ABAQUS user subroutines for the simulation of viscoplastic - loicz

Technical Note GKSS/WMS/01/5
internal report
ABAQUS user subroutines for the simulation of
viscoplastic behaviour including anisotropic damage
for isotropic materials and for single crystals
Weidong Qi, Wolfgang Brocks
June 2001

2
Institut für Werkstofforschung
GKSS-Forschungszentrum Geesthacht

0. Nomenclature 3
1. Introduction 7
2. The CDM-based anisotropic damage model 7
3. Unified models of BODNER-PARTOM and of CHABOCHE coupled with damage 10
4. Anisotropic creep model for cubic single crystals coupled with damage 11
5. User material routines and their applications 13
5.1 Circumferentially notched bar - CHABOCHE model coupled with damage 13
5.2 Plate containing a hole - BODNER-PARTOM model coupled with damage 19
5.3 Single crystal plate containing a hole - the anisotropic creep and damage model of BERTRAM,
OLSCHEWSKI & QI 21
5.4 TiAl turbine blade - CHABOCHE model coupled with damage 22
6. References 24
7. Appendices: ABAQUS-Inputfiles 26
7.1 Appendix 1: Circumferentially notched bar - CHABOCHE model coupled with damage 26
7.2 Appendix 2: Plate containing a hole - BODNER-PARTOM model coupled with damage 28
7.3 Appendix 3: Single crystal plate containing a hole - the anisotropic creep and damage model of
BERTRAM, OLSCHEWSKI & QI 31
7.4 Appendix 4: TiAl turbine blade - CHABOCHE model coupled with damage 33
4

0. Nomenclature
scalars
a, c, d material parameters for kinematic hardening in CHABOCHE's model
b material parameter for isotropic hardening in CHABOCHE's model
h(x) HEAVISIDE function
m material parameter for damage evolution, eq. (3b)
m1, m1 material parameters in the BODNER-PARTOM model
n material parameter in orientation function, eq. (6)
n creep exponent in CHABOCHE's model
p material parameter in eq. (6)
p accumulated effective plastic strain, internal variable in CHABOCHE's model
q material parameter in the definition of the damage-active stress, eq. (2)
r material parameter in CHABOCHE's model
r1, r2
material parameters in the BODNER-PARTOM model
A1, A2 material parameters in the BODNER-PARTOM model
B0
Ci
Di
D0
DI
DR
Ji
material parameter for damage evolution, eqs. (3b) and (4)
temperature-dependent material parameters (i = 1, 2, 3), eqs. (21a-e)
viscosities, temperature-dependent material parameters (i = 1, 2, 3), eqs. (21a-e)
material parameter in the BODNER-PARTOM model
maximum principal damage
critical value of maximum principal damage
scalar invariants (i = 1, 2, 3, 4)
K viscosity, material parameter in CHABOCHE's model
Ki
Ki
Li
material parameter (i = 1, 2, 3) in the BODNER-PARTOM model
temperature-dependent material parameters (i = 1, 2, 3), eqs. (21a-e)
viscosities, temperature-dependent material parameters (i = 1, 2, 3), eqs. (21a-e)
R(p) actual yield stress for isotropic hardening in CHABOCHE's model
R 0
initial yield stress in CHABOCHE's model
5

R ∞
Wi
Z I , Z D
Zij
6
saturated yield stress for isotropic hardening in CHABOCHE's model
specific work of inelastic strain in the BODNER-PARTOM model
internal variables for isotropic and kinematic hardening in the BODNER-PARTOM model
material parameters (i = 1, 2, 3; j = 1, 2, 3, 4), eqs. (22a,b)
β, βi material parameters for damage evolution, eqs. (3b, 4)
ε i
e
εi η i
eigen-values (i = 1, 2, 3) of total strain tensor E
eigen-values (i = 1, 2, 3) of elastic strain tensor E e
orientation function
φ(p) material function for kinematic hardening in CHABOCHE's model
φ ∞
ˆ
σ i
vectors
e i
c
ei saturated value of φ(p) , material parameter for kinematic hardening in CHABOCHE's model
eigen-values (i = 1, 2, 3) of damage-active stress tensor ˆ
S
orthogonal unit vectors (i = 1, 2, 3), reference base, e i ⋅ e j = δ ij
lattice vectors (i = 1, 2, 3)
e εe e
n , ni eigen-vectors (i = 1, 2, 3) of total and elastic strain tensors, E and E , respectively
i
s
n ˆ i
eigen-vectors (i = 1, 2, 3) of damage-active stress tensor ˆ
S
second order tensors
B internal variable for kinematic hardening in the BODNER-PARTOM model
D damage tensor
Da
E, E e
active damage tensor
total and elastic strain tensor
E + , E e+ positive projections of total and elastic strain tensors, E and E e , eqs. (11a,b)
E Ý i
inelastic strain rate tensor in unified models
H ε , H εe spectral tensors, eqs. (8a,b)
I second order identity tensor

S CAUCHY stress tensor
˜
S effective stress tensor
ˆ
S damage-active stress tensor
˜ ′
S deviator of the damage-active stress
Y D
thermodynamic force conjugate to damage tensor D
X backstress tensor
7

W internal variable, eq. (20b)
fourth order tensors
< 4>
Ai < 4>
I
< 4>
P , ε Pεe < 4>
R
< 4>
S
< 4>
T
operations
ab = a i b j e i e j
a ⋅ b = a ib i
8
material tensors (i = 1, 2, 3, 4, 5), eqs. (21a-e)
identity tensor
positive spectral projection operator for total and elastic strain tensor, eqs. (10a,b)
lattice tensor, eq. (5)
damage characteristic tensor, eq. (3a)
positive projection operator, eq. (12)
AB = A ij B kl e ie ke je l
A ⋅ B= A ijB jk e ie k
A : B= A ij B ji
tensor product of two vectors
scalar product of two vectors
tensor product of two (second order) tensors
scalar product of two (second order) tensors
double scalar product of two (second order) tensors
A = A 2 = A ij A ji EUKLIDean norm of second order tensor
< 4>
C : A = C A e e ijkl kl i j
double scalar product of a fourth and a second order tensor

1. Introduction
A CDM (continuum damage mechanics) based anisotropic damage model has been established
by QI and BERTRAM to describe the anisotropic development of material damage in single crystals [QI
& BERTRAM, 1997; QI, 1998; QI & BERTRAM, 1999; QI, BROCKS & BERTRAM, 2000] and in
isotropic material [QI & BROCKS, 2000a, b, c]. Using the effective stress concept of CDM, this model
can be coupled with any continuum mechanics model by introducing an adequately defined "effective
stress tensor". The resulting model is then able to describe the deformation behaviour with respect to
anisotropic material damage including lifetime predictions. The unified viscoplastic model proposed by
BODNER & PARTOM [1975] and by CHABOCHE [CHABOCHE & ROUSSELIER, 1983] for polycrystal
alloys and the creep model suggested by BERTRAM & OLSCHEWSKI [1996] for single crystal alloys,
respectively, are chosen for coupling with the damage model. The resulting models have been
implemented into subroutines of the FE-code ABAQUS as "user-defined material models" (UMAT) and
can be used to perform FE computations on structural components of poly and single crystals.
This report gives a brief description of the models and presents some results of FE-analyses
using the respective UMATs. Materials used for the analyses are the Ni-based superalloy IN738 LC,
the Ni-based single crystal SRR99 and a TiAl intermetallic alloy.
2. The CDM-based anisotropic damage model
Damage of materials is a progressive process ending in final fracture. A natural characteristic of
material damage is that the damage generally develops anisotropically. A second-order symmetric tensor
D is chosen in the present models as damage variable for the description of the anisotropic damage.
According to the effective stress concept of CDM, the effect of stresses and damage on the deformation
behaviour can be represented by an adequately defined effective stress. This effective stress tensor is
defined as
˜
S = (I − D) −1 2 ⋅S⋅ (I − D) −12 , (1)
9

where S is the CAUCHY stress tensor and I denotes the second-order identity tensor. Similarly, it is
supposed that the contribution of the stress and damage on the damage development can be represented
by a newly introduced damage-active stress defined analogously to eq. (1) as
10
ˆ
S = (I − D) −q ⋅S ⋅(I − D) −q =
3
σ σ
∑ σ ˆ ˆ
i n ˆ
i n i , (2)
i=1
σ
where q is a material parameter, σ ˆ and n ˆ
i i (i = 1, 2, 3) are the eigen-values and the corresponding
eigen-vectors of ˆ
S . From the point of view of linear irreversible thermodynamics the evolution law for a
second order symmetric damage tensor can be generally expressed as:
D Ý = S
: YD , (3a)
< 4>
where YD is the thermodynamic force conjugate to the damage tensor, and S
< 4>
characteristic tensor of rank four, respectively. If the fourth-order tensor S
is the damage
is symmetric and positive-
definite, the thermodynamic restrictions will be automatically satisfied [KRAJCINOVIC, 1983; GERMAIN,
NGUYEN & SUQUET, 1983; YANG, ZHOU & SWOBODA, 1999].
Motivated by the results of experimental investigations, it is assumed that only the tensile principal
damage-active stresses are responsible for the damage evolution and that damage grows perpendicularly
to the direction of the principal damage-active stresses. Thus, consider that damage may also develop
partly isotropically, the damage evolution law is assumed taking the following particular form:
< 4>
D Ý =
⎛
⎝
βII + (1 − β) I ⎞
⎠ :
ˆ
S
< 4>
where β, B0, m are material parameters. I
B 0
m
< 4>
=
⎛
⎝
βII + (1 − β) I ⎞
⎠ :
3 σ ˆ i σ σ
∑ n ˆ ˆ i n i , (3b)
i=1
B 0
m
denotes the fourth-order identity tensor, and 〈.〉 is the
MCCAULEY bracket, which equals one for positive arguments and zero else. Creep rupture is assumed
to take place when the maximum principal damage DI reaches a critical value DR.
For single crystal superalloys, the initial anisotropy must be considered. The following particular
form of damage law is taken for single crystals with cubic symmetry (F.C.C. and B.C.C. single crystals):

4>
with R
D Ý = β II + β I
1 2
+ β R 3
⎛
⎞
⎝
⎠ :
3 η ˆ iσ i
n ˆ σ ˆ σ ∑ i n i
(4)
⎡
ηi =
⎣
⎢
3
i=1
B 0
m
c c c c
= ∑ e ei ei ei , (5)
i
i=1
3
∑
j=1
σ c ( n ˆ i ⋅e j)
2n⎤
⎦
⎥
p
, (6)
c
where β1, β2, β3=(1−β1−β2), B0, p, n and m are material parameters, e j (j=1,2,3) are the lattice
vectors, and η i is an orientation function which satisfies the cubic symmetry.
Damage can also be inactive. Let us consider a single micro-crack embedded in an elastic material
with a tensile load perpendicular to the crack faces. If the load is reversed the crack will close and in a
one-dimensional case the material behaves as uncracked. This phenomenon is called “damage
deactivation” (not “healing”) in CDM. The damage still exists but the loading condition can render it
inactive. For the representation of this mechanism the phenomenological algorithm proposed by HANSEN
& SCHREYER [1995] can be used. In this method the microcrack opening/closing effect is introduced by
considering the spectral decomposition of the elastic strain tensor E e and the total strain tensor E
3
E e e εe εe
ε ε
= ∑ εi ni ni , E = ∑ εi ni ni , (7a, b)
i=1
3
i=1
e εe e e
where εi and εi are the eigenvalues, ni and ni are the corresponding eigenvectors of E and E,
respectively. Let the positive (tensile) spectral tensor corresponding to the elastic and to the total strain
be defined as
3
H εe e εe εe
= ∑ h(εi )ni ni , H ε ε ε
= ∑ h(εi )n ini
(8a, b)
i=1
respectively, with the modified Heaviside function
3
i=1
11

12
0 for x ≤ xm 1
h(x) =
2 1− cos π(x − x ⎧
⎫
⎪ ⎛ ⎡
m) ⎤ ⎞
⎪
⎨ ⎜
⎝ ⎣
⎢ xp − xm ⎦
⎥
⎟ for xm < x < xp ⎬
⎪
⎠
⎪
⎩ 1 for x ≥ xp ⎭
where xm and xp are two material parameters. The positive spectral projection operators (fourth-order
tensor) for the elastic and the total strains are defined as
Pεe = H εe H εe
, Pε = H ε H ε
respectively. The positive projection of the elastic and the total strain tensors are then given by
E e +
= Pee : E e , E + < 4>
= Pe (9)
(10a, b)
: E (11a, b)
respectively. By introducing a strain-based positive projection operator
< 4>
T
= I
< 4> < 4>
−
⎛
I − P ⎞
⎝ εe⎠
: I
⎛
− P ⎞
⎝ ε ⎠
a symmetric, so-called active damage tensor can be defined as
D = a T
: D (13)
Thus, the effective stress tensor and the damage-active stress tensor accounting for damage deactivation
are defined as
˜
S = (I − D a ) −1 2 ⋅ S ⋅(I − D a ) −1 2 , (14)
S ˆ = (I − Da) −q T −q
⋅S ⋅(I − Da ) , (15)
respectively.
If the effective stress tensor and the damage active stress tensor defined in (14) and (15),
respectively, are used instead of those defined in (1) and (2), the damage deactivation can be described.
(12)

3. Unified models of BODNER-PARTOM and of CHABOCHE coupled with
damage
For the description of viscoplastic behaviour a lot of unified models have recently been developed.
The main advantage of unified models compared to classical plasticity and creep models is the treatment
of all aspects of inelastic deformation behaviour including plastic flow under monotonic and cyclic
loading, creep and stress relaxation by a single inelastic strain quantity [OLSCHEWSKI et al., 1990]. The
total strain rate is decomposed into an elastic and an inelastic part by
E Ý = Ý E e + Ý E i
As the model proposed by BODNER & PARTOM [1975] and by CHABOCHE [CHABOCHE &
ROUSSELIER, 1983] are two popular models, they are chosen to be combined with the above damage
model.
The effective stress concept of CDM says that any constitutive equation for the damaged material
can be derived in the same way as for a virgin material if the stress tensor is replaced by an adequately
defined effective stress tensor. Following this concept, the BODNER-PARTOM model and the
CHABOCHE model can simply be extended to include material damage by replacing the stress tensor in
the respective constitutive equations by the effective stress tensor defined in the expressions (1) or (14).
The resulting models are summarized as follows:
(16)
13

flow rule:
E Ý i = p Ý
˜ S ′ − X
, p Ý =
S ˜ ′ − X
isotropic hardening rule:
14
CHABOCHE BODNER-PARTOM
3
2 ˜ S ′ − X − R(p)
K
Ý
R = b(R ∞ − R) Ý
p , R(p = 0) = R 0
kinematic hardening rule:
X Ý = c 3
aE Ý
2 i ⎡
−φ( p)Xp Ý ⎤ ⎡
− d
⎣
⎦ ⎣
⎢
3
2 X
a
− ωp
φ(p) = φ∞ − ( φ∞ − 1)e
r
⎤
⎦
⎥
X
3
2 X
where ˜ ′
S is the deviator of the damage-active stress.
n
E Ý i = 2D0 exp − 1 Z
2
I + ZD 3
2 ˜ ⎡ ⎛ ⎞
⎢ ⎜
⎝ S
⎟
′
⎣
⎢
⎠
Z D = B : ˜ S ′
S ˜ ′
Z Ý I = m1 K − 1 Z I ( ) Ý
⎡
B Ý = m2 ⎢ K3 ⎣
2n
⎤
⎥
⎦
⎥
S ˜ ′
S ˜ ′
Z
W − i A 1
I ⎡ − K ⎤ 2
⎢
⎣ K ⎥
1 ⎦
Z I (t = 0) = K 0
˜
S
˜
S
⎤
− B⎥
⎦
Ý
⎡ B ⎤
W i − A2 ⎣ ⎢ K1 ⎦ ⎥
W i = ˜ S ′ : Ý E i dτ
4. Anisotropic creep model for cubic single crystals coupled with damage
Starting from a rheological four-parameter BURGERS-model which consists of two springs and two
dampers, BERTRAM & OLSCHEWSKI [1993, 1996] used a projection method to construct an
anisotropic 3D model for the description of the creep behaviour of cubic single crystals at high
temperatures. Replacing the stress tensor by the effective-stress tensor (1) in the model, the constitutive
equations coupled with the damage model can be written as:
t
∫
0
r 2
r 1
B
B
(17)
(18)
(19)

4>
E Ý = A1
W Ý = A4 < 4>
: ˜ Ý
S + A2 < 4>
: ˜ Ý
S + A5 : ˜
< 4>
S + A3 :W , (20a)
: ˜ ( S − W)
, (20b)
where W is an internal tensor variable of rank two, and A i
tensors defined as:
< 4>
A1 =
A2 =
A3 =
A 4 =
A5 < 4>
where R
=
3
∑
i=1
3
∑
i=1
3
∑
i=1
3
∑
i=1
3
∑
i=1
1
C i + K i
< 4>
Pi 1 ⎛
⎜
Ci + Ki ⎝
1
C i + K i
K i
C i + K i
K i
C i + K i
< 4>
P1 = 1
3 II, P2 ,
Ci Di 1
D i
Pi C i
D i
Pi ,
< 4>
Pi + C i
L i
,
,
+ Ki ⎞ < 4>
⎟ P , i
Li ⎠
< 4> < 4>
= R − P1 , P3 = I
< 4>
< 4>
(i = 1, 2, 3, 4, 5) are fourth-order material
(21a-e)
− R , (21f-h)
is defined in eq. (5), Ci , Ki , Di , Li (i = 1, 2, 3) are temperature dependent material
parameters. Note that the viscosities Di and Li are also dependent on the applied stress. They are
assumed to have the following form:
⎛ 4 ⎞
⎛ 4 ⎞
D = i D0i exp ⎜ −∑ Zij J ⎟
j , L = i L0i exp⎜ −∑ ZijJ ⎟
j
(22a, b)
⎝ ⎠
⎝ ⎠
j=1
j=1
with the material parameters Zij (i = 1, 2, 3; j = 1, 2, 3, 4) and the following scalar invariants with cubic
symmetry
J 1 = σ 11 σ 22 +σ 22 σ 33 + σ 33 σ 11 ,
2 2 2 J = σ +σ23 + σ31 ,
2 12
J 3 = σ 11 σ 22 σ 33 ,
J 4 = σ 11 σ 12
2 2
2 2
( +σ13)+σ
22 σ23 + σ21
2 2
( )+ σ33( σ31 + σ32).
By the assumption that volume changes occur only elastically, it follows
(23a-d)
15

16
−1 −1
D1 = 0 , L1 = 0, Z i1 = 0 (i = 1, 2,3) . (24a-c)

5. User material routines and their applications
The material models mentioned above are implemented into the commercial FE code ABAQUS
as user-defined material model by writing the corresponding user subroutines, UMAT, see Table 1.
With help of these UMATs one can apply the models in an FE analysis of viscoplastic damage
behaviour of engineering components and structures. All the routines are written in the computer
language FORTRAN using the forward integration algorithm for numerical integration. Only isothermal
loading conditions have been considered and the damage deactivation has not been included in any of
the routines. Viscoplastic FE calculations are very time consuming. In the routines however, no
automatic time step control is used, so that there is a necessity to improve the respective algorithms. All
examples presented below are conducted using ABAQUS/Standard 5.8.
model UMAT
CHABOCHE model coupled with damage d-chaboche.f
BODNER-PARTOM model coupled with damage d-bodner.f
anisotropic creep and damage model of BERTRAM,
OLSCHEWSKI & QI
d-scsrr99.f
Table 1: Constitutive models and names of the respecitve UMATs
5.1 Circumferentially notched bar — CHABOCHE model coupled with damage
The material parameters of the CHABOCHE model of IN 738 LC have been determined by
OLSCHEWSKI et al. [1990] and their values at 850 °C are shown in Table 2. The material parameters of
the damage model were estimated by using numerical optimization methods to fit the creep data of the
three creep tests presented in the work of OLSCHEWSKI et al. [1990]. During this process the above
values of the parameters of the CHABOCHE model were kept constant. Table 3 shows a set of damage
parameters for IN 738 LC at 850 °C. Note that just three uniaxial tests are not sufficient for parameter
identification, so that the values given in Table 3 are only first estimates. For lack of biaxial test data, the
anisotropy parameter β can not be determined. Comparison of the experiments and the predictions by
the CHABOCHE model and by the coupled model with damage, respectively, using the material
17

parameters of Tables 2 and 3, and the damage evolution during the creep processes are shown in Fig.
1.
18

ε i
4
[%]
3
2
1
E 149650 MPa ν 0.33 a 311 MPa
K 397 MPa⋅h 1/n n 7.7 b 317
R0 153 MPa φ∞ 1.1 c 201
R∞ 0.0 MPa ω 0.04 r 3.8
d 81.72 MPa/h
Table 2: Material parameters of the CHABOCHE model for IN 738 LC at 850 °C.
β q B0 m DI
0.0 ∼ 1.0 0.4 613 MPa·h 1/m
14 0.07
Table 3: Material parameters of the damage model for IN 738 LC at 850 °C
Uniaxial Creep (IN738 LC, 850°C)
σ=335 MPa, Exp.
σ=392 MPa, Exp.
σ=410 MPa, Exp.
Chaboche-model
model with damage
0
0 20 40 time [h] 60
0.2
0.1
σ=410 MPa
σ=392 MPa
σ=335 MPa
0.0
0 20 40
time [h]
60
19

20
Fig. 1. Experiments and model predictions for inelastic strain (upper) and damage evolution
(lower diagramme)
Circumferentially notched bars are often used to investigate the influence of triaxial stress state on
the damage and fracture behaviour in creep processes. KOBAYASHI et al. [1998] reported the results of
their experimental studies on such bars. Several creep damage tests of pure aluminum were carried out.
The nucleation and growth of voids during the creep process were observed by means of scanning
electron microscopy and optical microscopy. They found out that only some portion of creep voids
actually appeared on the surface of the notch root, and that the lengths of creep voids beneath the notch
surface exceeded ten times the lengths of those appearing on the surface. They further found out that
under a relatively low load, many creep voids nucleated on a plane inclined at about 45° against the
tensile direction, and their coalescence formed a cone-type fracture surface. These results motivated the
simulation of the damage behaviour in such bars. As the test data of aluminium from which the material
parameters could have been estimated were not available, the alloy IN738 LC at 850 °C is used, again.
The stress concentration and the multiaxial stress distribution are dependent on the geometry of
the specimen. The same geometry as used in the work of KOBAYASHI et. al. [1998] is used. Fig. 2
shows the specimen geometry and the FE-mesh, where R = 5.0 mm, r0 = 1.0 mm and L = 10.0 mm.
Axisymmetric solid elements of type CGAX4 from the element library of ABAQUS are used for the FE
calculations. Fig. 3 shows the distributions of the maximum principal damage in the notch area for
β = 0.1, 0.5 and 1.0, respectively. The applied load is σ2 = 150 MPa. It can be clearly seen that the
most damaged area occurs beneath the notch surface in all cases. At the beginning of the creep process,
the maximum damage takes place on the notch root surface, where the maximum stress appears. During
the process, however, the location of the maximum damage moves away from the surface. The
ABAQUS input-file for β = 0.1 used for the present calculation is given in the Appendix 1.

)
a)
Fig. 2. a) Geometry of the specimen used by KOBAYASHI et al. [1998]
b) FE-mesh. R = 5.0 mm, r0 = 1.0 mm, L = 10.0 mm
Fig. 4. shows the contour plots of the second direction cosine of the principal directions
D D
corresponding to DI, n ⋅ e2 = cos∠ n I ( ,e2 I ), and of the maximum damage at t = 411 hours for
β = 0.1. The maximum local damage at this time reaches a value of 0.0643, immediately before the local
fracture takes place and meso-cracks may have been initiated; note that the critical value of damage is
Dc = 0.07. The values of the direction cosines at the location of maximum damage, DI, indicate that the
D
direction n of maximum damage coincides with the e2-direction, which means that the surfaces of
I
nucleated micro/meso-cracks will be perpendicular to the e2-direction, i.e. the loading axis.
21

22
Fig. 3. Contour plots of the maximum principal damage for β = 0.1, 0.5 and 1.0.
Other experimental investigations of KOBAYASHI et al. show that under relatively low loads, σ2, creep
voids nucleated on a plane inclined by about 45° against the tensile direction. Though pure aluminium
was used in these tests for which no material data exist, the experiments motivated the simulation of the
damage behaviour at a lower creep load of σ2 = 100 MPa for the present Ni-based alloy and β = 0.1.
The contour plots of the second direction cosine of the principal direction corresponding to DI,
D
cos∠( n ,e2 I ), at t = 11000 hours and the maximum damage distribution at t = 11000 and
12000 hours, respectively, are shown in Fig. 5. The values of the direction cosine at the location of
maximum damage, i.e. where local fracture will occur, is −0.7÷−0.8. That means that maximum principal
damage is inclined at about 45° against the tensile direction, which indicates that cracks may form a
cone-type fracture surface. Once the meso-crack has been formed or even local fracture has been taken
place, the local behaviour of the material will strongly depend on the shape and size of the crack. Further
experimental investigations are needed.

Fig. 4. Contour plots of direction cosine and max. damage at t = 411 hours for β = 0.1.
23

24
Fig. 5. Contour plots of direction cosine and of the maximum principal damage for β = 0.1.

5.2 Plate containing a hole — BODNER-PARTOM model coupled with damage
The material parameters of the BODNER-PARTOM model of IN 738 LC at 850 °C have also been
determined by OLSCHEWSKI et al. [1990], and their values at 850 °C are shown in Table 4. The
material parameters of the damage model are the same as listed in Table 3.
E 149650 MPa ν 0.33 K0 4.18 10 5 MPa
D0
8.82 10 9 h -1
n 0.289 K1 3.76 10 5 MPa
A1=A2 1.65 10 -7 MPa/h m1 0.581 K2 3.07 10 5 MPa
r1=r2 5.4 m2 0.344 K3 1.54 10 5 MPa
Table 4: Material parameters of the BODNER-PARTOM model for IN 738 LC at 850 °C.
In gas turbine blades with cooling channels, stress concentration occurs due at these channels. A
square plate with a central circular hole is therefore chosen as a model representation of the area of
blades where the air cooling channels are located. The FE model used for the calculation is shown in
Fig. 6. First, the plate is subjected to a creep load of σ3 = 180 MPa. After 40000 hours the maximum
damage reaches a value of about 0.1. A second load of σ2 = 180 MPa is then applied. There is only
one element in the thickness direction so that any gradient over the thickness can not be captured. The
three-dimensional 8-node linear brick continuum element with reduced integration, C3D8R, from the
element library of ABAQUS is used, and geometric non-linearity has been considered. Figs. 7 and 8
show the contour plot of the maximum principal damage after 40000 h and 98000 h, respectively. β is
assumed to be 0.5. Distribution of the maximum principal value of the strain and stress, after 40000 h
and 98000 h, are shown in the Figs. 9-12, respectively. The ABAQUS input-file used for the
computation is given in the Appendix 2.
25

26
Fig. 6. FE-mesh and loading condition
Fig. 7. Max. principal damage after 40000 h Fig. 8. Max. principal damage after 98000 h
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
5.3 Single crystal plate containing a hole — the anisotropic creep and damage model of
BERTRAM, OLSCHEWSKI & QI
The material parameters of anisotropic creep model for the single crystal SRR99 at 760 °C have
been estimated by BERTRAM and OLSCHEWSKI [1996]. The corresponding material parameters of the
damage model have been estimated by QI [1998]. Fig. 13 shows the applied FE model. The uniaxial
tensile load, σ2, is applied in the crystal direction [001]. Because of the symmetry, only 1/2 of the
thickness of the specimen has to be modelled. Three elements are used over the half-thickness to
capture the gradients in the thickness direction. The distribution of the maximum principal damage at
t = 34000 hours is shown in Fig. 14. For comparison, Fig. 15 shows the initiation of cracks at a cavity
in a prerafted single crystal CMSX-2 after 44 hours of creep at 850 °C and 520 MPa (creep life
fraction = 95%) [AI et al., 1990], indicating at least a qualitative coincidence of the damage loci between
numerical simulation and experiment. Contour plots of the strain ε22 and the stress σ22 at t = 34000
hours are shown in Figs. 16 and 17, respectively. The ABAQUS input-file used for the computation is
given in the Appendix 3.
27

28
Fig. 13. FE-mesh and loading condition

Fig. 14. Maximum. principal damage after 34000 h Fig. 15. Crack initiation at a cavity
Fig. 16. Max. principal strain after 34000 h Fig. 17. Max. principal stress after 34000 h
5.4 TiAl turbine blade — CHABOCHE model coupled with damage
A model turbine blade made of a TiAl intermetallic alloy developed at the GKSS Research
Centre is used as object of the FE-calculation. The blade has a length of 224 mm. The material
parameters of the CHABOCHE model have been estimated by MOHR [1999], as listed in Table 5. Table
6 gives the corresponding material parameters of the damage model used for the calculation. The
continuum element C3D4 of the element library of ABAQUS is used. The number of nodes is 1476 and
the number of elements is 4825. The blade is subject to centrifugal forces, only, at a constant rotation
speed of 40000 1/min; the density of TiAl is 3.8 g/cm 3 . Geometric non-linearity has been considered.
The distribution of the maximum damage at t = 9060 hours is shown in Fig. 18. It obviously ocurs at the
root of the blade which after all has not been modelled realistically. The calculation is just supposed to
29

proove that the model performs well also with large structures. The ABAQUS input-file used for the
computation is given in the Appendix 4.
30

E 150000 MPa ν 0.24 a 335 MPa
K 487 MPa⋅s 1/n n 15.3 b 207
R0 126 MPa φ∞ 0.0 c 35.4
R∞ 0.0 MPa ω 0.0 r 3.1
d 0.023 MPa/s
Table 5: Material parameters of CHABOCHE model for the TiAl at 700 °C.
β q B0 m
0.3 0.3 1500 MPa·h 1/m
Table 6: Material parameters of the damage model for the TiAl at 700 °C
14
Fig. 18. Maximum principal damage after 9060 h
31

6. References
AI, S.H.; LUPINC, V. and MALDINI, M. (1990): "Creep fracture mechanics in single crystal superalloys".
In: High Temperature materials for Power Engineering 1990, Proceedings of a Conference held in
Liège, Belgium, 24-27 September 1990. Part II (Eds. E. BACHELET et al.)
BERTRAM, A.; OLSCHEWSKI, J. (1993): "Zur Formulierung anisotroper linearer anelastischer
Stoffgleichungen mit Hilfe einer Projektionsmethode". ZAMM 73 (4-5), T401-403.
BERTRAM, A.; OLSCHEWSKI, J. (1996): "Anisotropic creep modeling of the single crystal superalloy
SRR99". J. Comp. Mat. Sci. 5, pp.12-16.
BODNER, S.R.; PARTOM, Y. (1975): "Constitutive equations for elastic-viscoplastic strain hardening
materials". J. Appl. Mech. 42, pp.385-389.
CHABOCHE, J.L.; ROUSSELIER, G. (1983): "On the plastic and viscoplastic constitutive equations". J.
Press. vess. technol. 105, pp.105-164.
GERMAIN, P.; NGUYEN, Q. S.; SUQUET, P. (1983): "Continuum Thermodynamics". J. Appl. Mech. 50,
pp.1010-1020.
KOBAYASHI, K.I.; IMADA, H.; MAJIMA, T. (1998): "Nucleation and growth of creep voids in
circumferentially notched specimens", JSME Int. J., Series A: Solid Mechanics and Material
Engineering, 41, 218-224.
KRAJCINOVIC, D. (1983): "Constitutive equations for damaging materials". J. Appl. Mech. 50, pp. 355-
360.
MOHR, R. (1999): "Modellierung des Hochtemperaturverhaltens metallischer Werkstoffe". Dissertation,
Technische Universität Hamburg-Harburg, GKSS 99/E/66.
OLSCHEWSKI, J.; SIEVERT, R.; MEERSMANN, J. and ZIEBS, J. (1990): "Selection, calibration and
verification of viscoplastic constitutive models used for advanced blading methodology". In: High
Temperature Materials for Power Engineering, Proceedings of a Conference held in Liège, Belgium,
24-27 September 1990 (Eds. BACHELET et al.), Kluwer Academic Publishers, pp.1051-1060.
QI, W.; BERTRAM, A. (1997): "Anisotropic creep damage modeling of single crystal superalloys".
Technische Mechanik. 17(4), pp.313-322.
QI, W. (1998): "Modellierung der Kriechschädigung einkristalliner Superlegierungen im
Hochtemperaturbereich". Dissertation, Technische Universität Berlin, Fortschritts-Berichte VDI Verlag
GmbH, Düsseldorf.
QI, W.; BERTRAM, A. (1998): "Damage modeling of the single crystal superalloy SRR99 under
monotonous creep". Computational Materials Science 13, pp.132-141.
33

QI, W. ; BERTRAM, A. (1999): "Anisotropic continuum damage modeling for single crystals at high
temperatures". Int. J. of Plasticity 15, pp.1197-1215.
QI, W.; BROCKS, W. (2000a): "A CDM-based approach to creep damage and component lifetime".
Proceedings of the Int. Conf. on Computational Engineering & Sciences, ”ICES‘2K” Ed. S. ATLURI),
21-25 August 2000, Los Angeles.
QI, W.; BROCKS, W. (2000b): "Simulation of anisotropic creep damage in engineering components".
Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering
”ECCOMAS 2000”, 11-14 September 2000, Barcelona.
QI, W.; BROCKS, W.; BERTRAM, A. (2000): A FE-analysis of anisotropic creep damage and
deformation in the single crystal SRR99 under multiaxial loads. Computational Materials Science 19
(2000), pp. 292-297.
YANG, Q.; ZHOU, W.Y.; SWOBODA, G. (1999): Micromechanical identification of anisotropic damage
evolution laws. Int. J. of Fracture 98, pp. 55-76.
34

7. Appendices: ABAQUS-Inputfiles
7.1 Appendix 1: Circumferentially notched bar — CHABOCHE model coupled with damage
*HEADING
circum. bar
a= 5.0, r0= 1.00, N-alfa=20, N-r=40, K= 6
*PREPRINT, ECHO=NO, MODEL=NO, HISTORY=NO
**
*RESTART, WRITE, FREQUENCY=2000
**
*NODE
1, 5.00000, 1.00000
101, 5.00000, 1.03927
201, 5.00000, 1.07542
...
5181, 5.00000, -7.75000
5281, 5.00000, -8.00000
5381, 5.00000, -8.25000
5481, 5.00000, -8.50000
5581, 5.00000, -8.75000
5681, 5.00000, -9.00000
5781, 5.00000, -9.25000
5881, 5.00000, -9.50000
5981, 5.00000, -9.75000
6081, 5.00000, -10.00000
*ELEMENT, TYPE=CGAX4, ELSET=solid
1, 2, 1, 101, 102
2, 3, 2, 102, 103
3, 4, 3, 103, 104
4, 5, 4, 104, 105
5, 6, 5, 105, 106
6, 7, 6, 106, 107
7, 8, 7, 107, 108
...
5978, 6078, 6079, 5979, 5978
5979, 6079, 6080, 5980, 5979
5980, 6080, 6081, 5981, 5980
**
** define node-set
**
*NSET, NSET=zplus, GENERATE
6001, 6021, 1
*NSET, NSET=zminus, GENERATE
6061, 6081, 1
**
** define element-set
**
*ELSET, ELSET=zlast, GENERATE
5901, 5920, 1
**
** define materials and UMA
**
*SOLID SECTION, ELSET=SOLID, MATERIAL=VISCOPLA
1.,
*MATERIAL, NAME=VISCOPLA
*USER MATERIAL, CONSTANTS=19
35

**
** IN738LC, 850 deg C
******** IN 738 LC, 850 C, ==== CH-model==== h, MPa
** E, nu, Ro, Q, b, c, a, phiinf
149650., .33, 153., -153., 317., 201., 311., 1.1
*** omega, d, r, n, K; q, beta, Bo
*** ( beta=1 -> isot. damage)
0.04, 81.72, 4.8, 7.7, 397.0, .4, 0.1, 613.
*** m, Dkey, Ckey Ckey ( Dkey < or = 0: do not consider damage)
***( Ckey must be –1.0, only UMAT-developer may change it!)
14., 1.0, -1.0
*DEPVAR
29
**
*USER SUBROUTINE, INPUT=/wms12/weiqi/umats/poly/d-chaboche.f
**
** loading !! the unit is hour !!
**
*STEP, INC=900000000, NLGEOM
*VISCO, CETOL=1.0E-6
0.0001, 0.001, , 0.0001
**
36

** auflagerung
**
*BOUNDARY, OP=NEW, TYPE=DISPLACEMENT
6061, 1, 2, 0.0
zminus, 2, 2, 0.0
**** === define amplitude for loading process
** 1s=0.00027778 h
*AMPLITUDE, TIME=TOTAL TIME, NAME=creep
0.0, 0.0, 0.001, 1.0, 9000000.0, 1.0
**
*DLOAD, OP=NEW, AMPLITUDE=creep
zlast, P3, -150.
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** step 2
**
*STEP, INC=900000000, NLGEOM
*VISCO, CETOL=1.0E-6
0.0001, 0.001, , 0.0001
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** step 3
**
*STEP, INC=900000000, NLGEOM
*VISCO, CETOL=1.0E-6
0.001, 1., , 0.01
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** step 4
**
*STEP, INC=900000000, NLGEOM
*VISCO, CETOL=1.0E-6
0.1, 90000000.0, , 0.1
**
*RESTART, WRITE, FREQUENCY=100
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
37

7.2 Appendix 2: Plate containing a hole — BODNER-PARTOM model coupled with damage
*HEADING
plate containing a crack
a= 5.0, r0= .50, N-alfa=10, N-r=15, K=12 D= .20
*PREPRINT, ECHO=NO, MODEL=NO, HISTORY=NO
*NODE
1, .20, .49846, .03923
5001, .00, .49846, .03923
101, .20, .53761, .04231
5101, .00, .53761, .04231
...
1380, .20, 3.47804, .00000
6380, .00, 3.47804, .00000
1480, .20, 4.16651, .00000
6480, .00, 4.16651, .00000
1580, .20, 5.00000, .00000
6580, .00, 5.00000, .00000
*ELEMENT, TYPE=C3D8I, ELSET=solid
*** 0 - PI/4
80, 5080, 80, 180, 5180, 5001, 1, 101, 5101
180, 5180, 180, 280, 5280, 5101, 101, 201, 5201
280, 5280, 280, 380, 5380, 5201, 201, 301, 5301
380, 5380, 380, 480, 5480, 5301, 301, 401, 5401
480, 5480, 480, 580, 5580, 5401, 401, 501, 5501
580, 5580, 580, 680, 5680, 5501, 501, 601, 5601
...
1477, 6477, 1477, 1577, 6577, 6478, 1478, 1578, 6578
1478, 6478, 1478, 1578, 6578, 6479, 1479, 1579, 6579
1479, 6479, 1479, 1579, 6579, 6480, 1480, 1580, 6580
**
** define node-set
**
*NSET, NSET=zlager0, GENERATE
6550, 6570, 1
*NSET, NSET=zlager1, GENERATE
1550, 1570, 1
*NSET, NSET=ylager0, GENERATE
6530, 6550, 1
*NSET, NSET=ylager1, GENERATE
1530, 1550, 1
**
** define element-set
**
*ELSET, ELSET=zlast, GENERATE
1410, 1429, 1
*ELSET, ELSET=ylast, GENERATE
1401, 1409, 1
1470, 1480, 1
**
** define materials and UMA
**
*SOLID SECTION, ELSET=SOLID, MATERIAL=VISCOPLA
1.,
*MATERIAL, NAME=VISCOPLA
*USER MATERIAL, CONSTANTS=19
**
** IN738LC, 850 deg C
******* BP-Model, units: h, MPa
********========================================
38

** E, nu, Do, n, K1, K2, K3, m1
149650., .33, 8.82E+9, 0.289, 3.76E+5, 3.07E+5, 1.54E+5, 0.581
*** m2, A, r, Unused, Ko; q, beta, Bo
*** ( beta=1 -> isot. damage)
0.344, 1.6524E+7, 5.4, 0.0, 4.18E+5, 0.4, 0.5, 613.
*** m, Dkey, Ckey Ckey ( Dkey < or = 0: do not consider damage)
***( Ckey must be –1.0, only UMAT-developer may change it!)
14., 1.0, -1.0
*DEPVAR
26
**
*USER SUBROUTINE, INPUT=/wms12/weiqi/umats/poly/d-bodner.f
**
**
*RESTART, WRITE, FREQUENCY=1000
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
0.0001, 0.001, , 0.0001
**
** auflagerung
**
39

*BOUNDARY, OP=NEW, TYPE=DISPLACEMENT
** 6550, 1, 3, 0.0
zLager0, 3, 3, 0.0
zLager0, 1, 1, 0.0
zLager1, 3, 3, 0.0
yLager0, 2, 2, 0.0
yLager1, 2, 2, 0.0
**** === define amplitude for loading process
*AMPLITUDE, TIME=TOTAL TIME, NAME=creep1
0.0, 0.0, 0.001, 1.0, 90000000.0, 1.0
**
*DLOAD, OP=NEW, AMPLITUDE=creep1
zlast, P2, -180.
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 2
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
0.001, 0.5, , 0.01
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 3
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
0.01, 49.5, , .5
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 4
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
1., 39950.00, , 4.
**
*RESTART, WRITE, FREQUENCY=400000
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 5
40

** ====== 2. load ein
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
.0001, .001, , .0001
**
**** === define amplitude for loading process
*AMPLITUDE, NAME=creep2
0.0, 0.0, 0.001, 1.0, 90000000.0, 1.0
**
*DLOAD, OP=MOD, AMPLITUDE=creep2
ylast, P5, -180.
*RESTART, WRITE, FREQUENCY=1500
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 6
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
.001, 0.1, , .01
41

**
*RESTART, WRITE, FREQUENCY=1000
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 7
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
.01, 49.50, , 1.
**
*RESTART, WRITE, FREQUENCY=100
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 8
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
1., 60000.00, , 5.
**
*RESTART, WRITE, FREQUENCY=20000
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 9
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
5., 1000000.00, , 5.
**
*RESTART, WRITE, FREQUENCY=200
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
42

7.3 Appendix 3: Single crystal plate containing a hole — the anisotropic creep and damage
model of BERTRAM, OLSCHEWSKI & QI
*HEADING
plate 10x10x0.5, hole radius 0.5, Non-symm
C3D8I, r15-h10-b10-t3, bias: r10-h1-b1-t2, I-DEAS 06-Mar-01
*NODE, SYSTEM=R
1, 3.5355339E-01,-3.5355339E-01, 2.5000000E-01
2, 4.2009962E-01,-4.2009962E-01, 2.5000000E-01
3, 5.0002275E-01,-5.0002275E-01, 2.5000000E-01
...
5294,-4.5000000E+00,-5.0000000E+00, 1.9248187E-01
5295,-4.5000000E+00,-5.0000000E+00, 1.1616359E-01
5296,-4.5000000E+00,-5.0000000E+00, 0.0000000E+00
*ELEMENT,TYPE=C3D8I ,ELSET=E0000001
1, 1, 2, 18, 17, 65, 66, 82, 81
2, 2, 3, 19, 18, 66, 67, 83, 82
3, 3, 4, 20, 19, 67, 68, 84, 83
4, 4, 5, 21, 20, 68, 69, 85, 84
5, 5, 6, 22, 21, 69, 70, 86, 85
...
*SOLID SECTION,ELSET=E0000001,MATERIAL=M0000001
*MATERIAL,NAME=M0000001
*USER MATERIAL, CONSTANTS=14
**
** SRR99, 760 deg C
**
***alfa0 alfa1 alfa2 B[MPa*h] n p m n1
1.0, 0.0, 0.5, 1442.0, 14.133, 0.45489, 51.852, -0.31326
*** Dcr, phi1, phi2, phi3 Dkey, Ckey
**** ( Dkey < or = 0: do not consider damage)
**** ( Ckey must be –1.0, only UMAT-developer may change it!)
0.9, 0.0, 0.0, 0.0, 1.0, -1.E8
*DEPVAR
36
**
*USER SUBROUTINE, INPUT=/wms12/weiqi/umats/single/d-scsrr99.f
**
** loading !! the unit is h !!
**
*RESTART, WRITE, FREQUENCY=1000
**
*STEP, INC=90000000, NLGEOM
*VISCO, CETOL=1.0E-10
0.0001, 0.001, , 0.0001
** auflagerung
*BOUNDARY,OP=NEW
BS000001, 1,, .00000E+00
BS000002, 2,, .00000E+00
3344, 1, 2, .00000E+00
3680, 1, 2, .00000E+00
4016, 1, 2, .00000E+00
BS000003, 3,, .00000E+00
BS000004, 1,, .00000E+00
BS000004, 3,, .00000E+00
BS000005, 2, 3, .00000E+00
3008, 1, 3, .00000E+00
**
** loading
43

**** === define amplitude for loading process
*AMPLITUDE, TIME=TOTAL TIME, NAME=creep1
0.0, 0.0, 0.001, 1.0, 90000000.0, 1.0
**
*DLOAD,OP=NEW, AMPLITUDE=creep1
** BS000006, P4, -100.
BS000007, P6, -350.
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 2
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
0.0001, 0.1, , 0.02
**
44

*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 3
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
0.02, 1.0, , 0.2
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 4
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
0.2, 49.0, , .5
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 5
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
1., 3950.00, , 1.
**
*RESTART, WRITE, FREQUENCY=1000
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEP
**
** Step 6
**
*STEP, INC=90000000
*VISCO, CETOL=1.0E-10
1., 1000000.00, , 1.
**
*RESTART, WRITE, FREQUENCY=200
**
*NODE FILE, FREQUENCY=0
*EL FILE, FREQUENCY=0
*NODE PRINT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
*PRINT, FREQUENCY=0
*END STEPT, FREQUENCY=0
*EL PRINT, FREQUENCY=0
45

*PRINT, FREQUENCY=0
*END STEP
46

7.4 Appendix 4: TiAl turbine blade — CHABOCHE model coupled with damage
*HEADING
SDRC I-DEAS ABAQUS FILE TRANSLATOR 10-Apr-00 13:53:27
units: mm, s, MPa isot. damage)
0.0, 0.023, 3.1, 15.3, 487.0, 0.3, 0.3, 1500.
*** m, Dkey, Ckey Ckey ( Dkey < or = 0: do not consider damage)
***( Ckey must be –1.0, only UMAT-developer may change it!)
47

14., 1.0, -1.0
*DEPVAR
29
**
*USER SUBROUTINE, INPUT=/wms12/weiqi/umats/poly/d-chaboche.f
**
** first loading, cycle 1
**
*STEP, INC=100000000, NLGEOM
***STATIC
*VISCO, CETOL=1.0E-10
.01, 100., , 0.1
**
**
*BOUNDARY,OP=NEW
12, 1, 3, .00000E+00
45, 1, 3, .00000E+00
46, 1, 3, .00000E+00
15, 1, 3, .00000E+00
fuss, 1, 3, .00000E+00
**
** rotations
**
-48-

*AMPLITUDE, NAME=CYCLEONE, DEFINITION=TABULAR, TIME=STEP TIME
0., 0., 10., .05415, 100., 1., 900000000., 1.
**
*DLOAD, OP=NEW, AMPLITUDE=CYCLEONE
blade, CENTRIF, 444000., 35., 0., 0., 0., 0., 1.
**entspricht Drehzahl von 40000/min oder 666/sec
**
*RESTART, WRITE, FREQUENCY=10000000
**
*NODE FILE, FREQ=0
*EL FILE, POS=INTEG, FREQ=0
S, E, SDV
*EL PRINT,POS=INTEG, FREQ=0
*NODE PRINT, FREQ=0
*PRINT, FREQ=0
*END STEP
***
***
***
*STEP, INC=100000000
***STATIC
*VISCO, CETOL=1.0E-10
0.1, 3600., , 10.
**
*EL FILE, POS=INTEG, FREQ=0
*NODE FILE, FREQ=0
*PRINT, FREQ=0
*END STEP
***
***
***
*STEP, INC=100000000
*VISCO, CETOL=1.0E-10
50., 720000., , 50.
*RESTART, WRITE, FREQUENCY=1440
**
*EL FILE, POS=INTEG, FREQ=0
*NODE FILE, FREQ=0
*PRINT, FREQ=0
*END STEP
***
***
**
*STEP, INC=100000000
*VISCO, CETOL=1.0E-10
200., 720000., , 200.
**
*RESTART, WRITE, FREQUENCY=360
**
*EL FILE, POS=INTEG, FREQ=0
*NODE FILE, FREQ=0
*PRINT, FREQ=0
*END STEP
***
**
*STEP, INC=100000000
*VISCO, CETOL=1.0E-10
400., 72000000., , 400.
**
*RESTART, WRITE, FREQUENCY=180
**
*EL FILE, POS=INTEG, FREQ=0
*NODE FILE, FREQ=0
-49-

*PRINT, FREQ=0
*END STEP
-50-