Characterization of the fracture behavior of polyethylene using measured cohesive curves. II. Variation of cohesive parameters with rate and constraint.

INTRODUCTION

In recent years, the cohesive zone model (CZM) has been put forward to address the limitation associated with linear elastic fracture mechanics in characterizing tough polymers. This is due to the extensive plasticity (plastic zone) known to develop ahead of the crack tip in these materials in which the K and J -fields underestimated the severity of the crack [1-3]. The origin of cohesive zone concept can be traced to Dugdale [4]; however, his work was perceived at that time as the crack-tip plastic zone analysis. Another notable early contribution to this technique came from Hillerborg et al. [5] who have pioneered the use of cohesive models in concrete by introducing the concept of fracture energy into the methodology. The principle behind the CZM is to idealize the fracture process in solids as occurring within thin layers confined to a line segment or a narrow strip, whereby the relationship between the cohesive force and the separating surfaces within the layers can be defined. It is common to assume that the cohesive stress depends only on the local opening and that a critical opening exists beyond which the cohesive stress vanishes. The material parameters used to describe the cohesive zone are the peak separation stress, [[sigma].sub.max] and the energy of separation, [GAMMA]. However, the critical separation [[delta].sub.break] between two separating surfaces when the traction stress has fallen to zero is not an independent parameter, because [GAMMA] [proportional] [[sigma].sub.max][[delta].sub.break], where the constant of proportionality depends on the shape of the traction-separation curve. Consequently, this relationship can be used to represent the material deterioration processes in the cohesive zone. In this manner, the macroscopic toughness of the structure can be described via the local failure criterion at the crack tip together with the interaction of this criterion with the surrounding material.

As the formulation of the CZM is relatively straightforward, the approach has become particularly attractive for simulating crack growth problems in which the failure process zones are long in one dimension but small in the perpendicular direction. However, the methodology constituted an area of research where the experimental investigations lag behind the computational work [6-11]. This is because for most test configurations, it is difficult to avoid crack growth or to obtain a uniform stress in the specimen. While characterizing the fracture properties of tough polyethylene, Duan and Williams [3] showed that these problems can be avoided by introducing a circumferential notch in a rectangular bar. As a result, the deformation mechanism (crazing) was confined in the ligament, resulting in a damage evolution that was uniform across the width of the specimen. For this reason, it was possible to postulate some form of local traction-separation relationship distinct from the properties of the bulk material to describe the behavior within the crack-tip damage zone [12, 13]. Subsequently, the scheme was extended to measure the cohesive laws as a function of rate in a range of polyethylenes. The technique is shown schematically in Fig. 1. The load cell was used for measuring the force whereas [delta] was obtained from a mechanical extensometer. Recently, Ting et al. [14, 15] enhanced the method by introducing a video (non-contact) extensometer to measure the material separation. The scheme has proved attractive in that it provided the physical link between the cohesive law and the local fracture process (or the specific experimental measurements).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

The cohesive parameters are also known to be affected by some macroscopic parameters such as crack-tip constraints resulting from geometry and the loading rate [16, 17]. Few studies in the literature have investigated in detail these effects experimentally. In the present study, cohesive parameters extracted from measured traction-separation curves using the circumferentially notched tensile specimens (CNT) as described in the preceding companion paper [18] are used to determine the macroscopic effects on the crack growth behavior.

EXPERIMENTAL

CNT specimens of the dimensions (1) 10 x 10 x 110 [mm.sup.3], (2) 16 x 16 x 110 [mm.sup.3], and (3) 20 x 20 x 110 [mm.sup.3] were cut from compression-molded plaques. The specimens were notched on a lathe using a single-point cutting tool of tip radius [less than or equal to] 20 [micro]m. The depth of notch was determined by calculating the ligament to bulk area (LBA) ratio at 10, 30, and 50%. Experiments were performed under constant displacement conditions on the Instron testing machine over the range of 0.005-50 mm/min. The results yield an in-situ traction-separation measurement [18]. The cohesive parameters extracted from the measured traction-separation properties of three copolymers and one homopolymer were investigated and the basic molecular features are given in Table 1. The materials were supplied by BP Chemical.

[FIGURE 7 OMITTED]

FEATURES OF THE FRACTURE SURFACES

The fracture features of each grade tested to failure under different LBA ratios and rate are shown in Figs. 2-5. In each figure, the fracture surfaces are arranged in an x-y coordinate according to their LBA ratio and test speed. The specimen thickness is 10 mm. Examination of the fracture surfaces revealed that, regardless of the material grade, the LBA ratio (constraint) and test speed have considerable influence on the failure mechanisms. It can be seen in Fig. 2 that the fracture surface of PE80 showed a transition from craze fibril failure at low LBA ratio and at high test speed to one which exhibited gross drawing and macroscopic necking at high LBA ratio and at low test speed. The fracture surface of PE100, on the other hand, shows a hollow neck appearance (Fig. 3) at high LBA ratio and at low test speed, indicating a shear mechanism near the surface and crazing on the interior. By comparison, the tip materials of BMPE experience higher fibril drawing with increasing LBA ratio at falling test speed (Fig. 4). The more blunted tip is likely to delay the onset of the crack growth process. By comparison, the HDPE, which is the only homopolymer examined here, showed a transition to localized fibrillation due to inward crack growth (Fig. 5). Based on the features observed, an increased in the LBA ratio is likely to invoke a ductile response, and hence, an increase in the separation energy. A similar effect can also be seen as the test speed reduces. On the other hand, a more brittle response can be expected in the low LBA ratio and high test speed regime. The changes in the fracture features are also manifested in the measured [GAMMA], [[sigma].sub.peak], and [[delta].sub.break]. However, the fracture characteristics cannot be simply explained using a single cohesive parameter. This is because the fracture properties are dependent not only on the area under the traction-separation curve but also on the actual cohesive zone parameters. For a given test condition, these parameters provided a unique description of the traction-separation behavior.

RESULTS AND DISCUSSION

Having identified the cohesive parameters, it is now possible to evaluate the variation in them with rate and constraint. For reasons of clarity, the results are organized in the following way. The [[sigma].sub.peak] data were divided into two groups: notch depth and size effects. Within each group, the data were plotted in the form of [[sigma].sub.peak] as a function of test speed. In each figure, there are six graphs; the first column consisted of three graphs which represent the notch depth effects while the second column shows the effect of specimen thickness. Under the first column, the first graph compares the data between the 10-mm specimens with different LBA ratios, while the 16- and 20-mm specimens are shown in the second and third graphs, respectively. The graphs are similarly arranged in the second column, except that the specimen thickness is varied while the LBA ratio remains constant.

Variation of Cohesive Stress With Constraint and Rate

Figure 6 shows the variation of [[sigma].sub.peak] with test speed of each material at various constraint conditions, along with the yield stress ([[sigma].sub.y]) data obtained from dumbbell specimens of 70 mm gauge length. As can be seen in Fig. 6, the [[sigma].sub.peak] and [[sigma].sub.y] of PE80 increase with the logarithm of test speed and both showed an almost linear rise in the stress value. Generally, the steep gradients of stress which exist at a notch would produce a deformation constraint so that the [[sigma].sub.peak] exceeds [[sigma].sub.y]. Therefore, the greater the degree of triaxial stress developed in the notch vicinity, the higher is the constraint, and hence, the higher would be the tensile stress required to induce deformation. As revealed in Fig. 6a-6c, the [[sigma].sub.peak] data of 10% LBA ratio (deep notch specimen) is much higher than that of the [[sigma].sub.y] (notch-free dumbbell sample). It can also be seen that the [[sigma].sub.peak] data of 10% LBA ratio specimen occupies an upper stress band, while the 50% LBA ratio specimen occupies a low stress band. The triaxial tensile stress state of the deep notch specimen (i.e. 10% LBA ratio) is obviously greater than that of the shallow notch specimens (30% and 50% LBA ratio), thereby giving rise to the higher [[sigma].sub.peak] values. In contrast, the variation in the [[sigma].sub.peak] values between specimens of different thickness is negligible, as shown in Fig. 6d-6f. This suggests that the [[sigma].sub.peak] is a material property, though a function of the constraint, in that it is independent of the specimen size. The [[sigma].sub.peak] trends in PE100, BMPE, and HDPE are similar to that of PE80.

Variation of Separation at Break With Rate and Constraint

Figures 7-10 show the variation of [[delta].sub.break] with test speed at various constraint conditions for PE80, PE100, BMPE, and HDPE, respectively. According to Fig. 7a, the [[delta].sub.break] values of 10% LBA ratio remain at an almost constant value across the speed range, while the 30% and 50% LBA ratio [[delta].sub.break] data showed an upward trend with falling test speed. However, the rise in the [[delta].sub.break] values is more significant in the 50% LBA ratio. It appears that the effect of an increase in ligament length (i.e. LBA ratio) is to reduce the plastic flow constraint. As the constraint is decreased, the notch-tip materials flow more readily to blunt the tip, as was reported in Ref. 19. However, the 50% LBA ratio samples seem to show signs of a low speed (<0.05 mm/min) plateau region. One might speculate that as the test speed is decreased further, a transition from ductile to brittle behavior may emerge as the value of [[delta].sub.break] begins to fall. On the other hand, the high plastic flow constraints in the 10% LBA ratio specimen tend to promote fracture (crazing) rather than flow, with the formation of homogeneous craze across the ligament throughout the speed range, as was seen also in the fracture surfaces in Fig. 2. At high loading rate (>10 mm/min), the [[delta].sub.break] values from different LBA ratio converge. It is likely that the craze growth is severely restricted because of fibrillar melting, as suggested by Leevers [19]. Pandya et al. [20] reported that crazing was suppressed as test rates were increased beyond 50 mm/min and showed a transition from ductile toward brittle behavior. The [[delta].sub.break] trends in Fig. 7b and 7c for specimen thickness of 16 and 20 mm respectively are similar to Fig. 7a.

Figure 7d-7f shows the variation of [[delta].sub.break] values with test speed between different sized specimens with identical LBA ratio. As can be seen from the figures, the difference in the [[delta].sub.break] values between the 10- and 16-mm specimens is not fully apparent, whereas the 20-mm specimens showed an upper bound [[delta].sub.break] values across the entire speed range. However, at a given test speed, the facture surfaces between the 20-mm and the smaller specimens did not reveal a difference in the failure mechanisms. In moving towards higher rates, the [[delta].sub.break] values of each thickness converge, suggesting a thermal-controlled material response.

The [[delta].sub.break] trends of PE100 in Fig. 8 resemble those seen in PE80. The notable difference is the rising trend in break separation values with falling test speed. The [[delta].sub.break] values of BMPE were also affected by the variation in notch depth, but the response was slightly different. It can be seen from the Fig. 9a-9c that at low test rates, the rise in the [[delta].sub.break] values with increasing LBA ratio is not as large as that in PE80 or PE100, indicating a greater plastic flow constraint in BMPE. By comparison, HDPE showed a transition from ductile to brittle behavior at the low speed regimes. The transition point becomes more pronounced with 50% LBA ratio in which a marked drop in the [[delta].sub.break] values, as seen in Fig. 10a-10c. On the other hand, the effect of size appears to be less noticeable on the [[delta].sub.break] values.

[FIGURE 8 OMITTED]

Figures 11-14 show the variation of [GAMMA] with test speed at various constraint conditions for PE80, PE100, BMPE, and HDPE respectively. As can be seen from the Fig. 11a, the energy values of the 10-mm specimen with a 50% LBA ratio occupy an upper energy band, while that of the 10% LBA ratio occupies a lower energy band. By comparison, the energy value of the 30% LBA ratio lies between the bands. The energy values of the 10% LBA ratio remained at a relatively constant value between the speeds of 0.005 and 50 mm/min. In contrast, the energy for 50% LBA ratio increases because of the changing deformation behavior, as was seen in the fracture surfaces in Fig. 2. The energy values are higher because of an increase in the [[delta].sub.break] values with falling test speed, as seen in Fig. 7a, even though the [[sigma].sub.peak] values in Fig. 6a showed a decline. In moving toward lower test rates (<0.05 mm/min), the fall in energy values was mainly due to the drop in the [[delta].sub.break] values. On the contrary, the difference in energy value is small at high test rates. Similar energy trends were observed with 30% LBA ratio. It would appear that there is a transition in behavior from being constraint dependent at the low-speed regime to a rate-dependent trend in the energy values at high rates. Increasing the specimen thickness to 16 mm (Fig. 11b) or 20 mm (Fig. 11c) produced similar energy trends.

Figure 11d-11f compares the energy values between specimens of different thickness. On the whole, the 20-mm specimen occupied an upper energy band, while the smaller sized specimens filled up the lower bands. In each figure, the same energy trends were observed between the 20 mm specimen and the smaller size specimen albeit with a slight difference in the energy values. This is due to the fact that more material becomes available for the plastic deformation energy absorption as the thickness is increased. However, the mechanics of deformation between the specimens are the same. By comparing the energy trends of size effects against the notch depth effects (Fig. 11a-11c), it becomes apparent that the variations in energy values are driven mainly by the changes in the notch depth.

[FIGURE 9 OMITTED]

One important observation to emerge from the foregoing discussion is the dependency of the energy values on the [[delta].sub.break] values at low speed regime. Comparison of the energy trends (Fig. 11) with the [[delta].sub.break] (Fig. 7) and the [[sigma].sub.peak] (Fig. 6) behavior shows that changes in energy are dominated by changes in the length of separation. As the energy transitions are driven primarily by changes in deformation mechanisms, they do not necessarily coincide with the stress. This is because any changes in the failure mechanism are not readily apparent from the slope of the stress-speed data which appears not to change with speed. It may be noticed that the trends in the energy values best reflect the changing features of the fracture surfaces in Fig. 2. Similarly, the energy trends of PE100 (Fig. 12), BMPE (Fig. 13), and HDPE (Fig. 14) were mainly governed by the effect of notch depth, which in turn depends on the [[delta].sub.break] trends, rather than being due to the effect of size.

Crack-Tip Constraint

In the preceding paragraphs, the results showed that the degree of geometrical constraint (plastic flow) acting at the crack /notch tip increases with an increase in notch depth (a decrease in LBA ratio). It is clear from the result presented in Fig. 6 that the high geometric constraint found in the 10% LBA ratio specimen (for a given thickness) elevated the degree of triaxial stress developed in the notch vicinity, causing yielding to occur at stress levels (i.e. [[sigma].sub.peak]) exceeding the uniaxial yield stress of the material. This is because it is more difficult to spread the yielded zone in the presence of triaxial stresses. On the other hand, the [[sigma].sub.peak] values fall with decreasing notch depth, signifying the reduction in the geometric constraint. One of the most noticeable features with the notch depth effects is to raise the [[delta].sub.break] values of the materials (Figs. 7-10). The increase in the [[delta].sub.break] values is due to a greater degree of crack-tip ductility which depends upon the size (volume) of the notch-tip plastic zone, which in turn depends upon the geometric constraint. It is known that the notch-tip plastic zone size increases with a decrease in constraint, thereby leading to a greater degree of crack-tip ductility. This was confirmed by the 50% LBA ratio sectioned specimen, which shows a larger whitened area surrounding the tips than those of the deep-notched specimen. For this reason, the fracture ductility increases (Figs. 11-14) as the crack tip is readily blunted, resulting in the increase of the [[delta].sub.break] values on the macroscopic scale. As the state of stress acting at the crack tip determines the geometric constraint, it is possible to quantify the notch-tip constraint level. Accordingly, a parameter called the crack-tip constraint factor, M, can be defined to describe the elevation of the applied stress at yield by an equation of the form:

M = [[sigma].sub.peak]/[[sigma].sub.y]. (1)

[FIGURE 10 OMITTED]

When M = 1, it would represent the plane stress case. A limiting M value, on the other hand, would denote a maximum constraint condition and represents a lower bound to the material toughness.

It can be seen from Fig. 15 that M values of PE80 decrease as the test speed increases, regardless of the LBA ratio and specimen thickness. On the other hand, comparison of the M values between specimens of different LBA ratios (Fig. 15a-15c) shows that the constraint factor is the highest for specimens with the 10% LBA ratio, while constraint is the lowest for specimens with a 50% LBA ratio. Therefore, as M increases (at the expense of the plastic zone size) with decreasing rate, one might expect a highly constrained failure, normally associated with cavitation and voiding (Fig. 2). On the other hand, a low M at low rates would be likely linked to a ductile fracture mode, owing to a much larger notch-tip plastic zone size which subsequently blunted the tip. These effects were accompanied by an increase in fracture toughness (Fig. 11). At high loading rates, the M values from different conditions converge. It would appear that the differences in deformation mechanisms observed due to different M values at low speeds become increasingly indistinguishable at high rates as was seen also in the fracture surfaces in Fig. 2. By comparison, the variation of M between specimens of different thickness is not significant (Fig 15d-15f), suggesting that M could be independent of the thickness effects. The same conclusions can be drawn for PE100, BMPE, and HDPE.

[FIGURE 11 OMITTED]

CONCLUSIONS

One important observation to emerge in the present work is that [GAMMA] is closely dependent on both the [[sigma].sub.peak] and the [[delta].sub.break]. Generally, at low constraint and rate conditions, the changes in the fracture features are manifested in the measured properties by an increase in achievable separation and a reduction in peak stress. Even though the peak stress decreases, the separation energy showed an increase because of the significant rise in the overall separation distance. On the other hand, the peak stress increases at high constraint and rate conditions, while the total separation distance and separation energy decreases. It thus showed that an accurate fracture assessment would require the description of all cohesive parameters.

[FIGURE 12 OMITTED]

A requirement of the results described here is to obtain an insight into how the traction curve for each material depends on constraint and rate, with a view to incorporating such effects into fracture models. From the detailed descriptions given, it is clear that there are patterns but the picture is rather complicated.

A basic parameter is the constraint factor, M, which is shown in Fig. 15. There is clear evidence that, as expected, M increases as the LBA ratio decreases. This occurs in all the materials at the low test speeds where M goes from about 2 at 50% to 2.5 at 10% LBA ratio. This rather modest change decreases the [[delta].sub.break] from about 12 to 2 mm for PE80, for example. Rather surprisingly, M decreases with rate and is about 1.7 for all materials at 50 mm/min. This is probably because of Poisson's ratio decreasing with rate since polymers tend to become more elastic at higher rate. A simple, elastic model of the constraint can be derived as follows. If [[sigma].sub.0] is the tensile stress in the bulk section and [^.[sigma]] is the stress in the ligament, then

[[sigma].sub.0] = [^.[sigma]]R (2)

where R = LBA ratio, and the hoop strain in the bulk is

[[epsilon].sub.[theta]] = -[[v[[sigma].sub.0]]/E] = -[[vR[^.[sigma]]]/E]. (3)

For the constrained ligament, one can assume that

[[sigma].sub.r] = [[sigma].sub.[theta]] = [[sigma].sub.c] (4)

where [[sigma].sub.r] is the radial stress, [[sigma].sub.[theta]] is the hoop stress, and [[sigma].sub.c] is the crazing stress, and that the hoop strain is the same as in the bulk, i.e.,

[FIGURE 13 OMITTED]

[[epsilon].sub.[theta]] = [1/E][[[sigma].sub.c] - v([^.[sigma]] + [[sigma].sub.c])] = -[vR[^.[sigma]]/E] (5)

and hence,

[[sigma].sub.c] = (v/[1 - v])(1 - R)[^.[sigma]]. (6)

For yielding of the ligament, we have,

[[sigma].sub.Y] = [^.[sigma]] - [[sigma].sub.c] = [^.[sigma]][1 - (v/[1 - v])(1 - R)] (7)

where [[sigma].sub.Y] is the tensile yield stress, and hence

M = [^.[sigma]]/[[sigma].sub.Y] = [([1 - 2v]/[1 - v]) + (v/[1 - v])R][.sup.-1]. (8)

Thus for high rate and elastic behavior v [right arrow] 0.3 and

M [right arrow] 7/[4 + 3R]

and is 1.6 for R = 0.1 and 1.3 for R = 0.5. At low rates v [right arrow] 0.4 say and

M [right arrow] 3/[1 + 2R] (9)

i.e., M = 2.4 and 1.5 for R = 0.1 and 0.5, respectively. Thus a rather modest change in v, i.e. 0.3 to 0.4, given the observed changes in M. Indeed, M may be taken as an indication of v, since

[FIGURE 14 OMITTED]

v = [M - 1]/[(2 - R)M - 1]. (10)

Since cavitation is the major failure mechanism involved in the failures discussed here it is appropriate to consider the hydrostatic stress in the ligament, [[sigma].sub.H], [e.g. Ref. 21]

[[sigma].sub.H] = [1/3]([^.[sigma]] + 2[[sigma].sub.c]) = [[^.[sigma]]/3][1 + [[2v(1 - R)]/[1 - v]]] (11)

and hence a hydrostatic stress factor is given by

H = [[sigma].sub.H]/[[sigma].sub.Y] = [M/3][[(1 - v) + 2v(1 - R)]/[1 - v]] = [1/3][[(1 - v) + 2v(1 - R)]/[(1 - 2v) + vR]] (12)

and on substituting for v from Eq. 10, we have

H = M - [2/3]. (13)

Thus H varies from about unity at high rates to almost two for R = 0.1 at low rates. The low rate data imply a power law dependence of the form

[[delta].sub.break] [proportional] [H.sup.-5] (14)

i.e., a very strong function. There is a concomitant dependence on rate of the form

[[delta].sub.break] [proportional] [dot.[delta].sup.-1/4] (15)

which has been observed previously [1]. The scatter in the [[delta].sub.break] data makes any more detailed fitting of the data too speculative. The rate dependence of [GAMMA] is largely controlled by [[delta].sub.break], but [^.[sigma]] increases with rate. Since [GAMMA] [proportional] [^.[sigma]], [[delta].sub.break] and the proportionality constant does not vary greatly from 0.5, [GAMMA] sometimes peaks but always falls to low values at high rates where [[delta].sub.break] dominates. The quantification of these effects must await a more detailed modeling of the cavitation processes in these rather ductile, rate-dependent materials.

[FIGURE 15 OMITTED]

ACKNOWLEDGMENTS

The authors are grateful to the Institute of Materials Research and Engineering of Agency for Science, Technology and Research (A*STAR) for hosting the completion of this study.

REFERENCES

1. M.K.V. Chan and J.G. Williams, Polymer, 24, 234 (1983).

2. I. Narisawa and M.T. Takemori, Polym. Eng. Sci., 29, 671 (1989).

3. D.M. Duan and J.G. Williams, J. Mater. Sci., 33, 625 (1998).

4. D.S. Dugdale, J. Mech. Phys. Solids, 8, 100 (1960).

5. A. Hillerborg, M. Modeer, and P.E. Petersson, Cem. Concr. Res., 6, 773 (1976).

6. V. Tvergavrd and J.W. Hutchinson, J. Mech. Phys. Solids, 40, 1377 (1992).

7. M.G.A. Tijssens, V.D. Giessen, and L.J. Sluys, Mech. Mater., 32, 19 (2000).

8. Q.D. Yang and M.D. Thouless, Int. J. Fract., 110, 175 (2001).

9. W. Li and T. Siegmund, Eng. Fract. Mech., 69, 2073 (2002).

10. M. Elices, G.V. Guinea, J. Gomez, and J. Planas, Eng. Fract. Mech., 69, 137 (2002).

11. J.G. Williams and H. Hadavinia, J. Mech. Phys. Solids, 50, 809 (2002).

12. K.C. Pandya and J.G. Williams, Polym. Eng. Sci., 40, 1765 (2000).

13. K.C. Pandya and J.G. Williams, Plast. Rubber Compos., 29, 439 (2000).

14. S.K.M. Ting, J.G. Williams, and A. Ivankovic, Proc. 12th Int. Conf. Def., Yield Fract. Polymers, 17 (2003).

15. S.K.M. Ting, J.G. Williams, and A. Ivankovic, "Effects of Constraint on The Traction-Separation Behaviour of Polyethylene," in Fracture of Polymers, Composites and Adhesives II, B.R.K. Blackman, A. Pavan, and J.G. Williams, editors, Elsevier Science, Oxford, 143 (2003).

16. T. Siegmund and W. Brocks, Int. J. Fract., 99, 97 (1999).

17. A. Ivankovic, K.C. Pandya, and J.G. Williams, Eng. Fract. Mech., 71, 657 (2004).

18. S.K.M. Ting, J.G. Williams, and A. Ivankovic, 46, 763 (2006).

19. P.S. Leevers, Int. J. Fract., 73, 109 (1995).

20. K.C. Pandya, E.Q. Clutton, and L.J. Rose, "The Influence of Molecular Structure on Impact and Stress Crack Performance in Polyethylene." (in preparation).

21. Y. Bao and T. Wierzbicki, Int. J. Mech. Sci., 46, 81 (2004).

S.K.M. Ting, J.G. Williams, A. Ivankovic

Department of Mechanical Engineering, Imperial College London, London SW7 2BX, United Kingdom

Correspondence to: J.G. Williams; e-mail: [e-mail]

Current address for S.K.M. Ting is Arkema-Kyoto Technical Center, SCB#3, Kyoto Research Park, 93 Chudoji Awatacho, Shimogyo-ku, Kyoto 600-8815, Japan.

Current address for A. Ivankovic is University College Dublin, Department of Mechanical Engineering, Engineering Building Belfield, Dublin 4, Ireland.

Contract grant sponsor: BP-Solvay Polyethylene.

TABLE 1. The properties of polyethylene studied. Where [rho] is the
density, SCB is the side chain branch density, and [M.sub.w] is the
weight average molecular weight.

                                               [M.sub.w]
PE types  [rho] (kg [m.sup.-3])  SCB (/1000C)  (g mol[.sup.-1])

PE80      940                    4.5           185,000
PE100     947                    2.5           310,000
BMPE      947                    1.5           290,000
HDPE      954                    0             355,000