Fatigue Life Prediction of High Temperature Components · PDF fileFatigue Life Prediction of...

Fatigue Life Prediction of High Temperature Components in Combustion Engines and Exhaust Systems Dr.-Ing. Thomas Seifert and Prof. Dr. Hermann Riedel Fraunhofer Institute for Mechanics of Materials IWM, Business Unit Component Safety, Freiburg, Germany

SYNOPSIS The aim of the work described in this paper is to provide a computational method for fatigue life prediction of high temperature components in combustion engines and exhaust systems. The method allows a substantial reduction of the number of bench tests. The fatigue life pre-diction is based on a law for microcrack growth, which is the predominant damage mechan-ism under the thermo-mechanical loading conditions of these components. In addition, a po-werful model for nonisothermal cyclic plasticity is employed, and an efficient laboratory test procedure is proposed for the determination of the model parameters. The models are effi-ciently implemented into finite element programs and are used to predict the fatigue life of a cast iron exhaust manifold and a sheet metal exhaust manifold. The simulated fatigue lives correspond very well to results of component tests where cracks are found at the predicted locations.

1 INTRODUCTION High temperature components in combustion engines and exhaust systems must withstand severe cyclic mechanical and thermal loads throughout their life cycle. The combination of thermal transients with mechanical load cycles results in a complex evolution of damage, leading to thermomechanical fatigue (TMF) of the material and, after a certain number of loading cycles, to failure of the component. Numerous expensive and time-consuming bench tests are necessary to find the appropriate design and material that ensure the integrity of the component for a whole product life. Hence, there is a demand for reliable computational me-thods allowing the calculation of the lifetime of the components and the optimization of the components via computer simulations. If the transient temperature fields in the components are known from calculations or mea-surements, computational methods for TMF life prediction comprise two main steps: In the first step, the transient temperature fields are prescribed in finite element calculations to compute the transient stress and strain fields. In the second step, the computed local stresses and strains enter a lifetime model to predict the fatigue life of the component. To obtain reliable stress and strain fields in the simulations, constitutive equations are needed for the description of time and temperature dependent cyclic plasticity. Existing mod-els in commercial finite element programs as ABAQUS/Standard [1] and ANSYS [2] do not consider combined time dependent behavior with kinematic hardening laws, although appro-

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priate models were developed in the past. Viscoplastic models due to Chaboche [3][4] can describe the essential phenomena, namely strain-rate dependency, creep, relaxation and recovery, as well as the Bauschinger effect, cyclic hardening and softening, mean stress relaxation and ratchetting. These models contain a larger number of parameters that must be adjusted to experimental data. On the one hand, the identification of the parameters requires deeper understanding of the models and, on the other hand, appropriate experimental data is often not available since it is expensive and time-consuming to generate. Both aspects are major obstacles that prevent the introduction of better constitutive models into industrial ap-plication. Reliable lifetime models for TMF problems are rare. Purely phenomenological lifetime mod-els generally cannot reproduce the dependency of damage on the stress-temperature-time history appropriately. Lifetime models are needed that refer to the actual damage mechan-isms. The cycle times of combustion engines and exhaust systems are relatively short, so that creep damage plays only a minor role. However, higher strain amplitudes arise in these applications due to the constrained thermal strains, so that cracks nucleate in an early stage of the lifetime. The lifetime limiting damage mechanism is therefore the growth of these cracks under cyclic loading. The crack growth rates depend on cycle times, strain ampli-tudes, mean stress, temperature and the environment [5][6][7]. A mechanism-based model for TMF fatigue life prediction of high temperature automobile components was proposed in [8]. The basic assumption of the model is that, according to the crack-tip blunting model [9][10], the increment in crack advance per loading cycle is corre-lated with the cyclic crack-tip opening displacement, da/dN = β ΔCTOD. Typically the factor β is ranging from 1/3 to 1 [11][12][13]. The correlation between fatigue lives predicted with the crack-tip blunting model and the fatigue lives measured in TMF test was studied in [13] with finite element simulations. It was found, that the crack-tip blunting model is a reasonable me-chanism-based approach to predict TMF lives. In this paper, a computational method is described which allows a substantial reduction of the number of bench tests. The methodology for calculating the lifetime of thermomechani-cally loaded components is described in detail in [14] and comprises constitutive equations describing time and temperature dependent cyclic plasticity, a mechanism-based model for fatigue life prediction, an efficient finite element implementation and an efficient experimental and numerical procedure to determine the model parameters. The paper is structured as follows: Next, the time and temperature dependent cyclic plasticity model is presented before the model for time and temperature dependent crack growth is described. Then, the models are adjusted and applied to predict the fatigue life of a D5S cast iron exhaust manifold and a 1.4301 sheet metal exhaust manifold. Finally, conclusions are drawn.

2 TIME AND TEMPERATURE CYCLIC PLASTICITY

2.1 Model equations The stresses are computed from

( ) TCTC

C mnklmnijklvp

klthklklijklij

&&&&& σεεεσ 1−

∂

∂+−−= , (1)

where Cijkl is the elasticity tensor. The thermal strain rate is given by the (differential) expan-sion coefficient αth and the temperature rate




ijthth

ij Tδαε && = . (2)

δij is the second order identity tensor The viscoplastic strain rate has a power law form:

eq

ijvpij p

ββ

ε &&23

= where n

Yeq

Kp

σβ −=& . (3)

βij = σ’ij -αij is the relative stress where σ’ij denotes the deviator of the stress tensor and αij is the backstress tensor. βeq is the von Mises equivalent stress value. The backstress tensor is the sum of two parts each following the evolution equation

)()(

)()()()()()()()( 1 k

ijk

kkij

kkij

kkvpij

kkij T

CTCRpC αααϕγεα &&&&∂∂

+−−= , 2,1=k . (4)

The functions

( ) pkss

kss

k k

e)()()()( 1 ωϕϕϕ −−+= , 2,1=k (5)

are introduced to model cyclic hardening or softening. Further the model contains the tem-perature dependent parameters:

• E and αth, describing the thermoelastic properties of the material • K and n, describing the viscous properties of the material • σY, C(k), γ(k), R(k), φss

(k) and ω(k), describing the plastic and hardening properties of the material

2.2 Remarks on the finite element implementation and parameter identification The increased complexity of the models demands efficient algorithms, so that the models can be applied in industrial practice to large scale finite element calculations. Also, it would be desirable that the algorithms are modular, so that model improvements can be worked in fast. Thus, algorithms which combine efficiency and modularity [15][16] are used for the im-plementation of the models into the finite element programs ABAQUS/Standard [1] and ANSYS [2]. Numerical methods are applied to identify the model parameters with a high level of automa-tion based complex low-cycle fatigue (LCF) tests [15]. The strain controlled loading history in these tests is shown in Figure 1. The loading history consists of a non-periodic part where different strain rates and hold times in tension and compression as well as different strain amplitudes are used. Following the non-periodic part, a periodic strain history is applied where the specimen is cycled until failure with constant strain rate and constant strain ampli-tude. With this complex loading program strain rate effects, stress relaxation and the cyclic hardening properties of the material can be investigated in a single experiment and sufficient information is generated for the determination of the parameters of the deformation model [17]. Thus, the temperature dependent parameters of the model can be determined at small cost. To predict stresses under non-isothermal loading conditions, the model parameter are interpolated linearly in temperature.




Figure 1: Loading history in complex LCF tests

3 MECHANISM-BASED MODEL FOR FATIGUE LIFE PREDICTION The crack-tip blunting model assumes that the increment in crack advance per loading cycle is correlated with the cyclic crack-tip opening displacement:

CTODdNda

Δ= β . (6)

The analytical, fracture mechanics based estimate 2ΔCTOD/a ≈ DTMF, where a is the crack length, was proposed in [8]. In case of semi-circular surface cracks and multiaxial but propor-tional loadings, the damage parameter DTMF has the functional form

( )tTFNE

D vpe

eCY

I

CY

effITMF ,

314.245.1

22,

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

ΔΔ

++

Δ= ε

σσσ

σσ

. (7)

Δσ and Δεvp are twice the stress and viscoplastic strain amplitudes of a saturated stress-strain hysteresis typically taken of a midlife cycle. The subscripts I and e denote the maxi-mum principal and the von Mises equivalent of the respective variable. The subscript eff indi-cates that the effect of crack closure is taken into account. The temperature dependent mod-el parameters are

• Young’s modulus E • the cyclic yield stress σCY, defined as the yield stress measured from the point of

strain rate reversal • the Ramberg-Osgood hardening exponent N of the cyclic stress-strain curve

They are determined form the stress-strain hysteresis at half lifetime. The temperature and time dependent function F in equation (7) is introduced to take an increased damage due to creep and hold times at higher temperatures into account. Integration of the crack growth law in equation (6) from an initial crack length a0 to the crack length at fracture af yields the expression for the number of cycles to failure

BTMFf DAN −= . (8)

A and B are introduced as model parameters, that are used to adjust the predicted fatigue lives to the fatigue lives measured in LCF and TMF tests. The underlying fracture mechanical model suggests that A = 2 ln(af/a0)/β, so that A is in the range from 8 to 24, and B = 1.




4 FATIGUE LIFE PREDICTION OF A D5S CAST IRON EXHAUST MANIFOLD

4.1 Adjustment of the cyclic plasticity model The Chaboche model is adjusted to complex LCF tests at room temperature, 300, 500, 650, 750 and 850 °C. Figure 2 shows the adjustment of the model to the results of LCF tests at room temperature, 650 and 850 °C. Only the complex part of the loading history is shown. The figure demonstrates that the model describes several aspects of the material behavior well: (1) the stress level decreases with increasing temperature, (2) strain rate sensitivity and stress relaxation become more pronounced with increasing temperature.

Figure 2: Experimentally measured stresses and adjusted model response at different

temperatures for D5S The adjusted model is validated with out-of-phase TMF tests. In the TMF tests the speci-mens were cycled between 200 and 800 °C. The thermal strain was either totally or partially suppressed by the testing machine. Tests with 0, 70 and 80 % compliance were carried out. Figure 3 shows that the experimental results of the TMF tests (symbols) with and without strain compliance can be predicted well by the model (lines).

4.2 Adjustment of the fatigue life model Figure 4 shows the measured numbers of cycles to failure plotted against the DTMF parame-ter. The blue symbols represent the LCF tests with the temperature and applied strain ampli-tude indicated at the respective symbol. The red symbols represent the TMF tests with the temperature range and the strain compliance indicated. It is apparent from figure 4 that all experimental results fall in a common scatter band when plotted against DTMF, except one of the room temperature LCF tests. The TMF test with 90 % strain compliance did not fail dur-ing the scheduled test duration. The outcome of the adjustment is A = 27.6 and B = 1, as suggested by the theory.




a)

b)

c)

Figure 3: Experimentally measured stresses and model prediction for TMF tests for D5S: a) without strain compliance, b) 70 % strain compliance and c) 80 % strain compliance




Figure 4: Experimentally measured cycles to failure and life time model prediction of the

LCF and TMF tests with D5S

4.3 Component simulation and comparison with experiments The models of the previous sections are applied in finite element calculations to predict the thermomechanical fatigue life of a cast exhaust manifold of a six cylinder diesel engine de-signed by BMW Steyr. The finite element mesh is shown in Figure 5. The manifold is mod-eled with three-dimensional continuum elements.

Figure 5: Finite element model of the BMW cast manifold In the simulation, the manifold flange was attached to the cylinder head by applying appro-priate bolt forces. Coulomb friction between the contact pair is assumed. The transient tem-perature distribution in the manifold was supplied by BMW Steyr and was obtained from computational fluid dynamics calculations and verified by test bench measurements. Figure 6 shows the result of the simulation for the D5S manifold together with experimental results from component tests. The critical regions that determine the service life turned out to be in the separating plate at the outlet flange of the manifold. The regions, which are prone to cracking, appear colored in Figure 6. The simulated result corresponds very well to the com-ponent tests, which show cracks at the predicted locations. The predicted number of cycles to failure is close to the measured value (significantly better than a factor two).




Figure 6: Regions of critical cycles to failure of the BMW cast manifold: experimental and simulation results

5 FATIGUE LIFE PREDICTION OF A 1.4301 SHEET METAL EXHAUST MANIFOLD

5.1 Adjustment of the cyclic plasticity model The Chaboche model is adjusted to complex LCF tests at room temperature, 550, 700, 850 and 950 °C. Figure 7 shows the adjustment of the model to the results of LCF tests at room temperature, 700 and 850 °C. Only the complex part of the loading history is shown. Rate dependent material behavior, relaxation during hold times and cyclic hardening can be mod-eled well. With increasing temperature the viscosity of the material increases.

Figure 7: Experimentally measured stresses and adjusted model response at different

temperatures for 1.4301 The adjusted model is validated with TMF tests. In the TMF tests the specimens were cycled between 200 and 700 °C as well as between 350 and 850 °C. The thermal strains were fully




constraint. Figure 8 shows that the experimental results of the TMF tests (symbols) can be predicted well by the model (lines).

a)

b)

Figure 8: Experimentally measured stresses and model prediction for TMF tests for 1.4301: a) 200 - 700 °C and b) 350 - 850 °C

5.2 Adjustment of the fatigue life model Figure 9 shows the measured numbers of cycles to failure plotted against the DTMF parame-ter. The red symbols represent the LCF tests with the temperature indicated at the respective symbol. The blue symbols represent the TMF tests with the temperature range indicated. It is apparent from Figure 9 that all experimental results fall in a narrow scatter band when plotted against DTMF. The outcome of the adjustment is A = 18.9 and B = 1.14, as suggested by the theory.




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Figure 11: Regions with critical cycles to failure of the ArvinMeritor sheet metal manifold: experimental and simulation results

6 CONCLUSION In this paper, a computational method for fatigue analysis of high temperature components is proposed. It allows a substantial reduction of the number of expensive and time-consuming bench tests. The key aspects of the methodology are (1) a time and temperature dependent cyclic plasticity model that is able to describe materials under non-isothermal cyclic loadings, (2) efficient algorithms to apply the model in finite element programs, (3) an efficient proce-dure to determine the model parameters and (4) a fracture mechanics based damage para-meter to predict thermomechanical fatigue lives. The developed methods are applied to two different materials, namely cast iron D5S and sheet metal 1.4301. The constitutive models are able to describe the deformation of these materials with high accuracy. With the fatigue life model, the measured cycles to failure of LCF as well as TMF tests are predicted in almost every case well within a factor of two. The adjusted models are employed to predict the TMF lives of exhaust manifolds. In both cases, the location of failure as well as the number of cycles to failure are in very good agreement with results obtained in bench tests.

7 REFERENCES [1] ABAQUS Theory Manual (2008), Version 6.8, Dassault Systèmes [2] ANSYS Theory Reference (2007), Release 11.0, ANSYS, Inc. [3] Chaboche, J. L. (1986), Time-independent constitutive theories for cyclic plasticity, Int.

J. Plast. 2, 149-188 [4] Chaboche, J. L. (1989), Constitutive equations for cyclic plasticity and cyclic viscoplas-

ticity, Int. J. Plast. 5, 247-302 [5] Sidey D. and Coffin, L. F. (1979), Low-cycle fatigue damage mechanisms at high tem-

perature, in Fatigue Mechanisms (Ed. J. T. Fong), ASTM STP 675, 528-658 [6] Riedel, H. (1987), Fracture at high temperatures, Springer-Verlag, Berlin, Heidelberg,

New York, 1st edition [7] Suresh, S (1991), Fatigue of materials, Cambridge University Press [8] Schmitt, W., Mohrmann, R., Riedel, H., Dietsche, A. and Fischersworring-Bunk, A.

(2002), Modeling the fatigue life of automobile components, in Fatigue 2002 - Proceed-ings of the Eighth International Fatigue Congress held 3 - 7 June 2002, Stockholm, Sweden (Ed. A. F. Blom), 781-788




[9] Laird, C. AND Smith, G. C. (1962), Crack propagation in high stress fatigue, Philosoph-ical Magazine 7, 847-857

[10] Pelloux, R. M. N. (1970), Crack extension by alternating shear, Engng. Fract. Mech. 1, 697-704

[11] Tvergaard, V. and Hutchinson, J.W. (2002), Crack growth per cycle by blunting and void growth, in Fatigue 2002 - Proceedings of the Eighth International Fatigue Con-gress held 3 - 7 June 2002, Stockholm, Sweden (Ed. A.F. Blom), 107-116

[12] Kiyak, Y., Fedelich, B., May, T. and Pfennig, A. (2008), Simulation of crack growth un-der low cycle fatigue at high temperature in a single crystal superalloy, Engng. Fract. Mech. 75, 2418-2443

[13] Schweizer, C., Seifert, T., Schlesinger, M. and Riedel, H. (2007), Korrelation zwischen zyklischer Rissspitzenöffnung und Lebensdauer, in DVM-Bericht 239: Proceedings of the 39. DVM-Arbeitskreis Bruchvorgänge; Bruchmechanik und Bauteilsicherheit, 237-346

[14] Seifert, T (2008), Computational methods for fatigue life prediction of high temperature components in combustion engines and exhaust systems, dissertation, Universität Karlsruhe (TH)

[15] Seifert, T., Schenk, T. and Schmidt, I. (2007), Efficient and modular algorithms in mod-eling finite inelastic deformations: objective integration, parameter identification and sub-stepping techniques, Comput. Methods Appl. Mech. Eng. 196, 2269-2283

[16] Seifert, T. and Maier, G. (2008), Linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm for large deformation inelasticity, Int. J. Numer. Meth. Engng. 75, 690-708

[17] Seifert, T. (2006), Ein komplexes LCF-Versuchsprogramm zur schnellen und günstigen Werkstoffparameteridentifizierung, in Proceedings of Werkstoffprüfung 2006 - Fort-schritte der Kennwertermittlung für Forschung und Praxis, 409-414




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