Predictive molding of elastomer aging effects on dynamic and static shear modulus.

Elastomers are used in applications for vibration isolation, motion accommodation and shock control. When designing components to provide vibration isolation and motion control, important physical properties are dynamic modulus, static modulus and tan delta. When elastomeric products are manufactured they are usually received by the customer in an optimum state for desired performance, however modulus and other properties change with time and temperature. Consequently, as elastomeric properties change from the original values used for the engineering design, performance is lost. As a result, the rate at which properties are lost will determine the shelf and service life of the part. By accounting for this property change in the design phase, customer satisfaction can be maintained with products which meet customer specifications for performance and life. Predictive techniques to characterize dynamic and static modulus change with time, temperature and formulation have been found that improve elastomer product design for optimum shelf and service life.

Elastomer property changes can be due to outgassing of residual materials left after the cure or to the migration of oils or other agents within the elastomer, but the most important property changes are caused by oxidation which results in additional crosslinking and scission of the elastomer (ref. 1). As crosslinking increases, modulus also increases and this is often referred to as "age stiffening." Whereas when scission occurs in polymers such as natural rubber, modulus decreases as chains are broken. Therefore, the change in modulus with time will be due to the difference between the rates of crosslinking and scission. With these oxidation reactions, the diffusion rate of oxygen becomes critical. At high temperatures, the diffusion rate is less than the oxidation rate which results in diffusion limited aging (ref. 2). With changing elastomer properties, engineers and chemists in the rubber industry are often asked to predict the service life of elastomeric products. Traditionally these estimations have been made with "rule of thumb" guesses, based on experience. Since aging affects part shelf and service life, it was of interest to define the rate of modulus change for elastomers. This would result in more accurate methods of estimating elastomer service life. A reasonable short term objective was to develop a predictive model for elastomer modulus change only due to heat aging as a function of time, temperature and elastomer composition (% polymer, % filler, % volatiles, etc.). A predictive model of modulus change with heat aging represents a first step toward an overall service life prediction.

Although many elastomeric properties are affected by heat, dynamic and static modulus were the subject of this study. A NR/BR compound heat aged under no load was an example of heat aging effects on dynamic and static modulus (figure 1). The modulus increased over time in ever decreasing increments for this compound. Although static and dynamic modulus have decreased with time in previous work (refs. 3 and 4). Much work has been done in the area of life estimation using time and temperature variables (refs. 5 and 6). Similar phenomena to figure I have been seen with elongation and time on tensile specimens. Equation I was an empirically derived formula which characterized short term (1,000 hours) changes in elongation to break.

[Elong.sub.o]/[Elong.sub.t] = exp ([bt.sup.1/2]) (1) where [Elong.sub.o] = original modulus; [Elong.sub.t] = elongation at time t; t = time; b = constant.

It is proposed that the sarne equation describes short term modulus change due to heat aging.

[E.sub.o]/[E.sub.t] = exp ([bt.sup.12]) or [E.sub.t]/[E.sub.o] = ([-bt.sup.1/2]) (2) where [E.sub.o] original modulus; [E.sub.t] = moudlus at t; t = time; b = constant.

With this equation, a straight line was obtained for shear modulus when the In ([G'.sub.t]/[G'.sub.o]) was plotted against [(t).sup.1/2]. In addition, the slope of the fitted line indicated the rate of modulus change due to heat aging, or the "heat resistance index" (ref. 5). This heat resistance index has been used for heat aging comparisons of elongation between elastomers (ref. 5). [G.sub.t]/[G'.sub.o] was used such that a "multiplier" could be found to predict the future shear modulus ([G'.sub.t]), based on the original shear modulus ([G'.sub.o]) with a given time and temperature.

[G'.sub.o] * (multiplier) = [G'.sub.t] (3) where: multiplier = f(time, temp.)

Models describing modulus change as a function of time were derived, at given temperatures for single elastomer formulations. A polybutadiene-based elastomer was used to confirm the Arrhenius temperature relationship with the heat resistance index "b" by comparing the various temperature models. Using the log of "b", a good fit was found with inverse temperature for polybutadiene elastomers of various original modulus (figure 2). A mathematical interpretation of the data relationship for shear modulus was then made, and is similar to models used for relative stress relationships (ref 7).

1n ([G'.sub.t]/[G'.sub.o]) = [b.sub.o] exp ([-E.sub.a]/RT) [t.sub.2] (4)

An expanded model further related elastomers by formulation similarity. Initial data analysis indicated that related elastomers had similar modulus changes. The incorporation of % polymer, % filler, % volatiles or other related variables into a single model was desired to avoid testing and modeling each individual elastomer formulation.

Experimental

The elastomeric properties used in this article are dynamic elastic shear modulus (G') and static shear modulus (G). Tan delta is also discussed and is the dynamic loss shear modulus (G") divided by the dynamic elastic shear modulus. These characteristics were measured at room temperature using a custom built servohydraulic test machine. This machine subjects a double lap shear specimen to a static or sinusoidal simple shear motion. For this analysis, dynamic elastic shear modulus measurements were taken at frequencies of 10 hertz and at strains of plus and minus 10% displacement. Static shear modulus was measured at a 25% displacement.

The double shear samples were initially cycled to a standard dynamic test profile, and was used throughout the experiment. The test profile consisted of six preflex cycles to 100% strain and was immediately followed by an increasing dynamic strain sweep. This preconditioning served to remove any "Mullins" effects from the heat aging analysis. Prior to any aging, the specimens were run again to the standard test profile to determine the original modulus at various strains and frequencies. These original moduli measurements served as the baseline for any change in modulus from oven aging. The samples were then oven aged for various periods of time. The test samples were aged in an upright position, with no load imposed on them. Duplicate samples were made, such that several temperatures were evaluated. After oven aging, the samples were allowed to return to room temperature. The samples were tested again on the servohydraulic machine, and then returned to the oven if longer aging was desired. Samples at different temperatures were not mixed, and cyclic testing was kept to a minimum to avoid the introduction of fatigue into the analysis.

Polybutadiene elastomers and natural rubber-polybutadiene blends were used in this study. The elastomers were carbon black filled at various levels, to achieve a wide range of moduli. The compounds are sulfur cured, and antioxidants and antiozonants are present in the elastomers. The elastomers are cured to at least a 90% cure level, as determined by rheometer characterization (Tc90). Double lap shear samples were cured at 425K for both elastomer types. The shipping container mounts were cured at 436K, and were also cured to at least a 90% cure level.

Results and discussion

Predictive model building

Based on equation 2 and the Arrhenius relationship, an initial hypothesized relationship for dynamic elastic shear modulus was derived.

ln (ln([G'.sub.t]/[G'.sub.o])) = a + b* ln([t.sup.0-5]) + c* l/T (5) where: a,b,c = constants

The hypothesized model explained the dynamic modulus change of a specific elastomer formulation at various temperatures and periods of time. As with other elastomeric products, vibration control products often use elastomer "series," where formulations use the same polymer and cure system, but vary modulus with filler loading. Therefore it was useful to interpolate information across an elastomer series as opposed to characterizing each modulus level within a series. In this manner, useful engineering data were more quickly generated. The most logical way to interpolate modulus during aging was to incorporate formulation variables into the model. The expanded model covered a series of related formulations.

ln (ln([G'.sub.t]/[G'.sub.o])) =

a + b* ln([t.sup.0.5]) + c* l/T + f (%P,%Fl, %F2, %V, [G.sub.o]) (6)

The best combination of variables found by regression analysis is shown in table 1. The original dynamic modulus ([G'.sub.o]) was a good independent variable since it was a function of weight percent polymer, filler and volatiles. Original modulus encompassed these three formulation variables into a single variable. This resulted in a more parsimonious regression equation, which was expected to be more accurate. With good regression statistics, this model's residuals and prediction-realization diagram (figure 3) were evaluated. Since [G'.sub.t] was the actual variable desired for prediction, all analyses used [G'.sub.t] predicted compared to [G'.sub.t] actual and residuals of [G'.sub.t] (actual) - [G'.sub.t] (predicted) as in figure 4.

[TABULAR DATA OMITTED]

One problem found in the prediction-realization diagram (figure 3) was a large bias when [G'.sub.t] was above 2.5 MPa. A second problem of heteroskedasticity was seen in the residuals versus [G'.sub.t] plot (figure 4). For large values of [G'.sub.t] a greater change in [G'.sub.t] was expected, which resulted in greater absolute residuals. Consequently, an ever increasing dispersion of residuals was expected with increasing values of [G'.sub.t]. Therefore a better way to analyze residuals was to use residuals as a percent of actual [G'.sub.t] (figure 5).

percent residual = residual* 100/[G'.sub.t] (7)

This put the residuals in terms of a percent error of [G'.sub.t], which was more reliable for measuring this elastomer "stiffening" phenomenon over large moduli ranges. With the percent residual vs. [G'.sub.t] method, the problem of heteroskedasticity was still present, as seen by the negative shift in the residuals (figure 5).

One phenomenon which was expected in long term aging was that of "oxygen diffusion limitation" (ref. 2) and the formation of an oxidized skin (ref 8). The oxidized skin formation continuously reduces the rate of oxygen diffusion and causes [G'.sub.t] to increase at a diminishing rate (ref. 2) . Since our ln(ln([G'.sub.t]/[G'.sub.o])) model predicted values of [G'.sub.t] that were too high (negative residual), the oxygen diffusion may have come into play. To correct the heteroskedasticity, methods of variable transformation or weighted variables are often helpful in modeling. Since the model was empirically correct with previous work (ref. 5), the preferred option was a weighted least squares regression model as opposed to an equation transformation. The weighted variable most likely to shift the model correctly was time. With the highest time values weighted most heavily, the model was best for time extrapolation. Table 2 contains the weighted least squares regression results.

  Table 2 - ln(ln(G'.sub.t]/[G'.sub.o])) regression weighted by
hours

Dependent variable is ln(ln([G'.sub.t]/[G'.sub.o]))
Number of observations: 141
Weighing series: hours

Variable   Coefficient  Std. error     T-stat.  2-Tail sig.
C          -15.679496   4.4843595   -3.4964850    0.000
LNSQRTHR    1.3347571   0.0895639    14.902839    0.000
INVTEMPK   -5095.4263   116.42971   -43.763970    0.000
LNGORIG    -4.2106070   0.6398522   -6.5805934    0.000
LNWTFIL     7.5804364   1.3683955    5.5396530    0.000

     Weighted statistics

R-squared           0.971466  Mean of dependent  -1.957813
Adjusted R-squared  0.970327  S.D. of dependent   3.314328
S.E. of regression  0.568027  Sum of squared      43.88108
Durbin-Watson stat  1.082300  resid               1157.573
Log likelihood     -117.7773  F-statistic

Where: C = constant

This regression had a higher [R.sup.2] and F ratio, but the SE (standard error) of regression and sum of squared residuals increased. An analysis of percent residuals and the prediction-realization diagram found that the model was only good for short extrapolation (100-300 hours). This conclusion was supported by the patterns present in the percent residuals, and the prediction realization diagram with a curvalinear relationship. Therefore a significant effect still remained outside the regression relationship.

Other weighted variables produced no better results, so a transformation of the model was used to improve the regression results. An initial assumption made about the original model (equation 6) was that [G'.sub.t]/[G'.sub.o]) and [G'.sub.o] were not correlated. They had an r = 0.65, but previous model building experience had shown that the variables [G'.sub.t]/[G'.sub.o] and [G'.sub.o]) sometimes caused collineatily problems. Knowing that [G'.sub.o] was a significant independent variable, a new equation was developed.

ln (ln([G'.sub.t])) =

a + b* ln([t.sup.0.5) + c* l/t + d*ln([G'.sub.o]) + f*in(wt. % total filer) (8) Where: a,b,c,d, and f = constants Equation 8 was used to develop a regression model which could predict modulus without leaving a pattern in the residuals. An important point to note for model analysis was that by changing the dependent variable, comparison of new statistical regression results with the original model's results was no longer possible. Another point of interest was that the significance of ln(wt % filler) dropped off, and the term was dropped from the equation. This was not surprising since [G'.sub.o] and total filler were correlated with an r = 0.73. The new regression, using ln(ln([G'.sub.t])) as the dependent variable, is shown in table 3. The residuals of the model had no pattern, a near normal distribution and a mean residual very close to zero (table 4). Before committing to this model, a comparison of this model to the previous [G'.sub.t]/[G'.sub.o], models was made to determine the best [G'.sub.t] forecasting model.

  Table 3 - ln(ln([G'.sub.t])) regression model

Dependent variable is ln(ln([G'.sub.t]))
Number of observations: 141

Variable  Coefficient  Std. error    T-stat.    2-Tail sig.
C          1.3311984   0.0271549    49.022488     0.000
LNSQRTHR   0.0322071   0.0019945    16.147564     0.000
INVTEMPK  -123.14751   5.1476841   -23.922895     0.000
LNGORIG    0.1321023   0.0030298    43.601307     0.000

R-squared           0.948912  Mean of dependent     2.050486
Adjusted R-squared  0.947793  S.D. of dependent     0.070818
S.E. of regression  0.016181  Sum of squared resid  0.035870
Durbin-Watson stat  0.741341  F-statistic           848.2081
Log likelihood      383.4308

kPA used for regression analysis ([G'.sub.t] & [G'.sub.o])

[TABULAR DATA OMITTED]

A way to evaluate models with different dependent variables was to make "ex-post" forecasts from six months of [G'.sub.t] data, and compare them to the actual values of [G'.sub.t] at the eight month data points with a root mean squared error (RMSE). The best model for interpolation and extrapolation forecasting was the ln(ln([G'.sub.t])) model (tables 5, 6 and 7). By incorporating the eight month ex-post data and 11 month values into the regression, the model in table 8 was used for future estimation. Residual data appeared to be random.

  Table 5 - error analysis of the ln(ln([G'.ub.t])) regression

ln(ln([G'.sub.t])) model:
Number of observations: 141

Series         Mean       S. D.      Maximum    Minimum
SQERROR        0.1535741  0.2777802  1.6055070  0.0000000
RMSE = 0.3919

Where: Error = [G'.sub.t] (actual) - [G'.sub.t] (pred.) in
Mpa; SQERROR
[error.sup.2]; RMSE = root mean squared error

  Table 6 - error analysis of the ln(ln([G'.sub.t]/[G'.sub.o]))
             regression model

ln(ln([G'.sub.t]/[G'.sub.o])) unweighted model:
Number of observations: 141

Series         Mean       S. D.      Maximum    Minimum
SQERROR        0.2983217  1.1341259  8.8038010  1.000E-06
RSME = 0.5462

Where: Error = [G'.sub.t] (actual) - [G'.sub.t] (pred.) in MPa;
SQERROR =
[error.sup.2]; RMSE = root mean squared error

  Table 7 - error analysis of the ln(ln([G'.sub.t]/[G'.sub.o]
                   weighted model

ln(ln([G'.sub.t]/[G'.sub.o])) weighted (hours) model:
Number of observations: 141

Series         Mean       S. D.      Maximum    Minimum
SQERROR        0.2331222  0.7813149  6.5913590  1.000E-06
RSME = 0.4828

Where: Error = [G'.sub.t] (actual) - [G'.sub.t] (pred.) in MPa;
SQERROR =
[error.sup.2]; RMSE = root mean squared error

  Table 8 - regression of ln(ln([G'.sub.t])) model with 11
                  months of data

Dependent variable is ln(ln([G'.sub.t]))
Number of observations: 156

Variable  Coefficient  Std. error    T-stat    2-Tail sig.
C         1.3614493    0.0263046    51.757064    0.000
LNSQRTHR  0.0329892    0.0017463    18.890713    0.000
INVTEMPK  -129.66984   5.0407398   -25.724367    0.000
LNGORIG   0.1303657    0.0029304    44,486791    0.000

R-squared           0.946936  Mean of dependent     2.052391
Adjusted R-squared  0.945889  S.D. of dependent     0.071291
S.E. of regression  0.016584  Sum of squared resid  0.041802
Durbin-Watson stat  0.759925  F-statistic           904.1568
Log likelihood      420.1690

([G'.sub.t] and [G'.sub.o] are in kPa

The [G'.sub.t]/[G'.sub.o] model supported previous work (ref. 5) as a good method for interpolation and short term estimation, but could not be used for long term time extrapolation because the actual to predicted modulus deviation due to "oxygen diffusion limitation" (ref 2) was too large. The [G'.sub.t] model was an improvement because it underestimated the modulus change and would eventually merge with an elastomer experiencing an ever decreasing rate of change due to diffusion limitation. Thus, the [G'.sub.t] model appeared to be the better long term aging model for modulus. However, both these models had inadequate prediction error for extrapolations greater than two months from the most recent data point. By predicting with both models and averaging their values, a much more empirically accurate model for extrapolation emerged.

Designed experiment for predictive aging technique

Another need for useful predictive models was the ability to quickly gather adequate data for time, temperature and modulus ranges. Data collection needed to be done faster while maximizing the predictive range. Utilizing statistical experimental design knowledge, a central composite design (ref. 9) was developed which generated modulus data with variables of time, temperature and initial modulus. To match the linear regression variables, the time, temperature and initial modulus were coded. Temperature and modulus remained unchanged, but the time variable used was the square root of time.

With a maximum exposure time of two months, oven age data for the NR/BR elastomer was generated. The regression analyses of the data were statistically acceptable. By comparing the actual versus the predicted modulus, it was seen that the "averaging" forecast technique provided useful predictions up to 5,800 hours, well beyond the experimental range of 1,344 hours. Quick estimation of modulus change was possible with useful extrapolation, using an averaging of the two models' modulus predictions.

A similar method of analysis was used for age estimations for static shear modulus at 25% strain. Similar regression statistics and estimations were achieved, based on the 20 run designed experiment with static modulus. Figure 6 displays the predicted versus actual measurements at 343K over time for the NR/BR blends at three different original moduli. The actual results for the low and middle modulus elastomers were close to the predicted lines, based on the model averaging technique. However, the high modulus elastomer was closer to the ln(ln([G.sub.t])) model than the averaged prediction. At the 5,800 hour test point, the high modulus sample cracked and gave erroneous information. The lower than expected value at 4,400 hours may contain an elastomer effect which occurs when the sample meets its stiffening threshold.

The same analysis method was used for the calculation and prediction of tan delta (tan delta = G"/G'). One problem that occurred was that the calculation of ln(ln(tan delta)) was impossible. Since tan delta was less than one, the double logarithm was incalculable. Therefore only the ln(ln(tan [delta.sub.t]/tan [delta.sub.o]) model was available for regression analysis. However as seen in figure 7, the tan delta change may either increase or decrease depending on the combined effect of aging on the G' and G". As a result, the tan delta change could be less than one and the double logarithm again would be incalculable. The current models are therefore unable to directly calculate tan delta. An attempt was made to model G", such that predicted G' and G" could be used to predict tan delta. However, the G" varied in a similar fashion to the tan delta. Thus, the prediction of tan delta can only be made on an individual formulation, once the general trend with time of the material has been established.

Utilizing a standard aging sample, shelf and service lifewere estimated at elevated temperatures over time in a cost effective manner. The models with the double shear test sample allowed for a comparison of elastomers, but have not directly impacted part service. But before samples could be used to represent parts, actual parts had to confirm the aging model's relevance to parts.

Application of predictive technique to product

A simple geometry part with high production volume was the J- 18106 series shipping container mount. "Shipping container mounts are for fragile, valuable products needing predictable, low to medium level protection (from shock loading). ... These shipping container mounts consist of two metal plates with an elastomer bonded between them. The composition and configuration of the elastomer determines the static and dynamic properties of the parts" (ref 10) . To serve their purpose, shipping container mounts need to maintain tight modulus control throughout their shelf and service lives. Therefore methods for predicting modulus change with time and temperature allow better estimation of maximum shelf and service life of shipping container mounts. The modulus characterization of shipping container mount elastomers can then be used to build a higher level of quality into the part.

Applying this technique to the J- 18106 shipping container mount resulted in performance interpolation and extrapolation at temperatures of 343K and 373K. In place of dynamic modulus, a static spring rate (or load/deflection) was used as the performance criteria. Pairs of shipping container mounts were tested in the shear direction. The method of experimentation was the same as the double shear samples, where parts were tested prior to oven aging, oven aged, allowed to return to room temperature, and tested. Returning to the original modulus model equation 2, the spring rate change was related to time at a given temperature for the middle modulus BR elastomer.

Using the middle modulus BR compound, the two regressions' predicted values in figure 8 described the spring rate change with lime at temperatures of 343K and 373K.

To compare the rates of aging for different geometries, the percent change of dynamic modulus (double shear specimen) and spring rate (Shipping container mount) were plotted together. Supporting the theories of oxygen driven aging, the larger shipping container mount stiffened at a slower rate than the double shear sample. To increase the usefulness of the double shear sample testing, correlations to part geometries should be made where predicted modulus is also a function of rubber thickness. The incorporation of rubber thickness into the model should be a diffusion relationship, such as Fiek's 1st and 2nd laws of diffusion (refs. 1 and 12). A modulus model that includes time, temperature, formulation, and rubber thickness would be extremely flexible and almost completely characterize the shelf life of elastomeric products.

Conclusions

* The change in modulus due to aging is a function of time, temperature, and formulation.

* The ln(ln([G'.sub.t]/[G'.sub.o])) model was found to predict modulus changes that were higher than actual values, failing to account for oxygen diffusion limitation.

* The ln(ln([G'.sub.t]/[G'.sub.o])) model was found to be a better statistical model with no residual patterns, but typically predicted modulus changes with lower than actual values.

* The best predictions of modulus change were determined by averaging the predictions arrived at by the ln(ln([G'.sub.t]/[G'.sub.o])) and ln(ln([G'.sub.t])) models.

* Useful modulus characterization was made with a central composite designed test plan, where the forecast range was at least four times the experimental region.

* The modulus prediction technique was useful for static shear modulus, dynamic elastic shear modulus, as well as spring rate properties.

* Predictive values for tan delta could only be made for specific elastomer compounds at a given temperature, once tan delta had been identified as increasing or decreasing with time or exposure.

Summary

The predictive techniques for modulus estimation with time, temperature and formulation was considered a successful model toward the development of a comprehensive elastomeric service life model. Time and temperature variables were incorporated into a single model. Elastomer formulation was also taken into consideration with the [G'.sub.o] variable, which summarized the effects of polymer, filler and volatile material. The model was parsimonious since it only used three to four independent variables, which are readily available to design engineers. The ln(ln([G'.sub.t]/[G'.sub.o]) model displayed its ability to predict modulus over short aging periods, whereas the ln(ln([G'.sub.t])) model was useful in the extreme cases of aging. By combining the methods, a comprehensive predictive technique was established over all time periods. One shortcoming of the model was the remaining error that was not explained by the model. With percent residual errors of [G'.sub.t] up to 30% there was some doubt cast on the extreme extrapolating power of the model. Additional independent variables should be evaluated in the future. Or these variables, the state of cure and secondary chemical reactions may explain more of the current equation's error. In addition, the effect of rubber thickness and aging under stress will be the next milestones toward the comprehensive elastomeric service life model. Using diffusion relationship, the effect of rubber thickness for simple pan geometeries could be incorporated to the modulus model. The effect of aging under stress could have a pronounced effect on modulus change, and must be addressed in order to characterize product service life. With the current modulus model and future expansions of this model, complete characterization of product service life will results in higher value added product designs.

References

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