Cohesive zone modelling of fracture in polybutene

$: Cohesive zone modelling of fracture in polybutene$
Engineering Fracture Mechanics 73 (2006) 2476–2485

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Cohesive zone modelling of fracture in polybutene

L. Andena a,*, M. Rink a, J.G. Williams b

a Dipartimento di Chimica, Materiali e Ingegneria chimica ‘‘G. Natta’’, Politecnico di Milano, Piazza Leonardo da Vinci 32,

20133 Milano, Italyb Department of Mechanical Engineering, Imperial College London, Exhibition Road, South Kensington,

London SW7 2AZ, United Kingdom

Received 30 October 2005; received in revised form 18 April 2006; accepted 18 April 2006Available online 21 June 2006

Abstract

Fracture properties of isotactic polybutene-1 have been investigated. Fracture tests have been conducted and relevantproperties at initiation have been determined according to linear elastic fracture mechanics. Two distinct fracture mecha-nisms have been identified, one of them causing partial instability during crack propagation. Numerical modelling has beenperformed using a cohesive zone approach. In particular, the identification of suitable cohesive laws has been tried usingparametric identification and two different experimental methods. Results suggest that two different cohesive laws may beneeded in order to describe crack initiation and crack propagation.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Polybutene; Fracture; Cohesive zone; Identification; FEM

1. Introduction

Isotactic polybutene-1 (PB) is increasingly being used for the manufacturing of piping systems to be used inheating and plumbing installations. According to the polybutene piping systems association (PBPSA), PB isconsidered to have many advantages over competitive and more traditional polymers, for example withrespect to its long-term mechanical performance at high temperatures. There are extensive studies on the tran-sition which occurs between its two crystalline forms (I and II) after melting [1,2]. However, the literature con-cerning the mechanical properties of PB is very scarce [3].

Cohesive zone modelling has proven to be a powerful method to describe fracture of adhesives and toughpolymers. For this reason it has been chosen to study the fracture behaviour of PB. The cohesive zoneapproach is used to describe the material behaviour in the zone which is ahead of the crack. To do so a con-stitutive law is defined, which correlates the stresses in this process zone (tractions) with the relevant openingdisplacement (separation). An open issue is the determination of cohesive laws able to describe differentmaterials. One way of doing it is by means of parameter identification, assuming a general shape for the

0013-7944/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2006.04.026

* Corresponding author. Tel.: +390223993207; fax: +390270638173.E-mail address: [email protected] (L. Andena).

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L. Andena et al. / Engineering Fracture Mechanics 73 (2006) 2476–2485 2477

law (i.e. linear, bi-linear, polynomial, etc.) and determining its parameters from experimental tests, using opti-mization procedures. Recently a direct measurement technique has been proposed by Williams [4–6]: the cohe-sive law can be obtained performing tensile tests on circumferentially notched specimens. The nearly uniformdistribution of stresses across the section allows for the determination of the tractions as a function of thelocally measured separation.

A new indirect method [7,8] has been more recently developed, which can be applied to any kind of fracturetest. It has been applied to the three point bending configuration in which the stresses across the section arenot uniform. Therefore, the proposed method uses an iterative procedure based on a finite element (FE)model. Although indirect, the identification procedure does not require an a priori definition of a shape forthe cohesive law as traditional parameter identification methods do.

2. Experimental details

The material investigated is a pipe grade PB kindly supplied in the form of pellets by Basell Polyolefins. Thepellets have been compression moulded into 170 · 120 · 10 mm plates. After cooling from the melt, PB crys-tallizes in form II, which is characterised by tetragonal symmetry. This form is unstable at room temperatureand spontaneously evolves into form I, which has an hexagonal lattice. To allow for completion of the tran-sition, specimens have been cut and machined at least 15 days after moulding [3], and then tested.

Pure Mode I (opening) conditions have been attained using Compact Tensile (CT), Single Edge NotchedBending (SENB) and Circumferentially Notched Tensile (CNT) configurations. Notches have been introducedby means of razor tapping, razor sliding and using a single point cutting tool on a lathe for CT, SENB andCNT respectively. For each configuration the most suitable notching technique has been chosen according togeometry requirements (e.g. axial symmetry) and laboratory expertise. Great care has been taken while per-forming the operation in order to ensure proper alignment of notches and to avoid damage to the specimen.

Fig. 1 shows a sketch of the three geometries while relevant dimensions are listed in Table 1. Tests havebeen performed at room temperature on screw-driven electro-mechanic dynamometers. Constant crosshead

Fig. 1. Configurations used for the fracture tests.

Table 1Nominal dimensions (in mm) of samples

Geometry w W l1 l2 h a R

CT 24 30 28.8 13.2 10 12 3.25SENB – 20 90 84 10 10 –CNT – – 3.4 10 – – –

2478 L. Andena et al. / Engineering Fracture Mechanics 73 (2006) 2476–2485

speeds of 1 and 10 mm/min have been used for CT and SENB specimens; in the case of CNT the tests havebeen run at 0.05 mm/min.

3. Results

For both CT and SENB configurations partially unstable crack propagation has been observed. The loadversus displacement curves are quite irregular after the peak load, with many small bumps and a few suddendrops (see Fig. 2). These drops were accompanied by clear ‘‘tick’’ sounds during the tests.

A high-resolution digital camera has been used to perform crack propagation measurements by takingshots at regular intervals during the tests. The pictures showed the formation of localized regions of highlystretched material along the crack path. The sudden rupture of these regions has been deemed responsiblefor causing small jumps in the propagating crack and the abrupt decreases of the load.

Fractured samples have been analysed using an optical microscope (see Fig. 3): two distinct kinds of behav-iour can be clearly identified. There is a general rough pattern characterised by a shiny appearance, and a fewdark marks, randomly distributed across the section, which are almost flat. These two separate set of featurescorrespond to different fracture mechanisms. A first one exhibits small scale ductility giving rise to the roughshiny surface. The other mechanism is clearly associated with the unstable crack propagation occurring whena localized stretched region fails: in fact the number and size of the dark marks have been correlated with thenumber and amount of the drops in the load versus displacement curves. This second mechanism is only activeduring the propagation stage.

The observed phenomenology was very similar for CT and SENB. The only difference was the presence, inthe latter case, of a small kink in the load trace at crack initiation: its origin is not clearly understood yet but ascan be seen in Fig. 4, it is associated to significant blunting at the crack tip, which was less evident on CTspecimens. Although this kink is a small feature on the overall load versus displacement curve, its existencemakes the application of the indirect method for the identification of cohesive laws difficult. This issue willbe further discussed in Section 6.

Linear elastic fracture mechanics (LEFM) critical parameters have been evaluated according to ISO13586[9]. Crack onset has been detected optically and critical values of 9 kJ/m2 for the energy release rate Gc and1.75 MPa

pm for the critical stress intensity factor KC have been determined; there is substantial agreement

between the two configurations (CT and SENB). The ISO standard specifies size criteria which need to be sat-isfied in order to ensure small scale yielding and plane strain conditions. Both criteria are usually simulta-neously satisfied if the specimens have standard dimensions and the condition is

h > 2:5 � KC

rY

� �2

ð1Þ

Fig. 2. Load versus displacement curves for two CT samples.

Fig. 4. Load versus displacement for a SENB sample. Detail of crack tip blunting at initiation.

Fig. 3. Fracture surface of a SENB sample.


where rY is the yield stress for the material, which is about 20 MPa for PB [3]. In the present case the thicknessh should be over 18 mm, which is significantly higher than the actual value of 10 mm. Therefore, size require-ments are not strictly satisfied due to the large extent of the plastic zone for this material. However, since thesame values for the critical fracture parameters are obtained, the different stress triaxiality ahead of the cracktip for CT and SENB does not play a significant role.

The calculation for Gc has been extended beyond crack initiation to get an elastic estimate of the propaga-tion energy release rate (see Fig. 5). It can be seen that after initiation G increases beyond Gc up to a valuewhich stays approximately constant during the propagation stage.

A thorough investigation of the influence of the sample geometry on fracture properties has also been con-ducted. Results are not discussed here but they have been presented in [10]. Rate effects have not beenobserved in the range of applied testing speeds (1–10 mm/min).

A preliminary set of tests on CNT samples has been conducted. Unstable crack propagation occurring afterthe peak load led to complete fracture before the whole traction-separation law could be observed. Thisprevented the application of the direct method for the identification of a cohesive law for polybutene. The

Fig. 5. Crack advancement and energy release rate versus displacement for a CT sample.


instability is very likely to be caused by the high compliance inherent in the deep notched specimens. Thenotch depth cannot be reduced in order to preserve the high level of constraint required by CNT tests [5].

4. Parametric identification

A first attempt to describe fracture data has been made using a cohesive model proposed by Hadavinia andother authors [11,12]. A cubic traction-separation law has been assumed

TableMater

YoungPoissoYield

T ¼ 27

4rmaxkð1� 2kþ k2Þ; ð2Þ

where T is the traction and

k ¼ uabove � ubelow

dc

; ð3Þ

uabove � ubelow is the normal displacement discontinuity at the interface. dc and rmax are the two parameterswhich fully define the cohesive zone behaviour. The fracture energy Gc can be derived through a simple inte-gration of the traction-separation law. It can be used in place of one of the other two parameters, to which it isrelated by the following equation:

Gc ¼9

16rmaxdc: ð4Þ

An elasto-plastic material model with a Mises yield surface has been used to describe the bulk material outsidethe cohesive zone. Relevant parameters are shown in Table 2 and they have been taken from results obtainedby Passoni [3]. The overall model has been implemented in a commercial finite element code.

A parametric study has been conducted to identify cohesive parameters from CT and SENB tests. The val-ues have been determined so as to obtain the best agreement between the outcome of the numerical simula-tions and the experimental load versus displacement curves. For both configurations it has been possible toidentify a set of parameters giving a very good agreement between numerical and experimental data, as shownin Fig. 6. However, the two sets differ in the values of both Gc and rmax. The a priori assumption of a cubic lawprevents the identification procedure from obtaining a cohesive law able to represent the intrinsic (i.e.

2ial parameters used in the finite element simulations

’s modulus 500 MPan’s ratio 0.3stress 20 MPa

Fig. 6. Comparison between experimental data and simulations with identified parameters.


independent of the testing configuration) material behaviour. Moreover, Gc values identified for both CT andSENB configurations significantly exceed the experimental value of 9 kJ/m2.

5. Indirect method

The a priori choice of a shape for the cohesive law can give rise to a potential transferability problem. Thecohesive model in this case is a purely phenomenological model which is determined by matching results of aparticular experiment with the numerical analyses. The consequence is a loss of generality; the extension to anarbitrary geometry is not guaranteed because the cohesive law does not necessarily describe the real physicalfracture processes. With this in mind, results obtained on CT and SENB tests using parametric identificationare not surprising.

A direct identification method could have been used as an alternative approach to determine the cohesivelaw but preliminary CNT tests on PB have not been successful yet.

A third approach has been considered, based on a hybrid experimental/numerical method which has beenpreviously applied to amorphous polymers [7,8]. This indirect method has been used for the determination of acohesive law from the SENB tests, without the need to assume a predefined shape. The method uses a finiteelement model in conjunction with experimental local separation and macroscopic load data.

The separation at the crack tip has been measured with a video extensometer. This device (a VE5000 byTrio Sistemi e Misure, Italy) can accurately follow the relative displacements of four markers placed in asquare pattern very close to the crack tip (as shown in Fig. 4). The crack tip opening displacement can thenbe obtained through an interpolation of the opening displacement of the two couples of markers above andbelow the crack tip.

Unlike with the CNT configuration, in the case of SENB it is not straightforward to derive the local trac-tions from the macroscopic load. The exact stress distribution along the process zone depends on the actualmaterial cohesive law, which is the objective of the identification procedure. Since local tractions at the cracktip cannot be directly measured, a numerical tool is needed. A finite element model of the SENB sample hasbeen used for this purpose, with interface elements placed along the crack path to implement the cohesive zonemodel. In order not to restrain the shape of the cohesive law, a linear stepwise function has been chosen. Pro-vided the number of linear segments is large enough, any shape can be described with good approximation.Our implementation used 30 segments, with a total of 60 parameters available (slope and length of eachsegment).

The actual identification procedure consists in an iterative scheme which uses experimental local separationand macroscopic load data up to crack initiation. Firstly, an initial slope for the cohesive law is guessed. Thisslope is positive as it is assumed to represent some kind of initial elastic response of the material inside thecohesive zone. A constant step size for the opening displacement at the crack tip node (CTOD) is chosenfor the model. The CTOD is the numerical equivalent of the measured local separation. The finite element


model is run using an arclength algorithm [13] to control the CTOD and get the macroscopic load as an out-put. After the first iteration this output is compared with the experimental load corresponding to a measuredseparation equal to the CTOD imposed in the FE model. If the two values differ, the segment slope is adjustedand the step repeated until the difference between experimental and numerical load is cut down to a specifiedtolerance. At this stage, the first segment of the cohesive law up to a separation value equal to the imposedCTOD has been identified. A new step is then performed, i.e. the CTOD is increased. If the increase is smallenough, only a very limited number of interface nodes close to the crack tip will increase their opening dis-placement beyond the CTOD value at the previous step: the behaviour of most of the interface nodes willbe described by the part of the cohesive law which has already been identified. Again, the slope of the‘‘new’’ segment of the cohesive law will be adjusted so that the load calculated with the FE model equalsthe corresponding experimental value. Then again the CTOD is increased and the procedure is repeated untilthe whole traction-separation law is identified, i.e. until the traction drops to zero. When this critical conditionis reached at the crack tip node, crack initiation occurs according to the FE model. The scheme of a singleiteration step is illustrated in Fig. 7. The cohesive law identified using this method is rate-independent butthe procedure may be applied to tests conducted at several speeds, thus deriving a set of rate dependent cohe-sive laws for each material. The same approach has been used in [6].

Fig. 8 shows the identified cohesive laws obtained for two different samples using the indirect method. Thetwo laws differ slightly; it is thought that the identification procedure could be improved by enhancing theaccuracy of the separation measurements. This can be done with better optics and lighting conditions. Nev-ertheless the associated fracture energy is almost identical. This value is in much better agreement with theexperimental Gc (9 kJ/m2) than the relevant value from the cohesive law obtained using parametric identifica-

Fig. 7. Iteration step for the determination of the traction-separation law: (a) situation at the beginning of the step; (b) the separation isincreased; (c) a slope is guessed for the new segment of the cohesive law; (d) the predicted macroscopic load is compared with theexperimental value; (e) the slope is adjusted to reduce the gap between predicted and measured load; (f) the procedure is repeated until thepredicted load equals the experimental value. A new step is performed.

Fig. 8. Comparison between cohesive laws identified using the indirect method and parametric identification on SENB samples(experimental Gc = 9 kJ/m2); the initiation line indicates the experimental value of the critical separation as measured by the videoextensometer.


tion on a SENB sample, also shown in Fig. 8. Apparently the segment size used for the purpose of the iden-tification is not very important, provided a sufficient number of segments is taken to avoid a coarse represen-tation of the cohesive law.

A video camera has been used to detect crack initiation time during the tests. If the plot of separation (mea-sured with the video extensometer) is entered with this initiation time, an experimental value of the criticalseparation can be obtained. This value, indicated by the dotted line in Fig. 8, is very small compared tothe specimen size. This justifies the use of a cohesive zone approach as in this case the cohesive energy (i.e.the area underlying the traction-separation curve) truly represents the energy release rate. The critical separa-tion predicted by the indirect method is quite accurate, especially when compared with dC of the SENB cubiclaw.

In the final part of the cohesive law identified with the indirect method there is a narrow peak which is clo-sely related to the kink observed at initiation on the experimental load curves. Since the identification proce-dure makes use of load data, it is clear that its final result is influenced by the kink, whose presence has notclearly been explained so far.

6. Discussion

The experimental load versus displacement curve and the one obtained with the FE model using the cohe-sive law identified with the indirect method have been compared in order to validate the proposed identifica-tion scheme. This comparison is significant since displacement data are not considered by this scheme whichonly uses load-separation data instead. As shown in Fig. 9 a very good agreement is obtained just up to thekink in the load trace, after which there is a steep decrease in the numerical curve that does not reproduce thebehaviour observed in the real experiment. A possible explanation may be given by looking back at the eval-uation of G shown in Fig. 5. Following crack initiation, the energy release rate increases up to a value which issignificantly higher, suggesting that two distinct levels of fracture energy may characterise initiation and prop-agation. As the identification procedure only considers data up to initiation, it cannot capture the propagationbehaviour.

This hypothesis is corroborated by the analysis of the simulations of the SENB tests performed using thecubic law obtained from parametric identification. As shown in Fig. 6 this law gives a good overall agreementbetween the experimental and calculated load versus displacement curves but this is quite obvious since thisagreement is the very criterion used for its identification. As the shape of load versus displacement curve islargely determined by the propagation stage, the parametric identification procedure is quite insensitive tothe initiation behaviour. In fact, the energy associated with this law is very close to the estimated propagation

Fig. 9. Comparison between experimental data and simulations with cohesive law identified using the indirect method.


value of the fracture energy. As a consequence, the cubic cohesive law’s parameters (fracture energy and crit-ical separation) strongly disagree with the corresponding experimental measurements at initiation. The lawidentified using the indirect method has proven significantly more accurate in describing crack initiation.

Ideally a single cohesive law should be able to capture the overall fracture behaviour as the apparent tough-ness increase observed during crack propagation would be caused by extensive plastic deformation occurringoutside the process zone [14]. However in the case of PB this instance has not been observed while two fracturemechanisms have been reported (see Section 3), one of them only active during propagation. For this reasonseparate cohesive laws should be used to reproduce the initiation and propagation behaviour. The differencebetween the two fracture energy levels is the net contribution associated to the formation of localized stretchedzones during crack propagation and their subsequent rupture. The complex interplay of the two mechanismsduring the propagation stage could require incorporation of rate effects both in the bulk and in the cohesivezone description.

7. Conclusions

The experiments performed on PB highlighted the existence of two fracture mechanisms. The main mech-anism is characterised by a small-scale ductile behaviour and a high surface roughness. During crack propa-gation a second mechanism is also active. It is associated to the formation of highly stretched localized regions.The sudden rupture of these regions causes partial instability during crack propagation and generates flatareas on the fracture surface. During propagation the fracture energy apparently increases to a value whichis higher than the critical value at initiation.

A cohesive zone approach has been adopted to model the fracture behaviour of PB. Three different meth-ods have been used for the purpose of identifying a suitable cohesive law for the material.

A direct measurement technique on CNT samples has not given any useful results because unstable crackpropagation occurred before completion of the tests. Nevertheless, these results are preliminary and furtherinvestigation is ongoing in order to clarify the reasons of this behaviour, possibly with the aid of numericalsimulations.

An alternative approach relies on numerical methods. In this work a finite element method has been used.Two identification techniques have been adopted: a purely numerical parametric identification and a morerecently developed hybrid technique. Parametric identification performed on the basis of CT and SENB experi-ments failed to identify an intrinsic cohesive law, as the result of the procedure depends on the testing config-uration. Provided different values of the cohesive parameters are chosen for the two configurations, theoverall experimental behaviour can be reproduced quite well using a two-parameter cubic law. However,the identified values of the critical fracture energy and separation performed at crack initiation do not agreewith experimental measurements. A significantly better agreement has been obtained by using the hybrid


technique on SENB samples. The cohesive law thus identified reproduces very well the experimental behaviourup to initiation, but fails to do so for the propagation stage.

The critical analysis of the observed fracture behaviour of PB and the results of the cohesive zone modellingapproach suggest that separate cohesive laws may be needed in order to give an accurate description of crackinitiation and crack propagation.

References

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Cohesive zone modelling of fracture in polybutene

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