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Application of the Essential Work of Fracture Concept to High Temperature Deformation in Polyoxymethylene.

C.J.G. PLUMMER [*][+]



R. W. LANG [**]

H. H. KAUSCH [*]

Essential Work of Fracture (EWF) tests have been carried out on double edge notched samples machined from injection molded sheets of commercial grades of polyoxymethylene homopolymer with different molecular weight averages. Most of the measurements were made at 100[degrees]C and over a range of test speeds in which polyoxymethylene is anticipated to undergo a macroscopic ductile-brittle transition with decreasing strain rate. The results reflect both the existence and the molecular weight dependence of this transition, and are argued to be valid in terms of the European Structural Integrity Society's EWF draft test protocol under certain test conditions. However, it is shown that the applicability of the test method used here becomes highly questionable for test speeds in the immediate vicinity of the transition, owing to the influence of the initial ligament length on the crack tip deformation mechanisms.


In room temperature tensile tests and for the usual range of strain rates, polyoxymethylene (POM) is not macroscopically ductile, insofar as failure is not preceded by substantial plastic necking. However, the strains to failure, thought to be accommodated by internal cavitation [1], can reach up to 70%, depending on the molecular weight, so that the effective toughness of POM remains comparable with that of other engineering thermoplastics. As the test temperature is raised, or the effective strain rate reduced [2, 3] (in static and dynamic fatigue tests, for example), there is an initial transition to plastic necking. accompanied by a decrease in cavitation. However, further increases in test temperature lead to suppression of global plasticity in un-notched samples, provided that the strain rate is sufficiently low, In contrast to the observed room temperature behavior, the strains to failure in this latter regime are of the order of a few percent, and we therefore refer to a "ductile-brittle transition."

This is of some practical concern, and indeed one might anticipate embrittlement to occur in POM at room temperature for sufficiently long times under low loads, although we have not so far been able to achieve such conditions consistently in our laboratory. Our investigations have therefore been concentrated on the influence of strain rate, and also the molecular weight, at a test temperature of 100[degrees]C. This temperature is sufficient to induce the ductile-brittle transition within a convenient range of strain rates and does not lead to significant chemical degradation of the samples over the times needed to complete the tests (based both on thermo-gravimetric data and on tensile tests on samples aged at 100[degrees]C over much longer times [4]).

In order to characterize the behavior of POM in the vicinity of the transition we have previously made use of slow crack growth measurements under static loading [5]. However, quantitative characterization of the fracture toughness using linear elastic fracture mechanics is not possible in the corresponding regimes of temperature and strain rate and for the available range of sample geometries, owing to the large plastic zone size in notched specimens and the consequent non-linearity of the force-displacement curves. One alternative approach to failure in relatively ductile materials which has gained considerable recent attention in the context of polymers [6-9], is based on the Essential Work of Fracture (EWF) [10]. In the present paper we report on our attempts to apply the EWF method to high temperature tensile fracture in POM in order to compare quantitatively the behavior of grades with different molecular weights. The extent to which the results reflect changes in deformation mechanisms underlying the ductile-brittle transition, and their validity in terms of the current European Structural Integrity Society's (ESIS) draft test protocol will be discussed [11].


The EWF test method and the concepts on which it is based have been extensively discussed in the references cited above and will only briefly be summarized here. It is assumed that the total work of fracture, [W.sub.f], can be separated into the essential work of fracture, [W.sub.e], and the non-essential work of fracture, [W.sub.p], such that

[W.sub.f] = [W.sub.e] + [W.sub.p] (1)

In the double edge notched tensile (DENT) geometry shown in Figure 1, this becomes

[frac{[W.sub.f]}{lt}] = [w.sub.f] = [w.sub.e] + [beta]l[w.sub.p] (2)

where [w.sub.e] ([[Jm.sup.-2]]) is the specific essential work of fracture required for the creation of unit area of crack face, and is considered to be a materials parameter for a given sample thickness, t. [w.sub.p] ([[Jm.sup.-3]]) accounts for the remainder of the energy dissipated up to total fracture of the sample and [beta] is a shape factor. By equating [w.sub.f] with the area under experimental force displacement curve divided by lt, and carrying out measurements at different l, [w.sub.e] can be obtained by a simple linear extrapolation. For such a procedure to be valid, [w.sub.e] and [beta][w.sub.p] should be independent of l. A necessary condition for this is generally held to be that the shapes of the projection of the plastic zone onto the specimen plane be similar for different l (implying similar force-displacement curves). The ESIS draft test protocol consequently recommends that l not be greater than twice the plastic zone size

[r.sub.p] = [frac{E[w.sub.e]}{[pi][[[sigma].sup.2].sub.y]}] (3)

(or W/3, depending on which is smaller). At the same time l should not be less than 3 to 5 times t in order to avoid the transition from plane stress to plane strain as l becomes comparable with t. To verify the consistency of the data it is also recommended to compare the maximum mean stress in the ligament, [[sigma].sub.max], with its average value for all l under consideration, [[sigma].sub.m], Data associated with [[sigma].sub.max] which differ by more than 10% from [[sigma].sub.m] should be rejected. As a further check, [[sigma].sub.max] may be compared with its theoretical value of 1.15[[sigma].sub.y], where [[sigma].sub.y] is the yield stress of an un-notched specimen measured at a test speed which gives a similar time to yield to that of the DENT specimen.

The materials used in the present study were commercial grades of POM homopolymer from DuPont de Nemours, with number average molecular weights, [M.sub.n], of 35, 41 and 66 kg/mol. They were supplied in the form of 100 X 100 [mm.sup.2] edge-gated injection molded plaques, with a nominal thickness of 1 mm. DENT specimens with the dimensions shown in Fig. 1 were machined from the plaques with the tensile axis both normal and parallel to the injection direction. However, unless stated otherwise, the results in what follows refer to specimens whose tensile axis is parallel to the injection direction. Pre-cracking was carried out by sliding a fresh razor blade along the notch tip just prior to testing. Tensile tests were performed using a screw-driven Zwick 1484 tensile test machine, equipped with an environmental chamber and an integrated frame-grabber, image analysis and numerical data acquisition package developed in-house using [Labview.sup.TM] software (National Instruments). The specimens were tested at spe eds in the range 0.002 to 2 min/min after tempering for one hour at the final test temperature.


The test speed suggested in the ESIS draft test protocol [11] would be of the order of 8 mm/min for the present geometry. For all the grades of POM investigated, crack propagation was partially unstable in tests at this speed and at 100[degrees]C, particularly for the largest l. However, at lower test speeds, stable crack propagation was consistently observed. Typical force-displacement curves are given for [M.sub.n] = 41 kg/mol and l = 8 mm for the different test speeds in Fig. 2a. As is often the case with semicrystalline polymers above their [T.sub.g], there was no sharp yield drop, and indeed POM deformed under these conditions in simple tension shows gradual drawing down over the whole gauge length rather than the formation of a well-defined neck. Figure 2b shows the EWF analysis for a range of l values. The specimens tested at 2, 0.2 and 0.002 showed comparable values of [w.sub.e] (between 9 and 11.5 [kJm.sup.-2]) although [beta][w.sub.p] (proportional to the slopes of the curves in Fig. 2b) was reduced by a factor of about 10 at the lowest test speed. On the other hand, at 0.02 mm/mm, the analysis led to [w.sub.e] = 50 [kJm.sup.-2].

Results from the EWF analysis for all three grades investigated are summarized in Fig. 3, which shows the derived parameters as a function of test speed. Figure 4 shows corresponding values of [[sigma].sub.max] plotted against the time at which [[sigma].sub.max] was reached during the test, and 1.15[[sigma].sub.y] from un-notched specimens for [M.sub.n] = 35 kg/mol and 66 kg/mol. Finally, Fig. 5 shows force-displacement curves for [M.sub.n] = 35 kg/mol and l = 8 mm along with video captures corresponding to the vertical bars on the former, illustrating the shape of the plastic zone and its evolution during the tests.


We begin this section by considering the evolution of [[sigma].sub.max] with l and the test speed. As shown in Fig. 4b, for the highest molecular weight, [[sigma].sub.max] and 1.15[[sigma].sub.y] were in reasonable agreement over the whole range of times to yield, these latter being taken to provide an effective measure of the local strain rate. The time to yield generally decreased with l for a fixed test speed and so the systematic decrease in [[sigma].sub.max] was assumed to reflect the strain rate sensitivity of [[sigma].sub.y]. Nevertheless, for the lowest molecular weight, [[sigma].sub.max] and 1.15[[sigma].sub.y] diverged significantly at the lowest test speeds, [[sigma].sub.max] decreasing more rapidly with increasing time to yield than [[sigma].sub.y].

The results for [M.sub.n] = 66 kg/mol were encouraging in that they suggested that full section yielding did occur prior to crack advance. However, taking [w.sub.e] = 10 [kJm.sup.-2], E = 1 Gpa and [[sigma].sub.y] = 30 MPa and using Eq 3, it is clear that the criterion l [less than] [r.sub.p] was not satisfied for l greater than about 7 mm for a test speed of 2 mm/ min. Experience with other materials has nevertheless indicated this to be an excessively conservative criterion [12] and in view of the similar shape of the force-displacement curves and the linearity of [w.sub.f] vs l, data for l up to 15 mm were retained in the analysis for this test speed. For [M.sub.n] = 66 kg/mol tested at 0.2 mm/min, on the other hand, there was significant curvature of [w.sub.f] vs l, so that whereas extrapolation for l between 3 and 14 mm gave [w.sub.e] of about 30 [kJm.sup.-2], extrapolation for l between 3 and 7 mm gave [w.sub.e] of about 14 [kJm.sup.-2]. It is this latter value which has been given in Fig. 3, alt hough the reasons for the curvature in this particular set of data were not clear, and it should be therefore be treated with caution. The situation was still less clear at lower test speeds owing to the onset of pronounced necking down of the plastically deformed region, with failure occurring by ductile tearing, accompanied by considerable work hardening in the force-displacement curve at large deformations. The absolute values of [w.sub.f] were about an order of magnitude greater than at the higher test speeds, as was [beta][w.sub.p] so that although [w.sub.f] vs l was apparently linear, the uncertainty in the derived [w.sub.e] value of 41 kg/mol was of the order of [w.sub.e] itself. This value was therefore considered invalid. At 0.002 mm/min there were also signs of ductile necking, although failure intervened at somewhat lower deformations, leading to a decrease in the measured [beta][w.sub.p]. Such behavior is interesting from a phenomenological point of view, as will be seen later, but in this case t ests were only carried out for two l values (4 and 11 mm) owing to the prohibitively long test times required. Curvature of [w.sub.f] vs l could therefore not be checked for and the measured value of We was again rejected.

For [M.sub.n] = 41 kg/mol tested at 2 and 0.2 mm/min, [[sigma].sub.max] and 1.15[[sigma].sub.y] were in agreement and [w.sub.f] vs l was linear in each case with the exception of the data for l = 15 mm and 2 mm/min. These were not included in the extrapolation (see Fig. 2) owing to the onset of unstable crack growth in the final stages of the test, which tended to truncate the force-displacement curves. This effect also occurred at somewhat lower l, but in this case made little difference to the overall value of [w.sub.f]. The same considerations led to the rejection of data points for l = 15 mm and [M.sub.n] = 35 kg/mol tested at 2 mm/min.

The behavior for both [M.sub.n] = 41 kg/mol and 35 kg/mol at the lowest test speeds contrasted sharply with that of both [M.sub.n] = 66 kg/mol and the same materials tested at higher speeds, in that [beta][w.sub.p] dropped by about an order of magnitude. The reason for this should be obvious from Fig. 5, where the evolution of the plastic zone, as reflected by the local stress whitening (or darkening in these transmitted light images), is shown for l = 8 mm at all the test speeds. At the highest test speed, failure occurred by the advance of a triangular crack through the plastically deformed ligament, leaving a roughly triangular stress whitened region after final failure. However, at the lowest test speed, the stress whitened zone remained almost planar and fibrillar material was visible on the crack faces. It is also significant that [[sigma].sub.max] was considerably less than 1.l5[[sigma].sub.y] under these conditions, which again suggests a change in mechanism (well developed fibrillar deformation zone s were not seen in unnotched specimens deformed under similar conditions, although small crack like features were visible in samples with [M.sub.n] = 35 kg/mol after deformation at the lowest test speeds). The resulting behavior was at least superficially similar to crack advance preceded by the formation of a craze, which occurs in many glassy polymers, and has been previously attributed to the long term low load brittle failure in POM described in the introduction [4].

As reflected by the data for [M.sub.n] 41 kg/mol tested at 0.002 mm/min, shown in Fig. 2b, [w.sub.f] was weakly dependent on l in the above regime and was thus highly insensitive to curvature in [w.sub.f](l). As can be seen from Fig. 5, there did appear to be a certain amount of crack propagation prior to full section yielding in so far as this latter was reflected by stress whitening across the whole ligament. However the force-displacement curves were similar for all l so that extrapolation was justified and the [w.sub.e] values for both [M.sub.n] = 41 kg/mol and 35 kg/mol at the lowest test speeds were retained.

For [M.sub.n] = 41 kg/mol and 35 kg/mol at intermediate test speeds, inspection of the samples indicated the range of l to be such that the decrease in effective strain rate with increasing l was sufficient to induce a transition between the mechanisms characteristic of high speeds (triangular crack) and low speeds (fibrillar deformation), with mixed behavior occurring in certain samples (note that a trend towards fibrillar deformation as the ligament length increases is contrary to what one would expect if the transition were provoked by a change in stress state, as opposed to a change in deformation rate). Thus it was no longer reasonable to expect a simple linear extrapolation to give meaningful values of [w.sub.p]. This is typified by the data for [M.sub.n] = 41 kg/mol and 0.02 mm/min. The sample with the shortest l showed a higher [w.sub.f] than samples tested at higher speeds, which one might attribute to an increase in ductility, the mode of failure being similar to that at higher speeds. On the other hand, specimens with longer l showed lower [w.sub.f] than at higher speeds, owing to the partial intervention of the fibrillar deformation mode, which restricted development of the plastic zone. The net result was a flattening of [w.sub.f](l) at large l, and large apparent values of [w.sub.e] derived using the extrapolation procedure. Although these values can scarcely be considered as materials parameters, it is possible to interpret the peak in the measured We in Fig. 3a and the sharp drop in [beta][w.sub.p] as reflecting the transition between the high and the low speed failure mechanisms. Moreover, it is possible that the drop in [beta][w.sub.p] for [M.sub.n] 66 kg/mol and 0.002 mm/min is also associated with this phenomenon (although this test speed was not sufficiently low for the transition to be complete). It may thus be inferred from the results in Fig. 3 that the transition moved to lower test speeds as the molecular weight was increased. This is consistent with the results of slow crack growth rat e measurements, which suggest that the formation and propagation of cracks in the fibrillar deformation zones are favored by low molecular weight [4, 5].

Those measurements of [w.sub.e] that were considered valid have been replotted in Fig. 6 as a function of [M.sub.n] and as a function of test speed. All of the valid [w.sub.e] were of similar order of magnitude, and varied only slowly with test speed. This is important to verify, since it is implicit in the EWF analysis that [w.sub.e] should be approximately independent of the variations in effective strain rate over the range of ligaments used for the linear extrapolations (variations in [[sigma].sub.max] and by inference, variations in [[sigma].sub.p] may be included within the shape factor [beta] provided that they are to a good approximation linearly dependent on l).

Finally, in Fig. 7 some nominal values of [w.sub.e] have been plotted against [t.sub.e], for tests carried out at both 100[degrees]C and 80[degrees]C. Here [t.sub.e] is an "essential time to fracture" and is obtained by extrapolating the time corresponding to the maximum stress, to zero ligament length, in order to obtain a value appropriate to [w.sub.e]. For both temperatures there was a clear peak which was associated with the ductile-brittle transition (as discussed above). Although the position of this peak was not precisely defined owing to the lack of data points, it had clearly moved to a longer time at the lower temperature. This is interpreted as reflecting the thermal activation of the creep processes presumed to lead to high temperature embrittlement. Assuming a simple Arrhenius expression for [t.sub.e] and given that it increased by about an order of magnitude when the temperature was reduced from 100[degrees]C to 80[degrees]C, an activation energy of 120 kJ/mol was inferred, which is consistent with that estimated from slow crack growth data [4, 5].


The present results show that the EWF method can be justified for high temperature failure in POM, provided that the test conditions do not coincide with those associated with the ductile-brittle transition. In other words, tests carried out at speeds either well below or well above that at which the transition occurs, may be argued to produce valid data. Nevertheless, if all the extrapolated values of [w.sub.e] were considered, including values which were invalid in terms of the EWF test protocol followed here, the transition could be identified with a clear peak in [w.sub.e] plotted as a function of test speed. This peak appeared to move to a longer characteristic time as the temperature decreased, consistent with the results of other investigations of high temperature brittle failure. The valid [w.sub.e] showed relatively little variation with test speed, but increased markedly with molecular weight. Given that fracture in POM is generally found to be strongly molecular weight dependent, this appears to support the interpretat ion of We as a materials parameter (at least for a given sample thickness).

(*.) Laboratoire de Polymeres Ecole Polytechnique Federale de Lausanne, CH-1015 Switzerland

(**.) Institut fur Werkstoffkunde und Werkstoffpr[ddot{u}]fung der Kunststoffe Montanuniversit[ddot{a}]t Leoben, Austria

(+.) To whom correspondence should be addressed.


(1.) J. H. Wendorff, Prog. Calloid. Polym. Scl. 66, 135 (1979).

(2.) C. J. G. Plummer, P. Begue1in, and H.-H. Kausch, Polym. Eng. Sci., 34, 318 (1994).

(3.) A. J. Lesser, Polym. Eng. Sci., 36, 2366 (1996).

(4.) P. Scarramuzzino. PhD thesis, EPF Lausanne (1998).

(5.) C. J. G. Plunimer. P. Scaramuzzino, and H.-H. Kausch, to appear in Polymer Engineering & Science.

(6.) G. Levita, L. Parisi, ana L. Lazzeri. J. Mat. Set., 29,4545 (1994).

(7.) S. Hasherni, J. Mat Set.., 28, 6178 (1993).

(8.) Y. M. Mai and P. Powell, J. Polym. Sci., 29, 785 (1991).

(9.) S. Saleemi and J. A Nairn, Polym. Eng. Sci., 30, 211 (1990).

(10.) B. Cotterell and J. K. Reddel, Int. J. Fract.. 13, 267 (1977).

(11.) European Structural Integrity Society, Draft Protocol for Essential Work of Fracture, Version 5 (1997).

(12.) J. Karger-Kocsis. T. Cezigany. and E. J. Moskala, Polymer, 39, 3939 (1998).
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Publication:Polymer Engineering and Science
Article Type:Statistical Data Included
Geographic Code:1USA
Date:Apr 1, 2000
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