Fatigue can be simulated for hyperelastic materials using Continuum Damage Mechanics by phasefield modelling (Miehe and Schänzel, 2014; Loew et al., 2020). In the work by Miehe and Schänzel (2014), using phase-field modelling, they were able to simulate crackinitiationaccording to Griffith’s Law, and in the work by Loew et al. (2020), fatigue damage was simulated with phasefield modelling incorporating viscous dissipation. However, in both works (Miehe and Schänzel, 2014) and (Loew et al., 2020), the modelling was done by using the CDM approach, which simulates damage at the microstructure level. This method gives extremely accurate results, but the computational power required and time will be very high. In CDM the degradation of material properties is described by the damage parameter, which is an internal variable governed by the damage evolution equation. To model damage evolution including initiation and propagation of microcracks, a very fine mesh is needed leading to a high computational load and time (Rodríguez et al., 2006). Due to the absence of internal variables, their critical threshold conditions, and evolution equations this energy limiter approach is simpler than CDM (Volokh, 2007a; Volokhh, 2008). Volokh (2007a) proposed the softening Hyperelasticity approach as a potential substitute for both the complex technique of damage mechanics including internal variables and the straightforward pointwise failure criterion of material strength. Volokhh (2008) used this approach to model failure in soft biological tissues and fracture of brittle materials. In this case, using energy limiters can be seen as a promising method for simulating failure in hyperelastic materials. Rather than using Continuum Damage Mechanics, using energy limiters is computationally less expensive and provides excellent accuracy. Volokh (2007b) introduced the concept of the critical failure energy to limit the strain energy. The material softening is controlled by this energy limiter. Energy limiters do not consider any changes to the microstructure of the material since the main energy limiter term Φ is calibrated from the stress strain response curve (Volokh, 2016). Any microstructural changes will be included in the energy limiter term Φ . Mythravaruni and Volokh (2018) have analysed the failure of rubber bearings under combined compression and shear, by enhancing the strain energy density with a limiter. The direction in which crack initiates, has been found by associating it with the loss of strong ellipticity. Trapper (2010) have studied the high-velocity penetration of projectile into a brittle plate by modelling the dynamic failure propagation using energy-limiter approach. Using the functional form of strain energy given in Volokh (2007b), and adopting a technique for element deletion, given in Trapper (2010), the strain softening in the bulk material can be modelled.

The main limitation of the energy limiter approach used in all these works is that this approach is suitable for monotonic loadings. Many researchers in the past have conducted studies to assess the change in the material response due to cyclic loading. Vinogradov et al. (2001), have performed experiments including vibrocreep tests and post cyclic stress strain teststo study the effects of temperature, frequencies, mean stress and stress amplitudes on the creep behaviour of Nylon 6/6 and polyvinylidene fluoride (PVDF). Tasdemir et al. (2023), developed data-driven constitutive model based on feed-forward neural networks trained with data obtained from random uniaxial stress controlled loading of titanium specimens. Sánchez-Santana et al. (2008), investigated the dynamic response of 6061-T6 aluminium alloy and AISI 4140T steel specimens with prior fatigue damage induced due to low cycle fatigue. Fang et al. (2008), performed experiments to study the degradation of tensile fracture properties of polycarbonate and acrylonitrile–butadiene–styrene (PC/ABS) due to cyclic loading. Galán López et al. (2011), studied the change in the microstructure and tensile properties of titanium alloy specimens subjected to different number of fatigue loading cycles. In all these studies, theenergy limiter approach has not been used to obtain the change in the mechanical response of elastomer blends due to fatigue cycles of loading and unloading.

Until now, all such experiments related to the energy limiter approach and material constant prediction by optimisation have only been done on pristine samples. None have been subjected to any sort of prior loading (dynamic or fatigue). The motivation for this work is that there has not been any investigation into the effect of fatigue loading on the material parameters used in constitutive models for elastomeric blends. Also, there has not been any study using the energy limiter approach in investigating the effect of fatigue loading on material parameters.

Initially, uniaxial tensile tests are performed on pristine sample and samples subjected to fatigue cycles and the result obtained for the stress-strain response of these samples is shown in Figure 1.

It is seen that the material is getting weaker with increasing number of cycles. The discrepancies in the initial loading part of the curves can be due to errors induced by clamping and UTM calibration.

Out of these seven uniaxial tensile tests, the tests conducted at 700 cycles, 1,200 cycles and 1933 cycles are used to validate the model. The model parameters are obtained by calibrating the model using the data for pristine, 1,100, 2,447 and 3,701 cycles.

A total of 8,640 linear brick elements (hexahedral elements) with reduced integration (C3D8R) elements are used in the simulation. Mesh density is high in the gauge section compared to the ends. The size of the elements in the ends is around 1.8 mm whereas in the gauge section, it is chosen to be around 0.5 mm. The simulation of the uniaxial tensile test was performed by fixing one end of the specimen and pulling the other end at an elongation rate of 500 mm/min (8.333 mm/s). Dynamic Explicit step has been used for analysis as this tensile test has been conducted at a high rate and explicit time integration is computationallyefficient, especially for short-duration, high-speed dynamic events, and requires less memory as it solves element-level equations without needing to calculate global stiffness and mass matrices.

During optimization, the value of m , D is kept constant at 200 and 0.001 MPa^-1 to simplify optimisations. Hence, the parameters for the Yeoh model are as given in Table 1. The stress-strain response from uniaxial tensile tests of specimens fatigued for different number of cycles shows the same high sharpness of transition to failure and the bulk modulus of the specimens is much higher than the shear modulus for all the specimens. m represents the suddenness of the transition to failure, i.e., an abrupt rupture of molecular bonds. A relatively high value has been selected here as it sufficiently captures the sudden failure that is experienced in the tensile tests. D is compressibility which is the inverse of bulk modulus, the values of these two parameters are kept constant at 200 and 0.001 MPa^-1 which could capture the experimental response reasonably well.

It is seen that there is also a decrease in the Modulus of the SBR/NR blends with the number of cycles. Here, the trend for all parameters is seen to vary exponentially with the Life factor (N/N_f) and hence can be written as shown in Table 3.

Using the above relations and values of D and m , the relation for strain energy density can be expressed as: W = 0.99 e − 2.18 N N f I 1 ¯ − 3 + 0.322 e 0.9267 N N f I 1 ¯ − 3 2 + 0.187 e 0.5394 N N f I 1 ¯ − 3 3 + 1 D J − 1 2 .

Consequently, for the strain energy density incorporating failure, Ψ , can be written in terms of the relations given above.

Finally, the engineering uniaxial stress component can be written as: σ 11 e n g g = 1 λ 1 exp − W m ϕ m 2 J 0.99 e − 2.18 N N f + 2 0.322 e 0.9267 N N f I 1 ¯ − 3 + 3 0.187 e 0.5394 N N f I 1 ¯ − 3 2 d e v B ¯ 11 + 2 D J − 1 .

The variation of lateral stretch obtained from the model response with uniaxial strain appliedduring the tensile test is plotted as shown in Figures 2, 3. The model response is also compared with the simulation result from ABAQUS.

Figures 3, 4 show that the lateral and axial strains do not have a fixed proportionality, meaning Poisson’s ratio changes with increasing strain, which is typical behaviour of rubber components due to nonlinear elasticity. Variation of lateral stretch in these figures shows that SBR/NR blends exhibit material softening which is evident from the deviation of Poisson’s ratio from 0.5 (incompressible behaviour) under large strains. The lateral stretch for the same amount of axial strain in specimens is constant with fatigue loading which is in line with the choice of constant compressibility (D) value.

Figures 5–11 show the comparison between the experimental result of tensile testing and the results obtained from MATLAB analysis and ABAQUS simulation for all samples. Out of these, Figures 5, 7, 10, 11 show the calibration of the model and Figures 6, 8, 9 show the validation of the model. From these figures, it can be seen that the developed model captures the experimental response quite well. The discrepancy with the ABAQUS simulation may be due to the relatively coarse mesh used in the simulation because of the limitation on node count. In Figure 11, the curves for experimental and simulated data are nearly parallel. The mismatch between them could be due to any calibration error of the UTM which may be due to residual stress in the load cell.

Although this study uses the energy limiter approach which has advantages compared to other approaches like CDM and the Strength of Materials approach, this study too has its limitations due to the experimental and simulation errors. Experimental errors include errors during the weighing and mixing of ingredients, the presence of foreign particles in the mix, errors during the rolling process, such as roll temperature, roll speed, and roller gap, temperature fluctuation during compression moulding, and errors during clamping and UTM calibration. Simulation errors include limitation on node count due to license terms affecting accuracy and as it is dynamic explicit analysis, the choice of mass scaling value may also affect the accuracy of the simulation.

From this work, it is seen that the developed constitutive model is capable of predicting the change in the hyperelastic response during uniaxial tensile fatigue as seen from the good match between the MATLAB and ABAQUS simulation results obtained for stress-strain response and experimental behaviour. Also, it is seen that the variation of all the model parameters is able to capture the degradation in stiffness of the material with the progression in fatigue cycles reasonably well.

From this work, it can be concluded that the application of the energy limiter approach can be used to predict changes in material response due to fatigue with excellent accuracy. Comparing the use of CDM to the energy limiter approach used in this work for simulating fatigue, it can be seen that the energy limiter approach is less computationally intensive and much easier to implement.