load sequence analysis in fatigue life prediction - tcsme.org · 1.introduction the assessment of...

LOAD SEQUENCE ANALYSIS IN FATIGUE LIFE PREDICTION

Moises Jimenez1,2, Jose Martinez2 and Ulises Figueroa31Technical Development, Volkswagen de México, México-Puebla, México

2Postgraduate Section, Mechanical Engineering, Instituto Politecnico Nacional, Distrito Federal, México3School of Mechanics, Material Science and Mechanical Engineering, ITESM-CEM, México

E-mail: [email protected]

Received July 2014, Accepted July 2015No. 14-CSME-79, E.I.C. Accession 3740

ABSTRACTIn this work, the load sequence effect is analyzed in fatigue test. One of the assumptions of the Miner’s ruleis that the total damage is equal to the sum of the damages absorbed; however, different models have beenproposed to take the effect of the load sequences under two load levels into account. To analyze this effect,a case study of a rear axle mounting bracket has been performed, analyzing six different sequences of threeload levels, defined as Low, Medium and High. A Finite Element Analysis was also performed using MSCTools. With these results and a series of test at constant amplitude, the component S-N curve was made. 24tests at room temperature were performed in order to evaluate the damage process. It was found that, undera block of three load levels, the sequence of each block has an effect in the total amount of damage underthe same number of cycles. With this information it is possible to improve the life prediction through themodification of the damage rule. The proposed model uses a factor which depends on the ultimate strengthand yield point. This is an advantage over other approaches, as the other models need additional dynamictests to obtain coefficients to perform the life prediction.

Keywords: fatigue test; load sequence; finite element analysis; damage rule.

ANALYSE DE CHARGE DANS LA FATIGUE SECUENCE PRÉVISION

RÉSUMÉDans ce travail, l’effet de la séquence de chargement est analysé à l’aide d’un test de fatigue. L’une dessuppositions de la règle de Miner est que le dommage total est égal à la somme des dommages absorbés ;cependant, différents modèles ont été proposés pour prendre l’effet des séquences de charge de moins dedeux niveaux de charge en compte. Pour analyser cet effet, une étude de cas d’un axe support de montage àl’arrière a été réalisée, analysant six séquences différentes de trois niveaux de charge, défini comme faible,moyen et élevé. Une analyse par éléments finis a également été réalisée en utilisant Outils de MSC. Avecces résultats et une série d’essais à amplitude constante, la courbe SN a été effectuée. Vingt-quatre tests àtempérature ambiante ont été réalisés afin d’évaluer le processus de dégâts. Il a été trouvé que, en vertu d’unbloc de trois niveaux de charge, la séquence de chaque bloc a un effet sur la quantité totale des dommagesdans le même nombre de cycles. Avec cette information, il est possible d’améliorer la prédiction de la vieutile à travers la modification de la règle de dommages. Le modèle proposé utilise un facteur qui dépend dela résistance à la rupture et le point de rendement. Ceci est un avantage par rapport à d’autres approches,car les autres modèles doivent inclure des essais dynamiques supplémentaires pour obtenir des coefficientspour effectuer la prédiction de vie.

Mots-clés : test de fatigue; la séquence de charge; analyse par éléments finis; règle de dommages.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 4, 2015 819

1. INTRODUCTION

The assessment of the fatigue life of components and structures is essential to prevent failures in the productlife, which is determined by the interaction between the strength of the component and its mission profile[1]. Fatigue damage increases with applied cycles in a cumulative manner [2]. Load profiles are based onthe load spectrum through a cycle counting method. The load profile defined can be whether of constantamplitude, or of variable amplitude; in both cases, the loads are compared with S-N curve through a damagerule. The linear damage rule is the most used damage mode, however, different authors reported that the lifeprediction can be nonconservative. The main point in this discussion is that in this model it is consideredthat the specimen under a sequence of load oscillations is equal to the sum of damages absorbed in eachoscillation during the load sequence.

It must be considered that the load interaction effect has led to propose different approaches, like themodel of Corten–Dolon and the Freudenthal–Heller approach. Both theories are based on the modificationof the S-N diagram, which is a clockwise rotation of the original S-N line around a reference point.

Another theory is the two-stage lineal damage rule, in which the fatigue is split into two processes: crackinitiation and crack propagation. Buic-Quoc performed a modification to his rule to take the load sequencesinto account. One of these models is a fiction load approach and the other is the ratio cycle modification.Gladskyi and Fatemi [3] have evaluated the effect of the sequence for axial and torsional loads. Harbouret al. [4] have analyzed the effects of variable amplitude loading of rubber specimens based on fatigue liferesults from experiments. Rognin et al. [5] have performed an analysis of the effect between load High-Low(H-L) and Low-High (L-H) for composites materials concluding that the change in the load sequence hasan effect in the damage amount, being < 1 for a sequence L-H, while for a sequence H-L is > 1. Tateishiet al. [6] proposed that the Miner’s life prediction for a Low cycle fatigue is greater than the experimentalshowing a nonconservative estimation. Yin et al. [7] concluded that, for the test with steps H-L, the sumof the damage relation is lesser than the one showing nonconservative predictions for this kind of loads.Ben-Amoz [8] proposed an accumulative damage model based on two boundaries for fatigue, concludingthat the load cycles H-L and L-H imply two different damage accumulation rules. Ghammuri et al. [9]based on the main fissure length, proposed that under a load sequence L-H, the Miner damage rule can beused to predict the fatigue residual life. There is still a great discrepancy between the experimental resultsand the prediction of the Miner’s rule for a load sequence H-L. On the other hand, Rejovitzky and Altus[10] demonstrated that, in fatigue tests of two levels, the fatigue life is longer for a load sequence L-H andshorter for a load sequence H-L. Dattoma et al. [11], proposed a model for a life prediction based on anonlinear continuum damage model, where the parameters of fatigue limit and the parameter of the modelare experimentally determined. Memom et al. [12] have evaluated the prediction of fatigue lives in elasto-plastic range using a load sequence H-L and L-H, concluding that there is a direct effect of loading sequenceon fatigue life. The experience shows that only in some cases and for some materials, these theories showgood correlation with experimental results. However, the coefficients have to be calculated for different loadconditions, limiting their application in engineering [13].

To evaluate the load sequence effects, the models had been proposed for two load levels. In addition tothis, it is necessary to perform additional dynamic tests to define the parameters of the models. In this work,a new damage model based on the modification of the Miner’s rule is proposed, which includes a factor totake the load sequence effects into account, results of different experiments defined a factor, which dependson the ultimate strength and yield point. The work flow is shown in Fig. 1.

820 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 4, 2015

Fig. 1. Work flow.

Fig. 2. Boundary conditions.

2. FINITE ELEMENT SIMULATION

A finite element simulation is performed in order to know the stress result of applying different loads onthe mounting bracket. The first step is the extraction of the midsurface and clean geometry. The boundaryconditions are defined applying a load in the central point of the rubber on the plane XZ and the spatialrestriction fixes the part on the four fastener positions using a spider (rigid elements) to connect the nodewith the mesh (Fig. 2).


Table 1. Finite element characteristics.Element size (mm) Shell elements

2 15,8223 7,1014 4,2125 2,7906 1,975

Table 2. Chemical composition in %.C max Si Max Mn max P max S max Al min Nb max Ti max V max0.120 0.500 1.600 0.025 0.015 0.015 0.090 0.150 0.200

Fig. 3. Stresses result.

The finite element model was built with shell elements (2D) and 1D for the rigid; to define the meshelements, different element sizes have been analyzed. Table 1 shows the characteristics of the model toperform a convergence analysis.

The material used is a steel S420MC and its mechanical properties are: Young’s modulus = 210,000 MPa,Poisson’s coefficient µ = 0.3, yield point = 467 MPa and ultimate tensile strength = 527 MPa [14]. Table 2shows the chemical composition of the material.

The size element to be used is 4 mm, as a result of its convergence. Stresses result of applying a force of14 kN, shown in Fig. 3. The maximum value is around the simulated washer of the fastener. Additional anal-yses were performed at different load levels for 10, 12 and 16, obtaining the following maximum stresses:151, 192 and 252 MPa respectively.

3. EXPERIMENTAL TEST

Fatigue tests are performed in different conditions. The test stand is shown in Fig. 4 and its main componentsare the device to fix the mounting bracket, the hydraulic actuator of ±125 mm, a load cell of 25 kN and thetemperature chamber.


Fig. 4. Set up for fatigue test.

Fig. 5. Result of test at constant amplitude.

Table 3. Cycles at each load level.16 kN 14 kN 12 kN

Nmedia 340639 580548 1348625Nadjusted 340650 580550 1349000

The first step is building a component S-N curve; to do so, 12 tests were performed at 10, 12, 14 and16 kN. The mean value was founded using Eq. (1). The results and the mean values are shown in Fig. 5.

µ =n

∑i=1

ni

Ni(1)

To evaluate the effect of the sequence loads, test were performed at different loads applying a portion ofdamage of 0.25 for 16 and 12 kN, and 0.5 portion of damage for 14 kN. With these information, the loads andthe cycles are Low 16 kN (85,163 cycles), Medium 14 kN (290,275 cycles) and High 12 kN (337,250 cycles)and the sequences are L-M-H, L-H-M, M-L-H, M-H-L, H-L-M and H-M-L.

All the fissures in the mounting bracket are in similar position, and the main difference is their length(Fig. 6).


Fig. 6. Position of fissure in mounting bracket.

Fig. 7. Results of sequence loads.

Table 4. Damage vs. cycles.16 kN 14 kN 12 kN ∑

D(Damage) 0.25 0.5 0.25 1N(Cycles) 85,163 290,275 337,250 712,688

The results of the length of fissures of these sequences are shown in Fig. 7. The evaluation of everysequence, eliminating the greater result of sequence 4 and the lesser result of sequence 5 for its scatterresults, is summarized in Table 5.

To evaluate the temperature effect on a load sequence of three load levels, additional tests have beenperformed at 35 and 45◦C, the results are shown in Figs. 8(a) and (b), respectively. In order to have auniform temperature in the specimen, previous to the test the temperature was applied by two hours.

In sequence 5 at 35◦C and sequence 6 at 45◦C the length of one specimen is larger than the others resultsat the same characteristics, however, the same sequences have results with a small length, reducing theaverage of the length of the fissure.


Table 5. Damage results for each sequence.Sequence Damage

1 L-M-H (12+14+16) kN < 12 L-H-M (12+16+14) kN 13 M-L-H (14+12+16) kN 14 M-H-L (14+16+12) kN < 15 H-L-M (16+12+14) kN > 16 H-M-L (16+14+12) kN � 1

Fig. 8. Results of sequence loads with temperature, (a) 35◦C and (b) 45◦C.

4. DAMAGE RULE

The assumptions for the Miner’s rule may be summarized as follows:

(a) The amount of damage absorbed by the material in any oscillation is determined only by the loadduring that oscillation.

(b) Each specimen can absorb the same amount of damage, and when that amount is attained, failureoccurs.

(c) The total damage absorbed by the specimen under a sequence of load oscillations is equal to the sumof damages absorbed in each oscillation during the sequence.

We introduce the following notation: W is the total damage at failure; wi is the amount of damage duringone repetition of load oscillation li; and Ni is the number of oscillations to failure under repeated applicationof the same load li, where there are only i = 1, . . . ,k possible loads considered. Under the assumptiondescribed, it is possible to define

W = NiWi for each i = 1, . . . ,k. (2)


Table 6. Damage with the proposed model.Sequence Damage

1 L-M-H (12+14+16) kN 0.982 L-H-M (12+16+14) kN 13 M-L-H (14+12+16) kN 14 M-H-L (14+16+12) kN 0.985 H-L-M (16+12+14) kN 1.019666 H-M-L (16+14+12) kN 1.0397

If a sequence of load is applied, which is different from oscillation to oscillation, and failure occurs afterload li was applied in ni and l2 in n2 and l3 in n3.

k

∑i

niwi =W. (3)

From Eqs. (2) and (3)k

∑i

ni

Ni= 1. (4)

However, according to the experimental results, the damage is not the same for all sequences. To improvethe life prediction, a new damage rule is proposed to take the load sequence effects into account under thenext assumption:

The effect of the load sequence has influence on the life prediction due to the change in a load block,modifying the damage process.

Based on the fact that the life prediction is based on the regression of this curve and taking into account theprevious work of the author, in which it has been reported that the ultimate strength can be used to estimatethe points S1000, Se in an S-N curve [15], the ultimate strength (Us) and the yield point (Ys) are used to finda parameter a for the factor.

a =logUs

logYs. (5)

Considering that the factor of Eq. (5) depends of the load sequence, we have that

f =

1 l1 > l2 < l3(MLH), l1 < l2 > l3(LHM),1/a l1 < l2 < l3(LMH), l1 < l2 > l3(MHL),a l1 > l2 < l3(HLM),a2 l1 > l2 > l3(HML).

(6)

Equation (2) is modified for the factor of Eq. (6), which depends on the sequence load

k

∑i

ni

Ni( f ) =W. (7)

For this analysis a = 1.0196. In Table 6, the damage with the proposed model is summarized.


5. CONCLUSIONS

According to the damage estimated with the proposed model (7) and the experimental results, the followingassumptions can be stated:

• It is believed that the level of each load in a block test has influence in the damage process.

• When the medium load is the first load, the result is approximately equal to one.

• When the low load is the first load, the result is less than one.

• When the high load level is the first in the load sequence, the result is greater than one.

• In a three-load level sequence test, it is believed that the first level of load has the highest influence onthe fatigue process, as the maximum degradation of the material starts in this stage.

• The effect of temperature increases the damage process; based on the analysis of the fatigue life resultsfrom these experiments is concluded that the effect of the loading sequence is more pronounced atroom that at elevated temperature, as proved in [16] for a sequence of two load levels.

• Finite element softwares like Patran/Nastran and MSC Fatigue are powerful tools, but they have to beapplied in the right way, calibrating the model. Their results have to be correlated with experimentaltests in order to give confidence to the final release of a design or mechanical evaluation.

ACKNOWLEDGEMENTS

The first author (JM) would like to acknowledge the support of the Mexican National Council for Scienceand Technology (CONACYT) for the scholarship for doctoral studies. He also thanks the MSC SoftwareCorporation for the grant of a Research Assist Program in the second quarter of 2012. Finally he is gratefulfor the use of the durability test area of Technical Development of Volkswagen, Mexico, and for their supportof this study.

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