# Measurement of the residual mechanical properties of crazed polycarbonate. II: Design of experiments analysis.

Stephen B. Clay (*)

The Design of Experiments (DOE) approach was used to build quantitative empirical models of the residual mechanical properties of crazed polycarbonate as functions of relative craze density, crazing stress, and strain rate. Crazing did not affect the yielding behavior of polycarbonate, but increasing the strain rate increased the yield stress according to the Eyring theory. The Eyring activation volume for yielding of crazed polycarbonate was measured to fall between reported values for conformational changes of a dilute solution of polycarbonate chains and yielding of uncrazed polycarbonate. Also, with as little as 1% relative craze density, the failure stress was approximately 10% lower and the ductility was, on the average, 50% lower than for uncrazed polycarbonate properties. It was also found that increasing the crazing stress from 40 to 45 MPa increased the failure stress and ductility for a given magnitude of relative craze density.

INTRODUCTION

The United States Air Force has been using polycarbonate as the structural ply In many of its advanced transparency systems since the 1970s. One of the most common causes for replacement of military aircraft transparencies is crazing, which is the formation of fibrillated crack-like defects in the presence of tensile stress. Crazes degrade both the optics and structural integrity of a polymer. To date, the effect of crazing on the mechanical properties of polycarbonate has been difficult to quantify since no standardized technique exists to quantify craze severity.

A recent study (1, 2) has qualitatively assessed the residual mechanical properties of crazed polycarbonate using a new technique to quantify crazing. One of the goals of this paper is to correlate the residual mechanical properties measured in the previous study to crazing stress, relative craze density, and strain rate. The Design of Experiments (DOE) approach will be employed to provide more valuable modeling results than the traditional approach of changing one factor at a time. The designed experiment approach requires more pre-experiment planning than traditional techniques, but increases the chances of obtaining a valid model that assigns the response to the proper input variables or interaction between input variables. The steps for methodically setting up a properly designed experiment have been well documented (3).

MATERIALS

The material used for this work was a commercial grade 6-mm-thick polycarbonate sheet supplied by McMaster Carr. The number average molecular weight as measured by GPC was 19,450 g/mol with a polydispersity index of 2.2. The glass transition temperature determined from DSC was found to be 147[degrees]C. The as-received yield stress and elastic modulus were measured to be 65 MPa and 2.3 GPa, respectively.

EXPERIMENTAL METHODS

Craze Testing

Craze testing was performed under constant load at 24[degrees]C and 85 [+ or -] 5% rh, as described in previous work (1, 2). The range of stress tested for this study was 40 to 45 MPa. While under stress, the relative craze density was measured with a reflective imaging system (1, 2) consisting of a CCD camera, backlight, and frame grabber. The craze growth charts were created using relative craze densities measured on the stored images with SigmaScan Pro[TM] image analysis software (4).

Residual Mechanical Property Measurement

The following procedure outlines the general steps to follow to measure the residual mechanical properties of crazed samples.

1) Conduct constant strain rate tensile tests to failure at the rate specified in the test matrix for a given sample.

2) Measure the yield stress, failure stress, elastic modulus and elongation for each sample.

For this paper, the yield stress was taken to be the maximum magnitude of stress before the constant drawing region. The failure stress was measured as the maximum stress obtained by the polycarbonate after the yield region just prior to fracture. The elastic modulus was calculated by measuring the slope of the initial linear portion of the curve. The measure of ductility was taken to be the final length of the necked portion of the gauge section after fracture normalized to the gauge section of 50 mm.

Design of Experiments

Two separate Design of Experiments analyses were conducted to build mathematical models for the residual mechanical properties of PC. A computer program called DOE KISS (5) was used to analyze the results of the craze test matrix. For a test matrix in which each factor is only tested at two levels, the program codes the minimum value to -1 and the maximum value to +1 while performing the analysis routines. The orthogonal nature of the coded matrix allows efficient statistical calculations to be performed on the collected data.

The first analysis was performed with a constant crazing stress of 40 MPa and included experimental variables of relative craze density and strain rate. The relative craze density varied from 0% to 1%, and the range for strain rate was 0.2 to 2 [min.sup.-1]. The two-factor, two-level values for the DOE model are shown in Table 1.

The second analysis included three factors each tested at two levels. The experimental factors include crazing stress, relative craze density, and strain rate. Table 2 displays the low and high levels for each factor.

For both two-factor and three-factor designed experiments, a full-factorial test matrix testing every possible combination of variables is the best approach. This approach results in four separate test conditions for the two factor matrix and eight separate conditions for the three factor design. Each test condition was repeated three times to provide variability information. No "aliasing" exists in full factorial designs, meaning that all two-way and three-way interactions are evaluated.

Both graphical and statistical analyses were performed on the results of the test matrix. The first analysis technique is to derive a predictive equation for the response using average effects of each factor. The predictive model output of DOE KISS for a two-level design results in the experimental response expressed in terms of a constant and a linear relationship with each factor and each interaction. The general form of the equation is given by

Y = [c.sub.0] + [c.sub.1]A + [c.sub.2]B + [c.sub.3]C + [c.sub.4]AB + [c.sub.5]AC + [c.sub.6]BC + [c.sub.7]ABC (1)

where Y is the predicted value of a given residual mechanical property, the c's are coefficients, A is the coded stress level (-1, + 1), B is the coded relative craze density (-1, +1), C is the coded strain rate (-1, +1), and multiplication of two main factors represents the interaction between the two main factors.

Often, a main factor or an interaction between main factors does not change the measured response significantly. For this reason, DOE KISS calculates the term P(2-tail), which is the probability that a term does not belong in the model. The confidence that a given term belongs in a model is given by

Confidence term belongs in model = (1 - P(2 - tail))*100% (2)

The rule of thumb given in the literature (3) is that if the value of P(2-tail) for a given factor is less than 0.05 then the term should be included in the model. The range of P(2-tail) from 0.05 to 0.10 is an intermediate zone. For this paper, each factor with a P(2-tail) [less than or equal to] 0.10 is included in the predictive model.

RESULTS AND DISCUSSION

Yield Stress (0% to 1% Relative Craze Density)

The first mechanical property of crazed polycarbonate to be considered in the two-factor model was yield stress. Table 3 shows the mean and standard deviation of the three repetitions of the four test runs.

Using the data in Table 3, the constant and coefficients for the predictive model for the yield stress of crazed polycarbonate were calculated and are shown in Table 4. The P(2-tail) values of each factor for the yield stress are also given.

The only terms shown in Table 4 that have P(2-tail) < 0.1 are the constant and the strain rate as shown in bold face. Therefore, the predictive model for the yield stress of crazed polycarbonate will contain only these two terms, as shown by

[[sigma].sub.y] = 67.0 + 1.44 [epsilon] (3)

where [[sigma].sub.y] is the yield stress in MPa and [epsilon] is the dimensionless coded strain rate. Recall that to use this equation, one must use - 1 for the low-level strain rate (0.2 [min.sup.-1]), +1 for the high-level strain rate (2 [min.sup.-1]), and an interpolated value between -1 and + 1 for strain rates between 0.2 and 2 [min.sup.-1].

These results correlate well with the qualitative results shown previously (1, 2). The previously reported qualitative results show that changing the relative craze density from 0% to 1% does not significantly affect the yield stress. They also showed a strong correlation between strain rate and yield stress, in that a strain rate of 2 [min.sup.-1] results in a significantly higher yield stress than a strain rate of 0.2 [min.sup.-1]. Equation 3 provides a quantitative model to the qualitative results previously reported.

Yield Stress (0% to 1% Relative Craze Density) Confirmation

The two-factor DOE model was developed to predict the mechanical properties of polycarbonate with relative craze densities from 0% to 1% and strain rates from 0.2 to 2 [min.sup.-1]. The first confirmation tests were conducted to determine the accuracy of the yield stress model shown in Eq 3. The strain rate was the only factor found to correlate with yield stress. Figure 1 shows the yield stress as a function of strain rate, with the solid line representing the predictive model and the data points depicting the average value of the confirmation tests. The endpoints represent the test conditions used to build the model with error bars showing one standard deviation of the data collected at each point.

As expected, the endpoints fit well to the predictive model since they were used to build the model. The yield stress at the midpoint is shown to be higher than predicted, with the difference between the predicted and average measured values being 0.8 MPa. These results show that the model produces a good prediction of the yield stress of crazed polycarbonate at the endpoints, with a slight underestimation at non-endpoint conditions, which is more desirable than overestimation.

Since the model shows that strain rate is the only important experimental factor in predicting the yield stress of crazed polycarbonate, the Eyring stress-rate model for thermally activated deformation processes will now be considered. The Eyring model (6) was developed to describe the effect of an applied stress on the deformation rate of a polymer as shown by e

e [approximately equal to] [e.sub.0]/2 exp (- [DELTA]H - v[sigma]/RT) (4)

where e is the strain rate, [e.sub.0] is a constant, v is an activation volume, [DELTA]H is the activation energy, R is the gas constant, T is the absolute temperature, and [sigma] is the applied stress. Solving for stress in Eq 4 and applying the model to the yielding behavior of polycarbonate produces (5)

[[sigma].sub.y] = RT/v[\.sub.v] ([DELTA][H.sub.y]/RT + ln 2[e.sub.y]/[e.sub.oy]) (5)

where [[sigma].sub.y] is the yield stress, R is the gas constant, T is the temperature. [v.sub.y] is an activation volume associated with yielding, [DELTA][H.sub.y] is the activation energy associated with yielding, and [e.sub.y] is the strain rate associated with yielding, and [e.sub.oy] is a constant. The Eyring equation for yield stress shown in Eq 5 will now be fit to the experimental data of Fig. 1 and shown in Fig. 2.

As shown in Fig. 2, the equation for the logarithmic data fit for yield stress in the range of 0 to 1% relative craze density is given by

[[sigma].sub.y] = 67.6 + 1.185 ln ([epsilon]) (6)

where [[sigma].sub.y] is the yield stress in MPa and [epsilon] is the strain rate in [min.sup.-1]. Salving for activation volume in Eq 5 in conjunction with Eq 6 gives

RT/v = 1.185 x [10.sup.6] Pa (7)

v = (1.38 x [10.sup.-23] N * m/K)(300K)/1.185 x [10.sup.6] N/[m.sup.2] = 3.49 x [10.sup.-27] [m.sup.3] = 3.49 [nm.sup.3] (8)

Equation 8 implies that chain segments occupying a volume of 3.49 [nm.sup.3] must work cooperatively for yielding to occur in crazed polycarbonate. Table 5 records the calculated value for the volume of an equivalent link for a polycarbonate chain in solution to be 0.48 [nm.sup.3] while the Eyring activation volume for uncrazed polycarbonate has been calculated to be 6.4 [nm.sup.3] by Haward and Thackray (7).

These results indicate that polycarbonate samples with relative craze densities up to 1% require more than seven times as much cooperative volume for yielding than a dilute solution of polycarbonate chains needs to change conformations. Also, a direct comparison of the crazed samples in this study to the uncrazed samples in the literature implies that uncrazed polycarbonate requires almost twice as much cooperative volume for yielding than crazed polycarbonate, but other unknown material properties may also play a role.

Since the yield stress is shown to follow the Eyring stress-rate model, a linear DOE model should not be expected to provide a good fit to the data far from the endpoints. If the presence of crazes affected the yielding behavior of polycarbonate, the Eyring model would need to be modified to account for the effect of crazing, but crazing was not shown to play a significant role in yielding.

Yield Stress (1% to 10% Relative Craze Density)

The yield stress of polycarbonate with relative craze densities in the range from 1% to 10% will now be considered. Test data are presented in Table 6 to show the effects of crazing stress and relative craze density on the yield stress. Each value represents the average from three tests with the standard deviation shown in parentheses.

Using the experimental data, the constant and coefficients for the predictive model for yield stress were calculated by the statistical software program with the results displayed in Table 7. P(2-tail) values for each factor are also given.

Similar to before, the only terms shown in Table 7 that have P(2-tail) < 0.1 are the constant and the strain rate. Therefore, the predictive model for yield stress of crazed polycarbonate will contain only these two terms, as shown by

[[sigma].sub.y] = 66.4 + 1.58 [epsilon] (9)

where [[sigma].sub.y] is the yield stress of the crazed sample in MPa and [epsilon] is the coded strain rate.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that increasing the crazing stress from 40 to 45 MPa and increasing the relative craze density from 1% to 10% produces a very slight decrease, but overall insignificant effect, on the yield stress. The previous results also showed that increasing the strain rate from 0.2 to 2 [min.sup.-1] increased the yield stress on the order of 3 MPa. Equation 9 implies that an increase in the strain rate from 0.2 to 2 [min.sup.-1] will increase the yield stress by 3.16 MPa since the coefficient shows the half effect of the factor. Therefore, Eq 9 provides a quantitative model to the qualitative results.

Yield Stress (1% to 10% Relative Craze Density) Confirmation

The three-factor DOE model was developed to predict the mechanical properties of polycarbonate with relative craze densities from 1% to 10%, crazing stresses from 40 to 45 MPa, and strain rates from 0.2 to 2 [min.sup.-1]. Confirmation tests were conducted to determine the accuracy of the yield stress model shown in Eq 9. The only factor found to correlate to yield stress was strain rate. Figure 3 shows the yield stress as a function of strain rate, with the solid line representing the predictive model, the data points depicting the average value of the confirmation tests, and the dashed line representing the logarithmic fit to the experimental data as done with the Eyring model above. The endpoints represent the test conditions used to build the model, with error bars showing one standard deviation of the data collected at each point. The number of non-endpoint confirmation tests at each strain rate ranged from 1 to 3.

As expected, the endpoints fit well to the predictive model since they were used to build the model. The yield stress at 0.4, 0.8, and 1.6 [min.sup.-1] are shown to be higher than predicted, with the largest difference between the predicted and measured values being about 1 MPa. These results show that the model slightly underestimates the yield stress, which is more desirable than overestimation.

These results are expected as shown in the two-factor DOE yield stress prediction above in which the Eyring rate effect governs the behavior. Since the Eyring data fit for polycarbonate with relative craze densities from 1% to 10% possesses an identical pre-logarithmic coefficient (1.185 MPa) as for the 0% to 1% crazed material, the activation volume in this craze range is also calculated to be 3.49 [nm.sup.3].

Failure Stress (0% to 1% Relative Craze Density)

Since the residual yield stress of crazed polycarbonate has already been quantitatively modeled and no correlation was found for the elastic modulus, the remainder of this paper attempts to model the failure properties of crazed polycarbonate. Crazing is expected to affect the failure properties more than the elastic and yield properties since fracture is highly dependent on flaws. The first failure property to be considered is failure stress, which will now be modeled in the range of 0% to 1% relative craze density. The mean and standard deviation of the three repetitions are given in Table 8.

Using the data in Table 8, the constant and coefficients for the predictive model for the failure stress of crazed polycarbonate were calculated similarly to the calculations above and are shown in Table 9. The P[2-tail) values of each factor for the failure stress are also given.

The only terms shown in Table 9 that have $2-tail) <0.1 are the constant and the relative craze density. Therefore, the predictive model for the failure stress of crazed polycarbonate will contain only these two terms, as shown by

[[sigma].sub.F] = 53.1 - 3.24 D (10)

where [[sigma].sub.F] is the failure stress in MPa and D is the dimensionless coded relative craze density.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that changing the relative craze density from 0% to 1% at a crazing stress of 40 MPa decreases the failure stress significantly. They also show that increasing the strain rate from 0.2 to 2 [min.sup.-1] does not produce significant changes in the failure stress of polycarbonate. Equation 10 provides a quantitative model to the qualitative results previously reported.

Failure Stress (0% to 1% Relative Craze Density) Confirmation

Next, the accuracy of the two-factor DOE model for failure stress will be evaluated. The only factor found to correlate to the failure stress of crazed polycarbonate was relative craze density, as shown in Eq 10. Figure 4 shows the failure stress as a function of relative craze density, with the solid line representing the predictive model and the data points depicting the average value of the confirmation tests. The endpoints represent the test conditions used to build the model, with error bars showing one standard deviation of the data collected at each point.

Figure 4 shows that the model developed for this paper accurately predicts the effect of relative craze density on the failure stress of crazed polycarbonate. The average values for both the endpoint conditions and the midpoint fit well with the prediction, but the magnitude of the error bars shows that there is significant scatter in the data with respect to the overall decrease in failure stress. This scatter is a result of the fact that failure properties depend on the distribution of flaws of various sizes and amounts, which is increased with increased crazing, but is highly variable from sample to sample. It should be noted that the Eyring rate effect is not seen in this model, since strain rate was not found to be a significant factor and crazing dominates the failure behavior. For this reason, a linear model is appropriate.

Failure Stress (1% to 10% Relative Craze Density)

Table 10 shows the effect of strain rate and relative craze density on the failure stress of crazed polycarbonate in the range from 1% to 10% relative craze density. Each value represents the mean from three tests, with the standard deviation in parentheses.

Using the experimental data, the constant and coefficients for the predictive model for failure stress were calculated similarly to the calculations above and are shown in Table 11. The P(2-tail) values of each factor for the failure stress are also given.

The only terms shown in Table 11 that have P(2-tail) < 0.1 are the constant, the crazing stress, and the interaction between the crazing stress and the strain rate. Therefore, the predictive model for the failure stress of crazed polycarbonate will contain only these three terms, as shown by

[[sigma].sub.f] = 52.8 + 3.l9 [[sigma].sub.c] - 1.57 [[sigma].sub.c]*[epsilon] (11)

where [[sigma].sub.f] is the failure stress of the crazed sample in MPa, [[sigma].sub.c] is the dimensionless coded crazing stress, and [[sigma].sub.c]*[epsilon] is the dimensionless interaction between the coded crazing stress and the coded strain rate.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that no strong correlation exists between relative craze density or strain rate and failure stress. They also showed a strong correlation between crazing stress and failure stress, in that a crazing stress of 40 MPa results in a failure stress significantly lower than a crazing stress of 45 MPa. Another observation from previous work that could have easily been overlooked is that when the crazing stress increases from 40 to 45 MPa, the effect that the strain rate has on the failure stress reverses. At the lower crazing stress, an increase in strain rate increases the failure stress, while at the higher crazing stress, an increase in the strain rate decreases the failure stress. Therefore, an interaction exists between crazing stress and strain rate. Equation 11 provides a quantitative model to the qualitative results previously reported.

Failure Stress (1% to 10% Relative Craze Density) Confirmation

Next, the three-factor DOE model for failure stress prediction will be evaluated. Both crazing stress and the interaction between crazing stress and strain rate were found to correlate to the failure stress, as shown in Eq 11. Table 12 shows the predicted values and the average of the measured values of failure stress at various crazing stresses and strain rates. The conditions of (40 MPa, 0.2 [min.sup.-1]), (40 MPa, 2 [min.sup.-1]), (45 MPa, 0.2 [min.sup.-1]), and (45 MPa, 2 [min.sup.-1]), represent the endpoints used to build the model.

The average difference between the predicted and measured values of failure stress in Table 12 was found to be approximately 1.5 MPa. Also, the measured values for failure stress at all three non-endpoint conditions are slightly higher than the predicted values, showing an underestimation of failure stress similar to what was seen in the yield stress model. Once again, underestimation of the failure stress is more desirable than overestimation.

The Eyring model has previously been applied to the fracture of polymers (6), similar to the yield stress as shown above. In this case, the failure stress has been shown to be a function of both crazing stress and strain rate, so the Eyring rate influence would be expected to predict failure stress values higher than the linear fit of the DOE model for a given strain rate far from the endpoints of 0.2 and 2 [min.sup.-1].

Ductility (0% to 1% Relative Craze Density)

The next property of crazed polycarbonate to be considered is the ductility in the range of 0% to 1% relative craze density. The measure of ductility is the ratio of the final length of the necked region of the fractured polycarbonate samples to the gauge length of 50 mm. The mean and standard deviations of the three repetitions for the ductility parameter are given in Table 13.

Using the data in Table 13, the constant and coefficients for the predictive model for the ductility of crazed polycarbonate were calculated similarly to the calculations above and are shown in Table 14. The P(2-tail) values of each factor for the ductility measurement are also given.

The only terms shown in Table 14 that have P(2-tail) < 0.1 are the constant and the relative craze density. Therefore, the predictive model for the ductility parameter will contain only these two terms, as shown by

[L.sub.f]/[L.sub.g] = 1.92 - 0.76 D (12)

where [L.sub.F] is the final length of the necked region of the sample in mm, [L.sub.g] is the gauge length of 50 mm, and D is the dimensionless coded relative craze density.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that changing the relative craze density from 0% to 1% at a crazing stress of 40 MPa decreases the ductility significantly. They also show that increasing the strain rate from 0.2 to 2 [min.sup.-1] does not produce significant changes in the ductility of polycarbonate. Equation 12 provides a quantitative model to the qualitative results previously reported.

Ductility (0% to 1% Relative Craze Density) Confirmation

The accuracy of the two-factor ductility model will now be evaluated. The only factor found to correlate to the ductility parameter of crazed polycarbonate was relative craze density, as shown in Eq 12. Figure 5 shows the ductility as a function of relative craze density, with the solid line representing the predictive model and the data points depicting the average value of the confirmation tests. The endpoints represent the test conditions used to build the model, with error bars showing one standard deviation of the data collected at each point.

Figure 5 shows that the model developed for this paper accurately predicts the effect of relative craze density on the ductility of crazed polycarbonate. The average values for both the endpoint conditions and the midpoint fit well with the prediction, but the magnitude of the error bars shows that there is a very large amount of scatter in the data with respect to the overall decrease in ductility. Similar to the failure stress model, this scatter is a result of the fact that failure properties depend on the highly variable distribution of flaws. Special care should be taken when using this model to predict the ductility or to set design loads for a crazed part. The large scatter at the midpoint shows that the actual value for the final length of the necked region of crazed polycarbonate could possibly be as small as one-third of the predicted value.

Ductility (1% to 10% Relative Craze Density)

Now, the ductility of crazed polycarbonate will be considered in the range of 1% to 10% relative craze density. The mean and standard deviations of the three repetitions for the ductility parameter are given in Table 15.

Using the experimental data, the constant and coefficients for the predictive model for the ductility parameter were calculated similarly to the calculations above and are shown in Table 16. The P(2-tail) values of each factor for the ductility measurement are also given.

The only terms shown in Table 16 that have P(2-tail) < 0.1 are the constant, the crazing stress, and the strain rate. Therefore, the predictive model for the final length of the necked region normalized to the gauge length will contain only these three terms, as shown by

[L.sub.f]/[L.sub.g] = 1.87 + 0.706 [[sigma].sub.c] - 0.316 [epsilon] (13)

where [L.sub.f] is the final length of the necked region of the crazed samples in mm, [L.sub.g] is the gauge length of 50 mm, [[sigma].sub.c] is the dimensionless coded crazing stress, and [epsilon] is the dimensionless coded strain rate. To graphically illustrate Eq 13, a three-dimensional plot of the ductility parameter with respect to the crazing stress and strain rate is given in Fig. 6, showing the dramatic decrease in ductility with a decrease in crazing stress and an increase in strain rate.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that no strong correlation exists between relative craze density between 1% and 10% and ductility. They also showed a strong correlation between crazing stress and ductility, in that a crazing stress of 40 MPa results in a significantly lower amount of ductility than a crazing stress of 45 MPa. A strong correlation was also qualitatively displayed between strain rate and ductility. The ductility was seen to decrease with increasing strain rate. Equation 13 provides a quantitative model to the qualitative results previously reported.

Ductility (1% to 10% Relative Craze Density) Confirmation

The accuracy of the ductility model will now be evaluated. The ductility parameter was found to correlate to both the crazing stress and the strain rate, as shown in Eq 13. Figure 7 shows a contour plot of the predictive model for ductility as a function of both important factors. The plot is broken up into three regions of 0.8 mm/mm increments, shown by the solid lines. The four endpoint conditions used to build the model are shown on the four corners of Fig. 7, along with four non-endpoint conditions.

As expected, the four endpoints fit well into the predictive model. The four non-endpoint conditions are shown to follow the correct trend, but three of the four measured values are slightly higher than the predicted values. These results show that the model slightly underestimates the final length of the necked region of crazed polycarbonate. Wide scatter exists in the ductility data because of the nature of the fracture dependence on flaw size and distribution, which varies significantly from sample to sample. This wide scatter is believed to be one of the causes of the differences between the measured and predicted values shown in Fig. 7. Also, taking into consideration the Eyring rate effect, the ductility parameter at a given non-endpoint strain rate would be expected to be higher than predicted by a linear fit, since ductility trends are similar to failure stress trends. Once again, an underestimation of the ductility is much more desirable than an overestimation.

SUMMARY

This paper was written to develop quantitative Design of Experiments (DOE) models based on previously published data for the residual mechanical properties of crazed polycarbonate. The yield stress, failure stress, and ductility were modeled as linear functions of crazing stress, relative craze density, and strain rate.

Yield Stress

The presence of crazing did not affect the yield stress, even for relative craze densities as high as 10%. It was found, however, that increasing the strain rate from 0.2 to 2 [min.sup.-1] increased the yield stress by approximately 4.5%. Also, confirmation tests showed that the yield stress does not have a linear relationship with strain rate but is governed by a logarithmic relationship, as described in the Eyring rate theory. The activation volume based on the Eyring theory was calculated and compared to literature values. It was determined that the magnitude of the activation volume of crazed polycarbonate falls between that of polycarbonate molecules in dilute solution and uncrazed polycarbonate.

Failure Stress

The failure stress was found to be much more affected by crazing than the yield stress. The failure stress decreased by more than 10% when going from uncrazed polycarbonate to samples containing 1% relative craze density, but increasing from 1% to 10% did not produce significantly more change. It was also found that the failure stress of a sample crazed at 40 MPa is more than 10% lower than for a sample crazed at 45 MPa.

Ductility

Samples with 1% relative craze density were found to possess on the average less than 50% of the ductility of uncrazed samples. It was also found that increasing the crazing stress from 40 to 45 MPa produced a significant increase in ductility, while increasing the strain rate from 0.2 to 2 [min.sup.-1] decreased the ductility.

Model Validity

The DOE models with strain rate as an important factor were found to produce a slight underestimation at non-endpoint conditions. These discrepancies were attributed to the nonlinear Eyring rate effect. All of the models in which rate was determined to be statistically insignificant were found to accurately predict the response. The data scatter in the ductility measurement was found to be large in comparison to the overall change. This variability was attributed to the ductility's strong dependence on flaw size and distribution, which varies significantly from sample to sample.

ACKNOWLEDGMENTS

The authors would like to acknowledge the Polymer Durability Group at Virginia Tech for technical support and facilities usage. Thanks are also due to Robert McCarty and the AFRL Next Generation Transparency program for providing funding and technical support throughout the scope of this project.

NOMENCLATURE

DOE = Design of Experiments

GPC = Gel Permeation Chromatography

DSC = Differential Scanning Calorimetry

RH = Relative Humidity

CCD = Charge Coupled Device

ASTM = American Society for Testing and Materials

PC = Polycarbonate

(*.) Corresponding author. Email: Stephen.clay@wpafb.af.mil

REFERENCES

(1.) S. B. Clay and R. G. Kander, "Measurement of the Residual Mechanical Properties of Crazed Polycarbonate. I: Qualitative Analysis." Polym Eng. Sci., this issue.

(2.) S. B. Clay, PhD dissertation. Virginia Polytechnic Institute and State University (2000) http://scholar.lib.vt.edu/theses/available/etd-08142000-13410044/.

(3.) S. R. Schmidt and R. G. Launsby, Understanding Industrial Designed Experiments, Air Academy Press, Colorado Springs, Colorado (1994).

(4.) SigmaScan Pro User's Manual, Jandel Corporation, San Rafael, Calif. (1995).

(5.) DOE Kiss: Keeping it Simple Statistically User's Guide, Air Academy Press, Colorado Springs, Colorado (1994).

(6.) I. M. Ward and D. W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, John Wiley & Sons, Inc., New York (1993).

(7.) R. N. Haward and G. Thackray, Proc. Roy. Soc., A302, 453 (1968).

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The Design of Experiments (DOE) approach was used to build quantitative empirical models of the residual mechanical properties of crazed polycarbonate as functions of relative craze density, crazing stress, and strain rate. Crazing did not affect the yielding behavior of polycarbonate, but increasing the strain rate increased the yield stress according to the Eyring theory. The Eyring activation volume for yielding of crazed polycarbonate was measured to fall between reported values for conformational changes of a dilute solution of polycarbonate chains and yielding of uncrazed polycarbonate. Also, with as little as 1% relative craze density, the failure stress was approximately 10% lower and the ductility was, on the average, 50% lower than for uncrazed polycarbonate properties. It was also found that increasing the crazing stress from 40 to 45 MPa increased the failure stress and ductility for a given magnitude of relative craze density.

INTRODUCTION

The United States Air Force has been using polycarbonate as the structural ply In many of its advanced transparency systems since the 1970s. One of the most common causes for replacement of military aircraft transparencies is crazing, which is the formation of fibrillated crack-like defects in the presence of tensile stress. Crazes degrade both the optics and structural integrity of a polymer. To date, the effect of crazing on the mechanical properties of polycarbonate has been difficult to quantify since no standardized technique exists to quantify craze severity.

A recent study (1, 2) has qualitatively assessed the residual mechanical properties of crazed polycarbonate using a new technique to quantify crazing. One of the goals of this paper is to correlate the residual mechanical properties measured in the previous study to crazing stress, relative craze density, and strain rate. The Design of Experiments (DOE) approach will be employed to provide more valuable modeling results than the traditional approach of changing one factor at a time. The designed experiment approach requires more pre-experiment planning than traditional techniques, but increases the chances of obtaining a valid model that assigns the response to the proper input variables or interaction between input variables. The steps for methodically setting up a properly designed experiment have been well documented (3).

MATERIALS

The material used for this work was a commercial grade 6-mm-thick polycarbonate sheet supplied by McMaster Carr. The number average molecular weight as measured by GPC was 19,450 g/mol with a polydispersity index of 2.2. The glass transition temperature determined from DSC was found to be 147[degrees]C. The as-received yield stress and elastic modulus were measured to be 65 MPa and 2.3 GPa, respectively.

EXPERIMENTAL METHODS

Craze Testing

Craze testing was performed under constant load at 24[degrees]C and 85 [+ or -] 5% rh, as described in previous work (1, 2). The range of stress tested for this study was 40 to 45 MPa. While under stress, the relative craze density was measured with a reflective imaging system (1, 2) consisting of a CCD camera, backlight, and frame grabber. The craze growth charts were created using relative craze densities measured on the stored images with SigmaScan Pro[TM] image analysis software (4).

Residual Mechanical Property Measurement

The following procedure outlines the general steps to follow to measure the residual mechanical properties of crazed samples.

1) Conduct constant strain rate tensile tests to failure at the rate specified in the test matrix for a given sample.

2) Measure the yield stress, failure stress, elastic modulus and elongation for each sample.

For this paper, the yield stress was taken to be the maximum magnitude of stress before the constant drawing region. The failure stress was measured as the maximum stress obtained by the polycarbonate after the yield region just prior to fracture. The elastic modulus was calculated by measuring the slope of the initial linear portion of the curve. The measure of ductility was taken to be the final length of the necked portion of the gauge section after fracture normalized to the gauge section of 50 mm.

Design of Experiments

Two separate Design of Experiments analyses were conducted to build mathematical models for the residual mechanical properties of PC. A computer program called DOE KISS (5) was used to analyze the results of the craze test matrix. For a test matrix in which each factor is only tested at two levels, the program codes the minimum value to -1 and the maximum value to +1 while performing the analysis routines. The orthogonal nature of the coded matrix allows efficient statistical calculations to be performed on the collected data.

The first analysis was performed with a constant crazing stress of 40 MPa and included experimental variables of relative craze density and strain rate. The relative craze density varied from 0% to 1%, and the range for strain rate was 0.2 to 2 [min.sup.-1]. The two-factor, two-level values for the DOE model are shown in Table 1.

The second analysis included three factors each tested at two levels. The experimental factors include crazing stress, relative craze density, and strain rate. Table 2 displays the low and high levels for each factor.

For both two-factor and three-factor designed experiments, a full-factorial test matrix testing every possible combination of variables is the best approach. This approach results in four separate test conditions for the two factor matrix and eight separate conditions for the three factor design. Each test condition was repeated three times to provide variability information. No "aliasing" exists in full factorial designs, meaning that all two-way and three-way interactions are evaluated.

Both graphical and statistical analyses were performed on the results of the test matrix. The first analysis technique is to derive a predictive equation for the response using average effects of each factor. The predictive model output of DOE KISS for a two-level design results in the experimental response expressed in terms of a constant and a linear relationship with each factor and each interaction. The general form of the equation is given by

Y = [c.sub.0] + [c.sub.1]A + [c.sub.2]B + [c.sub.3]C + [c.sub.4]AB + [c.sub.5]AC + [c.sub.6]BC + [c.sub.7]ABC (1)

where Y is the predicted value of a given residual mechanical property, the c's are coefficients, A is the coded stress level (-1, + 1), B is the coded relative craze density (-1, +1), C is the coded strain rate (-1, +1), and multiplication of two main factors represents the interaction between the two main factors.

Often, a main factor or an interaction between main factors does not change the measured response significantly. For this reason, DOE KISS calculates the term P(2-tail), which is the probability that a term does not belong in the model. The confidence that a given term belongs in a model is given by

Confidence term belongs in model = (1 - P(2 - tail))*100% (2)

The rule of thumb given in the literature (3) is that if the value of P(2-tail) for a given factor is less than 0.05 then the term should be included in the model. The range of P(2-tail) from 0.05 to 0.10 is an intermediate zone. For this paper, each factor with a P(2-tail) [less than or equal to] 0.10 is included in the predictive model.

RESULTS AND DISCUSSION

Yield Stress (0% to 1% Relative Craze Density)

The first mechanical property of crazed polycarbonate to be considered in the two-factor model was yield stress. Table 3 shows the mean and standard deviation of the three repetitions of the four test runs.

Using the data in Table 3, the constant and coefficients for the predictive model for the yield stress of crazed polycarbonate were calculated and are shown in Table 4. The P(2-tail) values of each factor for the yield stress are also given.

The only terms shown in Table 4 that have P(2-tail) < 0.1 are the constant and the strain rate as shown in bold face. Therefore, the predictive model for the yield stress of crazed polycarbonate will contain only these two terms, as shown by

[[sigma].sub.y] = 67.0 + 1.44 [epsilon] (3)

where [[sigma].sub.y] is the yield stress in MPa and [epsilon] is the dimensionless coded strain rate. Recall that to use this equation, one must use - 1 for the low-level strain rate (0.2 [min.sup.-1]), +1 for the high-level strain rate (2 [min.sup.-1]), and an interpolated value between -1 and + 1 for strain rates between 0.2 and 2 [min.sup.-1].

These results correlate well with the qualitative results shown previously (1, 2). The previously reported qualitative results show that changing the relative craze density from 0% to 1% does not significantly affect the yield stress. They also showed a strong correlation between strain rate and yield stress, in that a strain rate of 2 [min.sup.-1] results in a significantly higher yield stress than a strain rate of 0.2 [min.sup.-1]. Equation 3 provides a quantitative model to the qualitative results previously reported.

Yield Stress (0% to 1% Relative Craze Density) Confirmation

The two-factor DOE model was developed to predict the mechanical properties of polycarbonate with relative craze densities from 0% to 1% and strain rates from 0.2 to 2 [min.sup.-1]. The first confirmation tests were conducted to determine the accuracy of the yield stress model shown in Eq 3. The strain rate was the only factor found to correlate with yield stress. Figure 1 shows the yield stress as a function of strain rate, with the solid line representing the predictive model and the data points depicting the average value of the confirmation tests. The endpoints represent the test conditions used to build the model with error bars showing one standard deviation of the data collected at each point.

As expected, the endpoints fit well to the predictive model since they were used to build the model. The yield stress at the midpoint is shown to be higher than predicted, with the difference between the predicted and average measured values being 0.8 MPa. These results show that the model produces a good prediction of the yield stress of crazed polycarbonate at the endpoints, with a slight underestimation at non-endpoint conditions, which is more desirable than overestimation.

Since the model shows that strain rate is the only important experimental factor in predicting the yield stress of crazed polycarbonate, the Eyring stress-rate model for thermally activated deformation processes will now be considered. The Eyring model (6) was developed to describe the effect of an applied stress on the deformation rate of a polymer as shown by e

e [approximately equal to] [e.sub.0]/2 exp (- [DELTA]H - v[sigma]/RT) (4)

where e is the strain rate, [e.sub.0] is a constant, v is an activation volume, [DELTA]H is the activation energy, R is the gas constant, T is the absolute temperature, and [sigma] is the applied stress. Solving for stress in Eq 4 and applying the model to the yielding behavior of polycarbonate produces (5)

[[sigma].sub.y] = RT/v[\.sub.v] ([DELTA][H.sub.y]/RT + ln 2[e.sub.y]/[e.sub.oy]) (5)

where [[sigma].sub.y] is the yield stress, R is the gas constant, T is the temperature. [v.sub.y] is an activation volume associated with yielding, [DELTA][H.sub.y] is the activation energy associated with yielding, and [e.sub.y] is the strain rate associated with yielding, and [e.sub.oy] is a constant. The Eyring equation for yield stress shown in Eq 5 will now be fit to the experimental data of Fig. 1 and shown in Fig. 2.

As shown in Fig. 2, the equation for the logarithmic data fit for yield stress in the range of 0 to 1% relative craze density is given by

[[sigma].sub.y] = 67.6 + 1.185 ln ([epsilon]) (6)

where [[sigma].sub.y] is the yield stress in MPa and [epsilon] is the strain rate in [min.sup.-1]. Salving for activation volume in Eq 5 in conjunction with Eq 6 gives

RT/v = 1.185 x [10.sup.6] Pa (7)

v = (1.38 x [10.sup.-23] N * m/K)(300K)/1.185 x [10.sup.6] N/[m.sup.2] = 3.49 x [10.sup.-27] [m.sup.3] = 3.49 [nm.sup.3] (8)

Equation 8 implies that chain segments occupying a volume of 3.49 [nm.sup.3] must work cooperatively for yielding to occur in crazed polycarbonate. Table 5 records the calculated value for the volume of an equivalent link for a polycarbonate chain in solution to be 0.48 [nm.sup.3] while the Eyring activation volume for uncrazed polycarbonate has been calculated to be 6.4 [nm.sup.3] by Haward and Thackray (7).

These results indicate that polycarbonate samples with relative craze densities up to 1% require more than seven times as much cooperative volume for yielding than a dilute solution of polycarbonate chains needs to change conformations. Also, a direct comparison of the crazed samples in this study to the uncrazed samples in the literature implies that uncrazed polycarbonate requires almost twice as much cooperative volume for yielding than crazed polycarbonate, but other unknown material properties may also play a role.

Since the yield stress is shown to follow the Eyring stress-rate model, a linear DOE model should not be expected to provide a good fit to the data far from the endpoints. If the presence of crazes affected the yielding behavior of polycarbonate, the Eyring model would need to be modified to account for the effect of crazing, but crazing was not shown to play a significant role in yielding.

Yield Stress (1% to 10% Relative Craze Density)

The yield stress of polycarbonate with relative craze densities in the range from 1% to 10% will now be considered. Test data are presented in Table 6 to show the effects of crazing stress and relative craze density on the yield stress. Each value represents the average from three tests with the standard deviation shown in parentheses.

Using the experimental data, the constant and coefficients for the predictive model for yield stress were calculated by the statistical software program with the results displayed in Table 7. P(2-tail) values for each factor are also given.

Similar to before, the only terms shown in Table 7 that have P(2-tail) < 0.1 are the constant and the strain rate. Therefore, the predictive model for yield stress of crazed polycarbonate will contain only these two terms, as shown by

[[sigma].sub.y] = 66.4 + 1.58 [epsilon] (9)

where [[sigma].sub.y] is the yield stress of the crazed sample in MPa and [epsilon] is the coded strain rate.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that increasing the crazing stress from 40 to 45 MPa and increasing the relative craze density from 1% to 10% produces a very slight decrease, but overall insignificant effect, on the yield stress. The previous results also showed that increasing the strain rate from 0.2 to 2 [min.sup.-1] increased the yield stress on the order of 3 MPa. Equation 9 implies that an increase in the strain rate from 0.2 to 2 [min.sup.-1] will increase the yield stress by 3.16 MPa since the coefficient shows the half effect of the factor. Therefore, Eq 9 provides a quantitative model to the qualitative results.

Yield Stress (1% to 10% Relative Craze Density) Confirmation

The three-factor DOE model was developed to predict the mechanical properties of polycarbonate with relative craze densities from 1% to 10%, crazing stresses from 40 to 45 MPa, and strain rates from 0.2 to 2 [min.sup.-1]. Confirmation tests were conducted to determine the accuracy of the yield stress model shown in Eq 9. The only factor found to correlate to yield stress was strain rate. Figure 3 shows the yield stress as a function of strain rate, with the solid line representing the predictive model, the data points depicting the average value of the confirmation tests, and the dashed line representing the logarithmic fit to the experimental data as done with the Eyring model above. The endpoints represent the test conditions used to build the model, with error bars showing one standard deviation of the data collected at each point. The number of non-endpoint confirmation tests at each strain rate ranged from 1 to 3.

As expected, the endpoints fit well to the predictive model since they were used to build the model. The yield stress at 0.4, 0.8, and 1.6 [min.sup.-1] are shown to be higher than predicted, with the largest difference between the predicted and measured values being about 1 MPa. These results show that the model slightly underestimates the yield stress, which is more desirable than overestimation.

These results are expected as shown in the two-factor DOE yield stress prediction above in which the Eyring rate effect governs the behavior. Since the Eyring data fit for polycarbonate with relative craze densities from 1% to 10% possesses an identical pre-logarithmic coefficient (1.185 MPa) as for the 0% to 1% crazed material, the activation volume in this craze range is also calculated to be 3.49 [nm.sup.3].

Failure Stress (0% to 1% Relative Craze Density)

Since the residual yield stress of crazed polycarbonate has already been quantitatively modeled and no correlation was found for the elastic modulus, the remainder of this paper attempts to model the failure properties of crazed polycarbonate. Crazing is expected to affect the failure properties more than the elastic and yield properties since fracture is highly dependent on flaws. The first failure property to be considered is failure stress, which will now be modeled in the range of 0% to 1% relative craze density. The mean and standard deviation of the three repetitions are given in Table 8.

Using the data in Table 8, the constant and coefficients for the predictive model for the failure stress of crazed polycarbonate were calculated similarly to the calculations above and are shown in Table 9. The P[2-tail) values of each factor for the failure stress are also given.

The only terms shown in Table 9 that have $2-tail) <0.1 are the constant and the relative craze density. Therefore, the predictive model for the failure stress of crazed polycarbonate will contain only these two terms, as shown by

[[sigma].sub.F] = 53.1 - 3.24 D (10)

where [[sigma].sub.F] is the failure stress in MPa and D is the dimensionless coded relative craze density.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that changing the relative craze density from 0% to 1% at a crazing stress of 40 MPa decreases the failure stress significantly. They also show that increasing the strain rate from 0.2 to 2 [min.sup.-1] does not produce significant changes in the failure stress of polycarbonate. Equation 10 provides a quantitative model to the qualitative results previously reported.

Failure Stress (0% to 1% Relative Craze Density) Confirmation

Next, the accuracy of the two-factor DOE model for failure stress will be evaluated. The only factor found to correlate to the failure stress of crazed polycarbonate was relative craze density, as shown in Eq 10. Figure 4 shows the failure stress as a function of relative craze density, with the solid line representing the predictive model and the data points depicting the average value of the confirmation tests. The endpoints represent the test conditions used to build the model, with error bars showing one standard deviation of the data collected at each point.

Figure 4 shows that the model developed for this paper accurately predicts the effect of relative craze density on the failure stress of crazed polycarbonate. The average values for both the endpoint conditions and the midpoint fit well with the prediction, but the magnitude of the error bars shows that there is significant scatter in the data with respect to the overall decrease in failure stress. This scatter is a result of the fact that failure properties depend on the distribution of flaws of various sizes and amounts, which is increased with increased crazing, but is highly variable from sample to sample. It should be noted that the Eyring rate effect is not seen in this model, since strain rate was not found to be a significant factor and crazing dominates the failure behavior. For this reason, a linear model is appropriate.

Failure Stress (1% to 10% Relative Craze Density)

Table 10 shows the effect of strain rate and relative craze density on the failure stress of crazed polycarbonate in the range from 1% to 10% relative craze density. Each value represents the mean from three tests, with the standard deviation in parentheses.

Using the experimental data, the constant and coefficients for the predictive model for failure stress were calculated similarly to the calculations above and are shown in Table 11. The P(2-tail) values of each factor for the failure stress are also given.

The only terms shown in Table 11 that have P(2-tail) < 0.1 are the constant, the crazing stress, and the interaction between the crazing stress and the strain rate. Therefore, the predictive model for the failure stress of crazed polycarbonate will contain only these three terms, as shown by

[[sigma].sub.f] = 52.8 + 3.l9 [[sigma].sub.c] - 1.57 [[sigma].sub.c]*[epsilon] (11)

where [[sigma].sub.f] is the failure stress of the crazed sample in MPa, [[sigma].sub.c] is the dimensionless coded crazing stress, and [[sigma].sub.c]*[epsilon] is the dimensionless interaction between the coded crazing stress and the coded strain rate.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that no strong correlation exists between relative craze density or strain rate and failure stress. They also showed a strong correlation between crazing stress and failure stress, in that a crazing stress of 40 MPa results in a failure stress significantly lower than a crazing stress of 45 MPa. Another observation from previous work that could have easily been overlooked is that when the crazing stress increases from 40 to 45 MPa, the effect that the strain rate has on the failure stress reverses. At the lower crazing stress, an increase in strain rate increases the failure stress, while at the higher crazing stress, an increase in the strain rate decreases the failure stress. Therefore, an interaction exists between crazing stress and strain rate. Equation 11 provides a quantitative model to the qualitative results previously reported.

Failure Stress (1% to 10% Relative Craze Density) Confirmation

Next, the three-factor DOE model for failure stress prediction will be evaluated. Both crazing stress and the interaction between crazing stress and strain rate were found to correlate to the failure stress, as shown in Eq 11. Table 12 shows the predicted values and the average of the measured values of failure stress at various crazing stresses and strain rates. The conditions of (40 MPa, 0.2 [min.sup.-1]), (40 MPa, 2 [min.sup.-1]), (45 MPa, 0.2 [min.sup.-1]), and (45 MPa, 2 [min.sup.-1]), represent the endpoints used to build the model.

The average difference between the predicted and measured values of failure stress in Table 12 was found to be approximately 1.5 MPa. Also, the measured values for failure stress at all three non-endpoint conditions are slightly higher than the predicted values, showing an underestimation of failure stress similar to what was seen in the yield stress model. Once again, underestimation of the failure stress is more desirable than overestimation.

The Eyring model has previously been applied to the fracture of polymers (6), similar to the yield stress as shown above. In this case, the failure stress has been shown to be a function of both crazing stress and strain rate, so the Eyring rate influence would be expected to predict failure stress values higher than the linear fit of the DOE model for a given strain rate far from the endpoints of 0.2 and 2 [min.sup.-1].

Ductility (0% to 1% Relative Craze Density)

The next property of crazed polycarbonate to be considered is the ductility in the range of 0% to 1% relative craze density. The measure of ductility is the ratio of the final length of the necked region of the fractured polycarbonate samples to the gauge length of 50 mm. The mean and standard deviations of the three repetitions for the ductility parameter are given in Table 13.

Using the data in Table 13, the constant and coefficients for the predictive model for the ductility of crazed polycarbonate were calculated similarly to the calculations above and are shown in Table 14. The P(2-tail) values of each factor for the ductility measurement are also given.

The only terms shown in Table 14 that have P(2-tail) < 0.1 are the constant and the relative craze density. Therefore, the predictive model for the ductility parameter will contain only these two terms, as shown by

[L.sub.f]/[L.sub.g] = 1.92 - 0.76 D (12)

where [L.sub.F] is the final length of the necked region of the sample in mm, [L.sub.g] is the gauge length of 50 mm, and D is the dimensionless coded relative craze density.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that changing the relative craze density from 0% to 1% at a crazing stress of 40 MPa decreases the ductility significantly. They also show that increasing the strain rate from 0.2 to 2 [min.sup.-1] does not produce significant changes in the ductility of polycarbonate. Equation 12 provides a quantitative model to the qualitative results previously reported.

Ductility (0% to 1% Relative Craze Density) Confirmation

The accuracy of the two-factor ductility model will now be evaluated. The only factor found to correlate to the ductility parameter of crazed polycarbonate was relative craze density, as shown in Eq 12. Figure 5 shows the ductility as a function of relative craze density, with the solid line representing the predictive model and the data points depicting the average value of the confirmation tests. The endpoints represent the test conditions used to build the model, with error bars showing one standard deviation of the data collected at each point.

Figure 5 shows that the model developed for this paper accurately predicts the effect of relative craze density on the ductility of crazed polycarbonate. The average values for both the endpoint conditions and the midpoint fit well with the prediction, but the magnitude of the error bars shows that there is a very large amount of scatter in the data with respect to the overall decrease in ductility. Similar to the failure stress model, this scatter is a result of the fact that failure properties depend on the highly variable distribution of flaws. Special care should be taken when using this model to predict the ductility or to set design loads for a crazed part. The large scatter at the midpoint shows that the actual value for the final length of the necked region of crazed polycarbonate could possibly be as small as one-third of the predicted value.

Ductility (1% to 10% Relative Craze Density)

Now, the ductility of crazed polycarbonate will be considered in the range of 1% to 10% relative craze density. The mean and standard deviations of the three repetitions for the ductility parameter are given in Table 15.

Using the experimental data, the constant and coefficients for the predictive model for the ductility parameter were calculated similarly to the calculations above and are shown in Table 16. The P(2-tail) values of each factor for the ductility measurement are also given.

The only terms shown in Table 16 that have P(2-tail) < 0.1 are the constant, the crazing stress, and the strain rate. Therefore, the predictive model for the final length of the necked region normalized to the gauge length will contain only these three terms, as shown by

[L.sub.f]/[L.sub.g] = 1.87 + 0.706 [[sigma].sub.c] - 0.316 [epsilon] (13)

where [L.sub.f] is the final length of the necked region of the crazed samples in mm, [L.sub.g] is the gauge length of 50 mm, [[sigma].sub.c] is the dimensionless coded crazing stress, and [epsilon] is the dimensionless coded strain rate. To graphically illustrate Eq 13, a three-dimensional plot of the ductility parameter with respect to the crazing stress and strain rate is given in Fig. 6, showing the dramatic decrease in ductility with a decrease in crazing stress and an increase in strain rate.

These results correlate well with the qualitative results shown in previous work (1, 2). The previously reported results show that no strong correlation exists between relative craze density between 1% and 10% and ductility. They also showed a strong correlation between crazing stress and ductility, in that a crazing stress of 40 MPa results in a significantly lower amount of ductility than a crazing stress of 45 MPa. A strong correlation was also qualitatively displayed between strain rate and ductility. The ductility was seen to decrease with increasing strain rate. Equation 13 provides a quantitative model to the qualitative results previously reported.

Ductility (1% to 10% Relative Craze Density) Confirmation

The accuracy of the ductility model will now be evaluated. The ductility parameter was found to correlate to both the crazing stress and the strain rate, as shown in Eq 13. Figure 7 shows a contour plot of the predictive model for ductility as a function of both important factors. The plot is broken up into three regions of 0.8 mm/mm increments, shown by the solid lines. The four endpoint conditions used to build the model are shown on the four corners of Fig. 7, along with four non-endpoint conditions.

As expected, the four endpoints fit well into the predictive model. The four non-endpoint conditions are shown to follow the correct trend, but three of the four measured values are slightly higher than the predicted values. These results show that the model slightly underestimates the final length of the necked region of crazed polycarbonate. Wide scatter exists in the ductility data because of the nature of the fracture dependence on flaw size and distribution, which varies significantly from sample to sample. This wide scatter is believed to be one of the causes of the differences between the measured and predicted values shown in Fig. 7. Also, taking into consideration the Eyring rate effect, the ductility parameter at a given non-endpoint strain rate would be expected to be higher than predicted by a linear fit, since ductility trends are similar to failure stress trends. Once again, an underestimation of the ductility is much more desirable than an overestimation.

SUMMARY

This paper was written to develop quantitative Design of Experiments (DOE) models based on previously published data for the residual mechanical properties of crazed polycarbonate. The yield stress, failure stress, and ductility were modeled as linear functions of crazing stress, relative craze density, and strain rate.

Yield Stress

The presence of crazing did not affect the yield stress, even for relative craze densities as high as 10%. It was found, however, that increasing the strain rate from 0.2 to 2 [min.sup.-1] increased the yield stress by approximately 4.5%. Also, confirmation tests showed that the yield stress does not have a linear relationship with strain rate but is governed by a logarithmic relationship, as described in the Eyring rate theory. The activation volume based on the Eyring theory was calculated and compared to literature values. It was determined that the magnitude of the activation volume of crazed polycarbonate falls between that of polycarbonate molecules in dilute solution and uncrazed polycarbonate.

Failure Stress

The failure stress was found to be much more affected by crazing than the yield stress. The failure stress decreased by more than 10% when going from uncrazed polycarbonate to samples containing 1% relative craze density, but increasing from 1% to 10% did not produce significantly more change. It was also found that the failure stress of a sample crazed at 40 MPa is more than 10% lower than for a sample crazed at 45 MPa.

Ductility

Samples with 1% relative craze density were found to possess on the average less than 50% of the ductility of uncrazed samples. It was also found that increasing the crazing stress from 40 to 45 MPa produced a significant increase in ductility, while increasing the strain rate from 0.2 to 2 [min.sup.-1] decreased the ductility.

Model Validity

The DOE models with strain rate as an important factor were found to produce a slight underestimation at non-endpoint conditions. These discrepancies were attributed to the nonlinear Eyring rate effect. All of the models in which rate was determined to be statistically insignificant were found to accurately predict the response. The data scatter in the ductility measurement was found to be large in comparison to the overall change. This variability was attributed to the ductility's strong dependence on flaw size and distribution, which varies significantly from sample to sample.

ACKNOWLEDGMENTS

The authors would like to acknowledge the Polymer Durability Group at Virginia Tech for technical support and facilities usage. Thanks are also due to Robert McCarty and the AFRL Next Generation Transparency program for providing funding and technical support throughout the scope of this project.

NOMENCLATURE

DOE = Design of Experiments

GPC = Gel Permeation Chromatography

DSC = Differential Scanning Calorimetry

RH = Relative Humidity

CCD = Charge Coupled Device

ASTM = American Society for Testing and Materials

PC = Polycarbonate

(*.) Corresponding author. Email: Stephen.clay@wpafb.af.mil

REFERENCES

(1.) S. B. Clay and R. G. Kander, "Measurement of the Residual Mechanical Properties of Crazed Polycarbonate. I: Qualitative Analysis." Polym Eng. Sci., this issue.

(2.) S. B. Clay, PhD dissertation. Virginia Polytechnic Institute and State University (2000) http://scholar.lib.vt.edu/theses/available/etd-08142000-13410044/.

(3.) S. R. Schmidt and R. G. Launsby, Understanding Industrial Designed Experiments, Air Academy Press, Colorado Springs, Colorado (1994).

(4.) SigmaScan Pro User's Manual, Jandel Corporation, San Rafael, Calif. (1995).

(5.) DOE Kiss: Keeping it Simple Statistically User's Guide, Air Academy Press, Colorado Springs, Colorado (1994).

(6.) I. M. Ward and D. W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, John Wiley & Sons, Inc., New York (1993).

(7.) R. N. Haward and G. Thackray, Proc. Roy. Soc., A302, 453 (1968).

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Table 1. Coded Values for DOE Models. Low (-1) High (+1) Relative craze density (%) 0 1 Strain rate ([min.sup.-1]) 0.2 2 Table 2. Coded Values for DOE Models. Low (-1) High (+1) Crazing stress (MPa) 40 45 Relative craze density (%) 1 10 Strain rate ([min.sup.-1]) 0.2 2 Table 3. Mean and Standard Deviations of Yield Stress for 0% to 1% Relative Craze Density (Crazing Stress = 40 MPa). Strain Uncrazed Yield 1% Crazed Rate Stress Yield Stress ([min.sup.-1]) (MPa) (MPa) 0.2 65.6 (0.252) 65.5 (0.321) 2 68.2 (0.513) 68.6 (0.551) Table 4. Original Model for Yield Stress (0% to 1% Relative Craze Density) With All Factors Included in Regression. Coefficient Factor for Model (MPa) P(2 tail) Constant 67.0 0.0000 (A) Relative craze density 0.0917 0.4795 (B) Strain Rate 1.44 0.0000 AB 0.142 0.285 Table 5. Activation Volume of Polycarbonate. Volume of equivalent segment of chain in dilute solution ([nm.sub.3]) 0.48 Activation volume of crazed polycarbonate ([nm.sup.3]) 3.49 Activation volume of uncrazed polycarbonate ([nm.sup.3]) 6.4 Table 6. Mean and Standard Deviation of Yield Stress for 1% to 10% Relative Craze Density and 40 to 45 Mpa Crazing Stress. Strain Rate Yield Stress at Yield Stress at Yield Stress at ([min.sup.-1]) 40 MPa, 1% (MPa) 40 MPa, 10% (MPa) 45 MPa, 1% (MPa) 0.2 65.5 (0.321) 64.3 (2.44) 64.8 (0.573) 2 68.6 (0.551) 68.5 (0.561) 67.8 (0.943) Strain Rate Yield Stress at ([min.sup.-1]) 45 MPa, 10% (MPa) 0.2 64.6 (0.735) 2 67.0 (0.385) Table 7. Original Model for Yield Stress (1% to 10% Relative Craze Density) With All Factors Included in Regression. Coefficient Factor for Model (MPa) P(2 tail) Constant 66.4 0.000 (A) Crazing stress -0.350 0.119 (B) Relative craze density -0.295 0.184 (C) Strain rate 1.58 0.000 AB 0.0312 0.885 AC -0.244 0.268 BC 0.0513 0.812 ABC -0.191 0.381 Table 8. Mean and Standard Deviations of Failure Stress for 0% to 1% Relative Craze Density (Crazing Stress = 40 MPa). Strain Uncrazed 1% Crazed Rate Failure Stress Failure Stress ([min.sup.-1]) (MPa) (MPa) 0.2 54.6 (5.12) 49.4 (2.81) 2 58.1 (4.01) 50.4 (0.70) Table 9. Original Model for Failure Stress (0% to 1% Relative Craze Density) With All Factors Included in Regression. Coefficient Factor for Model (MPa) P(2 tail) Constant 53.1 0.0000 (A) Relative craze density -3.24 0.0135 (B) Strain rate 1.14 0.2987 AB -0.625 0.5598 Table 10. Mean and Standard Deviations of Failure Stress for 1% to 10% Relative Craze Density and 40 to 45 MPa Crazing Stress. Strain Rate Failure Stress at Failure Stress at Failure Stress at ([min.sup.-1]) 40 MPa, 1% (MPa) 40 MPa, 10% (MPa) 45 MPa, 1% (MPa) 0.2 49.4 (2.82) 47.4 (1.92) 57.5 (5.75) 2 50.4 (0.68) 51.3 (2.94) 56.0 (2.11) Strain Rate Failure Stress at ([min.sup.-1]) 45 MPa, 10% (MPa) 0.2 58.3 (4.77) 2 52.2 (1.93) Table 11. Original Model for Failure Stress (1% to 10% Relative Craze Density) With All Factors Included in Regression. Coefficient Factor for Model (MPa) P(2 tail) Constant 52.8 0.0000 (A) Crazing stress (MPa) 3.19 0.0002 (B) Relative craze density (%) -0.503 0.460 (C) Strain rate ([min.sup.-1]) -0.335 0.622 AB -0.259 0.702 AC -1.57 0.0308 BC -0.210 0.756 ABC -0.935 0.179 Table 12 Confirmation of Failure Stress Model (1% to 10% Relative Craze Density). Crazing Strain Predicted Failure Stress (MPa) Rate ([min.sup.-1]) Stress (MPa) 40 0.2 48.0 40 2 51.2 41 1.6 51.4 42.5 0.8 52.8 44.5 0.4 56.3 45 0.2 57.6 45 2 54.4 Crazing Measured Average [DELTA][[sigma].sub.F] Stress (MPa) Failure Stress (MPa) Meas-Pred (MPa) 40 46.7 -1.3 40 51.0 -0.2 41 52.3 +0.9 42.5 55.1 +2.3 44.5 60.0 +3.7 45 59.2 +1.6 45 54.1 -0.3 Table 13 Mean and Standard Deviations of Ductility Parameter for 0% to 1% Relative Craze Density (Crazing Stress = 40 MPa). Strain Uncrazed 1% Crazed Rate Ductility Parameter Ductility Parameter ([min.sup.-1]) (mm/mm) (mm/mm) 0.2 2.56 (0.54) 1.58 (0.78) 2 2.78 (0.44) 0.70 (0.06) Table 14 Original Model for Ductility Parameter (0% to 1% Relative Craze Density) With All Facotrs Included in Regression. Coefficient for Factor Model (mm/mm) P(2 tail) Constant 1.92 0.0000 (A) Relative craze density -0.76 0.0010 (B) Strain rate -0.16 0.3073 AB -0.28 0.1067 Table 15 Mean and Standard Deviations of Ductility Parameter for 1% to 10% Relative Craze Density and 40 to 45 MPa Crazing Stress. Strain Rate Ductility Parameter at Ductility Parameter at ([min.sup.-1]) 40 MPa, 1% (mm/mm) 40 MPa, 10% (mm/mm) 0.2 1.58 (0.78) 1.26 (0.58) 2 0.70 (0.05) 1.12 (1.08) Strain Rate Ductility Parameter at Ductility Parameter at ([min.sup.-1]) 45 MPa, 1% (mm/mm) 45 MPa, 10% (mm/mm) 0.2 2.92 (0.56) 3.00 (0.42) 2 2.60 (0.20) 1.80 (0.72) Table 16 Original Model for Ductility Parameter (1% to 10% Relative Craze Density) With All Factors Included in Regression. Coefficient for Factor Model (mm/mm) P(2 tail) Constant 1.87 0.0000 (A) Crazing stress 0.706 0.0000 (B) Relative craze density -0.080 0.5419 (C) Strain rate -0.316 0.0253 AB -0.0984 0.455 AC -0.0616 0.638 BC -0.0167 0.898 ABC -0.202 0.136

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Author: | Clay, Stephen B.; Kander, Ronald G. |
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Publication: | Polymer Engineering and Science |

Geographic Code: | 1USA |

Date: | Jan 1, 2002 |

Words: | 6866 |

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