Analysis of tensile test results for poly(acrylonitrile-butadiene-styrene) based on Weibull distribution.
Fracture resistance of polymers, often referred to as toughness, is an area of great interest to many research groups. Nature of "ductile" or "brittle" fracture is not absolute for a given polymer. Their transition may occur by varying temperature or loading rate in a relatively narrow range (1), within which a dramatic change of ductility or toughness occurs. Preliminary classification of polymer fracture is often based on mechanisms involved in the deformation, namely, crazing or shear yielding. The former is highly localized deformation mechanism. Its involvement often limits potential of the polymer in energy consumption for fracture. On the other hand, the latter activates a relatively large deformation zone, thus having the capability of increasing the energy consumption during fracture and showing high toughness. However, this concept is not fully applicable to rubber-modified polymers, as many polymers in this category, such as high-impact polystyrene (HIPS) (2-8), can show a wide range of ductility and fracture toughness in spite of crazing being the dominant deformation mechanism.
The above issue is further complicated by cavitation of rubber particles, which is suspected to play an essential role on the matrix deformation, thus the toughness value. Its actual effect is not clear, as some studies suggest that particle cavitation acts as the precursor for crazing (9-11), whereas the others for shear yielding (9), (12-14).
Another problem that aggravates complication of the issue is the techniques used for characterizing the deformation mechanisms. Most of the techniques are to characterize the mechanisms based on optical or electron microscopy (15-25). However, due to small amount of material used for this type of techniques, representation of the data for the general deformation mechanism has often been questioned. Concerns are also raised about sample preparation, especially for electron microscopy, as the preparation may generate artifacts, such as voids (26), which may not exist in the original sample. Although small-angle X-ray scattering (SAXS) can serve as an alternative technique that is not limited to small sample size (12), (13), (27-34), interpretation of the results is yet to reach a general agreement. For example, Ijichi et al. (29) show that after the yield point is reached in tensile testing of HIPS a distinct streak appears in the SAXS pattern, in the direction perpendicular to the tensile loading, which they attribute to scattering from craze fibrils. However, Magalhaes and Borggreve (13) by using the same technique, suggest that in this type of polymers, matrix crazing plays a minor role on the enhancement of plastic deformation. Instead, it should be cavitation-induced microscopic shear yielding that has caused significant increase in the toughness.
Another characterization technique that may provide quantitative information on the involvement of crazing or shear yielding in the deformation is through the measurement of volumetric strain, also known as dilatometry (35-42). The technique is based on detection of volume change during the deformation process. Because crazing is a void forming process, its occurrence may generate volume strain that should be distinctly different from shear yielding. Indeed. Xu and Tjong (42) based on results from the volume strain measurement, reported a transition from crazing to shear yielding by increasing the amount of high-density polyethylene (HDPE) in the polystyrene/HDPE blends. For rubber-modified polymers, however, particle cavitation that may act as a precursor for the shear yielding (14) can also increase the volumetric strain. Therefore, effectiveness of the technique to distinguish shear yielding from crazing can be greatly reduced.
This article will use Weibull distribution to characterize scattering of results from simple tensile test. It is well experienced that experimental data for polymers of high toughness often show more scattering than those of low toughness. To our knowledge, nature of the scattering has never been quantified. We believed that the data scattering is a generic phenomenon associated with the fracture behavior. Therefore, change of the data scattering is an indication of the change of the fracture behavior. Based on characterization of the data scattering, the article explores possibility of quantifying the ductile-brittle transition in rubber-modified polymers. ABS is chosen as the sample material to demonstrate the concept. Change of the deformation and fracture in ABS is through the change of crosshead speed for testing. Data collected from the tests are analyzed using Weibull distribution to quantify nature of the data scattering. With the information established, data from so-called "two-stage" tensile tests were collected. This type of tests uses two crosshead speeds in sequence, the first for damage generation and the second for specimen fracture. As to be shown in this article, results from the two-stage tensile tests depict the influence of existing damage on further damage development and its type in the fracture process.
BRIEF REVIEW OF WEIBULL ANALYSIS USED IN THE STUDY
Weibull analysis is a well-known statistical technique that was invented by Weibull (43) to describe data variation based on continuous probability distribution. Because the main concern in this study is the distinction between two types of damage, only two-parameter Weibull distribution function is considered for the data analysis. Three-parameter Weibull distribution was also considered in the preliminary study, but was found to show less difference with that based on the two-parameter Weibull distribution, thus not used.
For the two-parameter Weibull distribution, its cumulative distribution function (CDF) is expressed as
F(t) = 1 - exp[-[([t/[eta]]).sup.[beta]]] (1)
where [beta] and [eta] are shape and scale parameters, respectively, and F is the probability of failure (also known as unreliability) for up to a given t value that in this study can be ultimate tensile strength (UTS), extension at break, or total energy absorbed during the tensile test (named toughness hereafter).
The probability density function (PDF), [Florin], is determined through derivative of F with respect to t. That is,
[Florin](t) = ([[beta]/[eta]])[([t/[eta]]).sup.[[beta]-1]] exp [-[([t/[eta]]).sup.[beta]]] (2)
Note that the shape parameter [beta] controls the skewness of [Florin] and represents the trend variation of F. On the other hand, the scale parameter, [eta], represents the t value for F equal to 0.632.
Using [beta] and [eta], values of mean ([mu]) and standard deviation (SD) of the distribution can be expressed as (44)
[mu] = [eta][GAMMA](1 + [1/[beta]]) (3)
SD = [eta][[[GAMMA](1 + [2/[beta]]) - [[GAMMA].sup.2](1 + [1/[beta]])].sup.[1/2]] (4)
where function [GAMMA] is defined as
[GAMMA](z) = [[infinity].[integral] 0][y.sup.[z-1]] exp(-y)dy (5)
Because data collected in this study for the Weibull distribution analysis are discrete experimental measurements, it is required that unreliability ([F.sub.i]) be determined for each data point. Here, the concept of median ranks, first proposed by Johnson (45), was adopted to determine [F.sub.i]. That is, [F.sub.i] is defined as the probability for failure to occur from data point i onward in a ranked pool of size N at a confidence level of 50%. Based on this definition, [F.sub.i] should be determined using the following equation (44):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with inputs being the sample size N and the ranked order i. Although values of [F.sub.i] are available in handbooks, for convenience, an expression that gives a close approximation (44) was used to determine [F.sub.i], as shown in the equation below.
[F.sub.i] = [[i - 0.3]/[N + 0.4]] (7)
With [F.sub.i] calculated for each data point, [beta] and [eta] values were determined based on minimum variance of 1n(t) between the experimental measurements and the prediction from Eq. 1. That is, [beta] and [eta] values were determined using the following equations:
[beta] = [[N[N.summation over (i=1)][y.sub.i.sup.2] - [([N.summation over (i=1)][y.sub.i]).sup.2]]/[N[N.summation over (i=1)]([x.sub.i][y.sub.i]) - [N.summation over (i=1)][x.sub.i][N.summation over (i=1)][y.sub.i]]] (8)
[eta] = exp [([N.summation over (i=1)][x.sub.i] - [1/[beta]][N.summation over (i=1)][y.sub.i])/N] (9)
where [x.sub.i] = ln([t.sub.i]) and [y.sub.i] = 1n 1n[1/1-[F.sub.i]].
The corresponding correlation coefficient (R) for the linear regression is
R = [[N[N.summation over (i=1)][x.sub.i][y.sub.i] - [N.summation over (i=1)][x.sub.i][N.summation over (i=1)][y.sub.i]]/[[square root of N[N.summation over (i=1)][x.sub.i.sup.2] - [([N.summation over (i=1)][x.sub.i]).sup.2]][square root of N[N.summation over (i=1)][y.sub.i.sup.2] - [([N.summation over (i=1)][y.sub.i]).sup.2]]]] (10)
In view of the possibility of coexistence of two types of damage in the fracture process (as to be shown in the section Results and Discussion), 2-group, two-parameter mixed Weibull distribution was chosen for the data analysis, for which the overall CDF is expressed as
F(t) = [2.summation over (j=1)][P.sub.j][F.sup.j] (11)
where [P.sub.j] is the portion for the jth subpopulation and [F.sup.j] the CDF of the jth subpopulation, with [[beta].sub.j] and [[eta].sub.j] being the corresponding shape and scale parameters, respectively. Note that PDF of the above function is
[Florin](t) = [S.summation over (j=1)][P.sub.j][[Florin].sup.j] (12)
For our study, the total number of parameters, i.e., [P.sub.j], [[beta].sub.j], and [[eta].sub.j], that can be determined independently is 5, as the requirement of [2.summation over (j = 1)][p.sub.j]] = 1 has provided a constraint for [P.sub.j].
The above Weibull distribution analysis was carried out using commercial software, Weibull++ (version 7) from ReliaSoft Co.
Material and Specimen Preparation
ABS sheets of 3.2-mm thick were used in the study, supplied by McMaster-Carr (product number 8586K24), of which the basic physical properties are available from the supplier (46). The sheets were first cut into rectangular strips of 195 mm X 19 mm using a table saw, and then machined to tensile specimens using TensilKut (model 10-33 from Qualitest) using a template with dimensions according to ASTM D 638-08 Type I. Configuration and dimensions of the machined specimens are shown in Fig. 1.
[FIGURE 1 OMITTED]
Both simple and two-stage tensile tests were carried out using a hydraulic Material Testing System (MTS) machine (Model 810) at ambient temperature and pressure. To improve consistency of the test results, specimens were carefully mounted on the testing machine to avoid buildup of residual load before the start of the test. Load, stroke, and time were recorded during the test at a sampling rate of 50 Hz.
Maximum crosshead speed used in the study was 60 mm/min, equivalent to a strain rate of ~0.02 [s.sup.-1]. According to Inberg et al. (47), such a strain rate is insufficient to cause any noticeable temperature rise in the specimens, which has been supported by Steenbrink et al. (48) who reported that ABS specimens fractured at this strain rate did not show any relaxation on the fracture surface. Therefore, effect of adiabatic heating on the deformation has been ignored in the study.
Simple tensile tests were carried out in the stroke control mode. Three crosshead speeds of 5, 30, and 60 mm/min were used to investigate the effect of straining rate on the resistance of the damage to fracture, characterized by the values of UTS, extension at break, and toughness (defined as the energy consumption during the test, i.e., the area under the load-displacement curve). At least 42 specimens were tested at each crosshead speed, from which mean value and standard deviation were determined, and their scattering profile characterized using Weibull distribution. As to be shown later, those three crosshead speeds were chosen because the damage generated during the fracture process changes from one type (tiny strips) to the other (uniform whitening) with the increase of crosshead speed in this range. At 30 mm/min, a mixture of the two types of damage was generated, from which data were used to quantify their involvement in the fracture process.
Quantification of the two types of damage in the fracture process was further examined using results from two-stage tensile tests in which the first stage was used to generate damage and the second stage to characterize the "residual" properties (maximum strength, extension at break, and toughness). In this study, a crosshead speed of either 5 or 60 mm/min was applied at the first stage to generate the damage, and the other crosshead speed to evaluate the residual properties. For example, when the crosshead speed of 5 mm/min was used for the first stage test, to generate tiny strips in the specimens, the cross-head speed of 60 mm/min was used to measure the residual properties at the second stage. Stroke used at the first stage was varied to introduce different levels of damage in the specimens. Two testing scenarios used in this study are summarized in Table 1. For the two-stage tensile tests, at least 25 specimens were used for each test scenario. Data from the second stage tests were analyzed in the same way as those from the simple tensile tests.
TABLE 1.Test scenarios of the two-stage tensile tests. First stage Second stage Crosshead speed Prescribed Unload Crosshead (mm/min) stroke (mm) completely speed (mm/min) Scenario 1 5 2.6 60 3.1 6.0 Scenario 2 60 3.0 5 3.4 6.0
RESULTS AND DISCUSSION
Simple Tensile Tests
Typical appearance of specimens after the simple tensile test is presented in Fig. 2. The bottom photograph was fractured at 5 mm/min, and contains a large number of tiny strips in the vertical direction, i.e., perpendicular to the applied load. The specimen fractured at 60 mm/ min, the top photograph in Fig. 2, appears to contain a uniform whitening zone, with gentle necking in the middle section. At the medium crosshead speed of 30 mm/ min (the middle photograph in Fig. 2), the fractured specimen contains both features, i.e., a uniform whitening zone with necking in the middle section and tiny strips perpendicular to the loading direction at both ends. Coexistence of the two features at 30 mm/min was probably caused by the variation of strain rate introduced at this crosshead speed. It is believed that the initial strain rate introduced at 30 mm/min was high enough to generate the uniform whitening zone. However, as the necking evolved the strain rate gradually dropped to a level that was low enough to generate similar damage as that at 5 mm/min. i.e., tiny strips perpendicular to the loading direction. No necking was observed on any specimens tested at 5 mm/min.
[FIGURE 2 OMITTED]
The above two features of tiny strips and uniform whitening have been observed before in rubber-modified polymers, such as ABS and HIPS. In general, fracture with damage appearing as tiny strips exhibits much lower ductility and toughness than that with uniform whitening (2), (33), (49). In the past, matrix damage in rubber-modified polymers was commonly divided into two groups based on the deformation mechanism, i.e., crazing and shear yielding. Many works have been dedicated to characterizing the mechanism involved and the associated mechanical properties (e.g., (2), (13), (14), (29), (30), (33), (42), (48), (49). However, there may not be a direct relationship between mechanical properties (such as fracture toughness) and deformation mechanism. For example, Kuboki et al. (2) showed that even with crazing dominating the fracture process, the ductility and toughness can vary significantly, with the damage appearance changing from tiny strips to uniform whitening. Furthermore, it has been shown that variation of ductility and toughness for a rubber-modified polymer can be caused by the presence of intrinsic defects that are in the form of rubber gel (50) that has compositions similar to those for rubber particles (51). Therefore. the deformation mechanism itself, whether it is shear yielding or crazing, does not necessarily indicate the level of ductility or toughness for the polymer. In view of this, we use appearance of the damage, i.e., uniform whitening or tiny strips, to indicate difference of the deformation. As to be shown next, specimens with damage appearing as uniform whitening always show better ductility and toughness than those with tiny strips.
Typical load-displacement curves obtained from three crosshead speeds used in the testing are presented in Fig. 3. All curves show a significant load drop of over 10% immediately after the peak load, followed by extensive extension before the final fracture. Observation during the tests suggests that the damage, either crazing or shear yielding, appeared when the load drop occurred. Figure 3 also suggests that UTS, post-peak loading level, and extension at break all show a trend of increase with the increase of crosshead speed. Because all three factors contribute to the area under the load-displacement curve, toughness (defined as the area under the load-displacement curve) is also expected to increase with the increase of the crosshead speed.
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Variation of the mean values and their standard deviations for UTS, extension at break, and toughness are summarized in Fig. 4, and also given in Tables 2-4 for later comparison with results from the Weibull distribution analysis. Figure 4 suggests that all three properties increase with the increase of crosshead speed, though the increase of UTS is relatively small compared with the increase of extension at break and toughness. For example, by increasing the crosshead speed from 5 to 60 mm/min, the mean value of UTS increases by 9%, whereas those of extension at break and toughness by 93% and 104%, respectively.
TABLE 2. Summary of simple tensile test results for UTS. Crosshead speed (mm/min) Mean [+ or -] SD (MPa) Single-group Weibull [beta] [eta] (MPa) R Two-group mixed Weibull First [[beta].sub.1] subpopulation [[eta].sub.1] (MPa) [p.sub.1] Second [[beta].sub.2] subpopulation [[eta].sub.2] (MPa) [p.sub.2] R Crosshead speed (mm/min) 5 30 60 Mean [+ or -] SD (MPa) 36.7 [+ or -] 39.2 [+ or -] 40.2 [+ or -] 0.3 0.3 0.2 Single-group Weibull 189.7 159.4 250.7 36.8 39.3 40.3 0.95 0.96 0.99 Two-group mixed Weibull 358.9 247.8 300.7 36.5 39.0 40.1 0.50 0.50 0.49 324.0 320.5 434.8 37.0 39.5 40.4 0.50 0.50 0.51 0.98 0.98 0.99 TABLE 3. Summary of simple tensile test results for extension at break. Crosshead speed (mm/min) Mean [+ or -] SD (mm) Single-group Weibull [beta] [eta] (mm) R Two-group mixed Weibull First [[beta].sub.1] subpopulation [[eta].sub.1] (mm) [p.sub.1] Second [[beta].sub.2] subpopulation [[eta].sub.2] (mm) [p.sub.2] R Crosshead speed (mm/min) 5 30 60 Mean [+ or -] SD (mm) 11.8 [+ or -] 16.4 [+ or -] 22.9 [+ or -] 3.0 7.4 9.5 Single-group Weibull 4.5 2.3 2.5 12.9 18.6 25.9 0.98 0.99 0.99 Two-group mixed Weibull 30.1 24.3 3.9 15.4 16.3 26.3 0.28 0.18 0.53 5.0 1.9 1.7 11.4 19.0 25.0 0.72 0.82 0.47 0.98 0.94 0.96 TABLE 4. Summary of simple tensile test results for toughness. Crosshead speed (mm/min) Mean [+ or -] SD (J) Single-group Weibull [beta] [eta] (J) R Two-group mixed Weibull First [[beta].sub.1] subpopulation [[eta].sub.1] (J) [p.sub.1] Second [beta].sub.2] subpopulation [[eta].sub.2] (J) P2 R Crosshead speed (mm/min) 5 30 60 Mean [+ or -] SD (J) 13.7 (3.6) 19.7 (9.2) 28.0 (11.6) Single-group Weibull 4.2 2.2 2.4 15.1 22.3 31.8 0.98 0.99 0.99 Two-group mixed Weibull 22.7 21.5 4.8 18.1 19.8 32.1 0.23 0.18 0.38 4.4 1.9 1.9 13.7 22.8 31.3 0.77 0.82 0.62 1.00 0.95 0.97
[FIGURE 4 OMITTED]
In addition, data of UTS have a much smaller standard deviation than those of extension at break and toughness. For example, the standard deviation of UTS is less than 1% of the mean value, but those of extension at break and toughness are in the range from 25% to 47%. Hence, characteristics for the extension at break and toughness bear more variation with the change of crosshead speed than those for UTS. The results indicate that although UTS data are very consistent, showing less sensitivity to the change of damage type, those of the extension at break and toughness for uniform whitening have much larger standard deviation than those for tiny strips. Such characteristics are further analyzed in the following section using the Weibull distribution.
Because the study is concerned about two types of damage, two-group mixed Weibull distribution is considered for the data analysis. Note that in the two-group mixed Weibull distribution analysis, in addition to [beta] and [eta], portion for each subpopulation (P) also needs to be determined. The results are summarized in Tables 2-4. The corresponding CDF and PDF curves are presented in Figs. 5-7 for UTS, extension at break, and toughness, respectively. For the purpose of comparison, results using single-group Weibull distribution analysis are also presented, in Figs. 8-10 for the same mechanical properties as above. Curves in those figures arc arranged in such a way that the three curves on the left column arc CDF curves from different crosshead speeds (from the top to the bottom: 5, 30, and 60 mm/min), and those on the right column the corresponding PDF curves.
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Values of correlation coefficient (R) in Tables 2-4 indicate that both single-group and two-group mixed Weibull distribution can fit the test data with sufficient satisfaction. However, the two-group mixed Weibull distribution has the advantage of being able to provide a better fit to the data at the crosshead speed of 30 mm/min, as indicated by comparison of CDF curves for extension at break. Figs. 6b and 9b, and those for toughness. Figs. 7b and 10b. It has been mentioned earlier that at 30 mm/min, both tiny strips and uniform whitening are visible on the fractured specimens.
For UTS, [beta] values for fitting data at all three crosshead speeds are very large, based on either single- or two-group mixed Weibull distribution. The corresponding CDF curves, on the left column in Figs. 8 and 5, respectively, show a steep rise, and the PDF curves, on the right column in Figs. 8 and 5, contain relatively sharp and narrow peaks. Although two peaks exist in the PDF curves of Fig. 5, both peaks are sharp with nearly equal portion for the subpopulations. Therefore, it is believed that the two subpopulations for UTS belong to the same population. Overall, the analysis suggests that UTS of ABS increases with the increase of crosshead speed. Its distribution characteristics do not change much with the transition of damage from tiny strips to uniform whitening.
The only difference that can be detected among the UTS data at the three crosshead speeds is the [beta] value that is the smallest for the crosshead speed of 30 mm/min. Because [beta] reflects the steepness of the CDF curve, its relatively small value at 30 mm/min is possibly caused by the involvement of two damage types. In other words, distribution of UTS data at 30 mm/min consists of two subpopulations that may have slightly different peak positions of similar characteristics.
On the other hand, the Weibull distributions for extension at break and toughness show quite different characteristics. First of all, [beta] values for extension at break and toughness are much smaller than those for UTS. That is, CDF curves for extension at break and toughness do not rise as sharply as those for UTS, suggesting that the extension at break and toughness have a broader range of variation than UTS. In addition, the PDF curves in Figs. 6 and 7 (on the right column) indicate that characteristics for the data distribution change significantly with the increase of the crosshead speed. That is, with the increase of the crosshead speed from 5 to 60 mm/min (from top to bottom in each figure), the sharp peak gradually disappears and is replaced by a broad hump. The corresponding PDF curves from the single-group Weibull distribution, on the right column of Figs. 9 and 10, also show the general trend of increase of peak width with the increase of the crosshead speed. Note that judging from the CDF curves for extension at break and toughness at 30 mm/mm, the two-group mixed Weibull distribution. Figs. 6b and 7b, provide a better flit to the data than the single group Weibull distribution. Figs. 9b and 10b. This implies that two populations are involved in the data distribution at 30 mm/min, which is consistent with the observation of damage in Fig. 2.
Because distribution characteristics for extension at break are similar to those for toughness, the following discussion is only focused on data for extension at break using two-group mixed Weibull distribution. The PDF curves for extension at break at 5 and 60 mm/min, Fig. 6d and f, respectively, show very different characteristics. That is, Fig. 6d contains two relatively sharp peaks, whereas Fig. 6f shows a broad hump. Both sharp peaks in Fig. 6d are believed to represent the damage of tiny strips. They appear as different subpopulalions simply because of the use of two-group mixed Weibull distribution, which in a way is similar to the phenomenon of two peaks in the PDF curves for UTS in Fig. 5. Because the damage type that dominates the deformation at 5 and 60 mm/min is tiny strips and uniform whitening, respectively, it is reasonable to believe that the corresponding PDF curve provides the characteristics of data scattering when the particular damage type dominates the fracture process. Using this concept, the PDF curve for 30 mm/ min, Fig. 6e, which contains two peaks of different characteristics, indicates the coexistence of two damage types, that is, the sharp peak for tiny strips and the broad hump for uniform whitening, respectively.
The above analysis suggests that the PDF curve from the two-group mixed Weibull distribution can be used to detect damage types involved in the fracture process of ABS. Similar conclusions can be drawn from the toughness data, as shown in Fig. 7. In the following section. the above mixed Weibull distribution is applied to data from two-stage tensile tests, with the involvement of each damage type in the fracture process being quantified based on the portion for each subpopulation.
Two-Stage Tensile Tests
After identifying the characteristics of PDF curves for mechanical properties from simple tensile tests, the study proceeded to conduct two-stage tensile tests in which the first stage was to generate damage, i.e., tiny strips or uniform whitening, and the second stage to fracture the specimens. The objective was to understand the influence of pre-existing damage and its type on the deformation and fracture behavior of ABS. In the first test scenario, tiny strips were generated at the first stage using cross-head speed of 5 mm/min, and the degree of damage varied by changing the stroke from 2.6 to 6.0 mm. Similarly, in the second test scenario, the uniform whitening was generated at 60 mm/min at the first stage, with the stroke varying from 3.0 to 6.0 mm. Figure 11 depicts the damage generated at the first stage in the two test scenarios by loading the specimens to various strokes at 5 or 60 mm/min and then unloading. Note that the photographs in the bottom row of Fig. 11 (for crosshead speed of 60 mm/min) show some tiny strips at both ends of the uniform whitening zone. Generation of the tiny strips was possibly due to the decrease of crosshead speed at the end of the loading phase, even though the nominal crosshead speed was set at 60 mm/min. Such speed reduction is unavoidable without redesign of the clamp device of the testing system. Fortunately, existence of tiny strips at the end of the uniform whitening zone did not suppress the dominance of uniform whitening damage at the second stage, even though tiny strips are expected at 5 mm/ min in virgin specimens, as shown in Fig. 2.
[FIGURE 11 OMITTED]
Figure 12 presents the typical load-displacement curves from the first scenario of the two-stage tensile tests, i.e., by loading the specimens at 5 mm/min to a stroke of 2.6, 3.1, or 6.0 mm, unloading completely, and then reloading at 60 mm/min till fracture. The figure clearly shows a peak load in the curves generated from the second stage test, of which the magnitude decreases with the increase of the stroke applied at the first stage. In addition, the figure suggests that the initial stiffness of the curves from the second-stage test decreases with the increase of the stroke used at the first stage.
[FIGURE 12 OMITTED]
Typical load-displacement curves for the second test scenario, i.e., generating damage at 60 mm/min and then fractured at 5 mm/min, are shown in Fig. 13. Unlike Fig. 12. the curves in Fig. 13 from the second stage tests do not show any distinctive peak. Instead, the load rose quickly to a level and then maintained relatively constant till fracture. In addition, the loading level did not change much with the variation of the stroke used at the first stage. It was also noticed that even though those specimens were fractured at 5 mm/min, all specimens contain an extensive, uniform whitening zone that is similar to that shown by the top photograph in Fig. 2.
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Mean values and standard deviations of data obtained from the second stage tests for the above two test scenarios are summarized in Table 5. For the first scenario, Table 5 (first test scenario), the data suggest that increase of the stroke used at the first stage has caused the decrease of peak load, extension at break, and toughness determined from the tests at the second stage. The data also suggest that despite the extension at break determined from the second-stage test decreases with the increase of the stroke used at the first stage, the total extension that includes the stroke used at the first stage remains relatively constant, of around 14 mm, which is insensitive to the change of stroke used at the first stage.
TABLE 5. Mechanical properties of ABS in two-stage tensile tests with different strokes used at the first stage. (a) First test scenario Stroke applied 2.6 3.1 at first stage (5 mm/min) (mm) Mechanical Maximum 36.0 [+ or -] 0.2 35.2 [+ or -] 0.1 properties at strength (MPa) second stage (60 mm/min) Mean [+ or -] SD Elongation at 11.3 [+ or -] 3.3 10.9 [+ or -] 2.9 break (mm) Toughness (J) 14.1 [+ or -] 4.4 13.2 [+ or -] 3.9 (b) Second test scenario Stroke applied 3.0 3.4 at stage 1 (60 mm/min) (mm) Mechanical Maximum ~29 (a) ~29 (a) properties at strength (MPa) second stage (5 mm/min) Mean [+ or -] SD Elongation at 16.1 [+ or -] 4.9 19.7 [+ or -] 5.7 break (mm) Toughness (J) 18.5 [+ or -] 5.8 22.6 [+ or -] 7.1 (a) First test scenario Stroke applied at first 6.0 stage (5 mm/min) (mm) Mechanical properties at Maximum strength 34.5 [+ or -] 0.1 second stage (60 mm/min) (MPa) Mean [+ or -] SD Elongation at break 8.4 [+ or -] 2.4 (mm) Toughness (J) 9.5 [+ or -] 3.3 (b) Second test scenario Stroke applied at stage 1 6.0 (60 mm/min) (mm) Mechanical properties at Maximum strength ~29 (a) second stage (5 mm/min) Mean (MPa) [+ or -] SD Elongation at break 15.9 [+ or -] 6.9 (mm) Toughness (J) 17.6 [+ or -] 8.3 Refer to Table 1 for details of loading conditions for each scenario. (a) No distinctive peak load delected.
The second test scenario, however, generated a different trend, especially for the extension at break and toughness. Instead of monotonic decrease of their value with the increase of the stroke used at the first stage, data in Table 5 (second test scenario) suggest that both extension at break and toughness have the largest values from specimens that used the medium level of stroke at the first stage to generate the damage, i.e., with the stroke of 3.4 mm. Although causes of this trend are yet to be identified, it is believed that two factors may have influenced the extension at break and toughness determined from the second stage. One is the switch of damage appearance from tiny strips to uniform whitening due to the presence of uniform whitening damage introduced at the first stage, as mentioned earlier, and the other is the extent of damage introduced at the first stage. With the increase of the stroke used at the first stage, the amount of damage increased, thus decreasing the extension at break at the second stage. However, the increase of the stroke at the first stage also increases the amount of uniform whitening damage, thus encouraging its further development (instead of tiny strips) at the second stage to increase the value of extension at break. As a result of the two factors, the extension at break and toughness determined at the second stage are the largest from the medium level of the stroke used at the first stage.
It should be pointed out that even with the presence of damage the extension at break measured from the second stage in the second test scenario is still larger than that from the virgin specimens tested at 5 mm/min. It is also surprising to discover that no "peak load" is present in its load-displacement curve. Although causes for those phenomena are not clear now, it is speculated that those phenomena are relevant to the nature of the uniform whitening damage. Further study will be conducted to understand the phenomena.
Data from the two-stage tensile tests were also analyzed using two-group mixed Weibull distribution, to investigate possibility of quantifying the type of damage involved in the fracture process. Here, only analysis of toughness data from the second-stage tests is presented, as the trend is similar to that for the extension at break. Parameters for the two-group mixed Weibull distribution of the toughness data are presented in Table 6. and the corresponding PDF curves in Figs. 14 and 15 for the first and the second test scenarios, respectively. Note that parameters in Table 6 are arranged in such a way that the subpopulation with CDF curve of relatively sharp increase (i.e., representing the damage of tiny strips) is designated as the first subpopulation, and the other the second subpopulation.
TABLE 6. Values of parameters for two-group mixed Weibull distribution of toughness determined from the second stage lest. (a) First lest scenario Stroke used at the first stage (mm) 2.6 3.1 6.0 First subpopulation [[beta].sub.1] 10.7 11.3 28.6 [[eta].sub.1] (J) 14.8 11.7 7.8 [p.sub.1] 0.22 0.19 0.24 Second subpopulation [[beta].sub.2] 2.9 3.5 3.1 [[eta].sub.2] (J) 16.0 15.2 11.3 [p.sub.2] 0.78 0.81 0.76 R 0.99 0.97 0.92 (b) Second test scenario Stroke used at the first stage (mm) 3.0 3.4 6.0 First subpopulation [[beta].sub.1] 9.8 50.0 50.0 [[eta].sub.1] (J) 13.8 15.8 20.0 [p.sub.1] 0.34 0.30 0.06 Second subpopulation [p.sub.2] 4.0 5.2 2.1 [[eta].sub.2] (J) 23.6 27.6 19.9 [p.sub.2] 0.66 0.70 0.94 R 0.96 0.93 1.00 Refer to Table 1 for details of the test scenarios.
[FIGURE 14 OMITTED]
[FIGURE 15 OMITTED]
For the first test scenario, 5 mm/min and then 60 mm/min, Fig. 14 indicates that the peak above the broad hump in the PDF curve is gradually sharpened with the increase of the stroke introduced at the first stage. Based on the characteristics established from the simple tensile tests, this suggests that increase of the stroke introduced at the first stage has shifted the nature of the damage generated at the second stage from uniform whitening to tiny strips. Therefore, tiny strips can dominate the fracture process in ABS even at crosshead speed of 60 mm/min that only generates uniform whitening in the virgin specimens, provided that sufficient damage of tiny strips exists in the specimen. Based on the percentage of subpopulation for the relatively sharp peak in the PDF curves (the first sub-population in Table 6), amount of damage that shifts to tiny strips is not much affected by the increase of the stroke introduced at the first stage. However, the damage characteristics move toward those for tiny strips, thus causing decrease of the toughness.
For the second test scenario, 60 mm/min and then 5 mm/min. as shown in Fig. 15, the trend seems to be opposite. That is, the sharp peak in the PDF curve is gradually diminished with the increase of the stroke applied at the first stage. This suggests that the presence of the uniform whitening damage has suppressed the generation of tiny strips at the second stage. Such a phenomenon is probably related to nature of the damage for tiny strips and uniform whitening, which is being investigated by us using various characterization techniques.
Statistical analysis was applied in this study to understand the nature of variation in mechanical properties for a rubber-modified polymer, ABS, which include UTS, extension at break, and toughness (defined as the area under the load-displacement curve from the tensile test). Two types of damage, tiny strips and uniform whitening, were generated in simple tensile test by changing the crosshead speed. The study confirms that values of extension at break and toughness are sensitive to the damage type involved in the fracture process, but UTS is not. The results show that characteristics of PDF curve for extension at break and toughness consists of a relatively sharp peak when tiny strips dominate the fracture process, but a broad hump for uniform whitening. At a crosshead speed of 30 mm/min that generates both types of damage, application of two-group mixed Weibull distribution produces a PDF curve that contains characteristics of data scattering for both. Therefore, it is concluded that features of the PDF curve can be used to indicate damage type involved in the fracture processes, and fraction of the corresponding subpopulation to quantify the portion of damage for which the characteristics vary with the amount and type of damage present in the specimens.
In addition, two-stage tensile tests were used to evaluate the effect of existing damage on the fracture process of ABS. The results suggest that the presence of tiny strips encourages further development of tiny strips in the loading condition that should have uniform whitening as the dominant mechanism in virgin specimens. Likewise, the presence of uniform whitening damage suppresses the generation of tiny strips in the loading condition that should be in favor of the tiny strips in virgin specimens. Using two-group mixed Weibull distribution, the above trends are clearly depicted by the PDF curves.
Results from the two-stage tensile tests also show that the presence of uniform whitening has altered the load-displacement curve generated from the subsequent tensile test, resulting in the absence of peak load that was always observed from virgin specimens with either tiny strips or uniform whitening as the dominant damage. Such a phenomenon has not been expected, and not fully understood at this stage. However, this phenomenon is believed to be relevant to the presence of uniform whitening zone in the specimens, which affects the subsequent damage development in the fracture process.
The study concludes that by using two-group mixed Weibull distribution, two types of damage that may be involved in the fracture process of ABS can be clearly identified. Tiny strips that yield relatively brittle fracture give very consistent values for extension at break and toughness. On the other hand, the uniform whitening results in significant scattering in their values. In view of the strong correlation between characteristics of PDF curve and damage type, it is concluded that characteristics of data scattering can be used to identify the damage involved in the fracture process, and that due to large scattering for high extension at break and toughness values, a small number of tests for ductile ABS may not reflect the true nature of its mechanical properties.
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Correspondence to: P.-Y. Ben Jar; e-mail: firstname.lastname@example.org
Contract grant sponsor: Natural Sciences and Engineering Research Council of Canada (NSERC). University of Alberta (Graduate Studies and Research. Department of Mechanical Engineering).
Published online in Wiley Online Library (wileyonlinelibrary.com).
[C] 2011 Society of Plastics Engineers
P.-Y. Ben Jar, Jie Xu
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8
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|Author:||Jar, P.-Y. Ben; Xu, Jie|
|Publication:||Polymer Engineering and Science|
|Date:||Mar 1, 2011|
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