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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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