a shows the typical response expressed as stress versus number of loading cycles (N), both measured in the laboratory experiments and the FE simulations obtained for the circular shaped ROPS specimen. The figure shows the data over the first 100 cycles. As evident in the figure, there is a decrease in the compression stresses (σcomp
) resisted by the ROPS specimens, up to around 70 cycles in this example, whereas σcomp
values stabilized with minimal decrease or variation in the stress values with increasing N. Thus, the data from the 70th cycle has been used for comparison and analysis in this study. Overall, a good agreement was observed within the experimental and FE simulation data, with minor error (discussed later in this section).
4.2. ROPS Model Simulations
a–d show the stress–strain response of the circular ROPS specimens simulated using the hyperelastic functions for all four OPS contents. Figure also highlights the comparison of these simulation outputs with the experimental output. Observations from the figure clearly evince that all the SEFs match with the experimental curve—exhibiting similar concave upward pattern. However, for 10% and 20% OPS content alone, the Neo Hookean function represents a linear pattern. Similar patterns are evident irrespective of the varying OPS contents. The maximum compressive stress (σmax
), using the Yeoh function ranged between 590 and 1400 kN/m2
which is in good comparison with the experimental data (604–1438 kN/m2
). Likewise, the range of σmax
using other models are, 585–1370 kN/m2
for Arruda Boyce, 580–1370 kN/m2
for Marlow, 579–1538 kN/m2
for Moone, and 525–1185 kN/m2
for Neo Hookean functions. Overall, an increase in σmax
values was witnessed with increasing OPS contents, which also agrees with the observations from the experiments. For example, σmax
value for the ROPS specimens with 20% OPS was approximately 2.4 times higher than that of ROPS specimens without OPS content. Increasing σmax
values can be correlated to indicate higher absorption of the impact energy [28
]. This conceptually attributes to the distinct property of OPS which imparts increasing stiffness and hence improved strength and shock absorption. Table 3
tabulates the percentage error observed in the σmax
values estimated from the model simulation with that of the experiments. A lower error was observed in the case of the Yeoh model, thus justifying its suitability in simulating the stress–strain response of the ROPS and similar composites. Next to Yeoh, Marlow and Arruda Boyce functions were found preferable with an error less than 5%. The Mooney Rivlin function though exhibited error less than 7%, an overestimation of stress value was observed in ROPS composites with 20% OPS content, thus making this function only moderately suitable. The error with Neo Hookean model was relatively higher, thereby proving to be inappropriate for modelling ROPS composites to this end.
The comparison of percentage error values across different OPS contents reveals that ROPS composites with 20% OPS have higher error magnitudes in comparison to 0%, 5% and 10% OPS contents. This can also be related to the random orientation of OPS considered in this study. Thus, further studies are recommended to check for anisotropy at higher OPS contents and orientations. Figure 3
g,h show the contour plots of the Von Mises stress observed for a circular ROPS specimen when strain (ε) is approaching 50%. Figure 3
h reveals higher stress transfer directly below the loading ram, at the interface of the loading ram and ROPS’s surface, migrating gradually towards the internal zones. Further, deformations ranged from 0 for un-deformed material to 11.01 mm for deformed specimen. ROPS specimen did not show a specific failure type and maximum permanent deformation after 100 loading cycles was observed at 4%, which corresponds to 0.70 mm.
The main purpose of the ROPS composites considered in this study is to resist the energy generated due to impact or compressive type loads. Nonetheless, for validations purposes tensile testing of the ROPS composites were also carried out. The uniaxial stress stretch (tensile) behaviour of the ROPS composites were simulated using hyperelastic (aided with the Yeoh function) modelling which is evident as most suitable function for ROPS circular specimens. A dumbbell-shaped specimen and a circular specimen of ROPS was considered for this purpose. The geometry of the dumbbell shaped specimen used in the tensile test FE simulation for pure rubber was 150 (l
) × 25 (w
) × 3 (t
) mm, similar to procedure presented in Sahari and Maleque [15
]. The material parameters used for the simulation are listed in the table highlighted in Figure 4
. The loading procedure basically involved stretching the specimens to 1.5 times the original specimen length. Meshing involved the hybrid elements, followed by convergence analysis to get the most appropriate output. Figure 4
a shows the tensile stress for ethylene propylene diene monomer (EPDM) based rubber without OPS is 1.072 MPa. This value evidently is in the range of the tensile stresses of unfilled EPDM as presented in Milani and Milani [37
], thus validating the modelling procedure and reliability of the simulation results. Figure 4
c shows the tensile stress contours obtained for dumbbell shaped specimen with no OPS content. It should be noted from the work presented by Milani and Milani [37
] that the Mooney Rivlin simulations of the commercial rubbers with carbon black fillers showed stiff curves with quick increase in uniaxial tensile stress as compared to that of the unfilled rubber. However, simulations in this study shows an opposite behaviour with the pure rubber (with no filler/OPS) achieving higher strength more quickly as compared to its OPS counterparts.
The values of tensile stresses obtained for the circular ROPS composites are less than their compressive stress values, in spite of maintaining similar specimen dimensions. Quantitatively, the tensile stresses were approximately less than 0.5 times the compressive stresses for the same specimen. This is because, the effective area that resists the tensile force gets reduced with increasing OPS content. This can be correlated to perforations induced in steel plate, which are likely to reduce the force transmitted [38
]. The presence of OPS in rubber matrix reduces the tensile force transmitted in similar manner, since they can disperse only compressive loads and cannot transfer any tensile force. Hence, the maximum force at yield also gets reduced, leading to lower values of tensile stress. Moreover, studies employing fibre-rubber reinforcements also demonstrates similar reductions in tensile strength with increase in the fibre content [39
The suitability of the hyperelastic models used for circular ROPS specimens was further checked for possible stability limitations that might arise due to varying shape (surface) of the composites. It is for this reason that three additional shaped ROPS specimens (square, hexagon and octagon) were cast and tested in the laboratory. It should be noted that the similar procedure and methodology was adopted in the simulations and experiments conducted on all other shaped ROPS specimens as well.
a–d shows the comparison of stress–strain response simulated with that of the data measured in the experiments on square ROPS specimens with OPS content of 0%, 5%, 10% and 20% respectively. Observations from the figure highlight similar patterns of the simulation outputs in comparison with the experiments. Yeoh model was found to best illustrate the experimental data with only 5–6% error. Marlow and Arruda Boyce models occupy the next level with error ranging up to 9.6% and 17.80%, respectively. However, these error values are on the higher side as compared to that observed with the circular ROPS specimens (7%). This increase in error can be attributed to element distortions at the sharp edges, following which these elements upon experiencing excessive distortion may reach a scenario which cannot be modelled using hyperelasticity function. Mooney Rivlin function which was found to be stable for circular shaped ROPS was found unstable in the case of square shaped ROPS specimen. The material convergence of Mooney Rivlin is believed to have been influenced by the varying stress–strain conditions induced due to the sharp edges. This is also proved by the contour plots in Figure 5
f, where the maximum stress is concentrated at the sharp corners and not at the centre unlike in circular shaped ROPS specimens. The stress distribution along the edges were higher than that of the elements directly below loading ram. Other than this, the trends of poor predicting accuracy of Neo Hookean, increasing maximum stress values with increasing OPS content, and having higher error values for 20% OPS content ROPS specimens remained unchanged for the square shaped ROPS as well. Moreover, the maximum deformation recorded was 10.67 mm and the permanent deformation imparted is 4.44% (0.89 mm).
a–d shows the comparison of stress–strain response simulated with that measured in the experiments on hexagon shaped ROPS specimens with OPS content of 0%, 5%, 10% and 20%, respectively. The best estimation was observed to have been achieved considering the Yeoh function with a 3–6% error, followed by Marlow and Arruda Boyce functions with 15% and 22% error, respectively. These error values are higher than square shaped ROPS specimens, as the number of edges/corners is high in comparison to square specimens. Contour plots in Figure 6
f support this statement, which shows the maximum stress concentrations occurring at the sharp corners of the loading surface and along edges. The permanent deformation recorded was 3.8% (0.76 mm).
The behaviour of octagonal samples was very similar to that of observed in square and hexagon shaped ROPS specimen, except that the Marlow function was found to overestimate the σmax
values in ROPS with 20% OPS content. Figure 7
a–d shows comparison of simulated stress–strain response with that of the experimental data. Yeoh was the most appropriate with a 3–7% error, followed by the Marlow and Arruda Boyce functions with 16.7% and 20% error, respectively. The contour plots in Figure 7
f show the occurrence of σmax
at sharp corners which migrate further to the edges, similar to square- and hexagon-shaped specimens. Permanent deformation recorded was 4.3% (0.86 mm).
4.3. Effect of Shape on the Stress–Strain Response of ROPS Specimens
Overall observations from Figure 3
, Figure 4
, Figure 5
, Figure 6
and Figure 7
leads to an understanding that the increasing number of sharp corners and edges results in an increased error in the σmax
values predicted using the hyperelastic SEFs used in this study. This may be due to the element distortions, which is perhaps unavoidable with ROPS shapes containing edges and corners, though the mesh verification process revealed zero error. Nevertheless, the hyperelastic model aided by the Yeoh function was able to predict well the stress–strain response of the ROPS specimens for all shapes. This statement is justified based on the lower error in σmax
values (−7% to 0.7%) between predicted and experimental data. Mesh optimization considering different trials may be used to further reduce the error where required.
Another key factor, which is of major interest in this study, is the variations in stress distribution across the range of ROPS shapes considered. Especially, the corners and edges associated with the polygon (square, hexagon and octagon) shaped ROPS specimens revealed to have a major effect on the stress distribution pattern when compared to that in the circular shaped ROPS specimens. Literature also highlight the presence of corners and/or edges to increase the stress concentration in hexagon and octagon specimens, which are mostly located close to the corners as compared to that of circular and/or square shaped specimens [42
]. Having said this, there is yet one similarity observed in the stress distribution—higher stress concentrations are predominantly noted at close proximity to the loading surface than that at the lower portion or lateral surface of ROPS specimens. This can be explained with the transfer of stresses to the adjacent areas to have not yet initiated, particularly due to insufficient time owing to the loading frequency (0.8 Hz in this study). Nonetheless, higher stress concentrations at the circumference and edges may relate to the lateral movement (pushing effect) of OPS samples dispersed in the rubber matrix during maximum loading. Since the load transfer mechanism in the OPS (dome-shaped structures) is mainly by contact between the adjacent OPS, thereby leading to possible pushing effect, thus resulting in higher stresses at the specimen periphery.
The next important factor considered in this analysis was the variation in stress intensities. Though four different shapes were used in casting ROPS specimens, an equal loading area was devised to ensure similar loading transfer perhaps. This was considered essential to ensure equivalent stress intensities. However, the stress intensities obtained or observed were not similar and were found to vary with varying shapes of the ROPS. Octagon- and hexagon-shaped specimens did exhibit highest stress intensities, followed by square and circle shaped ROPS. This difference can be related to the effect of shape factor on the vertical stiffness of ROPS composites. The shape factor is typically defined as the ratio of loaded top area to the lateral surface area available for bulging. The shape factor plays an important role in determining the compression behaviour and the associated stress intensities [43
]. The effect of shape factors on the vertical stiffness of ROPS specimens should also be evaluated in order to understand the optimum range of shape factor that can help acquire the required range of vertical stiffness for the end application of the ROPS specimens.
a–d shows the variation of shape factor and the associated vertical stiffness of all of the four shapes used for the ROPS specimens in this study. Observations from the figures show that circular specimens with a shape factor of 0.48 have the lowest vertical stiffness value. While the increase in OPS content from 0% to 20% increases the vertical stiffness values of circular specimen. Nonetheless, this increase is consistently low in comparison with its counterpart polygon-shaped ROPS specimen. Square-shaped ROPS with a shape factor of 0.42 recorded stiffness values higher than those of circular samples and there was a continuous increase in the vertical stiffness value with the increase in OPS content. Likewise, hexagon-shaped specimens with shape factor of 0.46 recorded the highest vertical stiffness. The octagon-shaped specimen (shape factor of 0.475) showed vertical stiffness values similar to hexagon shaped specimen up to 10% OPS content. However, in the 20% OPS condition, a reduction in the octagon’s vertical stiffness was observed.
Hypothetically, within an optimum range of shape factors, the vertical stiffness is expected to increase with increasing shape factor values [43
]. This trend can be observed in the case of square- and hexagon-shaped specimens, while minor variation was observed in the octagon-shaped specimen. However, the circularly-shaped specimen, having the highest shape factor did not show such a trend, indicated by their lower vertical stiffness values. These observations perhaps indicate that the shape factor in the range of 0.42–0.47 can be considered as optimum for ROPS. Moreover, a typical elastomer with a higher shape factor is expected to behave differently as compared to that of the lower shape factor samples [44
]. Based on the trends observed in this study, it is apparent for the ROPS specimen with a shape factor greater than or equal to 0.48 can be classed as higher shape factor composites (SH) and all the polygonal-shaped specimens with shape factor values less than 0.47 can be classed as lower shape factor samples (SL). Another key observation which supports this classification is the high vertical stiffness occurring in SL samples as compared to that of SH samples, which is found compatible with earlier studies [42
]. SH samples are believed to have higher horizontal stiffness, which is yet to be studied in the case of the ROPS composites. The minor deviations observed in the octagon shaped specimen are due to its shape factor value approaching the SH type. From these stiffness and shape factors studied, the vertical stiffness values of polygonal shaped specimens are comparable to that of the carbon fibre-reinforced rubber samples [44