Ogden model for characterising and simulation of PPHR Rubber under different strain rates

ABSTRACT Pphr rubber with different Carbon percentage is extensively used in engineering applications such as tyres, in which it may be subjected to impact loading leading to high strain rate deformation. This causes large deformations to happen at a variety of strain rates of the material. For this reason, the Babylon industrial company in Iraq suggested a modification of the rubber material used in manufacturing tyres by using different percentages of Carbon black to study the mechanical properties of these materials over a range of loading conditions. Also, the high strain rate and with break strain dependent on their properties provides further reasoning for the research. As for this, the material being irradiated by five percentages of Carbon black makes it vital within the manufacturing process of specific tyres. This work aims to characterise the effects of Carbon percentage on the mechanical properties of pphr rubber. The material was irradiated by five different percentages of Carbon black of pphr rubber dose, Carbon black 30 pphr, Carbon black 40 pphr, Carbon black 50 pphr, Carbon black 60 pphr, and Carbon black 70 pphr. It is a mechanical response, which was studied with the help of uniaxial tensile tests with different strain rates. Previous experimental results are utilised within the Ogden’s Model to obtain the parameters for the simulation of the material response using the finite element method (FEM) for comparison purposes. Ogden’s Model with first, second, and third-order equations to provide the simulation of the tensile test curves is compared to the experimental curves. Ultimately, it was found that values up to N = 3 produced reasonable results with varying values of pphr and that any value higher only resulted in error within the simulation.


Introduction
Rubber is a crop from which a range of products can be produced that have vital applications to society in a variety of ways. Knowledge of the properties of rubber continues to expand. In time, additional manufactured goods will be produced by the rubber industry. In general, there are two kinds of rubber usually utilised in industry, natural rubber and synthetic rubber. The applications of natural rubber (NR) overall different productions are due to its mechanical properties such as high durability, low hysteresis, elasticity, and high toughness. In overall, NR is an unstructured material. However, when stretched NR can crystallise. This crystallisation gives characteristics such as abrasion resistance, tensile strength, and tear resistance to the mechanical properties of NR. The rubber combination is the main product of rubber commodity. Roughly speaking, all rubber compounds use Carbon black (CB) as a filler. Carbon black filler improves the physical properties of rubber, increasing strength, as well as strengthening vulcanisation. The results of the rubber compound can be useful in producing bearings, belts, tyres for cars.
The following research focused on the tensile strength due to its ability to demonstrate the capacity of plastic material to withstand maximum values of tensile stress whilst being stretched until the point of failure, which is defined as the point at which material goes from elastic to breaking point, the stress coordinate on the stress-strain curve at the end of rupture. The break stress and strain values are utilised to indicate the elastic behaviour as well as plastic behaviour of the material under specific amounts of loads. This can be predicted from the stress-strain diagram of the material from which the rupture point is defined. The reason for these five percentages of Carbon black pphr rubber doses is because these are the five percentages used within the tyres of the Public Company of tyres Babylon in Al-Najaf, as it was their request to investigate the characterisation and materials through simulations based on their tyres. The Babylon company planned to increase the stiffness of the beads in their tyres. The tire plies usually are composed of different materials such as Carbon black. A decision was made to deduce the best percentages of Carbon black and to link deductions to previous work in this respect which used different percentages of Carbon black. The ingredients of rubber used in this research compounds are SMR 20, SBR 1502, whole Tire Reclaim, PCTP-50, Zinc Oxide, Stearic acid Process oil, Phenolic Tack Resin, N660 Carbon black,6PPD, TMQ and Carbon black. The rubber materials used in this work have a wide application which can resist high strain rates before breaking. The flexibility, as well as its ability to absorb the dissipated energy found in these types of materials, encourages the designers to use them in engineering applications, such as automobile tyers, aircraft landing gear, belts, springs, hydraulic seals and many others uses. The main target of this research paper is to demonstrate through our results the potential for improving the understanding of problems encountered fitting theory to experimental data and their practical effect on the solution of boundary-value problems. The main target of this research paper is to demonstrate through our results the potential for improving the understanding of problems encountered adapting theory to experimental data and their practical effect on the solution of boundary-value problems. One possible outcome of this study is that, at least in the case of elastomeric materials, it is advisable to examine their behaviour first via mathematical models rather than by simulation models since the construction model that forms it can be designed to capture precisely the main qualitative features of the experimental data. At the same time, this work aims to produce quantitative predictions by introducing large numbers of material parameters whose physical significance is unclear. Carbon black (CB) that is used in this circumstance is a Carbon material nearest to its natural form being best produced by combustion crops which can be obtained from biomass or hydroCarbon products. Also, CB is the kind of filler generally used in making rubber compounds. Furthermore, CB fillers have a strengthening effect on mechanical and physical properties, specifically those with a small grain-scale (Pandey, Setua, and Mathur 2003;Mandal and Aggarwal 2001;Kim and Jeong 2005). On an industrial scale, the productions of tire compounds with CB as a filler aims to strengthen the bonds between compound-forming molecules. Additionally, CB can act as an active filler which belongs to a functional group that plays a part in supporting the bonds between molecules forming rubber products, together with the structure of CB prescribing the ideal composition of the filler in a polymer matrix. (Leblanc, Evo, and Lionnet 1994;Anqiang, Lianshi, and Yiyu 2003). The supplement of CB will affect compound behaviours, viscosity and strength of the compound. However, the usage of CB also has to overcome the difficulty of reducing stickiness. In general, the natural rubber is shown to have low mechanical properties. It is necessary to increase the value of natural rubber. One of the fillers is CB. Thus, CB filler is a material that can raise the tear resistance, hardness, abrasion resistance, and high breaking stress within of the goods produced Baah 2001, 2002;Siriwardena, Ismail, and Ishiaku 2002). As well as this, the rubber materials discussed are used in many engineering applications. Therefore, the investigation of mechanical properties within rubber has attracted many researchers. (Pusca, Bobancu, and Duta 2010) gave an overview of the mechanical properties of rubber materials and presented most of the industrial purposes and the re-use of scrap tyres and recycling of these elements to be used in making concrete. They highlighted the variables that affect the mechanical properties of rubber such as particle size and the surface of the elements, which have a direct effect on strength and stiffness. (Hussain, Jweeg, and Hamza 2010) presented an extensive explanation of the properties and application of elastomers materials and studied the large strains using the boundary element method. They analysed the techniques used in the manufacturing of elastomers with their mechanical properties in addition to the simulation of the elastomers. This is intended as a guide for the engineers working in this field and a research tool for the comparison of the mechanical properties with any new additional information published in the foreseeable future. (Santos and Batalha 2010) studied the characterisation of Polyacrylate rubber using the EB radiation and its effects on mechanical properties using UV curing of the polyacrylate rubber. They have shown that UV radiation affects the mechanical and thermal properties such that the amount of elongation decreases concurrently with the increase in strength. Ogden's model aligns ideally with the experimental findings, outputting the mechanical parameters of the material with a third-order degree. The thermal and mechanical properties were modified using UV for the first time. (Meunier et al. 2008) investigated the characteristics of the unfilled silicon rubber and conducted tensile, pure shear, and compression tests. They tried different models of silicon rubber behaviour and achieved a uniaxial tensile loading for a plate with holes using the finite element simulation. Hyperelastic models were suggested which have been proved to be effective by the comparison of the experimental and numerical results. 2007) The criteria of failure of the rubber applied by (Meunier et al. 2008;2007) using the strain energy principles. The failure limit was defined using a plane strain uniaxial specimen.

Experimental work
The experimental tests (Abdulkadhim 2010) were carried out using a tensile testing device. As shown in Figure 1, the tensile testing device consists of two parts. The first part is mechanical in order to conduct the test. The second part involves the printing of the results data of the test as the mechanical part consists of the hydraulic system that performs the tension according to the requirement of elongation and speed as well as a device for checking the values which are subsequently printed. The experimental tests consist of an axial tensile load applied to a standard specimen of rectangular cross-section Figure 2(a) with a different strain rate (mm/min) by the hydraulic tensile device shown in Figure 1, and this causes the specimen to elongate and completely fracture Figure 2(b). The tests for strength resulting from the stretching were carried out according to the Standard (ASTM D412, 2016), experiments were repeated five times, during which average results were observed and recorded.

Predicting Ogden's hyperelastic coefficients
Ogden's model is one of the applicable hyperelastic models for the simulations of these types of materials. The modelling has been commonly used to predict the mechanical properties of rubber as an elastomeric material when subjected to large deformations. Ogden model has the ability to the test data that can be employed directly in Abaqus TM. An agreement was reached, based on the experimental data previously gathered, to ascertain 800% of the tensioned length. In this research, the programmed fitting algorithm implemented in Abaqus TM, a commercial finite element software package for Ogden's model. has been employed within the constraints of the parameters of the model.
In this algorithm, the coefficients such as µ is related to a shear modulus, whilst α is related to material coefficients. Abaqus TM performs a leastsquares fit process to obtain the coefficients. Similarly, The Ogden model is used in Abaqus TM for the experimental data so that it is concurrent with the data used within Abaqus TM on top of the material curve fitting,' (Hameed, Jweeg, and Hussein 2009;Al-Shammari, Hussein, and Oleiwi 2017;Abbas et al. , 2019Al-Shammari et al. 2020). This procedure may minimise the relative error in stress, (Abdulridha et al. 2018;

Ogden's hyperelastic model
This section will briefly review the equations of the Ogden model for rubber materials which are incompressible isotropic and behave in a non-linear elastic manner, which is necessary for comparing the Ogden model with experiment tensile results. Thereafter, the data that has been taken into consideration in this work and is related to the results of experiments in homogenous deformation are compared Ogden  model, as discussed above, is highly suitable for incompressible materials, subjected to large deformations. Ogden's, (Ogden 1972;Ogden, Saccomandi, and Sgura 2004;Ogden 1997) research indicates how the most significant simulation of a hyperelastic model is the most critical parameter in the determination of the strain energy density function. In hyperelastic modelling, the role of the energy density depends typically on the strain tensor invariant. The hyperelastic model does not rely on the path because the stored elastic energy in the material is evaluated using the current state of strain. The hyperelastic Ogden function fitting is up to N=6 (N is the order of the polynomial). The strain energy density U function for an Ogden (Ogden, Saccomandi, and Sgura 2004;Twizell and Ogden 1983). Subsequently, the material is considered incompressible. The tensile test is used to estimate the form of U, where; λ i (i = 1,2,3) the principal extensions of the deformation and the principal stretch for an incompressible material satisfy the constraint (Arruda and Boyce 1993;Kim et al. 2012) is given by: As well, μ i and α i are empirically calculated material constants (describe the shear behaviour of the material). So, the initial shear modulus µ (Ogden, Saccomandi, and Sgura 2004;Ogden 1997) can result from the equation, So, when the material is under uniaxial tension loading the stretch ratio in the principal direction is defined by While the ratio of the other side is established as; In the principal stretch, λ i is related to the principal nominal strain ε i in the following equations, For each stress-strain test, Abaqus TM software will generate a mathematical relation for the stress in terms of the strain stretches. Since λ U ; the hyperelastic constants are known, assuming incompressibility. Hence, the test can be predicted only λ U (stretch ratio in the principal direction) . Furthermore, the tensile tests are used to indicate the function of W. Besides, for each stress-strain test, Abaqus TM software will yield an equation for the stress in terms of the strain. The tensile stress σ U calculate based on estimates according to, Thereafter, from the experimental test data, the material parameters are calculated through a least-squaresfit procedure, which minimises the relative error in stress. For the n nominal stress-nominal strain data pairs, the relative error estimates E is minimise as described below, In the case of the non-linear least-squares fit for the Ogden model, they are potentially non-linear within some of their parameters. Thus, a non-linear leastsquares-fit procedure is necessary. In this situation, Abaqus TM will use the Marquard-Levenberg algorithm, as suggested by Twizell and Ogden (Twizell and Ogden 1983). So, for this possibility of a non-linear model, the coefficients of the hypererlastic model were assumed as α i; , i ¼ 1, n where n is the number of data points, m = 2 and r is the iteration count. Subsequently, the coefficients can be estimated by iterating using the following equations: Here, E k is the vector of relative errors, where P ik is the derivative of the vector of relative errors for the coefficients α i . Furthermore, when the value ofγ ¼ 0, the Newton algorithm is obtained. Whilst for very high concentrations ofγ, the steepest descent method is accepted. Therefore, the Marquard-Levenberg algorithm represents a compromise between these two approaches: the value of γ is increased if the error occurs and is reduced otherwise. Finally, for non-linear least-squares fit for the Ogden model, AbaqusTM will initialise the parameters such as α 0 i , the μ 0 i which are already found with a linear least-squares fit. In the iterative procedure outlined above, the following derivatives are used by Equations (12 and13), (c ¼ À 0:5 (uniaxial loading)),

Numerical simulation
In this work, hyperelastic constitutive modelling is employed in Abaqus TM, with a single element type C3D8RH, cubic shape (Jebur 2013) (see Figure 3). Accordingly, the single element has been used to calculate the Ogden hyperelastic model coefficients (µ, α) and R (Relative root mean square error). The single element to do this is subjected to uniaxial tension to obtain predictions to compare directly with the experimental data, (Jweeg 1983;Al-Khayat et al. 2018;Kadhim, Chiad, and Takhakh 2018;Chiad, Al-Waily, and Al-Shammari 2018;Hussein et al. 2020;Jweeg, Alnomani, and Mohammad 2020). Similarly, the single tensile loading situation is described in one-step loading. In AbaqusTM, there are two choices for describing the material model. The first one can be calculated coefficients and may be entered using the order hyperelastic, model-name with the coefficients defined, on the next line. Alternatively, the second choice is that experimental data can be Referenced, as before, using the 'hyperelastic, model-name', 'test data input'(Uniaxial test data), (Jweeg, Hammoudi, and Alwan 2018a;Jweeg et al. 2018b; Al-Shammari and Al-Waily 2018; Al-Waily, Al-Shammari, and Jweeg 2020; Kader, Abed, and Al-Shammari 2020). Then, the parameters can be calculated prior to after finishing the analysis, (Jweeg and Said 1995; Neama, Al-Baghdadi, and Al-Waily 2018; Ismail, Abud Ali, and Al-Waily 2018; Al-Shammari, Zedan, and Al-Shammari 2018; Yousif, Resan, and Fenjan 2018). Subsequently, the investigation has implemented the tensile stress and strain test results and can be incorporated within a prediction for each tensile loading. Finally, these predictions stress-strain results should then be plotted against the experimental data, (Mahmood, Jweeg, and Rajab 1989;Abbas, Jweeg, and Al-Waily 2018;Jweeg et al. 2018c; Al-Shammari 2018; Al-Waily, Hussein, and Al-Roubaiee 2019; Abbod et al. 2020). It is essential after the analysis has completed running to open the file of calculated data with stable hyperelastic material, to confirm that the data prediction is accurate. Otherwise, the research should be repeated by changing the value of N (the order of polynomial).

Results and discussion
There were five levels of the strain rates requested within the simulation by the public company of Babylon in AlNajaf as they are the strain rates that will eventually be used in advanced simulations that lead to the manufacturing of the tyres. With respect to the Ogden parameters, through previous sources of research, it was concluded that the Ogden model could be deemed to be the most efficient model for this simulation, considering the material type, Carbon levels, and the varying strain rates.
In this work, the stress-strain behaviour of rubber, which behaves as hyperelastic material is predicted using the Ogden model with different percentages of Carbon black. The simulated results were extracted from the previous experimental results achieved by (Abdulkadhim 2010). Figure 4 (a-e) shows the observed and Ogden model for the strain rates 100, 200, 300, 400, and 500 mm/min using the paste of natural rubber with 30 pphr of Carbon black. It would appear that the Ogden model can be used as a tool for the characterisation of rubber materials and may also be used for design purposes. Figure 4. b shows some discrepancy, perhaps due to an experimental contrast accompanying the practical work. The distinction may not reach more than 10%. However, the Ogden parameters are shown in Table 1. These parameters are useful in predicting the failure stress of percentages of pphr 30 with strain rates which lie between 100 and 500 mm/min. Figure 5 shows the prediction of the Ogden model for pphr 40 of Carbon black with natural rubber. The strain rates used are 100, 200, 300, 400, and 500 mm/min, as shown in Figure 2a-to 2b, respectively. Again, the prediction coincides with the observed behaviour with some small discrepancies in some regions. The Ogden model may be trusted for estimating of the stress-strain models for the rubber material. However, the Ogden parameters are shown in Table 2.
The use of NR dough and Carbon black 60 pphr with strain rates 100, 200, 300, 400, and 500 mm/min are shown in Figure 7, and the Ogden parameters are shown in Table 4. The final model in this series of models is shown in Figure 8. In this case, 50 pphr with natural rubber and the same previous strain rates were used. Figure 8a -8b demonstrate some deviations which belong to the same previous conclusions stated above. However, the Ogden model parameters are shown in Table 5.
The effects of the Carbon black on the percentage of the breaking strain show a maximum of 10% difference between the strain rate of100 mm/ min and 500 mm/min and no trend was found for these results. Similar trends were found for the other Carbon black percentages and different strain rate results. The designer may choose the required paste according to the practical application. For example, if the user needs the material to be used as a damper element, the strain energy stored is the main target, i.e. the area under the stress-strain diagram is the main reason for the choice. Tabes 2, 3, 4, 5, and 6 indicate that using 70 Pphr with a strain rate 400 mm/min gives a maximum discrepancy of 12.7% compared to the Carbon black 30, 40, 50, and 60 Pphr. Also, the results have shown a stable error in the strain calculations using the Ogden model compared to experimental results. The discrepancy in stress levels using the Ogden model is higher than that in strain levels using the different carbon black values.

Conclusion
This work aimed to investigate the effects of adding additives to natural rubber for purposes of modifying its the strength and the stiffness by using carbon black with different percentages. The study included a systematic investigation of fitting the stress-strain relations of an experimental programme consisting of 20, 30, 40, 50, 60, and 70 Pphr Carbon black of the rubber material. The research focused on the determination of optimal material parameters in the Ogden strain energy function. The work also covers the estimation of the stress-strain of the samples using the Ogden model. It was proved that the Ogden model could be used as an indicator of the linear elastic materials and that some characteristics may be predicted. The broad scope and parametric study have been achieved using the models presented, employing the Ogden coefficients for rubber materials with different carbon black percentages. The Ogden model results show a stable error in strain compared to the experimental findings; however, using the Ogden model, acceptable results were obtained in both strain and stress parameters.

Data Availability Statement
The data that supports these findings are openly available at Zenodo at https://zenodo.org/record/4396263#. X-o-Y9j7TIU contained within the attached file.

Disclosure statement
No potential conflict of interest was reported by the author(s).