Fatigue Crack Growth of Rubber Under Variable ... - University of Toledo
Oral/Poster Reference: FT242
Published in the Proceedings of the 9th International Fatigue Congress, Atlanta, Georgia, May 2006
FATIGUE CRACK GROWTH OF RUBBER UNDER VARIABLE AMPLITUDE LOADING R. Harbour1, A. Fatemi1, W. V. Mars2 1
2
The University of Toledo, Toledo, OH, USA Cooper Tire and Rubber Company, Findlay, OH, USA
ABSTRACT Realistic loading conditions for rubber components are often more complex than the constant amplitude test signals typically used in simple laboratory testing. An understanding of the effects related to variable amplitude loading is necessary to improve the ability of designers to accurately predict the fatigue behavior of rubber components in service. A series of uniaxial fatigue crack growth experiments using variable amplitude input signals was conducted to evaluate the applicability of a linear crack growth prediction model. This model equates the crack growth rate for a variable amplitude signal to the sum of the constant amplitude crack growth rates associated with each component of the variable amplitude signal. The variable amplitude signals were selected to investigate the effects of varying R-ratio, load severity, and load sequence on the fatigue crack growth rate of pure shear test specimens with an initial crack. Constant amplitude fatigue crack growth experiments were also conducted to obtain the crack growth rates utilized in the linear prediction model for variable amplitude signals. In order to distinguish the effects of strain crystallization on fatigue crack growth behavior, two rubber compounds were included in the research project: one compound that strain crystallizes, natural rubber, and one that does not strain crystallize, styrene butadiene rubber (SBR). The linear crack growth prediction model was found to be an acceptable method of predicting the crack growth rate in most cases. KEYWORDS Rubber, Fatigue, Variable amplitude loading, Crack growth INTRODUCTION The nature of rubber to withstand large strains without being permanently deformed has made it a popular material choice for many manufactured products such as tires. This wide range of product usage means that rubber undergoes a large variety of loading conditions that need to be analyzed in order to fully understand the failure process of rubber. However, the complicated nature of rubber means that its behavior does not always conform to the general models used for other materials. The non-linear nature of rubber deformation along with inelastic effects such as the Mullins effect [1] make this process more complicated than that of a similar analysis on a material that behaves linearly. While many researchers have investigated the fatigue of rubber such as Mars and Fatemi [2,3], the majority of these cases have focused on constant amplitude loading conditions. Although constant amplitude test Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
signals are the most commonly used signals in fatigue research, most real world components do not experience constant amplitude loading. Since actual loading conditions vary over time and do not fit constant amplitude assumptions, crack growth prediction models based on constant amplitude testing often are not capable of accurately predicting the fatigue life of components that experience variable amplitude loading conditions. In order to improve the fatigue life of rubber products such as tires through the design process, it is imperative to first better understand the behavior of the rubber material in the component as it is exposed to realistic loading conditions. FATIGUE CRACK GROWTH BEHAVIOR Fatigue crack growth is defined as the increase in length of existing cracks in the material as the specimen is cyclically loaded. The most commonly used crack growth parameter for rubber is the energy release rate or tearing energy. This concept is based on the idea that crack growth is due to the conversion of stored potential energy to surface energy related to new crack surfaces [2]. Energy release rate T can be calculated as the change in stored mechanical energy dU per unit change in crack surface dA as shown in Eqn. 1. This method has been utilized in the analysis of rubber under cyclic loading with a great deal of success [4]. T = -dU / dA
(1)
Fatigue crack growth curves are used to graphically represent the fatigue crack growth behavior of a material. The crack growth behavior of rubber has been characterized into four distinct regions based on increasing energy release rates as illustrated below in Figure 1. These regions include an initial region of constant crack growth rate (Regime 1), a transition region of sharp increases in crack growth rate following the crack growth threshold (Regime 2), a region that follows a power-law relation (Regime 3) and a final region of unstable crack growth that leads to component failure. The fatigue crack growth curves in Figure 1 are for unfilled SBR (x) and unfilled natural rubber (o).
Figure 1: Fatigue crack growth curves for rubber [5]
Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
TEST METHODOLOGY Pure shear test specimens were used to generate the fatigue crack growth curves of rubber due to its energy release rate T being defined as the specimen height h times the strain energy density W of the specimen away from the crack as defined in Eqn. 2. Since the energy release rate does not depend on the crack length, the crack growth rate is constant for any size crack in the specimen at a given loading condition. The geometry of the pure shear test specimen is illustrated in Figure 2. The nominal dimensions of the specimens were a height of 12.5 mm with a thickness of 0.8-0.9 mm. The specimens were cut to a length of 150 mm to fit the dimensions of the gripping fixtures. T = Wh
(2)
Figure 2: Pure shear test specimen geometry [2,3] The test specimen was gripped along the long edges of the specimen to constrain the displacement of these edges of the test specimens. The uniaxial strain was produced in the specimen by fixing the lower grip and applying a displacement normal to the long edge of the specimen. The tests were conducted using a test frequency of approximately 5 Hz for all experiments. Since the thickness of the specimen is very thin with respect to other dimensions of the specimen, the effects of heat generation should not be significant. In order to study the effects of strain crystallization on fatigue crack growth behavior, the experiments were conducted using two filled rubber compounds. One compound that strain crystallizes, natural rubber, and one that does not strain crystallize, styrene butadiene rubber (SBR). The two materials are considered filled rubber compounds due to the addition of carbon black filler when the material is mixed. The initial cracks were introduced at the center edge of the specimen by utilizing a razor blade to cut the specimen. The initial cracks were cut to a minimum length of 25.4 mm to guarantee that the crack tip was free from any stress effects at the edge of the specimens. Since the resulting crack tip is sharper than the tip of a natural fatigue crack, the specimen was cycled prior to beginning an experiment to form a more natural crack tip. A similar process was utilized to produce a fresh crack tip in the situation that the crack tip became irregularly deformed such as changing directions or splitting into multiple cracks. These crack tip issues are common for pure shear test specimens [6]. In order to produce a steady-state stress level in the specimen during the experiments, pre-conditioning cycles were performed prior to each test at a slightly higher strain level. This process eliminated the cyclic softening in the rubber at the beginning of each test to ensure steady-state conditions. Crack growth was measured by monitoring the crack tip via a traveling microscope. By locating the crack tip at specific cycle counts during the experiments, the crack growth rate was obtained by dividing the crack growth increment by the number of elapsed cycles. Measurements were taken for approximately every 0.1 mm of crack growth until the crack growth for the given test condition reached a minimum of 0.5 mm. In order to calculate the energy release rate for a particular loading condition based on Eqn. 2, the strain energy density away from the crack tip for the given loading condition was multiplied by the specimen height. The strain energy density away from the crack was found by integrating the area under the stressstrain loading curve generated for an uncracked specimen.
Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
CONSTANT AMPLITUDE LOADING Experiments were conducted using constant amplitude signals to generate fatigue crack growth data that would be the basis for variable amplitude crack growth rate predictions. Tests were run at R-ratios of 0, 0.05, and 0.10 for each material where the R-ratio is defined by Eqn. 3. Loading conditions were selected to produce data points that fell in the linear third region of the fatigue crack growth curves as illustrated in Figure 1 where Tmax is the energy release rate at the maximum displacement while Tmin is the energy release rate at the minimum displacement. R = Tmin / Tmax
(3)
The constant amplitude crack growth rates for the natural rubber specimens are presented in Figure 3a as a function of the maximum energy release rate. Each individual R-ratio can be modeled via a power-law relation, but the relationship for each R-ratio is significantly different. The crack growth results show a significant drop in crack growth rate that results from a small increase in R-ratio. This decrease in crack growth rate indicates that a small minimum stress caused by an increased minimum displacement level can be very beneficial to the fatigue crack growth behavior of natural rubber. This same relationship between crack growth rates for various R-ratios in natural rubber has been previously documented [3,7,8]. The constant amplitude crack growth results for SBR are presented in Figure 3b. The experimental results for all R-ratios can be modeled by a single power-law relation. There is no significant effect on the fatigue crack growth behavior of the SBR as a function of minimum stress in the experimental conditions. 1.E-03 R=0
R=0
R = .05
R = .05
R = .10
R = .10 Crack Growth Rate (mm/cycle)
Crack Growth Rate (mm/cycle)
1.E-03
1.E-04
1.E-05
1.E-06 1.E-01
1.E+00
1.E+01
Energy Release Rate (KJ/m 2) (a)
1.E-04
1.E-05
1.E-06 1.E-01
1.E+00
1.E+01
Energy Release Rate (KJ/m 2) (b)
Figure 3: Constant amplitude crack growth curve for (a) NR and (b) SBR The constant amplitude fatigue crack growth rate results indicate that the R-ratio can have a drastic effect on the fatigue behavior of a component depending on the type of material. It has been shown that a straincrystallizing rubber compound such as natural rubber is very sensitive to changes in R-ratio while a compound that does not strain crystallize such as SBR is not affected by R-ratio. This conclusion clearly illustrates the importance of understanding the type of material being used when predicting fatigue behavior.
Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
VARIABLE AMPLITUDE LOADINGS AND PREDICTIONS The signals used during the variable amplitude crack growth experiments consisted of a repeating sequence of constant amplitude signal blocks. The variable amplitude test signals were selected to investigate specific aspects central to the fatigue behavior of rubber such as R-ratio, load severity, and load sequencing. Sample test signals for each category are presented in Figure 4 in terms of displacement d as a function of time t. The signals designed to investigate the R-ratio effect consisted of a sequence of cycles with the same peak strain level but with varying R-ratios as in signal A. The load severity test signals varied the peak strain level in the sequence while maintaining the same R-ratio as in signal B. The effect of load sequence was found by running signals with the same combination of cycles but switching the order of the cycles as illustrated by signals C and D.
Figure 4: Variable amplitude test signals Tests were conducted using multiple specimens for each test signal and the experimental crack growth rates were averaged for each signal. These experimental crack growth rates could then used to check the applicability of prediction models for variable amplitude loading. A linear prediction model analogous to Miner’s linear damage rule [9] was used to predict the crack growth rate for variable amplitude signals using the constant amplitude crack growth results as the basis for the predictions. This model did not differentiate between signals that were designed to investigate the effects of load sequencing. The linear prediction model equated the crack growth rate for the test sequence to be equal to the sum of the crack growth rates of each individual component of the block based on the data from the constant amplitude experiments. This prediction model assumed no interaction between the different components of the test signal. The prediction model can be represented as Eqn. 4 where the total crack growth rate of the block, r, is equal to the sum of the individual crack growth rates ri and the cyclic ratio of each component Ni. r = N1 r1 + N2 r2 + N3 r3 + …. + Ni ri
(4)
Since the results for fatigue crack growth experiments in rubber have a large amount of scatter in the data, it was necessary to account for this scatter when comparing experimental to predicted crack growth rates. By determining the degree of scatter in the constant amplitude data for each material, upper and lower bounds were calculated for each crack growth prediction. The upper and lower bounds varied from the predicted crack growth rate by approximately a factor of 2 in crack growth rate. Predicted crack growth rates were considered accurate if the experimental results fell between these bounds. The experimental crack growth rates were compared to the upper and lower predicted crack growth rate bounds for a select number of test signals in Figure 5 for natural rubber and Figure 6 for SBR. Test signal 1 was an R-ratio test that combined two different R-ratios similar to signal A in Figure 4. Test signal 7 was a load severity test that consisted of cycles at three peak strain levels with the same R-ratio as in signal B. Signals 10 and 11 each had the same combination of cycles, but applied in a different order to investigate the effects of load sequence similar to signals C and D. The experimental crack growth rates for the majority of the test signals fell within the predicted bounds for both materials. For a few cases such as test signal 1 in SBR, the experimental results were just outside of the predicted range. Due to the large amount of scatter inherent in fatigue crack growth data, these cases that just fall outside of the predicted bounds would be an acceptable approximation for the crack growth rate. Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
Low er Predicted Bound Average Experimental Rate
Crack Growth Rate (mm/cycle)
2.5E-04
Upper Predicted Bound
2.0E-04 1.5E-04 1.0E-04 5.0E-05 0.0E+00 1
7
10
11
Signal
Figure 5: Crack growth rate comparison for NR Low er Predicted Bound Average Experimental Rate
Crack Growth Rate (mm/cycle)
2.5E-04
Upper Predicted Bound
2.0E-04 1.5E-04 1.0E-04 5.0E-05 0.0E+00 1
7
10
11
Signal
Figure 6: Crack growth rate comparison for SBR The only test that showed significant differences between the predicted and experimental results included a dwell period of 5-10 seconds at the minimum stress level between the periods of cyclic loading as shown in Figure 7. This effect was most significant in the SBR specimens where the experimental crack growth rates were larger than the predicted crack growth rates by a factor of 10 on average. The worst-case scenario produced an experimental crack growth rate 30 times greater than the predicted crack growth rate. The effect of the dwell period was not as significant in the natural rubber with an average increase of only 2 times the predicted crack growth rate and a maximum increase of 4 to 5 times the predicted crack growth rate. Figure 8 shows the results for an average dwell period test in both materials.
Figure 7: Dwell period test signal
Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
Crack Growth Rate (mm/cycle)
1.6E-03
Low er Predicted Bound
1.4E-03
Average Experimental Rate
1.2E-03
Upper Predicted Bound
1.0E-03 8.0E-04 6.0E-04 4.0E-04 2.0E-04 0.0E+00 SBR
NR Signal
Figure 8: Crack growth rate for dwell period tests The linear prediction model has been shown to be a good first approximation method for the fatigue crack growth rate of variable amplitude signals in many cases, but it is important to recognize the limitations of the model. These limitations include signals with a dwell period or signals that involve step loading as shown by Sun et al [10]. It is also important to note the significance of material type when choosing a fatigue crack growth rate prediction model. CONCLUSIONS Pure shear test specimens were used to investigate the fatigue crack growth behavior of natural rubber and SBR. Strain-crystallizing rubbers such as natural rubber were shown to be very sensitive to changes in the R-ratio as the fatigue crack growth rates drastically dropped for natural rubber as the minimum stress level was gradually increased. This sensitivity to R-ratio was not present during tests performed on SBR, which does not strain-crystallize. The addition of a small minimum stress level can yield very beneficial effects to the fatigue crack growth behavior for certain rubber compounds while producing no change for other compounds. A series of variable amplitude fatigue crack growth tests using repeated block signals was conducted to check the applicability of a linear prediction model for crack growth rates in natural rubber and SBR. Tests were conducted to investigate the effect of R-ratio, load severity and load sequencing on fatigue crack growth rate. The linear prediction model was used to predict the fatigue crack growth rate of variable amplitude signals using data generated from constant amplitude tests for each material. It was found that the experimental crack growth rates fell within the bounds of the predicted crack growth rates for the majority of the test signals. However, it was shown that test signals that included a dwell period at the minimum stress level between periods of cyclic loading produced experimental crack growth rates that were significantly faster than the predicted crack growth rates for SBR. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Mullins, L. (1969). Rubber Chem. Technol. 42, 339. Mars, W. V. (2001). Ph.D. Dissertation, The University of Toledo, USA. Mars, W. V. and Fatemi, A. (2003). Fatigue Fract. Engng. Mater. Struct. 26, 779. Mars, W. V. and Fatemi, A. (2002). Int. J. Fatigue 24, 949. Lake, G. J. and Lindley, P. B. (1965). J. Appl. Polym. Sci. 9, 1233. South, J. T., Case, S. W., and Reifsnider, K.L. (2002) Mech. Mater. 34, 451. Lake, G. J. and Lindley, P. B. (1964). J. Appl. Polym. Sci. 8, 707. Mars, W. V. and Fatemi, A. (2003). J. Rubber Chem. Technol. 76, 1241. Miner, M. A. (1945). J. Appl. Mech. 6, 159. Sun, C., Gent, A., and Marteny, P. (2000). Tire Sci. Technol. 28, 196. Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
Published in the Proceedings of the 9th International Fatigue Congress, Atlanta, Georgia, May 2006
FATIGUE CRACK GROWTH OF RUBBER UNDER VARIABLE AMPLITUDE LOADING R. Harbour1, A. Fatemi1, W. V. Mars2 1
2
The University of Toledo, Toledo, OH, USA Cooper Tire and Rubber Company, Findlay, OH, USA
ABSTRACT Realistic loading conditions for rubber components are often more complex than the constant amplitude test signals typically used in simple laboratory testing. An understanding of the effects related to variable amplitude loading is necessary to improve the ability of designers to accurately predict the fatigue behavior of rubber components in service. A series of uniaxial fatigue crack growth experiments using variable amplitude input signals was conducted to evaluate the applicability of a linear crack growth prediction model. This model equates the crack growth rate for a variable amplitude signal to the sum of the constant amplitude crack growth rates associated with each component of the variable amplitude signal. The variable amplitude signals were selected to investigate the effects of varying R-ratio, load severity, and load sequence on the fatigue crack growth rate of pure shear test specimens with an initial crack. Constant amplitude fatigue crack growth experiments were also conducted to obtain the crack growth rates utilized in the linear prediction model for variable amplitude signals. In order to distinguish the effects of strain crystallization on fatigue crack growth behavior, two rubber compounds were included in the research project: one compound that strain crystallizes, natural rubber, and one that does not strain crystallize, styrene butadiene rubber (SBR). The linear crack growth prediction model was found to be an acceptable method of predicting the crack growth rate in most cases. KEYWORDS Rubber, Fatigue, Variable amplitude loading, Crack growth INTRODUCTION The nature of rubber to withstand large strains without being permanently deformed has made it a popular material choice for many manufactured products such as tires. This wide range of product usage means that rubber undergoes a large variety of loading conditions that need to be analyzed in order to fully understand the failure process of rubber. However, the complicated nature of rubber means that its behavior does not always conform to the general models used for other materials. The non-linear nature of rubber deformation along with inelastic effects such as the Mullins effect [1] make this process more complicated than that of a similar analysis on a material that behaves linearly. While many researchers have investigated the fatigue of rubber such as Mars and Fatemi [2,3], the majority of these cases have focused on constant amplitude loading conditions. Although constant amplitude test Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
signals are the most commonly used signals in fatigue research, most real world components do not experience constant amplitude loading. Since actual loading conditions vary over time and do not fit constant amplitude assumptions, crack growth prediction models based on constant amplitude testing often are not capable of accurately predicting the fatigue life of components that experience variable amplitude loading conditions. In order to improve the fatigue life of rubber products such as tires through the design process, it is imperative to first better understand the behavior of the rubber material in the component as it is exposed to realistic loading conditions. FATIGUE CRACK GROWTH BEHAVIOR Fatigue crack growth is defined as the increase in length of existing cracks in the material as the specimen is cyclically loaded. The most commonly used crack growth parameter for rubber is the energy release rate or tearing energy. This concept is based on the idea that crack growth is due to the conversion of stored potential energy to surface energy related to new crack surfaces [2]. Energy release rate T can be calculated as the change in stored mechanical energy dU per unit change in crack surface dA as shown in Eqn. 1. This method has been utilized in the analysis of rubber under cyclic loading with a great deal of success [4]. T = -dU / dA
(1)
Fatigue crack growth curves are used to graphically represent the fatigue crack growth behavior of a material. The crack growth behavior of rubber has been characterized into four distinct regions based on increasing energy release rates as illustrated below in Figure 1. These regions include an initial region of constant crack growth rate (Regime 1), a transition region of sharp increases in crack growth rate following the crack growth threshold (Regime 2), a region that follows a power-law relation (Regime 3) and a final region of unstable crack growth that leads to component failure. The fatigue crack growth curves in Figure 1 are for unfilled SBR (x) and unfilled natural rubber (o).
Figure 1: Fatigue crack growth curves for rubber [5]
Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
TEST METHODOLOGY Pure shear test specimens were used to generate the fatigue crack growth curves of rubber due to its energy release rate T being defined as the specimen height h times the strain energy density W of the specimen away from the crack as defined in Eqn. 2. Since the energy release rate does not depend on the crack length, the crack growth rate is constant for any size crack in the specimen at a given loading condition. The geometry of the pure shear test specimen is illustrated in Figure 2. The nominal dimensions of the specimens were a height of 12.5 mm with a thickness of 0.8-0.9 mm. The specimens were cut to a length of 150 mm to fit the dimensions of the gripping fixtures. T = Wh
(2)
Figure 2: Pure shear test specimen geometry [2,3] The test specimen was gripped along the long edges of the specimen to constrain the displacement of these edges of the test specimens. The uniaxial strain was produced in the specimen by fixing the lower grip and applying a displacement normal to the long edge of the specimen. The tests were conducted using a test frequency of approximately 5 Hz for all experiments. Since the thickness of the specimen is very thin with respect to other dimensions of the specimen, the effects of heat generation should not be significant. In order to study the effects of strain crystallization on fatigue crack growth behavior, the experiments were conducted using two filled rubber compounds. One compound that strain crystallizes, natural rubber, and one that does not strain crystallize, styrene butadiene rubber (SBR). The two materials are considered filled rubber compounds due to the addition of carbon black filler when the material is mixed. The initial cracks were introduced at the center edge of the specimen by utilizing a razor blade to cut the specimen. The initial cracks were cut to a minimum length of 25.4 mm to guarantee that the crack tip was free from any stress effects at the edge of the specimens. Since the resulting crack tip is sharper than the tip of a natural fatigue crack, the specimen was cycled prior to beginning an experiment to form a more natural crack tip. A similar process was utilized to produce a fresh crack tip in the situation that the crack tip became irregularly deformed such as changing directions or splitting into multiple cracks. These crack tip issues are common for pure shear test specimens [6]. In order to produce a steady-state stress level in the specimen during the experiments, pre-conditioning cycles were performed prior to each test at a slightly higher strain level. This process eliminated the cyclic softening in the rubber at the beginning of each test to ensure steady-state conditions. Crack growth was measured by monitoring the crack tip via a traveling microscope. By locating the crack tip at specific cycle counts during the experiments, the crack growth rate was obtained by dividing the crack growth increment by the number of elapsed cycles. Measurements were taken for approximately every 0.1 mm of crack growth until the crack growth for the given test condition reached a minimum of 0.5 mm. In order to calculate the energy release rate for a particular loading condition based on Eqn. 2, the strain energy density away from the crack tip for the given loading condition was multiplied by the specimen height. The strain energy density away from the crack was found by integrating the area under the stressstrain loading curve generated for an uncracked specimen.
Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
CONSTANT AMPLITUDE LOADING Experiments were conducted using constant amplitude signals to generate fatigue crack growth data that would be the basis for variable amplitude crack growth rate predictions. Tests were run at R-ratios of 0, 0.05, and 0.10 for each material where the R-ratio is defined by Eqn. 3. Loading conditions were selected to produce data points that fell in the linear third region of the fatigue crack growth curves as illustrated in Figure 1 where Tmax is the energy release rate at the maximum displacement while Tmin is the energy release rate at the minimum displacement. R = Tmin / Tmax
(3)
The constant amplitude crack growth rates for the natural rubber specimens are presented in Figure 3a as a function of the maximum energy release rate. Each individual R-ratio can be modeled via a power-law relation, but the relationship for each R-ratio is significantly different. The crack growth results show a significant drop in crack growth rate that results from a small increase in R-ratio. This decrease in crack growth rate indicates that a small minimum stress caused by an increased minimum displacement level can be very beneficial to the fatigue crack growth behavior of natural rubber. This same relationship between crack growth rates for various R-ratios in natural rubber has been previously documented [3,7,8]. The constant amplitude crack growth results for SBR are presented in Figure 3b. The experimental results for all R-ratios can be modeled by a single power-law relation. There is no significant effect on the fatigue crack growth behavior of the SBR as a function of minimum stress in the experimental conditions. 1.E-03 R=0
R=0
R = .05
R = .05
R = .10
R = .10 Crack Growth Rate (mm/cycle)
Crack Growth Rate (mm/cycle)
1.E-03
1.E-04
1.E-05
1.E-06 1.E-01
1.E+00
1.E+01
Energy Release Rate (KJ/m 2) (a)
1.E-04
1.E-05
1.E-06 1.E-01
1.E+00
1.E+01
Energy Release Rate (KJ/m 2) (b)
Figure 3: Constant amplitude crack growth curve for (a) NR and (b) SBR The constant amplitude fatigue crack growth rate results indicate that the R-ratio can have a drastic effect on the fatigue behavior of a component depending on the type of material. It has been shown that a straincrystallizing rubber compound such as natural rubber is very sensitive to changes in R-ratio while a compound that does not strain crystallize such as SBR is not affected by R-ratio. This conclusion clearly illustrates the importance of understanding the type of material being used when predicting fatigue behavior.
Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
VARIABLE AMPLITUDE LOADINGS AND PREDICTIONS The signals used during the variable amplitude crack growth experiments consisted of a repeating sequence of constant amplitude signal blocks. The variable amplitude test signals were selected to investigate specific aspects central to the fatigue behavior of rubber such as R-ratio, load severity, and load sequencing. Sample test signals for each category are presented in Figure 4 in terms of displacement d as a function of time t. The signals designed to investigate the R-ratio effect consisted of a sequence of cycles with the same peak strain level but with varying R-ratios as in signal A. The load severity test signals varied the peak strain level in the sequence while maintaining the same R-ratio as in signal B. The effect of load sequence was found by running signals with the same combination of cycles but switching the order of the cycles as illustrated by signals C and D.
Figure 4: Variable amplitude test signals Tests were conducted using multiple specimens for each test signal and the experimental crack growth rates were averaged for each signal. These experimental crack growth rates could then used to check the applicability of prediction models for variable amplitude loading. A linear prediction model analogous to Miner’s linear damage rule [9] was used to predict the crack growth rate for variable amplitude signals using the constant amplitude crack growth results as the basis for the predictions. This model did not differentiate between signals that were designed to investigate the effects of load sequencing. The linear prediction model equated the crack growth rate for the test sequence to be equal to the sum of the crack growth rates of each individual component of the block based on the data from the constant amplitude experiments. This prediction model assumed no interaction between the different components of the test signal. The prediction model can be represented as Eqn. 4 where the total crack growth rate of the block, r, is equal to the sum of the individual crack growth rates ri and the cyclic ratio of each component Ni. r = N1 r1 + N2 r2 + N3 r3 + …. + Ni ri
(4)
Since the results for fatigue crack growth experiments in rubber have a large amount of scatter in the data, it was necessary to account for this scatter when comparing experimental to predicted crack growth rates. By determining the degree of scatter in the constant amplitude data for each material, upper and lower bounds were calculated for each crack growth prediction. The upper and lower bounds varied from the predicted crack growth rate by approximately a factor of 2 in crack growth rate. Predicted crack growth rates were considered accurate if the experimental results fell between these bounds. The experimental crack growth rates were compared to the upper and lower predicted crack growth rate bounds for a select number of test signals in Figure 5 for natural rubber and Figure 6 for SBR. Test signal 1 was an R-ratio test that combined two different R-ratios similar to signal A in Figure 4. Test signal 7 was a load severity test that consisted of cycles at three peak strain levels with the same R-ratio as in signal B. Signals 10 and 11 each had the same combination of cycles, but applied in a different order to investigate the effects of load sequence similar to signals C and D. The experimental crack growth rates for the majority of the test signals fell within the predicted bounds for both materials. For a few cases such as test signal 1 in SBR, the experimental results were just outside of the predicted range. Due to the large amount of scatter inherent in fatigue crack growth data, these cases that just fall outside of the predicted bounds would be an acceptable approximation for the crack growth rate. Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
Low er Predicted Bound Average Experimental Rate
Crack Growth Rate (mm/cycle)
2.5E-04
Upper Predicted Bound
2.0E-04 1.5E-04 1.0E-04 5.0E-05 0.0E+00 1
7
10
11
Signal
Figure 5: Crack growth rate comparison for NR Low er Predicted Bound Average Experimental Rate
Crack Growth Rate (mm/cycle)
2.5E-04
Upper Predicted Bound
2.0E-04 1.5E-04 1.0E-04 5.0E-05 0.0E+00 1
7
10
11
Signal
Figure 6: Crack growth rate comparison for SBR The only test that showed significant differences between the predicted and experimental results included a dwell period of 5-10 seconds at the minimum stress level between the periods of cyclic loading as shown in Figure 7. This effect was most significant in the SBR specimens where the experimental crack growth rates were larger than the predicted crack growth rates by a factor of 10 on average. The worst-case scenario produced an experimental crack growth rate 30 times greater than the predicted crack growth rate. The effect of the dwell period was not as significant in the natural rubber with an average increase of only 2 times the predicted crack growth rate and a maximum increase of 4 to 5 times the predicted crack growth rate. Figure 8 shows the results for an average dwell period test in both materials.
Figure 7: Dwell period test signal
Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
Crack Growth Rate (mm/cycle)
1.6E-03
Low er Predicted Bound
1.4E-03
Average Experimental Rate
1.2E-03
Upper Predicted Bound
1.0E-03 8.0E-04 6.0E-04 4.0E-04 2.0E-04 0.0E+00 SBR
NR Signal
Figure 8: Crack growth rate for dwell period tests The linear prediction model has been shown to be a good first approximation method for the fatigue crack growth rate of variable amplitude signals in many cases, but it is important to recognize the limitations of the model. These limitations include signals with a dwell period or signals that involve step loading as shown by Sun et al [10]. It is also important to note the significance of material type when choosing a fatigue crack growth rate prediction model. CONCLUSIONS Pure shear test specimens were used to investigate the fatigue crack growth behavior of natural rubber and SBR. Strain-crystallizing rubbers such as natural rubber were shown to be very sensitive to changes in the R-ratio as the fatigue crack growth rates drastically dropped for natural rubber as the minimum stress level was gradually increased. This sensitivity to R-ratio was not present during tests performed on SBR, which does not strain-crystallize. The addition of a small minimum stress level can yield very beneficial effects to the fatigue crack growth behavior for certain rubber compounds while producing no change for other compounds. A series of variable amplitude fatigue crack growth tests using repeated block signals was conducted to check the applicability of a linear prediction model for crack growth rates in natural rubber and SBR. Tests were conducted to investigate the effect of R-ratio, load severity and load sequencing on fatigue crack growth rate. The linear prediction model was used to predict the fatigue crack growth rate of variable amplitude signals using data generated from constant amplitude tests for each material. It was found that the experimental crack growth rates fell within the bounds of the predicted crack growth rates for the majority of the test signals. However, it was shown that test signals that included a dwell period at the minimum stress level between periods of cyclic loading produced experimental crack growth rates that were significantly faster than the predicted crack growth rates for SBR. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Mullins, L. (1969). Rubber Chem. Technol. 42, 339. Mars, W. V. (2001). Ph.D. Dissertation, The University of Toledo, USA. Mars, W. V. and Fatemi, A. (2003). Fatigue Fract. Engng. Mater. Struct. 26, 779. Mars, W. V. and Fatemi, A. (2002). Int. J. Fatigue 24, 949. Lake, G. J. and Lindley, P. B. (1965). J. Appl. Polym. Sci. 9, 1233. South, J. T., Case, S. W., and Reifsnider, K.L. (2002) Mech. Mater. 34, 451. Lake, G. J. and Lindley, P. B. (1964). J. Appl. Polym. Sci. 8, 707. Mars, W. V. and Fatemi, A. (2003). J. Rubber Chem. Technol. 76, 1241. Miner, M. A. (1945). J. Appl. Mech. 6, 159. Sun, C., Gent, A., and Marteny, P. (2000). Tire Sci. Technol. 28, 196. Copyright (c) 2006 Elsevier Ltd. All Rights Reserved.
