Transition of neck appearance in polyethylene and effect of the associated strain rate on the damage generation.
Polyethylene (PE) is known for its excellent ductility, superb corrosion resistance and strong strain hardening ability , Because of those advantages, PE has been used for applications that require long-term durability, such as pipes for natural gas transportation. Although PE performance for those applications can be assessed using standard methods [2, 3], accuracy of the assessment is limited due to uncertainty about failure mechanisms that play a significant role only after many years in service.
For use of PE in applications such as pipeline for natural gas transportation, very strict requirements are specified to determine PE's suitability . But meeting those requirements does not warrant the time to failure to be longer than the designed lifetime which is often more than 50 years. This is partly due to the complex nature of the fracture behavior that is commonly referred to as slow crack growth (SCG) , Development of SCG was discovered from early tests on full-sized pipe specimens that were subjected to internal hydrostatic pressure for a period of more than 1 year [6-8], The tests show a transition in the relationship between hoop stress of the pipe and time to failure. Although the time to failure increases by decreasing the hoop stress, the rate of time increase is faster above a critical stress level than below. As a result, data obtained from laboratory tests at a high stress level, to shorten test duration, can lead to overestimate of the time to failure in service which is at a low stress level. Furthermore, fracture generated above the critical stress level is ductile, while below very brittle. At this stage, no laboratory test method has the capability to accurately identify the critical point and the corresponding stress level.
Because of the significant consequence from failure of pipes for natural gas transportation and length of the long duration involved for testing the full-sized specimens, many studies have been dedicated to search for short-term tests to characterize the transition to SCG development in a timeframe that is suitable for the laboratory testing. Most of those tests use small coupon specimens with sharp notches to initiate crack growth under constant loading [9-11]. Such tests have successfully generated a trend line with the transition phenomenon bearing some similarity to the transition obtained from the pipe testing  and one of the tests is now selected as a standard for PE testing . However, due to the use of the artificial notches, results from those tests cannot determine the critical stress level for the SCG development in service.
In addition to the above problem, the above laboratory tests do not always generate brittle fracture in a way similar to that observed in service. One coupon test method that successfully generates transition from ductile to brittle fracture was reported by O'Connell et al. , which also used specimens with sharp notches but under a controlled rate of stroke increase, instead of maintaining constant load. The study observed a ductile-to-brittle transition in fracture behavior by decreasing the crosshead speed and increasing temperature. In the extreme case of applying the crosshead speed of 0.005 mm/min at 110[degrees]C, a featureless fracture surface is generated which does not show any trace of plastic deformation or whitening. Such a brittle fracture behavior is very different from those reported in Refs. 14-18 in which whitening occurs at the notch tip before crack growth commences. It should be noted that specimens in the latter studies were always subjected to a constant load at a lower temperature (80[degrees]C or below).
Mechanism responsible for brittle fracture of notched specimens is known to be crazing. Lu and Brown  suggested that ductile-brittle transition from the notched specimens is due to the change from global shear deformation (ductile) to shear deformation of craze fibrils (brittle) . However, the explanation is not applicable to another ductile-brittle transition that occurs by changing the crosshead speed and temperature in the opposite direction . It should be noted that the two types of brittle fracture are very different, but to our knowledge, no attempts have been made to evaluate their relevance and difference.
Furthermore, strain rate at the notch tip is not zero even under constant loading. Rate of length increase in the craze fibrils has been well documented using microscopy [9, 14, 15, 17]. Although those studies did not quantify the strain rate applied to the craze fibrils, the strain rate can be estimated based on the rate of increase in the crack tip opening displacement (CTOD). Brown and Lu  reported that the rate of CTOD increase should be in the order of [10.sup.-4] mm/min for a notched specimen subjected constant loading. With the average CTOD of 0.015 mm, the corresponding strain rate for the craze fibrils should be about 1 x [10.sup.-4] [s.sup.-1], which can be generated on a notch-free specimen with a gauge length of 20 mm using a crosshead speed around 0.1 mm/min. As a result, deformation at a strain rate lower than 1 x [10.sup.-4] [s.sup.-1] should be studied using notch-free specimen at an appropriate crosshead speed.
In view of the time required in service for the SCG initiation and propagation, strain rate involved must be much lower than 1 x [10.sup.-4] [s.sup.-1]. However, most studies, even using notch-free specimens, still apply strain rates above 1 x [10.sup.-4] [s.sup.-1] [e.g., [20-22], The lowest strain rate that we could find in the literature is about 1 x [10.sup.-5] [s.sup.-1] . In view of lack of information for deformation at a low strain rate, work presented in this article is focused on mechanical testing of notch-free PE specimen at a crosshead speed from 5 to 0.001 mm/min, corresponding to an initial strain rate from 4 x [10.sup.-3] to 8 x [10.sup.-7] [s.sup.-1].
It should be pointed out that although tests conducted in our study do not maintain a constant strain rate the initial strain rate serves as a good indicator for the difference of the strain rates applied to the specimens, especially for deformation before the neck is developed.
The article will show that by reducing the crosshead speed, neck appearance changes from the usual opaque white (at a high crosshead speed) to translucent (at a low crosshead speed), with the transition occurring at the crosshead speed between 0.05 and 0.01 mm/min. Since the opaque white color is an indication of cavities existing in the specimen [24-26], damage that leads to cavitation at 1 mm/min could have been generated before the necking process. Using the crosshead speed of 1 and 0.001 mm/min, with the former leading to opaque white neck and the latter translucent neck, the article investigates effects of the damage on the mechanical properties of PE in small deformation. The article also investigates whether degradation of the mechanical properties, due to the presence of damage, can be detected at a strain level below 10%, i.e., before yielding of the material in the tensile test.
Specimens used for the mechanical testing are prepared from PE plaques of 10 mm thick, provided by NOVA Chemicals, of which molecular weight, molecular weight distribution, density and tensile strength are given in Table 1. The plates were compression-molded from pellets to ensure isotropy of the mechanical properties. Cylindrical specimens with dimensions given in Fig. 1, same as those used in ref. , were machined from those plates. Distance between the grips for testing is 40 mm, as marked in Fig. 1, within which most of the deformation occurs in the middle section of 20 mm in length and 6 mm in diameter. To ensure that necking was always initiated in the middle of the section where an extensometer was placed to monitor the change of diameter during the test, a small imperfection was introduced by reducing the diameter there by less than 2%. Note that despite the presence of such an imperfection, all specimens showed full neck development in the whole 20-mm section before the final fracture.
All tests were conducted using a universal testing machine (QUASAR 100) at room temperature. Eight crosshead speeds between 0.001 and 5 mm/min are selected, i.e., at the crosshead speeds of 1 x [10.sup.-n] and 5 x [10.sup.-n] mm/min with n = 0, 1, 2, and 3. The corresponding initial strain rate is in the range from 8 x [10.sup.-7] to 4 x [10.sup.-3] [s.sup.-1]. As the tests are under stroke control and PE is a polymer with highly nonlinear deformation, the initial strain rate represents the lower bound of possible strain rates generated during the test. However, in the strain range before the neck formation, the crosshead speed serves as a good indicator for the difference of the strain rates used in the study, and is used as the variable to depict the effect of strain rate on the mechanical properties.
Two types of mechanical tests are used in the study. One, named simple tensile test, applies a constant crosshead speed until the stroke reaches a preset value of 15 mm at which the neck has been fully developed. The other, named relaxation test, applies a constant crosshead speed to reach a preset stroke and then, the stroke is held constant for [10.sup.5] s (about 28 h) to relax the load. At the end of this holding period, the specimen is unloaded at 5 mm/min to zero load and then held at that stroke for a period of 7 x [10.sup.4] s. Schematic diagram for the relaxation test is shown in Fig. 2.
For both types of tests, changes of the load and diameter of the cross section are recorded for calculation of stress and area strain. The latter is defined as ln([A.sub.o]/A) where [A.sub.o] is the original cross-sectional area and A the cross-sectional area at the moment of measurement.
Note that the relaxation test is designed to determine the long-term mechanical properties of PE without the involvement of viscous components. A similar test scheme has been used by Strobl and coworkers [28, 29] for the same purpose. They show that due to the viscous nature of polymer, load drops immediately after stopping the crosshead movement during the loading cycle (during [t.sub.RL] in Fig. 2), and the load rises during the unloading cycle (during [t.sub.RU] in Fig. 2). They also show that by interrupting the test at the same strain level during the loading and unloading cycles, the two loads reach the same value eventually. The corresponding stress represents the true stress for the given strain without the viscous component. Such a test scheme enables the establishment of long-term stress-strain relationship that does not have any contribution from the viscous property. However, for PE used in this study, the time required for the two loading levels to meet each other is too long to be practical. Therefore, a period of [10.sup.5] and 7 x [10.sup.4] s was selected for relaxation during the loading and unloading cycles, respectively. In addition, the strain for introducing the relaxation during the loading cycle is not at the same level as that during the unloading cycle, to take advantage of the control function provided by the test machine so that the test can be interrupted for relaxation automatically when a preset condition is met. Nevertheless, by introducing the relaxation at different stroke values during the loading and unloading cycles, two stress-strain curves are established which correspond to the upper and lower bounds of the long-term stress-strain relationship which has the majority of the viscous components removed.
Since main interest for the current study is the effect of strain rate (varied by the change of crosshead speed) on the long-term deformation behavior, the relaxation test, with the loading scheme shown in Fig. 2, is used to assess the effect of strain rate on the long-term stress-strain relationship. Two crosshead speeds, 1 and 0.001 mm/min (mL in Fig. 2), were used to introduce the deformation because as to be shown in the next section, neck generated at the two crosshead speeds has distinctly different appearance.
RESULTS AND DISCUSSION
Simple Tensile Test
Figure 3 summarizes the relationship between engineering stress and stroke from simple tensile tests at different crosshead speeds. Here, stress is plotted as a function of stroke, instead of engineering strain, because neck formation invalidates the use of engineering strain to represent the deformation. However, special attention has been paid to the test setup to make sure that initial length between the grips is kept the same for all tests so that the stroke value reflects consistently the specimen elongation. Similar to that reported in the literature , peak stress decreases with the decrease in crosshead speed. In addition, decrease in the crosshead speed causes broadening of curve profile around the peak stress, and changes the peak profile from single-peak to double-peak with the two peaks further apart from each other.
In addition to the changes depicted in Fig. 3, necked section of the specimens shows a color transition from opaque white to translucent with the decrease in the crosshead speed from 0.05 to 0.01 mm/min, as presented in Fig. 4. Note that the translucent color is very similar to that of the original specimen.
Figure 5 shows a plot of the maximum engineering stress as a logarithmic function of crosshead speed, with the insert showing a zoom-in plot of the maximum engineering stress as a linear function of the crosshead speed below 0.01 mm/min at which color of the specimens no longer changes with the neck development. Neither of the two plots shows any trend of reaching a plateau value by decreasing the crosshead speed, but the insert plot indicates that the maximum engineering stress should be below 10.5 MPa at a crosshead speed close to zero, that is, below 50% of the tensile strength shown in Table I. The question is how low the maximum engineering stress could go when the crosshead speed reaches 0 mm/min. Answer to this question is searched using relaxation test, as discussed in the next section.
Key issue to answer the above question is to remove the viscous component of the stress, as its contribution is expected to diminish when the crosshead speed reaches zero. The idea is to conduct the relaxation tests at several strain levels, to collect the stress response to those strains without the viscous component. The corresponding stress-strain curve represents the long-term constitutive equation for deformation without the viscous component.
The above idea was first proposed and verified by Hong et al.  using poly(ethylene-co-12% vinyl acetate) that shows fairly fast stress relaxation, thus being able to reach a stress plateau at a given strain after a relatively short period of [10.sup.4] s. For PE used here, a preliminary study suggests that the stress continues to drop at a fixed stroke after 160 h, without any sign of reaching the plateau. Therefore, it is impractical to allow complete stress relaxation at each strain level. Instead, a relatively short period of [10.sup.5] s was chosen for the stress relaxation after the loading cycle, and 7 x [10.sup.4] s for the relaxation after the unloading cycle, which are represented by [t.sub.RL] and [t.sub.RU] in Fig. 2, respectively. Results from the preliminary study also indicate that the stress drop after the relaxation period of [10.sup.5] s (less than 28 h) is 67% of the stress drop for a relaxation period of 160 h. Therefore, the relaxation period of [10.sup.5] s for [t.sub.RL] is deemed a good compromise between removal of the viscous stress component and efficiency in the use of time. For [t.sub.RU], the period of 7 x [10.sup.-4] s was found to be sufficient for the same purpose.
Figure 6 gives an example of engineering stress-stroke plot from the relaxation test. Crosshead speed ([m.sub.L] in Fig. 2) used for the testing was 0.001 mm/min and the stroke ([[DELTA].sub.L]) 0.8 mm. The stress drop after the loading cycle and stress rise after the unloading cycle are similar to the phenomena reported by Hong et al. , except that in the current study, the two relaxation periods are not introduced at the same stroke and the final stress values differ by about 2-3 MPa. The latter is due to the relatively short relaxation periods that do not allow the stress to reach a plateau value. The former, as to be shown later, is due to the possibility of damage generation during the loading cycle if the specimen were loaded further from Point B to have [[DELTA].sub.U] = [[DELTA].sub.L]. Despite the difference, those stress values provides some interesting information, as discussed below, about PE's long-term response to deformation.
Figure 7 depicts variation of engineering stress after the relaxation as a function of stroke, at the crosshead speed ([m.sub.L] in Fig. 2) of 0.001 and 1 mm/min for (a) and (b), respectively. Again, stroke is used instead of engineering strain as the parameter to depict the change of engineering stress, due to the onset of neck formation which is apparent when the stroke is above 6 mm. As mentioned before, special attention has been paid to make sure that the specimens were gripped in the same position for the tests so that the stroke value reflects consistently the specimen elongation. Three sets of data are presented in each plot of Fig. 7. The upper round markers of small size represent stresses at the end of the relaxation period at [[DELTA].sub.L], the lower round markers of small size at [[DELTA].sub.U], and the middle square markers of large size the average of the above two stress values at the average of [[DELTA].sub.L] and [[DELTA].sub.U]. The plots suggest that after the stress relaxation, the maximum engineering stress at 0.001 mm/min is higher than that at 1 mm/min, for the average value of 8.8 and 8 MPa, respectively, and after the peak stress, around 4.6 mm of the average stroke, stress at 1 mm/min shows a tendency of decrease with the increase in stroke, but not for stress at 0.001 mm/min.
Some photographs for post-test specimens at 1 mm/ min are shown in Fig. 8. Neck whitening is visible only from the right photograph (with [[DELTA].sub.L] = 8 mm). Specimens tested at 0.001 mm/min have appearance very similar to that shown in Fig. 8 except for the specimen with [[DELTA].sub.L] = 8 mm which at 0.001 mm/min does not show any whitening.
To further evaluate mechanical behavior of the specimens at the two crosshead speeds, two specimens were selected for testing twice. In the first test, the two specimens were stretched to the same stroke of 1.8 mm, but at different crosshead speeds, 1 and 0.001 mm/min, respectively, following the testing scheme shown in Fig. 2. After the first test, for a period more than 3 weeks, those specimens were stretched again to 1.8 mm in the second test, but this time both were stretched at the crosshead speed of 0.001 mm/min. Plots of engineering stress versus stroke from the second test, along with one from a virgin specimen stretched at 0.001 mm/min, are shown in Fig. 9. The figure clearly suggests that after the first test at 1 mm/min, compliance of the specimen increases, which results in a stress at the stroke of 1.8 mm lower than that tested at 0.001 mm/min. Little difference was found between the virgin specimen and the specimen first tested at 0.001 mm/min.
Figure 10 compares curves of true stress versus area strain determined at the two crosshead speeds, using points at the end of the relaxation period in the loading cycle, i.e., at the end of [t.sub.RL] in Fig. 2, for [[DELTA].sub.L] in the range from 0.8 to 6 mm. Since the stress relaxation should have removed most of the viscous component of stress, the two curves in Fig. 10 represents the long-term relationship between stress and strain without much influence from the viscous component. The figure indicates that the curve for 1 mm/min lies below that for 0.001 mm/min, with the difference increasing with the increase in area strain. This trend is opposite to that expected by including the viscous stress component, as the viscous component should have made the curve for 1 mm/min above that for 0.001 mm/min. Since specimens tested at 1 mm/min generate a neck that changes the color to opaque white, it is reasonable to relate the color change to the mechanism that causes the stress-strain curve for 1 mm/min to be lower than that for 0.001 mm/min.
There are two possible causes for the stress whitening during the neck forming process. One is the consequence of dilatational deformation  that introduces cavities in the neck region [32-35]. Pae et al.  show that the whitening phenomenon can be completely removed at room temperature, by subjecting the whitening specimens to a very high hydrostatic pressure to close the cavities. Based on this idea, the neck whitening observed in our study is possibly due to sufficient hydrostatic tension generated during the neck forming process. As suggested by G'sell et al. , the hydrostatic tension can be generated in the externally concave region of the specimen during the necking. Since the level of the hydrostatic tension is proportional to the applied axial stress, our results suggest that the hydrostatic tension is high enough to induce cavitation in specimens tested at a crosshead speed above 0.1 mm/min. By reducing the crosshead speed to 0.05 mm/min or below, the hydrostatic tension is no longer sufficient to induce the cavitation, thus color of the neck remains translucent, not changing to opaque white.
It is also possible that both opaque white neck and translucent neck are caused by craze formation, with the former by the classical crazes that are developed in discrete regions but the latter by the so-called "delocalized" crazes  that consist of tiny pores of uniformly distributed through the whole volume. However, not much work is available in the literature about the concept of "delocalized" craze, and at this stage, it is not clear whether the delocalized craze is just another expression for the cavity generated by hydrostatic tension.
Although the above discussion provides plausible explanations for the change of neck appearance by decreasing the crosshead speed, it does not provide direct explanation for the drop of load-carrying capacity at a strain level as low as 3.7% (for a stroke of 0.8 mm), by increasing the crosshead speed from 0.001 to 1 mm/min, as shown in Figs. 9 and 10. At such a low level of deformation, there are two mechanisms that that have been widely accepted for damage in semicrystalline polymers. One is intralamellar slip and the associated dislocation generation, and the other is deformation in the interlamellar, amorphous region and the associated chain (tie molecule) movement. As pointed out by Bartczak and Galeski , due to the interleaved arrangement between the crystalline lamellae and the interlamellar amorphous regions, deformation damage in PE is deeply influenced by interaction of the two mechanisms, thus difficult for separate analysis. Nevertheless, results from the current study suggest that damage can be generated at a very small deformation level that is way before the neck is initiated.
Although it is yet to clarify the mechanism that is responsible for the reduction of PE's load-carrying capacity at the small level of deformation, results presented in this work indicate clearly that the change of crosshead speed, thus the change of strain rate, affects the long-term mechanical properties of PE. Since the change is detectable at very small strokes of 0.8 and 1.8 mm, corresponding to a strain level about 3.7 and 8.3%, respectively, which are close to the allowable deformation level for PE pipes in service, damage generated by loading PE products at a relatively high strain rate could be accumulated to shorten their service life. Therefore, possibility of generating this type of damage in service and resistance of PE to this type of damage should be considered for selection and analysis of PE products, to assure their reliability and durability in the long-term service.
An experimental study is presented to assess long-term strength of PE and influence of strain rate (in terms of crosshead speed) on the damage generation. The study is based on a newly discovered "deformation transition" in notch-free specimens, which occurs by decreasing the strain rate to a level that has rarely been studied in the past, but is realistic for long-term applications. The discovery leads to evaluation of the effect of strain rate on the long-term mechanical properties of PE, i.e., mechanical properties without the influence of the viscous component. The results suggest that without the contribution from the viscous component, long-term strength of PE is only about 50% of the strength determined from the standard test.
The study also shows that degradation of the mechanical properties can be detected at a strain level lower than 4%. Since deformation at such a strain level can occur during the pipe transportation and installation, or even during the service, and the strain rate introduced by a crosshead speed used in the study, 1 mm/min, is considered to be within a normal range for the service, possibility of generating this kind of damage may further reduce strength of the PE products. Therefore, long-term strength of virgin PE only represents the upper limit of its strength in applications such as for natural gas transportation. Proper assessment of long-term performance of PE products should include the loading conditions after the product fabrication. To our knowledge, such assessment is not considered in any method at present. Therefore, a new methodology is needed to quantify and monitor the damage evolvement in a PE product and its effect on the long-term performance.
Sincere appreciation is due to technical staff in the Department, particularly Campbell, Waege, and Bubenko for specimen preparation and Faulkner for fabrication of the extensometer used in the testing.
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P.-Y. Ben Jar
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8
Correspondence to: P.-Y. Ben Jar; e-mail: email@example.com Contract grant sponsor: Natural Sciences and Engineering Research Council of Canada (NSERC).
Published online in Wiley Online Library (wileyonlinelibrary.com).
TABLE 1. Material characteristics of PE used in this study. Weight-average Number-average molecular weight, molecular weight Branches [M.sub.w] (g/mol) 122 [M.sub.n] (g/mol) per 1000 C 73,074 30,391 3.4-4.2 Tensile Weight-average Density, [rho] strength molecular weight, (ASTM D792) Crystallinity (ASTM D638) [M.sub.w] (g/mol) (g/[cm.sup.-3]) (%) (MPa) 73,074 0.941 63.6 20.2 Standard methods used to measure density and tensile strength are provided in the table.
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|Author:||Jar, P.-Y. Ben|
|Publication:||Polymer Engineering and Science|
|Date:||Aug 1, 2014|
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