Numerical modeling of elastomeric seismic isolators for determining force–displacement curve from cyclic loading
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Abstract
The ideal performance of seismic isolating systems during the past earthquakes has proved them to be very useful in protecting structures against earthquakes. The cyclic loading experimental tests are an important part in the process of completing the design of the isolators, yet they are very expensive and time consuming. Using the accurate analytical modeling of hysteresis tests and knowing the limitations and the amount of error of the finite elements model and its effect on designing the isolated structure make it possible to reduce the financial and time expenses involved in designing seismic isolators along with experimental tests. In the present study, the cyclic loading of two different isolating systems, namely, the high damping rubber bearing (HDRB) and lead rubber bearing (LRB) have been modeled and analyzed in ABAQUS and the outcomes were compared with the experimental results attained by other researchers. Regarding the fact that the most important and complicated component of the elastomeric isolating system is rubber, it was modeled using various strain energy functions. Other factors affecting the finite elements models of elastomeric isolators were also studied. After comparing the effective stiffness of the experimental sample with the analytical model of HDRB, the Yeoh function had the best performance in determining the effective stiffness of the isolating system with an error of less than 7%. In studying LRBs, too, three types of bearings with different dimensions and lateral strain values were studied; the polynomial function in shear strain value of 150% had the best performance in estimating effective stiffness and damping with errors of less than 3% and 18%, respectively.
Keywords
Cyclic loading test High damping rubber bearing Lead rubber bearing Finite element analysis Strain energy function Analytical modelingIntroduction

design displacement;

effective stiffness in the design displacement;

amount of energy damping in each cycle at the design displacement.
 1.
Investigate the possibility of reducing the expenses of manufacturing isolators through modeling the hysteresis cycle tests.
 2.
Exactly know the effective factors in the resulting error and the contribution of each of them in that.
 3.
Learn about the performance of seismic isolating systems before running experimental tests.
 4.
Control the future experimental tests.
 5.
Have the ability to build some new seismic isolators and model their tests using the results of the present study.
In the past, researchers have used numerical methods as a seismic isolator analysis tool. In all of these researches, the main goal was to obtain precise and inexpensive models for the analysis of isolators by numerical methods (Asl et al. 2014; Ohsaki et al. 2015; Mishra et al. 2013; Talaeitaba et al. 2019).
Finite element modeling
Modeling the seismic isolators using the finite element software program is generally done in two ways. In the first, the whole isolating part and the structures under and over it are modeled in the form of concentrated mass, spring, and damper, then the whole system behavior is assessed. In the second method, however, only the isolating system is modeled and tests on it were performed (Suhara et al. 1992; Martelli et al. 1992).
Modeling methods of the parts
At the beginning of the analysis process, each element can generally be modeled in three forms: twodimensional, threedimensional, and axisymmetrical. Twodimensional modeling has a lot of limitations and was popular in the past decades regarding the hardware possibilities of those days (Imbimbo and De Luca 1998). Most cases of modeling now are threedimensional or axisymmetrical. Forni et al. state that although in axisymmetrical models the solution process takes less time, they will not be very accurate for shear strains of more than 150% and threedimensional models are more efficient for horizontal deformations (Talaeitaba et al. 2019).
Introducing materials
The most important step in modeling an elastomeric isolator is defining the materials especially rubber. In this section, the properties of the materials used in the model are explained.
Steel
Steel properties (Imbimbo and De Luca 1998)
Modulus of elasticity  210,000 MPa 
Poisson’s ratio  0.3 
Yield stress  240 MPa 
Lead
Lead properties (Doudoumis et al. 2005)
Modulus of elasticity  18,000 MPa 
Poisson’s ratio  0.43 
Yield stress  19.5 MPa 
Rubber
The elastomeric materials have an almost linear behavior in small strains; however, their behavior is highly nonlinear and elastic in large strains. This nonlinear behavior causes the material’s parameters including the shear modulus and elasticity modulus to change as the strain increases (Guo and Sluys 2008). The rubber’s shear modulus and damping depend on the load size, temperature changes, and the strain history (Charlton et al. 1993).
In the finite element program, materials whose stress–strain curve in large deformations is nonlinear and elastic are called hyperelastic materials. Polymers such as rubber are among these (APASmith 2007).
To model these materials, they can be assumed to be isotropic, isothermal, elastic, and incompressible. The effect of loading frequency and time on their behavior is also ignored (Salomon et al. 1999; Venkatesh and Srinivasa Murthy 2012).
Hyperelastic materials are described in terms of “strain energy potential” (U) which defines the strain energy stored in the material per unit of reference volume (volume in the initial configuration) as a function of the strain at that point in the material (APASmith 2007).
 1.
The stress–strain function of the model will not change for frequent loadings.
 2.
The stress–strain function is fully reversible.
 3.
The materials are assumed to be completely elastic with no permanent deformation.
There are several forms of strain energy potentials available in Abaqus to model approximately incompressible isotropic elastomers which are listed below. In the presented equations, I_{1}, I_{2}, and I_{3} are the deviatoric strain invariants. Strain energy functions are defined by these coefficients. Also, J^{e1} is the elastic volume ratio (APASmith 2007).
 1.
Uniaxial tension and compression test.
 2.
Equibiaxial tension and compression test
 3.
Planar shear test (also known as pure shear).
 4.
Volumetric tension and compression.
All tests must be done on the same material and at the same temperature. The most commonly performed experiments are uniaxial tension, uniaxial compression, and planar tension. After running these tests and determining the strain–stress relation in each of the above modes, the results are fed into the program and the program matches the results with the function; then the adapted curve is exhibited and the required coefficients are determined (APASmith 2007). In other words, for each of the above tests, the test is simulated with ABAQUS software, and then the required parameters of the simulation are extracted.
Modeling the high damping rubber bearing (HDRB)
The high damping rubber bearing under study is selected from the research done by Yoo et al. in the Korea Atomic Energy Research Institute (Doudoumis et al. 2005; Yoo et al. 2002).
Geometry
Geometric features of the HDRB (Doudoumis et al. 2005)
Diameter of isolator (mm)  125 
Thickness of rubber sheet (mm)  2.5 
Number of rubber sheets  12 
Total rubber thickness (mm)  30 
Initial shape factor  12 
Thickness of inner steel plates (mm)  1 
Number of steel plates  11 
Thickness of top and bottom loading plates (mm)  5 
Defining materials
Coefficients of the strain energy functions for the rubber in HDRBs
Function  \(C_{10}\)  \(C_{01}\)  \(C_{11}\)  \(C_{20}\)  \(C_{02}\)  \(C_{30}\) 
Mooney–Rivlin  0.428550  − 0.068869  –  –  –  – 
NeoHookean  0.232304  –  –  –  –  – 
Yeoh  0.197738  –  –  0.002146  –  0.000091 
Polynomial (N = 2)  0.195291  − 0.013261  0.000203  0.006484  − 0.000179  – 
Function  \(\mu\)  \(\lambda_{m}\)  \(a\)  \(\beta\)  –  – 
van der Waals  0.401558  4.381771  0.140441  0  –  – 
Arruda–Boyce  0.393941  3.592024  –  –  –  – 
Function  \(\alpha_{3}\)  \(\alpha_{2}\)  \(\alpha_{1}\)  \(\mu_{3}\)  \(\mu_{2}\)  \(\mu_{1}\) 
Ogden (N = 3)  0.092292  4.024120  0.266487  − 2.827823  0.063751  3.212091 
Afterwards to specify the amount of error in each model, the value of shear modulus for each function was compared with its experimental value.
Amount of error in calculating the model’s initial shear modulus in comparison to the experimental value
Function  Model’s shear modulus (MPa)  Experimental shear modulus (Salomon et al. 1999) (MPa)  Error percentage (%) 

Mooney–Rivlin  0.72  0.40  80.00 
NeoHookean  0.46  0.40  15.00 
Ogden (N = 3)  0.45  0.40  12.50 
Yeoh  0.40  0.40  0.00 
Arruda–Boyce  0.41  0.40  2.50 
Polynomial (N = 2)  0.38  0.40  − 5.00 
van der Waals  0.40  0.40  0.00 
Loading
According to the experimental processes on the model, in the first step, a vertical load of 50 kN was applied to the model uniformly distributed on top. In the second step, as the load exertion continues, a shear displacement of 60 mm was applied to the system. The amount strain due to this shear displacement was equal to 200%.
Meshing
Solution method
To analyze the finite elements model, the static general analysis was used; due to the high amount of displacement, the nonlinear geometry was also activated.
The analysis results
Comparing the calculated effective stiffness with the experimental results in HRDBs
Samples  \(F^{ + }\) (KN)  \(\Delta^{ + }\) (mm)  \(F^{  }\) (KN)  \(\Delta^{  }\) (mm)  \(K_{\text{eff}}\) (KN/mm)  Error percentage 

Experimental  9.770  63.151  − 10.712  − 61.766  0.164  – 
Mooney–Rivlin  17.279  60.000  − 17.040  − 59.121  0.288  75.72% 
NeoHookean  11.199  60.000  − 10.594  − 56.625  0.187  13.98% 
Ogden (N = 3)  10.833  60.000  − 10.124  − 56.625  0.180  9.60% 
Yeoh  10.555  60.000  − 9.861  − 56.625  0.175  6.78% 
Arruda–Boyce  10.535  60.000  − 10.465  − 59.625  0.176  7.08% 
Polynomial (N = 2)  11.252  60.000  − 10.392  − 56.625  0.186  13.19% 
van der Waals  10.663  60.000  − 10.589  − 59.625  0.178  8.36% 
What is concluded from the hysteresis loops and Table 6 is that the effective stiffness of the experimental model is generally less than that of the analytical models. The best result is for Yeoh function. Arruda–Boyce, van der Waals, and Ogden (N = 3) functions come next, respectively. These four functions have an error less than 10%.
Modeling the lead rubber bearings (LRB)
For lead rubber bearings (LRBs), three samples were modeled. The first model was chosen from the study of Doudoumis et al. (2005). The second and third models were selected from the paper presented by Nersessyan et al. (2001). The interpretation of the modeling process and the results are shown in the following.
Geometry
Geometrical specifications of the LRB models
Isolator characteristics  First model (Imbimbo and De Luca 1998)  Second model (Bergstrom 2002)  Third model (Bergstrom 2002) 

Diameter of top and bottom loading plates (mm)  601  450  180 
Thickness of top and bottom loading plates (mm)  31.8  23  15 
Diameter of top and bottom fixing plates (mm)  431  –  – 
Thickness of top and bottom fixing plates (mm)  25.4  –  – 
Diameter of rubber sheets (mm)  431  450  180 
Thickness of rubber sheets (mm)  9.5  4  3 
Number of rubber sheets  11  44  21 
Diameter of steel plates (mm)  431  450  180 
Thickness of steel plates (mm)  3  3  1 
Number of steel plates  10  43  20 
Diameter of lead core (mm)  116.8  90  25.4 
Height of lead core (mm)  185  305  83 
Lead core yield stress (MPa)  6  7  6 
Defining material
Regarding the fact that there are not any experimental results for quadruple tests on rubber for determining the coefficients of the strain energy functions, the required tests were done by the finite element software itself. To do this, the software instructions state that there must be the results of at least two tests. Regarding the fact that rubber is usually considered incompressible, there is no need to do the volumetric test. In this study, the tests that have been done for rubber are the uniaxial and planar shear tests.
The sample was modeled two dimensionally and the rubber was defined using the Arruda–Boyce function. To use this function, the initial shear modulus and the amount of locking stretch (\(\lambda_{m}\)) are needed. Regarding the amount of isolator’s strain and the vast range of numerical tests, 3 seems the proper value for \(\lambda_{m}\). This amount, which was determined by trial and error, was a premise that the speed and precision of the solution would be higher.
After analyzing the model, the strain–stress curve for rubber in its central zone was attained.
Coefficients of the strain energy functions for the first type of rubber in LRBs
Function  \(C_{10}\)  \(C_{01}\)  \(C_{11}\)  \(C_{20}\)  \(C_{02}\)  \(C_{30}\) 

Mooney–Rivlin  0.053708  − 0.670950  –  –  –  – 
NeoHookean  0.367465  –  –  –  –  – 
Yeoh  0.287802  –  –  0.001890  –  0.000424 
Polynomial N = 2  1.114475  − 0.912280  − 0.575830  0.1199180  0.470235  – 
Function  \(\mu\)  \(\lambda_{m}\)  \(a\)  \(\beta\)  –  – 

van der Waals  0.454676  7.637174  − 0.150100  0.000000  –  – 
Arruda–Boyce  0.569053  3.039270  –  –  –  – 
Function  \(\alpha_{3}\)  \(\alpha_{2}\)  \(\alpha_{1}\)  \(\mu_{3}\)  \(\mu_{2}\)  \(\mu_{1}\) 

Ogden N = 3  − 7.200500  7.747580  2.476592  0.00359  0.000318  0.517097 
Coefficients of the strain energy functions for the second and third types of rubber in LRBs
Function  \(C_{10}\)  \(C_{01}\)  \(C_{11}\)  \(C_{20}\)  \(C_{02}\)  \(C_{30}\) 

Mooney–Rivlin  1.636999  − 1.190500  –  –  –  – 
NeoHookean  0.410013  –  –  –  –  – 
Yeoh  0.311112  –  –  − 0.004411  –  0.000580 
Polynomial N = 2  1.684135  − 0.527640  − 0.866020  0.167200  0.713158  – 
Function  \(\mu\)  \(\lambda_{m}\)  \(a\)  \(\beta\)  –  – 

van der Waals  0.532218  7.586497  0.009330  0.000000  –  – 
Arruda–Boyce  0.546603  2.976734  –  –  –  – 
Function  \(\alpha_{3}\)  \(\alpha_{2}\)  \(\alpha_{1}\)  \(\mu_{3}\)  \(\mu_{2}\)  \(\mu_{1}\) 

Ogden N = 3  − 8.304410  8.322200  2.390980  0.000063  0.000134  0.527638 
Amount of error in calculating the initial shear modulus of the numerical models using the functions in comparison to the experimental value in LRBs
Function  Rubber type  Model’s shear modulus (MPa)  Experimental shear modulus (MPa) (Martelli et al. 1992; Garcia et al. 2005)  Error percentage (%) 

Mooney–Rivlin  First model  0.77  0.62  24.19 
Second and third model  0.89  0.59  50.85  
NeoHookean  First model  0.73  0.62  17.74 
Second and third model  0.82  0.59  38.98  
Ogden N = 3  First model  0.52  0.62  − 16.13 
Second and third model  0.53  0.59  − 10.17  
Yeoh  First model  0.58  0.62  − 6.45 
Second and third model  0.62  0.59  5.08  
Arruda–Boyce  First model  0.61  0.62  − 1.61 
Second and third model  0.59  0.59  0.00  
Polynomial N = 2  First model  0.40  0.62  − 35.48 
Second and third model  0.31  0.59  − 47.46  
van der Waals  First model  0.45  0.62  − 27.42 
Second and third model  0.53  0.59  − 10.17 
For the first rubber model, the Arruda–Boyce function had the least error followed by the functions of Yeoh, Ogden (N = 3), neoHookean, Mooney–Rivlin, van der Waals and polynomial (N = 2), respectively. Other than the functions of Mooney–Rivlin and neoHookean, all other functions estimated the initial shear modulus below the real value.
For the second and third models of rubber, too, the Arruda–Boyce function had the least error. After that came the functions of Yeoh, van der Waals, Ogden (N = 3), polynomial (N = 2), neoHookean and Mooney–Rivlin. The functions of Mooney–Rivlin, neoHookean and Yeoh have estimated the initial value of the shear modulus higher than its real value, yet the other functions have reached a lower value than the real one.
Loading
Vertical load and displacement applied to the LRB models
Meshing
Analysis results
In Figs. 16, 17, 18, 19, 20 and 21, the hysteresis loops of the strain energy functions are compared with the experimental results.
Calculated effective stiffness of the LRB models and the amount of its error compared with the experimental results in order of error percentage
Sample  First model  Second model  Third model  

K_{eff} (KN/mm)  Error percentage  K_{eff} (KN/mm)  Error percentage  K_{eff} (KN/mm)  Error percentage  
Experimental  1.474  –  0.636  –  0.187  – 
Mooney–Rivlin  1.838  19.80%  1.034  62.58%  0.367  95.52% 
NeoHookean  1.802  18.20%  0.972  52.83%  0.341  81.73% 
van der Waals  1.683  12.42%  0.839  31.92%  0.305  62.75% 
Arruda–Boyce  1.711  13.85%  0.808  27.04%  0.284  51.67% 
Ogden (N = 3)  1.658  11.10%  0.784  23.27%  0.278  48.06% 
Yeoh  1.655  10.94%  0.782  22.96%  0.265  41.37% 
Polynomial (N = 2)  1.570  6.11%  0.650  2.20%  0.245  30.58% 
In the first model with the shape factor of 11.3 and shear strain of 100%, the best performance belongs to the polynomial (N = 2) function with error of 6%. After that, the functions of Yeoh, Ogden models (N = 3), van der Waals and Arruda–Boyce with a nearly similar performance vary in error from 11 to 14%, respectively. The neoHookean and Mooney–Rivlin functions were the weakest with errors equal to 18% and 20%, respectively.
In the second model, the shape factor of the isolating system is 28.1 and the applied shear strain is 150%. Here too, the polynomial (N = 2) and an error less than 3% had the best performance followed by Yeoh and Ogden (N = 3) with 23% error. The Arruda–Boyce and van der Waals functions with 27% and 31% errors, respectively, come next. The neoHookean and Mooney–Rivlin functions have had the worst performances.
In the third model, the shape factor of the rubber layers is equal to 15 and its shear strain compared to the other two models is much higher (shear strain equals to 300%). Here too, the minimum error belongs to polynomial (N = 2) function which shows a 30% discrepancy in estimating the effective stiffness. The Yeoh function with 41% error comes next. After that, Ogden (N = 3), Arruda–Boyce, van der Waals, neoHookean and Mooney–Rivlin have the highest errors, respectively.
As seen, among the studied functions, the polynomial (N = 2) has the best performance.
Calculated effective damping of the LRB models and the amount of its error compared with the experimental results in order of error percentage
Sample  First model  Second model  Third model  

\(\beta_{\text{eff}}\)  Error percentage  \(\beta_{\text{eff}}\)  Error percentage  \(\beta_{\text{eff}}\)  Error percentage  
Experimental  0.319  –  0.318  –  0.11  – 
Mooney–Rivlin  0.268  − 15.99%  0.162  − 49.06%  0.076  − 31.03% 
NeoHookean  0.273  − 14.42%  0.173  − 45.60%  0.081  − 26.05% 
van der Waals  0.293  − 8.15%  0.200  − 37.11%  0.090  − 18.02% 
Arruda–Boyce  0.288  − 9.72%  0.208  − 34.59%  0.097  − 11.76% 
Ogden (N = 3)  0.297  − 6.90%  0.214  − 32.70%  0.099  − 9.65% 
Yeoh  0.298  − 6.58%  0.215  − 32.39%  0.105  − 4.22% 
Polynomial (N = 2)  0.314  − 1.57%  0.258  − 18.87%  0.112  1.97% 
In the first model, the polynomial function has an error about − 1.6%, and Yeoh, Ogden (N = 3), van der Waals and Arruda–Boyce with a similar performance have an error equal to − 6% to − 10%. The neoHookean and Mooney–Rivlin functions are the weakest in estimating the effective damping.
In the second model, too, the polynomial (N = 2) function and 18% error had the best estimation. After that come the Yeoh, Ogden (N = 3), Arruda–Boyce and van der Waals. Then, the neoHookean and Mooney–Rivlin with − 45% and − 49% have the highest error values, respectively.
The effective damping of the third model is less than the other models. Here too, the polynomial (N = 2) function and an error below 2% had the best result. Yeoh, Ogden (N = 3), Arruda–Boyce, van der Waals, neoHookean and Mooney–Rivlin come, respectively, after that.
Conclusion

High damping rubber bearing (HDRB)
Based on the available results of experimental tests conducted on the rubber of the isolators by other researchers and obtaining the stress–strain curve of each, the coefficients of the strain energy functions were derived. In Yeoh and van der Waals functions which estimated the shear modulus of the rubber without any error, the isolating system, too, had better results. Regarding these considerations, the best function is the Yeoh or the reduced polynomial function (N = 2) whose error in estimating the effective stiffness is below 7%.
 (a)
It is the property of hyperelastic materials that their force–displacement behavior is completely reversible, and regarding the fact that there is no plasticity in HDRBs, the derived hysteresis loop will be in the form of a line. Therefore, the amount of energy in each cycle and the damping cannot be determined.
 (b)
The behavior of HDRBs is highly nonlinear and the force–displacement behavior of the isolating system gains a softening attribute as the strain increases. As a result, in high strains the isolator’s behavior cannot be modeled in all stages of loading exactly like what happens during experimental tests.

Lead rubber bearings (LRB)
In the studied finite elements models, there were two types of error for LRBs: one caused by modeling the rubber tests which creates an error in determining the coefficients of the strain energy functions and another caused by modeling of LRBs.
In experimental models of LRBs, the stiffness and therefore the slope of the force–displacement curve decreases as the strain increases. However, in numerical modeling, the curve shows a linear behavior, therefore the error of modeling increases with the increase in strain. It seems the lower error in polynomial (N = 2) function is due to the lower value in the initial estimation for rubber’s shear modulus.
The hysteresis loop of the LRBs is bilinear. The initial slope follows the elasticity modulus of lead. The yield stress of lead indicates the ultimate strength in initial stiffness. The secondary stiffness is a function of the rubber’s stiffness in a way that the secondary stiffness is higher for higher stiffness of rubber which is derived from the strain energy function.
Regarding the fact that lead core has an elastoplastic behavior during the horizontal loading process, the damping of the isolating system can be determined.
In modeling LRBs, the best result is attained from the polynomial (N = 2) function. The effective stiffness of the isolating system can be estimated with an error less than 3% at 150% shear strain and with the shape factor of 28.1.
Notes
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