Fatigue crack growth of filled rubber under constant and variable amplitude loading conditions

10.1111/j.1460-2695.2007.01143.x Fatigue crack growth of filled rubber under constant and variable amplitude loading conditions RYAN J. HARBOUR 1 , A...
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10.1111/j.1460-2695.2007.01143.x

Fatigue crack growth of filled rubber under constant and variable amplitude loading conditions RYAN J. HARBOUR 1 , ALI FATEMI 2 and WILL V. MARS 3 1 Formerly

Graduate Research Assistant at The University of Toledo, Currently at Goodyear Tire and Rubber Company, 2 Professor, Mechanical, Industrial and Manufacturing Engineering Department, The University of Toledo, 2801 West Bancroft Street Toledo, Ohio 43606, USA, 3 Lead Engineer, Research Department, Cooper Tire and Rubber Company, 701 Lima Avenue, Findlay, Ohio 45840, USA Received in final form 22 March 2007

A B S T R A C T Service conditions experienced by rubber components often involve cyclic loads which

are more complex than a constant amplitude loading history. Consequently, a model is needed for relating the results of constant amplitude characterization of fatigue behaviour to the effects of variable amplitude loading signals. The issue is explored here via fatigue crack growth experiments on pure shear specimens conducted in order to evaluate the applicability of a linear crack growth model equivalent to Miner’s linear damage rule. 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 individual cycle. The variable amplitude signals were selected to show the effects of R-ratio (ratio of minimum to maximum energy release rate), load level, load sequence, and dwell periods on crack growth rates. In order to distinguish the effects of strain crystallization on crack growth behaviour, two filled rubber compounds were included: one that strain crystallizes, natural rubber, and one that does not, styrene-butadiene rubber. The linear crack growth model was found to be applicable in most cases, but a dwell effect was observed that is not accounted for by the model. Keywords fatigue; rubber; variable amplitude; crack growth rate. NOMENCLATURE

a = Crack length b = Power-law exponent C = Power-law coefficient h = Specimen height F = Fatigue crack growth exponent F 0 , F 4 = Material constants for fatigue crack growth exponent l = Specimen length N , N i = Cycles, per block i r, r i = Crack growth rate, per block i r c = Maximum fatigue crack growth rate R = Minimum to maximum energy release rate ratio t = Time T, T c , T max , T min = Energy release rate, critical, maximum, minimum T max,0 = Equivalent maximum energy release rate W , W max , W min = Strain energy density, maximum, minimum W L = Strain energy density from loading curve δ = Displacement

INTRODUCTION

Correspondence: Ali Fatemi. E-mail: [email protected]

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The nature of rubber to withstand large strains without being permanently deformed has made it a popular

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material choice for many manufactured products such as tyres. This wide range of product usage means that rubber undergoes a large variety of loading conditions that need to be analysed in order to fully understand the failure process of rubber. Realistic service histories for these rubber components almost always involve variable amplitude loading conditions. In order to improve the durability and the analytical methods used for structural integrity of rubber components, a better understanding of the effects of variable amplitude loading conditions on the fatigue behaviour of rubber is necessary. While many researchers have investigated the fatigue of rubber such as Mars and Fatemi,1–4 the majority focus on constant amplitude loading conditions. The research by Klenke and Beste5 on metal-rubber components subjected to single step test signals represented one of the first applications of Miner’s linear damage rule6 involving rubber. Sun et al.7 investigated the effects of load sequencing on fatigue life by using step-up and step-down experimental signals and found that Miner’s linear damage rule was not applicable under those conditions. Other studies that investigated aspects of variable amplitude fatigue behaviour include the research by Roland and Sobieski8 with annealing periods and a series of experiments by Mars and Fatemi9 that investigated the effects of initial overload periods on fatigue lives. A rainflow filtering process was also investigated by Steinwegger et al.10 as an attempt to reduce test times for both uniaxial and multiaxial variable amplitude test signals. The goal of the fatigue crack growth experiments described in this work was to investigate the effects of variable amplitude loading conditions on fatigue crack growth rates in filled rubbers. Since most actual service load histories are variable in nature, the effects associated with variable amplitude loading conditions can have significant implications in most applications. The variable amplitude test signals were selected to simulate some common aspects from actual load histories. The results from the variable amplitude crack growth experiments were used to determine the applicability of a linear crack growth rate prediction model that utilizes crack growth rate results from constant amplitude crack growth tests to predict variable amplitude fatigue crack growth rates. This paper begins with an overview of the experimental program including the materials and type of specimen used, the experimental procedures, and the scope of signals tested under both constant and variable amplitude loading conditions. Next, the crack growth rate results for the constant amplitude tests are presented along with a discussion of the observed R-ratio effects (ratio of minimum to maximum energy release rate) on fatigue crack growth rate. Then, for the variable amplitude experiments, the experimental crack growth rates are compared to predicted crack growth rates using a linear crack growth rate pre-

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diction model. The applicability of the model is discussed along with the introduction of an observed dwell effect on crack growth rates. E X P E R I M E N TA L P R O G R A M

Materials The fatigue crack growth experiments used specimens moulded from both filled natural rubber (NR) and filled styrene-butadiene rubber (SBR) compounds (recipes provided in Table 1). Figure 1 presents the monotonic stressstrain curves from pure shear specimens for both materials. The filler loadings were selected to produce roughly the same compound stiffness levels. Filled NR strain crystallizes,11 which refers to a phase transformation that some elastomers experience due to the application of strain, while filled SBR does not. Strain crystallization can significantly affect both the strength of the material and its fatigue properties. When in a crystalline state, a material can exhibit higher resistance to crack growth. Test specimen The fatigue crack growth experiments used a pure shear test specimen,12 as shown in Fig. 2 with a single edge crack. It is called a pure shear specimen since there is no strain along the length of the specimen, and therefore no change in the length. This type of specimen is commonly used for measuring fatigue crack growth rates in rubber due to its ability to produce constant crack growth rates regardless of crack length. The region of the specimen away from the crack tip and specimen edges undergoes a deformation state of pure shear. The most commonly used crack growth parameter for rubber is the energy release rate. The energy release rate T for the pure shear specimen is the specimen height, h, multiplied by the strain energy density in the specimen away from the crack, W , as shown in Eq. (1): T = Wh

(1)

Since the energy release rate is independent of the crack length for this specimen geometry, the crack growth rate is the same regardless of the location of the crack tip along the length of the specimen. A single test specimen can produce results for multiple crack growth tests since the crack growth is independent from crack length as long as the crack length and remaining specimen length are sufficiently longer than the specimen height of 12.5 mm. In order to satisfy these limitations, the test specimens had an initial crack length of at least 25.4 mm. Similarly, the specimen was no longer valid for experiments once the crack had grown to within 25.4 mm of the opposite edge of the specimen.

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Table 1 Recipes for filled NR and filled SBR compounds Filled NR

Filled SBR

Ingredient

PHR∗

% of Weight

NR SBR rubber cold, dry, NST (SBR 1502) Carbon black, N234 Carbon black, N650 Aromatic petroleum hydrocarbon oil Zinc oxide Resorcinol donor Stearic acid, rubber grade Sulfur, elemental Tert, butyl benzothiazole sulfenamide (TBBS) Polymerized 1,2 dihydro-2,2,4 trimethylquinoline (TMQ) CO Neodeconaoate Sulfur 20% naphthenic base oil Melamine formaldehyde resin on a silica carrier DCBS N-(Cyclohexylthio) phthalimide (PVI) Total parts per hundred rubber

100

53.7

∗ Parts

60 2 8 3 2

32.2 1.1 4.3 1.6 1.1

1 0.5 4.5 4.2 0.8 0.2 186.0

0.5 0.3 2.4 2.2 0.4 0.1 100.0

PHR∗

% of Weight

100 75

50.5 37.9

15 3

7.6 1.5

1 1.8 1.4 1

0.5 0.9 0.7 0.5

198.2

100.0

per hundred rubber, by weight.

the grips. The grips constrain the deformation of the long edges of the test specimens in all directions. Fixing the upper grip of a uniaxial, servohydraulic testing machine and applying a displacement normal to the long edge of the specimen produces strain in the specimen. The moulded specimens had nominal dimensions in the test section of 150 mm × 12.5 mm × 1 mm. Since the fatigue crack growth behaviour is independent of crack length, the length of the test specimen is not critical as long as it remains sufficiently long to maintain the pure shear state of strain in the specimen.

Test procedures

Fig. 1 Monotonic stress-strain behaviour of filled SBR and filled NR obtained from uncracked pure shear test specimens.

Specially designed grips hold the pure shear test specimens by locking onto beads of rubber that run along the long edges of the test specimens. This method of gripping the material reduces the possibility of cracks forming at

Tests were conducted in order of ascending peak strain level, so that the softening of the rubber due to the Mullins effect13 would be most nearly representative of typical conditions. The crack tip generated by the razor blade was sharper than the tip of a naturally occurring fatigue crack, and resulted in irregular fatigue crack growth behaviour during the first few applied cycles. The application of a few cycles prior to beginning a test produced a natural crack tip so that the fatigue crack growth behaviour of the specimen returned to its stable nature. A similar process was utilized to produce a fresh crack tip in the situation that the crack tip became irregularly shaped such as changing directions or splitting into multiple cracks. In order to reduce the effect of the initial cyclic softening that is characteristic of rubber behaviour, a number of cycles at a slightly higher strain were applied to the specimen

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Fig. 2 Pure shear test specimen geometry with a single edge crack.

prior to conducting any fatigue crack growth experiments. This process resulted in a steady-state stress level in the specimen that was easier to maintain at a consistent level during fatigue crack growth tests. A preconditioning level of 113% of the peak strain level was applied for 500 cycles prior to any crack growth tests being conducted at the given level. Based on previous experimental work, this degree of preconditioning maintained a relatively consistent ratio of peak loads between the preconditioning cycles and the subsequent test cycles for the entire experimental range of strain levels. A travelling microscope, capable of tracking the progression of the crack tip along the longitudinal axis of the specimen during the test, monitored crack growth. The experimental crack growth data consisted of the relative crack tip location recorded at specific cycle counts throughout the duration of the experiments. The crack length measurements were made while the machine continued to cycle the specimen. The crack tip locations were recorded at the same displacement level each time to guarantee that all of the measured crack growth increments were consistent. A minimum of 0.5 mm of crack growth per test was used to ensure that the inferred crack growth rate accurately captured the steady state crack behaviour of the material. Results from tests with crack length beyond 1 mm of crack growth confirmed that 0.5 mm was sufficient to obtain representative crack growth rates. Silver ink added to the surface of the specimen prior to placing the specimen in the test fixtures enhanced the crack tip visibility. The fatigue crack growth rate is the relationship between the change in crack length and the number of cycles that the specimen has experienced at the given strain level. The fitting of a linear relationship to the crack length data as a function of elapsed cycles yielded the average crack growth rate for the test as the slope of the linear fit. The calculation of the energy release rate for a pure shear test specimen required the calculation of the strain energy density away from the crack tip for each test condition. By definition, the strain energy density W is the area under the loading stress-strain curve for a stable cycle at a given testing condition. In order to obtain the necessary stressstrain hysteresis loops, an uncracked test specimen was used to capture the axial displacement and load data associated with each test condition. The uncracked test specimens were subjected to the same preconditioning cycles

as the cracked specimens. The stress-strain results from uncracked specimens were similar to those from cracked specimens. Numerical integration methods produced the area under the stress-strain curve from the experimental data. For non-fully relaxed conditions (R > 0), the peak strain energy density W max is the sum of the area under the loading stress-strain curve W L plus the strain energy density at the minimum strain level W min : Wmax = WL + Wmin

(2)

In order to account for the effects of stress relaxation and R-ratio on the stress-strain response, the values of strain energy density were calculated from the experimental stress-strain results for each R-ratio, as illustrated in Fig. 3. The area under the unloading stress-strain curve of a stable preloading cycle from the zero strain level to the minimum strain level provided the strain energy density at the minimum stain level.

Fig. 3 Illustration of the peak strain energy density calculation for R > 0 cycles in terms of the loading strain energy density W L and the minimum strain energy density W min .

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Test signals and effects evaluated

R-ratio

The constant amplitude fatigue crack growth experiments consisted of a series of tests at different peak strain levels for R-ratios of 0, 0.05, and 0.10 where the R-ratio is defined in terms of energy release rates as:

Test signals 1 through 5 in Fig. 4 involved combinations of different R-ratios while maintaining a constant peak strain level. Signals 1 and 2 consisted of 10 cycles at R-ratios of 0.05 and 0.10 sandwiched between single cycles at R-ratio of zero. Signals 3 and 4 increased the number of cycles at the R-ratios of 0.05 and 0.10 to 100 cycles. Signal 5 consisted of 10 cycles at an R-ratio of zero followed by 10 cycles at 0.05. The variation of the number of cycles between signals produced different predicted damage contribution ratios.

R=

Tmin Tmax

(3)

T max is the energy release rate at the maximum displacement and T min is the energy release rate at the minimum displacement. It is important to note that R-ratios defined in terms of stresses or strains do not match the R-ratios from energy release rates. The range of peak strains tested was 15 to 30% for filled SBR and 10 to 37.5% for filled NR. The variable amplitude experimental plan consisted of twelve test signals designed to investigate the effects of several typical factors that occur frequently in service load histories. These tested factors included: R-ratio, load level, load sequencing, and dwell periods. Each test signal consisted of blocks of constant amplitude cycles linked by short ramp periods (0.4 seconds) at their minimum axial positions to form test sequences. The ramp periods connected the minimum positions to avoid possible static crack growth at higher strain levels. The ramp duration was the shortest time that the test control software could consistently handle. The continuous repetition of the test sequence produced sustained crack growth. Table 2 outlines the basic formats of the variable amplitude test signals illustrated in Fig. 4. Both materials used the same basic test signals, but the applied peak strain levels differed between the materials.

Load level The effect of varying the peak strain level within a variable amplitude signal is an area of interest for rubber due to the Mullins effect. Test signals 6 through 9 in Fig. 4 varied the peak strain level while maintaining a constant R-ratio. Test signal 6 consisted of alternating blocks of 10 cycles for two peak strain levels with R-ratio of zero. Signal 7 added a third block of 10 cycles at a lower peak strain level also with an R-ratio of zero. Signal 8 involved the same three blocks as signal 7, but changed the sequence of application of the blocks. Test signal 9 consisted of 10 cycles for two peak strain levels with an R-ratio of 0.05. Load sequence The test signals designed to investigate the effect of load sequence involved the application of the same constant amplitude cycles in different orders. Test signals 7 and 8 as well as test signals 10 and 11 in Fig. 4 investigated the effects of load sequencing. Signal 10 consisted of 10

Table 2 Variable amplitude test signal structure Block 1 Test Signal

Peak

1 2 3 4 5 6 7 8 9 10 11 12

A A A A A A A A A A A A

∗∗ Peak ∗∗ Peak

Strain∗∗

Block 2 Cycles

R

Peak

1 1 1 1 10 10 10 10 10 10 10 5

0 0 0 0 0 0 0 0 0.05 0 0 0

A A A A A B B C B A A

Strain∗∗

Block 3 Cycles

R

Peak Strain∗∗

Cycles

R

10 10 100 100 10 10 10 10 10 10 10

0.05 0.10 0.05 0.10 0.05 0 0 0 0.05 0.05 0.10

A A A A

1 1 1 1

0 0 0 0

C B

10 10

0 0

A A

10 10

0.10 0.05

strain levels for NR: A – 32.5%, B – 30% and C – 20%. strain levels for SBR: A – 27.5%, B – 25% and C – 17.5%.

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Fig. 4 Variable amplitude test sequences designed to investigate the effects of R-ratios, load level, load sequencing and dwell periods on fatigue crack growth. Signals are plotted as displacement δ vs. time t. Test sequences are repeated to produce sustained crack growth.

cycles at the same peak strain level for R-ratios of 0, 0.05 and 0.10 in that order, while signal 11 reversed the order of application for the R-ratios greater than zero. Dwell period Test signal 12 consisted of 5 cycles at a peak strain level with an R-ratio of zero followed by a dwell period of 10 seconds at the minimum strain level of the applied cycles. This signal investigated the effects associated with a rest period between periods of cyclic loading. Extended block tests While the majority of the experiments used repeated block signals similar to the loading histories of components such as tires, a brief series of experiments investigated the behaviour of variable amplitude signals consisting of extended blocks as illustrated in Fig. 5. This approach allowed the crack growth rate to be measured at multiple points during a single block. The first extended block experiments involved cycles with the same peak strain level but different R-ratios similar to test signals 1 through 5. The experiments consisted of 5000 cycles at the peak strain level and an R-ratio of zero followed by a block of cycles

with the same peak strain level but a constant R-ratio of 0.05 or 0.1. The second series of extended block experiments (only conducted in filled SBR) consisted of cycles with a constant R-ratio of zero and different peak strain levels for each block similar to test signals 6 through 8. The experiments consisted of 5000 cycles at a peak strain level followed by a block of cycles at a constant lower peak strain level.

C O N S T A N T A M P L I T U D E R E S U LT S A N D R- R A T I O E F F E C T S

Cracks in both materials consistently grew at a constant rate during a given test, as expected. The experimental crack growth rates for multiple tests at a given loading condition were consistent within a factor of 3 in terms of crack growth rates for both materials. This degree of scatter in crack growth rate results is typical for rubber experiments. The initial series of constant amplitude tests conducted on both materials were for an R-ratio of zero based on energy release rate. Since the tests used displacement control, the test signals applied a slightly compressive displacement to the specimens to ensure that the specimens fully relaxed. Figure 6 presents the fatigue crack growth rate results of

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both materials for an R-ratio of zero as a function of the maximum energy release rate. The observed crack growth rates for both materials were consistent with the crack growth rate results previously reported for filled rubber by Lake and Lindley.14 A power-law relationship was fit to the data for each material in the form of: da = C (Tmax )b dN

Fig. 5 Extended block test signals designed to investigate R-ratio and load level effects. Signals are plotted as displacement δ vs. time t.

Fig. 6 Fatigue crack growth curve at R = 0 for filled NR and filled SBR. The dashed lines represent the scatter bounds for the given crack growth rate results.

(4)

where C and b are the power-law coefficient and exponent that relate the maximum energy release rate T max to the crack growth rate da/dN. The crack growth rate results for both materials, shown in Fig. 6, produce similar power-law relationships at an R-ratio of zero with filled SBR having a slightly steeper slope than filled NR (2.40 versus 2.19). Figure 7 plots the crack growth rate results for R-ratios of 0.05 and 0.10 along with the results for an R-ratio of zero for both materials. The power-law fits for all three R-ratios were almost identical for filled SBR, while significant differences were observed for the different R-ratios for filled NR. The crack growth rate results for filled NR showed a significant drop in crack growth rate due to a small increase in R-ratio. Based on the experimental findings of Lindley15 for gum NR, the power-law fits to the crack growth rate data for filled NR were constrained to converge at a single point. This point is defined by the critical energy release rate T c and the maximum fatigue crack growth rate r c . Using the value of critical energy release rate of 10 KJ/m2 proposed by Lindley for gum NR and later used by Mars and Fatemi16 for filled NR, the maximum fatigue crack growth rate was found according to the power-law fit for the filled NR to be 3.04 × 10−3 mm/cycle. The power-law fits for NR were significantly different for the three R-ratios as the slope of the fits increased with increasing R-ratio. The constant amplitude crack growth rate experiments in filled NR were consistent with the results from previous research conducted by Lindley15 on gum NR and Mars and Fatemi16 on filled NR. In all of these experiments, R-ratios greater than zero produced smaller crack growth rates than the rates for an R-ratio of zero. This indicates that the inclusion of a minimum displacement or load can produce beneficial effects in NR. This beneficial effect of R-ratio is characteristic of strain crystallizing rubbers. The constant amplitude crack growth experiments for SBR produced results that did not reflect the same R-ratio effect as NR exhibited. The inclusion of a minimum displacement or load produces no beneficial results in SBR. The significance of the R-ratio effect is most evident when comparing the fatigue crack growth rate behaviour between the two materials. It is important that the correct material model be chosen when predicting fatigue crack growth rates at R-ratios greater than zero in order to avoid

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Fig. 7 Fatigue crack growth rate curves at R = 0, 0.05, and 0.10 for (left) filled NR and (right) filled SBR.

errors in crack growth rate calculations. The inclusion of a minimum displacement or stress drastically affects the predicted crack growth rates in NR while producing no effect on predicted crack growth rates in filled SBR. Since the crack growth rate results for filled SBR overlap in Fig. 7, the fatigue crack growth behaviour could be represented as a single power-law relationship with similar constants to those in Fig. 6. For the case of the filled NR crack growth rate results, a single power-law relationship cannot model the rates for all of the R-ratios. In order to model all of the crack growth rates using a single relationship, the Mars-Fatemi model16 was used. This model is given by:   Tmax F (R) da (5) = rc dN Tc Here, the crack growth rate da/dN is defined in terms of the maximum energy release rate T max , the critical energy release rate T c , the maximum fatigue crack growth rate r c , and the power-law exponent F(R). The Mars-Fatemi model defines the exponential form of F(R) as: F (R) = F0 e F4 R

(6)

where F 0 and F 4 are material constants used to fit the model to crack growth rate data. The fit of the model in this work used the value of 10 KJ/m2 for the critical energy release rate to be consistent with the modelling

done previously by Lindley15 as well as Mars and Fatemi.16 The optimum values for F 0 and F 4 were 2.19 and 4.5 based on the crack growth rate data for filled NR from Fig. 7. This model can be used to reduce the crack growth rate data for all three R-ratios down to a single characteristic crack growth rate curve by plotting the crack growth data as a function of the equivalent maximum energy release rate T max,0 defined by Mars and Fatemi as: F (R) F

(1− FF(R) )

Tmax,0 = Tmax0 Tc

0

(7)

Figure 8 shows the plot of the crack growth rate results for filled NR in terms of the equivalent maximum energy release rate T max,0 along with the power-law relationship for an R-ratio of zero. The agreement in the results for the three R-ratios shows the ability of the Mars-Fatemi model to account for R-ratio effects on crack growth rate in NR.

V A R I A B L E A M P L I T U D E T E S T R E S U LT S A N D C R A C K G R O W T H R AT E P R E D I C T I O N S

A linear crack growth rate prediction model equivalent to Miner’s linear damage rule6 was used to predict the crack growth rate for the variable amplitude test signals using the constant amplitude crack growth results as the basis

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Fig. 8 Fatigue crack growth rate data for R = 0, 0.05, and 0.10 for filled NR using the Mars-Fatemi model with F 4 = 4.5 and F 0 = 2.19. The maximum equivalent energy release rate is defined in Eq. (7).

for the predictions. The linear prediction model equates the crack growth rate for the test sequence r to be equal to the sum of the crack growth rates r i of each individual component of the block based on the data from the constant amplitude experiments: r = N1 r1 + N2 r2 + N3 r3 + . . . . + Ni ri

(8)

where N i is the number of applied cycles for each component per sequence. Dividing the total crack growth rate r for the test sequence by the total number of cycles in the sequence yields the average crack growth rate per cycle. Since Miner’s linear damage rule does not account for the effects of load sequencing or the interaction between cycles, the linear prediction model does not differentiate between test signals designed to investigate these effects. Since the average and median crack growth rate results showed good agreement, the average crack growth rates were used in the analysis. The average crack growth rates for test signals 1 through 11 are plotted in Fig. 9. The amount of scatter present in the crack growth rate results for each signal was on the same order as the scatter observed in the constant amplitude results. Since each specimen was used for multiple tests, the order that the tests were conducted could have an effect on the crack growth

rate results. In order to investigate this issue, the sequence of application for the tests in each specimen was monitored. Comparison of these results showed no clear pattern of later tests producing higher or lower crack growth rates. Further analysis showed that the first test conducted in a specimen generally produced similar crack growth rates as the same tests conducted after applications of various other tests. This indicates that the sequencing of tests did not have a significant effect on the fatigue crack growth rate results. The average experimental crack growth rates for the variable amplitude test signals generally were on the order of 10−4 mm/cycle. These results were expected since the design of the test signals tried to produce similar average crack growth rates based on the linear prediction model in general. This approach to designing the variable amplitude test signals allowed for easier comparison of the variable amplitude crack growth rate results. While the average crack growth rates were similar, the components that make up the test signals covered a broader range of crack growth rates. Although the range of crack growth rates tested in this research was limited, the findings presented in this research are generally expected to be applicable to crack growth rates that fall in the power-law region of crack growth rate results. The constant amplitude fatigue crack growth rate curves provided the baseline data for the crack growth rate prediction model. 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. In order to account for the scatter in the data, scaling factors applied to the power-law fits of the data created upper and lower bounds to encompass all of the data as illustrated in Figure 6. After determining the degree of scatter in the constant amplitude data for each material, the linear crack growth rate prediction model determined the upper and lower predicted bounds for each test signal. The predicted bounds for the variable amplitude signals used the appropriate bounds from the constant amplitude crack growth rate results as inputs for the prediction model. These bounds represented approximately a factor of 2 difference in crack growth rate from the predicted crack growth rate. In order to compare the experimental and predicted crack growth rate results, the predicted ranges for crack growth rates are included with the average experimental crack growth rates shown in Fig. 9. The average experimental crack growth rates fell within the predicted bounds for the majority of the test signals in both materials. Signal 9 produced an average experimental crack growth rate slightly above the upper predicted bound for both materials. Test signals 1 and 2 also produced average experimental crack growth rates beyond the predicted upper

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Fig. 9 Comparison of the average experimental and predicted crack growth rates for variable amplitude test signals 1 through 11 in (top) filled NR and (bottom) filled SBR.

bound in filled SBR and near the top of the range in filled NR. For the few signals that produced experimental crack growth rates slightly outside of the predicted range, the differences were small enough that the error could be tolerated in many applications. The differences between the predicted and experimental crack growth rates were less than a factor of 2 in terms of crack growth rates for all of the test signals 1 through 11 in both materials except for signals 1 and 3 in filled SBR. The difference for these signals was less than a factor of 3. Figure 10 presents the comparison of the experimental and predicted crack growth results for signal 12 involving dwell period effects in both materials. The experimental

crack growth rate was slightly above the upper predicted bound for filled NR, but only by a factor of 2. However, the average experimental crack growth rate for signal 12 was significantly higher in filled SBR than the predicted crack growth rate. The average experimental crack growth rate was 10 times greater than the predicted value for filled SBR. Based on the significant effect of dwell periods on crack growth rates, additional tests were conducted to investigate the cause of the dwell effect. The effect on crack growth rates was observed to be as much as 30 times the constant amplitude crack growth rate for a single cycle alternating with a 10-second dwell period in filled SBR. Since the dwell period was at a near-zero stress level, the

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Fig. 10 Comparison of experimental crack growth rates for a dwell period test signal (10-second dwell) and the range of experimental constant amplitude crack growth rates for a peak strain of 27.5% in filled SBR and 32.5% in filled NR.

increases in crack growth rate were not a function of static crack growth. A proposed mechanism for the dwell period effect on crack growth rates is a time-dependent recovery in the rubber microstructure at the crack tip that produces a localized and temporary elevated stress-state. The influence of the length of the dwell period and the number of cycles between dwell periods on crack growth rates as well as a model for the dwell effect are discussed in another paper.17 The comparison of predicted and experimental crack growth rates in Fig. 11 for test signals 1 through 11 indicates that the linear crack growth rate model would be an acceptable approximation of variable amplitude fatigue crack growth rates for most cases. In general, the variable amplitude effects associated with R-ratio, load level, and load sequence did not significantly affect the applicability of the linear prediction model. The only test signal that drastically differed from the predicted crack growth rates was signal 12 in filled SBR. The linear crack growth rate model does not account for the increased crack growth rates associated with the dwell period. An increase in crack growth rate due to a dwell period could have serious implications if not properly accounted for in a crack growth rate prediction model. The average experimental crack growth rates for both materials under variable amplitude loading tended to be in the higher end of the predicted range of crack growth

Fig. 11 Comparison of experimental and predicted crack growth rates for variable amplitude test signals 1 through 11 in (top) filled NR and (bottom) filled SBR. The dotted lines represent factors of 2 and 3 from agreement.

rates. Although these experimental results fell within the predicted range, it was expected that the average experimental crack growth rates would be more evenly distributed throughout the range of predicted crack growth rates. This tendency for slightly higher crack growth rates could be attributed to dwell effects caused by the ramp periods between blocks. The effect in most signals would be small due to the number of cycles in the test signal reducing the relative effect on the overall crack growth rate and the short dwell period. The dwell period would have the largest effect on the experimental crack growth rates for signals 1 and 2 in filled SBR. Since test signals 1 and 2 consisted of only 12 cycles and a pair of ramp periods, the dwell effect would be more significant in these signals than in the other test signals that contained more cycles. Based on the model developed for the dwell effect17 on crack growth rates, an increase of approximately 4 to

 c 2007 The Authors. Journal Compilation  c 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 640–652

FAT I G U E C R AC K G R O W T H O F F I L L E D R U B B E R U N D E R C O N S TA N T A N D VA R I A B L E A M P L I T U D E

651

Table 3 Results for extended block tests Block 1

Block 2 R

N

Peak Strain

R

Max Stress (MPa)

Stress Ratio

CGR Ratioa

R-ratio extended block tests NR 5,000 32.5% NR 5,000 35% SBR 5,000 27.5% SBR 5,000 27.5%

0 0 0 0

8,000 20,000 8,000 10,000

32.5% 35% 27.5% 27.5%

0.05 0.10 0.05 0.10

1.35 1.46 1.75 1.72

0.15 0.20 0.11 0.17

1.02 0.96 0.85 1.14

Load level extended block tests SBR 5,000 27.5% SBR 5,000 27.5% SBR 5,000 27.5%

0 0 0

14,000 5,000 5,000

25% 17.5% 17.5%

0 0 0.05

1.47 1.10 1.10

0 0 0

0.89

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