialn n n n
trialn n n n n
q t
q q
q q t
ξ ξ γ
ξ ξ
ξ γ ξ
+ + +
+ +
+ + +
′ ′= −Π = −Π + ∆
′ ′= −Π = −Π =
′ ′= −Π + ∆ +Π
![Page 64: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/64.jpg)
Plastic loading: Yield function at time n+1
Nonlinear residual scalar equation on the plastic multiplier at time n+1
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 68
( ) ( ) ( )( )2 2 21 1 1 13 3 3
1
2
0
trialn n n n n n
n
f f t H tγ µ ξ γ ξ
γ η
+ + + +
+
′ ′= − ∆ + − Π + ∆ −Π
= >
( )
( ) ( )( )1 1 1 1
2 2 21 1 1 13 3 3
: 0
: 2 0
n n n n
trialn n n n n n
g g f
g f t H tt
γ γ η
ηγ µ ξ γ ξ
+ + + +
+ + + +
= = − =
′ ′= − ∆ + + − Π + ∆ −Π = ∆
![Page 65: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/65.jpg)
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
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 69
10, 0knk γ += =
1 1 1 0k k kn n ng Dg γ+ + ++ ∆ =
11 1 1
k k kn n nγ γ γ++ + += + ∆
![Page 66: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/66.jpg)
Newton-Raphson iterative solution algorithm
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 70
( ) ( )( )( )
( )
21 1 1 3
2 213 3
2 2 21 1 1 1 13 3 3
2 221 13 33
: 2
: 2
: 2
k trial kn n n
kn n n
k k k k kn n n n n n
k kn n n
g f t Ht
t
Dg H t t tt
t H tt
ηγ µ
ξ γ ξ
ηγ µ γ ξ γ γ
ηµ ξ γ γ
+ + +
+
+ + + + +
+ +
= − ∆ + + ∆
′ ′− Π + ∆ −Π
′′∆ = − + + ∆ ∆ − Π + ∆ ∆ ∆ ∆ ′′= − + Π + ∆ + + ∆ ∆ ∆
1 1 1 0k k kn n ng Dg γ+ + ++ ∆ =
![Page 67: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/67.jpg)
Consistent elastoplastic tangent operator The consistent elastoplastic tangent operator is computed taking the variation of the return mapping equation, yielding, where the variations of the trial stress tensor, plastic multiplier, and trial unit normal to the yield surface at time n+1, have to be computed.
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 71
1 1 1 1
1 1 1 1 1 1
2
2 2
trial trialn n n n
trial trial trialn n n n n n
td d d t t d
γ µ
γ µ γ µ+ + + +
+ + + + + +
= − ∆
= − ∆ − ∆
σ σ nσ σ n n
![Page 68: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/68.jpg)
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,
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 72
( ) ( )( )( )
( )
2 2 21 1 1 13 3 3
2 2 21 1 1 1 13 3 3
2 221 1 13 33
21 3
: 2 0
: 2
: 2 0
2
trialn n n n n n
trialn n n n n n
trialn n n n
n n
g f t H tt
dg df d t H t d tt
df d t t Ht
d t
ηγ µ ξ γ ξ
ηγ µ ξ γ γ
ηγ µ ξ γ
γ µ ξ
+ + + +
+ + + + +
+ + +
+
′ ′= − ∆ + + − Π + ∆ −Π = ∆ ′′= − ∆ + + − Π + ∆ ∆ ∆ ′′= − ∆ + Π + ∆ + + = ∆
′′∆ = + Π +( )1
221 133
trialn nt H df
tηγ
−
+ + ∆ + + ∆
![Page 69: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/69.jpg)
Substituting the variations shown before, the following discrete tangent constitutive equation can be obtained, where the consistent elastoplastic tangent operator at time n+1 is given by
and the following parameters have been introduced
J2 Plasticity Algorithms > J2 Rate Dependent Plasticity Algorithms
Nonlinear isotropic hardening
April 20, 2015 Carlos Agelet de Saracibar 73
1 1 1:epn n nd d+ + +=σ ε
1 1 1 1 112 23
ep trial trialn n n n nκ µδ µδ+ + + + +
= ⊗ + − ⊗ − ⊗
1 1 I 1 1 n n
( )( )1
1 1 12 221
13 33
2 2: 1 , : 1dev 2
nn n ntrial
n nn n
t
t Ht
µ γ µδ δ δηµ ξ γ
++ + +
++
∆= − = − −
− ′′+ Π + ∆ + +∆
σ q
![Page 70: Computational Solid Mechanicsagelet.rmee.upc.edu/master/Chapter 4. J2 Plasticity Algorithms v1.0... · Computational Solid Mechanics Computational Plasticity C. Agelet de Saracibar](https://reader031.vdocuments.us/reader031/viewer/2022021513/5b0475ff7f8b9a6c0b8dd8be/html5/thumbnails/70.jpg)
Contents 1. Introduction 2. J2 Rate independent plasticity algorithms
1. Return mapping algorithm 2. Consistent elastoplastic tangent modulus 3. Step by step algorithm 4. Nonlinear isotropic hardening
3. J2 Rate dependent plasticity algorithms 1. Return mapping algorithm 2. Consistent elastoplastic tangent modulus 3. Step by step algorithm 4. Nonlinear isotropic hardening
4. J2 Computational plasticity assignment
J2 Plasticity Algorithms > Contents
Contents
April 20, 2015 Carlos Agelet de Saracibar 74
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Implement in MATLAB the BE time-stepping algorithm for J2 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
J2 Plasticity Algorithms > J2 Computational Plasticity Assignment
J2 Computational plasticity assignment
April 20, 2015 Carlos Agelet de Saracibar 75
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For the perfect plasticity models, plot the stress11-strain11 and the dev[stress11]-strain11 curves.
For the linear isotropic/linear kinematic hardening models, plot the stress11-strain11 and dev[stress11]-strain11 curves showing the influence of the isotropic/kinematic hardening parameters
For the nonlinear isotropic hardening model, plot the stress11-strain11 and dev[stress11]-strain11 curves showing the influence of the exponential coefficient of the exponential saturation law
J2 Plasticity Algorithms > J2 Computational Plasticity Assignment
J2 Computational plasticity assignment
April 20, 2015 Carlos Agelet de Saracibar 76
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For the rate-dependent plasticity models, plot the stress11-strain11, dev[stress11]-strain11, and the stress11-time and dev[stress11]-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
J2 Plasticity Algorithms > J2 Computational Plasticity Assignment
J2 Computational plasticity assignment
April 20, 2015 Carlos Agelet de Saracibar 77
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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
J2 Plasticity Algorithms > J2 Computational Plasticity Assignment
J2 Computational plasticity assignment
April 20, 2015 Carlos Agelet de Saracibar 78