1. Introduction
Rubber fracture mechanics can be used to predict fatigue lifetime of an elastomer product, and the essential inputs are the crack growth rate law and the size of crack precursors [
1,
2,
3]. Our research investigation focuses on the microscopic crack precursors—also referred to as intrinsic flaws or defects—and how to effectively characterize their size distribution. Crack precursors can arise from undispersed filler agglomerates, regions of high crosslink density from incomplete dispersion of curatives, hard contaminants within the raw materials (e.g., dirt in natural rubber or grit in carbon black) or introduced during the processing of the rubber, and bubbles/voids within the material. Due to their heterogenous origins, crack precursors are not uniform in size or shape, and the distribution of crack precursor sizes in a rubber material leads to a distribution of failure properties, such as fatigue lifetime and tensile strength.
Direct observation of crack precursors was documented with microscopy by Huneau et al. [
4]. Carbon black filled natural rubber was fatigued, and then scanning electron microcopy (SEM) with energy-dispersive X-ray (EDX) elemental analysis was used to identify the chemical make-up of the sources for the noted micro-cracks. The majority of cracks were initiated from zinc oxide particles and carbon black agglomerates. We comment, however, that the images of presumed carbon black agglomerates in Figures 9 and 11 of that publication actually depict particles of ball coke, which is an impurity that can be produced in CB manufacturing [
5].
While particle-related influences like undispersed filler agglomerates can certainly be the source of larger crack starters in rubber compounds, even unfilled elastomers with simple cure systems have precursors that can be quantified from fatigue testing and strength measurements. Choi and Roland [
6] reported
c0 in the range from 10 to 29 μm for various types of natural rubber that were formulated only with an antioxidant and a peroxide for crosslinking the materials.
The statistical occurrence of precursors/defects within the actively deformed volume of a test specimen or real rubber product is an important aspect of durability. For the same rubber compound, the fatigue lifetime decreased as the test sample volume was increased—larger cylindrical dumbbell samples failed at lower number of loading cycles—in a study by Ludwig et al. [
7]. The corresponding precursor sizes for the three dumbbell shapes investigated were determined to be 100, 125, and 150 μm in order of increasing specimen test volume, reflecting the greater chance of bigger precursors occurring in larger sample volumes. The crack precursor size and its distribution are critical to elastomer reliability, but number density considerations are also important.
Ignatz-Hoover and coworkers [
8,
9,
10] developed an effective way of quantifying raw material effects and processing influences related to the dispersion of rubber ingredients including insoluble sulfur, vulcanization accelerators, and silica filler in various rubber compounds. Tensile strength of rubber is sensitive to the size of crack precursors, with larger precursors from poorly dispersed raw materials leading to lower strengths. They used Weibull statistics to fit the tensile strength (stress at break, σ
b) data from testing of numerous replicate rubber specimens, with 50 specimens determined to be an efficient population size for characterizing the failure distribution to give quantitative insights into dispersion effects [
10].
Our research further investigates the utility of tensile testing of 50 specimens to quantify the σb distribution in rubber, and we expand the approach to include assessment of the related crack precursor size (c0) distribution. This effective characterization of c0 distribution is key to fatigue lifetime modeling efforts to predict the reliability of rubber products, as will be discussed. The rubber compounds studied are based on carbon black (CB) filled styrene-butadiene rubber (SBR), with intentional poor mixing of CB and addition of glass microspheres used to emphasize the influence of inclusions on tensile strength and reliability of rubber. We compare the resulting crack precursor size distributions with defect size distributions measured using microscopy to illustrate that assessing c0 distribution from tensile strength testing is an important complementary approach for identifying the presence of minor amounts of large inclusions in rubber that cannot be detected by conventional filler dispersion measurements.
3. Results and Discussion
Four rubber compounds were created to study tensile strength distribution effects, and details of the formulations are given in
Table 1. We selected an ultra-clean grade of N550 carbon black with 0 ppm residue to avoid any potential influence from filler contaminants. Also, a fine particle grade of zinc oxide was used which has a surface area of 45 m
2/g, which is very similar to the N550 CB surface area of 40 m
2/g. The reference compound (Control) was a model CB-reinforced SBR, which was mixed using a standard laboratory mixing protocol. Poor mixing was simulated by adding 40% (20 phr) of the carbon black one minute before the end of the final (productive) mixing stage to produce a compound with poor CB dispersion (Poor Disp.). The final two compounds were generated by mixing the Control compound and then adding glass beads—solid microspheres with average diameter of 517 μm (0.517 mm) shown in
Figure 1—at two different concentrations on a two-roll mill after the productive mixing stage. The two levels of microspheres were 0.09 phr (Bead Low) and an eight times greater amount of 0.72 phr (Bead High), corresponding to volume averages of 0.78 bead and 6.24 beads per gauge section region of the specimen geometry used for tensile testing. The X-ray computed tomography (CT) scans presented in
Figure 2 confirmed: (1) Two distinct glass bead loadings for Bead Low and Bead High materials; (2) that the glass beads did not fracture during milling into the rubber compounds; and (3) the absence of large zinc oxide inclusions in the Control material.
For each material, 50 replicate tensile tests were conducted to characterize the failure populations. Examples of the stress-strain curves are shown in
Figure 3 and
Figure 4 for the Control and Bead Low materials, where it is immediately clear that there is a broader distribution of failure for the compound with the low concentration of glass microspheres compared to the reference compound. The failure distribution for the Bead Low compound spanned a very large range of σ
b, from 13.7 to 22.7 MPa, in contrast to the relatively narrow σ
b distribution for the Control, from 18.4 to 23.8 MPa. This is reflected in the standard deviations for σ
b in
Table 2 and
Figure 5. It is common to interpret such failure variances as “error bars” coming from combined material and test method variability considerations, but we emphasize that this information is providing insights into real crack precursor size distribution information, as will be shown. The average tensile strength decreased for the Poor Disp. (18.6 MPa), Bead Low (18.9 MPa), and Bead High (14.9 MPa) materials compared to the Control (21.1 MPa), as summarized in
Table 2. In contrast to σ
b, the critical tearing energy (tear strength,
Tc) was not affected by poor CB dispersion or the addition of the hard microspheres (
Table 2 and
Figure 5). In a tear test, the strain energy is focused on the pre-cut macroscopic crack in the sample rather than on the microscopic crack precursors within the rubber.
Tensile strength results from the 50 repeat tests were sorted from low to high values to produce fraction failed,
F, versus σ
b responses (
Figure 6a). The strength-reducing effects of poor carbon black dispersion and addition of the 0.5 mm glass inclusions are very evident in the results. The Bead Low material exhibits a bimodal σ
b population, which is reasonable because there was statistically less than one bead per test sample, so some specimens failed normally like the Control, and others had reduced strengths due to a large crack precursor from the presence of a glass microsphere in the gauge section of the specimen. The Bead High compound contained an average of over six glass microspheres per dumbbell gauge region, so the resulting failure distribution is fairly narrow, but shifted to much lower σ
b relative to the Control. The Poor Disp. compound has a σ
b distribution, which is generally shifted downward by about 2 MPa compared to the control, with a low σ
b tail that overlaps with the Bead High distribution. One important outcome from this exercise is the confirmation that the population size of 50 test specimens suggested by Wong et al. [
10] is sufficient to capture the variation in tensile strength within each material for these four diverse failure distributions. There are no significant gaps between experimental data points in
Figure 6a, and the results for each compound transition essentially smoothly from 0 to 1 on the fraction failed axis.
A combination of the tear testing and tensile testing results can be used to evaluate the crack precursor size distribution. The tearing energy or energy release rate,
T, depends on the crack size,
c, and the strain energy density,
W:
For simple tension deformation mode, the proportionality constant,
k, is approximately related to the tensile strain (ε) by [
2]:
These expressions can be used to determine the size of crack precursor,
c0. The quantity k W increases as a specimen is stretched during a tensile test, where
W is the integrated area under the stress–strain curve. When the product of this quantity and the largest crack precursor in the gauge section of the test specimen reaches the tearing limit of the material—the critical tearing energy (
Tc)—the sample ruptures. The breaking conditions in a tensile test,
Wb and ε
b, are thus linked to the tear strength,
Tc, of the rubber through
c0 (combining Equations (2) and (3) for
c =
c0):
Values of
c0 were accordingly determined for the 50 tensile test replicates for each compound, sorted from low to high, and plotted in
Figure 7a. The distinctly bimodal nature of the tensile strength population noted for the Bead Low compound is also observed for the crack precursor size distribution for this material. Investigating the fracture surfaces of the tensile specimens after testing revealed that the lower values of σ
b and larger
c0 for the Bead Low and Bead High materials were caused by the presence of the glass microspheres, and this is illustrated in
Figure 8 and
Figure 9. The inverse relationship between σ
b and
c0 can be noted by comparing their cumulative distribution functions,
F(σ
b) and
F(
c0), which are in reverse order for the four materials (
Figure 6a versus
Figure 7a), and this is also pointed out explicitly in
Figure 10. Lower tensile strengths come from larger precursors.
The stress at break and strain at break are commonly reported from standard tensile testing in most rubber laboratories, whereas
Wb is not typically included in a results summary for a material and raw data are often not readily available for integrating. For rubber compounds with typical stress-strain curve shapes, the following estimate for
Wb can be used in Equation (4) for situations where actual integrated stress–strain areas are not available:
This is the triangular area formed beneath a straight line drawn from the ε = 0, σ = 0 origin to the break point on the stress–strain plot. This approximation is quite good for the materials studied here, as verified in
Figure 11. Incorporating this estimate for
Wb in Equation (4) leads to the realization that tensile strength and tear strength are connected through crack precursor size:
c0 ~
Tc/σ
b.
Weibull statistics are commonly used to represent strength and fatigue lifetime results for many classes of materials including elastomers. As mentioned earlier, this was the approach of Ignatz-Hoover and coworkers [
8,
9,
10] for characterizing the σ
b distributions of rubber compounds for evaluating dispersion quality for fillers and additives. The Weibull cumulative distribution function (fraction failed,
F(
x)) expressions for a variable x in unimodal [
14,
15] and bimodal cases [
16,
17] are as follows:
The
xs is the characteristic value or scaling constant, m is the stretching exponent, and ϕ is the fraction (from 0 to 1) that separates the two components of a bimodal population. The reliability function (fraction survived,
R(
x)) is simply related to
F(
x):
The probability density function is the derivative of the cumulative distribution function:
The experimental data in
Figure 6a and
Figure 7a were fit using the Weibull
F(
x) function for
x = σ
b and
x =
c0, and the fitting parameters are summarized in
Table 3 and
Table 4. The Control data were successfully captured with a unimodal Weibull fit, but bimodal distributions were necessary for the other three materials. These fits are represented by the solid lines in
Figure 6a and
Figure 7a. The fitting results from
F(
x) were then used to produce the probability density
f(
x) curves for each material shown in
Figure 6b and
Figure 7b. The reference compound exhibits a narrow precursor distribution centered around
c0 = 180 μm, and adding the glass beads to the rubber produced a new population in the vicinity of 400 μm.
Conventional filler dispersion testing (
Table 5) did not detect the glass beads and yielded a fairly high dispersion index of 88.3 for the Poor Disp. compound wherein 40% (20 phr) of the carbon black was added just before the last minute of the final mixing stage. The IFM dispersion method analyzes an area of 825 μm × 825 μm of a cut rubber surface, which is repeated 10 times. This gives an overall probed area of 6.8 mm
2 in comparison to the total analyzed volume of 10,500 mm
3 in the gauge sections of the 50 tensile specimens. Characterizing tensile strength distribution is obviously an important complementary approach to traditional filler dispersion techniques for identifying the presence of minor amounts of large inclusions in rubber.
The observation that the addition of the 517 μm diameter microspheres results in a smaller size of about 400 μm for
c0 may be related to the expectation that the spherical glass beads introduce smooth, large diameter crack starters rather than sharp cracks with high stress concentrations [
18]. Also, based on microscopic diagnostics of crack initiation and growth from carbon black agglomerates, Huneau et al. [
4] proposed that rubber debonding first occurs at the poles of a precursor in a direction parallel to the loading, and, after additional fatigue, the crack direction eventually transitions to grow in the perpendicular direction to the load cycle input. This is an additional process that requires extra strain energy, and this may explain why the crack precursor size inferred from fatigue and strength measurements could be smaller in size than the actual physical initiator within the rubber. Other complexities may exist, including formation of nano- and micro-voids in front of a crack [
19,
20,
21].
Characterizing the precursor distributions in the manner shown here can be valuable for exploring rubber product reliability issues using elastomer fatigue simulations with critical plane analysis [
22,
23,
24]. A key input to such fatigue modeling is
c0, and knowing its distribution allows a comparison of the predicted lifetime, for example, due to the most probable
c0 versus worst case scenarios involving the large-
c0 tail of the distribution. Large precursors are only an issue if they end up in the areas of the product where crack growth driving forces are highest during use. Therefore, volume concentration and spatial statistics effects could be additionally included in modeling via Monte Carlo approaches by statistically mapping a population of precursors onto the rubber part geometry.