[Rosa; Nagode e Fajdiga] - Strain-Life Approach in Thermo-Mechanical Fatigue Evaluation of Complex...

8/10/2019 [Rosa; Nagode e Fajdiga] - Strain-Life Approach in Thermo-Mechanical Fatigue Evaluation of Complex Structures

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doi: 10.1111/j.1460-2695.2007.01154.x

Strain-life approach in thermo-mechanical fatigue evaluationof complex structures

U ROS RO SA, MARK O N AG O DE an d MATIJA F AJDIG A

University of Ljubljana, Faculty of Mechanical Engineering, Askerceva 6, SI-1000 Ljubljana, Slovenia

Received in final form 23 May 2007

A B S T R A C T This paper is a contribution to strain-life approach evaluation of thermo-mechanicallyloaded structures. It takes into consideration the uncoupling of stress and damage evalua-tion and has the option of importing non-linear or linear stress results from finite elementanalysis (FEA). The multiaxiality is considered with the signed von Mises method. Inthe developed Damage Calculation Program (DCP) local temperature-stress-strain be-haviour is modelled with an operator of the Prandtl type and damage is estimated by useof the strain-life approach and Skeltons energy criterion. Material data were obtainedfrom standard isothermal strain-controlled low cycle fatigue (LCF) tests, with linear pa-

rameter interpolation or piecewise cubic Hermite interpolation being used to estimatevalues at unmeasured temperature points. The model is shown with examples of constanttemperature loading and random force-temperature history. Additional research was doneregarding the temperature dependency of theKp used in the Neuber approximate for-mula for stress-strain estimation from linear FEA results. The proposed model enablescomputationally fast thermo-mechanical fatigue (TMF) damage estimations for randomload and temperature histories.

Keywords damage accumulation; elastoplasticity; finite element analysis; thermo-mechanical fatigue.

N O M E N C L A T U R E Amin= minimal cross-section

b =the fatigue strength exponent

c= the ductility exponentck= load influence factorD = damageD = elasticity matrixE=Youngs moduluse

= nominal straing= cyclic stress-strain curvei= data pointj= spring-slider segment indexk = temperature index

K= cyclic hardening coefficientKp

=limit load ratio

L = loadLP= full plastification forceLF= initial plastification force

Correspondence: U. Rosa, E-mail: [email protected]

808 c 2007 The Authors. Journal compilation c 2007 Blackwell Publishing Ltd.Fatigue Fract Engng Mater Struct 30, 808822


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S T R A I N - L I F E A P P R O A C H I N T H E R M O - M E C H A N IC A L F AT I G U E E V A L U AT I O N 809

n = index of the top data pointNf= number of cycles to crack initiation

n= cyclic hardening exponentPSWT= Smith-Watson-Topper parameter

R = load ratiorj= fictive yield stress

Rm= tensile strengthRp,0.2= cyclically stable yield stressS= nominal stress

t= timeT= temperaturej= the Prandtl density

r= fictive yield stress class widthT= temperature increment

Wp= dissipated energy per cycle = cycle strain range

= cycle stress range = strain tensor

=total strain

a= strain amplitudef= ductility coefficientj= spring-slider strain = stress tensor= total stress

a= stress amplitudee= pseudo elastic stressf= fatigue strength coefficient

m= mean stressSVM= signed von Mises stress

x,y,z= normal stress in X, Y and Z directionj= spring stress

xy,xz,yz= shear stress in XY, XZ and YZ plane

I N T R O D U C T I O N

For machines and components under variable multiaxialloading, fatigue evaluation is one of the most importantsteps in the design process. Appropriate material testingand simulation is the key to efficient life prediction.There are several requirements for making accurate

life predictions: a proper material model with materialparameters derived from testing, modelling of stressstrain response during cycle loading, a multiaxial fatigue

criterion, a proper damage accumulation model and com-ponent testing to evaluate the correlation between lifeprediction models and experiments.

In circumstances where service loads create complexstressstrain fields, the first challenge is to properly modelthe multiaxial fatigue. At present, there are several differ-ent approaches that can be divided in four major groups:

1 The approaches that calculate equivalent stress or strain13

in each node of the finite element model.

2 The use of stress or strain invariants.3

3 The critical plane approach (at first suggested by Brown

and Miller and later on modified by several researchers

(e.g. Socie, FatemiSocie and Papadopoulous,1,3,4) that

considers the critical plane in each selected node for life

estimation.

4 The energy criterion5,6 that takes into account the dissi-

pated energy during cycle loading.

Besides the four categories above, approaches can bedivided according to their field of usage or analyses re-quirements. Historically, the first division was betweenhigh-cycle (HCF) and low-cycle fatigue (LCF). Wohler7

pioneered the field of HCF and the development of stress-life approach with S/Ncurves, andlaterManson8 andCof-fin9 made substantial developments in the field of LCF

with the establishment of the strain-life approach. Thisclassic differentiation is still present nowadays mainly dueto the tendencies for elastic deformation in HCF and in-elastic deformation in LCF, as can be seen in the critical

c 2007 The Authors. Journal compilation c 2007 Blackwell Publishing Ltd.Fatigue Fract Engng Mater Struct 30, 808822


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810 U . R O S A et al.

plane approach in Ref. [3] or energy criteria in Ref. [5]. Animportant field of research concerns the development of aunifiedapproach to LCF and HCF, such as the energy cri-teria summarized in Ref. [5] or the unified approach basedon shakedown theories and dissipated energy proposed byConstantinescu et al.10

Materials exhibit different failure mechanisms that arecommonly divided into tensile-type failure and shear-typefailure. This division can be easily made in the criticalplane approach,where there aretensile critical plane mod-els, such as the tensile critical plane model of Socie,1,5

and shear critical plane models such as those of Brown-Miller,1,3 Fatemi-Socie3 and Socie.3

Regarding the analyses requirements, both propor-tional and non-proportional loading situations have tobe considered, as inappropriate models can lead to non-conservative life estimations.1,11 In commercial soft-

ware12,13 the signed von Mises stress method is frequentlyused for modelling the effects of proportional loading as it

generally gives conservative results by additionally takinginto account the stresses in the directions that do not con-tribute to damage. For simulations associated with non-proportional loading the critical plane approach is widelyused,12 especially the normal or shear strain model, theSocie tensile model and the WangBrown model. TheSocie tensile model, based on Smith-Watson-Topper,14

can give non-conservative results under non-proportionalloadings as shown in Ref. [1], whereas better agreement

with measurements has been obtained with the FatemiSocie approach. The WangBrown model for multiax-ial cycle counting and damage accumulation according to

Ref. [3] and Ref. [15] gives good correlation with experi-ments for multiaxial non-proportional loading. However,a general problem with critical plane methods is that thecomputational process can be very time-consuming, re-quiring damage calculation in all planes of each selectednode.

Improving computation times is an important issue inthe development of these methods, as can be seen in Liet al.2 and Amiable et al.16 Currently, calculations forsimple models can last as long as 1012 h or even fewdays.2,16 This is why fast computation methods like thoseproposed by Nagode and others1719 are under develop-ment. Choosing a method that gives the best compro-mise between computational speed and good correlation

with experiments is still an open question for the designengineer.The purpose of this paper is to extend the models already

developed by Nagode an others1719 to multiaxial statesfor use in computationally fast thermo-mechanical fatigue(TMF) life-predictions of complex 3D structures. Themethod uses the strain or stress-controlled rheologicalspring-slider model developed by Nagode and others thatenables elastoplastic material behaviour modelling with

an operator of the Prandtl type under constant or variabletemperature. It is based on isothermal strain-controlledLCF tests with the assumption that hysteresis loops arestabilized. The major drawback of the proposed model isthat it is basically uniaxial and that at the moment it iscapable of elastoplastic modelling with stabilized hystere-

sis loops without the consideration of cyclic creep, cyclichardening and cyclic softening.Multiaxiality is considered with the calculation of signed

von Mises stress in every node and damage is calculatedwith the Smith-Watson-Topper parameter14 and Minerlinear damage accumulation rule.20 Skeltons energy cri-terion6,21,22 is also applied as a damage indicator; it istemperature independent and based on the stabilized cy-cle. This approach has been successfully used in complexmultiaxial TMF analyses as shown by Amiable et al.16,Constantinescu et al.,23 Charklauk et al.24 and Thomaset al.25

A P P R O A CH T O T M F P R O B L E M S

Uncoupled analyses

In TMF evaluations, the thermal and mechanical loadingsacting on the specified structure must be considered. Inorder to obtain the temperature fields that are later ap-plied together with the mechanical loads in the structuralanalyses, an assumption must be made regarding the un-coupling of the thermal and structural analyses. Transientthermal analyses are performed in order to obtain the de-sired temperature fields for all load cases. The computed

temperature fields are then applied in combination withthe mechanical loading in the stressstrain finite elementanalysis (FEA) analyses.The second uncoupling regards the separation of the

stressstrain response from the damage calculation. Thestressstrain response together withthe temperaturefieldsare exported for the final damage evaluation into the de-

veloped Damage Calculation Program (DCP).This non-unified approach has been widely and success-

fully used in TMF evaluations with the Skelton energycriterion in the automotive industry.2325

Evaluation procedure

The proposed evaluation process is explained in detail be-low and includes the following important steps:

1 Calculation of the temperature fields for the load history

with transient thermal FEA.

2 Calculation of the stressstrain response for the turning

points in the given load history with FEA. The analysis

can be either linear or nonlinear.



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Fig. 1 Rheological spring-slider model.

3 From the FEA the stress and temperature histories for all

nodes are exported to the specifically developed DCP.

4 If the stresses have been computed with the linear FEAthe DCP estimates the elastoplastic stresses in the turning

points with the Neuber approximate formula.

5 In the DCP a stressstrain modelling of complete hystere-

sis loops is performed using the stress-controlled spring-

slider model.18

6 After the stressstrain modelling the rainflow cycle count-

ing is performed, by utilizing the cycle closure method

used in Ref. [18] and the calculation of the equivalent cy-

cle temperature.17,26

7 Damage estimation using the Smith-Watson-Topper pa-

rameter14 and the estimation of the dissipated energy is

performed.

Stress-controlled modelling of elastoplasticity

The stressstrain modelling in DCP is based on the stress-controlled serially connected spring-slider model18,27

shown in Fig. 1. It is capable of modelling elastoplastichardening solids and nonlinear kinematic hardening forisothermal or non-isothermal cases.

If isotropic hardening can be neglected, cyclically stable(or half-life) cyclic stressstrain curves (g) with Ramberg-Osgood18,28 relation

=g(, T) = E(T)

+

K(T)

1/n(T)(1)

are not necessary but are commonly used in fatigue analy-ses.T,E(T),K(T) andn(T) are temperature, the Youngmodulus, the cyclic-hardening coefficient and the cyclic-hardening exponent, respectively. There is no limitationhowever against using any other stressstrain relation thatexhibits elastoplastic behaviour with nonlinear kinematichardening.

Supposingthat cyclically stable cyclicstressstraincurvesare available, elastoplastic strain can be modelled with anoperator of the Prandtl type that connects several elemen-tary hysteresis operators and it is used for modelling realhysteresis phenomena.29

The proposed stressstrain modelling has been pre-

sented thoroughly by Nagode and others 18

, which iswhy only the final set of equations is listed below.The total elastoplastic strain can be expressed in a form

known as the operator of the Prandtl type18,29

(ti) =nr

j=0j(Ti)j(ti) (2)

for 0 t1 t2 ti , where Ti= T(ti) andj(ti) is the play operator with general initial value

j(ti)=max

(ti) rj,

min

(ti) + rj,

j(Ti1)j(T1)

j(ti1)

. (3)

Presumably, there is no residual stress initially, so j(0) =0 and(0) = 0. The Prandtl densitiesj(Tk) in range j=0, . . . , nrand k = 0, . . . , nT

j(Tk) =1

r(j+1(Tk) 2j(Tk) + j1(Tk)) (4)

are gained from the available isothermal cyclically sta-ble cyclicstressstrain curves, where1(Tk) = 0(Tk) = 0.Fictive yield stresses rj are usually dispersed equidis-tantly with constant fictive yield stress class width rbetween the zero stress and the maximal expected stress(Fig. 2).To speed up the computation, input histories of(t), T(t),

material parameters and the Prandtl densities are tabu-lated by setting the stress increment to rand choosinga temperature increment ofT. The tabulated materialparameters and the Prandtl densities are calculated onlyonce and stored before the,modelling process begin-ning at Eq. 2.

Stressstrain modelling

From the FEA results the signed von Mises stress12,13,30 is

calculated using Eq. 5 and exportedto the DCP where thestressstrain modelling with the stress-controlled modelis performed. The symbol SGNin Eq. 5 equals the signof the principal stress with the largest magnitude.

SVM(t)=SGN 1

2

(x y)2 + (x z)2

+ (y z)2 + 6

2xy + 2xz + 2yz1/2

.

(5)

Stresses can be computed with nonlinear or linear FEA.In the case of nonlinearly computed stresses, the same



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Fig. 2 Prandtl density assessment, stress control.

material model must be applied in both the FEA softwareand in the DCP program in order to obtain consistent re-sults. At the moment, due to the limitations of the spring-slider model elastoplastic kinemetic hardening materialmodels are allowed.

In order to achieve a computationally fast evaluationmethod, great care has been given to the linearly com-

puted stresses in combination with approximate formulaeto estimate the elastoplastic material behaviour. There areseveral approximate methods from which the frequentlyused12,18,31 Neuber approximate formula is applied,

= E(T)

e

2e

S/E(T) (6)

where the nominal stress and strains are defined as S=e/Kpand e =g(S, T).Kpis the limit load ratio andg(S,T) is the cyclic stressstrain curve as defined in Eq. 1. Thelimit load ratio gives the ratio between the limiting loadand the load when yielding starts. Its value is calculatedby,12

Kp=LPLF

, (7)

whereLP is the force that causes the full plastification ofthe analysed cross-section and LF is the force that startsthe plastification in the most stressed node. In our caseLPis computed as,

LP= Amin R p,0.2, (8)

whereAmin is the minimal cross-section of the specimenwhere the plastification begins and Rp,0.2 the yield stressfor the cyclically stable cyclic stressstrain curve.According to LMS Falancs Theory Manual Version 2.912

the useable range ofKpis between:Kp= 1 andKp= 30.The first value does not take into account the Neuber ap-

proximateformula,whichmeans that theinputstressesaredirectly used for the stressstrain modelling. This value issuitable when theinputstressesare calculated with nonlin-ear FEA or when extremely conservative predictions arerequired with the linearly calculated stresses. The second

value is used only in combination with linearly calculatedstresses when there is no influence from the plastic limitload.12

Due to temperature dependant mechanical characteris-tics of the material, some temperature dependency ofLPand LF values has also been expected. In an attempt toobtain better agreement between the results from linearand nonlinear FEA, a temperature dependantKphas been

introduced

Kp(T) =LP(T)

LF(T). (9)

This has been calculated for the maximal surface temper-ature for the constant temperature field cases shown inthe Examples chapter.The value Kp= 1 has been used for the evaluation of

the nonlinear analyses results and Kp= 30, Kp= 1 andthe calculatedKp(T) for the evaluation of linear analysesresults.At the moment, the proposed temperature dependency

ofKp(T) can be applied with constant temperatures only.

Damage estimation

An appropriate damage estimation method provides thekeyto an efficientlife-prediction. In theproposedmethod,the focus is on employing computationally fast and widelyaccepted approaches. This is the reason why the well-known Smith-Watson-Topper damage parameter1,11,14,32

has been applied as the first damage estimation method.It includes the influence of the stress and strain amplitudeand the mean stress on the damage and is in accordance

with physical tests in a variety of cases.11,12 Since the tem-perature duringan individual cycle canvary, theequivalentcycle temperature17,26 Te, as first proposed by Taira,

26 andthe cycle closure method, as describedby Nagodeand oth-ers 17,18, have been used. The cycle closure problem canbe observed in non-isothermal cases as can be seen for twosimulated conditions in Fig. 3 with temperature increaseand temperature decrease during one cycle.That is why a cycle closure method was introduced by

Nagode and others.18 Given that it is time consumingto estimate the true cycle closure point, a simple and



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Fig. 3 Left: cycle closure and temperature increase. Right: cycle

closure and temperature decrease.

conservative cycle definition has been applied as18

a=|from to|

2 , m=

from + to2

, a=|from to|

2 ,

(10)

where from denotes the starting point of the cycle andto its reversal point.

For each closed hysteresis loop, the Smith-Watson-

Topper damage parameter is calculated1,11,14,32

P2SWT= (a + m) aE(Te) (11)and the number of cycles to crack initiation is obtainedfrom1,11,14,32

P2SWT= 2f(Te)(2Nf)2b(Te)

+ f(Te)E(Te)f(Te)(2Nf)b(Te)+c(Te), (12)where f is the fatigue strength coefficient, b the fatiguestrength exponent, fthe ductility coefficient and c theductility exponent. The material parameters depend onthe equivalent cycle temperature and can be linearly orpiecewise cubic Hermite33 interpolated. The latter is veryefficient when there is information on the function valuesand the first derivatives values at a set of data points. Thenumber of cycles to crack initiation is used for damageestimation with the Miner linear damage accumulationrule.The Skelton energy criterion6,21,22 is gaining consider-

able importance in TMF evaluations.16,2325 It isbased onthe assumption that after the initial hardening or soften-ing the material reaches the stabilized state, in which the

part operates for the majority of its lifetime, and that thecumulated dissipated energy at stabilization can be con-sidered as a material constant used as a crack initiationcriterion.6,2325

In accordance with this, it has been considered that theload history is applied in the stabilized state and the dis-

sipated energy has been used for the identification of themost critical area in the evaluated structure. At the mo-ment in the DCP, the simplified equation16

Wp p (13)is used for calculating the dissipated energy per countedcycle. Through the calculation of the load history in thestabilized state it is linearly accumulated

Wp=

cycles

Wp,i (14)

and the valueWpused for the critical area identification.Both implemented damage assessment methods are

suitable for any combination of load and temperaturehistories.

E X A M P L E S D E F I N I T I O N S

The aim of the following section is to show that the above-described procedure can be successfully used in TMFanalyses of complex 3D structures. To enable DCP vali-dation, linear and nonlinear FEA with the renowned AN-SYS software have been performed and compared to theresults gained by DCP. The stressstrain fields obtainedfrom the nonlinear FEA have been used as reference to

evaluate the agreement between the material model im-plemented in the DCP and the material model used in

ANSYS.In order to facilitate the reduction of computational

times, great care has been put in to the linear FEA incombination with the Neuber approximate formula. Theinfluence ofKpupon stressstrain trajectory, damage anddissipated energy has also been studied.

Specimen and material

Figure 4 shows the modified 5 mm thick ASTM Inter-national34 specimen with an additional hole of diameter3 mm; boundary conditions and applied loads are alsoshown. The specimen was shortened for the FEA eval-uation in the fixation-gripping area to reduce the size ofthe model.The specimen is a commonly used flat specimen for stan-

dard LCF tests and is also suitable for tests at elevatedtemperatures. A hole was introduced in the centre of thesample in order to explore the stresses and strains aroundsuch a weak point inthe structure. With such a specimen, auniaxial test can be used to simulate complex stressstrain



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Fig. 4 Modified ASTM34 specimen with boundary conditions and

applied loads.

states and observe the stress-concentration phenomenaand easily predict the highly damaged areas. All tests werecarried out using the same finite element model, consist-ing of 7700 linear cubic and prismatic elements. The mesh

was refined around the hole. All analyses were performedwith a 2.8 GHz Pentium IV, 4 GB RAM, Windows XP

based workstation.For validation of the proposed approach, the following

material was chosen: Steel 10 CrMo 9 10 (heat treatment930C/1.5 h air, 710C/1.5 h air, 680C furnace) withRambergOsgood and Manson-Coffin-Morrow parame-ters (Table 1) obtained by Boller and others35 for 23, 300,400, 500 and 600C.

For the nonlinear analyses in the ANSYS program,the linearization of the RamberOsgood curve has beenmade with the multi-linear kinematic hardening Besselingmodel,13,36 which cannot exceed 20 input points per one

Table 1 Temperature dependant material parameters35

T(C) E(T) (MPa) K(T) (MPa) n(T) () f(T) (MPa) f(T) () b(T) () c(T) () Rp,0.2(T) (MPa)

23 210 000 842 0.118 736 0.266 0.065 0.0527 405300 204 100 691 0.102 675 0.782 0.064 0.6280 366400 187 800 681 0.102 638 0.424 0.066 0.6170 362500 184 800 497 0.077 473 0.320 0.051 0.6010 308600 162 000 327 0.057 316 0.576 0.038 0.6810 230

stress-strain curve. It represents the material behaviourwith a combination of various portions (or subvolumes),all subjected to the same total strain, but each having adifferent yield stress. Each subvolume has a simple stress-strain response but when these are combined, the modelcan represent kinematic hardening.13,36 The input points

for the material definition in nonlinear FEA are shownin Fig. 5. For the damage estimation, temperature depen-dentPSWTcurves have been calculated from the materialparameters in Table 1.

Loading

In fatigue evaluations with linear FEA and multiple loadchannels, the load superposition is used.11,12 For each ap-plied load direction, a unity loadLkis applied separately onthe structure and the tensor for the load influence factorckis calculated. This is later used for the stress calculation

concerning a specific time and location11,12

(t) =n

k=1ckLk(t). (15)

Considering the basic equation for stress calculation inlinear FEA37

= D( 0) + 0; D = f(T), (16)where the elasticity matrixD is temperature dependant,andsupposing that no initial stresses0are present, Eq.15and 16 can be rewritten for force with magnitude |L1| =L1= 1 in Eq. 17.1= D(1 0) = c1L1 c1, (17)

where the quantities related to the force are the stresstensor 1, the mechanical strain tensor 1 and the initialstrain tensor 0, which includes the influence of the tem-perature change. Taking, for example, a second force withthe same direction asL1and of different magnitude

L1= k L1 (18)with the same temperature change as in case of Eq. 17.Rewriting Eqs. 15 and 16 presents a correct formulation



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Fig. 5 Isothermal cyclically stable cyclic stressstrain curves; with

points used for the multi-linear curve in ANSYS program. Material

10 CrMo 9 10, heat treatment 930C/1.5 h air, 710C/1.5 h air,680C furnace.35

on the left side and the stresses formulated with Eq. 15 onthe right side.

1= D(1 0) = D(1 0) kL 1= c1kL1. (19)

From Eq. 19 it follows that, in general, load superposi-

tion cannot be used in thermo-mechanical situations dueto the simultaneous interaction of mechanical and ther-mal loads. However, load superposition can be used only

with the linear FEA if there are no initial stresses, and ifmechanical and thermal strain change from one time stepto another for the same factor. It is also not possible tosuperimpose stresses and strains obtained from separateforce applications with stresses and strains obtained fromseparate thermal loads. This is due to different elasticitymatrices, which in the first case is not temperature influ-encedD =f(T), but is indeed temperature influenced inthe second case D= f(T). The superposition is there-fore only possible for linearly computed pure mechanicalstresses and strains that are not influenced with tempera-ture changes.The later explanation leads to the conclusion that FEA

analyses have to be carried out for each change in temper-ature and force throughout the load history.

In the paper, two types of validation tests are shown.At first, this is done with constant temperature fields forone cycle at the following maximal surface temperatureson the heated region 20, 300, 400, 500 and 600 C (seeFig. 4). In these cases, analyses have been performed for

Fig. 6 Force and temperature histories with hold times; random

loading.

one reversed cycle with R= 1, firstly with force am-plitude L1= 12 kN and secondly with force amplitude

L2 = 25kN.ForalltemperaturesatthelowestforceL1,thegross stresses never exceed cyclic yield stress R p,0.2. Yield-ing is present only in the net section with stress peaks nearthe hole. The second loadL2leads to larger yield zones. Itshould be emphasized that the heating process producesdifferent temperature distributions in the cross-sectionscausing regions with higher temperature to yield first.

Secondly, the random force and temperature history(Fig. 6) has been applied with linear force and temper-ature changes between turning points. This is a syntheticload history with a combination of highlow heating ratesand hold times. Time dependent phenomena like heatingrates and strain rates do not influence elastoplastic mate-rial models at all. The temperature history is related to thehighest surface temperature on the specimen. The exactapplied force-temperature combination has high heatingrates and its exact path is rarely found in any industrialapplication, but is employed due to its complexity and

ability to demonstrate the broad applicability of the pro-posed model. The turning points define the number ofstructural analyses required, which amounts to a total of22 analyses for the 30 s history. The analysed force andtemperature loading combinations are shown with circlepoints in Fig. 6.

R E S U L T S

Damage and dissipated energy have been used to comparefatigue evaluations of nonlinear FEA and those of linear



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Fig. 7 Meshed specimen showing nodes 2288 and 7266.

FEA for differentKp-s. Due to different interactions be-tween mechanical and thermal strains, the location of themost damaged node can move if the temperature fieldchanges even if the mechanical force remains unchanged.For validation purposes, two nodes were selected (seeFig. 7). The most damaged node in the linear analysis at20C (node number 2288) and the node with the averagedamage value in the highly damaged area (node number

7266). Node 2288was also used forstressstrain trajectoryvalidation for the most complex random loading. Moredetailed analyses were made at 20 C in order to eliminateerrors caused by interpolation of temperature-dependantmaterial parameters.A complete set of damage and dissipated energy val-

ues for L1= 12 kN and random loading are given inAppendix A. The values for the second load level L2=25 kN and constant temperatures are tabulated in

Appendix B.

Stressstrain trajectories

Stressstrain trajectories were obtained directly from theexported signed von Mises stresses and the signed totalmechanical von Mises strains as provided by the nonlin-ear solutions in the ANSYS software. Stressstrain pathsobtained by DCP from nonlinear FEA stress results, as

well as those from linear FEA stresses for two distinctKp-s, are also presented. In the figures below, the follow-ing notation is used

1 Stresses and strains that were gained from nonlinear FEA.

Fig. 8 Stressstrain trajectories at node 7266; 20C,L1= 12 kN.

2 Stresses in turning points were taken from nonlinear FEA,

and these values were then processed by DCP to produce

complete stressstrain trajectories forKp= 1.3 Stresses in turning points were taken from linear FEA,

and these values were then processed by DCP to produce

complete stressstrain trajectories forKp= 30.4 Stresses in turning points were taken from linear FEA,

and these values were then processed by DCP to producecomplete stressstrain trajectories forKp(T) (see Eq. 9).

Stressstrain trajectories for L1 and L2 are given inFigs 8 and 9, respectively.The stressstrain trajectories and the temperature his-

tory at node 2288 for random loading are shown inFig. 10.

Nonlinear FEA

By comparing the Besseling material model13 built in AN-SYS with the DCP material model (see Figs 810)it can beseen that good agreement has been reached at the turningpoints. However, differences have arisen for the follow-ing reasons: Firstly, the trajectories drawn from the stressand strain values exported from ANSYS are influencedby the linearization of the RambergOsgood curves. Sec-ondly, in the FEA software, the equivalent stresses andstrains are computed from the complete stress and straintensors, whereas in DCP the strain is calculated with theproposed uniaxial model from the signed equivalent von

Mises stresses imported from ANSYS.



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Fig. 9 Stressstrain trajectories at node 7266; 20C,L2= 25 kN.

Fig. 10 Temperature, stress and strain history at node 2288;random loading.

Linear FEA

By comparing stressstrain trajectories first computedwith the linear FEA and then transformed with theNeuber approximate formula, the influence ofKp uponnonlinear material behaviour can be studied. For L1(Fig. 8), where negligible yielding has occurred, Kp= 30and Kp(T) gained from Eq. 9 result in nearly the same

curves. More pronounced differences were observed forL2. In this case, (Fig. 9) the calculated Kp(T) resulted inbetter agreement with the nonlinear FEA as Kp= 30. Itcan be concluded that at larger yielding, the influence of

Kpbecomes significant and may lead to non-conservativeresults in subsequent damage estimations for highKp.

For random loading, stresses were overestimated at cer-tain turning points, as shown in Fig. 10. Strain history Cobtained from linear FEA forKp = 30 is quite similar to Afrom nonlinear FEA, except for stresses which exceededthe yield stress. This is in agreement with previous obser-

vations for constant temperature and L2.

Damage and dissipated energy

The previous section provides the basis for damage anddissipated energy estimation. It is assumed that the dam-age and dissipated energy estimated with DCP based onnonlinear FEA are the best and considered as reference

values in relation to those obtained from the linear FEA.Visual comparison of the damage and dissipated energy

distributions for random loading and constant tempera-ture loading atTmax = 500C andL1 = 12 kN is given inFigs 11 and 12. Both linear and nonlinear FEA resultedin the same highly damaged areas, with more pronounceddamage around the hole being observed for the nonlinearFEA.The damage distributions for the random loading

(Figs 13 and 14) show two critical areas with a wider crit-ical region near the specimens constraint for the linear

Fig. 11 Damage distribution for constant temperature;Tmax=500C,L1= 12 kN, linear FEA,Kp= 30.



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Fig. 12 Damage distribution for constant temperature;Tmax=500C,L1= 12 kN, nonlinear FEA,Kp= 1.

Fig. 13 Damage distribution for random loading; linear FEA,Kp= 30.

FEA. The dissipated energy distribution is also used as anindicator of the most critical areas. As shown in Figs 15and 16, this approach gives the same critical regions asthose based on damage estimation.

By comparing the values given in Appendices A and B,it can be seen that for both load levels, an increase in thedamage and in the dissipated energy occurred at higher

Fig. 14 Damage distribution for random loading; nonlinear FEA,

Kp= 1.

Fig. 15 Dissipated energy distribution for random loading; linearFEA,Kp= 30.

temperatures. This happens both globally and also at node2288, which is in accordance with the temperature de-pendentPSWT and RambergOsgood parameters given in

Table 1. For the random loading, the maximal global dam-age and dissipated energy gained from linear FEA and

Kp= 30 are 4.45 times and 3.25 times higher as com-pared to the nonlinear FEA, respectively.



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Fig. 16 Dissipated energy distribution for random loading;

nonlinear FEA,Kp= 1.

Fig. 17 Comparison between estimated damages based on linear

and nonlinear FEA.

The comparison of the calculated damages for the nodenumber 2288 is given in Fig. 17. It can be seen that themajority of points are within the zone of five times thedamage value given in dashed lines. For L1, both Kp-sgive similar damage scores that are all on the conservativeside. For L2 and Kp= 30 non-conservative results were

Fig. 18 Comparison between estimated dissipated energies based

on linear and nonlinear FEA.

observed, whereas conservative damages were estimatedforKp(T).The influence ofKp at node 2288 is less pronounced

in the case of dissipated energy estimation, as shown inFig. 18. Except forL2andTmax = 600C, all other valuesare conservative and within five times the best fit line.The findings shown in Figs 17 and 18 are in accordance

with those described in the section Stressstrain trajecto-ries. The damages are more scattered than the dissipatedenergies. This is due to the fact that small errors in strainestimation lead to larger errors in damage estimations es-pecially for small values ofPSWT.

Computational time

In fatigue evaluation, the computational time is also of keyimportance in assessing the utility of the method. Com-putation time was measured for one reversed cycle with

R= 1, L2= 25 kN and Tmax= 600C, and also forthe random loading shown in Fig. 6. For the reversed cy-cle, the linear FEA took 46 s, whereas the nonlinear FEAlasted 140 s. The damage estimation took approximately7 s for linear and nonlinear FEA, as the input data for theDCP is the same for both cases. For the random load-ing, the linear FEA took 200 s to run, whilst the nonlineartook 2260 s. The corresponding damage estimation lastedapproximately 20 s. It should be noted that in the firstcase, the loading history had three turning points and inthe second case there were 22, but the damage estimationtime is higher only by a factor of 2.8. This is due to data



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preprocessing in the DCP that does not depend on thenumber of turning points.

C O N C L U S I O N S

The current paper is an extension of three previous pa-

pers by Nagode and others1719

and deals with the stress-controlled strain-life approach for the damage and dissi-pated energy estimation of arbitrary thermo-mechanicallyloaded 3D structures. It is a non-unified approach incor-porating the uncoupling of transient thermal calculations,stressstrain calculations and damage or dissipated energyestimations.The methoduses material data obtained fromstandard strain-controlled LCF tests carried out at dis-tinct but constant temperatures. The multiaxiality is con-sidered with the signed von Mises equivalent stress that isexported to the DCP, allowing computationally fast mate-rial behaviour modelling and damage or dissipated energyestimation. The main focus has been put on the applica-

tion of the proposed method in combination with stressescomputed using linear FEA. To improve the agreementbetween the nonlinear FEA and those from linear FEAin combination with the Neuber approximate formula,a temperature dependantKp(T) has been introduced. Ithas been established thatKp has significant impact uponthe estimated damage and the dissipated energy especiallyat larger yielding. This may lead to non-conservative es-timates for high Kp. However, if Kp approaches unity,conservative estimates can be expected. Linear FEA incombination with DCP is appropriate if yielding is notpronounced. This will be further investigated on the cold

part of the exhaust system. Linear FEA and DCP can onlybe used ifKpis properly selected.

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Table A1 Test results forL1= 12 kN and random loading

FEA Kp() DN2288() Wp(N2288)(105 J/mm3) Dmax () Wp,max(105 J/mm3)

Constant temperature field, Linear 30 9.75105 2.17 9.75105 2.17Tmax= 20C L1= 12 kN

Linear 1 8.79103 3.16101 8.79103 3.16101Linear 2.16 1.04

104 2.25 1.04

104 2.25

Nonlinear 1 2.68105 0.99 2.68105 0.99Constant temperature field, Linear 30 9.66105 2.51 1.07104 2.60Tmax= 300C L1= 12 kN

Linear 1 1.51102 8.48101 1.99102 8.98101Linear 1.97 1.05104 2.74 1.22104 2.86Nonlinear 1 3.17105 1.19 4.29105 1.41


Linear 1 2.51102 1.17102 2.74102 1.17102Linear 1.95 1.40104 2.94 1.40104 2.94nonlinear 1 3.84105 1.19 4.86105 1.44


Linear 1 4.09

102 1.10

102 6.79

102 1.44

102

Linear 2.06 2.49104 2.86 2.70104 3.12Nonlinear 1 1.07104 1.59 1.07104 1.59


Linear 1 1.52 8.88102 1.52 8.88102Linear 30 5.05104 3.42 5.05104 3.42Nonlinear 1 8.53105 1.11 4.26104 2.65

Random loading Linear 30 9.33104 1.40101 4.78103 5.33101Linear 1 2.14 1.81104 12 7.30105Nonlinear 1 1.07103 1.61101 1.07103 1.64101

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A P P E N D I X A : R E S U LT S F O R L1 A N D R A N D O M L O A D I N G



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A P P E N D I X B : R E S U LT S F O R L2

Table B1 Test results forL2= 25 kN

FEA Kp() DN2288() Wp(N2288)(105 J/mm3) Dmax() Wp,max(105 J/mm3)


Linear 1 1 3.49104 2.66 3.49104Linear 2.16 9.15102 1.26102 9.15102 1.26102Nonlinear 1 3.16102 6.74101 3.16102 6.74101




Linear 1 1 3.38105 1.06 3.38105

Linear 1.95 6.54101

9.61102

6.54101

9.61102

Nonlinear 1 5.62102 1.57102 7.27102 2.06102Constant temperature field, Linear 30 5.09103 1.97101 5.40103 1.99101Tmax= 500C L2= 25 kN



Linear 1 1 8.43106 5.11 1.96108Linear 30 8.97103 2.16101 9.73103 2.16101Nonlinear 1 3.64101 2.45102 4.84101 2.57102

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