Manual

manual

User Manual:

Open the PDF directly: View PDF .
Page Count: 21

Download
Open PDF In Browser	View PDF

NEML: Nuclear Engineering Material Library

Mark Messner

October 23, 2017

Chapter 1
Conventions
1.1

Mechanics

We use the Mandel notation to convert symmetric second and fourth order
tensors to vectors and matrices. The convention transforms the second order
tensor



σ11
 σ12
σ13

σ12
σ22
σ23


σ13

σ23  → σ11
σ33

σ22

√

σ33

2σ23

and, after transformation, a fourth order tensor










C1111
C1122
√C1133
√2C1123
√2C1113
2C1112

C1122
C2222
√C2233
√2C2223
√2C2213
2C2212

C1133
C2233
√C3333
√2C3323
√2C3313
2C3312

√
√2C1123
√2C2223
2C3323
2C2323
2C2313
2C2312

For symmetric two second order tensors
and

b̂

a

and

√

2σ13

√

2σ12

C becomes
√
√
√2C1113 √2C1112
√2C2213 √2C2212
2C3313
2C3312
2C2313
2C2312
2C1313
2C1312
2C1312
2C1212
b



(1.1)





.




(1.2)

and their Mandel vectors

â

the relation

a : b = â · b̂

(1.3)

expresses the utility of this convention. Similarly, given the symmetric fourth
order tensor

C and its equivalent Mandel matrix Ĉ contraction over two adjacent

indices

a=C:b
simply becomes matrix-vector multiplication

â = Ĉ · b̂.

1

(1.4)

1.2

Interfaces

NEML supports three interfaces into the material library.
1. update_ldF: Large deformations with kinematics expressed directly through
the deformation gradient

F.

2. update_ldI: Incremental large deformations with kinematics expressed
through the spatial velocity gradient l.
3. update_sd:

Incremental small deformations with kinematics expressed

through the small strain rate

ε̇.

A user dening a new material model selects one of these four interfaces when
implementing a new material model. NEML takes the interface the user elected
to implement and automatically translates the implemented interfaces to conform to the remaining two. The translation from interfaces 1 or 2 to interface
3 does not fundamentally change the implemented material model  it simply
applies the assumptions of small deformation kinematics. However, the conversion from interface 3 to interface 1 or 2 assumes a hypoelastic formulation using
the Jaumann objective stress rate. This can fundamentally and detrimentally
alter the material constitutive response [].
The library also provides several non-implementable interfaces:
1. update_warp3d: Interface linking the material library to the warp3d nite
element code.
2. update_umat_incremental: Interface for linking the material library to
ABAQUS Implicit via the UMAT interface and using the incremental
strains.
These interfaces will introduce similar errors as the small deformation to large
deformation translation step described above.

2

Chapter 2
Material models
2.1

Yield surfaces

2.1.1 Von Mises with isotropic hardening
q = [σF ]α = [ε̄p ]
r
r
2
2
f = J2 (σ) +
σF (ε̄p , T ) = ksk +
σF
3
3

Internal variables are

1
3 tr σIfor various yield strength functions.
The derivatives are:

with

s=σ−

∂f
∂σ
∂2f
∂σ∂σ
∂2f
∂σ∂q

=
=

s
ksk


1
1
s
s
1 − (I ⊗ I) −
⊗
ksk
3
ksk ksk

= 0

and

∂f
∂q
∂2f
∂q∂q
∂2f
∂q∂σ

r
=
=

2
3

0

= 0

2.1.2 Von Mises with isotropic and kinematic hardening
Internal variables are

q=



σF



X α = ε̄p
3

εp



r
f = J2 (σ + X) +

2
σF (ε̄p , T ) = ks + Xk +
3

r

2
σF .
3

The derivatives are:

∂f
∂σ
∂2f
∂σ∂σ
∂2f
∂σ∂q

s+X
ks + Xk


1
1
s+X
s+X
1 − (I ⊗ I) −
⊗
ks + Xk
3
ks + Xk ks + Xk



1
s+X
s+X
1−
⊗
ks + Xk
ks + Xk ks + Xk

=
=
=

and

∂f
∂q
∂2f
∂q∂q
∂2f
∂q∂σ

2.2

=

h q
"

=

s+X
ks+Xk

2
3

i

0
0

1
ks+Xk

1−

s+X
ks+Xk

"
=

#

0


⊗

s+X
ks+Xk


#

0
1
ks+Xk



1−

1
3

(I ⊗ I) −

s+X
ks+Xk

⊗

s+X
ks+Xk

Hardening Rules

2.2.1 Associative

2.2.1.1 Isotropic
Linear

σF (ε̄p , T ) = − (σ0 + K ε̄p )
∂σF
= −K
∂ ε̄p

Voce

σF (ε̄p , T ) = − [σ0 + R (1 − exp (−δ ε̄p ))]
∂σF
= −δR exp (−δ ε̄p )
∂ ε̄p

2.2.1.2 Kinematic
Linear

X (εp , T ) = −Hεp
dX
= −HI
dεp
4



2.2.2 Nonassociative

2.2.2.1 Frederick-Armstrong & Chaboche
A traditional model of this type uses an associative ow rule for the plastic
strain, with a yield surfaces expecting an isotropic hardening parameter and a
backstress.

The isotropic hardening rule can be selected from Section 2.2.1.1

while the backstress follows the formulation:

n
X

X=

Xi

i=1

−

Ẋi =

2
Ci n +
3

r

A Frederick-Armstrong model has

2
γi (ε̄p ) Xi
3

n = 1.
γ (ε̄p ):

So far the library has two forms for

!

r
γ̇ − Ai

3
a −1
kXi k i Xi
2

constant and a form proposed by

Chaboche in his book:

γ (ε̄p ) = γs + (γ0 − γs ) e−β ε̄p .

2.3

Viscoplastic models

2.3.1 Perzyna
This is implemented as associative viscoplasticity.

The user provides a yield

surface and a hardening rule from the list of associative models above.

The

formulation then uses the associative ow and hardening rules and the rate
function:


y (σ, α) =

where

g

g (f (σ, q (α)))
η

is some monotonic function.

power law for



The library currently only provides a

g:
g (f ) = f n .

2.3.2 Chaboche
The Chaboche viscoplastic model, as described in [], uses the ow rule associated to surface

f (σ, q),

a Chaboche backstress, described above, Voce isotropic

hardening, and the rate rule:

+n
r *
3 f (σ, q)
p
y (σ, q) =
.
2
2/3η
The NEML implementation allows for an arbitrary backstress and hardening
rule.

5

2.3.3 Yaguchi & Takahashi
This model is described by the equations:

ε̇p = nṗ
3 σ 0 − X0
2 J2 (σ 0 − X0 )
r
3 0
0
J2 (Y ) =
Y : Y0
2
n

J2 (σ 0 − X0 ) − σa
ṗ =
D
n=

X = X1 + X2

2
m−1
Ẋ1 = C1
(a10 − Q) n − X1 ṗ − γ1 J2 (X1 )
X1
3


2
m−1
a2 n − X2 ṗ − γ2 J2 (X2 )
Ẋ2 = C2
X2
3


Q̇ = d (q − Q) ṗ
= b (σas − σa ) ṗ
(
bh σas − σa ≥ 0
b =
br σas − σa < 0

σ̇a

= hA + B log10 ṗi

σas

The authors provided interpolation schemes for the model parameters in the
range

473 K < T < 873 K.

These parameters are hard-coded in the implemen-

tation, so this model takes no parameters.

2.4

Creep models

In general these models have the form

ε̇cr = f (σ, εcr , t, T )
so we need to keep in mind the explicit time dependence. The idea is to add
the integrated creep strain to a rate-independent plasticity model for a classical
non-unied approach. We solve the creep rate equation implicitly as

cr
εcr
n+1 = εn + f n+1 ∆tn+1
with the residual

cr
R = εcr
n+1 − εn − f n+1 ∆tn+1

6

and the Jacobian

J =I−

∂f
∆tn+1
∂εcr
n+1

which the user must provide the derivative for.
To solve the eventual outer nonlinear equation and to calculate the tangent
we need the additional derivative

dεcr
n+1
.
dσn+1
Play our usual game of

R = 0 =⇒ 0 = dR =

R = 0 =⇒ 0 = dR =
0=

∂R
∂R
∂R
: dσn+1 + cr : dεcr
: dtn+1
n+1 +
∂σn+1
∂εn+1
∂tn+1

∂R
∂R
∂R
∂R
: dσn+1 + cr : dεcr
: dtn+1 +
: dTn+1
n+1 +
∂σn+1
∂εn+1
∂tn+1
∂tn+1

dεcr
∂R
∂R
∂R
dtn+1
∂R
dTn+1
n+1
+ cr :
+
:
+
:
∂σn+1
∂εn+1 dσn+1
∂tn+1 dσn+1
∂tn+1 dσn+1

and pretending the time and temperature variation is small. Then

dεcr
n+1
=−
dσn+1



∂R
∂εcr
n+1

−1

∂R
∂σn+1

dεcr
∂f
n+1
= J −1
∆tn+1 .
dσn+1
∂σn+1
Therefore the whole model is dened by the function

∂f ∂f
∂f
∂σ , ∂t , and ∂T .

f

and derivatives

∂f
∂εcr ,

2.4.1 J2 creep
The above gives the specic framework, but we'll go further are assume that

ε̇cr = g (σeq , εeq , t, T )
with

3 s
2 σeq

1
s = σ − tr (σ)
3
r
3
σeq =
s:s
2

and

r
εeq =

2 cr cr
ε :ε .
3

7

Then we can dene the whole thing in terms of the nice scalar function

3 s
2 σeq
 


 

∂f
3
∂g
g
s
s
g
3
1
=
−
⊗
+
I
I − 1⊗1
∂σ
2 ∂σeq
σeq
σeq
σeq
σeq
2
3

 

∂f
s
∂g εcr
=
⊗
cr
∂ε
σeq
∂εeq εeq
f =g

∂f
∂g 3 s
=
∂t
∂t 2 σeq
∂g 3 s
∂f
=
∂T
∂T 2 σeq

2.4.2 Specic creep laws

2.4.2.1 Power law

n
Aσeq

g=
∂g
=
∂σeq
∂g
=
∂εeq
∂g
=
∂t

n−1
Anσeq

0
0

2.4.2.2 Norton-Bailey
g=
∂g
=
∂σeq
∂g
=
∂εeq
∂g
=
∂t

(m−1)/m

n/m
mA1/m σeq
(εeq )

(m−1)/m

n/m−1
nA1/m σeq
(εeq )

−1/m

n/m
(m − 1) A1/m σeq
(εeq )

0

8

g:

Chapter 3
Integration Algorithms
3.1

Elasticity

The equations are simply

where

C

σn+1 =

Cn+1 : εn+1

An+1 =

Cn+1

is some (possibly temperature dependent) elasticity tensor. This model

has no history variables.

3.2

Perfect plasticity
e

•

Trial state:εn+1

•

Evaluate

= εen , σn+1 = Cn+1 : εn+1 − εn + εen+1

f (σn+1 ).




If less than zero return, else do Newton iteration on

the scheme described here.
Equations:

εpn+1

with

σn+1

= εpn +

f (σn+1 , αn+1 ) =


= Cn+1 : εn+1 − εpn+1 .

∂fn+1
∆γn+1
∂σn+1

0

Use an unknowns:

x=



σn+1

∆γn+1



Residual:


R=

e
−εn+1 + C−1
n+1 : σn+1 + εn+1 − εn +
fn+1

Jacobian.

9

∂fn+1
∂σ ∆γn+1




=

R1
R2


.

"
J=

C−1
n+1 +

∂ 2 fn+1
∂σ 2 ∆γn+1
∂fn+1
∂σn+1

∂fn+1
∂σ

0

#


=

J11
J21

J12
J22



The algorithmic tangent is

−1
∂ 2 fn+1
∆γn+1
Φ=
+
∂σ 2
h
i h
i
n+1
n+1
Φ · ∂f∂σ
⊗ Φ · ∂f∂σ
An+1 = Φ −
h
i2
∂fn+1
∂fn+1
∂σ · Φ · ∂σ


3.3

C−1
n+1

General Newton algorithm for rate-independent
plasticity

For associative ow this algorithm has the interpretation of a closest point
projection.

3.3.1 Continuous equations
ε̇p

= γg (σ, α)

α̇

= γh (σ, α)

and the Kuhn-Tucker and consistency conditions

≥ 0

γ

f (σ, α) ≤ 0
γf (σ, α)

=

0

3.3.2 Newton integration scheme
•

Trial state:

•

Evaluate

αn+1 = αn ,εpn+1 = εpn , σn+1 = Cn+1 : εn+1 − εpn+1

f (sn+1 , αn+1 ).

If less than zero return, else do Newton iteration

on the scheme described here.
Equations:

with

σn+1 = Cn+1




εpn+1

=

αn+1

= αn + hn+1 ∆γn+1

f (σn+1 , αn+1 ) =


: εn+1 − εpn+1 .

10

εpn + gn+1 ∆γn+1
0

Residual:

 

−εpn+1 + εpn + gn+1 ∆γn+1
R1
R =  −αn+1 + αn + hn+1 ∆γn+1  =  R2  .
fn+1
R3


Jacobian. Helpful note is that

εpn+1 = εn+1 − C−1
n+1 : σn+1
∂σn+1
= −Cn+1
∂εpn+1


∂gn+1
∂σn+1 : Cn+1 ∆γn+1
∂hn+1
: Cn+1 ∆γn+1
− ∂σ
n+1
∂fn+1
− ∂σn+1 : Cn+1

−I −


J=

∂gn+1
∂αn+1 ∆γn+1
∂hn+1
−I + ∂α
∆γn+1
n+1
∂fn+1
∂αn+1

gn+1
hn+1
0





J11
 
 = J21
J31

J12
J22
J32

3.3.3 Algorithmic tangent
Consider the implicit function theorem applied to the residual at the nal iteration:



R εn+1 , εpn+1 , αn+1 , ∆γn+1 = 0
Provided several conditions are met (notably a non-singular Jacobian at this
point), we can solve for the derivatives

dεn+1
dεn+1
dεn+1
,
dαn+1 , and d∆γn+1 in terms of
dεp
n+1

the other derivatives. As it turns out, these other derivatives are parts of the
Jacobian we already computed for the Newton scheme.

dR1

=

dR2

=

dR3

=

∂R1
dεn+1 +
∂εn+1
∂R2
dεn+1 +
∂εn+1
∂R3
dεn+1 +
∂εn+1

∂R1
dεp +
∂εpn+1 n+1
∂R2
dεp +
∂εpn+1 n+1
∂R3
dεp +
∂εpn+1 n+1

∂R1
∂R1
dαn+1 +
d∆γn+1 = 0
∂αn+1
∂∆γn+1
∂R2
∂R2
dαn+1 +
d∆γn+1 = 0
∂αn+1
∂∆γn+1
∂R3
∂R3
dαn+1 +
d∆γn+1 = 0
∂αn+1
∂∆γn+1

0 =

p
∂R1
∂R1 dεn+1
∂R1 dαn+1
∂R1 d∆γn+1
+ p
+
+
∂εn+1
∂εn+1 dεn+1
∂αn+1 dεn+1
∂∆γn+1 dεn+1

0 =

∂R2
∂R2 dεn+1
∂R2 dαn+1
∂R2 d∆γn+1
+ p
+
+
∂εn+1
∂εn+1 dεn+1
∂αn+1 dεn+1
∂∆γn+1 dεn+1

0

∂R3
∂R3 dεn+1
∂R3 dαn+1
∂R3 d∆γn+1
+ p
+
+
∂εn+1
∂εn+1 dεn+1
∂αn+1 dεn+1
∂∆γn+1 dεn+1

p

p

=

Make some associations...

11


J13
J23 
J33


0=
where

K=

A
B

dεp
n+1
dεn+1 . Solve for




+

JKK
JEK

JKE
JEE

A=
and

"
B=

With

K
E



K:

K = JKK − JKE J−1
EE JEK
Here




−1

JKE J−1
EE B − A




∂R1
∂gn+1
=
: Cn+1 ∆γn+1
∂εn+1
∂σn+1

∂R2
∂εn+1
∂R3
∂εn+1

#

∂hn+1
∂σn+1 : Cn+1 ∆γn+1
∂fn+1
∂σn+1 : Cn+1

"
=

#
.

dεp
n+1
dεn+1 in hand we have

dσn+1
=C:
dεn+1

3.4



dεp
I − n+1
dεn+1



General integration algorithm for viscoplasticity

3.4.1 Continuous equations
The system is posed as:



σ̇ σ, q, ε̇, T, Ṫ , t


q̇ σ, q, ε̇, T, Ṫ , t

σ̇ =
q̇ =

3.4.2 Newton algorithm
R1 =
R2 =



−σn+1 + σn + σ̇ σn+1 , qn+1 , ε̇n+1 , Tn+1 Ṫn+1 , , tn+1 ∆tn+1 =


−qn+1 + qn + q̇ σn+1 , qn+1 , ε̇n+1 , Tn+1 , Ṫn+1 , tn+1 ∆tn+1 =

The Jacobian is simple:


J=

Jσσ
Jqσ

Jσq
Jqq

"


=

−I +

∂ σ̇n+1
∂σn+1 ∆tn+1

∂ q̇n+1
∂σn+1 ∆tn+1

12

∂ σ̇n+1
∂qn+1 ∆tn+1
n+1
−I + ∂∂qq̇n+1
∆tn+1

#
.

0
0

3.4.3 Algorithmic tangent
The tangent requires several unusual derivatives. It is formed as:

dR1 =
dR2 =

∂R1
: dσn+1 +
∂σn+1
∂R2
: dσn+1 +
∂σn+1

∂R1
: dqn+1 +
∂qn+1
∂R2
: dqn+1 +
∂qn+1

∂R1
: dε̇n+1 +
∂εn+1
∂R2
: dε̇n+1 +
∂εn+1

∂R1
: dTn+1 +
∂Tn+1
∂R2
: dTn+1 +
∂Tn+1

∂R1
: dtn+1 = 0
∂tn+1
∂R2
: dtn+1 = 0
∂tn+1

Divide through and substitute...

Jσσ : An+1 + Jσq :

dṪn+1
dqn+1
∂R1
∂R1
dTn+1
∂R1
∂R1
dtn+1
:
+
+
:
+
+
:
= 0
dε̇n+1
∂ ε̇n+1
∂Tn+1 dε̇n+1
d
ε̇
∂t
d
ε̇n+1
∂ Ṫn+1
n+1
n+1

Jqσ : An+1 + Jqq :

∂R2
∂R2
dTn+1
∂R2
dṪn+1
∂R2
dtn+1
dqn+1
+
+
:
+
:
+
:
= 0
dε̇n+1
∂ ε̇n+1
∂Tn+1 dε̇n+1
∂tn+1 dε̇n+1
∂ Ṫn+1 dε̇n+1

We will need to revisit this, but as an approximation assume:
1. There is no explicit time dependence or the time dependence is small
compared to the dependence on the physical variables.
2. The temperature change is gradual (need to consider this...).
With these assumptions:

dqn+1
∂R1
+
=
dε̇n+1
∂ ε̇n+1
dqn+1
∂R2
:
+
=
dε̇n+1
∂ ε̇n+1

Jσσ : An+1 + Jσq :

0

Jqσ : An+1 + Jqq

0

and this is basically our old algorithm:

∂ σ̇
∆tn+1
∂ ε̇n+1
∂ q̇
∆tn+1
∂ ε̇n+1

X=
Y=

dqn+1
+X=
dε̇n+1
dqn+1
:
+Y =
dε̇n+1

Jσσ : An+1 + Jσq :

0

Jqσ : An+1 + Jqq

0

An+1 = Jσσ − Jσq : J−1
qq : Jqσ


−1

: Jσq : J−1
qq : Y − X

And the actual tangent is just

Tn+1 = An+1

13

1
∆tn+1




3.4.4 Application to plasticity
Dene the rates as:

σ̇ (σ, q, ε, T, t) =



C : ε̇ − gγ γ̇ − gT Ṫ − gt

q̇ (σ, q, ε, T, t) =

hγ γ̇ + hT Ṫ + ht

The derivatives are then

∂ σ̇
∂σ
∂ σ̇
∂q
∂ q̇
∂σ
∂ q̇
∂q
∂ σ̇
∂ ε̇n+1
∂ q̇
∂ ε̇n+1



=
=
=
=




∂gγ
∂ γ̇
∂gT
∂gt
C: −
γ̇ − gγ ⊗
−
Ṫ −
∂σ
∂σ
∂σ
∂σ




∂gγ
∂ γ̇
∂gT
∂gt
C: −
γ̇ − gγ ⊗
−
Ṫ −
∂q
∂q
∂q
∂q


∂hγ
∂ γ̇
∂ht
∂hT
γ̇ + hγ ⊗
Ṫ +
+
∂σ
∂σ
∂σ
∂σ


∂hγ
∂ γ̇
∂hT
∂ht
γ̇ + hγ ⊗
+
Ṫ +
∂q
∂σ
∂q
∂q

=

C

=

0

14

Chapter 4
Meta-model algorithms
4.1

Combined plasticity and creep

The idea is to combine one of the base models described in the previous chapter
with some rate dependent creep law.
Start with two functions:

ep
ep
σn+1
εep
n+1 , εn , hn , ∆tn+1 , Tn+1




as dened in the previous chapter and

cr
cr
εcr
n+1 εn , σn+1 , ∆tn+1 , Tn+1




dened above. We want to make a combined update so that

εn+1 =

cr
εep
n+1 + εn+1

cr
σn+1
=

ep
σn+1
.

Combine these two equations into the residual





ep
cr
ep
ep
R = εep
n+1 +εn+1 εn − εn , σn+1 εn+1 , εn , hn , ∆tn+1 , Tn+1 , ∆tn+1 , Tn+1 −εn+1
which means we need to supplement the history variables of the elastic-plastic
model with

εep .

We want to solve this residual equation for

εep
n+1 using Newton's

method. The Jacobian of this residual is

dεcr dσn+1
dR
= I + n+1 ep
ep
dεn+1
dσn+1 dεn+1
where

dεcr
dσn+1
n+1
is the algorithmic tangent of the rate independent model and
dσn+1
dεep
n+1

is an output of the creep model. Finally, the overall algorithmic tangent is

dσn+1
=
dεn+1

dεcr
n+1
+
dσn+1

which is annoying but doable.

15



dσn+1
dεep
n+1

−1 !−1

4.2

Kocks-Mecking regime model

The goal of this meta-model is to switch between dierent material representations depending on the normalized thermal activation energy.
The model takes as parameters a collection of material models, as denined
in the previous chapter, a collection of activation energies, and the constants
required to compute the normalized activation energy:

g=

ε̇0
kT
ln .
µb3
ε̇

Let the material models be represented by the tuples of functions

[(σ1 , A1 , h1 ) , (σ2 , A2 , h2 ) , . . . , (σn−1 , An−1 , hn−1 ) , (σn , An , hn )]
and the cutos be

[g1 , g2 , . . . gn−1 ] .
The meta model rst computes the normalized activation energy of the update
from the total strain rate and temperature.

It then dispatches the call for a

stress update to the appropriate function through the rule:



(σ1 , A1 , h1 ) g ≤ g1
(σ, A, h) = (σi , Ai , hi )
gi−1 < g ≤ gi


(σn , An , hn ) g > gn−1 .

4.3

Continuum damage mechanics

4.3.1 Most generic
A damage model is a function

where

σn+1

and

0
σn+1
=

σ 0 (σn+1 , hn+1 , ωn+1 )

ωn+1 =

w (σn+1 , hn+1 , ωn+1 )

hn+1

are the stress tensor and history variables resulting from

an umodied stress update, using one of the models above, and

ωn+1 is a set
0
σn+1
, the

of damage variables. The model returns the damaged stress tensor
set of composite history variables

h0n+1 =

tangent

A0n+1 =



hn+1

ωn+1



and the modied

0
dσn+1
.
dεn+1

Of course these equations must be solved simultaneously with the original material update, complicating the stress update algorithm.

16

4.3.2 Scalar
A scalar damage model simples the general equations to

with

0
σn+1
=

(1 − ωn+1 ) σn+1

ωn+1 =

w (σn+1 , hn+1 , ωn+1 )

ωn+1 ∈ [0, 1].

The modied tangent is

A0n+1 = (1 − ωn+1 ) An+1 −

dωn+1
⊗ σn+1
dεn+1

4.3.3 Simplied scalar
In this simplied case the dependence on the history of the undamage model is
limited to the equivalent plastic strain. So:

0
σn+1
=

[1 − ωn+1 ] σn+1


ω̇ ωn+1 , σ 0n+1 , ε̄˙p
r
2
ε̇p : ε̇p
3

ω̇n+1 =
ε̄˙p =

4.3.3.1 Update
Our combined residual equation is



0
σn+1
− [1 − ωn+1 ] σn+1 (∆εn+1 )
=
−ωn+1 + ωn + ω̇n+1 ∆t
 0

ωn+1 .
x = σn+1

R1
R2

for unknowns








=

0
0



For convience we're going to assume we can additively decompose the update
into two parts:







ω̇ ωn+1 , σ 0n+1 , ε̄˙p = ω̇p ωn+1 , σ 0n+1 ε̄˙p + ω̇o ωn+1 , σ 0n+1
which makes the residual



R1
R2




=

0
σn+1
− [1 − ωn+1 ] σn+1 (∆εn+1 )
−ωn+1 + ωn + ω̇p ∆ε̄p + ω̇o ∆t




=

0
0



and the Jacobian



J 11
J 21

J 12
J22

"


=

I
∂ ω̇p
∂σ 0n+1 ∆ε̄p

+

∂ ω̇o
∂σ 0n+1 ∆t

Now the question is can we get

∆ε̄p

∂∆ε̄p

+ ω̇p ∂σ0

n+1

σn+1
∂ ω̇p
∂ ω̇o
∆ε̄
p + ∂ωn+1 ∆t − 1
∂ωn+1

without explicit knowledge of the plastic

strain?

r
ε̄˙p =

ε̇p = ε̇ − ε̇e = ε̇ − C −1 : σ
r


2
2
ε̇p : ε̇p =
ε̇ : ε̇ + C −1 : σ : C −1 : σ − 2ε̇ : C −1 : σ
3
3
17

#

4.3.3.2 Tangent

4.3.4 Chaboche
0
σn+1
=

ω̇n+1 =
φn+1 =
yn+1 =

sn+1 =
ε̄˙p =

(1 − ωn+1 ) σn+1
∂φn+1
∂yn+1
φ (ωn+1 , yn+1 , sn+1 , ε̄˙p )
"

2 #
2
σeq
2
σm
−
(1 + ν) + 3 (1 − 2ν)
2
σeq
2E (1 − D) 3
s

2
2
σm
(1 + ν) + 3 (1 − 2ν)
σep
3
σeq
r
2
ε̇p : ε̇p
3

18

Chapter 5
Implemented class hierarchy

19

20

Essentially each box in the ow chart corresponds to a node in the XML input le.

Figure 5.1: Structure of NEML. The diagram shows the available model combinations and guides the XML based input format.



---

Source Exif Data:

File Type                       : PDF
File Type Extension             : pdf
MIME Type                       : application/pdf
PDF Version                     : 1.5
Linearized                      : No
Page Count                      : 21
Producer                        : pdfTeX-1.40.15
Creator                         : TeX
Create Date                     : 2017:10:23 21:28:42-05:00
Modify Date                     : 2017:10:23 21:28:42-05:00
Trapped                         : False
PTEX Fullbanner                 : This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2015/dev/Debian) kpathsea version 6.2.1dev

EXIF Metadata provided by EXIF.tools