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International Journal of Fatigue 32 (2010) 769–779

Contents lists available at ScienceDirect

International Journal of Fatigue

journal homepage: www.elsevier.com/locate/ijfatigue

A new low cycle fatigue criterion for isothermal and out-of-phase thermomechanical loading

Taner Gocmez *, Ali Awarke, Stefan Pischinger

Institute for Combustion Engines, RWTH Aachen University, Germany

a r t i c l e

i n f o

a b s t r a c t

A new multiaxial low cycle fatigue criterion based on a damage parameter is presented as an attempt to condition energy based approaches for the general common use in isothermal and thermomechanical out-of-phase loading. The damage parameter is based on the dissipated plastic strain energy per cycle, with a stress correction factor to account for mean and maximum stress in?uences, and a thermal term to account for elevated temperature damages. With such a formulation, the low temperature damaging regime (dislocations glide) can be separated from the high temperature one (diffusion processes). Oxidation and creep damages are accounted for in an implicit manner. The prediction capability of the present criterion is compared with that of other classical lifetime criteria using data from uniaxial isothermal and out-of-phase thermomechanical fatigue (TMF) tests performed on three cast iron families which are relevant for different internal combustion engine components. The result is higher prediction accuracy and a capability to estimate TMF lifetime by calibrating on isothermal fatigue tests only, leading to a great potential cost saving for future test programs given similar materials and loading conditions. ? 2009 Elsevier Ltd. All rights reserved.

Article history: Received 29 May 2009 Received in revised form 24 September 2009 Accepted 5 November 2009 Available online 12 November 2009 Keywords: Low cycle fatigue Thermomechanical fatigue Lifetime criterion Damage parameter Cast irons

1. Introduction 1.1. Background Under thermomechanical fatigue conditions, the complex cycling may lead to damage contributions from fatigue, environmental degradation (oxidation), and creep [1,2] (Fig. 1). These damage mechanisms may act independently or in combination depending on various materials and operating conditions, such as maximum and minimum temperatures, temperature range, mechanical strain range, strain rate, the phasing of temperature and strain, dwell time, or environmental factors [3]. A well-accepted framework for the prediction of TMF life has been elusive [4]. The choice of an approach is often related to the availability of experimental data and the actual engineering application and loading conditions [5]. Attempts have been made to capture the above mentioned damaging mechanisms as in [6] by: relating measured mechanical ?elds and lifetime, without explicit consideration of the different damaging mechanisms (phenomenological models)

accounting explicitly for the different damaging mechanisms throughout damage laws (cumulative damage models) relating life to local inelastic strain at a crack tip (crack growth models) Classical approaches can be expressed under a general form of: /(e, ep, r,. . .) = f(Nf, a, b,. . .) [7,8], where Nf denotes the number of cycles to failure, a and b are material constants, e, ep and r are the response ?elds during a complete cycle which is generally assumed to be stabilized [9] (Table 1). The Manson–Cof?n criterion [10,11], which describes a linear relation between plastic strain amplitude and life on a log–log scale, stands for simplicity and robustness. The total strain approach [12,13] relates the total mechanical strain amplitude to life. The mean stress in?uence on life, which is observed in some materials such as GJL-300 [14], is accounted for in the Morrow [15] and the Smith–Watson–Topper (SWT) [16] approaches [17]. Ostergren [18] takes additionally the cycle frequency in?uence into account. These classical strain based approaches have been initially expressed for the uniaxial isothermal case. Engineering components such as engine cylinder heads are exposed to multiaxial anisothermal loading. The fatigue criteria should be compatible with this observation [19]. Bannantine [20], Fatemi and Socie [21] used transformation equations to capture the plane experiencing maximum damage (critical plane). Assuming these are suf?cient to pass for the multiaxial case, an approval for the anisothermal case can

* Corresponding author. Address: Institute for Combustion Engines, RWTH Aachen Schinkelstrasse 8, D-52062 Aachen, Germany. Tel.: +49 241 5689316; fax: +49 241 5689333. E-mail address: goecmez@vka.rwth-aachen.de (T. Gocmez). 0142-1123/$ - see front matter ? 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2009.11.003

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Nomenclature Nf Dep Demech number of cycles at failure plastic strain range mechanical strain range mean stress plastic strain amplitude total strain amplitude maximum stress ultimate tensile stress dissipated plastic strain energy per cycle dissipated plastic strain energy at failure due to monotonic loading melting temperature Tmax Tact E SCF ADIT OP TMF LCF th R2 SS maximum temperature reached during loading activation temperature for high temperature damages Young’s modulus stress correction factor thermal augmentation damage indicator out-of-phase thermomechanical fatigue low cycle fatigue hold time coef?cient of determination sum of square of error

rmean

Ea,p Ea,t

rmax rult

DWp Wp,u

Tmelt

hardly be justi?ed, as the material parameters can be temperature dependent [22]. Skelton [23,24] proposed energetic approaches showing a relation between dissipated energy per cycle and the number of cycles to failure. Energetic criterions overcome the previously mentioned dif?culties as they are readily adaptable to the multiaxial case and suitable for anisothermal loading, since they integrate the whole loading path and thus have temperature independent parameters [25]. TMF lifetime prediction approaches have promising potential when energy criteria are applied [19]. Yet, the important mean stress effect has not been taken into consideration. In the second class of approaches, where damage mechanisms have been considered separately, Halford et al. [27] proposed the strain range partitioning (SRP) method, while Neu and Sehitoglu [28] and Lemaitre and Chaboche [29] developed complex damage laws separating each of fatigue, oxidation and creep. The SRP method separates a priori the stress–strain cycle into creep and plastic parts and relates each to life. Although the method has proven to be reliable capturing all the ?rst order effects on TMF damage, its application on general variable amplitude TMF loading is ambiguous [30]. The reason lies in the fact that it is empirical and its proposed separation is not unique, as pointed out by Chaboche et al. [31,32]. The Sehitoglu damage model [28] takes mechanical fatigue, oxidation, and creep damages into account. The mechanical fatigue damage is computed using the classical total strain amplitude approach. The oxidation process is de?ned as a function of strain

range, strain rate, strain–temperature phasing, and oxidation kinetics. The creep damage is based on stress, temperature, strain–temperature phasing and time. Both damage terms include phasing factors to account for different damage contributions associated with different temperature–stress phasing [33]. A major drawback of this model is the need for extensive testing to identify its parameters [30,35]. The Chaboche damage model [29] is based on continuum damage mechanics principles, initially introduced by Rabotnov [36] and Kachanov [37]. The original model addresses only creep and fatigue failure mechanisms. The simultaneous introduction of creep and fatigue failure mechanisms is represented in the model by an incremental damage relation, which is integrated to obtain the total damage. This is a powerful feature of the Chaboche

Table 1 Classical lifetime prediction approaches. Approach Manson–Cof?n Total strain Morrow SWT Ostergren Energetic / f Parameter range 0:35 ? r0

f

Dep Demech Demech, rmean

e0f Nc f e

mech

ef < e0f < ef

0 c f Nf

e0f Nc f ?

r0 ?rmean

f

E

Nb f

E

?0.7 < c < ?0.5 Nb f

r0f % 3:5 rUTS

?0.14 < b < ?0.06

rmax, De

?c b r0f e0f ENb ? r0f2 N 2 f f b?k?1? AN f Bm

rmax, Demech R Dxp ? cycle r ? de

CN D f

D% b?c

Fig. 1. The different damages involved in thermomechanical loading [26].

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method, as it can accurately capture many intermediate effects in variable amplitude TMF loading histories. However, it is also the major drawback of the method as it requires substantial computational power [30]. Other concepts for lifetime prediction based on fracture mechanics theories have been developed and are more suitable for brittle materials [38]. Tomkins et al. [39] and Riedel and coworkers [40] take respectively crack propagation and crack closure effects into consideration. 1.2. Current work In this paper, a new lifetime prediction methodology is presented. The approach is empirical and based on the dissipated plastic strain energy per cycle as the main indicator of damage, with additional terms to account for stress in?uences and elevated temperature damages. The development of the model is explained in the ?rst section. In the second section, the prediction capability of the developed model is compared with that of other classical lifetime criteria using data from uniaxial isothermal and out-ofphase TMF tests performed on three cast iron families which are relevant for different internal combustion engine components. Finally, a discussion is enclosed along with concluding remarks. 2. Lifetime prediction methodology The elaborations in [22], showing that the dissipated energy per cycle is promising yet practical when it comes to predict commonly the lifetime of cast irons under isothermal LCF and anisothermal TMF loading, has been the motivation to develop an energy based approach. Nevertheless, energy approaches in their purest form, should be taken with a measure of skepticism if to be used as a general failure criterion. This has been deduced from experimental observations from [34], where LCF isothermal tests have been performed. A plot of the measured dissipated energy versus number of cycles to failure for different temperatures and strain amplitudes is shown for the GJS-700 cast iron in Fig. 2. We notice that the dissipated energy is a valid criterion for the temperatures between 20 °C and 200 °C, since the energy-life trend lines from both temperatures are almost matching. However, considering the tests at 350 °C and 500 °C, the energy-life trend lines for each temperature split apart such as the number of cycles to failure does not depend anymore alone on the energy but on the applied temperature as well. It is therefore not justi?ed to generalize to the TMF loading, that damage due to increased maximum temperature is correctly captured by the dissipated energy. Another drawback of the energy approach is its lack of explicit account for stress in?uences. In this work, we seek an improvement of the energy criterion. Although there have been some modeling attempts where fatigue oxidation and creep damages are accounted for separately, a high number of parameters with dif?cult identi?cation and exhaustive

testing is generally needed. Even the most complicated models are far from an accurate description of the underlying physical processes as elaborated in Fig. 3. The additional cost involved in pursuing complex models is not justi?ed in the point of view of the authors for a fast practical industrial environment. Instead, an empirical formulation with the following characteristics is preferred: low number of parameters association of parameters with measurable or reputed values based on physical observations high prediction capability

A lifetime formulation via the dissipated plastic strain energy per cycle corrected by a stress factor and augmented by a temperature dependent damage indicator is proposed as:

Nf ? A

c DW p ? SCF ? ADIT W p;u

?1?

where Nf is the number of cycles to failure, DWp is the dissipated plastic strain energy per cycle, Wp,u is the temperature dependent dissipated plastic strain energy at fracture during a monotonic tensile loading, SCF is the stress correction factor, ADIT is the temperature dependent augmentation damage indicator, A and c are material coef?cient and exponent respectively to be obtained by data regression analysis. The normalization of DWp by a temperature dependent Wp,u is a more suitable damage estimation than DWp alone, since a description of a temperature dependent pure fatigue damage is rendered possible. The stress correction takes the following form:

SCF ?

rmax rult ? m ? rmean

?2?

where rmax is the maximum tensile stress in the hysterisis curve, rmean is the mean stress in the hysterisis curve, rult is the ultimate tensile strength, m is a material constant. This correction aims to scale damages enhanced by rmax which accelerates crack propagations in an oxide layer (given the OP conditions), and to account for the mean stress in?uence, which also has an implicit acquaintance with the cycle frequency [4]. This is also observed on the TMF test data used in this work, where an increase in hold time is accompanied by an increase in mean stress. The material constant m is included in order to account for material wise variations in damage sensitivity with respect to mean stress. As shown in former work [37], mean stress effects on fatigue life are evident for pearlitic materials but almost not present for the ferritic materials. A non-linear relationship is modeled between damage and mean stress as shown in Fig. 4. As the temperature increases above a threshold (35–50% of Tmelt in K), the damage regime shifts to a dominant oxide layer and intergranular cracking. There is an abrupt change of damage rate and to our knowledge there is no physical proof that the dissipated energy alone can represent this damage modi?cation.

Fig. 2. Temperature dependent energy-life trend lines plotted for the LCF data of GJS-700.

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Fig. 3. Different interacting microstructural processes that can occur during thermomechanical loading. Tm: melting temperature.

For the above reason, the augmentation thermal damage indicator is added. By phenomenology, the indicator is typically near zero at low temperatures and starts to increase very fast above an activation temperature. The following power-law is thus proposed:

0 ? ? 1n T ?T act max e B C ADIT ? DW p @ A e

?T

T act melt

? 3?

where Tmax is the maximum temperature reached in a cycle, Tmelt is the melting temperature, Tact is the temperature above which thermally activated damages become prominent and can be approximated between 35% and 50% of Tmelt in K, n is a material exponent which controls the sensitivity of thermal damage to temperature as shown in Fig. 5.

Fig. 4. Evolution of the stress correction factor as a function of mean stress for GJS700.

Fig. 5. Evolution of the temperature dependent term.

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The temperature term is multiplied by DWp in order to couple thermal activated (creep, oxidation) and pure fatigue damages. This is justi?ed by the fact that the damage responsible microphysical processes interact. The ?nal equation is:

0 ? ? 1n 1c T ?T act rmax DW p B e max C C B DW p A A Nf ? A @ ? ? @ T W p;u rult ? m ? rmean ?MPa? ?T act melt e |??????????????????????????????????????????????????{z????????????????????????????????????????????????? ?} 0

DTMF?LCF

OP

? 4?

Fig. 6. Specimen geometry used in [34].

and contains four response variables (DWp, rmax, rmean, Tmax), three material properties (Wp,u, rult, Tmelt), and four through data regression identi?able parameters (A, c, n, m). The performed scaling (by Wp,u and rult) and the ranges of DWp under considered loading guarantee that their sizes do not affect their relative impact on damage while the scaling of the second term with 1 MPa?1 satis?es commensurability. 3. Application 3.1. Materials and test conditions As an application for the new lifetime prediction model, LCF and TMF test results from [34] are considered. These tests are performed on each of the nodular (GJS-700), vermicular (GJV-450) and lamellar (GJL-250) cast iron alloys (Table 2) using the specimen geometry shown in Fig. 6. The isothermal LCF tests are strain controlled with a frequency of 5 Hz. The mechanical strain amplitude varies from 0.1% to 0.5% such that the lifetime range is between 102 and 105 cycles, while the loading is kept symmetric. Loading temperatures ranges from room temperature (RT) to 500 °C. The specimens are heated without the application of external force and held at the desired temperatures for 60 s to ensure a uniform pro?le. For the anisothermal TMF tests the minimum temperature is ?xed to 50 °C while the maximum temperature varies between 375 °C and 475 °C. The heating rate is 10 K/s and achieved by inductive heating with a power of 5 KW. The cooling is performed by heat transfer throughout the cooled specimen mounting points with a target cooling rate of 10 K/s. The temperatures are controlled by the JUMO digital controller and measured by a Ni–Cr– Ni thermoelement located at the center of the specimen. Strains are measured using a capacitive strain gauge attached to the specimen. Mechanical to thermal strain ratios used are ?100%, ?120% and ?140%. Hold times of 60 s at the maximum temperature are also considered. Both isothermal and anisothermal tests have been performed on a servo hydraulic SCHENK test bench with a maximum force of 100 KN. The combined test conditions and the number of tests are shown in Tables 3 and 4. For further inquiries regarding test procedures and specimen speci?cations refer to [34]. 3.2. Results and discussions The damage induced in the material due to the loading described in the previous section is to be estimated using each of

the criteria plastic strain (Cof?n), total strain (Basquin–Cof?n), SWT (Smith–Watson–Topper), dissipated energy, and the current criteria stated in (4). The response of the material at half lifetime (Nf/2) is considered as stabilized and representative of the life. The plastic strain amplitude, total strain amplitude, maximum stress and mean stress are read directly from the measured stabilized hysteresis curve, while the plastic dissipated energy is calculated by integrating over the cycle:

DW p ?

Z

cycle

r ? de

?5?

A melting temperature (Tmelt) of 1200 °C and a thermal damage activation temperature of 350 °C, which corresponds to 0.425 ? Tmelt (mean value of the reputable range), are assumed for all materials. The material strength speci?c parameters, Wp,u and rult are obtained from monotonic tensile tests at different temperatures. The lifetime correlation curves using each of the above mentioned criterion and for each of the GJS-700, GJV-450, and GJL250 are shown respectively in Figs. 8–10. These ?gures contain also charts showing the improvement made when the classical energy approach is only corrected by the stress terms before including the thermal damage term (Figs. 8–10e). In order to compare the different approaches, two criteria should be considered: 1. The goodness of the ?t of the model considering all tests 2. The discrepancy between the LCF and LCF/TMF identi?ed trend lines The ?rst criterion is quanti?ed using the coef?cient of determination R2 which is calculated according to:

P ?y ? y0i ?2 R2 ? 1 ? P i ?yi ? y?2

?6?

where yi are measured damage parameters, y0i are calculated damage parameters, y is the mean of the measured damage parameters. The closer are the R2 values to 1, the better is the linear correlation between damage parameter and life on a log–log scale. Note that for the total strain approach, a linear regression is assumed for simpli?cation purposes although the model is non-linear. The comparison would remain valid since the nonlinearity of the total strain approach in the investigated lifetime range is not high. The second criterion is quanti?ed using a sum of the square of the normalized difference between the projection of the life value on the LCF trend line and the projection of the life value on the LCF/ TMF trend line:

Table 2 Material composition: elements as a percentage of weight. Material GJL-250 GJV-450 GJS-700 C 3.460 3.550 3.870 Si 2.030 2.190 1.600 Mn 0.650 0.410 0.510 P 0.030 0.220 0.030 S 0.100 0.010 0.003 Cr 0.244 0.034 0.860 Ni 0.116 0.036 0949 Mo 0.019 0.021 0.010 Cu 0.350 0.878 0.870 Al 0.002 0.003 0.007 Ti 0.024 0.003 0.008

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Table 3 Isothermal loading conditions and number of specimens. Material Temperature (°C) Frequency (Hz) Strain amplitude [%] 0.1 GJS-700 RT 200 350 500 RT 200 350 500 RT 200 350 500 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 2 0.15 0.2 0.25 0.3 2 2 2 2 2 2 2 3 2 1 3 2 1 1 1 0.4 1 2 2 05 2 2 2

2 2 2 3 2 2 2 2 2

GJV-450

GJL-250

Table 4 Thermomechanical loading conditions and number of specimens. Material Temperature range (°C) Heating rate (K s?1) Restraint ratio 100 Hold time at Tmax (s) 0 GJS-700 50–400 50–425 50–450 50–475 50–400 50–425 50–450 50–350 50–375 50–400 50–425 10 10 10 10 10 10 10 10 10 10 10 3 2 3 2 2 3 2 2 2 2 60 2 2 1 2 1 2 3 3 2 2 2 2 0 60 0 60 120 140

GJV-450

GJL-250

TMF=LCF ? yLCF 1 X yi i SS ? TMF n yi =LCF

!2 ? 7?

TMF=LCF

where n is the total number of tests, yi is the damage parameter calculated from the trend line through TMF and LCF test data,

yLCF is the damage parameter calculated from the trend line through i LCF test data. This quantity (SS) shows how accurate can the lifetime of the combined LCF/TMF tests be predicted using only information from LCF tests.

Fig. 7. Calibration of m and n at the maximum coef?cient of determination (R2) value (GJS-700 example).

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Fig. 8. Comparison between different lifetime prediction criteria for the GJS-700 spherical cast iron. Circular and polygon shapes represent respectively isothermal LCF and TMF test data. Blurry and solid lines are the resulting ?t to respectively the LCF tests alone and the LCF/TMF tests combined. The SS values are measures of discrepancy between these two ?ts while the R2 values are measures of the agreement of the criteria to all measurements.

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Fig. 9. Comparison between different lifetime prediction criteria for the GJV-450 spherical cast iron. Circular and polygon shapes represent respectively isothermal LCF and TMF test data. Blurry and solid lines are the resulting ?t to respectively the LCF tests alone and the LCF/TMF tests combined. The SS values are measures of discrepancy between these two ?ts while the R2 values are measures of the agreement of the criteria to all measurements.

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Fig. 10. Comparison between different lifetime prediction criteria for the GJL-250 spherical cast iron. Circular and polygon shapes represent respectively isothermal LCF and TMF test data. Blurry and solid lines are the resulting ?t to respectively the LCF tests alone and the LCF/TMF tests combined. The SS values are measures of discrepancy between these two ?ts while the R2 values are measures of the agreement of the criteria to all measurements.

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The model parameters were identi?ed by regression, once using the LCF data alone and once using both of the LCF and TMF data: the parameters n and m control the coef?cient of determination (R2) of the model and are identi?ed by solving the optimization problem of maximizing the value R2 (minimizing – R2) using a quasi-Newton method Fig. 7. A and c are next identi?ed (keeping m and n ?xed) by solving the linear regression problem of ?tting a line (in logarithmic domain) into the DTMF-LCF/OP-Nf scatter using the least square method. The identi?ed coef?cients and exponents for each criterion are summarized in Tables 5 and 6. For the GJS-700, the new approach showed superior to each of the plastic strain, total strain, SWT and plastic strain energy approaches as shown in Fig. 8. From the correlation point of view and compared to the plastic strain energy approach alone (R2 = 0.4923, Fig. 8d), achieved are slight improvements when using the corrected version (R2 = 0.5836, Fig. 8e) taking stress effects into account, and considerable improvements reaching 83% when using the ?nal developed criteria taking into account thermal effects additionally (R2 = 0.9023, Fig. 8f). These effects are lacking in the referred classical approaches resulting in a large scatter. Considering the agreement between LCF and LCF–TMF trend lines, the SWT approach shows an excellent match (SS = 0.0018, Fig. 8c). The total strain along with the new criterion show acceptable agreements, while the plastic strain and plastic strain energy criterion lead to a discrepancy error of one order higher (compared to the latter) as can be inferred by comparing the SS values in Fig. 8. Even though the SWT show an excellent LCF and LCF–TMF trend line convergence, the lifetime correlation is extremely worsened by the high temperature (500 °C) isothermal LCF tests where the life under these conditions is greatly overestimated (factor > 10). The reason is due to the decrease of the maximum stress with increasing temperature. Still, the use of the SWT parameter would be very reliable and simple when the application temperature is lower than 350 °C. The new modi?ed energy approach manifests itself by far as the most suitable for the common LCF–TMF lifetime prediction, as it results in the highest correlation values (R2 = 0.9023) and a very good LCF and LCF–TMF trend lines convergence (SS = 0.0339) as summarized in Fig. 8f. For the GJV-450, improvement in predictions compared to all considered classical approaches are achieved (Fig. 9). From correlation point of view the total strain (R2 = 0.6478) and the SWT (R2 = 0.5892) approaches show comparable relatively bad values as can be deduced respectively from Figs. 9b and c. For the plastic strain and plastic strain energy approaches, the damage temperature dependency is less obvious since the LCF test data (Figs. 9a and d) do not show a wide trend line separation such as shown

for the GJS-700 material (Fig. 8a and d). As a result, lifetime predictions are acceptable using the plastic strain and plastic strain energy approaches. The new modi?ed energy approach reveals nevertheless the best prediction capability (R2 = 0.9411, Fig. 9f) with a 9.36% improvement in correlation compared to its classical energy counterpart approach (Fig. 9d). Note that, the improvements are mainly achieved by the temperature term as the stress correction alone did not lead to considerable positive effects (1.8%, Fig. 9e) like in the case for GJS-700 (18.5%, Fig. 8e). This is justi?ed by the fact that the considered GJV-450 specimens have partially a ferritic matrix which reduces the sensitivity to the mean stress. Considering the agreement between LCF and LCF–TMF trend lines, the SWT shows the worst match (SS = 0.0164, Fig. 9c) which, considering also its correlation value, makes it the worst candidate for this material. The total strain approach shows the best match (SS = 0.0002, Fig. 9b) but with its bad correlation, it is not a favorable approach. The plastic strain (Fig. 9a), plastic strain energy (Fig. 9d) and the current approach (Fig. 9f) show small SS values of the same order. As a result, given again its higher prediction and comparable match between LCF and LCF–TMF trend lines the proposed approach remains favorable (Fig. 9f). For the GJL-250 improvement in lifetime predictions are still realized but not considerable as before (Fig. 10). Unlike the previous two cast irons, GJL-250 does not show a temperature dependency of the lifetime curves which can be deduced by observing for the classical approaches in Fig. 10, how the LCF test data with different temperatures converge to one trend line. From correlation point of view, the plastic strain approach (Fig. 10a) shows the best ?t (R2 = 0.8549), followed in this order by the energy (Fig. 10d, R2 = 0.7863), the SWT (Fig. 10c, R2 = 0.7646), and the total strain (Fig. 10b, R2 = 0.7496) approaches. The current criterion (Fig. 10f) shows improvement (5.5%) in comparison to the classical energy approach (Fig. 10d), mainly due to the stress correction factor (Fig. 10e) while the addition of the thermal term have negligible in?uence (Fig. 10f). This again correlates with the fact that GJL250 has a dominant pearlitic matrix (high stress in?uence), and the fact that its strength does not vary over the range of considered temperatures. Considering the agreement between LCF and LCF– TMF trend lines, the plastic strain approach (Fig. 10a) shows the best SS values (0.0006) while the remaining approaches show similar values. It is important to mention that the measured TMF test data for this material exhibit stress–strain hysterisis loops with slightly intersecting heating–cooling curves near the tips (near the maximum and minimum stress), which will result in an unrealistic negative energy increments during the cycle. As GJL-250 is

Table 5 Identi?ed material coef?cients and exponents for the different approaches considered. Criterion parameter GJS-700 GJV-450 GJL-250 Cof?n Basquin SWT Plastic strain energy DWp 157.54Nf?0.670 92.028Nf?0.755 5.705Nf?0.652 DTMF-LCF (4)

OP

ea,p

0.068Nf?0.590 0.055Nf?0.666 0.005Nf?0.542

ea,e

0.002Nf?0.017 0.002Nf?0.032 0.005Nf?0.155

rmax?ea,t

7.65Nf?0.215 3.88Nf?0.214 1.82Nf?0.296

103.560Nf?0.902 153.210Nf?1.160 2.799Nf?0.723

Table 6 Material properties with identi?ed mean stress sensitivity (m) and thermal damage exponent (n). E in GPa 20–500 °C GJS-700 GJV-450 GJL-250

a

rult in MPa 20–500 °C

680–485 500–300 250–200a

Wp,u in MPa 20–500 °C 24.3–22.1 15.4–10.7 1.3–1.4a

n 4.28 5.84 8.92

m 1.94 1.42 2.01

173–137 163–130 130–113a

For GJL-250 maximum temperature was 450 °C.

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brittle exhibiting small dissipated energy per cycle, such a measurement noise affects, for this material, the prediction quality of any energy based approach. The Cof?n approach ends up to be the most suitable since the data points needed to measure the plastic strain amplitude are far from the corrupted measurements region. To conclude, the proposed approach led to higher lifetime correlations (up to 83% improvement for GJS-700) while the lifetime trend lines of the LCF tests are consistent with those of LCF and TMF tests together. With such a prediction quality, future test programs given similar materials and loading conditions may by conducted using isothermal LCF tests (0.1–5 Hz) till failure for the lifetime model calibration, and TMF tests (0.005–0.02 Hz with 5– 10 K/s heating and cooling rate) till stabilization for the material constitutive model calibration and/or validation. By doing so, the costly TMF tests can be limited to approximately 20 cycles, i.e., until stabilization occurs, and the lifetime test can be accelerated under isothermal conditions, with specimen geometry (e.g., v-shaped) being another variable to further accelerate the LCF failure. A potential cost saving is thus evident with the proposed approach. A remaining issue is the evaluation of the stress term involved in the new equation given a multiaxial loading condition. The tests considered in this work were performed with a uniaxial loading and the stresses were directly measured in the dominant loading direction. A dominant direction relevant to thermomechanical fatigue can be de?ned in a multiaxial case using the critical cutting plane approach. Multiaxial LCF–TMF tests need to be performed in order to validate the above proposition. 4. Summary The aim of this work was to present a new lifetime prediction methodology capable of predicting life under general LCF–TMF loading while calibration is performed using merely LCF test data. A conditioning of the dissipated plastic strain energy approach is sought by accounting for stress and high temperature damage effects. The result is an empirical formula with four adjustable parameters. Data from tests performed on three different cast iron materials have been used, and the new approach proved to be superior to all classical ones. Higher lifetime correlations (up to 83% improvement) are achieved while the lifetime trend lines of the LCF tests are consistent with those of LCF and TMF tests together. With such a prediction quality, future test programs with signi?cantly less costs can be conducted by using TMF tests till stabilization for the material constitutive model calibration and/or validation and isothermal LCF tests till failure for the lifetime model calibration, given materials and loading conditions similar to those in this work. The generalization of damage parameter to multiaxial case via critical plane approach remains invalidated due to lack of multiaxial test data. References

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