A simple fatigue methodology for filled natural rubbers under positive and negative R ratios

This paper suggests a simple methodology to optimize the uniaxial fatigue predictions for filled natural rubbers. In this study, the stress ratio R effect on rubber fatigue life is investigated numerically. For this purpose, a Finite Element model is developed to determine both amplitude and mean stress values in the critical zone of the considered specimen. The suggested model incorporates the R ratio effect using an equivalent stress amplitude. It is proven that the new model can efficiently describe the fatigue life for a large R -value interval with higher accuracy.


Introduction
Elastomer parts are frequently employed in industrial functions related to safety and anti-vibration.Mechanical strength is a crucial condition for ensuring that these products function properly in a dynamic operating atmosphere over their entire service life.Adequate prediction of an item's service life presupposes the relevance of the calculation model used at the product design stage.
Notwithstanding the strong industry need for rubber parts, this topic remains poorly discussed in the literature; there is little published documentation and fewer quantitative predictors.The limited investigations focused mostly on the parameters unifying multiaxial fatigue tests under constant-amplitude loading conditions and zero R-ratios [1][2][3][4][5][6].
With regard to the rupture failure of rubbers under cyclic loading, there are two basic approaches.The first one is the crack nucleation approach, in which fatigue life is considered as the cycles needed to initiate a specific crack size, and, the second one is the crack growth approach, in which fatigue life is defined as the cycles required for a pre-existing crack to enlarge until breaking.
• Stress-based fatigue criteria: Bourchak and Aid [11] used maximum stress and strain energy density as fatigue parameters to calculate the fatigue of polymer hoses.[12] used the maximum principal Cauchy stress as a strong predictive parameter for the fatigue life of rubber.Moreover, Brunac et al [13] put forward a predictive model relying on the Dang Van's stress.Moreover, Li et al [14] studied the fatigue of rubber bridge bearings by means of stress.
The above fatigue predictors are sufficient to unify the fatigue life of rubbers for simple loading cases with a zero R load ratio, but, not sufficient for loading cases with a non-zero (negative or positive) load ratio.
Many researchers studied the R ratio load effect on the fatigue life of rubbers.In fact, Saintier et al [6] examined this effect on the fatigue life of natural rubbers.It is found that for non-relaxing loading conditions (i.e.> R 0 such that R is the load ratio defined by = R S S min max ) and for increasing mean stress S , m the fatigue life increases.For relaxing ( < R 0 ) and increasing mean stress S m loading conditions, the fatigue life decreases.In 2011, Poisson et al [23] looked at the stress-load ratio impact on the fatigue life of the polychloroprene rubber using the Haigh diagram.In their graph presentation, the amplitude and the mean of the Piola-Kirchhoff PK1 principal stress were examined.Le Cam et al [24] in their study explored the various damage mechanisms in natural rubber in terms of the load ratio.The considered mechanical quantity was the principal nominal strain.A heuristic approach is developed considering the R-ratio effect for natural rubbers using an equivalent stress amplitude.which is the contribution of this paper.For this purpose, numerous uniaxial tensile/ compression fatigue tests for a cylindrical dumbbell-type specimen with a set of stress ratios are explored in this study.Finite element analysis (FEA) is performed to determine the mean and amplitude stress values at critical zones.
Then, two models based, respectively, on mean stress S m and stress amplitude S a are examined for a range of stress ratio values.The correlation shows that the model based upon the stress amplitude exhibits the highest R 2 correlation coefficient only for positive and zero -R ratio loads, i.e.R 0.  The S a based model and for loads with ratios < R 0, as well as, for the S m based model whatever the R ratio value, the R 2 correlation coefficient is low.
Therefore, a model based on an equivalent stress amplitude taking into consideration the R ratio is suggested.Finally, the effectiveness of such a model is tested and discussed for the R ratio stress range.A good correlation between experimental and numerical results is observed.

Finite element model
For this dumbbell cylindrical specimen, in the [7], the Mooney-Rivlin constitutive model is used for modelling the hyperelastic behavior of the rubber.The Mooney-Rivlin strain energy function is expressed as the following equation: Where = I j 1,2 are the first and second invariants of the Cauchy-Green right strain tensor, C ij are the material constants evaluated by fitting data from three independent strain states, simple tension, planar tension and     [7].Two finite element models are generated for the considered specimen: a three-dimensional solid model and another axisymmetric model.For the 3D model, a moderately fine mesh with approximately 28,980 hexahedral linear C3D8IH elements is selected.Concerning the axisymmetric model, the sample is meshed with about 1103 CGAX4H twist elements, with bilinear four-node and hybrid with constant pressure.
Figure 2 shows the loads and boundary conditions (BC) applied to both the (a) 3D-specimen and (b) axisymmetric model.The sine displacement is set on the sample's top surface, while the bottom one is entirely encased.
The applied sine wave is described as follows: Where w is frequency equaling 5 Hz, and, d a and d m are, successively, the amplitude and mean displacement defined by the following formulas: Figure 3 illustrates the simulation results of both 2D and 3D models, simulated with ABAQUS software, in terms of the second Piola-Kirchhoff stress tensor and the logarithmic strain tensor LE for the second load case.In addition, the 2D and 3D results show good agreement with each other.This comparison indicates a slight loss of accuracy, but a significant reduction in calculation time when the 3D model is converted to 2D.Consequently, the axisymmetric model is selected for the remainder of the work.
Figure 4 depicts the variation of the applied displacement in time, and their corresponding stress responses at three different R ratio values, > R 0, < R 0 and R null.The measurements are taken in the most critical zone of the specimen.It is obvious that S-responses are proportional to the applied loads.
For every studied case, the amplitude and the mean values for Piola-Kirchhoff's second stress tensor, noted S a and S m respectively, are recorded and listed in table 3.

Constant life haigh diagrams
Haigh diagrams, also referred to as constant-life plots, are graphical representations of a constant-amplitude load for a specified lifetime.These graphs can be plotted in a variety of formats, depending on the parameters defined to identify the constant-amplitude cycle.The fixed amplitude cycle can be selected by using two of the following variables: In this section, the parameters selected to present the Haigh diagrams are S , m S a and R. Iso-duration curves are employed to explore the structural relationships existing between these aforementioned parameters and fatigue life N .m These isolines are generated by the linear least-squares method.The fitting function is a bi-parameter polynomial function x and y, with degree 1, given by the following equation: The fitting degree is measured by the R 2 correlation coefficient.The function's parameters are summarized in table 4.
Iso-duration curves are ascending curves, whereby S m increases with R.Moreover, for a fixed R value (e.g.= R 0 as shown in figure 5(b)), fatigue life decreases as mean stress increases.
Figure 6 is similarly a drawing presenting the N f lifetime in terms of both parameters R and S .a Figure 6(a) is depicts N f versus the two other parameters in a 3D graph, and, figure 6(b) is a contour plot showing iso lines drawn according to the stress ratio R and the stress amplitude S .a The contour graph depicting the isovia curves are obtained by the same method utilized previously.The adapted fitting function has the same form as equation (5).The fitting coefficients are precised in table 5.
It is shown that the iso-curves are decreasing, whereby S a decreases with R.Moreover, for a fixed R value (e.g.= R 0 as shown in figure 6(b)), fatigue life decreases as stress amplitude increases.Figure 7 depicts the axial Haigh stress plot for natural rubber.The x-axis denotes the mean part of the axial component of the second Piola-Kirchhoff stress tensor, and the amplitude part is given on the y-axis.Looking at this chart, the tensile fatigue behavior of natural rubber can be divided into two different zones, which are separated by the = R 0 curve.In the first area (Area A), fatigue life decreases with increasing R ratio, and the the  iso-duration curves tend to be parallel to y-axis.In the second area (Area B), it is reversed, the fatigue life increases with increasing R ratio, and the the iso-duration curves tend to be parallel to x-axis.
The observed N f dispersion in figures 4-6 is linked to the R N f lifetime ratio, reported in table 2, which equals to the ratio / N N .
A custom equation used to adjust N f as a function of S m and S a takes the following form: =q e 4.66 5 and the related correlation coefficient is = R 0.98.

Fatigue lifetime prediction with sophisticated models
It is clear from the preceding that the load ratio has a major effect on the fatigue life of rubber.
To investigate this effect, two predictive models based successively on S a and S m for different intervals of R are proposed.The goodness of fitness is calculated of such models is calculated using the R 2 correlation coefficient.

Fatigue life time models based on S a
Commonly, the damage parameter P and the fatigue lifetime of rubbers N f are related through a power law relationship expressed as follows: Where a and g are material parameters determined by the least-squares method.
Considering the loading cases where R 0  and the amplitude stress S a as damage parameter, the correlations between N f and S a are plotted in figure 8.The equation ( 8) is written as:    For load cases < R 0, the correlation between N f and S m is shown in figure 11, and equation (8) can be put as follows: 7 12

A fatigue life prediction model using an equivalent stress parameter
According to the above analysis, it is noticeable that the stress amplitude model (equation ( 9)) using R 0  is the most suitable among the other described models (equations ( 10)-( 12)) for correlating fatigue life.Indeed, the correlation coefficient of this model is the highest one = R 0.99. 2 To extend this model to load cases with < R 0, a generalized model incorporating the load ratio is still required.The algorithm shown in figure 12 illustrates the suggested method.

Equivalent stress parameter determination S eq
In the context of phenomenological fatigue life prediction, the major objective is to identify the most appropriate damage parameter that offers the best correlation with fatigue life.The study focused on establishing a solid relationship between the equivalent damage parameter and fatigue life.Tao et al [5] investigated three approaches essentially based upon strain, stress and strain energy density.In these three approaches, an extra average stress/strain term was introduced into the equivalent damage parameter to take account of the R effect.In a similar vein, Wang et al [7] considered a strain damage parameter.
In this work, an equivalent stress damage parameter, S , eq expressed as a function of the R ratio, is given by a simple power law given as follows Where h and n are the material parameters and and, assuming For fully reversed fatigue tests, i.e. = - R 1, = S 0, m so equation (13) can be written as: = - S S eq a 1 where the superscript -1 identifies the totally reversed tests.
For = R 0, = = S S S , m a a 0 equation ( 13) is transformed as follows: Using equation (16), any uniaxial fatigue loading condition, described as function of stress amplitude S a and stress ratio R, can be converted into an equivalent stress amplitude S a 0 corresponding to a zero stress ratio.

Proposed fatigue life model and its application
For zero R conditions indicated in table 3, an appropriate fatigue lifetime model can be determined and expressed by a simple power law according to the following formula: 95 17 with R 2 = 0.95 As figure 13 reveals, a high correlation between the damage parameter based on S a 0 and the measured fatigue time, N , f can be observed (R 2 exceeds 0.9).Consequently, equation (17) could be considered as an effective fatigue prediction model for null R conditions.

Equivalent stress amplitude calculation
From figures 8 and 9, it is remarkable that the R < 0 loadings have a major effect on the fatigue life of the material.Table 6 summarizes the N f fatigue lives, the R ratios and the correspondent equivalent stress amplitude S a 0 values calculated from equation (17).
From equation (16) one can write The = a 0.3and = b 2.768 are material parameters determined using the non-linear least-squares method with = R 0.99 2 as mentioned in figure 14.Then, by substituting the values of a and b in equation (18), the expression between S a 0 and S a will be given as follows: The predictive fatigue models based on the equivalent stress amplitude for R < 0 load conditions are defined by relations (17) and (19), which are as follows:  Figure 15 depicts the correlation between experimentally measured and numerically calculated fatigue lives.Figure 15(a).Displays this correlation with the equivalent stress model as suggested, while figure 15(b).Displays this correlation with the equivalent strain model as suggested in [7].
Figure 15 presents the experimental average lifetime values ignoring the scattering of experimental fatigue lives, which is accounted for and presented in figure 16. Figure 16(a).Shows the correlation, taking into account the dispersion of N f fatigue times, obtained with the proposed equivalent stress model, while figure 16(b).exhibits it using the proposed method in [7].
The results of both approaches show good correlations, particularly for load cases with negative and zero ratios (i.e.= R 0 and < R 0. In fact, all estimated service life values, for these conditions, fall within the scatter bands of factor 2. For > R 0, the proposed approach exhibits a lower dispersion than that proposed in [7].Indeed, for the deterministic model, the predicted results in figure 15(a) fall within a dispersion band of 6, and in figure 15(b), the predictions are within dispersion bands of 9. Whereas for the dispersed model, the predicted results in figure 16(a) lie within a dispersion band of 6, and in figure 16(b), the predictions are in scatter bands of 8.
These results prove the effectiveness of the proposed model for all loading cases over the entire R ratio range.

Conclusions
The present paper proposes a simple methodology for determining the effect of the R-ratio on the fatigue life of natural rubbers.The proposed approach needs two adjustable parameters to be obtained before the method can be used.It is based on the second Piola-Kirchhoff stress tensor calculated by means of the FE analysis method.
A range of conclusions can be derived from this work: (i) By correlating the fatigue results with various damage parameters, it is clear that natural rubber fatigue is highly dependent on the R.
(ii) Predictive models based upon the mean and the amplitude of the second Piola-Kirchhoff stress tensor are considered.These two models' effectiveness is measured using the R 2 correlation coefficient.The amplitude model shows the highest R 2 for zero and positive load ratios.while the mean-based predictive model displays a moderately low R 2 throughout the entire R range.
(iii) A predictive fatigue model based upon equivalent stress amplitude is proposed.This suggested model takes into account the R-ratio influence and provides very satisfactory results for all R ratio load values.

Figure 2 .
Figure 2. Mesh, load and BC for the (a) 3D and (b) axisymmetric model.

Figure 3 .
Figure 3. Stress-strain comparison between 3D realistic and 2D axisymmetric models for the second load case ( = d 36 max and = d 0 min

Figure 4 .
Figure 4. Variation over time of 2nd Piola-Kirchhoff stress for three values of R ratio. f

Figure 5 .
Figure 5. Constant life diagrams as function of R ratio and mean stress S m (a) main plot (b) contour plot.

Figure 5
Figure5is a drawing depicting the N f lifetime in terms of both parameters S m and R. Figure5(a).depicts the main 3D plot of the three involved parameters.The other drawing, figure 5(b). is essentially a contour plot showing iso-life lines determined as a function of stress ratio R and mean stress S .mThese isolines are generated by the linear least-squares method.The fitting function is a bi-parameter polynomial function x and y, with degree 1, given by the following equation:

Figure 6 .
Figure 6.Constant life diagrams as function of R ratio and stress amplitude S a (a) main plot (b) contour plot.

Figure 7 .
Figure 7. Constant life diagrams as function of mean stress S m and amplitude stress S a (a) main plot (b) contour plot (c) detailed contour plot.

Figure 8 .
Figure 8. Correlation of stress amplitude S a as damage parameter with fatigue lifetime N f for R 0  loading cases.

Figure 9 .2Figure 10 .
Figure 9. Correlation of stress amplitude S a as damage parameter with fatigue lifetime N f for < R 0 loading cases.

Figure 11 .
Figure 11.Correlation of mean stress S m as damage parameter with fatigue lifetime N f for < R 0 loading cases.

4. 2 .
Fatigue life time models based on S m For load cases with R 0  and the average stress S m is taken as the damage parameter, correlations involving N f and S m are given in figure 10.The equation (8) is then written as follows:

Figure 12 .
Figure 12. S eq and N f calculation algorithm.

Figure 13 .
Figure 13.Correlation of stress amplitude S a 0 as damage parameter with fatigue lifetime N f for = R 0.

Figure 15 .
Figure 15.Comparison of the correlation of the (a) proposed approach (b) existing approach [7] between deterministic experimental and predicted fatigue lives.

Figure 16 .
Figure 16.Comparison of the correlation of the (a) proposed approach (b) existing approach [7] between dispersed experimental and predicted fatigue lives.

Table 1 .
[7]stituents of the rubber compound used in the test specimens[7].

Table 3 .
The computed S a and S m stress values and measured fatigue life under various loading conditions.

Table 5 .
Linear model coefficients given by equation (6) for the assumed fit.

Table 4 .
Linear model coefficients given by equation (5) for the assumed fit.

Table 6 .
S a 0 equivalent stress amplitude calculation for