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Materials at High Temperatures
ISSN: 0960-3409 (Print) 1878-6413 (Online) Journal homepage: https://www.tandfonline.com/loi/ymht20
Procedures for handling computationally heavy
cyclic load cases with application to a disc alloy
material
Daniel Leidermark & Kjell Simonsson
To cite this article: Daniel Leidermark & Kjell Simonsson (2019) Procedures for handling
computationally heavy cyclic load cases with application to a disc alloy material, Materials at High
Temperatures, 36:5, 447-458, DOI: 10.1080/09603409.2019.1631587
To link to this article: https://doi.org/10.1080/09603409.2019.1631587
© 2019 The Author(s). Published by Informa
UK Limited, trading as Taylor & Francis
Group.
Published online: 20 Jun 2019.
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Procedures for handling computationally heavy cyclic load cases with
application to a disc alloy material
Daniel Leidermark and Kjell Simonsson
Division of Solid Mechanics, Linköping University, Linköping, Sweden
ABSTRACT
The computational efficiency in analysing cyclically loaded structures is a highly prioritised
issue for the gas turbine industry, as a cycle-by-cycle simulation of e.g. a turbine disc is far too
time consuming. Hence, in this paper, the efficiency of two different procedures to handle
computational expansive load cases, a numerical extrapolation and a parameter modification
procedure, are evaluated and compared to a cycle-by-cycle simulation. For this, a local
implementation approach was adopted, where a user-defined material subroutine is used
for the cycle jumping procedures with good results. This in contrast to a global approach
where the finite element simulation is restarted and mapping of the solution is performed at
each cycle jump. From the comparison, it can be observed that the discrete parameter
modification procedure is by margin the fastest one, but the accuracy depends on the
material parameter optimisation routine. The extrapolation procedure can incorporate stabi-
lity and/or termination criteria.
ARTICLE HISTORY
Received 11 April 2018
Accepted 3 June 2019
KEYWORDS
Cycle jumping; cyclic
response; computational
efficiency; gas turbine disc
alloy; user-defined material
subroutine
Introduction
Gas turbines (both for propulsion and power genera-
tion) will by necessity continue to play a central role
in order to reach a more sustainable energy and
resource usage system for the future. There is
a strong need for making these machines more effi-
cient than today, which calls for higher combustion
temperatures. Furthermore, with the increasing
amount of renewable energy sources, which are
inherently intermittent, the running profile of sta-
tionary power generating machines has to change to
more cyclic operation, as they will be used for balan-
cing the grid. In this context, more efficient cooling,
material characterisation and life assessment models
are of importance.
In order to accurately predict the life of compo-
nents subjected to isothermal or thermomechanical
fatigue loading situations, it is of importance to cor-
rectly predict the local stress-strain history. A fatigue
life evaluation process can in principle be performed
using finite element (FE) analysis to simulate every
load cycle with respect to time until failure. However,
even with the computational power of today,
a complete cycle-by-cycle analysis is generally far
too time-consuming, with regard to the often com-
plex geometry of the component, material behaviour
and thousands of load cycles, and thus, a cycle jump-
ing scheme needs to be invoked.
The usage of a cycle jumping procedure will speed-
up the computational evaluation in an FE-analysis,
where a couple of cycles are evaluated and the
material state is updated (cycle jump) with respect
to a large number of cycles. Thus, from
a computational point of view, the gain lies in the
number of cycles that need to be carried out. The
success of such a procedure relies on the fact that
even if the different fields may momentarily vary
rapidly or in-homogeneously within each cycle (or
sequence), their values at a specific instant/point of
the cycle vary slowly with respect to the cycle num-
ber. However, it is important that an accurate predic-
tion is maintained. There is a vast entity of published
work in the field of cycle jumping procedures, each
refining the implementation, adding new unique fea-
tures or just using the tool as a ‘black-box’for speed-
ing-up their computations. Different areas are
touched, as well as materials, but a common denomi-
nator is cyclic loadings. For further details regarding
cycle jumping and relevant application areas, see e.g.
[1–9].
The most direct way to accomplish a cycle jump-
ing scheme is to base it on a Taylor expansion, where
a chosen number of full/complete load cycle simula-
tions at cycle Nprovide the basis for the prediction of
the state at cycle NþΔN. Furthermore, by evaluating
the influence of additional full cycle simulations at
cycle N, see e.g.[10–13], or by evaluating some full
cycles at cycle NþΔN, see e.g.[14,15], a posteriori
error estimation can be obtained for the step
N!NþΔN. Thus, in an adaptive context, an
unsuccessful step may then be redone with a smaller
ΔNand/or higher degree of approximation (based on
CONTACT Daniel Leidermark daniel.leidermark@liu.se
MATERIALS AT HIGH TEMPERATURES
2019, VOL. 36, NO. 5, 447–458
https://doi.org/10.1080/09603409.2019.1631587
© 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/
by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered,
transformed, or built upon in any way.
more full cycle simulations at cycle N). A different
numerical approach than the Taylor extrapolation
scheme exists in Abaqus [16], the direct cyclic algo-
rithm. This method allows for a direct way to obtain
the stabilised state, based on a modified Newton
method in conjunction with Fourier series of the
solution and the residual.
An alternative approach for reducing the computa-
tional effort is to artificially change the material para-
meters, such that the stress-strain history can be
obtained by a substantially reduced number of full cycle
simulations. This may be done by a ‘continuous’cycle
scaling approach, see e.g. Brommesson et al.[17], where
the constitutive parameters are optimised to accommo-
date the reduced number of cycles, or by a ‘point wise/
discrete’approach, where the constitutive parameters are
optimised to reflect the behaviour at selected instances,
typically only midlife, see e.g. Hasselqvist [18]. In the first
approach, a normal type of elasto-plastic analysis is per-
formed for a reduced number of load cycles but with
modified material parameters, while in the second case
a couple of initial cycles are carried out with virgin
material data, followed by a change of material para-
meters (midlife), and a couple of new load cycles with
the new data (a procedure which can directly be gener-
alisable to more discrete points). The most obvious ben-
efit of the latter discrete approach is that a much simpler
constitutive description may be adopted, as e.g.cyclic
hardening/softening or ageing does not need an evolu-
tion description.
Even though cycle jumping is not a new issue in the
literature, focus has mainly been placed on the Taylor
expansion paradigm. Furthermore, hardly any work has
focused on comparing the two basic approaches, and
theirrelativeprosandcons,whichistheaimofthe
present work. More specifically, the present study inves-
tigates how well the discrete material parameter modifi-
cation approach captures the cyclic behaviour of a simple
component with a stress raiser under strain-controlled
cyclic loading, and which speed-ups that can be achieved
with respect to a basic extrapolation (Taylor approxima-
tion) approach (without error control/adaptivity) in an
FE-context. The geometry has deliberately been chosen as
simple as possible in order not to let a complex geometry
obscure (influence) the analysis and comparison, but still
encompassing the type of inhomogeneous fields of stress
and strain found at local stress raisers in gas turbine
components. Material properties and observed cyclic
behaviourforacommondiscalloymaterial,takenfrom
the literature, was used in the evaluation.
Constitutive model
As mentioned previously, components in a gas tur-
bine are exposed to severe loading conditions, due to
the cyclic nature of loading arising from e.g. the
repeated starts and stops for a stationary gas turbine
balancing the power grid or the many take-offs and
landings for an aircraft engine operating midrange
distances. Under these circumstances a component
such as a turbine disc will experience thermomecha-
nical fatigue [19–21], low-cycle fatigue [22–24], high-
cycle fatigue [25], creep [26], mean stress relaxation
[27], creep-fatigue crack growth [28], dwell crack
growth [29]etc., and thus, an appropriate constitutive
model needs to be utilised to account for the beha-
viour of the material. Focusing attention on disc
alloys, as a non-linear hardening behaviour is gener-
ally adopted, see e.g.[24,30–34]. The constitutive
model adopted in this work is based on the non-
linear kinematic hardening law proposed by Ohno
and Wang [35,36] in conjunction with a saturated
isotropic hardening law, cf. Chaboche [37]. In what
follows, all tensors are presented in index notation,
where second-order tensors are quantified by upper-
case Roman or Greek-letters and scalar-valued para-
meters are defined by lower-case Roman and Greek-
letters. The following yield function is employed
f¼σvM
eq ^
σij Bij
rσY(1)
where ^
σij represents the deviatoric stress tensor and
Bij is the back-stress tensor. The drag-stress (isotropic
hardening) is described by r,σYis the initial yield
limit and σvM
eq is the von Mises equivalent stress,
defined as
σvM
eq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
2^
σij Bij
^
σij Bij
r(2)
The evolution law of the plastic strain tensor is
defined by the flow rule based on the adopted yield
function as
_
εp
ij ¼_
λ@f
@σij
¼_
λ3
2
^
σij Bij
σvM
eq
(3)
where _
λis the plastic multiplier. Furthermore, it can
also be observed that no viscous effects are included
in the study, as the aim is to analyse and compare
cycle jumping procedures.
Based on Chaboche [38], the total back-stress may
be additively decomposed by several back-stress
terms Bij ¼P
NB
k¼1
Bk
ij to increase the accuracy, where
the following evolution law for each individual term
in this work is taken to be given by
_
Bk
ij ¼2
3ck_
εp
ij γk
bk
wk
mk_
εp
pqBk
pq
bk
*+
Bk
ij (4)
with the definition bk¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
3
2Bk
ijBk
ij
q. The material para-
meters ckand γkgovern the linear and the recovery
term, respectively,
hiis the Macaulay bracket and
finally the value of mkcontrols the ratcheting or
mean stress relaxation rate due to the non-linearity
448 D. LEIDERMARK AND K. SIMONSSON
of the power function. For mk!1the model yields
no ratcheting or mean stress relaxation when bkis
below the critical state wk¼ck=γk, thus reverting
back to a pure linear hardening model. With decreas-
ing values an increasing ratcheting or mean stress
relaxation is obtained.
An additive decomposition is also employed for
the total drag-stress, r¼P
Nr
k¼1
rk, to correlate to the
different stages of the cyclic evolution behaviour.
Hence, the evolution law of the saturated isotropic
hardening law from Chaboche [37] for each indivi-
dual term is given by
_
rk¼akqkrk
ðÞ
_
λ(5)
in which the material parameter akdescribes the rate
of cyclic hardening/softening and where the satura-
tion is directly included in the model by the material
parameter qk.
The above presented constitutive model has been
implemented as a user-defined material subroutine in
FORTRAN, to be used in an FE-context. It was based
on the incremental total strain tensor (Δεtot
ij ) as input,
and subsequently, only the back-stress and drag-
stress components need to be stored as history vari-
ables and reused at the beginning of the next time-
step.
Material parameters
As no experiments have been performed within this
study, the material parameters were quantified based
on available literature data of an appropriate disc
alloy. Gustafsson et al.[32] performed isothermal
cyclic experiments on the disc alloy IN718 at
400C and specified material parameters for an
Ohno-Wang model with three back-stress terms,
which was coupled with three linear isotropic hard-
ening terms. The use of three back-stress terms
(NB¼3) in the constitutive model and values for
the back-stress parameters, Young’s modulus,
Poisson’s ratio and initial yield limit were adopted
from that work. As a different isotropic hardening
evolution law was adopted in this work, a new set of
parameters needed to be defined. Hence, the values of
the drag-stress parameters, using three terms
(Nr¼3), were obtained by an optimisation of the
maximum and minimum stress data from the experi-
ments in Gustafsson et al.[32] using the built-in
lsqnonlin function in MATLAB [39]. Based on the
three back-stress and drag-stress terms in this work,
a satisfactory prediction of the material response with
respect to the experimental hysteresis loop and cyclic
softening, cf.[32], was obtained. The material para-
meters can be found in Table 1. It is to be noticed
that even though the loading and material in this
work have been chosen to represent turbine disc
applications the adopted cycle jumping procedures
do have a general applicability to other materials
and applications.
Evaluated cycle jumping procedures
As advertised earlier, the numerical extrapolation
method and the discrete material parameter modifi-
cation method will be evaluated and compared. The
two approaches and implementation aspects are pre-
sented in detail below.
Implementation framework
Independently of the type of applied cycle jumping
scheme, different implementation frameworks can be
adopted. One way is to define/implement a global
framework surrounding the FE-analysis, which expli-
citly extracts internal state variables and global
responses after each finished time step to extrapolate
these forward. This can be done by saving the
extracted variables in an external database to handle
the large quantity of previous and updated variables,
mapping these onto the FE-model to get the corre-
sponding internal state variables and global responses
(e.g. stress state, displacements, contact conditions . . .)
for the updated state, and restarting the FE-analysis.
Another way, which has been implemented and used
in this study, is a so called local implementation
approach. Here, the available user-defined material
subroutine enables a direct control of the adopted
cycle jumping procedure. As the cycle jumping proce-
dure can be directly implemented into the user-
defined subroutine as an add-on, in close connection
to the material model, the history variables keep track
of the previous cycles state to be used in the cycle
jump. Hence, everything is handled on the local level
in the user-defined material subroutine with respect to
the element. This local approach of implementation
Table 1. Material parameters for the constitutive model.
Parameter Value Units
σY864:20 MPa
E187 GPa
ν0:32 –
c1370:23 GPa
c2147:01 GPa
c334:36 GPa
γ14776:87 –
γ2987:20 –
γ3171:52 –
m112 –
m212 –
m312 –
a11622:78 –
a21229:46 –
a32:607 –
q199:04 MPa
q235:007 MPa
q365:609 MPa
MATERIALS AT HIGH TEMPERATURES 449
gives a more straightforward application of the cycle
jumping procedure compared to a global implementa-
tion approach, see Figure 1. On the downside, only
static or discrete stepping of ΔNcan be used in the
local approach for the numerical extrapolation cycle
jumping scheme (described below) for FE-models con-
sisting of multiple elements/integration points, as the
user-defined material subroutine is applied per inte-
gration point and the variables in the response set are
related to that specific integration point. Hence, an
adaptive (dynamic) approach which is able to either
increase or decrease ΔNbased on e.g. the stress state
would incline that different extrapolation step sizes are
generated locally within the FE-model, and such
a dynamic method is thus not applicable locally (per
integration point). This can be handled by a global
approach as all responses will be available and any
kind of control measure (adaptive ΔN, accuracy and/
or termination control) can be adopted due to the
global state. An alternative possible approach for the
local implementation can be to define a certain cycle
jump size for all elements in the beginning of the
initial softening and later switch to another size when
the rapid initial softening has diminished. This can, of
course, be done continuously over the elapsed time of
the simulation, by discretely increasing the jumping
size with a number of performed cycles at distinct
times.
Extrapolation method
The first approach is the numerical extrapolation
method, where the common denominator is the use of
a Taylor expansion of the internal state variables and
global responses. The response set FNobtained in the
FE-simulation at cycle N, containing internal state vari-
ables and global responses, are extrapolated ΔNcycles
forward, according to the standard Taylor expansion
FNþΔN¼X
m
n¼0
1
n!
@ðnÞFN
@NðnÞΔNn(6)
Based on this type of extrapolation, Lin et al.[10]
evaluated components undergoing cyclic thermal
loading combined with constant mechanical loading.
They defined a number of criteria when the cycle
jumping was to initiate and terminate, based on the
steady-state condition as well as acceptable stress
change, rate of ratcheting and change in damage.
A further study was performed by Johansson and
Ekh [11], focusing on the accuracy of the solution,
where they adopted an adaptive extrapolation proce-
dure with respect to the received error. An extrapola-
tion procedure based on linear shape functions was
presented by Wang et al.[14], where the use of linear
shape functions was motivated by the stability of
these compared to polynomial functions, which are
more accurate but very sensitive to the size of extra-
polation step and in need of more supporting points.
To account for erroneous results in the new extra-
polated state, a backward extrapolation was used to
control the accuracy by a set of conventional cycles
from the extrapolated state and the relative error was
compared to a defined limit. The above implementa-
tion was further enhanced by Kontermann et al.[15],
by introducing a multi-parallel processing capability.
In the present study the above described local imple-
mentation approach has been used, where an updated
user-defined material subroutine also includes subrou-
tines which saves the variable values for each cycle,
reads these saved variables and extrapolates the vari-
ables. The procedure is based on a first-order Taylor
expansion, setting m¼1 in Equation (6), hence
FNþΔN¼F
NþdFN
dNΔN¼F
NþFNFN1
1ΔN
ð7Þ
where dN¼1 is due to that only one cycle separates
the two time-frames of the saved variables, and in this
case, the response set contains the stress tensor, the
three individual back-stress tensors and the three
Global approach
FE
Extraction Database
Cycle jump
Mapping
Restart
Local approach
FE
Umat
Cycle jump
Hist. variables
Figure 1. Local and global implementation approaches for the cycle jumping procedures.
450 D. LEIDERMARK AND K. SIMONSSON
drag-stresses, hence F¼F σij;Bk
ij;rk
hi
. These seven
fields are the only variables that locally vary over the
time-step and need to be extrapolated, other variables
are generated within the iterative process of the user-
defined material subroutine.
The observed response from the cyclic experi-
ments displays a smooth softening (mean stress
relaxation) behaviour, beginning with a steep descent
that stabilises with an increased number of cycles, see
Gustafsson et al.[32]. Based on this, the first four
extrapolation steps are performed with a ΔNequal to
2 and onward followed by 10. This is due to stability
reasons with the steep descent, as a ‘too’large step
might generate an unstable response. Note that the
aim of this paper is not to evaluate the continuous
accuracy or to develop an innovative extrapolation
procedure; hence, no consideration has been spent on
these matters. Instead, focus is on simplicity and
comparison of the two basic approaches.
Parameter modification method
The second approach, the discrete material parameter
modification method, is fast and straightforward. In
this method, no extrapolation is performed, and by
this, no stability or accuracy controls can or need to
be utilised. Furthermore, it is an appropriate proce-
dure to use from a post-processing fatigue life evalua-
tion point of view, as only the stable final state
(midlife) is of interest. A proposed procedure,
according to Hasselqvist [18], is as follows:
(1) Perform two load-cycles with virgin material
parameters. During these cycles, redistribution
of stress and strain occurs in the cyclically
loaded structure.
(2) Perform one load-cycle with half-way to mid-
life material parameters.
(3) Perform one load-cycle with midlife material
parameters.
(4) A stabilisation of the cyclic response is assumed,
and e.g. the fatigue life can be evaluated.
Note that midlife refers to a stable state condition, in
which the response is no further changed. One can of
course control that a stable cycle has been achieved
by comparison to experiments and/or prior knowl-
edge, and if not stable, perform a re-evaluation of the
material parameters and re-run the analysis. The
evaluated procedure adopted in this study is slightly
different from the one described above. The load-
cycle at half-way to midlife (point 2) is not per-
formed, as the target is to obtain a stable response
at midlife, thus minimising the computational effort.
This was both due to that no half-way to midlife cycle
was specified in Gustafsson et al.[32] and from
a speed-up perspective, the half-way to midlife is
not of interest. Furthermore, to avoid issues with
too rapid changes in the parameters, an incremental
transition of the parameters was set up during the
load-cycle following the initial two. An accumulative
transition which smooths the change during the load-
cycle was employed accordingly to
xnþ1¼1θðÞxnþθ~
x(8)
where xare the parameters to be modified in the
expressions of the internal variables, which are initi-
ally set to the values from Table 1 and incrementally
transitioned to the midlife material parameters ~
xwith
respect to the transition parameter θ. The transition
parameter is based on the ratio of the actual time step
and the total load-cycle time (peak to peak) with
a scale factor, as
θ¼βΔt
Tcycle
(9)
The scale factor βwas set to 8 due to that this enable
with certainty that the midlife material parameters
are obtained prior to the onset of the final load-
cycle. During this load-cycle, equilibrium is obtained
with respect to the modified parameters, and then
afinal load-cycle to find the stable midlife response
is performed. The local implementation approach
was also used here, where the constitutive user-
defined material subroutine was updated to modify
the material parameters after the two first cycles,
according to Equation (8). Hence, the following
steps have been applied in this study:
(1) Perform two load-cycles with virgin material
parameters. During these cycles, redistribution
of stress and strain occurs in the cyclically
loaded structure.
(2) Perform one load-cycle where the midlife
material parameters are incrementally intro-
duced and find equilibrium.
(3) Perform one load-cycle with the midlife mate-
rial parameters, where the cyclic response is
assumed to be stable.
For the above cycle jumping procedure a new set of
material parameters is needed for the midlife cycle.
These were obtained by parameter optimisation in
which the midlife hysteresis from the experiment
was compared to the obtained response from an FE-
analysis. The midlife cycle was identified as cycle 300
from the mean stress relaxation behaviour displayed
in the paper by Gustafsson et al.[32]. The parameters
that influence the stress-strain state at midlife are the
MATERIALS AT HIGH TEMPERATURES 451
back-stresses and drag-stresses. For simplicity, the
values of the exponents mkand the rate of softening
akwere kept constant, using the previously defined
values. Furthermore, a restriction was placed on qkto
be more negative than the initial values as a lower
maximum stress state is present at the midlife cycle
compared to the first couple of cycles. The new values
describing the midlife cycle are given in Table 2.
A version of the above-presented parameter modifi-
cation procedure could be to use a down-scaled (less
complicated) constitutive model. This saves even more
computational effort, and also requires fewer material
parameters. Such a constitutive model could, for
instance, be defined without a drag-stress term or as
a perfect plastic model, and would only require modified
back-stress terms or a decrease of the initial yield stress to
account for the decrease in mean stress relaxation. To
find equilibrium with the latter approach the yield stress
needs to be decreased incrementally (successively), as
a too large decrease in yield stress would render numer-
ical problems and the algorithm will not likely find
f¼0, cf.Figure 2.Thisisavoidedintheabove-
presented parameter modification procedure by the
inclusion of the load-cycle to find the new equilibrium
during which the midlife material parameters are
introduced.
Simulation and evaluation
Evaluated component
In order to evaluate the two-cycle jumping procedures,
acomponent with a stress/strain raising feature was to
be chosen, as that is the typical critical location for
fatigue crack initiation. From this point of view, it is
vital that the correct cyclic response isobtained and that
the computational time in an industrial context is low,
saving development costs and effort. As the component
geometry was to have a general relevance, and as it was
not to add extra complications or move the focus from
the aim of the study, a too detailed or application
specific geometry was ruled out. Instead, an unidirec-
tionally loaded plate with a central hole was chosen as
the object of study, see Figure 3.Furthermore,
a reference simulation was also carried out based on
the standard cycle-by-cycle evaluation, for which no
cycle jumping procedure was adopted. This reference
cycle-by-cycle simulation is used to compare the accu-
racy of the two evaluated cycle jumping procedures.
The FE-model of the plate was generated only
accounting for a quarter of the plate by using symmetry
boundary conditions, see Figure 3(b), thus preventing
translation in the horizontal direction on the edge
between A to B and in the vertical direction on the
edge along C to D. The plate is subjected to the cyclic
load Δδ¼0:6mm (Rε¼εmin=εmax ¼0). The FE-
model consists of a mapped mesh with 228 fully inte-
grated 8-noded brick elements (one element through
the thickness) and was analysed in the FE-software LS-
DYNA [40], version R7, using an implicit solution
technique coupled with the three implemented user-
defined material subroutines (one for each solution
approach). The response in the highly loaded element
in the top of the hole (at B) was extracted during the
entire loading sequence. The response at this point will
be the most damaging to the structure from a fatigue
point-of-view, as cyclic loading produces considerable
plastic deformation which for a real application
Table 2. Material parameters for the midlife cycle.
Parameter Value Units
c1550:0 GPa
c293:172 GPa
c350:839 GPa
γ12359:802 –
γ2680:625 –
γ375:847 –
q1288:523 MPa
q2119:991 MPa
q3240:0 MPa
Increment
ε
σ
ε
σ
)b)a
Figure 2. The different approaches to find equilibrium, a)
incrementally decreasing the yield stress for a perfect plasti-
city down-scaled constitutive model or b) perform one load-
cycle for a more complex constitutive model.
Length = 200mm T hick ness =1mm
Radius =10mm
W idth =80mmΔδΔδ
A
B
DC
Δ
δ
a) b)
Figure 3. The a) evaluated plate with the centred hole, and the used b) FE-model with symmetry boundary conditions.
452 D. LEIDERMARK AND K. SIMONSSON
eventually might lead to crack initiation and subsequent
crack growth. Thus, it is of high importance that
a correct response is obtained in locations like this
when designing gas turbine components subjected to
cyclic loading conditions.
The FE-simulations were performed on a Linux-
based cluster, equipped with four Intel(R) Xeon(R)
CPU E5 2650 v2 @ 2:60GHz eight-core processors.
Eight cores were used in each simulation, giving
equal conditions for the comparison. The consumed
computational time (CPU time) for each of the three
different simulations can be found in Table 3.
Loading sequence
The loading sequence is different for the two investi-
gated cycle jumping procedures, see Figure 4. For the
material parameter modification procedure, the load-
ing is conveyed by a sequence of 4:5 cycles, in which
the first on-loading (0:5 cycle, defined as cycle 0) is
not included in the evaluation due to initial plasticity
and redistribution of the stress state. After two
unloading-loading cycles (1 2), the set of the mod-
ified material parameters based on the midlife prop-
erties are incrementally introduced during the full
new initial cycle (0) to find equilibrium. Finally,
a last cycle is performed to obtain a stable hysteresis
loop with the modified parameters.
When it comes to the extrapolation procedure,
only two consecutive cycles could have been per-
formed, but due to numerical stability, three conse-
cutive cycles (1 23) were used. Furthermore, the
initial on-loading (0) is not included here either, for
the same reasons as discussed above. Then, an extra-
polation step is performed generating new values of
the variables in the response set, in which the load is
kept constant to yield equilibrium at the new initial
step (0). The above-described loading sequence is
then repeated until the stabilised state at midlife is
obtained. In the case of the local implementation
approach, no stability condition can be present, as
no global stability can be determined, and thus the
analysis was terminated at N¼300 (midlife). Lastly,
the reference cycle-by-cycle simulation was per-
formed with continuous loading and unloading
until 300 cycles were reached. The same amount of
cycles were evaluated in all cases for comparative
reasons.
Results and discussion
From the simulations, it can be seen that a cycle jumping
procedure will lower the computational effort pro-
foundly, see Table 3. Depending on which type of cycle
jumping procedure that is used, different levels of effi-
ciency are obtained. As can be seen in Figure 5, the good
accuracy of the different simulation methods is note-
worthy. However, the extrapolation method does not
exactly predict the same increasing maximum and mini-
mum stress levels after about 50 cycles compared to the
reference simulation, see Figure 5(c–d).Areasoncanbe
thelargerextrapolationstepscomparedtoduringthefirst
four cycle jumps that increases the uncertainties with the
extrapolation method. A second-order extrapolation
approach or an adaptive extrapolation stepping approach
with convergence criterion might generate better resolu-
tion of the predictions, but from a global perspective,
a good agreement is achieved with this simple extrapola-
tion method, cf.Figure 5(a). Moreover, the parameter
modification cycle jumping procedure generates
a midlife response that is in-line with both the extrapola-
tion method and the reference ‘cycle-by-cycle’simula-
tion. A small shift ‘upwards’can be noticed at midlife, but
it is only by a few MPas, cf.Figure 5. Further, the large
compressive state seen in Figure 5(d) for the modification
method is the reversed loading peak in the cycle prior to
midlife. This peak is not to be mistaken for the midlife
response, as during this cycle the new set of material
parameters are introduced and equilibrium is acquired.
Thegoodresponseisfurtherenhanced when comparing
the hysteresis loops at the midlife cycle of the different
simulations, see Figure 6. One can note that the shape of
the hysteresis loops differs. This is due to that the new
parameters in the modification procedure has been
obtained based on the actual midlife hysteresis from the
experiment in Gustafsson et al.[32], and the two other
methods are dependent on the constitutive model to
generate the hysteresis loops based on evolution with
the initially defined parameters. Hence, a discrepancy is
obtained due to the accurateness of the constitutive
model,anditislikelythatasmalldeviationwillalways
be present as the reference and extrapolation models will
not precisely predict the midlife cycle as good as the new
set of parameters used in the parameter modification
Table 3. The CPU time for each simulation.
Simulation CPU time Relative time
Reference 89h 56min 49s 56:02
Extrapolation 25h 33min 32s 15:92
Modification 1h 36min 20s 1
N
Δδ
012 01
Modify
N
Δδ
ΔN
01230123
Extrapolate
a)
b)
Figure 4. Loading sequence for a) parameter modification
and b) extrapolation.
MATERIALS AT HIGH TEMPERATURES 453
method does. In addition, the difference can also be due
to that a different approach might have been employed
when calibrating the midlife material parameters com-
pared to the initial material parameters defined in
Gustafsson et al.[32]. These might be reasons for the
slightly different response at the midlife cycle.
Furthermore, all data points were not explicitly available
and a puzzling operation had to be performed to acquire
the material response from the experiments. This could,
of course, generate errors in the material parameters,
which might be further accentuated in the FE-
simulation. Another, close-related issue is the choice of
the adopted constitutive model and the ongoing amount
of back-stress and drag-stress terms. The model has been
chosen based on industrial relevance, and the number of
terms gives a good representation of the material beha-
viour. Of course, one could have used more terms to
enhance the correlation to the observed material beha-
viour or less to minimise the calibration process (practical
fortheparametermodification method). It can also be
seen in Figure 6 that the strain range of the hysteresis
loops does not match between the different simulations,
especially the extrapolation method. This can be due to
that the accumulated plastic strain locally in the element
of interest (at B in Figure 3(b))isdifferent, and that the
element becomes more severely deformed with each
cycle. The more performed cycles generate a larger plastic
strain range in subsequent hysteresis loops. Moreover,
the horizontal and inclined lines for the modification and
extrapolation method in Figure 5, are purely graphical.
The stress states during these time intervals are not gen-
erated in the material model, but they give an estimate of
theincreaseordecreaseofthestressstatebetweenthe
skipped cycles, especially for the extrapolation method. In
thecaseofthemodification method, the stress state is
kept constant during the skipped cycles, and then one
load-cycle is performed with the introduction of the new
material parameters to acquire equilibrium, cf. Figure 2,
thus the horizontal line.
The ability to speed-up the computational time is of
essence, as one load-cycle takes approximately 18min,
implying that every cycle counts. From the performed
simulations, the parameter modification cycle jumping
procedure is the fastest one, see Table 3. The compu-
tation time is reduced by a factor of 15:92 compared to
the extrapolation method and 56:02 compared to the
reference cycle-by-cycle simulation. Though, one
needs to keep in mind that no evolution of the internal
variables is updated within the modification method,
and thus one lack knowledge about e.g. the continuous
-1000
-500
0
500
1000
Ref. Mod. Ext.
Stress MPa
Cycles
295 296 297 298 299 300
1040
1060
1080
1100
1120
1140
Ref. Mod. Ext.
Max stress MPa
Cycles
0 50 100 150 200 250 300
0 50 100 150 200 250 300
0 50 100 150 200 250 300
1070
1080
1090
1100
1110
1120
1130
1140
1150
1160
1170
1180
Ref. Mod. Ext.
Max stress MPa
Cycles
-1110
-1105
-1100
-1095
-1090
-1085
-1080
-1075
-1070
Ref. Mod. Ext.
Min stress MPa
Cycles
a) b)
c)
d)
Figure 5. Stress versus cycle response from the three FE-simulations, displaying a) the entire stress history, b) the maximum
stress at midlife, c) the maximum stress and d) the minimum stress evolution.
454 D. LEIDERMARK AND K. SIMONSSON
‘cycle-by-cycle’damage accumulation to be used in,
for instance, a fatigue evaluation. Moreover, it is also
to be mentioned that a fatigue evaluation is often
performed by evaluating the obtained hysteresis loop
with respect to e.g.aCoffin-Manson type of expres-
sion, and this is easily done with the modification
method at the stable midlife cycle. Furthermore, it is
stressed that these responses are a result of the specific
load-case and current cycle jumping procedures. With
this specified, the pros and cons of the two cycle
jumping methods are given in Table 4.
To speed-up the extrapolation procedure
a dynamic ΔNcould be used, as discussed earlier
regarding the local and global implementation
approach. A dynamic extrapolation procedure was
investigated where ΔNwas increased at discrete
times, but due to a stability problem this approach
was not further investigated and a static step was used
throughout the simulation (except for the first four
extrapolation steps). The reason for the stability issue
refers to a too large extrapolation step in the area
where the maximum stress starts to increase during
the loading sequence. It can be observed in Figure 7
that the discrepancy between the extrapolated state
and following stabilising cycle becomes large at spe-
cific times (steep slope), which will be further
enhanced by a large extrapolation step. This can
result in a response that the non-linear solver is not
able to handle. An adaptive extrapolation stepping
functionality might render a solution for this problem
using a global implementation approach. In addition,
it is further stressed that no intention was to develop
a robust dynamic approach, where the steps could be
changed as a function of the response or the time, or
an as good as possible extrapolation jumping scheme.
The aim is just to compare with the modification
method. Of course, there are many innovative
approaches that handle the continuous accuracy, sta-
bility and speed-up process in a far more better way,
see e.g.[10]or[2]. It is also to be mentioned, that as
the parameter modification method generates no his-
tory evolution; consequently, no stress evolution is
present as can be seen in Figure 7, also cf.Figure 5(c)
. Hence, the observed drift in stress for the reference
and extrapolation simulations can never be obtained
with the parameter modification-jumping procedure,
as it is a discrete method evaluating the behaviour at
the specific midlife cycle where the constitutive para-
meters have been evaluated.
The local implementation approach saves compu-
tational time in the extrapolation cycle jumping pro-
cedure with respect to a global approach. This is due
to that the FE-simulation do not have to be restarted
and all internal variables do not have to be mapped
on the mesh for each cycle. But in the local imple-
mentation approach, it will be difficult to have
a global termination criterion in which the stability
is monitored, as the neighbouring element might
generate a stress state that does not meet the control
criteria, and the one you are analysing might do so.
Hence, a contradiction has arisen, and the criteria is
not generally applicable for the entire FE-model. This
is a lot easier in a global approach with a tolerance
measure, as the extrapolation procedure is executed
after a completed time-step in the FE-analysis instead
of during the time-step, where an accuracy control
and termination criterion can, for instance, be based
on the global stress state, cf.Figure 8. However, in
Figure 7 it can be seen that the global maximum
stress is slowly increasing with each cycle after,
approximately, the 85th cycle. Hence, such a global
termination criteria might generate unwanted conse-
quences, and thus one needs to account for the small
increase in a potential termination criterion of such
character or base the criterion on some other entity.
As one of the main driving forces to speed-up, the
computational effort is related to fatigue, and predic-
tion of the fatigue life of an industrial component, the
choice of fatigue identification parameter is highly
dependent on the chosen cycle jumping method.
For the extrapolation procedure, all the history is
0 0.005 0.01 0.015 0.02
-1000
-500
0
500
1000
Ref. Mod. Ext.
Stress MPa
Strain range
Figure 6. Hysteresis loops at the midlife cycle.
Table 4. Pros and cons for the two different cycle jumping
procedures.
Method Pros Cons
Modification + Very fast, only
4:5 cycles
–No evolution history
+ Simple –Evaluation of two material
parameter sets
+ Stable
Extrapolation + Evolution history –Static extrapolation step (local)
= slow
+ One material
parameter set
–Complex implementation
(global)
+ Access to any
hysteresis loop
–May give stability issues for
large extrapolation steps
MATERIALS AT HIGH TEMPERATURES 455
present and the definition of the fatigue parameter
can be chosen relatively freely (depending on the
application). A traditional post-processing, extracting
amplitudes, ranges or max and min values, cf. Ince
and Glinka [41], can be adopted as well as
a continuously accumulated fatigue damage para-
meter over the cycles, see e.g. Leidermark et al.[42].
In the case of the parameter modification procedure,
a cyclic accumulative approach is not an option, as
the intermediate cycles have been skipped. Secondly,
the type of adopted constitutive model at the jumped
state is of great importance. Here, a down-scaled
material model, e.g. perfect plastic or linear kinematic
behaviour, will not generate the accurate non-linear
shape of the hysteresis. Hence, if a dissipative energy
criterion is considered as fatigue identification para-
meter, cf. Cruzado et al.[43], then differences com-
pared to a complex non-linear material description
will be present in the fatigue life predictions. On the
other hand, a fatigue identification parameter
dependent on, for instance, ranges can still be
adopted in this case.
An important aspect from a design perspective is
that a highly accurate response might not necessarily
be the target. It might be that only a fast response is
needed to get a first picture of the stress/strain state,
and from this, the design may be changed and re-
evaluated. This promotes the parameter modification
procedure over the extrapolation method, even
though one has to evaluate two sets of material para-
meters which is time-consuming. On the other hand,
one does not necessarily have to perform all the
extrapolation steps to acquire a response that is sui-
table to use in an evaluation. Instead, one can, as
discussed above, settle for a response that gives
a satisfying response during the design process to
get a first estimate. To sum it up, use the fast discrete
parameter modification method iteratively in the
early stages of the design process. Then, perform
more accurate calculations at the end of the process
using the full extrapolation method accounting for
various stability conditions, accuracy controls and
termination criteria.
Conclusions
Based on the above-presented cycle jumping study
for a simple component, with a relevant disc alloy
material description, the following conclusions may
be drawn:
(a) A large difference in computational time
between the three simulation approaches can
be observed. The more cycles computed the
30025020015010050
1092
1094
1096
1098
1100
1102
1104
1106
1108
1110 Ref. Mod. Ext.
Max stress MPa
Cycles
Figure 7. The maximum stress versus cycles for the different simulations.
Δσmax
Δσmax <tolerance
σmax
N
Figure 8. Tolerance measure with respect to the global max-
imum stress from each cycle as the termination criterion.
456 D. LEIDERMARK AND K. SIMONSSON
more effort is needed. Here, the discrete para-
meter modification cycle jumping procedure is
with margin the fastest one.
(b) The parameter modification cycle jumping
procedure is very simple and stable, but one
needs to evaluate two sets of material para-
meters, which can be hard and, in itself, time-
consuming. However, a down-scaled constitu-
tive model can be used for the midlife cycle
with fewer material parameters, reducing this
effort.
(c) In the extrapolation cycle jumping procedure
the evolution history is obtained, but it may be
unstable for large extrapolation step sizes. An
adaptive extrapolation stepping functionality
with a global implementation approach might
be a way to solve this.
(d) The local implementation approach gives
direct control of the cycle jumping procedure,
and no mapping of the variables in the
response set or restarts of the FE-analysis
need to be performed. However, a global
implementation approach can be used to
enable control over the accuracy, stability con-
ditions and termination criterion of the cycle
jumping procedure with respect to e.g. the
global stress state.
(e) During the initial design process, the para-
meter modification method offers fast and reli-
able results for a first estimate by using
minimal resources.
Acknowledgments
The study has received funding from the Clean Sky 2 Joint
Undertaking under the European Union’s Horizon 2020
research and innovation programme under grant agree-
ment No 686600.
Disclosure statement
No potential conflict of interest was reported by the
authors.
Funding
This work was supported by the Cleansky [686600];
Cleansky [686600];
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458 D. LEIDERMARK AND K. SIMONSSON
