ABAQUS user subroutines for the simulation of viscoplastic - loicz
ABAQUS user subroutines for the simulation of viscoplastic - loicz
ABAQUS user subroutines for the simulation of viscoplastic - loicz
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Technical Note GKSS/WMS/01/5<br />
internal report<br />
<strong>ABAQUS</strong> <strong>user</strong> <strong>subroutines</strong> <strong>for</strong> <strong>the</strong> <strong>simulation</strong> <strong>of</strong><br />
<strong>viscoplastic</strong> behaviour including anisotropic damage<br />
<strong>for</strong> isotropic materials and <strong>for</strong> single crystals<br />
Weidong Qi, Wolfgang Brocks<br />
June 2001
2<br />
Institut für Werkst<strong>of</strong><strong>for</strong>schung<br />
GKSS-Forschungszentrum Geesthacht
0. Nomenclature 3<br />
1. Introduction 7<br />
2. The CDM-based anisotropic damage model 7<br />
3. Unified models <strong>of</strong> BODNER-PARTOM and <strong>of</strong> CHABOCHE coupled with damage 10<br />
4. Anisotropic creep model <strong>for</strong> cubic single crystals coupled with damage 11<br />
5. User material routines and <strong>the</strong>ir applications 13<br />
5.1 Circumferentially notched bar - CHABOCHE model coupled with damage 13<br />
5.2 Plate containing a hole - BODNER-PARTOM model coupled with damage 19<br />
5.3 Single crystal plate containing a hole - <strong>the</strong> anisotropic creep and damage model <strong>of</strong> BERTRAM,<br />
OLSCHEWSKI & QI 21<br />
5.4 TiAl turbine blade - CHABOCHE model coupled with damage 22<br />
6. References 24<br />
7. Appendices: <strong>ABAQUS</strong>-Inputfiles 26<br />
7.1 Appendix 1: Circumferentially notched bar - CHABOCHE model coupled with damage 26<br />
7.2 Appendix 2: Plate containing a hole - BODNER-PARTOM model coupled with damage 28<br />
7.3 Appendix 3: Single crystal plate containing a hole - <strong>the</strong> anisotropic creep and damage model <strong>of</strong><br />
BERTRAM, OLSCHEWSKI & QI 31<br />
7.4 Appendix 4: TiAl turbine blade - CHABOCHE model coupled with damage 33<br />
4
0. Nomenclature<br />
scalars<br />
a, c, d material parameters <strong>for</strong> kinematic hardening in CHABOCHE's model<br />
b material parameter <strong>for</strong> isotropic hardening in CHABOCHE's model<br />
h(x) HEAVISIDE function<br />
m material parameter <strong>for</strong> damage evolution, eq. (3b)<br />
m1, m1 material parameters in <strong>the</strong> BODNER-PARTOM model<br />
n material parameter in orientation function, eq. (6)<br />
n creep exponent in CHABOCHE's model<br />
p material parameter in eq. (6)<br />
p accumulated effective plastic strain, internal variable in CHABOCHE's model<br />
q material parameter in <strong>the</strong> definition <strong>of</strong> <strong>the</strong> damage-active stress, eq. (2)<br />
r material parameter in CHABOCHE's model<br />
r1, r2<br />
material parameters in <strong>the</strong> BODNER-PARTOM model<br />
A1, A2 material parameters in <strong>the</strong> BODNER-PARTOM model<br />
B0<br />
Ci<br />
Di<br />
D0<br />
DI<br />
DR<br />
Ji<br />
material parameter <strong>for</strong> damage evolution, eqs. (3b) and (4)<br />
temperature-dependent material parameters (i = 1, 2, 3), eqs. (21a-e)<br />
viscosities, temperature-dependent material parameters (i = 1, 2, 3), eqs. (21a-e)<br />
material parameter in <strong>the</strong> BODNER-PARTOM model<br />
maximum principal damage<br />
critical value <strong>of</strong> maximum principal damage<br />
scalar invariants (i = 1, 2, 3, 4)<br />
K viscosity, material parameter in CHABOCHE's model<br />
Ki<br />
Ki<br />
Li<br />
material parameter (i = 1, 2, 3) in <strong>the</strong> BODNER-PARTOM model<br />
temperature-dependent material parameters (i = 1, 2, 3), eqs. (21a-e)<br />
viscosities, temperature-dependent material parameters (i = 1, 2, 3), eqs. (21a-e)<br />
R(p) actual yield stress <strong>for</strong> isotropic hardening in CHABOCHE's model<br />
R 0<br />
initial yield stress in CHABOCHE's model<br />
5
R ∞<br />
Wi<br />
Z I , Z D<br />
Zij<br />
6<br />
saturated yield stress <strong>for</strong> isotropic hardening in CHABOCHE's model<br />
specific work <strong>of</strong> inelastic strain in <strong>the</strong> BODNER-PARTOM model<br />
internal variables <strong>for</strong> isotropic and kinematic hardening in <strong>the</strong> BODNER-PARTOM model<br />
material parameters (i = 1, 2, 3; j = 1, 2, 3, 4), eqs. (22a,b)<br />
β, βi material parameters <strong>for</strong> damage evolution, eqs. (3b, 4)<br />
ε i<br />
e<br />
εi η i<br />
eigen-values (i = 1, 2, 3) <strong>of</strong> total strain tensor E<br />
eigen-values (i = 1, 2, 3) <strong>of</strong> elastic strain tensor E e<br />
orientation function<br />
φ(p) material function <strong>for</strong> kinematic hardening in CHABOCHE's model<br />
φ ∞<br />
ˆ<br />
σ i<br />
vectors<br />
e i<br />
c<br />
ei saturated value <strong>of</strong> φ(p) , material parameter <strong>for</strong> kinematic hardening in CHABOCHE's model<br />
eigen-values (i = 1, 2, 3) <strong>of</strong> damage-active stress tensor ˆ<br />
S<br />
orthogonal unit vectors (i = 1, 2, 3), reference base, e i ⋅ e j = δ ij<br />
lattice vectors (i = 1, 2, 3)<br />
e εe e<br />
n , ni eigen-vectors (i = 1, 2, 3) <strong>of</strong> total and elastic strain tensors, E and E , respectively<br />
i<br />
s<br />
n ˆ i<br />
eigen-vectors (i = 1, 2, 3) <strong>of</strong> damage-active stress tensor ˆ<br />
S<br />
second order tensors<br />
B internal variable <strong>for</strong> kinematic hardening in <strong>the</strong> BODNER-PARTOM model<br />
D damage tensor<br />
Da<br />
E, E e<br />
active damage tensor<br />
total and elastic strain tensor<br />
E + , E e+ positive projections <strong>of</strong> total and elastic strain tensors, E and E e , eqs. (11a,b)<br />
E Ý i<br />
inelastic strain rate tensor in unified models<br />
H ε , H εe spectral tensors, eqs. (8a,b)<br />
I second order identity tensor
S CAUCHY stress tensor<br />
˜<br />
S effective stress tensor<br />
ˆ<br />
S damage-active stress tensor<br />
˜ ′<br />
S deviator <strong>of</strong> <strong>the</strong> damage-active stress<br />
Y D<br />
<strong>the</strong>rmodynamic <strong>for</strong>ce conjugate to damage tensor D<br />
X backstress tensor<br />
7
W internal variable, eq. (20b)<br />
fourth order tensors<br />
< 4><br />
Ai < 4><br />
I<br />
< 4> <br />
P , ε Pεe < 4><br />
R<br />
< 4><br />
S<br />
< 4><br />
T<br />
operations<br />
ab = a i b j e i e j<br />
a ⋅ b = a ib i<br />
8<br />
material tensors (i = 1, 2, 3, 4, 5), eqs. (21a-e)<br />
identity tensor<br />
positive spectral projection operator <strong>for</strong> total and elastic strain tensor, eqs. (10a,b)<br />
lattice tensor, eq. (5)<br />
damage characteristic tensor, eq. (3a)<br />
positive projection operator, eq. (12)<br />
AB = A ij B kl e ie ke je l<br />
A ⋅ B= A ijB jk e ie k<br />
A : B= A ij B ji<br />
tensor product <strong>of</strong> two vectors<br />
scalar product <strong>of</strong> two vectors<br />
tensor product <strong>of</strong> two (second order) tensors<br />
scalar product <strong>of</strong> two (second order) tensors<br />
double scalar product <strong>of</strong> two (second order) tensors<br />
A = A 2 = A ij A ji EUKLIDean norm <strong>of</strong> second order tensor<br />
< 4><br />
C : A = C A e e ijkl kl i j<br />
double scalar product <strong>of</strong> a fourth and a second order tensor
1. Introduction<br />
A CDM (continuum damage mechanics) based anisotropic damage model has been established<br />
by QI and BERTRAM to describe <strong>the</strong> anisotropic development <strong>of</strong> material damage in single crystals [QI<br />
& BERTRAM, 1997; QI, 1998; QI & BERTRAM, 1999; QI, BROCKS & BERTRAM, 2000] and in<br />
isotropic material [QI & BROCKS, 2000a, b, c]. Using <strong>the</strong> effective stress concept <strong>of</strong> CDM, this model<br />
can be coupled with any continuum mechanics model by introducing an adequately defined "effective<br />
stress tensor". The resulting model is <strong>the</strong>n able to describe <strong>the</strong> de<strong>for</strong>mation behaviour with respect to<br />
anisotropic material damage including lifetime predictions. The unified <strong>viscoplastic</strong> model proposed by<br />
BODNER & PARTOM [1975] and by CHABOCHE [CHABOCHE & ROUSSELIER, 1983] <strong>for</strong> polycrystal<br />
alloys and <strong>the</strong> creep model suggested by BERTRAM & OLSCHEWSKI [1996] <strong>for</strong> single crystal alloys,<br />
respectively, are chosen <strong>for</strong> coupling with <strong>the</strong> damage model. The resulting models have been<br />
implemented into <strong>subroutines</strong> <strong>of</strong> <strong>the</strong> FE-code <strong>ABAQUS</strong> as "<strong>user</strong>-defined material models" (UMAT) and<br />
can be used to per<strong>for</strong>m FE computations on structural components <strong>of</strong> poly and single crystals.<br />
This report gives a brief description <strong>of</strong> <strong>the</strong> models and presents some results <strong>of</strong> FE-analyses<br />
using <strong>the</strong> respective UMATs. Materials used <strong>for</strong> <strong>the</strong> analyses are <strong>the</strong> Ni-based superalloy IN738 LC,<br />
<strong>the</strong> Ni-based single crystal SRR99 and a TiAl intermetallic alloy.<br />
2. The CDM-based anisotropic damage model<br />
Damage <strong>of</strong> materials is a progressive process ending in final fracture. A natural characteristic <strong>of</strong><br />
material damage is that <strong>the</strong> damage generally develops anisotropically. A second-order symmetric tensor<br />
D is chosen in <strong>the</strong> present models as damage variable <strong>for</strong> <strong>the</strong> description <strong>of</strong> <strong>the</strong> anisotropic damage.<br />
According to <strong>the</strong> effective stress concept <strong>of</strong> CDM, <strong>the</strong> effect <strong>of</strong> stresses and damage on <strong>the</strong> de<strong>for</strong>mation<br />
behaviour can be represented by an adequately defined effective stress. This effective stress tensor is<br />
defined as<br />
˜<br />
S = (I − D) −1 2 ⋅S⋅ (I − D) −12 , (1)<br />
9
where S is <strong>the</strong> CAUCHY stress tensor and I denotes <strong>the</strong> second-order identity tensor. Similarly, it is<br />
supposed that <strong>the</strong> contribution <strong>of</strong> <strong>the</strong> stress and damage on <strong>the</strong> damage development can be represented<br />
by a newly introduced damage-active stress defined analogously to eq. (1) as<br />
10<br />
ˆ<br />
S = (I − D) −q ⋅S ⋅(I − D) −q =<br />
3<br />
σ σ<br />
∑ σ ˆ ˆ<br />
i n ˆ<br />
i n i , (2)<br />
i=1<br />
σ<br />
where q is a material parameter, σ ˆ and n ˆ<br />
i i (i = 1, 2, 3) are <strong>the</strong> eigen-values and <strong>the</strong> corresponding<br />
eigen-vectors <strong>of</strong> ˆ<br />
S . From <strong>the</strong> point <strong>of</strong> view <strong>of</strong> linear irreversible <strong>the</strong>rmodynamics <strong>the</strong> evolution law <strong>for</strong> a<br />
second order symmetric damage tensor can be generally expressed as:<br />
D Ý = S<br />
<br />
: YD , (3a)<br />
< 4><br />
where YD is <strong>the</strong> <strong>the</strong>rmodynamic <strong>for</strong>ce conjugate to <strong>the</strong> damage tensor, and S<br />
< 4><br />
characteristic tensor <strong>of</strong> rank four, respectively. If <strong>the</strong> fourth-order tensor S<br />
is <strong>the</strong> damage<br />
is symmetric and positive-<br />
definite, <strong>the</strong> <strong>the</strong>rmodynamic restrictions will be automatically satisfied [KRAJCINOVIC, 1983; GERMAIN,<br />
NGUYEN & SUQUET, 1983; YANG, ZHOU & SWOBODA, 1999].<br />
Motivated by <strong>the</strong> results <strong>of</strong> experimental investigations, it is assumed that only <strong>the</strong> tensile principal<br />
damage-active stresses are responsible <strong>for</strong> <strong>the</strong> damage evolution and that damage grows perpendicularly<br />
to <strong>the</strong> direction <strong>of</strong> <strong>the</strong> principal damage-active stresses. Thus, consider that damage may also develop<br />
partly isotropically, <strong>the</strong> damage evolution law is assumed taking <strong>the</strong> following particular <strong>for</strong>m:<br />
< 4><br />
D Ý =<br />
⎛<br />
⎝<br />
βII + (1 − β) I ⎞<br />
⎠ :<br />
ˆ<br />
S<br />
< 4><br />
where β, B0, m are material parameters. I<br />
B 0<br />
m<br />
< 4><br />
=<br />
⎛<br />
⎝<br />
βII + (1 − β) I ⎞<br />
⎠ :<br />
3 σ ˆ i σ σ<br />
∑ n ˆ ˆ i n i , (3b)<br />
i=1<br />
B 0<br />
m<br />
denotes <strong>the</strong> fourth-order identity tensor, and 〈.〉 is <strong>the</strong><br />
MCCAULEY bracket, which equals one <strong>for</strong> positive arguments and zero else. Creep rupture is assumed<br />
to take place when <strong>the</strong> maximum principal damage DI reaches a critical value DR.<br />
For single crystal superalloys, <strong>the</strong> initial anisotropy must be considered. The following particular<br />
<strong>for</strong>m <strong>of</strong> damage law is taken <strong>for</strong> single crystals with cubic symmetry (F.C.C. and B.C.C. single crystals):
4><br />
with R<br />
D Ý = β II + β I<br />
1 2 <br />
+ β R 3 <br />
⎛<br />
⎞<br />
⎝<br />
⎠ :<br />
3 η ˆ iσ i<br />
n ˆ σ ˆ σ ∑ i n i<br />
(4)<br />
⎡<br />
ηi =<br />
⎣<br />
⎢<br />
3<br />
i=1<br />
B 0<br />
m<br />
c c c c<br />
= ∑ e ei ei ei , (5)<br />
i<br />
i=1<br />
3<br />
∑<br />
j=1<br />
σ c ( n ˆ i ⋅e j)<br />
2n⎤<br />
⎦<br />
⎥<br />
p<br />
, (6)<br />
c<br />
where β1, β2, β3=(1−β1−β2), B0, p, n and m are material parameters, e j (j=1,2,3) are <strong>the</strong> lattice<br />
vectors, and η i is an orientation function which satisfies <strong>the</strong> cubic symmetry.<br />
Damage can also be inactive. Let us consider a single micro-crack embedded in an elastic material<br />
with a tensile load perpendicular to <strong>the</strong> crack faces. If <strong>the</strong> load is reversed <strong>the</strong> crack will close and in a<br />
one-dimensional case <strong>the</strong> material behaves as uncracked. This phenomenon is called “damage<br />
deactivation” (not “healing”) in CDM. The damage still exists but <strong>the</strong> loading condition can render it<br />
inactive. For <strong>the</strong> representation <strong>of</strong> this mechanism <strong>the</strong> phenomenological algorithm proposed by HANSEN<br />
& SCHREYER [1995] can be used. In this method <strong>the</strong> microcrack opening/closing effect is introduced by<br />
considering <strong>the</strong> spectral decomposition <strong>of</strong> <strong>the</strong> elastic strain tensor E e and <strong>the</strong> total strain tensor E<br />
3<br />
E e e εe εe<br />
ε ε<br />
= ∑ εi ni ni , E = ∑ εi ni ni , (7a, b)<br />
i=1<br />
3<br />
i=1<br />
e εe e e<br />
where εi and εi are <strong>the</strong> eigenvalues, ni and ni are <strong>the</strong> corresponding eigenvectors <strong>of</strong> E and E,<br />
respectively. Let <strong>the</strong> positive (tensile) spectral tensor corresponding to <strong>the</strong> elastic and to <strong>the</strong> total strain<br />
be defined as<br />
3<br />
H εe e εe εe<br />
= ∑ h(εi )ni ni , H ε ε ε<br />
= ∑ h(εi )n ini<br />
(8a, b)<br />
i=1<br />
respectively, with <strong>the</strong> modified Heaviside function<br />
3<br />
i=1<br />
11
12<br />
0 <strong>for</strong> x ≤ xm 1<br />
h(x) =<br />
2 1− cos π(x − x ⎧<br />
⎫<br />
⎪ ⎛ ⎡<br />
m) ⎤ ⎞<br />
⎪<br />
⎨ ⎜<br />
⎝ ⎣<br />
⎢ xp − xm ⎦<br />
⎥<br />
⎟ <strong>for</strong> xm < x < xp ⎬<br />
⎪<br />
⎠<br />
⎪<br />
⎩ 1 <strong>for</strong> x ≥ xp ⎭<br />
where xm and xp are two material parameters. The positive spectral projection operators (fourth-order<br />
tensor) <strong>for</strong> <strong>the</strong> elastic and <strong>the</strong> total strains are defined as<br />
<br />
Pεe = H εe H εe <br />
, Pε = H ε H ε<br />
respectively. The positive projection <strong>of</strong> <strong>the</strong> elastic and <strong>the</strong> total strain tensors are <strong>the</strong>n given by<br />
E e + <br />
= Pee : E e , E + < 4><br />
= Pe (9)<br />
(10a, b)<br />
: E (11a, b)<br />
respectively. By introducing a strain-based positive projection operator<br />
< 4><br />
T<br />
<br />
= I<br />
< 4> < 4><br />
−<br />
⎛<br />
I − P ⎞<br />
⎝ εe⎠<br />
: I<br />
⎛<br />
− P ⎞<br />
⎝ ε ⎠<br />
a symmetric, so-called active damage tensor can be defined as<br />
D = a T<br />
<br />
: D (13)<br />
Thus, <strong>the</strong> effective stress tensor and <strong>the</strong> damage-active stress tensor accounting <strong>for</strong> damage deactivation<br />
are defined as<br />
˜<br />
S = (I − D a ) −1 2 ⋅ S ⋅(I − D a ) −1 2 , (14)<br />
S ˆ = (I − Da) −q T −q<br />
⋅S ⋅(I − Da ) , (15)<br />
respectively.<br />
If <strong>the</strong> effective stress tensor and <strong>the</strong> damage active stress tensor defined in (14) and (15),<br />
respectively, are used instead <strong>of</strong> those defined in (1) and (2), <strong>the</strong> damage deactivation can be described.<br />
(12)
3. Unified models <strong>of</strong> BODNER-PARTOM and <strong>of</strong> CHABOCHE coupled with<br />
damage<br />
For <strong>the</strong> description <strong>of</strong> <strong>viscoplastic</strong> behaviour a lot <strong>of</strong> unified models have recently been developed.<br />
The main advantage <strong>of</strong> unified models compared to classical plasticity and creep models is <strong>the</strong> treatment<br />
<strong>of</strong> all aspects <strong>of</strong> inelastic de<strong>for</strong>mation behaviour including plastic flow under monotonic and cyclic<br />
loading, creep and stress relaxation by a single inelastic strain quantity [OLSCHEWSKI et al., 1990]. The<br />
total strain rate is decomposed into an elastic and an inelastic part by<br />
E Ý = Ý E e + Ý E i<br />
As <strong>the</strong> model proposed by BODNER & PARTOM [1975] and by CHABOCHE [CHABOCHE &<br />
ROUSSELIER, 1983] are two popular models, <strong>the</strong>y are chosen to be combined with <strong>the</strong> above damage<br />
model.<br />
The effective stress concept <strong>of</strong> CDM says that any constitutive equation <strong>for</strong> <strong>the</strong> damaged material<br />
can be derived in <strong>the</strong> same way as <strong>for</strong> a virgin material if <strong>the</strong> stress tensor is replaced by an adequately<br />
defined effective stress tensor. Following this concept, <strong>the</strong> BODNER-PARTOM model and <strong>the</strong><br />
CHABOCHE model can simply be extended to include material damage by replacing <strong>the</strong> stress tensor in<br />
<strong>the</strong> respective constitutive equations by <strong>the</strong> effective stress tensor defined in <strong>the</strong> expressions (1) or (14).<br />
The resulting models are summarized as follows:<br />
(16)<br />
13
flow rule:<br />
E Ý i = p Ý<br />
˜ S ′ − X<br />
, p Ý =<br />
S ˜ ′ − X<br />
isotropic hardening rule:<br />
14<br />
CHABOCHE BODNER-PARTOM<br />
3<br />
2 ˜ S ′ − X − R(p)<br />
K<br />
Ý<br />
R = b(R ∞ − R) Ý<br />
p , R(p = 0) = R 0<br />
kinematic hardening rule:<br />
X Ý = c 3<br />
aE Ý<br />
2 i ⎡<br />
−φ( p)Xp Ý ⎤ ⎡<br />
− d<br />
⎣<br />
⎦ ⎣<br />
⎢<br />
3<br />
2 X<br />
a<br />
− ωp<br />
φ(p) = φ∞ − ( φ∞ − 1)e<br />
r<br />
⎤<br />
⎦<br />
⎥<br />
X<br />
3<br />
2 X<br />
where ˜ ′<br />
S is <strong>the</strong> deviator <strong>of</strong> <strong>the</strong> damage-active stress.<br />
n<br />
E Ý i = 2D0 exp − 1 Z<br />
2<br />
I + ZD 3<br />
2 ˜ ⎡ ⎛ ⎞<br />
⎢ ⎜<br />
⎝ S<br />
⎟<br />
′<br />
⎣<br />
⎢<br />
⎠<br />
Z D = B : ˜ S ′<br />
S ˜ ′<br />
Z Ý I = m1 K − 1 Z I ( ) Ý<br />
⎡<br />
B Ý = m2 ⎢ K3 ⎣<br />
2n<br />
⎤<br />
⎥<br />
⎦<br />
⎥<br />
S ˜ ′<br />
S ˜ ′<br />
Z<br />
W − i A 1<br />
I ⎡ − K ⎤ 2<br />
⎢<br />
⎣ K ⎥<br />
1 ⎦<br />
Z I (t = 0) = K 0<br />
˜<br />
S<br />
˜<br />
S<br />
⎤<br />
− B⎥<br />
⎦<br />
Ý<br />
⎡ B ⎤<br />
W i − A2 ⎣ ⎢ K1 ⎦ ⎥<br />
W i = ˜ S ′ : Ý E i dτ<br />
4. Anisotropic creep model <strong>for</strong> cubic single crystals coupled with damage<br />
Starting from a rheological four-parameter BURGERS-model which consists <strong>of</strong> two springs and two<br />
dampers, BERTRAM & OLSCHEWSKI [1993, 1996] used a projection method to construct an<br />
anisotropic 3D model <strong>for</strong> <strong>the</strong> description <strong>of</strong> <strong>the</strong> creep behaviour <strong>of</strong> cubic single crystals at high<br />
temperatures. Replacing <strong>the</strong> stress tensor by <strong>the</strong> effective-stress tensor (1) in <strong>the</strong> model, <strong>the</strong> constitutive<br />
equations coupled with <strong>the</strong> damage model can be written as:<br />
t<br />
∫<br />
0<br />
r 2<br />
r 1<br />
B<br />
B<br />
(17)<br />
(18)<br />
(19)
4><br />
E Ý = A1 <br />
W Ý = A4 < 4><br />
: ˜ Ý<br />
S + A2 < 4><br />
: ˜ Ý<br />
S + A5 : ˜<br />
< 4><br />
S + A3 :W , (20a)<br />
: ˜ ( S − W)<br />
, (20b)<br />
where W is an internal tensor variable <strong>of</strong> rank two, and A i<br />
tensors defined as:<br />
< 4><br />
A1 =<br />
<br />
A2 =<br />
<br />
A3 =<br />
<br />
A 4 =<br />
<br />
A5 < 4><br />
where R<br />
=<br />
3<br />
∑<br />
i=1<br />
3<br />
∑<br />
i=1<br />
3<br />
∑<br />
i=1<br />
3<br />
∑<br />
i=1<br />
3<br />
∑<br />
i=1<br />
1<br />
C i + K i<br />
< 4><br />
Pi 1 ⎛<br />
⎜<br />
Ci + Ki ⎝<br />
1<br />
C i + K i<br />
K i<br />
C i + K i<br />
K i<br />
C i + K i<br />
< 4><br />
P1 = 1 <br />
3 II, P2 ,<br />
Ci Di 1<br />
D i<br />
<br />
Pi C i<br />
D i<br />
<br />
Pi ,<br />
< 4><br />
Pi + C i<br />
L i<br />
,<br />
,<br />
+ Ki ⎞ < 4><br />
⎟ P , i<br />
Li ⎠<br />
< 4> < 4><br />
= R − P1 , P3 = I<br />
<br />
< 4><br />
< 4><br />
(i = 1, 2, 3, 4, 5) are fourth-order material<br />
(21a-e)<br />
− R , (21f-h)<br />
is defined in eq. (5), Ci , Ki , Di , Li (i = 1, 2, 3) are temperature dependent material<br />
parameters. Note that <strong>the</strong> viscosities Di and Li are also dependent on <strong>the</strong> applied stress. They are<br />
assumed to have <strong>the</strong> following <strong>for</strong>m:<br />
⎛ 4 ⎞<br />
⎛ 4 ⎞<br />
D = i D0i exp ⎜ −∑ Zij J ⎟<br />
j , L = i L0i exp⎜ −∑ ZijJ ⎟<br />
j<br />
(22a, b)<br />
⎝ ⎠<br />
⎝ ⎠<br />
j=1<br />
j=1<br />
with <strong>the</strong> material parameters Zij (i = 1, 2, 3; j = 1, 2, 3, 4) and <strong>the</strong> following scalar invariants with cubic<br />
symmetry<br />
J 1 = σ 11 σ 22 +σ 22 σ 33 + σ 33 σ 11 ,<br />
2 2 2 J = σ +σ23 + σ31 ,<br />
2 12<br />
J 3 = σ 11 σ 22 σ 33 ,<br />
J 4 = σ 11 σ 12<br />
2 2<br />
2 2<br />
( +σ13)+σ<br />
22 σ23 + σ21<br />
2 2<br />
( )+ σ33( σ31 + σ32).<br />
By <strong>the</strong> assumption that volume changes occur only elastically, it follows<br />
(23a-d)<br />
15
16<br />
−1 −1<br />
D1 = 0 , L1 = 0, Z i1 = 0 (i = 1, 2,3) . (24a-c)
5. User material routines and <strong>the</strong>ir applications<br />
The material models mentioned above are implemented into <strong>the</strong> commercial FE code <strong>ABAQUS</strong><br />
as <strong>user</strong>-defined material model by writing <strong>the</strong> corresponding <strong>user</strong> <strong>subroutines</strong>, UMAT, see Table 1.<br />
With help <strong>of</strong> <strong>the</strong>se UMATs one can apply <strong>the</strong> models in an FE analysis <strong>of</strong> <strong>viscoplastic</strong> damage<br />
behaviour <strong>of</strong> engineering components and structures. All <strong>the</strong> routines are written in <strong>the</strong> computer<br />
language FORTRAN using <strong>the</strong> <strong>for</strong>ward integration algorithm <strong>for</strong> numerical integration. Only iso<strong>the</strong>rmal<br />
loading conditions have been considered and <strong>the</strong> damage deactivation has not been included in any <strong>of</strong><br />
<strong>the</strong> routines. Viscoplastic FE calculations are very time consuming. In <strong>the</strong> routines however, no<br />
automatic time step control is used, so that <strong>the</strong>re is a necessity to improve <strong>the</strong> respective algorithms. All<br />
examples presented below are conducted using <strong>ABAQUS</strong>/Standard 5.8.<br />
model UMAT<br />
CHABOCHE model coupled with damage d-chaboche.f<br />
BODNER-PARTOM model coupled with damage d-bodner.f<br />
anisotropic creep and damage model <strong>of</strong> BERTRAM,<br />
OLSCHEWSKI & QI<br />
d-scsrr99.f<br />
Table 1: Constitutive models and names <strong>of</strong> <strong>the</strong> respecitve UMATs<br />
5.1 Circumferentially notched bar — CHABOCHE model coupled with damage<br />
The material parameters <strong>of</strong> <strong>the</strong> CHABOCHE model <strong>of</strong> IN 738 LC have been determined by<br />
OLSCHEWSKI et al. [1990] and <strong>the</strong>ir values at 850 °C are shown in Table 2. The material parameters <strong>of</strong><br />
<strong>the</strong> damage model were estimated by using numerical optimization methods to fit <strong>the</strong> creep data <strong>of</strong> <strong>the</strong><br />
three creep tests presented in <strong>the</strong> work <strong>of</strong> OLSCHEWSKI et al. [1990]. During this process <strong>the</strong> above<br />
values <strong>of</strong> <strong>the</strong> parameters <strong>of</strong> <strong>the</strong> CHABOCHE model were kept constant. Table 3 shows a set <strong>of</strong> damage<br />
parameters <strong>for</strong> IN 738 LC at 850 °C. Note that just three uniaxial tests are not sufficient <strong>for</strong> parameter<br />
identification, so that <strong>the</strong> values given in Table 3 are only first estimates. For lack <strong>of</strong> biaxial test data, <strong>the</strong><br />
anisotropy parameter β can not be determined. Comparison <strong>of</strong> <strong>the</strong> experiments and <strong>the</strong> predictions by<br />
<strong>the</strong> CHABOCHE model and by <strong>the</strong> coupled model with damage, respectively, using <strong>the</strong> material<br />
17
parameters <strong>of</strong> Tables 2 and 3, and <strong>the</strong> damage evolution during <strong>the</strong> creep processes are shown in Fig.<br />
1.<br />
18
ε i<br />
4<br />
[%]<br />
3<br />
2<br />
1<br />
E 149650 MPa ν 0.33 a 311 MPa<br />
K 397 MPa⋅h 1/n n 7.7 b 317<br />
R0 153 MPa φ∞ 1.1 c 201<br />
R∞ 0.0 MPa ω 0.04 r 3.8<br />
d 81.72 MPa/h<br />
Table 2: Material parameters <strong>of</strong> <strong>the</strong> CHABOCHE model <strong>for</strong> IN 738 LC at 850 °C.<br />
β q B0 m DI<br />
0.0 ∼ 1.0 0.4 613 MPa·h 1/m<br />
14 0.07<br />
Table 3: Material parameters <strong>of</strong> <strong>the</strong> damage model <strong>for</strong> IN 738 LC at 850 °C<br />
Uniaxial Creep (IN738 LC, 850°C)<br />
σ=335 MPa, Exp.<br />
σ=392 MPa, Exp.<br />
σ=410 MPa, Exp.<br />
Chaboche-model<br />
model with damage<br />
0<br />
0 20 40 time [h] 60<br />
0.2<br />
0.1<br />
σ=410 MPa<br />
σ=392 MPa<br />
σ=335 MPa<br />
0.0<br />
0 20 40<br />
time [h]<br />
60<br />
19
20<br />
Fig. 1. Experiments and model predictions <strong>for</strong> inelastic strain (upper) and damage evolution<br />
(lower diagramme)<br />
Circumferentially notched bars are <strong>of</strong>ten used to investigate <strong>the</strong> influence <strong>of</strong> triaxial stress state on<br />
<strong>the</strong> damage and fracture behaviour in creep processes. KOBAYASHI et al. [1998] reported <strong>the</strong> results <strong>of</strong><br />
<strong>the</strong>ir experimental studies on such bars. Several creep damage tests <strong>of</strong> pure aluminum were carried out.<br />
The nucleation and growth <strong>of</strong> voids during <strong>the</strong> creep process were observed by means <strong>of</strong> scanning<br />
electron microscopy and optical microscopy. They found out that only some portion <strong>of</strong> creep voids<br />
actually appeared on <strong>the</strong> surface <strong>of</strong> <strong>the</strong> notch root, and that <strong>the</strong> lengths <strong>of</strong> creep voids beneath <strong>the</strong> notch<br />
surface exceeded ten times <strong>the</strong> lengths <strong>of</strong> those appearing on <strong>the</strong> surface. They fur<strong>the</strong>r found out that<br />
under a relatively low load, many creep voids nucleated on a plane inclined at about 45° against <strong>the</strong><br />
tensile direction, and <strong>the</strong>ir coalescence <strong>for</strong>med a cone-type fracture surface. These results motivated <strong>the</strong><br />
<strong>simulation</strong> <strong>of</strong> <strong>the</strong> damage behaviour in such bars. As <strong>the</strong> test data <strong>of</strong> aluminium from which <strong>the</strong> material<br />
parameters could have been estimated were not available, <strong>the</strong> alloy IN738 LC at 850 °C is used, again.<br />
The stress concentration and <strong>the</strong> multiaxial stress distribution are dependent on <strong>the</strong> geometry <strong>of</strong><br />
<strong>the</strong> specimen. The same geometry as used in <strong>the</strong> work <strong>of</strong> KOBAYASHI et. al. [1998] is used. Fig. 2<br />
shows <strong>the</strong> specimen geometry and <strong>the</strong> FE-mesh, where R = 5.0 mm, r0 = 1.0 mm and L = 10.0 mm.<br />
Axisymmetric solid elements <strong>of</strong> type CGAX4 from <strong>the</strong> element library <strong>of</strong> <strong>ABAQUS</strong> are used <strong>for</strong> <strong>the</strong> FE<br />
calculations. Fig. 3 shows <strong>the</strong> distributions <strong>of</strong> <strong>the</strong> maximum principal damage in <strong>the</strong> notch area <strong>for</strong><br />
β = 0.1, 0.5 and 1.0, respectively. The applied load is σ2 = 150 MPa. It can be clearly seen that <strong>the</strong><br />
most damaged area occurs beneath <strong>the</strong> notch surface in all cases. At <strong>the</strong> beginning <strong>of</strong> <strong>the</strong> creep process,<br />
<strong>the</strong> maximum damage takes place on <strong>the</strong> notch root surface, where <strong>the</strong> maximum stress appears. During<br />
<strong>the</strong> process, however, <strong>the</strong> location <strong>of</strong> <strong>the</strong> maximum damage moves away from <strong>the</strong> surface. The<br />
<strong>ABAQUS</strong> input-file <strong>for</strong> β = 0.1 used <strong>for</strong> <strong>the</strong> present calculation is given in <strong>the</strong> Appendix 1.
)<br />
a)<br />
Fig. 2. a) Geometry <strong>of</strong> <strong>the</strong> specimen used by KOBAYASHI et al. [1998]<br />
b) FE-mesh. R = 5.0 mm, r0 = 1.0 mm, L = 10.0 mm<br />
Fig. 4. shows <strong>the</strong> contour plots <strong>of</strong> <strong>the</strong> second direction cosine <strong>of</strong> <strong>the</strong> principal directions<br />
D D<br />
corresponding to DI, n ⋅ e2 = cos∠ n I ( ,e2 I ), and <strong>of</strong> <strong>the</strong> maximum damage at t = 411 hours <strong>for</strong><br />
β = 0.1. The maximum local damage at this time reaches a value <strong>of</strong> 0.0643, immediately be<strong>for</strong>e <strong>the</strong> local<br />
fracture takes place and meso-cracks may have been initiated; note that <strong>the</strong> critical value <strong>of</strong> damage is<br />
Dc = 0.07. The values <strong>of</strong> <strong>the</strong> direction cosines at <strong>the</strong> location <strong>of</strong> maximum damage, DI, indicate that <strong>the</strong><br />
D<br />
direction n <strong>of</strong> maximum damage coincides with <strong>the</strong> e2-direction, which means that <strong>the</strong> surfaces <strong>of</strong><br />
I<br />
nucleated micro/meso-cracks will be perpendicular to <strong>the</strong> e2-direction, i.e. <strong>the</strong> loading axis.<br />
21
22<br />
Fig. 3. Contour plots <strong>of</strong> <strong>the</strong> maximum principal damage <strong>for</strong> β = 0.1, 0.5 and 1.0.<br />
O<strong>the</strong>r experimental investigations <strong>of</strong> KOBAYASHI et al. show that under relatively low loads, σ2, creep<br />
voids nucleated on a plane inclined by about 45° against <strong>the</strong> tensile direction. Though pure aluminium<br />
was used in <strong>the</strong>se tests <strong>for</strong> which no material data exist, <strong>the</strong> experiments motivated <strong>the</strong> <strong>simulation</strong> <strong>of</strong> <strong>the</strong><br />
damage behaviour at a lower creep load <strong>of</strong> σ2 = 100 MPa <strong>for</strong> <strong>the</strong> present Ni-based alloy and β = 0.1.<br />
The contour plots <strong>of</strong> <strong>the</strong> second direction cosine <strong>of</strong> <strong>the</strong> principal direction corresponding to DI,<br />
D<br />
cos∠( n ,e2 I ), at t = 11000 hours and <strong>the</strong> maximum damage distribution at t = 11000 and<br />
12000 hours, respectively, are shown in Fig. 5. The values <strong>of</strong> <strong>the</strong> direction cosine at <strong>the</strong> location <strong>of</strong><br />
maximum damage, i.e. where local fracture will occur, is −0.7÷−0.8. That means that maximum principal<br />
damage is inclined at about 45° against <strong>the</strong> tensile direction, which indicates that cracks may <strong>for</strong>m a<br />
cone-type fracture surface. Once <strong>the</strong> meso-crack has been <strong>for</strong>med or even local fracture has been taken<br />
place, <strong>the</strong> local behaviour <strong>of</strong> <strong>the</strong> material will strongly depend on <strong>the</strong> shape and size <strong>of</strong> <strong>the</strong> crack. Fur<strong>the</strong>r<br />
experimental investigations are needed.
Fig. 4. Contour plots <strong>of</strong> direction cosine and max. damage at t = 411 hours <strong>for</strong> β = 0.1.<br />
23
24<br />
Fig. 5. Contour plots <strong>of</strong> direction cosine and <strong>of</strong> <strong>the</strong> maximum principal damage <strong>for</strong> β = 0.1.
5.2 Plate containing a hole — BODNER-PARTOM model coupled with damage<br />
The material parameters <strong>of</strong> <strong>the</strong> BODNER-PARTOM model <strong>of</strong> IN 738 LC at 850 °C have also been<br />
determined by OLSCHEWSKI et al. [1990], and <strong>the</strong>ir values at 850 °C are shown in Table 4. The<br />
material parameters <strong>of</strong> <strong>the</strong> damage model are <strong>the</strong> same as listed in Table 3.<br />
E 149650 MPa ν 0.33 K0 4.18 10 5 MPa<br />
D0<br />
8.82 10 9 h -1<br />
n 0.289 K1 3.76 10 5 MPa<br />
A1=A2 1.65 10 -7 MPa/h m1 0.581 K2 3.07 10 5 MPa<br />
r1=r2 5.4 m2 0.344 K3 1.54 10 5 MPa<br />
Table 4: Material parameters <strong>of</strong> <strong>the</strong> BODNER-PARTOM model <strong>for</strong> IN 738 LC at 850 °C.<br />
In gas turbine blades with cooling channels, stress concentration occurs due at <strong>the</strong>se channels. A<br />
square plate with a central circular hole is <strong>the</strong>re<strong>for</strong>e chosen as a model representation <strong>of</strong> <strong>the</strong> area <strong>of</strong><br />
blades where <strong>the</strong> air cooling channels are located. The FE model used <strong>for</strong> <strong>the</strong> calculation is shown in<br />
Fig. 6. First, <strong>the</strong> plate is subjected to a creep load <strong>of</strong> σ3 = 180 MPa. After 40000 hours <strong>the</strong> maximum<br />
damage reaches a value <strong>of</strong> about 0.1. A second load <strong>of</strong> σ2 = 180 MPa is <strong>the</strong>n applied. There is only<br />
one element in <strong>the</strong> thickness direction so that any gradient over <strong>the</strong> thickness can not be captured. The<br />
three-dimensional 8-node linear brick continuum element with reduced integration, C3D8R, from <strong>the</strong><br />
element library <strong>of</strong> <strong>ABAQUS</strong> is used, and geometric non-linearity has been considered. Figs. 7 and 8<br />
show <strong>the</strong> contour plot <strong>of</strong> <strong>the</strong> maximum principal damage after 40000 h and 98000 h, respectively. β is<br />
assumed to be 0.5. Distribution <strong>of</strong> <strong>the</strong> maximum principal value <strong>of</strong> <strong>the</strong> strain and stress, after 40000 h<br />
and 98000 h, are shown in <strong>the</strong> Figs. 9-12, respectively. The <strong>ABAQUS</strong> input-file used <strong>for</strong> <strong>the</strong><br />
computation is given in <strong>the</strong> Appendix 2.<br />
25
26<br />
Fig. 6. FE-mesh and loading condition<br />
Fig. 7. Max. principal damage after 40000 h Fig. 8. Max. principal damage after 98000 h<br />
Fig. 9. Max. principal strain after 40000 h Fig. 10. Max. principal strain after 98000 h
Fig. 11. Max. principal stress after 40000 h Fig. 12. Max. principal stress after 98000 h<br />
5.3 Single crystal plate containing a hole — <strong>the</strong> anisotropic creep and damage model <strong>of</strong><br />
BERTRAM, OLSCHEWSKI & QI<br />
The material parameters <strong>of</strong> anisotropic creep model <strong>for</strong> <strong>the</strong> single crystal SRR99 at 760 °C have<br />
been estimated by BERTRAM and OLSCHEWSKI [1996]. The corresponding material parameters <strong>of</strong> <strong>the</strong><br />
damage model have been estimated by QI [1998]. Fig. 13 shows <strong>the</strong> applied FE model. The uniaxial<br />
tensile load, σ2, is applied in <strong>the</strong> crystal direction [001]. Because <strong>of</strong> <strong>the</strong> symmetry, only 1/2 <strong>of</strong> <strong>the</strong><br />
thickness <strong>of</strong> <strong>the</strong> specimen has to be modelled. Three elements are used over <strong>the</strong> half-thickness to<br />
capture <strong>the</strong> gradients in <strong>the</strong> thickness direction. The distribution <strong>of</strong> <strong>the</strong> maximum principal damage at<br />
t = 34000 hours is shown in Fig. 14. For comparison, Fig. 15 shows <strong>the</strong> initiation <strong>of</strong> cracks at a cavity<br />
in a prerafted single crystal CMSX-2 after 44 hours <strong>of</strong> creep at 850 °C and 520 MPa (creep life<br />
fraction = 95%) [AI et al., 1990], indicating at least a qualitative coincidence <strong>of</strong> <strong>the</strong> damage loci between<br />
numerical <strong>simulation</strong> and experiment. Contour plots <strong>of</strong> <strong>the</strong> strain ε22 and <strong>the</strong> stress σ22 at t = 34000<br />
hours are shown in Figs. 16 and 17, respectively. The <strong>ABAQUS</strong> input-file used <strong>for</strong> <strong>the</strong> computation is<br />
given in <strong>the</strong> Appendix 3.<br />
27
28<br />
Fig. 13. FE-mesh and loading condition
Fig. 14. Maximum. principal damage after 34000 h Fig. 15. Crack initiation at a cavity<br />
Fig. 16. Max. principal strain after 34000 h Fig. 17. Max. principal stress after 34000 h<br />
5.4 TiAl turbine blade — CHABOCHE model coupled with damage<br />
A model turbine blade made <strong>of</strong> a TiAl intermetallic alloy developed at <strong>the</strong> GKSS Research<br />
Centre is used as object <strong>of</strong> <strong>the</strong> FE-calculation. The blade has a length <strong>of</strong> 224 mm. The material<br />
parameters <strong>of</strong> <strong>the</strong> CHABOCHE model have been estimated by MOHR [1999], as listed in Table 5. Table<br />
6 gives <strong>the</strong> corresponding material parameters <strong>of</strong> <strong>the</strong> damage model used <strong>for</strong> <strong>the</strong> calculation. The<br />
continuum element C3D4 <strong>of</strong> <strong>the</strong> element library <strong>of</strong> <strong>ABAQUS</strong> is used. The number <strong>of</strong> nodes is 1476 and<br />
<strong>the</strong> number <strong>of</strong> elements is 4825. The blade is subject to centrifugal <strong>for</strong>ces, only, at a constant rotation<br />
speed <strong>of</strong> 40000 1/min; <strong>the</strong> density <strong>of</strong> TiAl is 3.8 g/cm 3 . Geometric non-linearity has been considered.<br />
The distribution <strong>of</strong> <strong>the</strong> maximum damage at t = 9060 hours is shown in Fig. 18. It obviously ocurs at <strong>the</strong><br />
root <strong>of</strong> <strong>the</strong> blade which after all has not been modelled realistically. The calculation is just supposed to<br />
29
proove that <strong>the</strong> model per<strong>for</strong>ms well also with large structures. The <strong>ABAQUS</strong> input-file used <strong>for</strong> <strong>the</strong><br />
computation is given in <strong>the</strong> Appendix 4.<br />
30
E 150000 MPa ν 0.24 a 335 MPa<br />
K 487 MPa⋅s 1/n n 15.3 b 207<br />
R0 126 MPa φ∞ 0.0 c 35.4<br />
R∞ 0.0 MPa ω 0.0 r 3.1<br />
d 0.023 MPa/s<br />
Table 5: Material parameters <strong>of</strong> CHABOCHE model <strong>for</strong> <strong>the</strong> TiAl at 700 °C.<br />
β q B0 m<br />
0.3 0.3 1500 MPa·h 1/m<br />
Table 6: Material parameters <strong>of</strong> <strong>the</strong> damage model <strong>for</strong> <strong>the</strong> TiAl at 700 °C<br />
14<br />
Fig. 18. Maximum principal damage after 9060 h<br />
31
6. References<br />
AI, S.H.; LUPINC, V. and MALDINI, M. (1990): "Creep fracture mechanics in single crystal superalloys".<br />
In: High Temperature materials <strong>for</strong> Power Engineering 1990, Proceedings <strong>of</strong> a Conference held in<br />
Liège, Belgium, 24-27 September 1990. Part II (Eds. E. BACHELET et al.)<br />
BERTRAM, A.; OLSCHEWSKI, J. (1993): "Zur Formulierung anisotroper linearer anelastischer<br />
St<strong>of</strong>fgleichungen mit Hilfe einer Projektionsmethode". ZAMM 73 (4-5), T401-403.<br />
BERTRAM, A.; OLSCHEWSKI, J. (1996): "Anisotropic creep modeling <strong>of</strong> <strong>the</strong> single crystal superalloy<br />
SRR99". J. Comp. Mat. Sci. 5, pp.12-16.<br />
BODNER, S.R.; PARTOM, Y. (1975): "Constitutive equations <strong>for</strong> elastic-<strong>viscoplastic</strong> strain hardening<br />
materials". J. Appl. Mech. 42, pp.385-389.<br />
CHABOCHE, J.L.; ROUSSELIER, G. (1983): "On <strong>the</strong> plastic and <strong>viscoplastic</strong> constitutive equations". J.<br />
Press. vess. technol. 105, pp.105-164.<br />
GERMAIN, P.; NGUYEN, Q. S.; SUQUET, P. (1983): "Continuum Thermodynamics". J. Appl. Mech. 50,<br />
pp.1010-1020.<br />
KOBAYASHI, K.I.; IMADA, H.; MAJIMA, T. (1998): "Nucleation and growth <strong>of</strong> creep voids in<br />
circumferentially notched specimens", JSME Int. J., Series A: Solid Mechanics and Material<br />
Engineering, 41, 218-224.<br />
KRAJCINOVIC, D. (1983): "Constitutive equations <strong>for</strong> damaging materials". J. Appl. Mech. 50, pp. 355-<br />
360.<br />
MOHR, R. (1999): "Modellierung des Hochtemperaturverhaltens metallischer Werkst<strong>of</strong>fe". Dissertation,<br />
Technische Universität Hamburg-Harburg, GKSS 99/E/66.<br />
OLSCHEWSKI, J.; SIEVERT, R.; MEERSMANN, J. and ZIEBS, J. (1990): "Selection, calibration and<br />
verification <strong>of</strong> <strong>viscoplastic</strong> constitutive models used <strong>for</strong> advanced blading methodology". In: High<br />
Temperature Materials <strong>for</strong> Power Engineering, Proceedings <strong>of</strong> a Conference held in Liège, Belgium,<br />
24-27 September 1990 (Eds. BACHELET et al.), Kluwer Academic Publishers, pp.1051-1060.<br />
QI, W.; BERTRAM, A. (1997): "Anisotropic creep damage modeling <strong>of</strong> single crystal superalloys".<br />
Technische Mechanik. 17(4), pp.313-322.<br />
QI, W. (1998): "Modellierung der Kriechschädigung einkristalliner Superlegierungen im<br />
Hochtemperaturbereich". Dissertation, Technische Universität Berlin, Fortschritts-Berichte VDI Verlag<br />
GmbH, Düsseldorf.<br />
QI, W.; BERTRAM, A. (1998): "Damage modeling <strong>of</strong> <strong>the</strong> single crystal superalloy SRR99 under<br />
monotonous creep". Computational Materials Science 13, pp.132-141.<br />
33
QI, W. ; BERTRAM, A. (1999): "Anisotropic continuum damage modeling <strong>for</strong> single crystals at high<br />
temperatures". Int. J. <strong>of</strong> Plasticity 15, pp.1197-1215.<br />
QI, W.; BROCKS, W. (2000a): "A CDM-based approach to creep damage and component lifetime".<br />
Proceedings <strong>of</strong> <strong>the</strong> Int. Conf. on Computational Engineering & Sciences, ”ICES‘2K” Ed. S. ATLURI),<br />
21-25 August 2000, Los Angeles.<br />
QI, W.; BROCKS, W. (2000b): "Simulation <strong>of</strong> anisotropic creep damage in engineering components".<br />
Proceedings <strong>of</strong> <strong>the</strong> European Congress on Computational Methods in Applied Sciences and Engineering<br />
”ECCOMAS 2000”, 11-14 September 2000, Barcelona.<br />
QI, W.; BROCKS, W.; BERTRAM, A. (2000): A FE-analysis <strong>of</strong> anisotropic creep damage and<br />
de<strong>for</strong>mation in <strong>the</strong> single crystal SRR99 under multiaxial loads. Computational Materials Science 19<br />
(2000), pp. 292-297.<br />
YANG, Q.; ZHOU, W.Y.; SWOBODA, G. (1999): Micromechanical identification <strong>of</strong> anisotropic damage<br />
evolution laws. Int. J. <strong>of</strong> Fracture 98, pp. 55-76.<br />
34
7. Appendices: <strong>ABAQUS</strong>-Inputfiles<br />
7.1 Appendix 1: Circumferentially notched bar — CHABOCHE model coupled with damage<br />
*HEADING<br />
circum. bar<br />
a= 5.0, r0= 1.00, N-alfa=20, N-r=40, K= 6<br />
*PREPRINT, ECHO=NO, MODEL=NO, HISTORY=NO<br />
**<br />
*RESTART, WRITE, FREQUENCY=2000<br />
**<br />
*NODE<br />
1, 5.00000, 1.00000<br />
101, 5.00000, 1.03927<br />
201, 5.00000, 1.07542<br />
...<br />
5181, 5.00000, -7.75000<br />
5281, 5.00000, -8.00000<br />
5381, 5.00000, -8.25000<br />
5481, 5.00000, -8.50000<br />
5581, 5.00000, -8.75000<br />
5681, 5.00000, -9.00000<br />
5781, 5.00000, -9.25000<br />
5881, 5.00000, -9.50000<br />
5981, 5.00000, -9.75000<br />
6081, 5.00000, -10.00000<br />
*ELEMENT, TYPE=CGAX4, ELSET=solid<br />
1, 2, 1, 101, 102<br />
2, 3, 2, 102, 103<br />
3, 4, 3, 103, 104<br />
4, 5, 4, 104, 105<br />
5, 6, 5, 105, 106<br />
6, 7, 6, 106, 107<br />
7, 8, 7, 107, 108<br />
...<br />
5978, 6078, 6079, 5979, 5978<br />
5979, 6079, 6080, 5980, 5979<br />
5980, 6080, 6081, 5981, 5980<br />
**<br />
** define node-set<br />
**<br />
*NSET, NSET=zplus, GENERATE<br />
6001, 6021, 1<br />
*NSET, NSET=zminus, GENERATE<br />
6061, 6081, 1<br />
**<br />
** define element-set<br />
**<br />
*ELSET, ELSET=zlast, GENERATE<br />
5901, 5920, 1<br />
**<br />
** define materials and UMA<br />
**<br />
*SOLID SECTION, ELSET=SOLID, MATERIAL=VISCOPLA<br />
1.,<br />
*MATERIAL, NAME=VISCOPLA<br />
*USER MATERIAL, CONSTANTS=19<br />
35
**<br />
** IN738LC, 850 deg C<br />
******** IN 738 LC, 850 C, ==== CH-model==== h, MPa<br />
** E, nu, Ro, Q, b, c, a, phiinf<br />
149650., .33, 153., -153., 317., 201., 311., 1.1<br />
*** omega, d, r, n, K; q, beta, Bo<br />
*** ( beta=1 -> isot. damage)<br />
0.04, 81.72, 4.8, 7.7, 397.0, .4, 0.1, 613.<br />
*** m, Dkey, Ckey Ckey ( Dkey < or = 0: do not consider damage)<br />
***( Ckey must be –1.0, only UMAT-developer may change it!)<br />
14., 1.0, -1.0<br />
*DEPVAR<br />
29<br />
**<br />
*USER SUBROUTINE, INPUT=/wms12/weiqi/umats/poly/d-chaboche.f<br />
**<br />
** loading !! <strong>the</strong> unit is hour !!<br />
**<br />
*STEP, INC=900000000, NLGEOM<br />
*VISCO, CETOL=1.0E-6<br />
0.0001, 0.001, , 0.0001<br />
**<br />
36
** auflagerung<br />
**<br />
*BOUNDARY, OP=NEW, TYPE=DISPLACEMENT<br />
6061, 1, 2, 0.0<br />
zminus, 2, 2, 0.0<br />
**** === define amplitude <strong>for</strong> loading process<br />
** 1s=0.00027778 h<br />
*AMPLITUDE, TIME=TOTAL TIME, NAME=creep<br />
0.0, 0.0, 0.001, 1.0, 9000000.0, 1.0<br />
**<br />
*DLOAD, OP=NEW, AMPLITUDE=creep<br />
zlast, P3, -150.<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** step 2<br />
**<br />
*STEP, INC=900000000, NLGEOM<br />
*VISCO, CETOL=1.0E-6<br />
0.0001, 0.001, , 0.0001<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** step 3<br />
**<br />
*STEP, INC=900000000, NLGEOM<br />
*VISCO, CETOL=1.0E-6<br />
0.001, 1., , 0.01<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** step 4<br />
**<br />
*STEP, INC=900000000, NLGEOM<br />
*VISCO, CETOL=1.0E-6<br />
0.1, 90000000.0, , 0.1<br />
**<br />
*RESTART, WRITE, FREQUENCY=100<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
37
7.2 Appendix 2: Plate containing a hole — BODNER-PARTOM model coupled with damage<br />
*HEADING<br />
plate containing a crack<br />
a= 5.0, r0= .50, N-alfa=10, N-r=15, K=12 D= .20<br />
*PREPRINT, ECHO=NO, MODEL=NO, HISTORY=NO<br />
*NODE<br />
1, .20, .49846, .03923<br />
5001, .00, .49846, .03923<br />
101, .20, .53761, .04231<br />
5101, .00, .53761, .04231<br />
...<br />
1380, .20, 3.47804, .00000<br />
6380, .00, 3.47804, .00000<br />
1480, .20, 4.16651, .00000<br />
6480, .00, 4.16651, .00000<br />
1580, .20, 5.00000, .00000<br />
6580, .00, 5.00000, .00000<br />
*ELEMENT, TYPE=C3D8I, ELSET=solid<br />
*** 0 - PI/4<br />
80, 5080, 80, 180, 5180, 5001, 1, 101, 5101<br />
180, 5180, 180, 280, 5280, 5101, 101, 201, 5201<br />
280, 5280, 280, 380, 5380, 5201, 201, 301, 5301<br />
380, 5380, 380, 480, 5480, 5301, 301, 401, 5401<br />
480, 5480, 480, 580, 5580, 5401, 401, 501, 5501<br />
580, 5580, 580, 680, 5680, 5501, 501, 601, 5601<br />
...<br />
1477, 6477, 1477, 1577, 6577, 6478, 1478, 1578, 6578<br />
1478, 6478, 1478, 1578, 6578, 6479, 1479, 1579, 6579<br />
1479, 6479, 1479, 1579, 6579, 6480, 1480, 1580, 6580<br />
**<br />
** define node-set<br />
**<br />
*NSET, NSET=zlager0, GENERATE<br />
6550, 6570, 1<br />
*NSET, NSET=zlager1, GENERATE<br />
1550, 1570, 1<br />
*NSET, NSET=ylager0, GENERATE<br />
6530, 6550, 1<br />
*NSET, NSET=ylager1, GENERATE<br />
1530, 1550, 1<br />
**<br />
** define element-set<br />
**<br />
*ELSET, ELSET=zlast, GENERATE<br />
1410, 1429, 1<br />
*ELSET, ELSET=ylast, GENERATE<br />
1401, 1409, 1<br />
1470, 1480, 1<br />
**<br />
** define materials and UMA<br />
**<br />
*SOLID SECTION, ELSET=SOLID, MATERIAL=VISCOPLA<br />
1.,<br />
*MATERIAL, NAME=VISCOPLA<br />
*USER MATERIAL, CONSTANTS=19<br />
**<br />
** IN738LC, 850 deg C<br />
******* BP-Model, units: h, MPa<br />
********========================================<br />
38
** E, nu, Do, n, K1, K2, K3, m1<br />
149650., .33, 8.82E+9, 0.289, 3.76E+5, 3.07E+5, 1.54E+5, 0.581<br />
*** m2, A, r, Unused, Ko; q, beta, Bo<br />
*** ( beta=1 -> isot. damage)<br />
0.344, 1.6524E+7, 5.4, 0.0, 4.18E+5, 0.4, 0.5, 613.<br />
*** m, Dkey, Ckey Ckey ( Dkey < or = 0: do not consider damage)<br />
***( Ckey must be –1.0, only UMAT-developer may change it!)<br />
14., 1.0, -1.0<br />
*DEPVAR<br />
26<br />
**<br />
*USER SUBROUTINE, INPUT=/wms12/weiqi/umats/poly/d-bodner.f<br />
**<br />
**<br />
*RESTART, WRITE, FREQUENCY=1000<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
0.0001, 0.001, , 0.0001<br />
**<br />
** auflagerung<br />
**<br />
39
*BOUNDARY, OP=NEW, TYPE=DISPLACEMENT<br />
** 6550, 1, 3, 0.0<br />
zLager0, 3, 3, 0.0<br />
zLager0, 1, 1, 0.0<br />
zLager1, 3, 3, 0.0<br />
yLager0, 2, 2, 0.0<br />
yLager1, 2, 2, 0.0<br />
**** === define amplitude <strong>for</strong> loading process<br />
*AMPLITUDE, TIME=TOTAL TIME, NAME=creep1<br />
0.0, 0.0, 0.001, 1.0, 90000000.0, 1.0<br />
**<br />
*DLOAD, OP=NEW, AMPLITUDE=creep1<br />
zlast, P2, -180.<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 2<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
0.001, 0.5, , 0.01<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 3<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
0.01, 49.5, , .5<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 4<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
1., 39950.00, , 4.<br />
**<br />
*RESTART, WRITE, FREQUENCY=400000<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 5<br />
40
** ====== 2. load ein<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
.0001, .001, , .0001<br />
**<br />
**** === define amplitude <strong>for</strong> loading process<br />
*AMPLITUDE, NAME=creep2<br />
0.0, 0.0, 0.001, 1.0, 90000000.0, 1.0<br />
**<br />
*DLOAD, OP=MOD, AMPLITUDE=creep2<br />
ylast, P5, -180.<br />
*RESTART, WRITE, FREQUENCY=1500<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 6<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
.001, 0.1, , .01<br />
41
**<br />
*RESTART, WRITE, FREQUENCY=1000<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 7<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
.01, 49.50, , 1.<br />
**<br />
*RESTART, WRITE, FREQUENCY=100<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 8<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
1., 60000.00, , 5.<br />
**<br />
*RESTART, WRITE, FREQUENCY=20000<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 9<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
5., 1000000.00, , 5.<br />
**<br />
*RESTART, WRITE, FREQUENCY=200<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
42
7.3 Appendix 3: Single crystal plate containing a hole — <strong>the</strong> anisotropic creep and damage<br />
model <strong>of</strong> BERTRAM, OLSCHEWSKI & QI<br />
*HEADING<br />
plate 10x10x0.5, hole radius 0.5, Non-symm<br />
C3D8I, r15-h10-b10-t3, bias: r10-h1-b1-t2, I-DEAS 06-Mar-01<br />
*NODE, SYSTEM=R<br />
1, 3.5355339E-01,-3.5355339E-01, 2.5000000E-01<br />
2, 4.2009962E-01,-4.2009962E-01, 2.5000000E-01<br />
3, 5.0002275E-01,-5.0002275E-01, 2.5000000E-01<br />
...<br />
5294,-4.5000000E+00,-5.0000000E+00, 1.9248187E-01<br />
5295,-4.5000000E+00,-5.0000000E+00, 1.1616359E-01<br />
5296,-4.5000000E+00,-5.0000000E+00, 0.0000000E+00<br />
*ELEMENT,TYPE=C3D8I ,ELSET=E0000001<br />
1, 1, 2, 18, 17, 65, 66, 82, 81<br />
2, 2, 3, 19, 18, 66, 67, 83, 82<br />
3, 3, 4, 20, 19, 67, 68, 84, 83<br />
4, 4, 5, 21, 20, 68, 69, 85, 84<br />
5, 5, 6, 22, 21, 69, 70, 86, 85<br />
...<br />
*SOLID SECTION,ELSET=E0000001,MATERIAL=M0000001<br />
*MATERIAL,NAME=M0000001<br />
*USER MATERIAL, CONSTANTS=14<br />
**<br />
** SRR99, 760 deg C<br />
**<br />
***alfa0 alfa1 alfa2 B[MPa*h] n p m n1<br />
1.0, 0.0, 0.5, 1442.0, 14.133, 0.45489, 51.852, -0.31326<br />
*** Dcr, phi1, phi2, phi3 Dkey, Ckey<br />
**** ( Dkey < or = 0: do not consider damage)<br />
**** ( Ckey must be –1.0, only UMAT-developer may change it!)<br />
0.9, 0.0, 0.0, 0.0, 1.0, -1.E8<br />
*DEPVAR<br />
36<br />
**<br />
*USER SUBROUTINE, INPUT=/wms12/weiqi/umats/single/d-scsrr99.f<br />
**<br />
** loading !! <strong>the</strong> unit is h !!<br />
**<br />
*RESTART, WRITE, FREQUENCY=1000<br />
**<br />
*STEP, INC=90000000, NLGEOM<br />
*VISCO, CETOL=1.0E-10<br />
0.0001, 0.001, , 0.0001<br />
** auflagerung<br />
*BOUNDARY,OP=NEW<br />
BS000001, 1,, .00000E+00<br />
BS000002, 2,, .00000E+00<br />
3344, 1, 2, .00000E+00<br />
3680, 1, 2, .00000E+00<br />
4016, 1, 2, .00000E+00<br />
BS000003, 3,, .00000E+00<br />
BS000004, 1,, .00000E+00<br />
BS000004, 3,, .00000E+00<br />
BS000005, 2, 3, .00000E+00<br />
3008, 1, 3, .00000E+00<br />
**<br />
** loading<br />
43
**** === define amplitude <strong>for</strong> loading process<br />
*AMPLITUDE, TIME=TOTAL TIME, NAME=creep1<br />
0.0, 0.0, 0.001, 1.0, 90000000.0, 1.0<br />
**<br />
*DLOAD,OP=NEW, AMPLITUDE=creep1<br />
** BS000006, P4, -100.<br />
BS000007, P6, -350.<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 2<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
0.0001, 0.1, , 0.02<br />
**<br />
44
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 3<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
0.02, 1.0, , 0.2<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 4<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
0.2, 49.0, , .5<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 5<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
1., 3950.00, , 1.<br />
**<br />
*RESTART, WRITE, FREQUENCY=1000<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEP<br />
**<br />
** Step 6<br />
**<br />
*STEP, INC=90000000<br />
*VISCO, CETOL=1.0E-10<br />
1., 1000000.00, , 1.<br />
**<br />
*RESTART, WRITE, FREQUENCY=200<br />
**<br />
*NODE FILE, FREQUENCY=0<br />
*EL FILE, FREQUENCY=0<br />
*NODE PRINT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
*PRINT, FREQUENCY=0<br />
*END STEPT, FREQUENCY=0<br />
*EL PRINT, FREQUENCY=0<br />
45
*PRINT, FREQUENCY=0<br />
*END STEP<br />
46
7.4 Appendix 4: TiAl turbine blade — CHABOCHE model coupled with damage<br />
*HEADING<br />
SDRC I-DEAS <strong>ABAQUS</strong> FILE TRANSLATOR 10-Apr-00 13:53:27<br />
units: mm, s, MPa isot. damage)<br />
0.0, 0.023, 3.1, 15.3, 487.0, 0.3, 0.3, 1500.<br />
*** m, Dkey, Ckey Ckey ( Dkey < or = 0: do not consider damage)<br />
***( Ckey must be –1.0, only UMAT-developer may change it!)<br />
47
14., 1.0, -1.0<br />
*DEPVAR<br />
29<br />
**<br />
*USER SUBROUTINE, INPUT=/wms12/weiqi/umats/poly/d-chaboche.f<br />
**<br />
** first loading, cycle 1<br />
**<br />
*STEP, INC=100000000, NLGEOM<br />
***STATIC<br />
*VISCO, CETOL=1.0E-10<br />
.01, 100., , 0.1<br />
**<br />
**<br />
*BOUNDARY,OP=NEW<br />
12, 1, 3, .00000E+00<br />
45, 1, 3, .00000E+00<br />
46, 1, 3, .00000E+00<br />
15, 1, 3, .00000E+00<br />
fuss, 1, 3, .00000E+00<br />
**<br />
** rotations<br />
**<br />
-48-
*AMPLITUDE, NAME=CYCLEONE, DEFINITION=TABULAR, TIME=STEP TIME<br />
0., 0., 10., .05415, 100., 1., 900000000., 1.<br />
**<br />
*DLOAD, OP=NEW, AMPLITUDE=CYCLEONE<br />
blade, CENTRIF, 444000., 35., 0., 0., 0., 0., 1.<br />
**entspricht Drehzahl von 40000/min oder 666/sec<br />
**<br />
*RESTART, WRITE, FREQUENCY=10000000<br />
**<br />
*NODE FILE, FREQ=0<br />
*EL FILE, POS=INTEG, FREQ=0<br />
S, E, SDV<br />
*EL PRINT,POS=INTEG, FREQ=0<br />
*NODE PRINT, FREQ=0<br />
*PRINT, FREQ=0<br />
*END STEP<br />
***<br />
***<br />
***<br />
*STEP, INC=100000000<br />
***STATIC<br />
*VISCO, CETOL=1.0E-10<br />
0.1, 3600., , 10.<br />
**<br />
*EL FILE, POS=INTEG, FREQ=0<br />
*NODE FILE, FREQ=0<br />
*PRINT, FREQ=0<br />
*END STEP<br />
***<br />
***<br />
***<br />
*STEP, INC=100000000<br />
*VISCO, CETOL=1.0E-10<br />
50., 720000., , 50.<br />
*RESTART, WRITE, FREQUENCY=1440<br />
**<br />
*EL FILE, POS=INTEG, FREQ=0<br />
*NODE FILE, FREQ=0<br />
*PRINT, FREQ=0<br />
*END STEP<br />
***<br />
***<br />
**<br />
*STEP, INC=100000000<br />
*VISCO, CETOL=1.0E-10<br />
200., 720000., , 200.<br />
**<br />
*RESTART, WRITE, FREQUENCY=360<br />
**<br />
*EL FILE, POS=INTEG, FREQ=0<br />
*NODE FILE, FREQ=0<br />
*PRINT, FREQ=0<br />
*END STEP<br />
***<br />
**<br />
*STEP, INC=100000000<br />
*VISCO, CETOL=1.0E-10<br />
400., 72000000., , 400.<br />
**<br />
*RESTART, WRITE, FREQUENCY=180<br />
**<br />
*EL FILE, POS=INTEG, FREQ=0<br />
*NODE FILE, FREQ=0<br />
-49-
*PRINT, FREQ=0<br />
*END STEP<br />
-50-