A Fracture Mechanics Approach for the Prediction of the Failure

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A fracture mechanics approach for the prediction of the failure

time of polybutene pipes

L. Andena a,*, M. Rink a, R. Frassine a, R. Corrieri b

a Dipartimento di Chimica, Materiali e Ingegneria Chimica G. Natta, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italyb Basell Poliolefine Italia, PT&C ARC, G. Natta R&D, P.le P.to Donegani 12, 44100 Ferrara, Italy

a r t i c l e i n f o

Article history:

Received 31 October 2008

Received in revised form 11 May 2009

Accepted 13 October 2009

Available online 17 October 2009

Keywords:

Polybutene

Fracture mechanics

Timetemperature superposition

Pipes

a b s t r a c t

In this work two grades of Isotactic polybutene-1 with a different degree of isotacticity

have been investigated; fracture tests have been performed at various temperatures and

testing speeds on DCB and SENB samples. Optical methods have been used to record crack

advancement.

Results of the tests have been interpreted using the fracture mechanics framework; a

timetemperature superposition scheme has been adopted to describe crack propagation

behaviour over several decades of time-scale. An analytical model has been applied to pre-

dict the lifetime of pressurised pipes from experimental fracture data. There is good agree-

ment between model predictions and experimental data obtained from full-scale tests on

real pipes.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

There are several areas in which isotactic polybutene-1 (i-PB1) finds application thanks to its good thermal and mechan-

ical properties: the packaging industry, hot-melt adhesives, tanks for various domestic appliances. In Europe and Asia i-PB1

also became during the past years one of the preferred materials to be used for the manufacturing of hot and cold water

plumbing and heating piping systems. i-PB1 offers many advantages in terms of easy, fast installation with a reduced num-

ber of joints and connectors compared to much stiffer conventional plumbing materials (such as metals). i-PB1shares with

more traditional polyolefins good resistance to chemicals and environmental stress cracking in addition to its excellent creep

properties even at high temperatures.

In the literature there are many works concerning i-PB1s crystallization behaviour (e.g. [1]) and the subsequent transi-

tion which occurs between its two crystalline forms (I and II) [2,3]. Fewer works involve its mechanical properties, with

widely different approaches. For example, AFM investigation has been used recently to study crazing at the micrometricand nanometric scales [4]. Cohesive zone modelling (CZM), a phenomenological approach which proved to be a powerful

method to describe fracture of adhesives and tough polymers [5,6], has been adopted to describe mode I fracture of i-PB1

[7] and different methods have been used to identify cohesive zone parameters. It was shown, however, that i-PB1exhibits

a complex fracture behaviour, previously unreported in the literature, with partial instability arising during crack propaga-

tion and this limited the effectiveness of CZM in reproducing crack initiation and propagation. Although yielding of i-PB1 has

not been extensively studiedper se, a better understanding of the damage mechanisms preceding crack initiation could sup-

port the investigation of the fracture behaviour of i-PB1.

0013-7944/$ - see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfracmech.2009.10.002

* Corresponding author. Tel.: +39 0223993207; fax: +39 0270638173.

E-mail address: [email protected] (L. Andena).

Engineering Fracture Mechanics 76 (2009) 26662677

Contents lists available at ScienceDirect

Engineering Fracture Mechanics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n g f r a c m e c h
http://dx.doi.org/10.1016/j.engfracmech.2009.10.002mailto:[email protected]://www.sciencedirect.com/science/journal/00137944http://www.elsevier.com/locate/engfracmechhttp://www.elsevier.com/locate/engfracmechhttp://www.sciencedirect.com/science/journal/00137944mailto:[email protected]://dx.doi.org/10.1016/j.engfracmech.2009.10.002


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A key issue in the use of i-PB1 for pressurised pipe applications is the evaluation of their lifetime as related to creep crack

growth. This phenomenon consists in the initiation and subsequent slow growth of a crack originating from a surface flaw.

The materials resistance to this type of fracture is hard to characterise, as the lifetime of pressurised pipes in typical oper-

ating conditions may exceed 100 years. Therefore it is common practice to perform accelerated tests at high temperature andextrapolate data to predict pipes lifetime [8]. The schematic diagram shown in Fig. 1 illustrates the outcome of a typical full-

scale test on i-PB1 [9]; two distinct regions can usually be recognised. For high hoop stress values (region A), failure occurs

due to ductile yielding of the material when the stress in the pipe wall exceeds the yield stress of the material. The term

ductile failure is used as large deformations can generally be observed when the pipe cross-section yields before fracture;

however, this is not the case for i-PB1 pipes which fail without exhibiting ballooning phenomena, which are quite common

in this regime for other polyolefins. At lower values of the applied hoop stress (region B) creep crack growth occurs and fail-

ures in this region are termed as brittle. This field is more interesting from the application point of view as pipe failures typ-

ically take place under this regime. The main drawback of this kind of test is their long duration (i.e. 12 years) and high cost.

Fracture mechanics (FM) can provide an alternative, useful approach. With FM it is possible to characterise fracture prop-

erties of a given material from laboratory tests and use them to predict the lifetime of any manufactured article. In [10] FM

has been used to study fracture of two grades of i-PB1performing creep tests at high temperature on SENB specimens. The

tests lasted for several weeks, thus granting a significant time saving when compared with full-scale tests on pipes. The

authors also developed an analytical model able to predict pipes lifetime and a promising comparison with the referencecurves shown in [9] was made.

A similar approach has been followed in the present work, performing fracture tests on laboratory specimens. Yet in this

study a constant displacement rate rather than a constant load has been applied. This allows a further, significant reduction

in testing times which in this study ranged between a few seconds and a couple of hours these times are much shorter than

those required by creep tests, not to speak of full-scale tests on pipes. In addition to that, tests have been carried out with

varying speed and temperature in order to ascertain the influence of these variables on the general fracture behaviour ofi-

PB1 and especially on crack stability.

Finally, pipe predictions have been obtained using the model developed in [10] and they have been validated against data

obtained from full-scale tests.

2. Theoretical background

Fracture mechanics data was analysed in terms of the stress intensity factor at the crack tip Kfor any given crack size a. Inthe present work only mode I (opening) conditions were considered.

Several authors, including Williams [11] and Schapery [12], suggested possible approaches to extend linear elastic FM to

viscoelastic materials. Under certain simplifying assumptions, Williams derived the following relationships between the

stress intensity factor K, the initiation time ti and the crack speed _a:

ti B Kp 1_a A Kq 2

in which A, B, p and q are material properties which generally depend on external conditions, such as the temperature.

Following Schapery [13] it is recognised that _a depends on the current value of K but not on its past values and for this

reason Eq. (2) applies for any loading history. It becomes thus possible to determine crack propagation parameters from any

convenient loading history in a laboratory test and use the obtained data to predict the behaviour of any manufactured item.

By combining Eqs. (1) and (2) a prediction of the lifetimetf

of a pipe under constant pressure with wall thicknesss

can be

obtained:

h

Log tf

ductile failure

Log

brittle failureA

B

Fig. 1. Schematic diagram of hoop stress vs. lifetime for a polymeric pressurised pipe.

L. Andena et al. / Engineering Fracture Mechanics 76 (2009) 26662677 2667
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tf ti tp B Kp0 Zsa0

da

A Kq 3

where tp represents the time required for a crack of initial size a0 to propagate across the wall thickness after initiation oc-

curs. Before crack initiation, as the crack size remains constant and equal to a0, the stress intensity factor is also constant and

equal to K0.

In the present work Eq. (3) has been used, neglecting the initiation time ti in the evaluation of tf. This leads to a conser-

vative prediction of the pipe lifetime.

3. Experimental details

The materials investigated are two pipe grades of i-PB1 kindly supplied in the form of pellets by Basell Polyolefins. The

two grades will be called PB1 and PB2, with PB2 having a higher degree of isotacticity and consequently crystallinity. Full-

scale testing run by the producer on pipes made from both materials showed that PB2 offers a better resistance to creep

crack growth. The greater degree of crystallinity of PB2 has also been reported to increase elastic modulus and yield stress

[10]. The tensile behaviour of i-PB1 is characterised by the absence of strain localisation and necking.

The pellets were compression moulded into 170 120 10 mm plates. After cooling from the melt, i-PB1 crystallizes in

form II, which is characterised by tetragonal symmetry. This form is unstable at room temperature and spontaneously

evolves into form I, which has an hexagonal lattice. To allow for completion of the transition, specimens were cut and ma-

chined at least 15 days after moulding [10], and then tested.

Fracture experiments under pure mode I conditions have been run on double cantilever beam (DCB) and single edgenotch bending (SENB) samples, shown in Fig. 2. Relevant dimensions are listed in Table 1. SENB configuration was used only

on preliminary tests on PB2, before moving onto DCB which grants a more stable crack propagation. Also, the longer liga-

ment of DCB specimens grants the acquisition of more data and extended fracture surfaces (see Table 1).

Notches in the case of SENB were made by means of razor sliding. The same apparatus could not be used for DCB, due to

the larger dimensions: the samples were first cut using a saw and then a razor blade was pushed into the material. On both

configurations the final root radius of the notches was about 13lm. The use of two different notching techniques can induce

a different degree of damage in the area surrounding the notch tip. This may in turn lead to a different behaviour at crack

initiation; however this is not very important for this study in which only crack propagation has been considered.

2h

Bn

P

P

W

aB

W

P

2h

a

Bn B

Fig. 2. DCB (above) and SENB (below) samples used for the fracture tests.

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Side grooves were also introduced in order to guide crack advancement along the notch plane. Great care was taken dur-

ing specimen preparation in order to ensure proper alignment of notch and grooves.Tests were performed on an Instron 1185R5800 screw-driven electro-mechanical dynamometer fitted with an environ-

mental chamber. Constant crosshead speeds of 1, 10 and 100 mm/min were used for the tests run at temperatures of 23 C,

50 C, 70 C and 90 C For every testing condition (sample geometry, speed, temperature) at least two specimens were tested.

Crack advancement was monitored using a photo-camera (at 1 mm/min) or a video-camera (at 10 and 100 mm/min) with a

calibration gauge applied on the specimens. ImageJ software was used to process the captured images.

4. Evaluation of the stress intensity factors

The stress intensity factor Khas been evaluated for both testing configurations from the measurements of load and crack

length recorded during the tests. In the case of SENB specimens the widely known formula:

K

f

a=W

P

B W0:5

4

has been used, in which f(a/W) is a non-dimensional shape factor [14].

For the DCB configuration the formula:

K 2ffiffiffi3

p P aB h1:5

5

is generally used. However, Eq. (5) works well only for a/h > 70, which is not the case for the samples tested in the present

study. This has been discussed in [15] where an alternative formula by Kanninen is proposed, in which the accuracy of the

simple beam theory is improved using EulerBernoulli beam theory together with a Winkler foundation. The resulting equa-

tion applies for a/h > 2:

K 2ffiffiffi3

p P aB h1:5

1 0:64ha

6

These expressions are derived according to linear elastic fracture mechanics (LEFM). To ensure validity of LEFM, smallscale yielding and plane strain conditions should be fulfilled. This can be guaranteed if the specimens meet appropriate size

criteria: the size of the plastic zone around the crack tip shall be significantly smaller than the specimen dimensions, i.e. the

thickness B, the crack length a and the ligament length (Wa). The characteristic length of the plastic zone, rp, can be esti-

mated from the following equation [15]:

rp KCrY

27

in which KC and rY are the material fracture toughness and yield stress respectively. For both i-PB1 grades rp is approxi-

mately 8 mm [7], a value which is comparable with the specimen thickness. However, the influence of thickness on the frac-

ture properties of PB1 and PB2 has already been investigated on SENB samples in [16] and no effect has been reported in the

range between 5 and 20 mm. An effect of the ligament width has been observed instead, with a decrease of the toughness for

small values of (Wa

); this was already reported by Hashemi and Williams in [17] and it can be explained considering the

constraint that such a small ligament size exerts on the plastic zone which, as a consequence, is not free to fully develop.

Table 1

Nominal dimensions of DCB and SENB samples.

DCB SENB

2 h 45 mm 2 h 80 mm

W 150mm W 20mm

a 4575 mm a 10mm

B 10 mm B 10mm

Bn 6 mm Bn 8 mm

a 60 a 60U 8 mm

Table 2

Parameters of the pipe model.

s 2 mm

R0 22mm

a0 50lm

e 1



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However the presence of frequent and/or large crack jumps limits the usefulness of this approach. Yet even when instabil-

ities occur there are large load drops but K values remain almost constant because a increases (see Fig. 7). Moreover, the

combination of stable and unstable crack propagation gives rise to an average crack speed which can be determined by a

linear fit of crack length vs. time data. The average K and da/dtvalues are reproducible within tests performed in the same

conditions (temperature and speed) and they have been used in the following analysis, as they are believed to truly represent

the materials behaviour. However, this approach has the obvious drawback of generating only a single data point for each

test.

Figs. 8 and 9 show Kvs. da/dtdata at the various temperatures for PB1 and PB2 respectively. According to Eq. (2), Kvs. da/

dtcurves are expected to be straight lines on a bilogarithmic scale. It is hard to detect a single slope of the data for all tem-

peratures. Moreover, data at 50 C do not fall on straight line.

Fig. 4. Fracture surfaces of PB1 and PB2 samples tested at 1, 10 and 100 mm/min and 23 C.


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Data at different temperatures can be interpreted using timetemperature superposition, a reduction scheme which is

widely accepted in the literature concerning polymers. An example of its application to Kvs. da/dtfracture data can be found

in [18,19]. Basically one temperature is selected as a reference temperature; data points belonging to another temperature

are shifted along the crack speed axis until they superimpose with the reference curve. The process is repeated for the next

temperature and so on, until all data merge on a single master curve at the reference factor. The time shift factor a23C

T T

required for each temperature is usually reported on an Arrhenius plot as a function on the reciprocal of temperature and

the slope of this plot can be related to the activation energy of the mechanical process involved.

This scheme was applied to PB1 and PB2 data in Figs. 8 and 9 and a Kvs. da/dtmaster curve at 23 C was obtained for both

materials, as shown in Fig. 10. The shift factor a23C

T T was found to be the same: this quantity seems to be independent of

the materials crystallinity. A similar result was found in [10] for the shift factor related to relaxation modulus and yield

stress. Values of a23C

T T are reported on an Arrhenius plot in Fig. 11, where a linear dependence on the reciprocal of tem-

perature can be observed. In Fig. 10 a knee is clearly visible between crack speeds of about 103 to 102 mm/s, indicating a

transition between two regions with a different slope in the Kvs. da/dtcurve. Data analysis reveals that the two regions are


0 600 1200 1800 2400 3000 3600 4200 4800 54000.0

0.5

1.0

1.5

2.0

2.5

3.0

avg da/dt

K

Crack length

Time (s)

K(MPa*

m1/2)

avg K

40

50

60

70

80

90

100

110

120

130

140PB2 23C 1mm/min

Crack

length(mm)

Fig. 7. Typical K and crack length vs. time curves for two DCB samples having different initial crack lengths of 45 and 75 mm; dashed lines indicate the

average values of K and da/dt determined during the analysis.



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-5 -4 -3 -2 -1 0 1

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

PB1

logK(MPam

)

log da/dt (mm/s)

stable

crack

propagation

partially

unstable

crack

propagation

PB2

Fig. 10. K vs. da/dt master curves for PB1 and PB2 at 23 C. Dashed lines represent the slopes for the stable and partially unstable regimes.

-5 -4 -3 -2 -1 0 1-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

DCB23CDCB50CDCB70C

PB1

logK(M

Pam

)

log da/dt (mm/s)

Fig. 8. K vs. da/dt data at 1, 10 and 100 mm/min for PB1.

-5 -4 -3 -2 -1 0 1-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40PB2

DCB23CDCB50CDCB70CSENB50CSENB70CSENB90C

logK(MPam

)

log da/dt (mm/s)

Fig. 9. K vs. da/dt data at 1, 10 and 100 mm/min for PB2.



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characterised by a distinct behaviour: for K values below the knee stable crack propagation is observed while the values

above are associated to partially unstable propagation. In the latter region the crack speed is much more sensitive to the

applied K, i.e. an increase of K will cause an increase of da/dtwhich is larger than in the case of purely stable propagation.It is found then that the simple power-law model detailed in Section 2 can be used to describe propagation data for a given

mechanism but the transition between the two mechanisms needs to be correctly taken into account according to the effec-

tive K range for a given application.

Direct comparison of the two master curves clearly shows that crack propagation on PB1 is faster than PB2 for any level of

applied K. This result is in good agreement with results of full-scale testing on pipes, as mentioned earlier at the beginning of

Section 3. The present fracture mechanics approach can be used to rank fracture resistance of different materials with an

enormously reduced effort.

6. Prediction of pipes lifetime

Eq. (3) can be used as the basis of a simple analytical model able to predict lifetime of polymeric pipes; in the present

study the contribution of initiation time to the total failure time was neglected, as previously done in [10]. This is only a first

approximation giving conservative predictions that were compared against experimental data obtained from full-scale tests

on pipes.

In order to apply the model one needs to properly define the geometry of both the pipe and the initial defect, and use a

suitable shape factor; relevant material parameters (namely A and q) need to be known as well.

A pipe geometry analogous to that sketched in Fig. 12 was considered: a semi-circular flaw was assumed to be situated at

the pipe inner surface, lying on a radial plane. Physical dimensions were chosen according to the actual dimensions of the

pipes used by Basell for full-scale tests on i-PB; they are listed in Table 2. Experimental observations indicate that fracture

always initiates at the inner surface and quality controls performed prior to pipe testing excluded the presence of defects

larger than 50 lm: the location and size of the initial defect were chosen accordingly.

When internal pressure is applied to a pipe pure mode I conditions are generated at the crack tip of a radial defect. The

latter was assumed to propagate keeping a semi-circular shape: in this way K could be calculated as a function of the hoop

stress and the defect size by using the same shape factor (taken from [20]) throughout all the analysis.

Pipe lifetimes were evaluated for different levels of applied hoop stress (up to 20 MPa) by integrating Eq. (3) for a crack

growing from the initial defect size a0 to the wall thickness s.

The choice of which material parameters to use is not straightforward, since both materials exhibit a transition in the Kvs.

da/dtmaster curve. However, even for the highest level of applied stress (20 MPa) Kvalues ranged from 100.66 (0.2 MPa m,

for aa0) up to 100.48 ((3.0 MPa m, when as). By looking at Fig. 10 it is obvious that in these conditions K values lie in the

stable propagation region for most of the pipe life. Therefore, predictions were made considering A and q as obtained from

the stable part of each of the two master curves at 23 C, thus extrapolating the stable behaviour to the whole K range.

Experimental data from full-scale tests on pipes performed by Basell were available at 23, 70 and 95 C and the same tem-

peratures were considered in the model. The shift factor shown in Fig. 11 was used to obtain the K vs. da/dtcurve at 70 C;

the curve at 95 C was generated by applying a shift factor obtained by linear extrapolation.

A comparison between model predictions and full-scale experiments is shown in Fig. 13. The FM model predictions are

shown as the lines on the right and they should represent the region of brittle failure for the two materials. However, in the

time-scale considered most of the experimental pipe failures were reported as being ductile. Actually a fair estimate of the

failure time in this region can be obtained by simply considering time to yield data for each stress level and temperature.

Time to yield was calculated from yield stress vs. time curves reported in [10].

Fig. 11. Shift factor of K vs. da/dt curves as a function of temperature for both materials studied.

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Only a few pipes corresponding to data points at the highest failure times (greater than 10 4 h) presented a failure mech-

anism which was reported as mixed, indicating a transition towards the brittle region; the experimental curves show a

hint of a knee for these data. This is where the curves describing the brittle region are expected to intersect those for ductile

Fig. 12. Cracked pipe model considered for the prediction of pipe lifetimes.

1

10

100

101

102

103

104

105

106

107

108

109

experiment PB2

experiment PB1

95C

70C

23C

model PB2

model PB1

time (h)

stress(MPa)

Fig. 13. Comparison between model predictions and experimental data from full-scale tests on pipes at 23, 70 and 95 C.



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failure and indeed this is what happens for model predictions at 23 and 70 C, with remarkably good agreement. At 95 C

predicted curves seem to slightly overestimate lifetimes for both materials, nevertheless the overall agreement is substan-

tially good and shows that this approach can be used to obtain reliable estimates of pipe lifetimes. Moreover, the model cor-

rectly reproduces the different behaviour of PB1 and PB2, with curves of the latter slightly above those of the less resistant

material as it is observed on full-scale tests.

It may be surprising that such a good agreement has been obtained despite neglecting initiation times: this should lead to

conservative estimates of pipe lifetimes. However the model predictions strongly depend on the initial flaw size. A different

value of this parameter would cause an horizontal shift of the predicted curves, as discussed in [10].

This research is continuing with the aim of studying crack initiation for these materials and future developments of the

model will take it into account as well.

7. Conclusions

Fracture properties of two pipe grades of polybutene have been studied performing experiments on DCB and SENB con-

figurations. The existence of two mechanisms of stable and partially unstable crack propagation has been observed, as pre-

viously reported in [7]. The effect of testing speed and temperature on crack stability has been investigated and a transition

from stable to partially unstable crack propagation has been detected on both materials. It has been found that higher testing

speeds and lower temperatures promote the occurrence of instabilities.

The combined effect of testing speed and temperature fits well into a timetemperature superposition scheme and crack

propagation master curves could be obtained for both materials. The two curves are characterised by a bilinear trend with aknee separating the two regions of stable and partially unstable crack propagation.

The analysis performed using the fracture mechanics approach gave two main results:

1. Direct comparison of the crack propagation master curves can in most cases (unless they intersect) give a ranking of dif-

ferent materials with respect to their creep crack growth resistance. In the present case the more crystalline grades curve

lies above the other materials, thus indicating a slower crack speed for any value of the applied stress.

2. Quantitative predictions of manufactured items lifetime may be obtained by using crack propagation data in

conjunction with simple models based on FM and the different performance of the materials investigated has been

evaluated.

The analysis and the predictions agree well with experimental data obtained by the materials supplier from full-scale test

on pipes. Fracture mechanics therefore can be used to perform accelerated testing and proves itself to be a quick, inexpensive

and reliable method to evaluate the long-term performance of different materials.

Acknowledgements

The authors wish to thanking Evaristo Odinolfi for his precious support in performing DCB tests and analysing the data

and Mr. Oscar Bressan for the specimen preparation.

References

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[16] Andena L. Frassine R. Rink M. Roncelli M. Thickness effect on fracture behaviour of polybutene. In: Proceedings of 4th ESIS TC4 conference Les

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A Fracture Mechanics Approach for the Prediction of the Failure