Procedia Engineering 133 ( 2015 ) 655 – 668

Available online at www.sciencedirect.com

1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of CETIMdoi: 10.1016/j.proeng.2015.12.647

ScienceDirect

6th Fatigue Design conference, Fatigue Design 2015

Effect of Constitutive Model on Thermomechanical Fatigue Life Prediction

Jianghui Maoa*, Carlos C. Engler-Pinto Jr.a, Tie Lib, Jerry Hsieha, Xuming Sua aFord Motor Company, 2101 Village Rd. Daerborn, MI 48124,USA

bFord Motor Company, Dunton, SS13-SS16, Great Britain

Abstract

In this paper, thermomechanical fatigue life (TMF) prediction is performed on an automotive exhaust manifold using in-house postprocessor code (Hotlife) with four different constitutive models: elastoplastic model with kinematic hardening, two-layer viscoplastic model, and two unified elasto-viscoplastic models based on the equations proposed by Sehitoglu and Chaboche, respectively. Commercial FEA software Abaqus has been used for the analysis. The first two models are already available in Abaqus; the Chaboche and Sehitoglu models were implemented using a user material subroutine (UMAT, available in Abaqus). The stress and strain histories calculated by these four models are output to Hotlife for post-processing. The results from the prediction are then compared to the experimental TMF life obtained from a component thermal fatigue bench test. Result show the Hotlife calculations based on the two implemented elasto-viscoplastic models yield better correlation with test results. © 2015 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of CETIM.

Keywords: thermomechanical fatigue, constitutive model, elastoplasticity, two-layer viscoplasticity, elasto-viscoplasticity, fatigue damage model.

* Corresponding author. Tel.: +1(313)248-4002; fax: +1(313)390-0514.

E-mail address: jhm538@gmail.com, jmao9@ford.com

© 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of CETIM

656 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

1. Introduction

The effect of combined thermal and mechanical loading in engine components is generally referred to as thermomechanical loading. The steady increase in specific power output for fuel and emissions considerations has subjected automotive powertrain components to severe thermomechanical fatigue (TMF) loadings [1].

The design against thermomechanical fatigue of automotive engine components very often involves a preliminary

monotonic and cyclic stress-strain analysis, followed by a fatigue crack initiation or propagation criteria to determine the durability of the component. Hence, the correct choice of cyclic plastic or viscoplastic equations may play a very important role in the final results, especially when the interaction between thermal and mechanical loading could significantly alter the material behavior (creeping and relaxation, etc) [2].

It is well recognized that accurate descriptions of temperature effects are difficult to obtain by general plastic

constitutive equations. For temperatures close to room temperature, experiments [2] have shown that engine component materials exhibit very limited rate-dependency, and the its behavior could be correctly captured by constitutive equations based on general time-independent plasticity. At elevated temperatures, however, time-dependent or rate-dependent material behavior takes place and becomes dominant. In that case, a viscoplastic model has to be used. In this paper, five different elastoplastic and viscoplastic constitutive equations based on multi-kinematic hardening rules are employed for the TMF durability simulation of a cast iron exhaust manifold. The impact of the stress-strain behavior obtained for these different constitutive models is then investigated using the TMF damage model proposed by Sehitoglu [3,4].

2. A general constitutive equation framework and implementation details for each model

In what follows, the general format of plasticity and viscoplasticity used in this investigation are outlined, followed with a more detailed formulation for each of the models used.

2.1. Generalized constitutive model

The constitutive models used are all based on the infinitesimal strain theory. The total strain ij is partitioned into three parts as:

Tij

inij

eijij (1a)

where eij is the elastic strain tensor; in

ij is the inelastic strain tensor and Tij is the thermal strain tensor. In non-

unified viscoplasticity theories, the he inelastic strain can be further divided as:

crij

pij

inij (1b)

where pij corresponds to the rate-independent plastic strain and cr

ij is the creep strain. For elastoplastic models, 0cr

ij and pij

inij ( in

ij corresponds to plastic strain). For unified viscoplasticity theories, there is no distinction between plastic and creep strains – only in

ij exists, representing the combined effect of rate-independent deformation (plastic deformation) and rate-dependent deformation (creep and relaxation).

For all models, inelastic flow is governed by the inelastic flow equation f in the stress space, expressed as:

RKXJff ijij ,),( (2)

where ij is the stress deviator; ijX is the back stress tensor, K is the drag stress and R defines the evolution of the elastic limit. J defines an invariant function in stress space; the von Mises invariant is typically used for isotropic

657 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

material, ijijij SSSJ 2/3)( . For models without an elastic domain, like the Sehitoglu viscoplastic model [6-7] and the Sandia viscoplastic model [8-9], the term R is ignored (R=0).

In the associated plasticity framework (the flow potential is identical within the yield surface), the normality

assumption reads:

ij

ij

inij n

f (3)

where inij is the inelastic strain rate tensor, is the inelastic multiplier. In case of rate-independent plasticity, is

determined by the consistency condition of plastic flow 0ff . In the rate-dependent (viscoplasticity) case, f is the viscous part of the stress (or overstress) and is defined as a given function of f .

)(1 fF (4)

The expression of function 1F varies. Sehitoglu viscoplastic model [6-7] employed a step function differentiating plastic dominated and creep dominated deformation; the Sandia viscoplastic model [8-9] proposes a hyperbolic sine function; the Chaboche viscoplastic model uses a power law [10-11]. It should be noted that is identical to the norm of inelastic strain rate 2/1)3/2( in

ijinij

inij in the framework of associated plasticity.

As in Eq. 5, the back stress (center of yield surface) is decomposed as a sum of independent variables, each of

which obeys the same rule with different material parameters as in Eq. 6, and it usually contains a linear hardening term, a dynamic recovery term, a static recovery term, and temperature rate term.

k

kijij XX )( (5)

),,( )(2

)( TXFX kij

inij

kij (6)

Similarly to the back stress equations, the isotropic hardening equation may also include three terms to represent linear hardening, dynamic recovery, static recover, or temperature rate change, as show in Equation 7 below:

),,(3 TRFR inij (7)

2.2. Implementation for each constitutive model

2.2.1 Elastoplastic model

The elastoplastic model used in the present investigation has already been incorporated in Abaqus [12]. The user has the option to choose different versions of hardening rules depending on the material used and the application. Chosen here for comparison is the combined hardening rule including isotropic hardening and kinematic hardening. The yield function in this model reads:

RXJf ijij )( (8)

where R represents isotropic hardening, ),( TRR pij , and ijX is the back stress, representing the center of yield

surface. In the model setup, two keyword cards will be activated in material definition: *EL ASTI C and

658 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

*PL ASTI C,HARDENI NG=COMBI NED. They define the variation in modulus of elasticity and yield stress with temperature, respectively.

2.2.2 Two-layer viscoplastic model

Also available in Abaqus [13] for metal creep behavior is the power law model as shown in Eq. 9:

mnij tqA~cr (9)

where crcrcr 3/2 ijijij is the uniaxial equivalent creep strain rate, q~ is the uniaxial equivalent deviatoric stress, t is the total time, A, n, and m are temperature dependent material parameters. The model is invoked in Abaqus by the command *CREEP,L AW=T I ME. Eq. 9 determines the creep strain of material, the plastic strain will be calculated the same as elastoplastic model described in section 2.2.1. The total inelastic strain is hence the summation of these two strains cr

ijp

ijinij .

2.2.3 Chaboche unified viscoplastic model

Equation 4 is particularized in the Chaboche unified viscoplasticity model as

n

ijijinij

K

RXJ )( (10)

where K is the drag stress, treated here as a temperature dependent material parameter. The Macaulay bracket “< >” results in zero if the expression inside the bracket is less than zero. In this manner, the elastic and viscoplastic domains can be differentiated. The back stress in the Chaboche unified viscoplastic model is formulated as:

TXT

C

CXXJpXCX k

ijk

k

kij

mkijk

kijk

inijk

kij

k )()(1)(*)()( 1)]([

3

2 (11)

The first term in the right hand side is the linear hardening term, it’s a linear function of inelastic strain rate; the second term is dynamic recovery, it governs the nonlinear hardening behavior; the third term represents time-dependent static recovery, it only becomes effective at low strain rate; the last term represents the temperature rate effect, the influence of rapid or slow heating will be reflected by this term. Similarly to Nouailhas et al. [14], the time recovery of isotropic hardening has been described as:

TRT

Q

QT

b

bQRQRRQbR r

m

rrin

ij

11)()(

1 (12)

where the asymptotic value rQ can ensure incomplete recovery, kR when 0inij , representing initial yielding.

This model reproduces fairly well the dependency between stress-range and hold time in cyclic relaxation tests, under a constant strain range [15].

2.2.4 Sehitoglu unified viscoplastic model

Equation 4 is expressed in the Sehitoglu unified viscoplastic model as:

659 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

K

XJAF ijijin

ij

)(4 (13a)

where A is a function of temperature (Eq. 13b), and 4F depends on the operative deformation mechanism, such as power-law creep or plasticity (Eq. 13c).

)273(

exp=TR

HAA

g

c (13b)

0.1 when1)(

exp

0.1hen w )(

)(24

KK

XJ

KK

XJ

K

XJF n

ijij

n

ijij

ijij

1

(13c)

where, cA , 1n and 2n are material parameters, H is the thermal activation energy for creep and gR is the universal gas constant. In the model proposed by Sehitoglu, the size of the yield surface is controlled by the drag stress K (Eq. 14a-14c), which evolves in a hardening/recovery form as described by the equations below.

TTKTKrTKhK KK ),(),(),( (14)

inijsatHK KKBTKh )(),( (14a)

)(),( recRK KKBTKr (14b)

where satK and recK are temperature dependent material parameters. The evolution of the back stress is described by the following expressions:

ijSin

ijin

ijDin

ijij XTrTrhX ),(),,()(3

2 (15a)

)( ijXJ (15b)

0 when))(exp(

0 when))(exp()(

3

3

021

021

ijinij

a

ijinij

a

Xaa

Xaah (15c)

einij

dinij

D cTr )()(),,( (15d)

where 1a , 2a , 3a , d and e are temperature dependent material parameters. One thing to be noted is that there is only one back stress term in the proposed Sehitoglu model, hence 1k in Eq. 5.

2.2.5 Summary

All the above four models have the capability to describe the material inelastic flow. The elastoplastic (EP) model is time-independent; it will not reflect the strain rate effect experienced by materials at elevated temperatures. In this study, the test data from the highest available strain rate (5x10-3/s) has been used to fit one of the elastoplastic models, designated here as EP1. Test data from a lower strain rate (5x10-5/s) was used for fitting a second elastoplastic model, designated as EP2.

The other three models being investigated can describe strain rate effects. The Two-Layer model is the simplest

in formulation, and also is the easiest one to identify the model parameters, as it is based on the non-unified plasticity theory.

660 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

The Chaboche and Sehitoglu models are quite similar: both use a flow rule to describe the material inelastic flow,

a back stress equation describing the evolution of the yield stress surface center, and governing equations for yield surface size evolution. The major differences between these two models include:

Although both models are based on the unified viscoplasticity theory, the Sehitoglu model differentiates plasticity-dominant and creep-dominant regions by adopting two distinct flow equations.

Still in regards to the flow rule, the Chaboche model contains an elastic domain and the Sehitoglu does not; therefore, foe the Sehitoglu formulation, inelastic flow starts right away when material is loaded.

The effect of temperature on the flow rule is accounted by Sehitoglu using an Arrhenius equation; the Chaboche model needs to be fitted for different temperatures and the material parameters are then interpolated to account for the temperature effect.

Chaboche explicitly accounts for isotropic hardening through means of the isotropic hardening resistance parameter (R), which describes the change in size of the yield surface; in the Sehitoglu model, the formulation for the drag stress is mainly responsible for the isotropic hardening representation.

A temperature rate term is included in the Chaboche back stress formulation, but not in the model proposed by in Sehitoglu.

3. Thermomechanical fatigue damage model for durability assessment

The TMF model proposed by Sehitoglu et. al. [3,4] has recently been reviewed as a comprehensive damage model including effects of fatigue, creep and oxidation [16]. The damage per cycle is thus obtained by summing all three mechanisms.

creepoxfattot DDDD (16)

Depending on the temperature, strain, and phasing, all three of these damage mechanisms could be operating. In isothermal room temperature fatigue test, fatigue damage plays a major role. If the temperature is high enough, oxidation and creep are readily activated. Equation 16 can be rewritten in terms of fatigue failure life, fN , assuming linear damage summation:

creepoxfat

1111

ffff NNNN (17)

The fatigue damage term is represented by strain-life equation [5]

cff

bf

f NNE

)2()2(2

fatfatmech (18)

where f is the fatigue strength coefficient, E is modulus of elasticity, b is the fatigue strength exponent, f is the fatigue ductility coefficient, and c is the fatigue ductility exponent.

For the oxidation-fatigue damage, assuming that cracks initiate from the oxidized surface, the additional damage

is described as:

/1

1/2mech

/1

effox

0

ox

)(21

p

cr

f KB

h

N (19)

661 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

where crh is the critical crack length. ox is oxidation phasing factor, effpK is the effective oxidation constant. B ,

, and are constants. For creep-fatigue damage, Sehitoglu has proposed the following equation:

ct

mH

RTH dtKAeD0

21)/(creepcreep ]/)[( (20)

where creep is creep phasing factor, is the effective stress, H is the hydrostatic stress. 1 and 2 are scaling factors which represent the relative amount of damage occurring in tension and compression. It is worth noting that all variables are scalar, i.e. the model is originally developed for unidirectional case. In this study, a critical plane criterion [17-19] is used to extend the model to a three-dimensional state: stress and strain tensors are rotated to the critical plane, and then components with maximum values are used for damage calculation; for additional details on the implementation, the reader is referred to Reference [19].

4. Finite element implementation

The Abaqus finite element solver has been used in the present investigation. The elastoplastic and the two-layer models are alreasy implemented in Abaqus. For the Chaboche and Sehitoglu models, user material subroutines (UMAT) have been written, as illustrated in Figure 1. At the beginning of an increment, internal variables at the end of previous increment will be passed into the UMAT as “known information”, such as ij , ij and p , etc. Together with the current increments of strain, time and temperature determined by Abaqus, the UMAT will perform an iterative calculation to determine a convergent solution for the current increment. Then, the updated internal variables will be used to form the consistent tangent modulus (Jacobian Matrix) for the Abaqus main program, so that the increments of strain and temperature of next increment could be determined. For additional information on the UMAT implementation, the reader is referred to References [20-22].

Fig. 1 Flow chart of finite element implementation process

knownincrement current at etc. , , , ijijij X

etc , , , nnij

nij

nij pX

toleranceppgpg nnn 1)('/)(NO

YES

updatedincrement current of end at the etc. , , , ijijij X

Jacobian Matrix ijklJ updated

ABAQUS

Main program

1nn interationnext

pgpgpp nnnn

)('/)(1

givenincrement next for , , tTij

662 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

4.1. Constitutive model comparison –uniaxial behavior

All models were fitted exclusively to isothermal low-cycle fatigue tests performed at different strain rates; the models were subsequently correlated to non-isothermal TMF tests. Figure 2a and 2b illustrate the models ability to represent the material behavior at 600°C at two different strain rates. It is obvious that the elastoplastic models can only represent well the behavior for the strain rates in which they have been fitted. The three viscoplastic models can represent the strain rate effect reasonably well, although the two-layer model is prone to some instability as there is an abrupt transition between plasticity and creep dominated domains. Figure 2c gives an summary overview of the three viscoplastic models, showing the stress amplitude as a function of strain rate at different temperatures for fixed strain amplitude tests (0.5 % strain amplitude isothermal tests). The Chaboche model, which is fitted independently for each temperature, yields the best overall correlation, but the Sehitoglu models also provides a very reasonable representation of the material viscoplastic behavior.

The non-isothermal TMF correlation is illustrated in Figure 3. The test condition (total strain, mechanical strain

and temperature as a function of time) is shown in Figure 3a; the resultant hysteresis in plotted in Figure 3b, compared to the predictions from the different models. It is obvious that model EP1, which is fitted from tests performed at high strain rates, cannot represent the TMF material behavior, as the the TMF strain rate is approximately 1.7x10-4 s-1. All other models provided reasonable representation of the hysteresis, the two-layer model showing some instabilities in the initial viscoplastic deformation points.

(a)

-400

-300

-200

-100

0

100

200

300

400

-6.00E-03 -4.00E-03 -2.00E-03 0.00E+00 2.00E-03 4.00E-03 6.00E-03

Stre

ss (

MP

a)

Strain

600°C 5x10-3/s

EXP. Two-layer Chaboche

Sehitoglu EP1 EP2

663 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

(b)

(c)

Fig. 2 Model correlation with isothermal test data: (a) 600°C, 5x10-3/s; (b) 600°C, 5x10-5/s; (c) rate-dependence

-400

-300

-200

-100

0

100

200

300

400

-6.00E-03 -4.00E-03 -2.00E-03 0.00E+00 2.00E-03 4.00E-03 6.00E-03

Stre

ss(M

Pa)

Strain

600°C 5x10-5/s

EXP. Two-layer ChabocheSehitoglu EP1 EP2

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+01 1.00E+02 1.00E+03

Stra

in R

ate

Stress Amplitude @ 0.5% Strain

EXP Two-layer Sehitoglu Chaboche

300⁰

490⁰600⁰700⁰800⁰

400⁰

664 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

(a)

(b)

Fig. 3 Model correlation with non-isothermal test data: (a) test condition and (b) resulting hysteresis.

5. Results and discussions

This paper investigates the application of the different constitutive models previously described in the simulation of a cast iron exhaust manifold test. Figure 4 shows the FEA mesh for the model, which consists of 27900 nodes and 69025 3D elements. The exhaust manifold is fastened to an aluminum engine head (only a small portion is modelled) using steel bolts. Gaskets for bolt connection are ignored since these local areas are not of interest in the current study. Constraints are applied to the engine head. The simulations are performed in Ford High Performance Computer server.

600

650

700

750

800

850

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

00 20 40 60 80 100 120

Tem

p(C

)

Stra

in

Time (Sec)

Tot strain Mech.strain temp

-210

-110

-10

90

190

290

-1.2E-02 -1.0E-02 -8.0E-03 -6.0E-03 -4.0E-03 -2.0E-03 0.0E+00

Stre

ss (

MP

a)

Strain

Stress vs. Mech. Strain EXP.Two-layerSehitogluChabocheEP1EP2

665 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

The FEA analysis is performed in two sequential steps: heat transfer analysis and stress analysis. The purpose of the first analysis is to determine the transient temperature profile for the whole model. The temperature history is then inputted into the second step for the calculation of stress and strain histories using the five material models described in Section 2. Five full cycles are simulated, including heat up and cooling down, and the last cycle is used for durability analysis using the damage model described in section 3. Also shown in Figure 4 is the critical location (location in which a fatigue crack developed during the test, for an average fatigue life of 460 cycles).

Fig. 4 exhaust manifold model

The predicted von Mises stress contours at the end of simulations are illustrated in Fig. 5. With the exception of model EP1 (material parameters are fitted from highest strain rate of 5x10-3/s), all model predictions are very similar in regard to stress distribution. The von Mises predicted at the critical location is between 350 and 480MPa.

Fig. 5 Contour of von Mises stress (after 5 cycles) on critical area predicted by the material models

Figure 6 depicts the von Mises stress predicted at the 2nd integration point of element 836700, which is located in the manifold critical area. When maximum temperature reaching up to 750°C, von Mises predicted is about 370 MPa, with minimum of 340 MPa by Sehitoglu model, and maximum of 410MPa by the two-layer model.

(a) (b)

(c) (d)

(e) (a) EP1; (b) EP2; (c) Sehitoglu; (d) Chaboche; (e) Two-layer

666 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

Fig. 6 von Mises stress (5th cycle) predicted at critical location (element 836700 integration point 2)

Stress and strain predicted by the five models have been post-processed by an in-house code based on the

Sehitoglu damage model. The calculated total fatigue lives are shown in Fig. 7. Also shown in Fig. 7 is the experiment with the critical (crack) location indicated. The critical location is correctly captured by the damage assessment for all the constitutive models. The total fatigue life predicted at the critical node based on the five constitutive models are shown in Table 1, together with the partial damage predicted by the Sehitoglu model for the 3 different damage modes being considered. From Table 1, it can be concluded that the elastoplastic models resulted in the most inaccurate damage calculations, while the Sehitoglu and Chaboche models yielded the best correlation with the experimental fatigue life (460 cycles).

Fig. 7 Fatigue life prediction at critical location using results from the selected models (a)EP1, (b)EP2, (c) Sehitoglu, (d) Chaboche, (e) Two-layer, (f) critical location from Experiment

0

100

200

300

400

500

600

700

800

0

50

100

150

200

250

300

350

400

450

0 200 400 600 800 1000 1200

Tem

pera

ture

(C

)

von

Mis

es s

tres

s (M

Pa)

Time (s)

EP1 EP2 Two-layerSehitoglu Chaboche TEMP

(a)

(c)

(b)

(d)

(e) (f)

667 Jianghui Mao et al. / Procedia Engineering 133 ( 2015 ) 655 – 668

Table 1 Fatigue life predicted at node 25048

Model Tot_Life_Ext Fat_Life Env_Life Cre_Life

EP1 2.22E+03 1.07E+04 2.84E+03 1.90E+05

EP2 2.14E+03 4.57E+03 4.10E+03 2.91E+05

Two-layer 1.50E+03 1.98E+03 6.22E+03 4.00E+05

Sehitoglu 6.29E+02 2.10E+03 1.15E+03 4.06E+03

Chaboche 7.62E+02 1.90E+03 1.68E+03 5.22E+03

6. Conclusions

The thermomechanical fatigue damage model proposed by Sehitoglu has been used to evaluate the TMF life of an automotive exhaust manifold made of HiSiMo. The TMF life predictions are based on the stress and strain predicted by five different constitutive models: two elastoplastic models, a non-unified two-layer viscoplastic model, and two unified viscoplastic models proposed by Chaboche and Sehitoglu.

The elastoplastic models resulted in the worst correlation. This was expected as these models cannot represent the

material behavior (creep and stress relaxation) at elevated temperatures. These models were included in this study for comparison purposes as they are easy to fit, computationally efficient, and are usually readily available and already implemented in commercial FEA software.

The two-layer viscoplastic model, which can be obtained from independent creep and low-cycle fatigue tests,

resulted in an improved prediction and could be used in the simulation of high temperature components in the absence of better unified viscoplastic models.

Overall, the results obtained from the unified viscoplastic models (Chaboche and Sehitoglu) were very similar

and resulted in the best correlation with the observed TMF life. The Chaboche model was more computationally efficient then the Sehitoglu model in the present study due to its simpler formulation.

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