Multiaxial fatigue criterion for polypropylene – Automotive applications

International Journal of Fatigue 32 (2010) 1389–1392

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Technical note

Multiaxial fatigue criterion for polypropylene – Automotive applications A. Berrehili a, Y. Nadot a,*, S. Castagnet a, J.C. Grandidier a, C. Dumas b a b

Laboratoire de Mechanique et de Physique des Materiaux, LMPM, ENSMA, 1 Avenue Clément Ader, BP 40109-86961 Futuroscope Chasseneuil, France RENAULT S.A.S, 1 Avenue du Golf, 78288 Guyancourt Cedex, France

a r t i c l e

i n f o

Article history: Received 24 February 2009 Received in revised form 18 September 2009 Accepted 19 January 2010 Available online 25 January 2010

a b s t r a c t Multiaxial fatigue behaviour of polypropylene pipes is investigated under tension and torsion loading with or without mean stress. Fatigue limit is experimentally determined and compared to self heating curve method. A multiaxial fatigue criterion is proposed and shows that the fatigue behaviour of this semi-crystalline polymer seems to be governed by the von Mises maximum stress. Ó 2010 Elsevier Ltd. All rights reserved.

Keywords: Polymers Self heating curve High-cycle fatigue

1. Introduction The fatigue behaviour of semi-crystalline polymers is crucial for the design of polymer-based structural applications. Such materials exhibit a complex behaviour due to time-dependent mechanisms and thermomechanical coupling. Scientific literature widely focused on the craze and crack propagation, and the competition with plasticity, often based on tension–compression experiments and mainly performed to compare materials [1–11]. Fatigue tests have been mainly conducted in a uniaxial framework, under stress or strain controlled sinusoidal tension–tension or tension–compression waveforms. Only a few works were carried out in the range of high-cycle fatigue and biaxial loading [12–15] or flexural [7,8,16] loading mode. So far, there are only a few studies on the multiaxial endurance fatigue behaviour of semi-crystalline thermoplastics. Multiaxial fatigue life prediction of engineering materials has been a challenging task for over past decades. Indeed, an effective multiaxial fatigue criterion is needed for practical applications since most of the load-bearing components are used under multiaxial loading conditions. Again, such criteria have not been proposed for semi-crystalline polymers. Only a few works have reported fatigue lifetime prediction in un-notched polyolefines. The tensile endurance limit at fixed mean strain and variable positive stress-ratio R was shown to be reasonably described by the applied stress amplitude in case of fatigue life thermal failure [17]. In this work, failure occurred by necking and the tensile endurance limit was defined as the stress level for fatigue life between 106 * Corresponding author. E-mail address: [email protected] (Y. Nadot). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.01.008

and 107 cycles. Under close loading conditions and in similar materials considered as elastic–plastic, the fatigue life criterion could also be formulated from the accumulated plastic strain [18]. The polymer tested in the present study is a polypropylene devoted to automotive applications. Therefore, this study is focused on multiaxial behaviour with or without mean stress. It is also proposed to compare two methods to estimate the fatigue limit: standard S–N curves and self heating method. The paper is separated into three parts: – The first part presents the experimental device. The fatigue sample is a thin extruded pipe. Special grips are developed to perform tests under tension, torsion or combined loading on the same sample (i.e. obtained from the same process, material and with a unique geometry). – The second part presents experimental results: S–N curves are depicted and analyzed for both tension and torsion loading. Self heating curves are also analyzed. – The third part is devoted to the mechanical analysis of experimental results by the mean of multiaxial fatigue criteria. Basic fatigue criteria are tested and a more suitable criterion for PP is finally proposed.

2. Experimental details 2.1. Material The material is a polypropylene (PP) for automotive applications. It is a semi-crystalline thermoplastic. Differential Scanning Calorimetry (DSC) experiments performed under N2 sweep at

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10 K/min in a TA Instrument DSC-2920 apparatus give a glass transition temperature of 10 °C, a melting temperature of 166 °C, and a degree of crystallinity about 41% (as calculated from the melting peak heat flow area divided by the 100% crystalline melting enthalpy 149 J/g [19]).

2.2. Experimental device A special experimental device has been developed for the multiaxial fatigue testing of thermoplastics. As illustrated in Fig. 1, experiments were performed on a PP provided by Renault S.A.S in the form of 56 mm external diameter and 3 mm thick extruded pipes, aimed at providing a single geometry for all multiaxial tests without microstructure differences enhanced by different processes. Multiaxial fatigue samples (total length 180 mm; gauge length 50 mm) were machined from the pipes with a very smooth curvature radius, as depicted in Fig. 1b. The internal diameter was unchanged (49.9 mm on average) all along the tubular sample. The minimal external diameter was 54.9 mm and the minimal thickness was 2.5 mm. The initial minimal cross-section area, averaged from 30 measurements (i.e. 3 diameter and thickness measurements in each of a 10 tubular samples series), was 411.5 ± 13.4 mm2. For the fatigue tests, tubular samples were mounted on special mechanical grips with the 100 mm gauge length l° distance between each other. These grips used to perform tests under tension, torsion or combined loading on the same sample. All multiaxial fatigue tests were performed at room temperature on a servo-hydraulic INSTRON 1343 biaxial testing machine. Force controlled tests with a triangular wave function were carried out in tension and torsion. Due to the viscous behaviour of the polymer, loading conditions inducing a constant strain-rate all along the test are chosen, i.e. a triangular signal. Tests have been conducted at a constant frequency 1 Hz, chosen to minimize self heating and thermal failure. Besides, an air cooling system is used to maintain the air inside the pipe at room temperature, and to minimize self heating by increasing the heat exchanges between the pipe wall and the air. The air enters in the pipe at the top of the grips, circulates inside it thanks to the pressure of injection (0.5 bar), and goes out through holes machined into the lower vice grip. Besides, vice grips are continuously cooled by a constant debit of cold water (to isolate at least the device from the important heating of the jack during the test). As for the ambient environment, we did not take particular precautions to protect the sample

(a)

(b)

of the humidity. However PP is neither known to be hydrophilic nor to be affected by water-induced degradation processes at room temperature. In the present study, temperature change on the specimen surface is continuously measured by three thermocouples type K, all of them located at the middle of the pipe. Two thermocouples are located at the inner (point A in Fig. 1) and outer surface (point B in Fig. 1) of the specimen (the contact between the thermocouple and the pipe is maintained thanks to the elasticity of the thermocouple), and the third one measures the temperature of the air inside (point O in Fig. 1) the pipe (24 ± 1 °C). The temperature is controlled within ±1 °C. The stress-ratio R (defined as the minimum stress divided by the maximum stress over the fatigue cycle) was 0 or 1. All fatigue tests for the unbroken specimens were arbitrarily stopped between 105 and 4  105 cycles. Thus, the fatigue limit (fatigue strength) corresponded to the stress level at an arbitrary number of cycles of 105 cycles. It does not mean that the fatigue limit corresponded to an asymptotic behaviour: we have not enough experimental results in the range of high-cycle fatigue to state this. Under tension (R = 1 and 0) and torsion (R = 1) the material fractured into two parts while under torsion (R = 0) it failed by buckling.

3. Fatigue test results 3.1. S–N curves under tension and torsion The usual S–N curves obtained under tension and torsion at R = 1 and 0 are presented in Fig. 2 on a linear–log scale with normalized stress values. For tests performed under tension below the fatigue limit (pipes did not fracture), no significant temperature raise is observed. For tests leading to a low number of cycles to failure, the temperature first increases by about 3 °C before reaching a nearly constant value after 300 cycles, and drastically increasing in the last 100 cycles. The fatigue limit under tension rD (amplitude of the applied stress: rD = (rmax  rmin)/2) at 1 Hz and R = 1 is arbitrary normalized to 1 for further comparison. Following this normalization, the experimental value of the tension fatigue limit at 1 Hz and R = 0 is 0.50, as illustrated in Fig. 2. This last result proves that the mean stress has a significant influence on the amplitude fatigue limit under tension. Under torsion, the evolution of the temperature is strictly the same as under tension. Pipes submitted to the lowest stress

z r 2.5

O θ

A B Thermocouple Gage section 1/2 Sample

90

49.9 56 Tension Torsion Fig. 1. Experimental device: (a) general view with contact temperature measurement by thermocouple and (b) detail of sample and grip system.

Amplitude stress (MPa)

A. Berrehili et al. / International Journal of Fatigue 32 (2010) 1389–1392 Tension, R = -1 Tension, R = 0 Torsion, R = -1

– Torsion (R = 1) SN rD = 0.58; SHC rD = 0.64. – Torsion (R = 0) SN rD = 0.33; SHC rD = 0.35.

Torsion, R = 0

1

0.58 0.5 0.33

100

1000

10000

100000

1000000

N (number of cycles to failure) Fig. 2. S–N curve under tension and torsion loading at R = 1 and 0. Normalized stress values.

amplitude do not fracture. The normalized values of the fatigue limit are 0.58 and 0.33 at R = 1 and R = 0 respectively. Like under tension, these tests reveal an influence of the mean shear stress on the fatigue limit. 3.2. Fatigue limit determinate from self heating curve Due to the low frequency of 1 Hz, tests described above are very time consuming. Additional tests based on a self heating curve [20] are also carried out to determine the fatigue limit using a single sample loaded step by step. This method has the advantage of being performed in a shorter time compared with traditional S–N curve, and has been used for metallic materials, giving interesting results [21]. We apply such a methodology to PP pipes to determine the fatigue limit under different loading conditions at R = 1 and 0. According to this method, the pipe is loaded at a given stress amplitude (below the estimated fatigue limit) for 200 cycles, and the temperature is recorded at the end of the 200-cycles stage. The same sample is further loaded at higher stress amplitude for 200 cycles more with a new temperature measurement at the end of the step. This procedure is repeated until the pipe failure. The number of cycles imposed (200) at each step is chosen arbitrarily, minor than the lower number of cycles to failure measured during S–N tests. At the end of the test, Fig. 3 can be plotted and the fatigue limit is supposed to be given by the change of evolution of DT [21]. As shown in the figure, DT remains nearly constant before a rapid increase but this changing behaviour is not so easy to determine from the curve. The fatigue limit obtained by the self heating curve (SHC) is always higher than the fatigue limit given by the S–N curve: – Tension (R = 1) SN rD = 1.00; SHC rD = 1.07. – Tension (R = 0) SN rD = 0.50; SHC rD = 0.57.

Temperature variation ΔT (°C)

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8 7 6 5 4 3 2 1 0

Tension R = -1 Torsion R = -1 Tension R = 0 Torsion R = 0

Similar results are obtained for the different load cases: the fatigue limit given by the SHC method is about 6–14% higher than the experimental SN results. This preliminary investigation on the relevance of self heating methods for this material must be completed and extended to other loading conditions. Before deeper investigation, it suggests that self heating methods would be of interest for fast estimation of the fatigue limit in such material. Nevertheless, the following points should be study in order to validate the SHC method on this material: – Measurement of the temperature (thermocouple). The measure by IR camera can be a good mean to avoid problems with onsurface measurement. – The fact that the self heating curve is not always stabilized after 200 cycles. It may be necessary to increase this duration. – This method is often used for metals (even if it is not relevant for all metals). Has it a sense for polymers subjected to other dissipative mechanisms? Is it relevant to treat this curve in such way? 4. Multiaxial fatigue criterion Various multiaxial fatigue criteria have been proposed in the past decades. They can be classified within three main groups: stress-based, strain-based and energy-based criteria. The so-called critical plane approach [22–25], which has become popular in recent years, can be regarded as special cases of stress-based or strain-based approaches, depending on the chosen parameters. These criteria have been widely tested on different metallic materials but not on semi-crystalline polymers. Initiation fatigue criteria (i.e. Sines (1) and Crossland (2)) are used to analyze fatigue limit results from the previously presented multiaxial tests.

qffiffiffiffiffiffiffi J 2;a þ aSi rH;moy 6 bSi qffiffiffiffiffiffiffi J 2;a þ aCr rH;max 6 bCr

0.3

0.5 0.6

1.07

1.4

Normalized amplitude stress (MPa)

Fig. 3. Fatigue limit determination using temperature measurements during self heating under tension and torsion at R = 1 and 0.

ð2Þ

where J2,a is the amplitude of the second invariant of the stress tensor, rH,moy the mean value of the first invariant of the stress tensor, and rH,max the maximum value of the first invariant of the stress tensor. For each criterion, parameters a and b can be calculated from two given fatigue limits. The comparison between the predicted fatigue limit and the experimental value shows that these criteria are not very good with mean stress prediction: error is up to 50%. Indeed, as shown in Fig. 2, we can note that the fatigue limit is divided roughly by 2 when we apply a mean stress. This last result proves that the mean stress has a significant influence on the amplitude fatigue limit. Thus, it could be interesting to test a criterion based on the maximum stress rather than the amplitude stress. Thus, we test the von Mises stress because monotonic tension, shear, and compression tests performed on a PP in the same temperature range are well correlated by the von Mises criterion [26]. The following criterion is proposed:

qffiffiffiffiffiffiffiffiffiffiffi J 2;max 6 bPP 0

ð1Þ

ð3Þ

J2,max is the maximum of the second invariant of the stress tensor and bPP can be calculated from one given fatigue limit, for instance with torsion test at R = 1. Thus, the criterion (3) can be validated with the torsion test at R = 0, and the tension tests at R = 0 and R = 1.

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– Tests conducted under tension and torsion at R = 1 and 0 show that the multiaxial fatigue behaviour of this PP is governed by the von Mises maximum stress. It is therefore possible to propffiffiffiffiffiffiffiffiffiffiffi pose the following criterion J2;max 6 bPP . This criterion is identified using only one fatigue limit. The validation on three other load cases gives good results (max error: 15%). Acknowledgements Authors would like to gratefully acknowledge S. Bergamo and RENAULT S.A.S for technical and financial support. References Fig. 4. Equivalent maximum stress vs. fatigue life.

The comparison between the predicted fatigue limit and the experimental value (S–N curve) shows that for loading cases without or with any mean shear stress, the proposed criterion (3) is satisfying (max error: 15%). This result is also illustrated in Fig. 4 where S–N curves are plotted using equivalent stress (3). This result shows that the maximum shear energy seems to be the first order governing parameter for multiaxial fatigue of PP. It must be underlined that further investigations are necessary to extend the current proposed criterion to more complicated proportional or no proportional loading conditions. 5. Conclusions Several conclusions can be drawn from this study on the multiaxial fatigue behaviour on PP: – Self heating curve method used to determine the fatigue limit leads to slightly non conservative estimations for PP (tension, torsion, R = 1 and 0).

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