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Fatigue behavior and modeling of thermoplastics including temperature and mean stress effects.

INTRODUCTION

Thermoplastics presenl many advantages in terms of applications in many industries, including automotive and aerospace. Many components made of these materials are subjected to cyclic loading and. therefore, prone to fatigue failure as the most common mechanical failure mode of structural components. While well-established models for fatigue behavior of metallic alloys exist, their applicability to the modeling of polymeric materials is uncertain. The objective of this study was to evaluale and model the fatigue behavior of Iwo thermoplastic polymers.

Specimens were subjected to stress-controlled fatigue tests. The effects of testing temperature, mean stress, cycling frequency, cutting direction with respect to the mold flow, and specimen thickness were investigated. Based on incremental step cyclic tests cyclic stress-strain curves were also developed. This allowed for the calculation of strain-life fatigue properties and strain-life fatigue curves as well. Prediction of S-N fatigue properties based on tensile strength was explored and compared with existing predictions used for steels. Various mean stress parameters. some of which are commonly used for metallic materials, were filled to the experimental data to evaluate their applicability for polymeric materials. The effect of ratcheting or cyclic creep due to the applied mean stress in fatigue tests was also examined.

In this article, first a brief literature review is presented. followed by description of the experimental procedures and materials tested for this investigation. Then experimenta) data and properties from the fatigue tests are presented, including models and lits which account for the effect of mean stress. Last, conclusions from the findings of this investigation are presented. Tensile and creep behaviors and modeling of the thermoplastics used in this study are the subjeci of another publication [l]. It should be mentioned that the approach used for fatigue analysis in this study is a macro-mechanics approach and based on crack initiation or nucleation.

Brief Literature Review

Some general information on thermoplastic polymers and their mechanical behavior characteristics is presented in (1). In this brief review some of the previous studies on the effects considered in this work with regard to fatigue testing of thermoplastic polymers are presented. Widely recognized polymer fatigue testing standards have not been available until recently (2). As a result, fatigue test specimen geometry, manufacturing, and loading were found to vary significantly between the studies in the literature.

In a study of fatigue behavior several thermoplastic by Janssen et al. (3) specimens were manufactured by injection molding. They annealed the polycarbonate samples for 72 h at 120[degrees]C to mitigate stress accelerated physical ageing during testing. Constable et al. (4) conducted fatigue testing on PMMA and PVC extruded rod and cast sheet in cantilever bending. Zhou and Mailick (5) used a 40 wt% talc-filled polypropylene homo-polymer made into plates by injection molding and produced dog-boneshaped specimens from piales in iwo directions, parallel to and normal to the flow direction. The results showed an increase in fatigue life for longitudinal specimens at room temperature.

Li et al. (6) conducted fatigue tests of polycarbonate under load control in tension-tension with a constant test frequency of 5 Hz at 55[degrees]C. Measurements of the sample temperature were made using a thermocouple glued to the surface of the specimens and volume changes as a function of applied cycles were measured using strain gages mounted parallel and perpendicular to the applied load. From the strain gage readings, when the stress was zero, they determined the cyclic residual volume as a function of the number of cycles to determine the change in cyclic modulus. The results indicated significant softening of the material throughout testing.

Many studies have been conducted to investigate the effects of mean stress in fatigue testing of metals. Compared with fully reversed fatigue tests, a tensile mean stress generally has a negative effect on fatigue life. One of the predominant equations used in determining fatigue strength with mean stress used for metals is the Modified Goodman equation (7). given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [S.sub.a] is the stress amplitude. [S.sub.m] is the mean stress,,[S.sub.u] is the ultimate tensile strength, and [S.sub.Nf] is the fully reversed fatigue strength ai [N.sub.f] cycles. A similar equation to the Modiiied Goodman equation is the Soderberg equation. where the yield strength is used instead of ultimate tensile strength.

Another equation is the stress version of the Smith. Watson, and Topper (SWT) equation for stress-life analysis (8):

[S.sub.[N.sub.t]] = sqrt[S.sub.a]([S.sub.a] + [S.sub.m]) (2)

Walker equation (9) represents another model of mean stress effect for metals, given by:

[S.sub.[N.sub.t]] = [([S.sub.a]+[S.sub.m]).sup.1-[gamma]][([S.sub.a]).sup.[gamma]] (3)

In this equation, [gamma] is a material dependent mean stress sensitivity factor. In the special case where [gamma] = 0.5, the Walker equation reverts to the SWT equation.

In a study (10) of polystyrene under cyclic loading, when the mean stress was increased while holding the stress amplitude constant, the fatigue life was reduced. If the mean stress was increased during testing while holding a constant maximum stress, the fatigue resistance actually improved. For high mean stress values, crack initiation occurred at several closely spaced crazes and the smooth slow-growth fracture region surrounding the source changed its character to a step-like appearance. In a study of fatigue crack growth rate in polycarbonate, James et al. (11) evaluated stresses ahead of a crack tip and took account of the elastic-plastic boundary stresses induced by the presence of the crazed region that surrounds a fatigue crack.

The decrease in fatigue life caused by mean stress may not only be due to fatigue, but also due to creep and ratcheting, particularly at high mean stress and/or at elevated temperatures. Crawford and Benham (12) performed a study relating the effects of mean stress and creep rupture for an acetal copolymer at room temperature. They concluded that mean stress is significantly more important in polymers than in metals due to the fact the polymers are susceptible to creep at much lower temperatures. Using the creep rupture strength ([S.sub.c]) at the lime equivalent to the applied fatigue cycles, a relationship between mean stress and alternating stress was proposed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

This model is a modified version of the Goodman-Boiler mean stress model (13), which has a form similar to the Modified Goodman equation (1), bui using creep rupture strength as opposed to the ultimate tensile strength. Equation (4) was applied to experimental dala at room temperature with stress ratios. R = [S.sub.min]/[S.sub.max], of 1 to 0.6 at low frequencies and resulted in good predictions of the effect of mean stress. Operating at low frequencies reduced the self-healing effect and the effects of creep damage. Increasing the frequency to the point where temperature rise did not stabilize resulted in thermal induced failure that deviated significantly from the model.

In a recent experimental study by Shariati et al. (14), the effect of mean stress in fatigue tesis of polyacetal was investigated using both the stress-based and strainbased approaches. Their study also incorporated the effect of ratcheting in tests with positive stress ratios. The model applied in their study was developed from a unified approach for mean stress proposed by Kujawski and Ellyin (15). It is based on the assumption that fatigue damage is caused by strain energy rate. A simple power law for the mean stress effect was then expressed (14),(15) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where n and [eta] are material constants. In their study, the specific case of n = 1 was explored, which reduces Eq. 5 to the following form:

[S.sub.[N.sub.t]] = [S.sub.a]+[eta][S.sub.m] (6)

The model fit to the experimental data for a variety of positive and negative stress ratios showed that [eta] = 0.655 best fits the experimental data.

Fully reversed fatigue tests (R = -1) at elevated temperatures may be difficuli to conduct for polymers due to the fact that specimens may buckle in compression due to softening. Therefore, R ratios greater than zero (i.e., tension-tension tests) are generally used. Due to the tensile mean stress in this case, however, the material may experience damage due not only to fatigue, but also due to the effects of creep or ratcheting. Son si no and Moosbrugger (16) evaluated the reduction in fatigue resistance due to elevated temperatures for polyamide (nylon). In an experimental study of temperature effects on fatigue life, Furue and Shimamura (17) conducted alternating bending tests at room and elevated (50[degrees]C) temperatures for several thermoplastics. A significant decrease in fatigue life was observed for all of the materials tested at elevated temperatures. Two main parameters evaluated were the temperature rise in the specimens and the decrease in rigidity. They suggested that thermoplastics could be categorized into several groups based on their reduction in rigidity and self-healing during testing.

Fatigue behavior of polymers includes a broad range of other topics, for example, transient cyclic deformation response during cyclic loading. Ayoub et al. (18) developed a visco-hyperelastic model to capture the time-dependent mechanical response of elastomeric materials under cyclic loading. The model was shown to adequately capture the important features of the observed siressstrain curves in SBR rubber. Drozdov (19) developed constitutive equations for cyclic viscoelastoplasticity of polymer composites, which were shown to correctly predict creep test dala and dependencies of maximum and minimum strains per cycle on number of cycles in fatigue tests. These aspects, however, are out of the scope of the present study.

EXPERIMENTAL PROGRAM AND PROCEDURES

This section discusses the experimental program and procedures used in this study. One of the materials used was an impact polypropylene copolymer, here referred to as PP, with a melting point of about 170[degrees]C. Common applications of this material in automobiles include interior trim, exterior trim, and large thin-walled parts. This material was used in ils pure, unreinforced (neat) form. Another material used was a compounded, here referred to as PO. This material is designed for automotive applications that require energy management combined with ductility, stiffness, and impact resistance over a broad temperature range. This material is considered to be a polypropylene-elastomer blend where the elastomers make up about 25% of the material by weight and contribute to elasticity. It is reinforced with magnesium silicate (talc) mineral, which makes up aboui 30% of the material by weight to improve stiffness. Carbon black is also present and makes up about 1% of the material by weight. The melting point of this material is greater than 120[degrees]C. Compared to the neat polypropylene material tested, the PO is considerably softer and exhibits a slippery, opaque surface finish. Additional details for the iwo materials used can be found in (1).

An optimized specimen geometry was designed for usage in fatigue tests, as shown in Fig. 1. Both materials were injection molded into 20 cm [times] 10 cm [times] 3.8 mm rectangular plaques. The material plaques were then first cut into I inch wide strips in both the transverse and lon-gitudinal directions, relative to the mold (low direction. Specimens were then machined using a CNC milling machine. Excess material was removed with line grit sandpaper and all corners were smoothed without damaging the surface finish.

Fatigue tests were performed with both materials to investigate the effects of mold llow direction, testing temperature, R ratio or mean stress, testing frequency, and specimen thickness. Incremental step cyclic deformation tests were also conducted with both materials under the R = -1 and R = 0.1 stress ratios to generate cyclic stress-strain curves, to develop a relationship between the stress amplitude in load-controlled tests and the slrain amplitude in the specimen gage section.

Fatigue tests were performed using a servo-hydraulic testing machine. Hydraulic grips raled at 30 kN were used for the majority of room temperature lests, while 5 kN pneumatic grips were used for high and low temperatures. Prior to testing, the load train components were aligned using a strain-gaged specimen. For cyclic deformation lests, a mechanical extensometer was used for strain measurement. Cold and elevaled temperature lests were performed in an environmental chamber. This chamber employs an electronic heating element and a liquid nitrogen cooling system. To observe the temperature rise and profile throughout the specimen gage seclion during testing at room temperature, a thermal imaging camera was used.

Although a new ASTM tesi standard for evaluation of fatigue resistance of plastic materials subjected to uniaxial stress now exists (2), this standard was nol used since it was not yet available at the time of this study. This new ASTM standard applies to uniaxial fatigue tests where the strains and stresses are relatively elastic. It addresses specimen depth, tesi frequency, and various specimen preparation techniques. At present, there is no ISO equivalent to this standard.

Incremental step tests were performed with both materials at room temperature and 85[degrees]C for stress ratios of R = -1 and R = 0.1. These tests were conducted with a mechanical extensometer under load control to determine the strain amplitude behavior at various stress amplitude levels. The stress amplitude and stable strain amplitude values are then plotted to develop the cyclic stress-strain curves. In addition to evaluation of cyclic softening, the cyclic stress-strain curve allows for relating the stress amplitude in load-controlled fatigue tests to sirain amplitude in the gage section to allow construction of strain-life curves.

Fatigue tests were performed at room, 85[degrees]C. and -40[degrees]C temperatures. In order to shorten the grip length and avoid buckling during fully reversed tests, specimens were gripped into the transition section at a consistent location (see Fig. 1) where the width of the specimen is 15 mm. Failure was defined as the number of cycles applied until the specimen fractured. Stress amplitudes were chosen to produce laligue lives between [10.sup.3] and [10.sup.6] cycles.

To study the effect of mean stress, in addition to fully reversed (R = - 1 ) tests, fatigue tests were also performed with a tensile mean stress with R =0.1 and R = 0.3, These tests were performed at room temperature. 85[degrees]C and -40[degrees]C. To evaluate thickness effect, specimen thicknesses of 3.0 mm and 3.8 mm were used. To evaluate the effect of cycling frequency, limited tests were conducted for given stress levels at several frequencies at room temperature. Except in tests where frequency effect was evaluated. frequency of cycling for room temperature tests was chosen such that the temperature rise during the lest was less than 8[degrees]C for both materials. The maximum temperature rise due to self-heating was monitored and recorded during most fatigue tests at room temperature. The displacement amplitude and displacement mean of the loading actuator were measured throughout every fatigue tesi to observe the softening of the materials with cycling.

FATIGUE TESTS RESULTS AND DISCUSSION

This section presents and discusses the results and analysis of the load-controlled fatigue tests performed, as detailed in the experimental program. Test data were lilted with a line on log-log scale to develop stress versus cycles to failure (S-N) curves in the form represented by the Basquin equation:

[S.sub.a] = A[([N.sub.t]).sup.B] (7)

where,[S.sub.a] is the stress amplitude, A is the intercept at [N.sub.f] = 1, B is she slope of the lit, and [N.sub.f] represents the cycles to failure. Run-oul test data or data at stress levels with a run-oul test were not included in the lits. A summary of the tesi conditions and stress-life fatigue properties with intercept A and slope ti in Eq. 7 is reported in Table 1 for both materials.

Mold Flow Direction Effect

One effect considered was that of mold flow direction, or the culling direction of the test specimens, relative to the injection mold flow. Specimens were machined in directions longitudinal and transverse to the mold flow direction. Due to the isotropy of both materials, the effect of mold flow was minimal for both materials tested at room temperature and at 85[degrees]C under the fully reversed (R = - 1) and R = 0.1 conditions. Displacement amplitude versus cycles also indicated similarities in cyclic softening throughout the test for both cutting directions for each material at each temperature. The mid-life hysteresis loops were also nearly identical. Therefore, the majority of fatigue tests were performed for specimens cut in the transverse direction due to the lack of evidence of a mold flow direction effect.

Thickness Effect

To evaluate thickness effect, specimens for both materials were tested with thicknesses of 3.8 and 3.0 mm at room temperature. In order to avoid buckling of the thinner test specimens, the tests were performed under the R = 0.1 condition, except for some limited additional tests for PP under the R = -1 condition at stress levels that did not result in specimen buckling. Based on the experimental data obtained, thickness in the range evaluated was not observed to affect the fatigue behavior of either material. Displacement amplitude and displacement mean versus cycles and versus normalized cycles as well as the mid-life stress versus displacement hysteresis loops for the two thicknesses were also identical for each material and at each R ratio. Therefore, most of the tests for each material were performed with the 3.8 mm thickness specimens.

TABLE 1. Summary of stress-life (S-N) fatigue properties of PO and PP.

Thickness  Flow      Temp.      Stress        A              [S.sub.f]
(mm)       Dir.     [degrees]C)     Ratio (R)     (MPa)    B      (MPa)

PO

3,8        T        Room         -1           19.15   -0.067    7.63

3.8        L        Room         -1           22.79   -0.084    7.18

3.8        T        Room          0.1         10.53   -0.073    3.82

3.8        T        Room          0.3          6.98   -0.053    3.35

3.8        T          85          0.1          2.82   -0.06     1.22

3.8        L          85          0.1          2.69   -0.052    1.31

3.8        T         -40          0.1         20.38   -0.042   11.41

3.8        T         -40          0.3         13.84   -0.026    9.61

3.0        T        Room          0.1          9.68   -0.064    3.99

PP

3.8        T        Room         -1           29.52   -0.059   13.07

3.8        L        Room         -1           28.84   -0.058   12.98

3.8        T        Room          0.1         12.8    -0.041    7.3

3.8        L        Room          0.1         14.71   -0.054    6.96

3.8        T        Room          0.3          9.48   -0.038    5.58

3.8        T          85         -1           16.84   -0.069    6.49

3.8        T          85          0.1          5.94   -0.051    2.92

3.8        L          85          0.1           7.81  -0.078    2.59

3.8        T         -40          0.1          24.2   -0.035   14.88

3.8        T         -40          0.3          17.96  -0.029   12.05

3.0        T        Room         -1            24.72  -0.041   13.99

3.0        T        Room          0.1          14.93  -0.054   7.09

[S.sub.a] = A[([N.sub.f]).sup.B] and [S.sub.f] is defined at
[N.sub.f]= [10.sup.6].


Cycling Frequency Effect

High testing frequency can result in material sell'-heating which can lead to cyclic softening and premature failure. Specimens were tested at relatively low frequencies at room temperature in fatigue testing to avoid this effect. To verify that small changes in frequency did not signilicantly affect fatigue test results, some tests were performed at one half and double the frequency of the baseline tests. The increase in the specimen surface temperature was monitored throughout test using the thermal imaging camera. For this evaluation, the PO material was tested under the R = -1 condition and the PP material was tested under the R = 0.1 condition. Some additional tests were performed at frequencies three to four times that of the baseline tests.

Fatigue lives for different frequencies for both materials are shown in a bar graph form in Fig. 2. Running lests at one half or double the baseline frequencies show no effect on the fatigue life for either material. The rise in temperature at double the frequency was slightly higher ([Tilde]6[degrees]C) than the observed temperature rise in the baseline tests (3[degrees]C). However, both materials experienced significantly shorter fatigue lives at a frequency three to four times the baseline tests, as shown in Fig. 2. Recorded temperature rise of the tests at 12 Hz for the R = 1 condition for both materials had no appareilt indication of stabilization, resulting in temperature increases in excess of 30[degrees]C and premature failure. Additional experimental data for the effect of temperature ris due to self-heating during cyclic loading at different frequencies and mathematical modeling of this effect as a function of the applied stress and frequency is the subject of another publication.

Test Temperature Effect

To evaluate the effect of testing temperature, both materials were tested at 85[degrees]C and -40[degrees]C. Due to softening of the PO material at elevated temperatures, tests were only conducted under the R = 0.1 condition for this material at 85[degrees]C to avoid buckling.

Figures 3 and 4 show the temperature effect at different R ratios for PO and PP. respectively. At R = 0.1, stress levels used to produce similar lives to the room temperature tests were about three times lower at 85[degrees]C and about 2.5 times higher at -40[degrees]C for both materials (see Figs. 3a and 4b). Under the R = 0.3 condition, stress levels at -40[degrees]C used to produce similar lives to the room temperature tests were about two times higher for both materials, as shown in Fig. 3b for PO and Fig. 4c for PP. Under the R = - 1 condition at 85[degrees]C for PP, four orders of magnitude reduction in fatigue life was observed, as compared to at room temperature, see Fig. 4a. These results indicate the drastic effect of testing temperature on fatigue behavior of both materials.

CYCLIC STRESS-STRAIN AND STRAIN-LIFE FATIGUE PROPERTIES AND CURVES

Incremental step tests at selected stress ampliludes were used in load-conlrolled lests to obtain the stable strain amplitudes. The tests were performed in blocks of increasing stress level. with sufficient number of cycles at each stress level for the strain amplitude to stabilize. From these tests, the cycl ic elastic modulus ([S.sub.a]) the 0.2% offset cyclic yield strength ([S.sub.'.sub.y]). the cyclic strength coeffcient ([E.sup.']). and the cyclic strain hardening exponent ([n.sup.']) were obtained. These values for each test condition are included in Table 2. Fits of the true stress versus true plastic cyclic strajn for obtaining [K.sup.'] and [n.sup.'] are shown in Fig. 5.

TABLE 2. Summary of strnin-life lallgue propcr1ics for PO and PP at
room temperature for R = 1 tests in the longitudinal direction.

          [E.sup.']  [[S].sub.y.sup.']  [K.sub.']
Material  (MPaJ)     (MPa)              (MPa)

PO             2762              10.04      34.82

pp             1744              15.12      51.23

                    [[sigma].sub.f.sup.']
Material  n.sub.']  (MPa)                  [[Sigma].sub.f.sup.']

PO           0.199                   20.5                  0.062

pp           0.193                  30.75                  0.071

Material       b      c   2[N.sub.t]

PO        -0.067  -0.335       3,026

pp        -0.059  -0.306         281


Stable cyclic stress-strain curves and data are superimposed with the experimental monotonic curve of each material in Fig. 6. The R = - 1 tests indicate little cyclic softening, whereas the R = 0.1 tests exhibit softening due to the maximum stress being much higher than the stress amplitude. A Ramberg-Osgood type curve is used to mathematically model the cyclic stress-strain curve of both materials in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Using the cyclic stress-strain equation to determine strain amplitudes based on the stress amplitudes from load-controlled fatigue lests, strain-life fatigue curves were developed for fully reversed condition (R = 1). The strain-life faligue curve is given in the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where the first term represents elastic deformation and the second term represents plastic deformation. The fatigue strength coefficient ([[sigma].sub.f.sup.']) and exponent (b) are determined with a power fit of the true stress amplitude ([[sigma].sub.a]) versus reversais to failure (22[N.sub.f]) data, represented by:

[sigma.sub.a] = [[sigma].sub.f.sup.'](2[N.sub.f])(10)

This is analogous to Eq. 7. In Eq. 10. the fatigue strength coefficient ([[sigma].sub.f.sup.']) is related to A in Eq. 7 by A = [[sigma].sub.f.sup.'][[(2)].sup.h]ft. In terms of the relalion between slopes h and B. these fitting constants are identical (B = b).

Plastic strain amplitude for each test is calculated from:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Plastic strain amplitude versus reversals to failure data power tit then yields the fatigue ductility coefficient ([[Sigma].sub.f.sup.']) and exponent (c) properties. represented by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

The strain-life fatigue properties for both materials are included in Table 2. Figure 7 shows the strain amplitude versus reversals to failure curves for both materials at room temperature. Data points for the fully reversed fatigue tests with strain amplitudes calculated from stress amplitudes from Eq. 8 are also superimposed on these curves. The elastic and plastic curves and data are also shown. The intersection of the elastic and plastic curves is called the transition fatigue life (2[N.sub.t]) and is calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

For lives less than 2[N.sub.t], the deformation is mainly plastic, whereas for lives greater than 2[N.sub.t], the deformation is mainly elastic. The PO material experienced this transition after about 3000 reversals which is significantly longer than for the PP material, with a transition fatigue life of about 300 reversals. This indicates that PO experienced a higher degree of plastic deformation compared to PP during most of the fatigue tests.

PREDICTION OF FATIGUE BEHAVIOR FROM TENSILE PROPERTIES

A widely used method to predict fully reversed (R = - 1 ) S-N fatigue behavior for metals based on tensile properties, in the absence of experimental fatigue data, was evaluated for application to polymers. For steels, the S-N line is often approximated by connecting the tensile strength ([S.sub.u]) at one cycle or 0.9i[S.sub.u] at [10.sub.3] cycles to 0.5i[S.sub.u] at [10.sub.6] cycles with a straight line in a log-log plot (20). To evaluate these approximations for thermoplastics, coefficient A in Eq. I. fatigue strength at [10.sub.3] cycles ([S.sub.1000]), and faligue strength at [10.sub.6] cycles ([S.sub.f]) are plotted versus 5U in Fig. S for the iwo materials tested in both directions and at the three temperatures. These correlations suggest A = 1.25([S.sub.u]), ([S.sub.1000]) = 0.8 ([S.sub.u]), and [S.sub.f] = 0.5([S.sub.u]). Therefore, analogous to the aforementioned estimations for metallic materials, these predictions of S-N line properties for polymers are not unreasonable. The prediction value of the intercept (A) at 1.25([S.sub.u]) is higher than the value of [S.sub.u] proposed for steels, while the predicted value of [S.sub.1000] at 0.8([S.sub.u]) is slightly lower than the 0.9([S.sub.u]) predicted for steels. The fatigue strength at [10.sub.6] cycles ([S.sub.f]) is identical to that of 0.5([S.sub.u]) for steels.

MEAN STRESS EFFECT AND MODELING

The effect of mean stress on the fatigue life of both PO and PP was evaluated at room temperature, 85[degrees]C. and -40 [degrees]C. The R = 0.1 ratio produces a mean stress which is 122% of the stress amplitude (i.e., [S.sub.m] = 1.225[S.sub.a]) and the R = 0.3 ratio produces a mean stress which is 186% of the stress amplitude (i.e. [S.sub.m] = 1.86[S.sub.a].

Figure 9 shows the effect of tensile mean stress for PO and PP at room temperature. As can be seen from this figure. both materials exhibit a significant reduction in faligue life in comparison to the fully reversed tests. Figure 10 shows the effect of mean stress or stress ratio at -40[degrees]C for both materials and Fig. 11 shows the effect of mean slress for PP at 85[degrees]C.

As discussed in the literature review section, various models have been proposed for quantifying the effect of mean stress for metallic materials and the applicability of several of these models to polymeric materials has been explored in some experimental studies. In this study, several mean stress correction models were evaluated. For each model, an equivalent fully reversed stress amplitude ([S.sub.Nf]) is computed, which is then used in Eq. 7 to determine the fatigue life.

The Goodman-Boiler mean stress model (13) which uses creep rupture strength ([S.sub.c]) instead of ultimate tensile strength in Modified Goodman Eq. 1, as well as the Crawford and Benham model (12) given by Eq. 4, were used with both material mean stress data at room and 85[degrees]C temperatures. Neither model provided satisfactory correlations of the mean stress data for either material, Correlations of the mean stress data with the Goodman-Boiler mean stress model are shown in part (a) of Figs. 12 thru 16 for each material and test temperature.

The Modified Goodman Eq. 1 was also applied to the mean stress data of both materials and this common model did not provide good correlations of the mean stress data either. Soderberg equation, which is the same as the Modified Goodman equation but replacing tensile strength by yield strength ([S.sub.y]), resulted in even less accurate correlations than the Modified Goodman model.

Another form of the Modified Goodman mean stress parameter was also used, which contains a correction factor exponent ([alpha]) for the mean stress term, given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The value of experimentally obtained correction factor exponent ([alpha]) was about 0.75 for the two materials and three temperatures. However, although this equation provided a better correlation to the experimental data compared to the Modified Goodman equation, the correlations obtained were not as good as those obtained from Eqs. 3 or 6 discussed next. A value of [alpha] = 2 in Eq. 14 results in a parabola, referred to as the Gerber parabola, sometimes used for metallic materials (20).

A simple model is the maximum stress, where [S.sub.max] = [S.sub.m] + [S.sub.a]. This model generally over-predicted (i.e., nonconservative) the effect of mean stress at room temperature, but showed a reasonable correlation at -40[degrees]C for both materials. A variation of this model was also used, as presented by Eq. 6, where [eta] is the mean stress sensitivity factor determined by the best line fit to the fully reversed data. The results of this model are shown in part (b) of Figs. 12 thru 16 for both materials at the three testing temperatures. As can be seen from these figures, this mean stress model yielded very good results for both materials and at all three testing temperatures.

The Walker model with the mean stress sensitivity factor ([gamma]) given by Eq. 3 was another mean stress model applied to the data in this study. The correlation results of this model are shown in part (c) of Figs. 12 thru 16. This model predicts the effect of mean stress very well for both materials at all three testing temperatures. A constant value of [gamma] = 0.5 in the Walker equation results in the SWT parameter given by Eq. 2. However, the SWT parameter did not yield predictions that correlated well with the experimental data for either material and at any of the test temperatures.

Due to the fact that the [eta] mean stress sensitivity factor and the [gamma] value in the Walker equation vary with material and temperature, it was attempted to predict these values as a function of tensile strength ([S.sub.u]), as shown in Fig. 17. With the exception of PP at 85[degrees]C, all other cases of [eta] and [gamma] values appear to have a relatively strong correlation with [S.sub.u]. at least for the room temperature and -40[degrees]C data. These correlations are represented by:

[eta] = 3.1 X [10.sup.-3] ([S.sub.u]) + 0.71 ([R.sup.2] = 0.77) (15)

[gamma] = -2.2 X [10.sup.-3] ([S.sub.u]) + 0.19 ([R.sup.2] = 0.86) (16)

where [S.sub.u] is in MPa.

An issue mentioned earlier in connection with tensile mean stress is cyclic ratcheting or creep, which can cause significant additional damage to fatigue damage. This damage increases with increasing the R ratio) and/or the temperature. To quantify the contribution of creep damage in fatigue tests, a simple linear damage rule proposed by Manson and Halford (21) which combines creep damage and fatigue damage in a linear fashion was used, given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

In this equation, the applied cycles (N) as a fraction of cycles to failure ([N.sub.f]) in a fatigue test is added to the time of the test (t) as a fraction of the time to rupture from a creep test ([t.sub.R]) for an applied stress equal to the mean stress of the given fatigue test. According to this model, failure occurs when the sum of the damage becomes equal to one.

Using stress versus time to creep rupture data and fits reported for the two materials in this invesiigation reported in (1). the fraction of creep damage in each mean stress fatigue test was computed based on Eq. 17. The damage due to creep was found to be negligible (less than 2% of the total damage) at room temperature. At 85[degrees]C, damage due to creep was more significant and ranged between about 10 and 30% for both materials. As a general trend, the longer life fatigue tests experienced the most damage due to creep, in spite of the lower mean stress, due to the longer duration of the test. A shortcoming of Eq. 17 is that it determines creep damage and fatigue damage by linear addition of the two. The combination of creep and fatigue damage may exhibit synergistic behavior, where the sum of creep and fatigue damage is more than that predicted by this equation.

SUMMARY AND CONCLUSIONS

This study investigated fatigue behavior of two thermoplastic polymers under a variety of testing conditions, Load-controlled fatigue tests were performed at three different temperatures (room, 85[degrees]C, and -40[degrees]C), and in two directions. The effects of specimen thickness, testing frequency and stress ratio or mean stress were evaluated. An empirical model for prediction of fatigue behavior from tensile strength was developed. Several mean stress models were evaluated for their ability to account for the mean stress effect on fatigue behavior.

On the basis of the experimental results and the analysis conducted, the following conclusions can be made:

1. The effect of mold flow direction as well as specimen thickness in the range evaluated on fatigue behavior was found to be minimal for both materials.

2. Test frequencies in fatigue tests producing temperature rise in excess of about 9[degrees]C for either material resulted in significantly shorter lives due to self-healing.

3. The effect of test temperature in fatigue was significant for both materials, where the resulting lives in both low and high cycle fatigue were significantly higher at -40[degrees]C and significantly lower at 85[degrees]C, compared to room temperature.

4. S-N fatigue properties including the S-N line intercept coefficient, fatigue strength at [10.sup.6] cycles, and fatigue strength at [10.sup.3] cycles showed proportional relationships with ultimale tensile strength of both materials. The intercept can be approximated at 1.25([S.sub.u]), fatigue strength at [10.sup.3] cycles can be estimated at 0.8([S.sub.u]), and fatigue strength at [10.sup.6] cycles can be estimated at 0.5([S.sub.u]).

5. Incremental step cyclic deformation tests can be used to determine the cyclic stress-strain curve, which in turn could be used to determine strain-life fatigue properties and curves from the S-N curves. A Rain-herg-Osgood type relaiion accurately represented the cyclic stress-strain curves of both materials.

6. The effect of mean Stress or stress ratio was significant for both materials. Several mean stress parameters were evaluated to model this effect. Commonly used Modified Goodman and Goodman-Boiler equations were not able to correlate the mean stress daia for the two thermoplastics. Two models that fitted the mean stress data best for both materials were the Walker model with a correction factor [gamma] and another model with a mean stress sensitivity factor ([eta]). Both [gamma] and [eta] were found to correlate with ultimate tensile strength.

7. A contribution to damage in fatigue tests at 85[degrees]C for both materials was due to creep damage, which became more significant at longer faligue lives. Creep damage in the mean stress tests conducted was insignificant at room temperature for both materials.

NOMENCLATURE

A                     Fatigue strength coefficient

B, b                  Fatigue strength coefficient

c                     Fatigue ductility exponent

E'                    Cyclic modulus of elasticity

K'                    Cyclic strength coefficient

n'                    Cyclic strain hardening exponent

N                     Number of cycles

[N.sub.f]             Cycles to failure

[N.sub.t]             Transition faligue life

R                     Stress ralio

[S.sub.1000]          Fatigue strength at [10.sub.3] cycles to failure

[S.sub.a]             Alternaiing stress

[S.sub.c]             Creep rupture strength

[S.sub.e]             Fatigue strength at [10.sub.6] cycles to failure


[S.sub.eq]            Equivalent stress

[S.sub.m]             Mean stress

[S.sub.max]           Maximum stress

[S.sub.min]           Minimum stress

[S.sub.[N.sub.f]]     Fully reversed fatigue strength

                       at [N.sub.f] cycles
[S.sup.'[.sub.f]]     0.2% offset cyclic yield strength

[S.sub.u]             Ultimate tensile strength

t                     Time


[t.sub.r]             Time to rupture in creep test

[gamma]               Walker equation mean stress correction factor

[[epsilon].sub.a]     True strain amplitude

[[epsilon]'[.sub.f]]  Faligue duclility coefficient

[[epsilon].sub.p]     Plastic strain

[[sigma].sub.a]       True stress amplitude

[[sigma]'.sub.a] Fatigue strength coefficient


ACKNOWLEDGMENTS

Technical assistance of Dr. A.K. Khosrovenh and Mr. Charles Buchler of General Motors is appreciated.

REFERENCES

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Stephen R. Mellott, Ali Fatemi

Mechanical, Industrial and Manufacturing Engineering Department, The University of Toledo, Toledo, Ohio 43606

Correspondence to: A. Fatemi; e-mail: afatemi@eng.ulolcdo.edu Contract gram sponsor: General Mulors in Warren, Michigan.

DOI 10.1002/pen.23591

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

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Author:Mellott, Stephen R.; Fatemi, Ali
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Date:Mar 1, 2014
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