# Effect of mean strain on the cyclic deformation and stress relaxation in polypropylene.

INTRODUCTION

Cyclic deformation and stress relaxation after cyclic preloadings in polypropylene have been studied in a push-pull tension-compression mode by using a closed-loop, electrohydraulic, servocontrolled testing machine (1, 2). The data were compared with theoretical results analyzed by using a linear viscoelastic model (3, 4) and an overstress model (5, 6). An equilibrium stress and a viscosity function were treated in the overstress theory. The calculated results based on the overstress theory agree well with experiment in both the stress-strain and the stress-relaxation curves. It was concluded that the overstress theory explained the nonlinear viscoplastic behavior of polypropylene. On the other hand, it can be seen that the linear viscoelastic model explained the viscoelastic characteristics of polypropylene despite the solution of the constitutive equation constructed by the simple model, which consisted of a Maxwell unit and Hookean spring in parallel (3, 4).

Kitagawa et al. (7) reported that from the torsional tests of polypropylene at low strain rates, at room temperature, the stress-strain behavior was sensitive to strain rates and a slight work softening was seen at all strain rates. They also compared the data with theoretical results calculated from Krempl's overstress model (8) and explained well the viscoelastic-plastic characteristics of polypropylene.

It has been reported that stress relaxation after simple tension and cyclic preloading indicates complex features and reveals differences in the deformation of molecular chains in polypropylene, as subjected to different cyclic preloadings (9); the stress-strain curves and stress-relaxation curves under three mean strains were fairly different. Moreover, it was concluded from the results of the stress-strain curves and stress-relaxation curves, and scanning electron microscopy (SEM) fractographs, that significant microstructural changes occurred under cyclic preloading.

The aim of this study is to clarify the effect of mean strain on mechanisms of cyclic deformation in polypropylene. The stress-strain tests and stress-relaxation tests were performed under various sets of strain rate, strain amplitude, and mean strain by using an electrohydraulic, servocontrolled testing machine at room temperature.

EXPERIMENTAL

Materials

The polypropylene used was a commercial stock described as Grade J105 and manufactured by Ube Industries, Ltd. Crystal modification was the mono-clinic [Alpha]-form and the apparent density was 907 kg/[m.sup.3] at 18 [degrees] C. The degree of crystallinity calculated from the density was 56% (10). The melt flow index was 0.005 kg/10 min and the molecular weight was 285,000, estimated from the melt flow index (11). Details on characterization of the as-received material are given elsewhere (12).

Sample Preparation

The as-received material was machined to rod specimen with a lathe. The surface of the straight part was polished on emery papers as there were visible flaws and scratches. The gauge length of the specimen was 150 mm and the diameter of the straight part was 10 mm (2). The microstructure of the cross section of the straight part was homogeneous: This fact was confirmed by a direct observation of the spherulite deformation by polarizing microscopy (12). The mean spherulite diameter of the as-received PP was 180 [[micro]meter] (13, 14). A detailed description of the dimensions of the testing specimen has been given (2).

Testing System

Cyclic stress-strain curves and stress-relaxation curves after cyclic preloadings were obtained by use of a closed-loop, electrohydraulic, servocontrolled testing machine. Specimens were tested in a push-pull tension-compression mode under strain control at room temperature (18 to 23 [degrees] C). Load and displacement data after each specimen were fed to a computer (NEC, Co., PC 9801) and stored on floppy disks after data processing. The details of the testing procedure have been reported elsewhere (2).

Conditions of Cyclic Loading

Table 1 lists the conditions of cyclic preloading. The strain rate [Mathematical Expression Omitted], number of cycles N, strain width [Delta][Epsilon], and mean strain [[Epsilon].sub.m] are taken as parameters. The stress-strain curves were obtained at three mean strains: [[Epsilon].sub.m] = 1.0%, 2.0%, and 4.0%. The tests were performed at three strain widths of [Delta][Epsilon] = 3.0%, 5.0%, and 7.0%.

Stress-Relaxation Tests

The stress-relaxation tests were performed after three numbers of cycles (N = 35, 40, and 50). The tests were made after different predetermined strain rates ([Mathematical Expression Omitted], 100, and 1000 [micro][s.sup.-1] where [micro] = microstrain units) with a relaxation period of 1200 s duration.

[TABULAR DATA FOR TABLE 1 OMITTED]

RESULTS AND DISCUSSION

Stress-Strain Behavior

Figures 1 through 3 show the stress-strain curves for a mean strain [[Epsilon].sub.m] = 2% at different strain widths [Delta][Epsilon] of [Delta][Epsilon] = 3%, 5%, and 7%, respectively. In the cyclic deformation tests performed here, stress-relaxation tests were made at the numbers of cycles N = 35, 45, and 50. The hysteresis loops at N = 35, 45, and 50 in Fig. 1 are irregular in shape, and the tensile portion decreases in size as the number of cycles increases. On the other hand, the compressive portions at N = 35, 45, and 50 are identical in size. It is reported that the stress-strain curves at strain amplitudes from [+ or -] 1.5% to [+ or -] 5% indicated a propeller-like shape; these loops changed into the steady-state response as the number of cycles was increased up to N = 30 to 50 (2). In the case of [Delta][Epsilon] = 5% [ILLUSTRATION FOR FIGURE 2 OMITTED] and [Delta][Epsilon] = 7% [ILLUSTRATION FOR FIGURE 3 OMITTED], the feature of the tensile portions is quite similar to that of [Delta][Epsilon] = 3% [ILLUSTRATION FOR FIGURE 1 OMITTED]; however, the feature of the compressive portions is very different. The marked increase in stress value in compressive side is observed. This fact is because of different strain widths; i.e., the hysteresis loop at [Delta][Epsilon] = 3% is within the tensile region. but the hysteresis loop at [Delta][Epsilon] = 5% and [Delta][Epsilon] = 7% lie in both the tensile region and the compressive region (or negative strain region).

Figure 4 shows the stress-strain curves at a strain rate of 1000 [micro]/s for mean strains [[Epsilon].sub.m] = +2.5%, 0%, -2.5% after a number of cycles N = 10. An unusual propeller-like shape of the hysteresis loop labeled b In Fig. 4 is observed in the first ten cycles, in marked contrast to the behavior of metals (15). These loops change into the steady state response as the number of cycles is increased up to N = 30 to 50. A slight work-softening can be seen in the initial number of cycles for the mean strain of [[Epsilon].sub.m] = 0%. The stress-strain curves under three mean strains are fairly different from each other. For the mean strain of [[Epsilon].sub.m] = 0% the magnitude of the peak tensile stress is smaller by [approximately] 10 MPa than that of the peak compressive stress, and the mean stress, defined as the mean of the peak tensile stress and the compressive one, is shifted to the compressive side. On the other hand, for [[Epsilon].sub.m] = +2.5% and [[Epsilon].sub.m] = -2.5% the mean stress is shifted in the preloading direction. The stress-strain curves for the mean strains of [[Epsilon].sub.m] = +2.5%, 0%, and -2.5% are very different from those of different strain widths of [Delta][Epsilon] = 3%, 5%, and 7% at the mean strain of [[Epsilon].sub.m] = 2%.

Effect of Mean Strain on Stress Level

The experimental data for the stress-strain curve at three mean strains of [[Epsilon].sub.m] = 1%, 2%, and 4% at a strain width [Delta][Epsilon] = 3% after a number of cycles of N = 35 are depicted in Fig. 5. The reason for the choice of the number of cycles of N = 35 is that the stress-strain curve tended to reach a steady state as the number of cycles increased and that the steady state behavior was seen at a number of cycles N = 30 to 50 (2, 9). The stress value of the tensile portion for three mean strains remains constant but the value of the compressive portion decreases as the mean strain increases.

Figures 6 and 7 show the stress-strain curve at mean strains of [[Epsilon].sub.m] = 1%, 2%, and 4% after a number of cycles of N = 35 at strain widths of [Delta][Epsilon] = 5% and [Delta][Epsilon] = 7%, respectively. The overall behaviors of the hysteresis loop shapes are fairly different from each other; i.e., the tensile portion behavior is the same as that of the strain width of [Delta][Epsilon] = 3% but the compressive portion behavior is different in the conditions of strain width.

Figures 8 through 10 show the stress vs. mean strain at three strain widths of 3%, 5%, and 7%, respectively, at different numbers of cycles. At a strain width [Delta][Epsilon] of [Delta][Epsilon] = 3%, the stress level changes little with mean strain. For strain widths of [Delta][Epsilon] = 5% [ILLUSTRATION FOR FIGURE 9 OMITTED] and [Delta][Epsilon] = 7% [ILLUSTRATION FOR FIGURE 10 OMITTED], however, the minimum stress levels at three numbers of cycles increase with increasing mean strain, in contrast to the behavior at maximum stress level.

The data of the stress-strain curves for uniaxial simple tension at different strain rates [Mathematical Expression Omitted] of [Mathematical Expression Omitted], 100, 1000, and 10000 [micro]/s have been reported for the spherulite nature of polypropylene (9). In the elastic range the value of stress depends strongly on strain rate; i.e., the magnitude of stress increases with increasing strain rate above the yield point of [approximately] [Epsilon] = 0.7%. The strain range below [Epsilon] = 0.7% indicates an elastic deformation, which is independent of strain rate. Accordingly, for mean strain rates of [[Epsilon].sub.m] = 3%, 5%, and 7%, both the maximum stress and mean strain are beyond the yield point of [Epsilon] = 0.7%, i.e., the deformation behavior in the tensile side of the cyclic mode is nonlinear.

Stress Width and Mean Strains

Figures 11 through 13 show the stress width (peak-to-peak stress) vs. mean strain at three strain widths of 3%, 5%, and 7%, respectively, at different numbers of cycles. For three strain widths [ILLUSTRATION FOR FIGURES 11 THROUGH 13 OMITTED], the stress width decreases with increasing mean strain. In addition, the stress width is influenced significantly by mean stress as the strain width is large. The mean stress is defined as the mean of the peak tensile stress and the compressive one. At a strain width of 7% [ILLUSTRATION FOR FIGURE 13 OMITTED], the magnitude of the stress width for three numbers of cycles (N = 35, 45, and 50) is the same at a mean strain of 4%.

Stress-Relaxation Behavior

The experimental data for stress relaxation curves at a mean strain of 2%, at a strain width of 5%, are depicted in Fig. 14. The drop of stress decreases with an increasing number of cycles. In the case of the width of 5%, it was reported (9) that the stress-relaxation curves for [[Epsilon].sub.m] = 0% and +2.5% show the same behavior, and these are different from the curve for [[Epsilon].sub.m] = -2.5%; the behavior of the stress-strain and stress-relaxation results were related to microstructural changes in the molecular chains. In addition, the relaxation curve for [[Epsilon].sub.m] = -2.5% did not reach the equilibrium stress at 1200 s. The same behavior is observed for N = 50 (case a in [ILLUSTRATION FOR FIGURE 14 OMITTED]). The discrepancy in the results of the relaxation tests is due to the effect of the difference in strain rates tested here.

Figures 15 and 16 show the stress-relaxation curves at a mean strain of 2% at a strain rate of 1000 [micro]/s and 10 [micro]/s, respectively, after a number of cycles of N = 50. The stress drop increases with increasing strain width for both the strain rates of 1000 [micro]/s and 10 [micro]/s. The magnitude of the stress drop of strain rate of 1000 [micro]/s is larger than that of 10 [micro]/s. In Fig. 16, a little magnitude of stress drop is observed in contrast to that of Figs. 15 and 16. Hence it is concluded that the magnitude of stress drop after cyclic preloadings is influenced significantly by strain width as strain rate increases. It is seen that the drop of stress decreases as an increase in mean strain [ILLUSTRATION FOR FIGURES 14 THROUGH 16 OMITTED]. This is due to the effect of changes of stress level, as shown in stress-strain behavior.

The linear viscoelastic model has explained the vis-coelastic characteristics of polypropylene despite the solution of the constitutive equation constructed by the simple three-element model (3, 4). On the other hand, the calculated stress-strain curves and the calculated stress-relaxation curves were determined from a constitutive equation based on an overstress theory in which an equilibrium stress and a viscosity function were treated (5, 6). The calculated results agreed well with experiment. It was concluded that the overstress theory explained the nonlinear viscoelastic-plastic behavior of polypropylene. The results calculated by these models were based on the experimental data of a zero mean strain ([[Epsilon].sub.m] = 0). In the experimental facts of stress-relaxation behavior at three mean strains ([[Epsilon].sub.m] = 0, -2.5, and +2.5%) and at a strain width of 5% (or strain amplitude of [Delta][Epsilon]/2 = [+ or -] 2.5%), the significant microstructural changes occurred for the polypropylene samples after cyclic preloading of N = 50 (9). Therefore, the microstructural changes must be considered in order to analyze the stress-relaxation behavior of the effects of different mean strains. However, the theoretical treatments of a nonlinear model of viscoelasticity seem to be difficult; in particular, the analysis of the effect of mean strain on the structural changes is complicated.

CONCLUSIONS

The effect of mean strain on the cyclic deformation and stress-relaxation behavior of polypropylene was investigated by use of an electrohydraulic, servocon-trolled testing machine. A set of cyclic loading and stress relaxation was repeated three times; i.e., the stress-relaxation tests were made at the numbers of cycles N of N = 35, 45, and 50. A marked increase in stress value in the compressive side was observed. This is because of different strain widths; i.e., the hysteresis loop at [Delta][Epsilon] = 3% is within the tensile region, but the hysteresis loops at [Delta][Epsilon] = 5% and [Delta][Epsilon] = 7% lie in both the tensile region and the compressive region. The overall behaviors of the hysteresis loop shapes are fairly different; i.e., the tensile portion behavior is the same as that of the strain width of [Delta][Epsilon] = 3%, but the compressive portion behavior is different in the conditions of strain width. For strain widths of [Delta][Epsilon] = 5% and 7%, the minimum stress levels at three numbers of cycles increase with increasing mean strain, in contrast to the behavior of maximum stress level. At a strain width of 7%, the magnitude of the stress width for three numbers of cycles (N-35, 45, and 50) is the same at a mean strain of 4%. It is concluded from the results of stress-relaxation tests that the magnitude of stress drop after cyclic preloadings is influenced significantly by strain width as strain rate increases.

ACKNOWLEDGMENTS

The author is indebted to Professor K. Kaneko for helpful discussions throughout these studies and also to Dr. H. Yumoto for encouragement and suggestions.

REFERENCES

1. T. Ariyama, M. Takenaga, K. Yamagata, A. Kasai, and K. Kaneko, Rept. Prog. Polym. Phys. Jpn., 33, 311 (1990).

2. T. Ariyama, Polym. Eng. Sci., 33, 18 (1993).

3. T. Ariyama, M. Sakuma, M. Takenaga, K. Yamagata, A. Kasai, and K. Kaneko, Rept. Prog. Polym. Phys. Jpn., 34, 209 (1991).

4. T. Ariyama, Polym. Eng. Sci., 33, 1494 (1993).

5. T. Ariyama, M. Sakuma, K. Sakamoto, and K. Kaneko, Rept. Prog. Polym. Phys. Jpn., 35, 319 (1992).

6. T. Ariyama, M. Sakuma, and K. Kaneko, Trans. JSME, Part A, 58, 113 (1992).

7. M. Kitagawa, T. Mori, and T. Matsutani, J. Polym. Sci: Part B: Polym. Phys., 27, 85 (1989).

8. E. Krempl, Trans. ASME, J. Eng. Mater. Tech, 101, 380 (1979).

9. T. Ariyama, J. Mater. Sci., 28, 3845 (1993).

10. G. Farrow, Polymer, 2, 409 (1960).

11. P. Parrini, Macromol Chem., 38, 27 (1967).

12. T. Ariyama and M. Takenaga, Polym. Eng. Sci., 31, 1101 (1991).

13. T. Ariyama, M. Takenaga, T. Nakayama, K. Yamagata, and N. Inoue, Rept. Prog. Polym. Phys. Jpn., 30, 337 (1987).

14. T. Ariyama and M. Takenaga, Polym. Eng. Sci., 32, 705 (1992).

15. K. Kaneko and T. Ariyama, Int. J. Plasticity, 5, 421 (1989).

Cyclic deformation and stress relaxation after cyclic preloadings in polypropylene have been studied in a push-pull tension-compression mode by using a closed-loop, electrohydraulic, servocontrolled testing machine (1, 2). The data were compared with theoretical results analyzed by using a linear viscoelastic model (3, 4) and an overstress model (5, 6). An equilibrium stress and a viscosity function were treated in the overstress theory. The calculated results based on the overstress theory agree well with experiment in both the stress-strain and the stress-relaxation curves. It was concluded that the overstress theory explained the nonlinear viscoplastic behavior of polypropylene. On the other hand, it can be seen that the linear viscoelastic model explained the viscoelastic characteristics of polypropylene despite the solution of the constitutive equation constructed by the simple model, which consisted of a Maxwell unit and Hookean spring in parallel (3, 4).

Kitagawa et al. (7) reported that from the torsional tests of polypropylene at low strain rates, at room temperature, the stress-strain behavior was sensitive to strain rates and a slight work softening was seen at all strain rates. They also compared the data with theoretical results calculated from Krempl's overstress model (8) and explained well the viscoelastic-plastic characteristics of polypropylene.

It has been reported that stress relaxation after simple tension and cyclic preloading indicates complex features and reveals differences in the deformation of molecular chains in polypropylene, as subjected to different cyclic preloadings (9); the stress-strain curves and stress-relaxation curves under three mean strains were fairly different. Moreover, it was concluded from the results of the stress-strain curves and stress-relaxation curves, and scanning electron microscopy (SEM) fractographs, that significant microstructural changes occurred under cyclic preloading.

The aim of this study is to clarify the effect of mean strain on mechanisms of cyclic deformation in polypropylene. The stress-strain tests and stress-relaxation tests were performed under various sets of strain rate, strain amplitude, and mean strain by using an electrohydraulic, servocontrolled testing machine at room temperature.

EXPERIMENTAL

Materials

The polypropylene used was a commercial stock described as Grade J105 and manufactured by Ube Industries, Ltd. Crystal modification was the mono-clinic [Alpha]-form and the apparent density was 907 kg/[m.sup.3] at 18 [degrees] C. The degree of crystallinity calculated from the density was 56% (10). The melt flow index was 0.005 kg/10 min and the molecular weight was 285,000, estimated from the melt flow index (11). Details on characterization of the as-received material are given elsewhere (12).

Sample Preparation

The as-received material was machined to rod specimen with a lathe. The surface of the straight part was polished on emery papers as there were visible flaws and scratches. The gauge length of the specimen was 150 mm and the diameter of the straight part was 10 mm (2). The microstructure of the cross section of the straight part was homogeneous: This fact was confirmed by a direct observation of the spherulite deformation by polarizing microscopy (12). The mean spherulite diameter of the as-received PP was 180 [[micro]meter] (13, 14). A detailed description of the dimensions of the testing specimen has been given (2).

Testing System

Cyclic stress-strain curves and stress-relaxation curves after cyclic preloadings were obtained by use of a closed-loop, electrohydraulic, servocontrolled testing machine. Specimens were tested in a push-pull tension-compression mode under strain control at room temperature (18 to 23 [degrees] C). Load and displacement data after each specimen were fed to a computer (NEC, Co., PC 9801) and stored on floppy disks after data processing. The details of the testing procedure have been reported elsewhere (2).

Conditions of Cyclic Loading

Table 1 lists the conditions of cyclic preloading. The strain rate [Mathematical Expression Omitted], number of cycles N, strain width [Delta][Epsilon], and mean strain [[Epsilon].sub.m] are taken as parameters. The stress-strain curves were obtained at three mean strains: [[Epsilon].sub.m] = 1.0%, 2.0%, and 4.0%. The tests were performed at three strain widths of [Delta][Epsilon] = 3.0%, 5.0%, and 7.0%.

Stress-Relaxation Tests

The stress-relaxation tests were performed after three numbers of cycles (N = 35, 40, and 50). The tests were made after different predetermined strain rates ([Mathematical Expression Omitted], 100, and 1000 [micro][s.sup.-1] where [micro] = microstrain units) with a relaxation period of 1200 s duration.

[TABULAR DATA FOR TABLE 1 OMITTED]

RESULTS AND DISCUSSION

Stress-Strain Behavior

Figures 1 through 3 show the stress-strain curves for a mean strain [[Epsilon].sub.m] = 2% at different strain widths [Delta][Epsilon] of [Delta][Epsilon] = 3%, 5%, and 7%, respectively. In the cyclic deformation tests performed here, stress-relaxation tests were made at the numbers of cycles N = 35, 45, and 50. The hysteresis loops at N = 35, 45, and 50 in Fig. 1 are irregular in shape, and the tensile portion decreases in size as the number of cycles increases. On the other hand, the compressive portions at N = 35, 45, and 50 are identical in size. It is reported that the stress-strain curves at strain amplitudes from [+ or -] 1.5% to [+ or -] 5% indicated a propeller-like shape; these loops changed into the steady-state response as the number of cycles was increased up to N = 30 to 50 (2). In the case of [Delta][Epsilon] = 5% [ILLUSTRATION FOR FIGURE 2 OMITTED] and [Delta][Epsilon] = 7% [ILLUSTRATION FOR FIGURE 3 OMITTED], the feature of the tensile portions is quite similar to that of [Delta][Epsilon] = 3% [ILLUSTRATION FOR FIGURE 1 OMITTED]; however, the feature of the compressive portions is very different. The marked increase in stress value in compressive side is observed. This fact is because of different strain widths; i.e., the hysteresis loop at [Delta][Epsilon] = 3% is within the tensile region. but the hysteresis loop at [Delta][Epsilon] = 5% and [Delta][Epsilon] = 7% lie in both the tensile region and the compressive region (or negative strain region).

Figure 4 shows the stress-strain curves at a strain rate of 1000 [micro]/s for mean strains [[Epsilon].sub.m] = +2.5%, 0%, -2.5% after a number of cycles N = 10. An unusual propeller-like shape of the hysteresis loop labeled b In Fig. 4 is observed in the first ten cycles, in marked contrast to the behavior of metals (15). These loops change into the steady state response as the number of cycles is increased up to N = 30 to 50. A slight work-softening can be seen in the initial number of cycles for the mean strain of [[Epsilon].sub.m] = 0%. The stress-strain curves under three mean strains are fairly different from each other. For the mean strain of [[Epsilon].sub.m] = 0% the magnitude of the peak tensile stress is smaller by [approximately] 10 MPa than that of the peak compressive stress, and the mean stress, defined as the mean of the peak tensile stress and the compressive one, is shifted to the compressive side. On the other hand, for [[Epsilon].sub.m] = +2.5% and [[Epsilon].sub.m] = -2.5% the mean stress is shifted in the preloading direction. The stress-strain curves for the mean strains of [[Epsilon].sub.m] = +2.5%, 0%, and -2.5% are very different from those of different strain widths of [Delta][Epsilon] = 3%, 5%, and 7% at the mean strain of [[Epsilon].sub.m] = 2%.

Effect of Mean Strain on Stress Level

The experimental data for the stress-strain curve at three mean strains of [[Epsilon].sub.m] = 1%, 2%, and 4% at a strain width [Delta][Epsilon] = 3% after a number of cycles of N = 35 are depicted in Fig. 5. The reason for the choice of the number of cycles of N = 35 is that the stress-strain curve tended to reach a steady state as the number of cycles increased and that the steady state behavior was seen at a number of cycles N = 30 to 50 (2, 9). The stress value of the tensile portion for three mean strains remains constant but the value of the compressive portion decreases as the mean strain increases.

Figures 6 and 7 show the stress-strain curve at mean strains of [[Epsilon].sub.m] = 1%, 2%, and 4% after a number of cycles of N = 35 at strain widths of [Delta][Epsilon] = 5% and [Delta][Epsilon] = 7%, respectively. The overall behaviors of the hysteresis loop shapes are fairly different from each other; i.e., the tensile portion behavior is the same as that of the strain width of [Delta][Epsilon] = 3% but the compressive portion behavior is different in the conditions of strain width.

Figures 8 through 10 show the stress vs. mean strain at three strain widths of 3%, 5%, and 7%, respectively, at different numbers of cycles. At a strain width [Delta][Epsilon] of [Delta][Epsilon] = 3%, the stress level changes little with mean strain. For strain widths of [Delta][Epsilon] = 5% [ILLUSTRATION FOR FIGURE 9 OMITTED] and [Delta][Epsilon] = 7% [ILLUSTRATION FOR FIGURE 10 OMITTED], however, the minimum stress levels at three numbers of cycles increase with increasing mean strain, in contrast to the behavior at maximum stress level.

The data of the stress-strain curves for uniaxial simple tension at different strain rates [Mathematical Expression Omitted] of [Mathematical Expression Omitted], 100, 1000, and 10000 [micro]/s have been reported for the spherulite nature of polypropylene (9). In the elastic range the value of stress depends strongly on strain rate; i.e., the magnitude of stress increases with increasing strain rate above the yield point of [approximately] [Epsilon] = 0.7%. The strain range below [Epsilon] = 0.7% indicates an elastic deformation, which is independent of strain rate. Accordingly, for mean strain rates of [[Epsilon].sub.m] = 3%, 5%, and 7%, both the maximum stress and mean strain are beyond the yield point of [Epsilon] = 0.7%, i.e., the deformation behavior in the tensile side of the cyclic mode is nonlinear.

Stress Width and Mean Strains

Figures 11 through 13 show the stress width (peak-to-peak stress) vs. mean strain at three strain widths of 3%, 5%, and 7%, respectively, at different numbers of cycles. For three strain widths [ILLUSTRATION FOR FIGURES 11 THROUGH 13 OMITTED], the stress width decreases with increasing mean strain. In addition, the stress width is influenced significantly by mean stress as the strain width is large. The mean stress is defined as the mean of the peak tensile stress and the compressive one. At a strain width of 7% [ILLUSTRATION FOR FIGURE 13 OMITTED], the magnitude of the stress width for three numbers of cycles (N = 35, 45, and 50) is the same at a mean strain of 4%.

Stress-Relaxation Behavior

The experimental data for stress relaxation curves at a mean strain of 2%, at a strain width of 5%, are depicted in Fig. 14. The drop of stress decreases with an increasing number of cycles. In the case of the width of 5%, it was reported (9) that the stress-relaxation curves for [[Epsilon].sub.m] = 0% and +2.5% show the same behavior, and these are different from the curve for [[Epsilon].sub.m] = -2.5%; the behavior of the stress-strain and stress-relaxation results were related to microstructural changes in the molecular chains. In addition, the relaxation curve for [[Epsilon].sub.m] = -2.5% did not reach the equilibrium stress at 1200 s. The same behavior is observed for N = 50 (case a in [ILLUSTRATION FOR FIGURE 14 OMITTED]). The discrepancy in the results of the relaxation tests is due to the effect of the difference in strain rates tested here.

Figures 15 and 16 show the stress-relaxation curves at a mean strain of 2% at a strain rate of 1000 [micro]/s and 10 [micro]/s, respectively, after a number of cycles of N = 50. The stress drop increases with increasing strain width for both the strain rates of 1000 [micro]/s and 10 [micro]/s. The magnitude of the stress drop of strain rate of 1000 [micro]/s is larger than that of 10 [micro]/s. In Fig. 16, a little magnitude of stress drop is observed in contrast to that of Figs. 15 and 16. Hence it is concluded that the magnitude of stress drop after cyclic preloadings is influenced significantly by strain width as strain rate increases. It is seen that the drop of stress decreases as an increase in mean strain [ILLUSTRATION FOR FIGURES 14 THROUGH 16 OMITTED]. This is due to the effect of changes of stress level, as shown in stress-strain behavior.

The linear viscoelastic model has explained the vis-coelastic characteristics of polypropylene despite the solution of the constitutive equation constructed by the simple three-element model (3, 4). On the other hand, the calculated stress-strain curves and the calculated stress-relaxation curves were determined from a constitutive equation based on an overstress theory in which an equilibrium stress and a viscosity function were treated (5, 6). The calculated results agreed well with experiment. It was concluded that the overstress theory explained the nonlinear viscoelastic-plastic behavior of polypropylene. The results calculated by these models were based on the experimental data of a zero mean strain ([[Epsilon].sub.m] = 0). In the experimental facts of stress-relaxation behavior at three mean strains ([[Epsilon].sub.m] = 0, -2.5, and +2.5%) and at a strain width of 5% (or strain amplitude of [Delta][Epsilon]/2 = [+ or -] 2.5%), the significant microstructural changes occurred for the polypropylene samples after cyclic preloading of N = 50 (9). Therefore, the microstructural changes must be considered in order to analyze the stress-relaxation behavior of the effects of different mean strains. However, the theoretical treatments of a nonlinear model of viscoelasticity seem to be difficult; in particular, the analysis of the effect of mean strain on the structural changes is complicated.

CONCLUSIONS

The effect of mean strain on the cyclic deformation and stress-relaxation behavior of polypropylene was investigated by use of an electrohydraulic, servocon-trolled testing machine. A set of cyclic loading and stress relaxation was repeated three times; i.e., the stress-relaxation tests were made at the numbers of cycles N of N = 35, 45, and 50. A marked increase in stress value in the compressive side was observed. This is because of different strain widths; i.e., the hysteresis loop at [Delta][Epsilon] = 3% is within the tensile region, but the hysteresis loops at [Delta][Epsilon] = 5% and [Delta][Epsilon] = 7% lie in both the tensile region and the compressive region. The overall behaviors of the hysteresis loop shapes are fairly different; i.e., the tensile portion behavior is the same as that of the strain width of [Delta][Epsilon] = 3%, but the compressive portion behavior is different in the conditions of strain width. For strain widths of [Delta][Epsilon] = 5% and 7%, the minimum stress levels at three numbers of cycles increase with increasing mean strain, in contrast to the behavior of maximum stress level. At a strain width of 7%, the magnitude of the stress width for three numbers of cycles (N-35, 45, and 50) is the same at a mean strain of 4%. It is concluded from the results of stress-relaxation tests that the magnitude of stress drop after cyclic preloadings is influenced significantly by strain width as strain rate increases.

ACKNOWLEDGMENTS

The author is indebted to Professor K. Kaneko for helpful discussions throughout these studies and also to Dr. H. Yumoto for encouragement and suggestions.

REFERENCES

1. T. Ariyama, M. Takenaga, K. Yamagata, A. Kasai, and K. Kaneko, Rept. Prog. Polym. Phys. Jpn., 33, 311 (1990).

2. T. Ariyama, Polym. Eng. Sci., 33, 18 (1993).

3. T. Ariyama, M. Sakuma, M. Takenaga, K. Yamagata, A. Kasai, and K. Kaneko, Rept. Prog. Polym. Phys. Jpn., 34, 209 (1991).

4. T. Ariyama, Polym. Eng. Sci., 33, 1494 (1993).

5. T. Ariyama, M. Sakuma, K. Sakamoto, and K. Kaneko, Rept. Prog. Polym. Phys. Jpn., 35, 319 (1992).

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Author: | Ariyama, Takashi |
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Publication: | Polymer Engineering and Science |

Date: | Sep 1, 1995 |

Words: | 2751 |

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