Article Determining the Compression-Equivalent of SBR-Based Rubber Material Measured in Tensile Mode Using the Finite Element Method

Sahbi Aloui 1,* and Mohammed El Yaagoubi 2

1 Netzsch Gerätebau GmbH, Schulstraße 6, 29693 Ahlden, Germany 2 MS-Schramberg GmbH & Co. KG, Max-Planck-Straße 15, 78713 Schramberg, Germany; [email protected] * Correspondence: [email protected]

Abstract: A timesaving characterization approach of rubber materials in compression using the finite element method (FEM) is presented. Rubber materials based on styrene butadiene rubber (SBR) are subjected to tensile and compression tests. Using the neo–Hooke, Mooney–Rivlin and Yeoh material models, a compression-equivalent deformation of the SBR samples is derived from the tensile testing. The simulated state is then compared with the experimental results obtained from compression measurement. The deviation in the strain energy density between the measurements and the simulations depends on the quality of the fitting.

Keywords: compression-equivalent deformation; finite element method; Neo-Hooke; Mooney- Rivlin; Yeoh  

Citation: Aloui, S.; El Yaagoubi, M. Determining the 1. Introduction Compression-Equivalent Rubber materials are used due to their high in almost all technical areas. Deformation of SBR-Based Rubber Rubber materials owe this to the ability to store deformation energy and to release it back, Material Measured in Tensile Mode if necessary. This behavior is due to imminent restoring forces, which are generated from Using the Finite Element Method. the stored energy and act back in the direction of the rest position [1]. Appl. Mech. 2021, 2, 195–208. Tires represent by far the largest industrial use of rubber materials. During the drive, https://doi.org/10.3390/ at least three deformation modes act on the tire. The tire tread is compressed under the applmech2010012 effect of the weight load. Furthermore, the tire apex, the sidewalls and the tire bead are bent, when they come into contact with the road surface. During steering maneuvers, shear of Received: 12 February 2021 the tire tread and sidewalls occurs [2–4]. For safe drive behavior and performance reasons, Accepted: 21 March 2021 Published: 23 March 2021 a detailed, usually cost-intensive and time-consuming characterization of the different tire’s components in various measurement modes is necessary [5–7]. All requirements

Publisher’s Note: MDPI stays neutral placed on the tire, such as safety, comfort, handling and economic aspects, mostly depend with regard to jurisdictional claims in on the tire tread. For this reason, the precise characterization of the mechanical behavior of published maps and institutional affil- the tire tread is of central importance for the tire industry [8–11]. iations. Sealing elements are also used in technical applications to prevent mass transfer between two components [12,13]. They undergo continuous changes in operating systems due to load variation and environmental conditions. Characterization of mechanical properties like hardness, rebound and compressive strength is a critical issue in order to prevent damage development in sealing elements. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. Independently of the application, there is no getting around a very time-consuming This article is an open access article and cost-intensive characterization of the rubber materials. Simulations using finite ele- distributed under the terms and ment method contribute to reducing the development time and characterization costs by conditions of the Creative Commons analyzing and optimizing the material selection. Real testing conditions can be shortened Attribution (CC BY) license (https:// or even partly avoided. creativecommons.org/licenses/by/ The simulation of the mechanical behavior of rubber materials is a major challenge due 4.0/). to the wide structural diversity of rubber materials themselves. Different material models

Appl. Mech. 2021, 2, 195–208. https://doi.org/10.3390/applmech2010012 https://www.mdpi.com/journal/applmech Appl. Mech. 2021, 2 196

with different degrees of complexity are available and are more or less able to display the non-linear material behavior of rubber, characterized by softening, residual deformation and hysteresis. The material models can be categorized into two groups. Physically-based material models such as the tube model [14,15] or dynamic flocculation model [16,17], which try to capture the material behavior based on physical processes causing the structural changes. Phenomenological material models such as the hyperelastic models [18–23], which describe the material behavior without necessarily knowing the physical reasons for what is really happening locally. The accuracy of the simulation depends on the selection of the material model. However, all material models have limitations that must be taken into account and only provide meaningful results within their range of validity [24–26]. In the frame of this study, hyperelastic material models are used because the simulation effort required to describe the material behavior is small compared to physical material models. Additionally, they are implemented in the common finite element method (FEM) programs. The neo-Hooke model is sufficient for approximate numerical calculations. The model contains only one fit parameter, proportional to shear modulus and displays the length change in different spatial directions. The Mooney–Rivlin model contains two fit parameters and considers the surface area change in addition to the length change in different spatial directions. In order to increase the accuracy of the prediction, the Yeoh model is used. With its three fit parameters, the Yeoh model could be more suitable for modeling strong non-linear effects of rubber material. In this article, a time-saving characterization approach of styrene butadiene rubber (SBR)-based rubber material is presented. The focus is on determining the compression- equivalent deformation state measured in tensile mode using FEM and comparing it to real experimental data obtained from compression mode.

2. Theoretical Basics 2.1. Basics The continuum mechanics describes the kinematics of deformable bodies without detailed knowledge of their physical and chemical inner structures. A deformable body is divided into finite elements so that it can be described as a continuous set of material points. Since soft materials are discussed in the paper, the finite strain theory is considered. In the reference configuration, the time t = 0, a material point A0 has the following coordinates 3 X = ∑ Xiei, (1) i=1

where X is its position vector and Xi are the coordinates in the three directions in space defined by the coordinate vector ei with i = {1, 2, 3}. The position of the material point A0 is subject to change if the mechanical load is applied. In the instantaneous configuration at t > 0, the material point, denoted as A1, has the following coordinates:

3 x = ∑ xiei, (2) i=1

where x is the new position vector and xi are the coordinates in the same coordinate system ei. The displacement vector u determines the new position in space of the material point A0 using its position vectors in the two configurations according to

u = x − X. (3)

Figure1 shows the reference and current configuration of the material points A0 and B0 in the same coordinate system ei as well as the Equation of motion Φ(x, t). Appl. Mech. 2021, 2, FOR PEER REVIEW 3

𝒖=𝒙−𝑿. (3) Appl. Mech. 2021, 2 197 Figure 1 shows the reference and current configuration of the material points 𝐴 and 𝐵 in the same coordinate system 𝒆𝒊 as well as the Equation of motion Φ(𝑥, 𝑡).

Figure 1. Reference and current configuration of the material points 𝐴 and 𝐵 in the same coor- Figure 1. Reference and current configuration of the material points A0 and B0 in the same coordinate dinate system 𝒆𝒊. as well as the Equation of motion Φ(𝑿, 𝑡). system ei as well as the Equation of motion Φ(X, t). Φ(𝑿, 𝑡) 𝒆 Φ(X, t) describesdescribes the the trajectory trajectory of of material material points points in in the the coordinate coordinate system system e𝒊i.. It It is is a a continuous,continuous, bijective bijective function and,and, therefore,therefore, hashas an an inverse inverseΦ − Φ1(x)(.𝒙 The). The latter latter ensures ensures the themapping mapping of theof the material material points points from from the th referencee reference to the to currentthe current configuration configuration and and vice viceversa. versa. The The Equation Equation of motion of motion and and its inverse its inverse take take the followingthe following form form [27, 28[27,28]:]: 𝒙=Φ(𝑿) (4) x = Φ(X) (4) 𝑿=Φ(𝒙). (5) X = Φ−1(x). (5) The local deformation of a material point results from the local length and angle changesThe between local deformation the material of line a materialelements. point The second-order results from deformation the local length tensor and 𝑭angle and itschanges inverse between 𝑭 are the defined material as follows: line elements. The second-order deformation tensor F and its inverse F−1 are defined as follows: 𝜕 𝑭=∂ Φ(𝑿) (6) F = 𝜕𝑿Φ(X) (6) ∂X 𝜕 𝑭 = Φ(𝒙). (7) 𝜕𝒙∂ F−1 = Φ−1(x). (7) The deformation is stretching or shear.∂ Thex right Cauchy–Green tensor 𝑪 describes the distortionThe deformation of the material is stretching points. or 𝑪 shear. is positive The right semi-definite, Cauchy–Green symmetric, tensor refersC describes to the referencethe distortion configuration of the material and is points.as follows:C is positive semi-definite, symmetric, refers to the reference configuration and is as follows: 𝑪=𝑭 ∙𝑭. (8) C = FT·F In the principal stretch ratios representation .𝑭 and 𝑪 take the following form: (8)

In the principal stretch ratios representation𝜆 00F and C take the following form: 𝑭=0𝜆 0 (9)   λ100𝜆0 0 F = 0 λ 0 (9) and  2  0 0 λ3

and  2  λ1 0 0 2 C =  0 λ2 0 , (10) 2 0 0 λ3 Appl. Mech. 2021, 2 198

where λi are the principal stretch ratios in the dimension i. The principal stretch ratio in one dimension is λ = 1 + ε = l/l0, (11)

where ε is the strain, l is the actual length of the sample and l0 is the initial length of the sample. From the right Cauchy–Green tensor C, the three invariants I1, I2 and I3 are derived as follows: 2 2 3 I1 = λ1 + λ2 + λ2, (12) 2 2 2 2 3 2 I2 = λ1λ2 + λ2λ3 + λ2λ1, (13)

I3 = λ1λ2λ3 (14)

I1 informs about the length change, I2 about the surface change and I3 about the volume change. For incompressible materials, I3 is equal to 1. This identity means that the volume is conserved. Therefore, if the principal stretch ratio λi in the direction of deformation, under uniaxial tension or compression deformation, has the following form:

λ1 = λ. (15)

Then applied in the cross-sectional plane √ λ2 = λ3 = 1/ λ. (16)

2.2. Hyperelastic Material Models The aim of the work is to determine the correlation between the strain amplitudes in uniaxial tensile and compression mode using the finite element method. In order to compare the mechanical behavior of the rubber samples in the different load modes, it is necessary to ensure that the mechanical state present in the material is the same. This precondition is based on the intrinsic properties of the material itself. Rubber materials usually show softening, regardless of load mode stress, when they are subjected to constant strain or creep behavior when they are subjected to a constant force. The pure consideration of the strain or stress amplitude does not necessarily lead to the same mechanical state within the sample. Other effects, like temperature, crosshead speed or frequency may also play a role. Besides the measurement settings, the mechanical state also depends on polymer chains, their molecular weight, filler type, filler content and other additives. That is why we deliberately did not use the absolute values of mechanical stress or strain for the simulation. However, the strain energy density function W is defined as the integral area below the stress–strain curve. Equal areas stand for the same strain energy density functions. This represents a more suitable approach to characterize the mechanical state within rubber samples. The strain energy density function also considers the mechanical prehistory of the sample. The comparison of mechanical stresses or strains can lead to misstatement. It can happen that the peak values of the mechanical stress or strain are the same, but the course of the stress–strain curves is completely different. This in turn leads to different deformation states or different strain energy density functions. Since the strain energy density function is calculated from the area below the stress–strain curve, equal areas stand for the same strain energy density functions and then for the same mechanical state within the rubber material. Various hyperelastic material models well describe the non-linear viscoelastic behavior of rubber material at large deformations. They are defined by their strain energy density function W, which depends on the first, second and third invariant of the Green–Cauchy Appl. Mech. 2021, 2 199

tensor. In this study, an isotropic, incompressible material behavior is assumed. The first Piola–Kirchhoff stress tensor P is determined as follows. ∂W  −1 P = 2 FT , (17) ∂C

−1 where FT is the inverse of FT. For determining the correlating between the strain amplitudes in tensile and compression, only the first Piola–Kirchhoff stress in the main loading directions is considered. In order to perform the adjustment and the simulation of filled rubber behavior, three appropriate material models were chosen. The material model neo–Hooke (NH) has only one fit parameter that is proportional to the shear modulus. The strain energy density only depends on the first invariant of the right Cauchy–Green tensor, which describes the length change in different spatial directions. That is what is needed to describe the measurements performed in tensile and compression. The Yeoh model (Y) also depends on the first invariant of the right Green–Cauchy tensor. With three fit parameters, the Yeoh model is more appropriate for modeling of strong non-linear effects. In contrast to the neo–Hooke and Yeoh, the Mooney–Rivlin model (MR) depends on the first and the second invariant of the right Cauchy–Green tensor. Besides length change in different spatial directions, the Mooney–Rivlin model takes into consideration the surface area change. Additionally, two fit parameters offer a more degree of freedom for the modeling of non-linear effects than the neo–Hooke model. The strain energy density function WNH for a neo-Hookean material is [18,19]

WNH = C1·(I1 − 3), (18)

where C1 is a material constant and I1 is the first invariant (trace) of the right Cauchy–Green deformation tensor (Equation (12)). According to Equations (15) and (16), the strain energy density becomes as follows:

 2  WNH = C1· λ + 2/λ − 3 . (19)

The corresponding first Piola–Kirchhoff stress is

 2 PNH = 2 C1· λ − 1/λ . (20)

For a Mooney–Rivlin material, the strain energy density function WMR is [20,21]

WMR = C1·(I1 − 3) + C2·(I2 − 3), (21)

where C2 is a material constant and I2 is the second invariant of the right Cauchy–Green deformation tensor (Equation (13)). The corresponding first Piola–Kirchhoff stress is

 2 PMR = 2 (C1 + C2/λ) λ − 1/λ . (22)

Yeoh materials have a strain energy density function WY as follows [22,23]

2 3 WY = C1·(I1 − 3) + C2·(I1 − 3) + C3·(I1 − 3) , (23)

where C3 is a material constant. The corresponding first Piola–Kirchhoff stress is

 2  2  2   2  PY = 2 λ − 1/λ C1 + 2 C2 λ + 2/λ − 3 + 3C3 λ + 2/λ − 3 . (24)

The following constraints are taken during the fit process into account. The fit pa- rameter C1 in the neo-Hooke model, also present in Mooney–Rivlin and Yeoh models, Appl. Mech. 2021, 2 200

Appl. Mech. 2021, 2, FOR PEER REVIEW 6

is for consistency reason with the theory of in the limit of small strains strict positive. In this context, C1 is proportional to shear modulus, which has for rubber a the neo-Hooke model strict positive. This constraint permits a negative value for 𝐶. The non-zero positive value. In the Mooney–Rivlin model, C1 + C2 is for the same reason as for fit parameter 𝐶 in Yeoh model is freely selected the neo-Hooke model strict positive. This constraint permits a negative value for C2. The fit parameter C in Yeoh model is freely selected 2.3. Numerical Investigation3 2.3. NumericalThe fitting Investigation of the measurement data from the tensile experiment was automatically performedThe fitting using of Abaqus. the measurement The fit parameters data from are the determined tensile experiment based on was a method automatically of least squares.performed The using accumulated Abaqus. Therelative fit parameters error between are determined the fitted and based measured on a method mechanical of least stresssquares. at the The same accumulated strain is relativeminimized. error The between fitting the is controlled fitted and measuredusing the mechanicalDrucker stability stress check,at the samewhich strain presupposes is minimized. that the The tangential fitting is controlledmaterial stiffness using the is positive Drucker definite. stability check, whichThe presupposes fitting process that starts the tangential by importing material the nominal stiffness stress–strain is positive definite. values into Abaqus. Then,The the fittingmaterial process model starts and bythe importing load condit theions nominal are chosen, stress–strain Afterwards, values the into evaluation Abaqus. starts,Then, thedetermining material modelthe specific and the material load conditions constants and are chosen,the goodness Afterwards, of fit. the evaluation starts,To determining determine the the stress–strain specific material curve constants in the compression and the goodness mode, simulations of fit. were car- ried outTo determineeach time using the stress–strain one of the three curve hyperelastic in the compression material models mode, in simulations Abaqus. were carriedFor out the each simulation, time using a 3D one deformable-body of the three hyperelastic was used material for the rubber models sample in Abaqus. and two rigid Forbody the models simulation, were used a 3D for deformable-body the sample holders. was used for the rubber sample and two rigidA body static models finite element were used analysis for the was sample carri holders.ed out. A 1 mm long element edge of type C3D8HA staticmesh finite was used element to mesh analysis the wassample. carried This out. type A of 1mm finite long element element mesh edge is ofwidely type usedC3D8H for meshincompressible was used material to mesh thebehavior. sample. The This boundary type of conditions finite element were mesh defined is widely at the referenceused for incompressiblepoint of the lower material sample behavior. holders. The At boundary this point, conditions all degrees were of freedom defined atwere the prevented.reference point At the of reference the lower point sample of the holders. upper sample At this holder, point, allthe degrees displacement of freedom in the wereload directionprevented. was At specified. the reference The pointrotations of the and upper the remaining sample holder, displacement the displacement degrees of in freedom the load weredirection prevented. was specified. Figure The2 shows rotations a schemati and thec remainingdiagram of displacement the finite element degrees model of freedom used duringwere prevented. the FEM simulation Figure2 shows highlighting a schematic the sample diagram geometry of the finite and elementthe boundary model condi- used tion.during the FEM simulation highlighting the sample geometry and the boundary condition.

Figure 2. SchematicSchematic diagram diagram of ofthe the finite finite element element model model used used duri duringng the finite the finite element element meth methodod (FEM) (FEM) simulation simulation high- lightinghighlighting the sample the sample geometry geometry and the and boundary the boundary conditions. conditions.

AtAt the contact surface between the sample and the sample holders, the friction was defineddefined by by two two components: components: the the normal normal and and tangent tangent one. one. The Thenormal normal component component of fric- of tionfriction ensures ensures that thatthe sample the sample holder holder does no doest penetrate not penetrate the sample. the sample. The tangential The tangential compo- nentcomponent of the friction of the friction describes describes the relative the relative sample sample movement movement on the on sample the sample holder. holder. In the In the simulation, the normal component was assumed a “hard contact”. This would prevent simulation, the normal component was assumed a “hard contact”. This would prevent penetration. The tangential component of the friction was defined as “frictionless”. This penetration. The tangential component of the friction was defined as “frictionless”. This means that the coefficient of friction is zero. The last assumption was made because there

Appl. Mech. 2021, 2 201

means that the coefficient of friction is zero. The last assumption was made because there is no precise knowledge about changes in the coefficient of friction during the measurement. The force-displacement curve and the resulting stress–strain curve were evaluated at the reference points of the sample holders. In order to determine the compression-equivalent deformation of the SBR sample measured in tensile mode, the following steps are performed: i. calculate the stress–strain curve from tensile measurements. ii. use Abaqus to fit the experimental stress–strain curves using the hyperelastic material models neo-Hooke (NH), Mooney–Rivlin (MR) and Yeoh (Y). The first Piola–Kirchhoff stress P, as well as the fit parameters, are determined. iii. with the obtained fit parameters derive the strain energy density function W as a function of the principal stretch ratio λ for the uniaxial tension and compression deformation using the hyperelastic material models mentioned above. iv. simulate the first Piola–Kirchhoff stress P for compression measurement using W. v. compare P to the stress–strain curve from compression measurements.

3. Experiments and Testing 3.1. Samples A high-performance tread compound based on SBR (from Lanxess Germany) ex- tended with 37.5 parts of highly aromatic oil and filled with 70 phr carbon black N 234 (from Orion Engineered Carbon Germany) is prepared. Besides the zinc oxide (ZnO) and stearic acid, the antioxidants N-(1,3-dimethylbutyl)-N0-phenyl-p-phenylenediamine (6PPD) and 2,2,4-Trimethyl-1,2 dihydrochi-nolin (TMQ) are added. A protective wax for rubber compounds against cracking is also used. The SBR is sulfur-cured. The crosslinking system is additionally composed of two accelerators: N-cyclohexylbenzthiazol-2-sulfenamide (CBS) and 1,3-diphenylguanidine (DPG). The full recipe is presented in Table1.

Table 1. Recipe of the high-performance tread compound.

Component Amount [phr] SBR 1712 100.00 Aromatic oil 37.50 N 234 70.0 ZnO 3.00 Stearic acid 2.00 Antioxidant 6PPD 2.00 Antioxidant TMQ 2.00 Wax 2.00 Sulfur 1.75 Accelerator CBS 1.00 Accelerator DPG 0.40

For the measurement in tensile mode, 40 mm long and 10 mm wide strip specimens were punched out from a 2 mm thick cured SBR sheet. For compression tests, cylindrical samples with a diameter of 17.5 mm and a height of 25 mm were prepared.

3.2. Mechanical Analysis A crosshead speed of 10 mm/min is used to enable quasi-static investigations. The rubber materials are deformed up to 200% in a tensile mode in order to attain the non-linear material behavior. In compression, the samples were subjected to a strain of 35%. Appl. Mech. 2021, 2, FOR PEER REVIEW 8

Appl. Mech. 2021, 2 202

Three samples are examined in each test mode. The high-force dynamic-mechanical analyzerThree DMA samples GABO are examinedEPLEXOR in 500 each N testfrom mode. Netzsch The high-forceGerätebau dynamic-mechanicalwas used. The DMA analyzerGABO EPLEXOR DMA GABO 500 N EPLEXOR is a dynamic-mechanical 500 N from Netzsch analyzer Gerätebau with two was independent used. The DMAdrives GABOfor the EPLEXOR generation 500 of Nstatic is a and dynamic-mechanical dynamic forces. A analyzer servomotor with generates two independent the static drives forces forand the an generation electrodynamic of static shaker and dynamicsystem genera forces.tes A the servomotor dynamic generatesforces. The the two static drives forces are andindependently an electrodynamic controlled shaker of each system other. generates With a maximal the dynamic deformation forces. Therange two of drives60 mm, are the independentlyDMA GABO EPLEXOR controlled 500 of each N can other. also Withbe used a maximal as a universal deformation tester. range of 60 mm, the DMA GABO EPLEXOR 500 N can also be used as a universal tester. 4. Results and Discussion 4. Results and Discussion The stress–strain curves for the SBR filled with 70 phr carbon black N234 measured in tensileThe stress–strain are shown in curves Figure for 3. the SBR filled with 70 phr carbon black N234 measured in tensileFigure are shown 3 shows in Figure three 3stress–strain. curves suggesting an excellent reproducibility of the measurementFigure3 shows data three with stress–strain a slightly curves small suggestingstatistical deviation, an excellent due reproducibility to material inhomo- of the measurementgeneity. One datarepresentative with a slightly stress–strain small statistical curve of deviation, the SBR sample due to material filled with inhomogeneity. 70 phr N 234 Oneis fitted representative with Abaqus stress–strain using the neo-Hooke curve of the (NH), SBR sampleMooney–Rivlin filled with (MR) 70 phrand NYeoh 234 (Y) is fitted mod- withels. The Abaqus respective using thefit curves neo-Hooke are shown (NH), in Mooney–Rivlin Figure 4. (MR) and Yeoh (Y) models. The respective fit curves are shown in Figure4. The curve fitting shows different course along the measured stress–strain curve. The curve fitting shows different course along the measured stress–strain curve. Compared to the neo-Hooke and Mooney–Rivlin models, the Yeoh model matches the Compared to the neo-Hooke and Mooney–Rivlin models, the Yeoh model matches the most suitable curvature of the stress–strain curves for SBR samples filled with 70 phr N234 most suitable curvature of the stress–strain curves for SBR samples filled with 70 phr N234 in tensile. in tensile. The mean squared error was used as a criterion for the fitting quality of the three The mean squared error was used as a criterion for the fitting quality of the three hyperelastic material models. The smaller the mean squared error, the better the fitting hyperelastic material models. The smaller the mean squared error, the better the fitting quality. The determined mean squared error of the fit curves and the measurement curve quality. The determined mean squared error of the fit curves and the measurement curve in tensile are listed in Table 2. in tensile are listed in Table2. The Yeoh model provides the smallest mean squared error and thus provides the best The Yeoh model provides the smallest mean squared error and thus provides the best fit. The neo-Hooke comes off the worst. fit. The neo-Hooke comes off the worst. ThisThis stepstep aimsaims toto determine determine accordingaccording toto EquationsEquations (19),(19), (21)(21) and and (23) (23) the the material material constants 𝐶, 𝐶 and 𝐶. The obtained fit parameters are summarized in Table 3. constants C1, C2 and C3. The obtained fit parameters are summarized in Table3.

4 Sample 1 Sample 2 Sample 3 3

2 (MPa) P

1

0 0 50 100 150 200 ε (%)

FigureFigure 3. 3.Stress–strain Stress–strain curvescurves measuredmeasured inin tensiletensile forfor styrenestyrene butadienebutadiene rubberrubber (styrene(styrene butadienebutadiene rubberrubber (SBR) (SBR) filledfilled withwith 7070 phrphr carboncarbon blackblack N234 at room temperature and and a a crosshead crosshead speed speed of of 1010 mm/min. mm/min.

Appl.Appl. Mech. Mech.2021 2021, 2, 2, FOR PEER REVIEW 2039

4 Measurement data NH fit MR fit 3 Y fit

2 (MPa) P

1

0 0 50 100 150 200 ε (%)

FigureFigure 4. 4.Fit Fit of of aa representativerepresentative stress–strain stress–strain curve curve of of the the SBR SBR filled filled with with 70 70 phr phr N234 N234 with with Abaqus Abaqus usingusing neo-Hooke neo-Hooke (NH), (NH), Mooney–Rivlin Mooney–Rivlin (MR) (MR) and and Yeoh Yeoh (Y) (Y) models. models.

TableTable 2. 2.Mean Mean squared squared error error of of the the measurement measurement curve curv ine in tensile tensile and and the the fit curvesfit curves using using neo-Hooke, neo- Mooney–RivlinHooke, Mooney–Rivlin and Yeoh and models. Yeoh models.

Neo-HookeNeo-Hooke Mooney–RivlinMooney–Rivlin YeohYeoh MeanMean squared squared error error 0.027 0.027 0.017 0.017 0.006 0.006

Table 3. Fit parameters of a stress–strain curve of the SBR filled with 70 phr N234 using neo- TableHookean, 3. Fit Mooney–Rivlin parameters of a stress–strainand Yeoh models. curve of the SBR filled with 70 phr N234 using neo-Hookean, Mooney–Rivlin and Yeoh models. 𝑪 𝑪 𝑪 𝟏 𝟐 𝟑 C1 [MPa][MPa] C2 [MPa][MPa] C3 [MPa] Neo-HookeNeo-Hooke 0.60020.6002 - - - - Mooney–Rivlin 0.7216 −0.2837 - Mooney–Rivlin 0.7216 −0.2837 - Yeoh 0.5621 −0.0052 0.0014 Yeoh 0.5621 −0.0052 0.0014

The material constants 𝐶, 𝐶 and 𝐶 are used to calculate the strain energy den- sity Thefunction material 𝑊 according constants toC1 Equations, C2 and C (18),3 are (20) used an tod (22) calculate for neo-Hooke, the strain energyMooney–Rivlin density functionand YeohW materialsaccording respectively. to Equations The (18), strain (20) and energy (22) density for neo-Hooke, function Mooney–Rivlin 𝑊 as a function and of Yeohthe principal materials stretch respectively. ratio 𝜆 The for the strain uniaxial energy tension density and function compressionW as adeformation function of state the principalis plotted stretch in Figure ratio 5. λ for the uniaxial tension and compression deformation state is plottedIn inFigure Figure 5,5 .the characterized principal stretch ratio 𝜆=1represents the state with- out Indeformation Figure5, the because characterized this implies principal a zero-strain stretch ratio 𝜀λ = = 10 accordingrepresents theto Equation state without (11). deformationAbove this limit, because the thisSBR impliessample ais zero-strainsubjected toε a= uniaxial0 according tensile to strain. Equation Below (11). this Above limit, thisthe limit,SBR sample the SBR is sample subjected is subjected to a compression to a uniaxial deformation. tensile strain. In Belowreal applications, this limit, the rubber SBR samplesamples is can subjected be compressed to a compression up to 50% deformation. corresponding In real to a applications, principal stretch rubber ratio samples of 𝜆= can0.5. be As compressed an incompressible up to 50% materialcorresponding, the principal to a principal stretch ratio stretch 𝜆=0 ratio is of forλ rubber= 0.5. Asout an of incompressiblereach. material, the principal stretch ratio λ = 0 is for rubber out of reach. TheThe strain strain energy energy density density functions functionsW, calculated𝑊, calculated as a functionas a function of the principalof the principal stretch ratiostretchλ from ratio the𝜆 from fit parameters the fit parameters obtained obtained in Table 1in using Table the 1 using neo-Hooke, the neo-Hooke, Mooney–Rivlin Mooney– andRivlin Yeoh and models, Yeoh models, show a show quite a similar quite simila curver shapecurve shape for the for tensile the tensile area. Thearea. deviations The devia- become obvious between the single curves at high loads even though the general trend tions become obvious between the single curves at high loads even though the general

Appl. Mech. 2021, 2 204 Appl. Mech. 2021, 2, FOR PEER REVIEW 10

remains preserved. For the compression area, the Mooney–Rivlin curve deviates already at trend remains preserved. For the compression area, the Mooney–Rivlin curve deviates small strains from the tighter neo-Hooke and Yeoh curves. already at small strains from the tighter neo-Hooke and Yeoh curves. 4

WNH

WMR

3 WY

2 Compression Tensile (MPa) W

1

0 0.5 1.0 1.5 2.0 2.5 3.0 λ (-)

FigureFigure 5. 5.StrainStrain energy energy density density function function 𝑊 W as asa function a function of the of the stretch stretch ratio ratio 𝜆 forλ for SBR SBR filled filled with with 7070 phr phr carbon carbon black black N234. N234. In the following, the Yeoh graph is discussed in detail. The conclusions obtained are In the following, the Yeoh graph is discussed in detail. The conclusions obtained are also valid for the Mooney–Rivlin and neo-Hooke graphs. Only the numerical values vary. also valid for the Mooney–Rivlin and neo-Hooke graphs. Only the numerical values vary. The principal stretch ratio λ = 0.8 corresponds to a compression deformation of 20%. The principal stretch ratio 𝜆 = 0.8 corresponds to a compression deformation of 20%. The corresponding strain energy density function W is 0.08 MPa. The same W can also be The corresponding strain energy density function 𝑊 is 0.08 MPa. The same 𝑊 can also assigned to the principal stretch ratio λ = 1.23. This means, for the same strain energy be assigned to the principal stretch ratio 𝜆 = 1.23. This means, for the same strain energy density function W, the SBR sample can also be stretched to a deformation of 23% instead density function 𝑊, the SBR sample can also be stretched to a deformation of 23% in- of the compression deformation discussed above. Additional two strain energy density steadfunctions of the Wcompressionare selected: deformation 0.04 MPa and discusse 0.16 MPa.d above. Additional two strain energy den- 𝑊 0.04 0.16 sity functionsThe selected are strain selected: energy values MPa cover and a wide MPa. load range for real applications. Up to 30%The compressionselected strain deformation energy values represents cover a a wi typicalde load load range during for applicationsreal applications. for rubber Up tomaterials 30% compression like seals deformation or tires. represents a typical load during applications for rubber materialsThe like correlated seals or strains tires. in the uniaxial tension and compression are determined for the threeThe models correlated used strains and are in summarized the uniaxial tension in Table and4. compression are determined for the three models used and are summarized in Table 4. Table 4. Determination of the compression-equivalent strain of SBR measured in tensile mode from the Tablestrain 4. energy Determination density function of the compression-equivalentW calculated using the strain neo-Hooke, of SBR Mooney–Rivlin measured in tensile and Yeoh mode models. from the strain energy density function 𝑊 calculated using the neo-Hooke, Mooney–Rivlin and Yeoh models. Strain Energy Density Compression Strain Tensile Strain Function W [MPa] [%] [%] Strain Energy Density Func- Compression− Strain Tensile Strain tion W0.04 14.5 16.0 [%] [%] Neo-Hooke [MPa]0.08 −20.0 23.0 0.04 −14.5 16.0 0.16 −27.5 33.0 Neo-Hooke 0.08 −20.0 23.0 0.16 0.04 −27.5− 16.033.0 17.0 Mooney–Rivlin 0.04 0.08 −16.0− 23.017.0 24.2 Mooney–Rivlin 0.08 −23.0 24.2 − 0.16 0.16 −32.0 32.035.0 35.0 0.04 0.04 −14.5− 14.516.0 16.0 Yeoh 0.08 −20.0 23.0 Yeoh 0.08 −20.0 23.0 0.16 −27.5 34.0 0.16 −27.5 34.0

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It can rightly be claimed that for the same strain energy density function W, indepen- It can rightly be claimed that for the same strain energy density function 𝑊, inde- dently of the hyperelastic model used, the sample can be either stretched or compressed dependingpendently onof the deformationhyperelastic direction.model used, It should the sample be noted, can that be dueeither to thestretched incompressibil- or com- itypressed constraint depending of rubber on the absolutedeformation value direction. of the attained It should strain be innoted, tensile that is alwaysdue to the larger in- thancompressibility that in compression. constraint of rubber the absolute value of the attained strain in tensile is alwaysTo checklarger thesethan findings,that in compression. three stress–strain curves in compression mode for SBR filled with 70To phrcheck carbon these blackfindings, N234 three are performed.stress–strain The curves test in results compression are shown mode in Figure for SBR6. filled with 70 phr carbon black N234 are performed. The test results are shown in Figure 6 0 Sample 1 Sample 2 Sample 3 –0.5

–1.0 (MPa) P

–1.5

–2.0 –30 –20 –10 0 ε (%)

FigureFigure 6.6. Stress–strainStress–strain curves measured in compression for for SBR SBR filled filled with with 70 70 phr phr carbon carbon black black N N 234234 atat roomroom temperaturetemperature andand aa crossheadcrosshead speed speed of of 10 10 mm/min. mm/min.

SimilarSimilar toto stress–strainstress–strain measurementmeasurement inin tensile,tensile, FigureFigure6 6 confirms confirms an an excellent excellent repro- repro- ducibility of the measurement. A small statistical deviation is however inevitable due to ducibility of the measurement. A small statistical deviation is however inevitable due to material inhomogeneity. material inhomogeneity. Figure7 illustrates a representative stress–strain curve of the SBR sample in compres- Figure 7 illustrates a representative stress–strain curve of the SBR sample in compres- sion mode and the corresponding simulated curves using the neo-Hooke, Mooney–Rivlin sion mode and the corresponding simulated curves using the neo-Hooke, Mooney–Rivlin and Yeoh models derived from measurement data in tensile. and Yeoh models derived from measurement data in tensile. Compared to tensile measurements, the simulated stress–strain curves in compression Compared to tensile measurements, the simulated stress–strain curves in compres- mode using the Mooney–Rivlin model differs considerably from the results obtained using sion mode using the Mooney–Rivlin model differs considerably from the results obtained the neo-Hooke and Yeoh models, which are almost comparable. using the neo-Hooke and Yeoh models, which are almost comparable. The simulated stress–strain curve for compression differs quite clear from the experi- The simulated stress–strain curve for compression differs quite clear from the exper- mental measurement in contrast to tensile results, even for the Yeoh model, which fits the imental measurement in contrast to tensile results, even for the Yeoh model, which fits the measurement results performed in tensile. measurementThe determined results mean-squaredperformed in tensile. errors of the fit and the measurement curves in com- pressionThe aredetermined listed in Tablemean-squared5. errors of the fit and the measurement curves in com- pressionThe bestare listed fit quality in Table is achieved 5. with the neo-Hoke model. The Mooney–Rivlin model providesThe best the poorfit quality fit quality. is achieved The with fit quality the neo-Hoke of the Yeoh model. model The Mooney–Rivlin is comparable tomodel the neo-Hookeprovides the model. poor fit quality. The fit quality of the Yeoh model is comparable to the neo- HookeThe model. measured stress–strain curve is significantly stiffer than the three simulated curves. This behaviorThe measured is due tostress–strain the assumption curve made is signif for theicantly boundary stiffer conditions than thein three the simulation. simulated Forcurves. the uniaxialThis behavior compression, is due ato homogeneous the assumption deformation made for statethe boundary is assumed; conditions neglecting in thethe edgesimulation. effects andFor the friction uniaxial between compression, the sample a homogeneous and the sample deformation holder. state is assumed; neglectingFrom thethe rawedge data effects of the and compression friction between test, thethe areasample above and the the corresponding sample holder. stress– strainFrom curve the represents raw data the of strainthe compression energy density test, functionthe area aboveW. the corresponding stress– strain curve represents the strain energy density function 𝑊.

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𝑊 calculated from the raw data for compression and tensile tests as well as the stand- W calculated from the raw data for compression and tensile tests as well as the ard deviation are shown in Table 6. standard deviation are shown in Table6.

0

–0.5

–1.0 (MPa) P Measurement data –1.5 NH fit MR fit Y fit –2.0 –30 –20 –10 0 ε (%)

Figure 7. Representative stress–strain curve of the SBR filled with 70 phr N234 measured in compres- Figure 7. Representative stress–strain curve of the SBR filled with 70 phr N234 measured in compressionsion mode and mode the and corresponding the corresponding simulated simu curveslated curves using neo-Hookeusing neo-Hooke (NH), (NH), Mooney–Rivlin Mooney– (MR) Rivlinand Yeoh (MR) (Y) and models Yeoh derived(Y) models from derived measurement from measurement performed inperformed tensile. in tensile.

TableTable 5. 5. MeanMean squared squared error error of ofthe the measurement measurement curv curvee in compression in compression and and the fit the curves fit curves using using neo-Hooke,neo-Hooke, Mooney–Rivlin Mooney–Rivlin and and Yeoh Yeoh models. models.

Neo-HookeNeo-Hooke Mooney–RivlinMooney–Rivlin Yeoh Yeoh Mean sqauredsqaured error error 0.0332 0.0332 0.300 0.300 0.0501 0.0501

Table 6. Strain energy density function 𝑊 calculated from the raw data for compression and ten- sileTable tests. 6. Strain energy density function W calculated from the raw data for compression and ten- sile tests. Tensile Compression 𝑾 from 𝑾 from Com- Deviation TensileStrain Compressionstrain TensileW from pressionW from Deviation[%] Strain Strain Tensile Compression [%] [%] [MPa] [MPa] [%] [%]16.0 −[%]14.5 [MPa]0.066 [MPa]0.073 −9.59 Neo-Hooke 16.023.0 −20.014.5 0.116 0.066 0.126 0.073 −7.949.59 Neo-Hooke 23.033.0 −27.520.0 0.204 0.116 0.225 0.126 −9.337.94 33.017.0 −16.027.5 0.072 0.204 0.085 0.225 −−15.299.33 Mooney–Rivlin 17.024.2 −23.016.0 0.124 0.072 0.161 0.085 −−22.9815.29 Mooney– 24.235.0 −32.023.0 0.223 0.124 0.296 0.161 −−24.6622.98 Rivlin 16.0 −14.5 0.066 0.073 −9.59 35.0 −32.0 0.223 0.296 −24.66 Yeoh 23.0 −20.0 0.116 0.126 −7.94 16.0 −14.5 0.066 0.073 −9.59 34.0 −27.5 0.213 0.225 −5.33 Yeoh 23.0 −20.0 0.116 0.126 −7.94 The material models34.0 neo-Hooke−27.5 and Yeoh show 0.213 a deviation 0.225below 10% between−5.33 the measured and simulated strain energy density function 𝑊 for all strain amplitudes. The Yeoh Themodel material even shows models a reduced neo-Hooke deviation and Yeoh with show increasing a deviation strain amplitudes. below 10% betweenThe de- viationthe measured of the simulated and simulated value for strain compression energy density at 27.5% function is onlyW aboutfor all5%. strain This behavior amplitudes. is The Yeoh model even shows a reduced deviation with increasing strain amplitudes. The

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deviation of the simulated value for compression at 27.5% is only about 5%. This behavior is due to a better match of the stress–strain curves for SBR samples filled with 70 phr N234 in tensile. These results are within the simulation inaccuracies related to the simple assumptions made-no edge effects and no friction between the sample and the sample holder-and are in line with our expectations. It is worthy to note, that the deviation between the measured and simulated strain energy density function W using the Yeoh model decreases with increasing the strain. The Mooney–Rivlin model delivers a deviation between the measured and the simu- lated strain energy density function W larger than 15%. These unsatisfactory results are already guessed due to the poor fit quality compared to neo-Hooke and Yeoh models, observed in Figure7.

5. Conclusions The aim of this study is to simulate from experimental data measured in tensile how the stress–strain curves in compression mode look like and subsequently to compare them with real test results. The way to match the tensile and compression modes together is based on hyperelastic models for rubber materials. First, neo-Hooke, Mooney–Rivlin and Yeoh models are used to fit experimental data measured in tensile. From that, the strain energy density function W is calculated and then extrapolated to compression mode. The first Piola–Kirchhoff stress, and thus a “simulated” stress–strain curve in compression mode are derived. The simulations show taking into account the degree of simplicity of the material models used as well as non-consideration of friction and the edge effects a good agreement to the result measured in compression mode. It is true, none of the implemented models exactly captures the real behavior of the SBR material in compression. However, the deviation remains smaller than 10% for the neo-Hooke and Yeoh models. The deviation even decreases with increasing the strain using the Yeoh model implying a better capture of the measurement results at large displacements. This makes the Yeoh model more appropriate than neo-Hooke and Mooney–Rivlin models to describe the large non-linear behavior of rubber materials. With a deviation of up to 25%, the Mooney–Rivlin model is performing poorly in comparison to neo-Hooke and Yeoh models. The relationship achieved between the strain amplitudes in tensile and compression tests using Yeoh and neo-Hooke material models provides better adjustment curves to measurement results compared to Mooney–Rivlin material model. Mooney–Rivlin overestimates the strain amplitudes in compression due to the poor prediction of the stress–strain curve in compression from experimental results. Therefore, a good relationship between the tensile and compression modes is estab- lished. This serves to save time and money. For this purpose, the neo-Hooke model would be enough.

Author Contributions: Investigation, S.A.; measurements, S.A.; validation, M.E.Y.; Finite element method, M.E.Y., writing—review and editing, S.A. and M.E.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. Conflicts of Interest: The authors declare no conflict of interest. Appl. Mech. 2021, 2 208

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