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Material deformation and fatigue behavior characterization for elastomeric component life predictions.

INTRODUCTION

Fatigue failure is a typical service failure mode resulting from alternating and repeated loads over a period of time. Although elastomeric components are commonly used in many industries and their field of application has broadened in recent years, there is very limited research related to fatigue failure aspects of elastomeric components, as compared with metallic components.

Elastomers are highly nonlinear elastic materials with large deformations. Therefore, linear elastic and small strain assumptions typically used for metals do not apply to elastomers. As a result, continuum mechanics parameters such as deformation gradient, stretch ratio, and Green-Lagrange strain components are used in rubber mechanics analysis. The Mullin's effect, which is associated with initial cyclic softening, is another challenging characteristic when dealing with elastomers.

To be able to predict life of elastomeric components, several inputs are necessary. These include the component geometry, relevant material properties, and the loading history. For analysis, two basic steps are often required. First, stress and/or strain analysis is used, typically by finite element method, to obtain stress/or strain histories at critical location(s) from the applied load history. Then, fatigue life prediction analysis is performed based on a fatigue damage quantification parameter and damage accumulation model using the critical location stress and/or strain history.

Mars and Fatemi (1) conducted a literature study on different approaches associated with fatigue analysis for rubber. Fatigue failure process is divided into two distinct phases, crack nucleation phase in which crack nucleates in regions that were initially crack free, and crack growth phase when the nucleated cracks grow to the point of failure or rupture. They identified maximum principal strain (or stretch) and strain energy density as two broadly used damage quantification parameters for the nucleation phase. Although it is commonly observed that cracks initiate on a plane normal to the maximum principal tensile strain direction, the strain energy density criterion applied as a scalar criterion can not account for this preferred orientation. For the fatigue crack growth stage, a fracture mechanics analysis is used based on energy release rate. Mars and Fatemi also discuss the relationship between crack nucleation and crack growth approaches by using an integrated power-law model which is also proposed in (1).

Rubber can have a wide range of mechanical properties by changing the compound formulation and manufacturing process. Strain crystallization has been shown to have beneficial effect on fatigue life at moderate or high strain levels. The addition of carbon black to rubber compound could strengthen the material against fatigue failure and drastically change its mechanical properties. Antide-gradants are added to rubber compounds to avoid the deleterious effects of oxygen and ozone. Vulcaniztion is used in thermosetting elastomers to create covalent bonds or crosslinks between adjacent polymer chains. Crosslink density determines the physical properties of rubber, with higher crosslink density resulting in increased stillness and reduced hysteresis. Compound stiffness has a direct effect on energy release rate. Filled rubbers show more dissipative mechanical responses at both high strains because of network chain breakage and strain crystallization, and at small or moderate strains under alternating loading. Hysteresis is an important aspect when evaluating fatigue properties in terms of deformation, crack nucleation, or crack growth.

Kim and Jeong (2) investigated natural rubber compound with three different filler types to examine the effect of carbon black on fatigue life. They concluded that for larger carbon black agglomerates, separation of fillers from rubber matrix could relatively easily occur resulting in shorter fatigue life, compared to smaller size fillers. Santangelo and Roland (3) verified that double network of natural rubber (NR) has a higher modulus than single network of equal crosslink density. Fatigue life of double networks NR was also found to be as much as a factor of 10 higher than conventionally crosslinked NR, although tensile strength was slightly lower.

Stevenson (4) conducted a study of fatigue crack growth of rubber in compression with a cylindrical specimen. He concluded that crack growth in compression occurs at an approximately constant rate and is restricted to the outer regions of the test specimen with high local shear strains. Legorju-jago and Bathias (5) showed that a mean stress in tension improves the fatigue behavior by crystallization of the stretched bonds in pure tension cycles. On the other hand, a minimum stress in compression seriously damages the material.

Abraham et al. (6) investigated the effect of minimum stress and stress amplitude on fatigue life of nonstrain crystallizing elastomers such as ethylene propylene (EPDM) and styrene--butadiene (SBR) rubbers. They used cylindrical dumbbell specimens and concluded that increasing minimum stress with constant stress amplitude can increase the service life by a factor of 10 despite the increase in maximum stresses.

To account for [R.sub.[epsilon]] ratio (minimum to maximum strain ratio) effect on either crack nucleation or crack growth life, Mars and Fatemi (7) proposed a phenomenological model. The ability of the model to represent data for different R-ratios was shown to be reasonable based on their own experimental data, as well as data of Lindley, Cad-well et al. (7). Harbour et al. (8) also utilized this model to represent their nonzero fatigue crack growth rate data.

Environmental effects can significantly affect fatigue life of elastomers. High temperature has a detrimental effect both on crack nucleation life and crack growth rate. This effect is intensified in amorphous rubber. Ozone could shorten the crack growth life due to reaction with carbon bonds at the crack tip. Presence of oxygen increases the fatigue crack growth rate. Mars and Fatemi discussed the aforementioned effects including temperature, ozone, oxygen, and electrical charges in detail in (9).

On the basis of the aforementioned discussion, the material properties needed for elastomeric component fatigue analysis and life prediction include deformation behavior under different stress states, crack initiation life, and crack growth rate properties. To obtain each of these material characterizations, specific experimental procedures and data analysis techniques are used. The objective of this article is to demonstrate how such properties can be obtained from simple specimen geometries and experimental techniques. The material used for this demonstration is a filled natural rubber (NR) used in an automobile cradle mount, containing 20.8% carbon black and 9.5% plasticizer, provided in 1-mm-thick sheets. For each type of characterization, first the experimental procedure is described, followed by data analysis, and mathematical model representation of the obtained results.

DEFORMATION BEHAVIOR CHARACTERIZATION

Monotonic and cyclic deformation curves properties are typically needed for FE modeling and stress--strain relations for fatigue analysis. Phenomena associated with elastomers, such as Mullins effect, can also be evaluated.

Extension ratio or stretch ratio ([lambda]) is commonly used for finite strain analysis and is defined as the ratio of the extended length (L) to the original length ([L.sub.0]):

[lambda] = L/[L.sub.0] (1)

The relationship between engineering strain (hi and stretch ratio is given by:

[lambda] = [epsilon] + 1 (2)

As the rubber is an incompressible material, there is no change in volume under deformation, [DELTA]V = 0. This results in the determinant of the deformation gradient to be one, which in turn results in the following relation for principal stretch ratios:

[[lambda].sub.1][[lambda].sub.2][[lambda].sub.3] = 1 (3)

The stress states of simple tension and planar tension are often used to characterize the deformation behavior. Simple tension specimens can be cut by using specimen cutting dies according to ASTM standard D4482-99 (10). The test specimen used in this study had an hourglass shape with 1mm thickness and dimensions shown in Fig. 1a. Specimen preparation conditions are described in the aforementioned ASTM standard. For planar tension deformation and crack growth tests, a wide rectangular of 1 mm thickness with dimensions shown in Fig. 1b is used. For crack growth specimen, a precrack is cut at mid-length, as shown in Fig. lb, but for cyclic deformation test no precrack is cut.

Figure 2 shows the stretch and stress states for each specimen condition under uniaxial loading. For simple tension condition, uniaxial state of stress is present, with a multiaxial stretch state with the longitudinal stretch value of [lambda] and two equal transverse stretch values of [[lambda].sup.-1/2] satisfy the incompressibility condition. Planar tension specimen is under plane stress condition with longitudinal and transverse stresses. The stretch in the width dimension is 1, indicating there is no strain in this direction. A stretch of [lambda] in loading direction and [[lambda].sup.-1] exist in the out of plane direction. It should also be mentioned that due to the incompressibility condition, Poisson's ratio is 0.5 and the initial modulus of elasticity (E) is approximately equal to three times the shear modulus (G), E = 3G.

To determine the stress-strain behavior of the material, tests were conducted in displacement control. The applied displacement was then converted into gauge section strain and the measured load from the load cell, which had a maximum capacity of 5 kN, was used for stress calculation. The monotonic deformation curve is shown in Fig. 3, indicating a markedly nonlinear behavior.

A cyclic incremental test on simple tension specimen in displacement control with the sinusoidal waveform was performed to obtain the cyclic stress-strain curve. Because of Mullin's effect, stabilized cycle data is used for each strain level (i.e., after about 20 applied cycles) and all tests are performed in ascending order of displacement amplitude. Figure 4a shows the resultant hysteresis loops, where the area between loading and unloading curves represents energy dissipation during the cycle.

Hyperelastic constitutive models are used to represent the nonlinear elastic deformation curve. A Fictitious curve for simple tension condition is obtained by using a least squared fit polynomial to the end points of stress and strain from each loop (11). This curve is depicted in Fig. 3 and the mathematical representation of maximum engineering stress ([S.sub.max]) versus maximum engineering strain ([e.sub.max]) for simple tension condition is given by:

[S.sub.max] = 0.17 [e.sub.max.sup.3] - 0.80 [e.sub.max.sup.2] + 2.03 [e.sub.max] (4)

The superimposed plot of monotonic and cyclic simple tension curves illustrate that using monotonic properties in a cyclic loading application can underestimate the level of strain which can he present. It is, therefore, important to obtain and use cyclic material properties in fatigue life analysis and applications.

Cyclic incremental step tests were also conducted with the planar tension specimen. Figure 4b shows the hysteresis loops from the 128th cycle from each incremental step. The 128th cycle is used as the stable cycle, subsequent to initial softening due to Mullins effect. As can be seen, after the initial straining the material does not return to zero strain at zero stress due to some degree of permanent deformation. A fitted curve for planar tension condition is also depicted in Fig. 3. The nominal stress-strain relationship for cyclic planar tension test is given by:

[S.sub.max] = 0.85 [e.sub.max.sup.3] - 2.46 [e.sub.max.sup.2] + 2.79 [e.sub.max] (5)

According to Fig. 3, cyclic simple tension deformation curve is less stiff than cyclic planar tension at strain levels below 0.6. This is consistent with other deformation behavior characterization studies done by Sharma (12) and Mars-Fatemi (13). However, at higher strain values the simple tension curve is stiffer than planar tension specimen curve. This could be mainly due to the higher amount of strain crystallization in simple tension, which in turn causes higher stiffness. In simple tension specimen there is no lateral constraint to cause specimen thinning and therefore the polymer network chains could become sufficiently aligned. The choice of the curve to use in FE simulations depends on the stress state at the critical location of the component being analyzed.

The initial softening phenomenon, also called the pre-stretch or preconditioning effect, is widely referred to as the Mullin's effect (14). Subsequent loadings of equal or lesser magnitude rapidly tend towards a steady state, nonlinear elastic response. This effect is considered to be a consequence of the breakage of links inside the material and both filler-matrix and chain interaction links are involved in the phenomenon (15). The Mullins effect is transient and is exhibited primarily by filled rubbers (16).

Figure 5a provides an illustration of the Mullin's effect as the stress level drops for each successive loop. The stress-strain loop stabilizes after 3-30 cycles of loading for most elastomers. For a higher maximum strain, the initial softening is larger, as can be seen from Fig. 5b. This figure also shows that by increasing peak strain, it takes a longer time for the material to show a stabilized response. Therefore, due to the load history dependence associated with the Mullin's effect, peak loading should be a key consideration in fatigue analysis of rubbers, in addition to the load amplitude and the mean load.

CRACK INITIATION TESTING AND DATA REPRESENTATION

Cracks often nucleate from pre-existing flaws in the compound (i.e., such as undispersed carbon black agglomerates), due to processing defects (such as contaminations, voids, and molding flaws), or at stress concentrations. In the context of mechanical design and fatigue analysis, crack initiation life is referred to the life involved in growing a crack from the preexisting flaw to a small macro-crack typically on the order of 1mm. Strain is commonly used for crack initiation life analysis as an ideal fatigue damage quantification parameter, due to ease of determining displacements in elastomers.

The specimen geometry used in crack initiation tests is the geometry shown in Fig. 1a. This type of specimen is designed in such a way that when a crack grows to a length on the order of 1 mm, there would be little remaining life to fracture. Therefore, these tests characterize fatigue life for a crack on the order of 1 mm, although the criterion for defining fatigue nucleation life in the test is the number of cycles in which the specimen ruptures completely according to ASTM standard D 4482 (10). Multiple specimens can be used for each test. The test setup with five specimens used in this study is shown in Fig. 6.

Tests were performed in displacement control and three strain [R.sub.[epsilon]] ratios of 0.02, 0.1 and 0.2 were chosen for testing. As the specimen can not support compression load, these [R.sub.[epsilon]] ratios represent tension-tension loading condition. For each [R.sub.[epsilon]] ratio, three levels of maximum strain were tested. The testing frequency used for all conditions was 1 Hz, except those with peak strain values higher than 2.75 which utilized testing frequencies of 0.5 Hz.

Test conditions and result from these tests are tabulated in Table 1. The ratio of maximum fatigue life to minimum fatigue life in each testing condition ([N.sub.f.max]/[N.sub.f.min]) which is an indicator of data scatter is also shown in this Table. This ratio for all crack initiation tests was in the range of 1.11 and 2.58, which is quite reasonable for elastomeric material fatigue tests.

A mean value for each strain ratio and maximum strain was calculated and used in curve fitting and data analysis. The reason for using a mean life value for each test condition rather than direct fit of all the data is so that there is equal contribution of each test condition in test data analysis. Figure 7 shows raw data of maximum strain versus crack nucleation life for all tests at different [R.sub.[epsilon]] ratios. It is observed that the power trend lines shown fit the crack initiation data well. The best-fit lines use the least squares fit method with fatigue life as the dependent variable. The fatigue life equations based on the different strain R ratios (denoted by [R.sub.[epsilon]]) are given by:

[[epsilon].sub.max] = 28.9 [([N.sub.f]).sup.-0.30] for [R.sub.[epsilon]] = 0.02 (6)

[[epsilon].sub.max] = 17.4 [([N.sub.f]).sup.-0.30] for [R.sub.[epsilon]] = 0.01 (7)

[[epsilon].sub.max] = 9.41 [([N.sub.f]).sup.-0.30] for [R.sub.[epsilon]] = 0.02 (8)

From Fig. 7 it can be seen that the effect of [R.sub.[epsilon]] = 0.2 cycles is to increase the nucleation life. The life improvement is very significant at low strain, about an order of magnitude, but less important at high strain, less than a factor of two. This is in contrast with metals where a tensile mean load has detrimental effect on fatigue life. This life improvement in natural rubber is mainly due to strain crystallization. By applying a nonzero minimum strain, rubber does not come back from crystalline state to amorphous rubbery state. This crystalline state in which the polymer chains are aligned highly in the loading direction increase resistance to crack growth, therefore fatigue life would be longer.

The influence of strain ratio can be estimated by an empirical relationship by the Mars-Fatemi model (7) that relates fatigue failure at a given life for [R.sub.[epsilon]] > 0 conditions to fatigue failure at the same life for [R.sub.[epsilon]] [approximately equal to] 0 condition by the following relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

G(R) = -0.35 - 258[R.sup.3] (10)

where [[epsilon].sub.max,R] is the maximum engineering strain at a given R ratio, [[epsilon].sub.max,0] is the equivalent maximum engineering strain at R [approximately equal to] 0, G(R) is the power law exponent, and [[epsilon].sub.e] = 6.25. Figure 8 shows correlation of all crack nucleation data on a single plot for all strain ratios. The equivalent R = 0 maximum scram ([[epsilon].sub.max.0]) is then given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

TABLE 1. Crank nucleation test conditions and result

[[epsilon].sub.min]  [[epsilon].sub.max]  [R.sub.[epsilon]]

0.02                                1.00               0.02
0.04                                2.00               0.02
0.06                                3.06               0.20
0.09                                0.95               0.10
0.16                                1.60               0.10
0.33                                3.33               0.10
0.36                                1.80               0.20
0.55                                1.75               0.20
0.75                                3.75               0.20

[[epsilon].sub.min]  [[epsilon].sub.a]  Freq  Number of       Mean
                                        (Hz)      tests  [N.sub.t]
                                                          (cycles)

0.02                              0.49     1          4     68,528
0.04                              0.98     1          5      6,682
0.06                               1.5   0.5          5      1,704
0.09                              0.43     1          5    103,940
0.16                              O.72     1         10     18,321
0.33                              1.49   0.5          5        712
0.36                              0.72     1          5    107,966
0.55                               1.1   0.5          5     18,641
0.75                               1.5   0.5          5        976

[[epsilon].sub.min]   Standard  [N.sub.f,max]/[N.sub.f,min]
                     deviation

0.02                    12,145                         1.41
0.04                       275                         1.11
0.06                       666                         2.58
0.09                     6,904                         1.18
0.16                     3,295                         2.03
0.33                        32                         1.11
0.36                    30,989                         2.37
0.55                     4,095                         1.79
0.75                       201                         1.75


Once [[epsilon].sub.max,0] is determined for a given [R.sub.[epsilon]] loading condition, it is then used in Eqs, 6-8 to calculate the fatigue crack initiation life.

CRACK GROWTH TESTING AND DATA REPRESENTATION

Crack growth testing of elastomers typically uses a precracked planar tension specimen, as shown in Fig. lb with a precrack length of 25 mm. While the original application of fracture mechanics approach to rubber was to predict static strength, in the late 1950s, Thomas extended the approach to analyse the growth of cracks under cyclic loads in natural rubber (17). Since rubber is a nonlinear elastic material, energy release rate is often used as the crack driving parameter, rather than the stress intensity factor which is typically used for linear elastic nominal material behavior, such as in metals. This parameter relates the global specimen loading to the local stress and strain fields at the crack tip, and is also applicable to finite strain materials, such as elastomers.

Energy release rate is the change in the stored mechanical energy dU, per unit change in new crack surface area c/A, also often called the tearing energy T, and given as:

T = - dU/dA (12)

Under cyclic loading, the maximum energy release rate achieved during a cycle is related to the crack growth rate.

Experimental Procedure

Because of a simple relationship between energy release rate and strain energy density and the fact that energy release rate is independent of crack length in planar tension specimen, this specimen is an ideal choice for fatigue crack growth experiments. In the planar tension specimen, shown in Fig. 1b, energy release rate depends only on the strain energy density (W) remote from the crack and specimen edges, and the specimen height (h):

T = Wh (13)

This relation makes the planar tension specimen geometry quite practical for use in a fatigue crack growth characterization of elastomers. In this specimen, a state of plane stress under uniaxial tension is present, as shown in Fig. 2.

The strain energy density (W) is the area under the loading stress-strain curve for a stable cycle for each peak strain level. The energy release rate and crack growth rate are independent of crack length in the planar tension specimen. Stress-strain hysteresis loops from an uncracked test specimen can be used to obtain the test signal parameters for the fatigue crack growth rate. Numerical integration methods (such as Simpson's rule), as explained in the ASTM standard D4482-99 (10), can be used to calculate the area under the stress-strain curve from the experimental data.

TABLE 2. Constant amplitude fatigue crack growth test conditions.

[[epsilon].sub.min]  [[epsilon].sub.min]  [R.sub.E]  [R.sub.W]

0                                   0.40          0          0
0                                   0.45          0          0
0                                   0.50          0          0
0                                   0.55          0          0
0                                   0.60          0          0
0                                   0.65          0          0
0                                   0.40          0          0
0                                   0.45          0          0
0.04                                0.45       0.09       0.05
0                                   0.50          0          0
0.05                                0.50       0.09       0.05
0.09                                0.50       0.17       0.10
0                                   0.55          0          0
0.05                                0.55       0.10       0.05
0.10                                0.55       0.18       0.10
0                                   0.60          0          0
0.06                                0.60       0.10       0.05
0.11                                0.60       0.18       0.10
0                                   0.65          0          0
0.07                                0.65       0.10       0.05
0.12                                0.65       0.19       0.10

[[epsilon].sub.min]  dulclN      [T.sub.max]     Freq.
                     (mm/cycle)  (kJ/[m.sup.2])  (Hz)

0                    2.82E - 04            2.63      5
0                    3.71E - 04            3.13      5
0                    4.87E - 04            3.66      4
0                    8.40E - 04            4.23      3
0                    9.76E - 04            4.84    2.5
0                    1.51E - 03            5.49      2
0                    3.60E - 04            2.63      5
0                    4.70E - 04            3.13      5
0.04                 1.77E - 04            3.13      5
0                    6.30E - 04            3.66      4
0.05                 4.70E - 04            3.66      4
0.09                 4.43E - 05            3.66      4
0                    8.86E - 04            4.23      3
0.05                 6.14E - 04            4.23      3
0.10                 5.48E - 05            4.23      3
0                    1.05E - 03            4.84    2.5
0.06                 6.93E - 04            4.84    2.5
0.11                 6.85E - 05            4.84    2.5
0                    1.19E - 03            5.49      5
0.07                 9.69E - 04            5.49      2
0.12                 1.01E - 04            5.49      2


For a constant displacement or strain, strain energy density is constant and so the crack growth rate would be constant with repeated cycles of the same amplitude. For non-fully relaxed conditions (R > 0), the peak strain energy density ([W.sub.max]) is the sum of the area under the loading stress-strain curve ([W.sub.L]) plus the strain energy density at the minimum strain level ([W.sub.min]):

[W.sub.max]. = [W.sub.L] + [W.sub.min] (14)

Control parameters for crack growth test with planar tension specimen are the amplitude and mean values of the applied displacement. Figure 9 presents the maximum strain energy density as a function of maximum engineering strain obtained from uncracked test specimens. Tests were performed with both fully relaxing conditions ([R.sub.W] = 0), as well as at three [R.sub.W] ratios of 0, 0.05, and 0.10, while changing peak strain values from 40 to 65%, as listed in Table 2.

To precondition the material to avoid transient Mullin's effect, stable deformation curves were defined at the 128th cycle of planar tension uncracked specimen, where the peak stress was 10% lower than the first cycle stress (see Fig. 5b). For the cracked specimen 110% of maximum strain was used as preconditioning load for 500 cycles at each strain level used to minimize the transient deformation response as well as to produce a natural crack tip.

A single test specimen can produce results for multiple crack growth tests as long as the crack length and remaining specimen length are sufficiently longer than the specimen height of 16.83 mm. Because of the transient softening, however, it is necessary to conduct tests on the same specimen in ascending order of maximum strain level to make sure that the higher levels of load would not affect the lower load level test results. The desired range of fatigue crack growth rates were between [10.sup.-6] and [10.sup.-3] mm/cycle. Using a cycling frequency of 5 Hz, it takes about 5 h to grow the crack 0.1 mm at a crack growth rate of [10.sup.-6] mm/cycle.

A traveling microscope which can track the growth of the crack tip along the crack line can be used to measure the crack growth. Crack growth data based on a minimum interval of ~0.1 mm of crack growth was used for measurements. After obtaining several data points for each test condition, the R-ratio was changed by increasing the minimum displacement. After all R-ratio test data were obtained, the value of peak strain was then increased to obtain crack growth rate at the next strain level. If the crack grew irregularly or at an inclined angle, recutting was performed by a razor blade, followed by some initial cycles to initiate a natural crack tip.

Test Results and Representation Including R-Ratio Effect

Fatigue crack growth rate (da/dN) is obtained by fitting a linear relationship to the crack length versus cycles data and determining the slope of the linear fit. Crack growth data fits at different energy release rate ratios and peak strain levels are shown in Fig. 10. The test conditions used for the constant amplitude fatigue crack growth experiments produces crack growth rates in the region of crack growth that can be characterized by a power-law relation. The coefficient and exponent of this relation are calculated from [R.sub.T] = 0 data as:

da/dN = 4 x [10.sup.-5][([T.sub.max]).sup.2.0] (15)

Note that for planar tension specimen, due to the relation given by Eq. 13, [R.sub.T] = [R.sub.W]. In strain crystallized rubbers, such as the natural rubber used in this study, increasing the minimum strain or energy release rate ratio has a significant beneficial effect on fatigue life compared to [R.sub.T] = 0 condition, as fatigue crack growth rate decreases. This behavior is depicted in Fig. 11 where the fits to data show that increasing R-ratio causes slower crack growth rate and, therefore, being beneficial to fatigue life. The power law equations for the two energy release rate ratios higher than zero are obtained as:

da/dN = 1 x [10.sup.-5][([T.sub.max]).sup.2.73] for [R.sub.T] = 0.05 (16)

da/dN = 3 x [10.sup.-6][([T.sub.max]).sup.1.99] for [R.sub.T] = 0.10 (17)

By using Mars-Fatemi model (7), it is possible to reduce fatigue data taken at various levels of R ratio to a single equivalent R = 0 curve. This is achieved by plotting the crack growth rates against the equivalent R = 0 maximum energy release rate, [T.sub.(max, 0)]. The Mars-Fatemi model is given by:

da/dN = [r.sub.c][([T.sub.max]/[T.sub.c]).sup.F(R)] (18)

where [T.sub.c] = 10 kJ/[m.sup.2] is based on Lindley's estimate for fatigue crack growth of natural rubber (7). The critical crack growth rate for this condition is defined as [r.sub.c] = 0.004 mm/cycle based on R = 0 crack growth equation. The equivalent R = 0 maximum energy release rate ([T.sub.(max, 0)]) is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

This form of F(R) is used when there are limited data points and it adds just one variable to be calculated to the material properties obtained from R = 0 analysis. On the basis of Lindley's model and using energy release rate ratios of 0 and 0.10 constants, the values of [F.sub.0], obtained from R = 0 power law fit data for this material, and F4 are calculated as 2 and 8.27, respectively. Figure 12 shows correlation of all fatigue crack growth (FCG) data on a single plot for different [R.sub.T] ratios by the use of Mars-Fatemi model.

Relationship Between Crack Growth and Crack Initiation Approaches

In a single edge cut simple tension specimen geometry the energy release rate depends on the gauge section strain energy density W, crack length a, and a stretch dependent parameter k (18):

T = 2kWa (21)

By combining the power-law fatigue crack growth rate relation (Eq. 15) and the above equation and then integrating the resulting equation, the following relationship is obtained (1):

[N.sub.f] = 1/(F - 1) 1/B[(2kW).sup.F] [1/[a.sub.0.sup.(F - 1)] - 1/[a.sub.f.sup.(F - 1)]] (22)

where B and F are the fatigue power-law coefficient and exponent for R = 0 condition, respectively. If the initial flaw size [a.sub.0] is much smaller than the critical flaw size, [a.sub.f], the life becomes nearly independent of the critical flaw size:

[N.sub.f] = 1/(F - 1) 1/B[(2kW).sup.F] 1/[a.sub.0.sup.(F - 1)] (23)

Effective flaw sizes in the range of 0.02-0.06 mm were observed in a study by Lake and Lindley, which covered different polymer types, and various fillers, curatives and other compounding variables I191. In order to compare the relationship between crack initiation and crack growth approaches, Eq. 23 was used to obtain fatigue life based on the crack growth approach and to compare with the fatigue life based on the crack initiation approach. By assuming [a.sub.0] = 0.02 mm, utilizing B = 4 x 105 and F = 2 as crack growth rate coefficient and exponent. respectively. in Eq. 23, and by assuming k [approximately equal to] 2 (in the stretch range studied), relatively good agreement (i.e., about a factor of two in fatigue life) between the R = 0 crack initiation life and crack growth life is obtained, as shown in Fig. 13. Therefore, the crack growth approach could be used as a total life approach, based on growth of preexisting flaws to failure. Correlation between crack nucleation life obtained from simple tension specimen and crack growth life obtained from planar tension specimen was studied by Mars and Fatemi (16) and they also found good agreement between the results. It should be mentioned that a change of initial assumed crack length from 0.02 mm to 0.04 mm resulted in very small change in the obtained results.

If [N.sub.(f, [infinity])] represents the fatigue life associated with a long final crack size (i.e., a crack size much longer than the initial crack size of 0.02 mm), the following relation based on Eq. 22 is obtained (20):

[N.sub.f]/([N.sub.f], [infinity]) = [1/[a.sub.0.sup.(F - 1)] - 1/[a.sub.f.sup.(F - 1)]]/(1/[a.sub.0.sup.(F - 1)]) = 1 - [([a.sub.0]/a.sub.f).sup.(F - 1)] (24)

The failure crack nucleation size used for this study is [a.sub.f] = 1 mm. Therefore, for the initial flaw size in the range observed by Lake and Lindley, it is estimated that the observed initiation life is in the range of 94-98% of the life if crack grew in an infinitely wide specimen. Thus, the crack initiation results obtained are nearly geometry independent.

CONCLUSIONS

A key ingredient of fatigue analysis and life prediction of elastomeric components is relevant material properties. These properties include deformation behavior under different stress states, crack initiation life, and crack growth rate properties. Specific experimental procedures and data analysis techniques were presented and discussed to obtain each of these material characterizations.

Two simple specimen geometries can he used for deformation and fatigue behavior characterizations of elastomers: simple tension and planar tension specimens. In the simple tension specimen a uniaxial state of stress with a multiaxial stretch state is present, while the planar tension specimen is under plane stress condition with longitudinal and transverse stresses. As monotonic and cyclic deformation behaviors can be vastly different, it is important to obtain and use cyclic deformation properties in fatigue life analysis and applications. Cyclic incremental tests on simple tension and/or planar tension specimen can be performed to obtain the stabilized cyclic stress-strain curve. The choice of the curve to use in FE simulations depends on the stress state at the critical location of the component being analyzed.

Because of the load history dependence associated with the Mullin's effect, peak stress (or strain) should be a key consideration in fatigue analysis of elastomers, in addition to the stress (or strain) amplitude and the mean stress. Fatigue crack initiation tests can be performed with the simple tension specimen geometry in displacement control and with different strain ratios ([R.sub.E] ratios) representing tension-tension loading conditions. Best fit lines in log-log scale, where the fatigue life is treated as the dependent variable, represent maximum strain versus crack nucleation life for each strain ratio. The effect of [R.sub.E] > 0 is to increase the nucleation life and is significant at low strains. This is in contrast to metals and this effect in natural rubber can be attributed to strain crystallization.

Energy release rate is often used as the crack driving parameter in characterizing elastomers fatigue crack growth behavior. A precracked planar tension specimen is typically used for fatigue crack growth tests since energy release rate is independent of crack length for this specimen geometry. Therefore, a single specimen can produce results for multiple crack growth tests with different loading conditions and [R.sub.T] ratios. A power-law relation can be used to describe crack growth rates in terms of the maximum energy release rate. The crack growth approach could be used as a total life approach, based on growth of pre-existing flaws to failure.

The Mars-Fatemi model can be used to correlate test results from different R ratio conditions. This model can be used to obtain an equivalent maximum strain in crack initiation tests, or an equivalent maximum energy release rate in crack growth tests, to account for the effect of R ratio.

ACKNOWLEDGMENTS

PAULSTRA provided the molded rubber sheet material for the study.

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Correspondence to: Ali Fatemi; e-mail: afatemi@eng.utoledo.edu

Contract grant sponsor: Chrysler Group LLC.

DOI 10.1002/pen.23125

Published online in Wiley Online Library (wileyonlinelibrary.com).

[C] 2012 Society of Plastics Engineers

Touhid Zarrin-Ghalami, All Fatemi

Mechanical, Industrial and Manufacturing Engineering Department, The University of Toledo, Toledo, Ohio 43606
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Author:Zarrin-Ghalami, Touhid; Fatemi, Ali
Publication:Polymer Engineering and Science
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Date:Aug 1, 2012
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