Statistical analysis of the strength and lifetime under tension of crystalline polymeric solids

The ductile fracture behavior under uniaxial tension of melt-crystallized isotactic polypropylene specimens at room temperature was investigated from a statistical point of view. Each tensile test was performed more than one hundred times and statistical data for the breaking point were obtained under each tensile condition. The probability distribution curves of the fracture time and strength approximately followed Gaussian statistics at lower tensile speeds, but changed to a Weibull function at higher-speed tests. Additionally, with increasing tensile speed the mean and standard deviation of the fracture time decreased linearly. The toughness, which is the total area under the stress-strain curves, was found to be independent of the tensile conditions, indicating that fracture toughness is a criterion for fracture under tension.

strength distribution function. Their work focused on probabilistic formulations for practical assessments of uncertainties in strength predictions using the classical weakest link theory [6]. The behavior of brittle fracture was analyzed using Weibull distribution functions [7]. However, few statistical studies have been performed on the ductile fracture behavior of commodity plastics in typical tensile tests [8].
In this work, isotactic polypropylene (iPP) was chosen as a model plastic material. The ability to predict the fracture points of such iPP materials under uniaxial extension is of critical importance to determine reliability and efficiency for practical use. Therefore, it is useful to find the characteristics of fracture behavior of iPP that relate directly to the service performance of the material. Mechanical experiments such as the uniaxial elongation test are useful to determine optimal conditions for drawing of the plastics. In this work, we examined the probability functions for tensile breakage data such as fracture time, strength, and toughness of iPP films under various tensile conditions.

Experimental
Commercial grade iPP with M w =4.010 5 and M w /M n =5 was used in this work. The iPP films were prepared by compression molding at 230C for 5 min, and quenching in boiling water (100C). The film thickness was controlled to be about 200 μm. The crystallinity of the films was 59% (estimated from density) and their melting temperature was 162 o C (estimated from differential scanning calorimetry). Double-edge notched specimens were cut from films for the tensile measurements. We employed a double-edge notched specimen with gauge length L 0 = 4 mm, 4-mm width and 200-μm thickness. The uniaxial elongation tests for these specimens were repeated more than 100 times at room temperature under each test condition. The tensile speed was varied from 15 mm/min to 70 mm/min. Additionally, tensile tests were performed at higher tensile speeds of 140 and 170 mm/min. The time to fracture was read out from the tensile load-time data. The tensile strength was determined by dividing the tensile load at the breaking point by the initial cross-sectional area.
The data from the tensile tests were analyzed statistically by taking the class size to be 10, corresponding to the square of the sample size. Distribution analysis of the fracture data was carried out using OriginPro 8 commercial software (OriginLab Co. Ltd, USA).

Probability distribution functions of fracture time and strength
All stress-elongation time curves were measured more than 100 times with fixed crosshead speed. The ten stress-time curves measured at 15-mm min −1 crosshead speed are shown in figure 1. The iPP specimens showed a distinct yield point as a maximum in the stress-strain curves beyond the elastic limit. When using notched specimens, the neck constitutes the entire portion of the gauge length immediately after the neck is initiated [1]. Thus, the necked region in the stress-strain curves can be neglected. Therefore, the yield region after the initial elastic region is followed by a virtually complete strain-hardening deformation, and as a result the stress linearly increases as elongation proceeds after the yield process. As seen in this figure, the fracture time has a wide scatter, and thus the probability distribution data for the fracture time as well as the fracture stress (strength) were fit to a typical Gaussian distribution (figure 2).
The shape of the curve of the tensile stress versus elongation time (figure 1) is roughly composed of a yield peak, hardening region, and a negligible necking region. Therefore, we found a linear relationship between stress and elongation time in the hardening region near the breaking point: where G p is the slope of the stress-time curve in the hardening region and ω p is the intercept on the stress axis. The fluctuation of G p is very small compared with the time to fracture, t B , and ω p , with G p = Consequently, the tensile strength is linearly transferred over the time to fracture, t B , offset by the ω p values. When we consider the fluctuation of ω p as a noise fluctuation, this process can be treated as a well-known Kalman filter problem [9]. Because the noise in ω p is a result of fluctuation of the load cell and specimen thickness, ω p is independent of t B . Furthermore, the fact that the probability density function of the fracture strength can be fit by a Gaussian distribution implies that ω p is Gaussian. If the relation between the stress and elongation time closely followed a concave or convex curve, the

Tensile speed dependence
With increasing tensile speed, fracture time decreased and the dispersion in the PDF curves narrowed. Furthermore, the mean fracture time B t and its standard deviation t  are proportional to the inverse of the tensile speed v, given by figure 3). Consequently, the PDF curves of the fracture time can be written in terms of the reciprocal quantity Here  is the intrinsic time. In the tensile tests, the elongation velocity v is above zero so that the intrinsic time  has a time horizon. Hence, as   v , the limit of the intrinsic time is given by = 0.
Here we can assume the initial condition t B = 0 when = 0, according to the straight line to the origin.
The PDF satisfies the partial differential equation This is a Kolmogorov forward equation [11], in which the fracture behavior in the ductile experiment can be regarded as a diffusion process in which fracture events correspond to particle diffusion, intrinsic time (=1/v) corresponds to the diffusion time, and the fracture time t B corresponds to the diffusion distance.    (1), revealing the mean and variance of strength: Considering that the noise fluctuation ω p can be caused by experimental errors such as fluctuations of the load cell and specimen thickness, p  and 2   are likely to be independent of L 0 . Hence, the mean of the strength is predicted to be proportional to the gauge length and its variance is independent of the gauge length. The concept of toughness is defined in several ways, one of which is the tensile toughness defined as  The tensile toughness, which is the energy per volume required to break a specimen, is a measure of the ability of a material or fabricated article to withstand load application in the form of tensile resistance. Interestingly, not only the mean value but also the standard deviation of the fracture toughness were almost independent of the tensile speed as shown in figure 5. This independence implies that fracture toughness is an intrinsic characteristic and a criterion for fracture under the present tensile conditions. As shown in figure 3, the mean fracture time and its standard deviation were proportional to the reciprocal tensile speed. The fracture events at various speeds were determined by the fracture toughness, and the toughness values were constant, independent of the tensile speed. So as speed is increased, the fracture time required to reach the criteria of toughness will be shorter. Therefore, the mean fracture time is proportional to the reciprocal of the tensile speed. The stretched sample consists of a number of oriented cavities (see figures 9 and 10). As the tensile speed is increased, the time period between fracture events shortens, therefore the fracture time standard deviation becomes narrower with higher tensile speed. Figure 6 compares the PDF curves of fracture time measured at higher tensile speeds of 140 and 170 mm/min. The fracture time data at a crosshead speed of 140 mm/min were symmetric and fit to a typical Gaussian distribution with a skewness value around zero, similar to the fracture time data measured at tensile speeds ranging from 15 to 70 mm/min. The symmetric fracture time scatter for the 170-mm/min data leans to the left with a skewness value of 1.39, and the solid curve is the nest normalized Weibull function given by  where m is the shape parameter and  is the scale parameter of the distribution. The transition of the PDF curve from a Gaussian to a Weibull function occurred at a tensile speed between 140 and 170 mm/min. This transition seems to be associated with the transition from ductile to semi-brittle [13]. Also, the mean values of the fracture time at higher tensile speeds fall on the same line v t B / 0   plotted by the inverse of the tensile speed v -1 , with the empirical relationship obtained at the tensile speed range from 15 to 70 mm/min.

Higher tensile speed
The standard deviation of fracture toughness data measured at speeds of 170 and 140 mm/min are 43 and 51 Jm -3 , respectively; larger than all the values of toughness measured at lower speeds (about 25 Jm -3 ). However, there is no clear difference in the mean values of the fracture toughness for the specimens stretched at lower and higher tensile speeds.

Cavity and void observations
The intensity of small angle X-ray scattering (SAXS) in deformed semicrystalline polymers such as iPP is known to be a superposition of contributions from scattering on periodic structures (crystalline-amorphous) and much more intense scattering from voids [14]. For specimens stretched beyond the yield point, the scattering profile ascribed to the periodic structure is concealed behind the scattering signals from voids [15]. Figure 7 shows SAXS 2D patterns for the drawn specimens stretched beyond the yield point in the stress-time curve at speeds of 45 and 170 mm/min. All the drawn directions are horizontal. According to these data, for all specimens the signals ascribed to a large fraction of voids suddenly appeared at smaller angles, consisting of two intense round lobes centered on the median in a peanut shell shape. This pattern corresponds to the signature from larger heterogeneities elongated with the long axis oriented perpendicular to the applied load [16]. Moreover, the intensity of scattering tends to increase at lower tensile speeds, indicating that the void size and volume show a positive dependence on the tensile speed. Figure 8 shows SAXS patterns for fracture specimens taken from the zone in the vicinity of the fracture cross-section drawn up to 170% of nominal strain. A rhombus or elliptical scattering pattern is seen in the vertical direction for the fracture specimens drawn at 45 or 170 mm/min tensile speed, respectively. According to figures 7 and 8, we find that as elongation proceeds, the orientation of the voids changed from perpendicular to parallel to the stretching direction. Further deformation led the cavities to align along the deformation direction (maximum scattering is in the direction perpendicular to deformation). These patterns show that the cavities have been reoriented at a local strain of around 170%. [17]. To characterize the difference in features between the ductile and semi-brittle fractures, we performed scanning electron microscopic (SEM) imaging of the surface zone of specimens failed at speeds of 170 and 45 mm/min. To distinguish the different influences of intense and low tensile speeds, figure 9 shows SEM images of the surface zone of specimens failed at speeds of 170 and 45 mm/min. Similar void clusters separated by damaged and undamaged walls are detected perpendicular to the tensile direction on the surfaces for both kinds of specimens. There are more numerous and broader voids over the surface for the specimen stretched at 170 mm/min, compared with the one stretched at 45 mm/min. The same results can be visualized by polarized optical microscopy ( figure 10). This means that, for specimens stretched at various tensile speeds, the fracture behavior is dominated by the propagation of voids and cavities. For similar void types, the opening and propagation speeds of the voids and cavities is proportional to the tensile speed, and thus the mean fracture time is inversely proportional to the tensile speed. When the specimen is stretched at extremely high tensile speed, voids and cavities coalescence as large voids appear and then propagate instantly, reducing the specimen lifetime. Thus, the probability of fracture increases, and the fracture time probability density distribution leans to the left.

Conclusions
In this work, the statistical nature of tensile fracture for iPP materials under different tensile speeds was sd. Data collected from more than one hundred tensile tests were analyzed using Gaussian and Weibull distribution curves to quantify the nature of the data scattering. We have drawn the following conclusions: 1: The probability distribution curves of fracture time, strength, strain and toughness can be fit by Gaussian distribution functions. 2: The fracture time is inversely proportional to the tensile speed. The fracture time monotonously increases with increasing gauge length of the specimen. 3: The average fracture strength is more sensitive to gauge length than to tensile speed. Longer gauge length leads to larger average strength. 4: Toughness is an intrinsic characteristic, independent of tensile speed and gauge length, and is a possible criterion for fracture. 5: At crosshead speeds above 170 mm/min, the probability distribution function of the fracture time changed from a Gaussian to a Weibull distribution.
These fracture time distributions make it possible to predict precise safety factors based on the probability of fracture events. Consequently, we can define the service limitations of the material. This will provide important information for materials design, to improve the reliability and lifetime of the materials under various conditions.