Abstract Fatigue lifetime of HDPE structures such aspipes is recognized to show a large scatter. This studyaims to compare different statistical methods and dis-tributions, in order to give convenient modeling of ten-sile and fatigue test results of commercially availablepolyethylene compression molded sheets. The medianrank, the maximum likelihood and the Kolmogorov-Smirnov fitting are compared for the estimation ofWeibull parameters. The choice of the best distribu-tion to fit fatigue lifetime is discussed on the basis ofthe goodness-of-fit results. It is found that whether thethree-parameter distributions of Weibull and lognor-mal types are suitable for lifetime prediction, the two-parameter Weibull is more conservative for probabilis-tic fatigue design.
R. Khelif (�)LaMI—UBP & IFMA, Campus de Clermont-Ferrand,BP 265, 63175, Aubière Cedex, Francee-mail: [email protected]
R. Khelif · K. ChaouiLR3MI, Mechanical Engineering Dept., Badji MokhtarUniversity, BP 12, Annaba 23000, Algeria
A. ChateauneufLGC—UBP, Complexe Universitaire des Cézeaux,BP 206, 63174 Aubière Cedex, France
Keywords Polyethylene · Fatigue · Data scatter ·Tensile strength · Lifetime distribution
1 Introduction
Polyethylene pipes are extensively used for the trans-portation and distribution of natural gas, with over90% of the new piping installations using High Den-sity Polyethylene (HDPE). One striking examplethat highlights the importance of controlling HDPEstrength can be found in the aftermath investigations ofthe Kobe earthquake in 1995 during which many firesand explosions from damaged gas pipelines causedconsiderable damage to life and properties [1].
However, there are few indications on HDPE pipefatigue resistance especially under extreme serviceconditions. In addition to internal and external pres-sures, the HDPE gas pipelines are subjected to dete-rioration processes such as creep, fatigue damage andcrack growth. In spite of the apparently simple struc-ture, the pipe analysis leads to very specific workingconditions: (i) The pipes often work under cyclic load-ing (fluctuations in internal pressure) with a significantstress ratio; (ii) The pipes accumulate a high amount ofenergy in the fluid under pressure, which is to say inthe form of elastic strain energy. This stored energyallows for initiation and development of tube crackgrowth over important lengths in different ways: brit-tle, semi-brittle and ductile failures; and (iii) The pipes
present high probability of occurrence of defects inparticular by their manufacturing process and assem-bly techniques.
One of the most dangerous ruptures that a pipe mayundergo is the “stress cracking”. In practice, poly-ethylene pipes do not support static loading exclu-sively, but they are also subjected to periodic andcyclic actions of constant, random or intermittent fre-quencies, sometimes combined with creep. Hence, fa-tigue strength must be taken into account in any designapproach.
In general, fatigue can be defined as a phenomenonthat takes place in components and structures sub-jected to time-varying external loading and that man-ifests itself in the deterioration of the material abilityto carry an applied load that is well below the elasticlimit. Today, the fatigue phenomenon is known to orig-inate in the local damage of the material or, in otherwords, in the sliding of atomic layers. As indicated byexperimental observations, at low stress levels (high-cycle fatigue), the crack initiation period may con-sume a significant percentage of the usable fatigue life,whereas at high stress amplitude (low-cycle fatigue),fatigue cracks start to develop in the early cycles[2–4].
Plastic pipes used for gas and water distributionare continuing to be the subject of many studies thathighlight various behavior aspects in terms of servicelifetime [5]. In order to allow for engineering designand lifetime prediction, many tests of long-term be-havior such as fatigue, creep and environmental stresscracking (ESC) are reported although this informationis still not sufficiently developed in the literature. Theaim of dynamic fatigue tests is to quantitatively definethe endurance limit of polyethylene resins subjectedto periodic loading; the endurance limit is defined asthe maximum amplitude which can be supported foran infinite number of cycles without apparent rupture.Long-term testing methods differ mainly according tothe type, the nature and the loading scheme.
To describe the fatigue strength, Wöhler curves arethe most common model for practical use and design.The log-scale leads to a straight line between the stressamplitude and the number of cycles to failure. Whencrack is initiated, the analysis of crack growth historyshows that there is a log-linear relationship betweenthe apparent crack size and the lifetime. Also, there isa linear relationship between the crack growth rate andthe crack length or the stress intensity factor range inmode I (�KI ) when plotted in a log–log scale.
In the literature, many works have been undertakenon fatigue crack growth. Accelerated fatigue testswere performed on circumferentially notched HDPEbars [6–8]. Kasakevich et al. [9] presented a compar-ative study on crack propagation in HDPE under fa-tigue and creep loading conditions. This analysis isfocused on the interaction between creep and fatigueand discusses the validity of fatigue as an acceleratedlaboratory test for long-term field failure under creepconditions.
The correlation between creep and fatiguestrengths, undertaken by Parsons et al. [10], showedthat MDPE was much more creep resistant thanHDPE, but MDPE pipes was much more sensitive tostrain rate in fatigue. The mechanisms of failure ofexternally notched sections of polyethylene pipe un-der pressure have been investigated by Reynolds etal. [11], who confirmed that the origin of branchingfracture features lies in the localized shear deforma-tion.
An experimental investigation was designed by Ki-ass et al. [12] to establish the distribution of mechan-ical properties throughout a HDPE gas pipe wall. Thevariability of mechanical properties within the pipewall revealed the complexity of hierarchical structurein HDPE and such an approach is intended to con-tribute in understanding pipe long-term behavior andassociated brittle failure.
Unfortunately, the large majority of HDPE fatiguestudies concerns the crack propagation, which as-sumes that stress concentration and crack already ex-ists. However, in most of pipeline parts, fatigue mayoccur without any previously formed crack. In thiscase, fatigue produces accumulated material damage,which latterly leads either to crack nucleation or tolarge deformation, depending on the stress levels. Infact, the total lifetime is the sum of both damage andpropagation stages; in many cases the fatigue dam-age represents more than 90% of the overall fatiguelifetime. For moderate and low stress variations, thedamage time is much larger than the crack propaga-tion time. That is why, it is very important to considerfatigue of uncracked specimens. Due to strong ran-domness of the fatigue behavior, the statistical char-acterization of fatigue lifetime is an important issue inthe literature, where the final goal concerns structuralsafety problems. The analysis of fatigue data aims todetermine the best statistical distribution and parame-ters of the relevant variables, allowing to better fit theexperimental observations.
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The most well-known and popular distribution isthe 3-parameter Weibull distribution function. Severalworks have been carried out to characterize the life-time distribution for metal structures, especially steeland aluminum. Schijve [4] compared three distribu-tion functions for metal fatigue structures, with atten-tion to very low failure probabilities. He discussedthe different practical problems for which statisticsare important and the necessity to perform full-scaleservice-simulation tests for realistic prediction. Tobiaset al. [13] proposed the use of the Maximum Likeli-hood Method and a Bayesian analysis for the Weibullparameters and the corresponding confidence interval.Wu et al. [14] discussed the scatter of fatigue crackgrowth for aluminum alloys, on the basis of new testsconfirming that material inhomogeneity is the originof randomness. Unfortunately, no equivalent works areactually available for HDPE fatigue statistical distrib-utions.
In this paper, a statistical study is carried out on fa-tigue lifetime of HDPE, where different probabilisticdistribution models have been compared. The speci-men manufacturing scatter is also considered in thetest procedure. After characterizing the material ten-sile strength, the fatigue tests are performed in orderto define the probabilistic S-N curves for HDPE. Thestatistical analysis allows us to determine the lifetimedistribution type and parameter to be used for safe de-sign of systems.
2 Polyethylene specimens
The machining of test specimens is carried out from300×300×5 mm sheets, purchased from GoodfellowCo. of Lille (France), with the nominal characteristicsindicated in Table 1. It should be noted that the max-imum operating temperature for this resin is between55 and 120 °C. The temperature of deflection to heatis 75 °C for 0.4 MPa and 46 °C for 1.8 MPa.
The geometry of test specimens is drawn from theStandard NF IN ISO 527 (type 1BA), as indicatedin Fig. 1, with the dimensions given in Table 2. Themilling process is used for specimen machining, wherespecial care has been considered regarding the visco-plastic behavior of such a material. During machining,it is necessary to reduce the maximum warping, whichhas been produced by cutting out a series of strips of27 mm wide. The manufacturing unit is a three-axes
Table 1 Physical and mechanical properties of HDPE
Property Value
Density (kg/m3) 950
Water absorption (24 h basis) <0.01%
Thermal expansion factor (106 K−1) 100–200
Poisson’s ratio 0.46
Friction factor 0.29
Rockwelll hardness (shore) D 60–73
Elastic modulus (GPa) 0.5–1.2
Impact resistance, IZOD (J/m) 20–210
Yield stress (MPa) 15–40
Table 2 Normalized specimen dimensions
Designation Dimension (mm)
Width at the ends, w0 10
Testing width, w 4 or 5
Thickness, B 5
Small radius, R 14
Large radius, R0 25
Length of the gauged part, L 30
Initial free distance, D 58
Overall specimen length, L0 115
References length G 25
Fig. 1 Specimen shape according to ISO 527
machining center of HURON type, allowing us to cutthe specimens in the plane (x-y) with a feed rate of0.2 mm/rotation and a cutting speed of 60 m/min.
3 Fatigue testing
The MTS 858 Elastomer Test System, shown in Fig. 2,has been used for fatigue testing, where the loadingis applied by hydraulic actuators. It allows us to per-form fatigue tests in the range ±15 kN with frequen-cies up to 200 Hz. In order to describe the whole S-N
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Fig. 2 Fatigue testing machine
curve, 42 specimens have been prepared to be testedunder 6 stress levels, each with 7 tests. The HDPEfatigue life is characterized by applying a sinusoidalloading of frequency 5 Hz with the load ratio R = 0.1.Knowing that the HDPE elongation at rupture lies be-tween 500% and 1000%, the tests are carried out foran elongation over 200% of the specimens. Havingthe ultimate strength determined by static testing (i.e.33 MPa), it is possible to determine the force rangeto be applied for fatigue tests. It has been chosen toapply first a maximum force of 600 N (correspondingto a nominal stress of 24 MPa for a 25 mm2 of cross-sectional area, corresponding to 72% of the ultimatestrength), then to decrease it gradually.
As we could not automatically stop the testing ata given elongation under load control, the tests arestopped when the machine elongation span is reached.In practice, most of the specimens did not break at theend of the test; therefore, the number of cycles is de-termined for the same elongation for all specimens, inorder to compare data with the same strain levels. Evenif the specimen did not break at the specified deforma-tion, the remaining lifetime is not useful for practicaluse.
4 Results and discussion
Fatigue tests are now performed for six loading lev-els: 300, 400, 450, 500, 550 and 600 N. At each level,seven specimens have been tested. That makes 42specimens of nominal thickness of 5 mm. The use of
Fig. 3 Typical tested specimens for 550 N and 600 N
a test frequency less of equal to 5 Hz ensures that hys-terical heating of polyethylene is low and thus, there isvery little influence on the obtained lifetime.
Figure 3 shows typical deformations undergone bythe specimens. For high load amplitudes (more than500 N), the specimens break while for lower ampli-tudes, they undergo large strains without rupture; thiscould be explained by higher capacity for molecularchain reorganization under low and moderate straincycles.
Table 3 gives the test results in terms of number ofcycles at the specified elongation (defined as 200%);the data are censured at 1 million of cycles and thenominal stress is computed on the basis of the averagecross-sectional width for each loading level. It is ob-served that the average curve presents the traditionalform for thermoplastics. For example, at the load levelof 400 N, the number of cycles of 52105 correspondsto an elongation of 60 mm that is to say about 200% ofthe useful length. The elongation of 100% (i.e. 30 mm)is reached at 46710 cycles; the difference in lifetimeis about 10%. Contrary to steel, there is no horizon-tal asymptote to define the endurance limit σD . Evenbeyond 106 cycles, the curve presents a small slope,which is not surprising for thermoplastics.
The results of the statistical distribution of the 210measured widths show a mean width of 5.432 mm anda standard deviation of 0.208 mm; i.e. the coefficientof variation is 3.8%, which is conformal to practicalmanufacturing tolerances. As the fatigue lifetime isgoverned by the minimum width, the corresponding
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Table 3 Basic statistical parameters calculated starting from the 6 applied loads levels
distribution leads to a mean value of 5.225 mm with astandard deviation of 0.109 mm (i.e. 2% of coefficientof variation); it can be seen that the specified width of5 mm is satisfied as a minimum value for all speci-mens. It is noted that the nominal stresses mentionedin Table 3 correspond to the mean value of the speci-men width.
The coefficients of variation ranges from 37% to71%, illustrating the very large scatter of fatigue re-sults, contrary to static tensile tests. The scatter of thefatigue results is accepted today as an integrated phys-ical aspect of the phenomenon, whose origins dependon [15]:
– Material morphology (inclusions, structure hetero-geneity, residual stresses, etc.);
– Specimen integrity (differences in surface quality,dimensional tolerances, etc.);
– Testing conditions (specimen alignment on the ma-chine, adjustment of load and frequency, specimenheating, environment effects, etc.).
Figure 4 shows the test points in the S-N space. In thelog-scale, the data fitting is carried out for linear andpower functions, where the solutions are given by:
S = 35.6 − 1.85 ln(N) (1)
or
NS8.6 = 6.7 × 1014. (2)
As the number of tests is limited for each load level,classical parameter estimates are not well adapted, asthey are valid only for large sample size. For this rea-son, a complementary analysis is carried out to checkthe robustness of the parameter estimates as well asthe confidence intervals at the level of 95%. Moreover,
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Table 4 Parameter estimates and confidence intervals by three different methods
Fig. 5 Resampling and Bootstrap distributions for the lifetime at loading level of 600 N
the statistical analysis should take into considerationthe censored data at the loading level of 300 N, whichis carried out in this study on the basis of Bayesianlikelihood function. Two methods are applied to takeaccount for small sample estimates: the resamplingtechnique consisting of re-evaluating the estimates byremoving randomly one (or more) experiment(s) andthe Bootstrap method where a virtual distribution of10000 samples is generated from the initial sample byusing the resampling technique. The Bootstrap tech-nique has shown to give the narrowest confidence in-tervals for the given data. Moreover, the standard errorin this technique is lower than in the other methods.Table 4 shows the obtained mean and standard devia-
tion, as well as the confidence intervals for the differ-ent loading levels. The confidence intervals are glob-ally narrower for the Bootstrap method. At the loadinglevel of 600 N (i.e. 22.09 MPa), the comparison be-tween Resampling and Bootstrap distributions, illus-trated in Fig. 5, shows that the Bootstrap method givesmore regular lifetime distribution.
For reliability-based design purposes, it is neces-sary to define an appropriate probabilistic distribu-tion for fatigue lifetime. Usually, Weibull and lognor-mal distributions are proposed in the literature, but thechoice of appropriate distribution is far from being aneasy task. Three methods have been compared: linear
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Table 5 Weibull parameter estimates by linear fitting
Load (N) 300 400 450 500 550 600
Stress (MPa) 10.04 11.32 14.30 15.58 18.56 22.75
β 1.600 2.732 1.496 2.354 2.312 1.228
η 733819 60270 41304 12900 3332 2378
Table 6 Maximum likelihood estimates for Weibull distributions
Load (N) 300 400 450 500 550 600
Stress (MPa) 10.04 11.32 14.30 15.58 18.56 22.75
β 1.400 2.774 1.5744 2.0311 2.167 1.0336
η 674110 54255 33 216 12147 3015 2173
K-S test 0.177 0.154 0.100 0.162 0.120 0.201
β 0.794 0.407 0.6786 0.826 0.821 2.5984
η 382050 17381 24623 5228 1700 2747
γ 201390 35236 11667 4725 1428 −481.7
K-S test 0.166 0.213 0.124 0.145 0.175 0.170
fitting on appropriate Weibull paper, Maximum likeli-hood estimate and Kolmogorov-Smirnov fitting.
Table 5 gives the Weibull distribution parameterson the basis of the median rank method, allowing to fita straight line in appropriate logarithmic scales. Themedian rank is computed by:
F̂i = ri − 0.3
n + 0.4(3)
where ri is the rank of the number of cycles (Ni) and n
is the total number of data (i.e. n = 7 in our case). Thepoints are then plotted as: ln(Ni) versus ln(ln( 1
1−F̂ i)).
The fitting of these points leads to the Weibull para-meters. Whether the parameter η has decreasing evo-lution, the shape parameter β does not follow a spe-cific monotonic evolution (it could be considered to bemonotonic decreasing unless for loads 300 and 450 N,which is due to the large scatter of the fatigue data). Itis noted that the stresses mentioned in Tables 5–7 cor-respond to the minimum value of the specimen width.
Table 6 indicates the maximum likelihood esti-mates for two- and three-parameter Weibull distrib-utions. For the two-parameter Weibull, the estimatesare more or less close to those obtained by the lin-ear fitting model; the shape parameter β is globallylower for high loading levels (note that negative shiftparameter cannot have physical interpretation). Con-
trary to what could be expected, for most of the points,the two-parameter Weibull fits better the Kolmogorov-Smirnov (K-S) test than the three-parameter distribu-tion.
In order to search for a better fitting, we havecompared six different distributions (Table 7): Nor-mal, Lognormal, Three-parameter lognormal, Two-parameter Weibull and Three-parameter Weibull. Un-less for normal and lognormal where the estimatescannot be modified, the parameters of the other distrib-utions are searched by an iterative procedure allowingto minimize the Kolmogorov-Smirnov test (K-S) val-ues. In other words, the search is oriented towards theparameters that give the best fitting of the K-S test.
For two-parameter Weibull, Fig. 6 compares thedistributions obtained by the different approaches, atloading levels of 450 and 600 N. It seems that theMedian rank approach leads to the longest tail at theright side; i.e. this method expects much longer life-times than the other methods. On the other hand,the Kolmogorov-Smirnov based fitting leads to largerprobability drops for high fatigue lifetimes. The Max-imum likelihood seems to be between the two othersituations. Concerning the lower distribution tails, theMaximum likelihood method seems to give more con-servative results than the other methods, which is verysuitable for design purposes of HDPE structures.
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Table 7 Kolmogorov-Smirnov test based estimates of six different distributions
In Table 7, the distributions are ordered accordingto their goodness-of-fit. The three-parameter Weibullgives globally the best fitting, but the differencebetween the first three distributions: two-parameterWeibull, three-parameter Weibull and three-parameterlognormal, is not very significant. Alternatively, log-normal and normal distributions are not convenient forfitting the fatigue lifetime of HDPE structures; it is to
be noted that classical lognormal distribution is com-monly used for this issue.
Figure 7 compares the best three distributions(Weibull with two and three parameters and lognor-mal with three parameters). Although the right-sidetails are similar, very large differences are observedfor the left-side tails. That is why much care should beconsidered when choosing appropriate distribution in
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Fig. 7 Comparison of different lifetime distributions
Fig. 8 Fatigue lifetime quantiles at 10% for the different models
the reliability-based design and assessment of HDPEstructures, especially for pressurized piping systems.
For design purposes, it is recommended to definethe lifetime quantiles at 10%. Figure 8 shows these10% fatigue curves for the three models: Lognormalwith two parameters and Weibull with two and threeparameters. The expression of these curves are as fol-lowing:
Lognormal 3P: NS8.5 = 2.2 × 1014,
Weibull 2P: NS8.5 = 1.6 × 1014,
Weibull 3P: NS8.6 = 3.2 × 1014.
(4)
It can be observed that the Lognormal and Weibullwith three parameters give almost the same quantilecurve. The Weibull with two parameters leads to moreconservative curve, which could be recommended fordesign purposes.
5 Conclusion
In this study, a probabilistic characterization of HDPEand fatigue lifetime has been carried out. The fatiguestrength shows very large scatter. An expression in thepower type form has led to more appropriate repre-sentation of fatigue data, compared to linear fitting in
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the log-scale. The statistical analysis has shown thatfatigue lifetime cannot be suitably modeled by lognor-mal distribution as usually adopted in the literature.Two- and three-parameter Weibull distribution, as wellas three-parameter lognormal, can be adequately ap-plied for probabilistic lifetime characterization. More-over, the median rank and the maximum likelihoodmethods do not lead to the best goodness-of-fit testresults, and hence they should not blindly applied inpractice. According to the 10% quantile fitting, it isrecommended to use the two-parameter Weibull dis-tribution for design and reliability assessment of ther-moplastic gas distribution piping systems, as it leadsto more conservative products.
Acknowledgements The authors wish to strongly acknowl-edge the Franco-Algerian cooperation program (Program BAF)for financially supporting this PhD work. The experimentalworks have been carried out at both LR3MI (Annaba, Algeria)and LaMI (IFMA, Clermont-Ferrand, France).
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