Research Article

Advances in Mechanical Engineering2017, Vol. 9(5) 1–12� The Author(s) 2017DOI: 10.1177/1687814017699352journals.sagepub.com/home/ade

Effects of the bulk compressibility onrubber isolator’s compressive behaviors

Wenku Shi, Guozheng Liu and Zhiyong Chen

AbstractThrough combined experimental and finite element analysis, the effect of bulk compressibility on rubber isolators wasstudied. First, the strain energy functions for rubber materials were studied, and the volumetric term was described byPoisson’s ratio. Second, parameter identification tests of rubber specimens were performed to obtain basic materialproperties of strain energy functions. It has shown that bulk compressibility has an important effect on the vertical stiff-ness of rubber isolators. When the value of Poisson’s ratio is 0.499, the results of simulation and experiment are in goodagreement. Based on the finite element analysis model, the stress of the rubber isolator was studied and the structurewas improved. In addition, Poisson’s ratio could be used to modify the finite element analysis model of rubber isolatorsand give rise to the accuracy of the simulation result.

KeywordsRubber pad, compressibility, finite element analysis, structure improvement, hyperelastic constitutive models

Date received: 29 September 2016; accepted: 20 February 2017

Academic Editor: Filippo Berto

Introduction

Rubbers are widely applied in engineering fields, suchas automobile tires, seals, engine mounts, bridge bear-ings, and vibration absorbers. These materials are usu-ally regarded incompressible to facilitate the analysis,although they are slight compressive in reality.

For the rubber isolator with rubber pads bonded torigid supports, the deformation of rubber is highly con-strained, where the bulk compressibility plays an impor-tant role. In the literature, some researchers have studiedthe rubber pad by finite element (FE) modeling.1–4 Intheir studies, rubber pads are usually assumed to be iso-tropic and impressible. And the influence of rubber com-pressibility on the simulation has not been investigated.There are many analytical studies on the effect of bulkcompressibility of rubber isolators, such as Chalhouband Kelly,5,6 Constantinou and Kartoum,7 Tsai andLee,8 and Pinarbasi et al.9 But in most of these studies,the rubber is considered as a linear material. There aretwo parameters to describe rubber compressibility: bulkmodulus and Poisson’s ratio (PR).

The bulk modulus K is a classic and direct para-meter to describe the compressibility of hyperelasticmaterials, and the bulk modulus means the ratiobetween the hydrostatic pressure and the volumechange. The bulk modulus is determined by the hydro-static compression experiment, which is difficult andthere is no test standard,10 although the experimentshave been performed by many researchers, such asPenn,11 Wood and Martin,12 and Christensen andHoeve.13 There is an approximate linear relationbetween the pressure P and the volume change (1� i3)

P=K(1� i3), i3 = l1l2l3 ð1Þ

where lK = 1, 2, 3 are principle stretch ratios (1 + strain).

State Key Laboratory of Automotive Simulation and Control, Jilin

University, Changchun, China

Corresponding author:

Zhiyong Chen, State Key Laboratory of Automotive Simulation and

Control, Jilin University, Changchun 130022, P.R. China.

Email: chen_zy@jlu.edu.cn

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/

open-access-at-sage).

PR v is defined as the ratio between the lateral andaxial strains in uniaxial tension or compression.14 Itwas originally created for rigid materials, such as steeland concrete whose deformations are small.15 Someauthors contend that PR is not a constant, and theyuse ‘‘lateral contraction ratio’’ or ‘‘Poisson function’’ todefine the ‘‘PR’’ of rubber-like materials, for it canchange with the deformation.16–18 Some values for PRof rubber have been used in the literature. For example,Lindley19 has pointed out a value of 0.4997 for naturalrubber and Bradley et al.20 thought the PR of solidrubber ranges from 0.49 to 0.5.

By the classic equations of elastic materials, there isa relation between K and v21

v=3K � 2m

6K + 2mð2Þ

where m is the shear modulus. For rubbers, the bulkmodulus is much greater than the shear modulus,K � m, and PR v approaches 0.5. For incompressiblematerials, the value of PR is 0.5.

In this article, PR was used to describe the volu-metric term of rubber’s strain energy functions, whoseparameters were derived from uniaxial compressionand tension tests. Based on the strain energy functions,the vertical stiffness of the rubber isolator was simu-lated by the commercial FE software ABAQUS. TheFE model was modified by adjusting the value of PR.When PR is 0.499, the simulation result is in goodagreement with test data. By the modified FE model,the structure of rubber isolator was improved to meetthe design request.

Constitutive models with compressibility

Rubbers are assumed isotropic, hyperelastic, andalmost incompressible. Their large deformation beha-viors are determined by the constitutive modelsdescribed by the strain energy function W. And it is afunction that means the stored strain energy on unitinitial volume fraction.22 The strain energy function Whas many different forms and is usually made up withthe invariants Ii or the principal stretches li. The rela-tions between Ii and li are

I1 =l12 +l2

2 +l32, I2 = l1

2l22

+ l12l3

2 +l22l3

2, i3[l12l2

2l32

ð3Þ

Because of the compressibility of hyperelastic mate-rials, such as rubbers, the strain energy function W issplit into two parts C and G as follows

W =C(l1, l2, l3)+G(i3), or W =C(�I1,�I2)+G(i3)

ð4Þ

where the first part C is the deviatoric term whichmeans the strain energy necessary to change the shapeof rubbers. The second part G is the volumetric term,which means the strain energy necessary to change thevolume. If rubber is assumed incompressible, the volu-metric part G of the strain energy function could beneglected. G is a function of the volume ratio i3 and itis assumed that G(i3 = 1)= 0. �I1 and �I2 are the firstand second modified invariants, respectively23

�I1 =I1

i32=3

,�I2 =I2

i34=3

ð5Þ

For typical hyperelastic material models, polynomialmodels are often used, where the strain energy functionhas the form

W (�I1,�I2, i3)=XN

i+ j= 1

Cij(I1 � 3)i(I2 � 3)j

+XN

i= 1

1

Di

(i3 � 1)2i

ð6Þ

Cij and Di are the material constants determined byfitting to the test data. Values of N. 2 are rarely used,because it is difficult to fit such a large number of mate-rial properties from experimental data. Whatever thevalue of N, the shear modulus m and the initial bulkmodulus K are decided only by the first-order polyno-mial coefficients24

m= 2(C10 +C01),K =2

D1

ð7Þ

With reference to equations (2) and (7), the coeffi-cient D1 of the volumetric term G in strain energy func-tions could be expressed as

D1 =3(1� 2v)

2(v+ 1)(C10 +C01)ð8Þ

So the strain energy function is

W (�I1,�I2, i3)=XN

i+ j= 1

Cij(�I1 � 3)i(�I2 � 3)j

+2(v+ 1)(C10 +C01)

3(1� 2v)(i3 � 1)2

ð9Þ

Mooney–Rivlin model

Mooney–Rivlin model25 is the polynomial model oforder N=1. From equations (6) and (9), its strainenergy function is

2 Advances in Mechanical Engineering

WMooney =C10(�I1 � 3)+C01(�I2 � 3)

+2(v+ 1)(C10 +C01)

3(1� 2v)(i3 � 1)2

ð10Þ

Mooney–Rivlin model is most commonly used. Butit cannot predict the large deformation behavior of rub-bers and is often used in small deformation state.

Neo-Hookean model

Neo-Hookean model26 is a simplified form of Mooney–Rivlin model, which only has the first invariant �I1

WNeo�Hookean =C10(�I1 � 3)+2(v+ 1)C10

3(1� 2v)(i3 � 1)2 ð11Þ

Yeoh model

Yeoh model27 is proposed not to use the second invar-iant and it has a form more complicated than the for-mer two models

WYeoh =C10(�I1 � 3)+C20(�I1 � 3)2 +C30(�I1 � 3)3

+1

D1

(i3 � 1)2 +1

D2

(i3 � 1)4 +1

D3

(i3 � 1)6

ð12Þ

When the volume change is very small, the value ofi3 is close to 1 and the value of (i3 � 1) approaches 0.(i3 � 1)4 and (i3 � 1)6 are negligible compared to(i3 � 1)2. So the volumetric term G of W is simplifiedto only one part containing the coefficient D1, and W is

WYeoh =C10(�I1 � 3)+C20(�I1 � 3)2 +C30(�I1 � 3)3

+2(v+ 1)C10

3(1� 2v)(i3 � 1)2 ð13Þ

Ogden model

Ogden model28 is different from polynomial models, asits strain energy function is expressed in terms of princi-pal stretches

W (�l1, �l2, �l3, i3)=XN

i= 1

2mi

ai2(�l1

ai + �l2ai + �l3

ai )

+XN

i= 1

1

Di

(i3 � 1)2i

ð14Þ

where �li = li=i31=3, N , mi, and ai are the material para-

meters derived from test data. There are useful relationsbetween these parameters29

m=XN

i= 1

mi,K =2

D1

ð15Þ

From equation (2), D1 could be expressed by mi andai. For Ogden N=3 model

D1 =3(1� 2v)

(m1 +m2 +m3)(1+ v)ð16Þ

Same as Yeoh model, the volumetric term G of W issimplified to only one part containing the coefficientD1, and the strain energy function of Ogden model is

WOgden =XN

i= 1

2mi

ai2(�l1

ai + �l2ai + �l3

ai)

+(m1 +m2 +m3)(1+ v)

3(1� 2v)(i3 � 1)2

ð17Þ

According to Holzapfel,30 the test data are in goodagreement with the theoretical results in Ogden N=3model. So this model was used in this article.

Parameter identification tests

For the strain energy functions (equations (6) and(14)), the parameters Cij, ai, and mi are derived fromthe experimental data, while D1 is mainly determinedby PR as we discussed before. To get the material para-meters of strain energy functions, there are many typesof tests to choose. The uniaxial tensile test is commonlyused. For rubber-like materials, their behavior wasmore different in compression than in tension. So theuniaxial compression test was also applied in this arti-cle, and it is equal to the biaxial tensile test whoseexperimental conditions are stricter. There are alsomany other tests, such as the planar tests and volu-metric test. Due to limitations of the experimental con-ditions in our laboratory, only uniaxial tension andcompression tests were taken to get the stress–straindata of rubber specimens. The shore hardness of therubber specimen is 61. Uniaxial tension and compres-sion tests were performed on a universal test machinein laboratory air, and the air temperature was 25�C.

Uniaxial tension test

In Figure 1, the dumbbell specimens were cut downfrom a rubber sheet with thickness 2mm, and the widthof the middle area of specimens is 6mm. According toISO 37:2005,31 uniaxial tension tests were performed ata universal mechanical testing machine equipped with acomputerized control and measuring systems of exten-sion and force. In Figure 2, the extensometers were usedto measure the extension of the test area, and the loadforce was measured by load cells inside the machine.The load–displacement history was recorded and con-verted into stress–strain data during the tensile test. Forthe repeatability and consistency, three specimens weretested and the averages of test data were calculated.

Shi et al. 3

Uniaxial compression test

In Figure 3, cylindrical specimens were used in thecompression test. The dimensions of it are 29mm dia-meter and 12.5mm thickness. Same as the tension tests,

three specimens were tested, and the test was in accor-dance with ISO 7743:2011.32 In Figure 4, the specimenwas compressed between two rigid plates. To make thespecimens free to expand in the radial direction duringthe compression, the surfaces of top and bottom plateswere coated with silicone oil to reduce the friction.Even very small levels of friction would significantlyaffect the measured stiffness, as Day and Miller33 dis-cussed in their article, the compression test must betaken carefully.

Fitting constitutive models

Nominal (or engineering) stress–strain data wereobtained from tension and compression tests. And thedata were inputted into the FE code ABAQUS, in

Figure 1. Dumbbell specimens.

Figure 2. Uniaxial tension test.

Figure 3. Cylindrical specimens.

Figure 4. Uniaxial compression tests.

4 Advances in Mechanical Engineering

which the constitutive models were fitted in the leastsquare method.34 Four strain energy functions,Mooney–Rivlin equation (10), neo-Hookean equation(11), Yeoh equation (13), and Ogden equation (17),were chosen.

The fitted curves of different strain energy functionsare shown in Figure 5. The curves of Ogden N=3 andYeoh strain energy functions are in good agreement withtest data in all ranges of strain. Neo-Hookean andMooney–Rivlin models can only describe rubber behavioraccurately at low strain, and the fitted curves deviate fromthe experimental curves in large values of the stretch.

To evaluate these four constitutive models, the rela-tive error e of the stress between experimental and the-oretical value was calculated

e=se � st

se

����

����3 100% ð18Þ

where st is the theoretical value and se is the experi-mental value. The relative errors of different constitu-tive models are shown in Figure 6. At low strain oftension and compression, the relative errors of all con-stitutive models are higher, and it is in agreement withOgden.28 As shown in Figure 6, the relative error e ofYeoh and Ogden constitutive models is smaller thanMooney–Rivlin and neo-Hookean models, which

means the former two models are more accurate todescribe the hyperelastic behavior of rubber materialthan the latter two models.

Through a least-squares-fit procedure minimizingthe relative error in stress, the parameters Cij, ai, andmi of the strain energy functions were derived from theexperimental data, while the parameter D1 which deter-mines the compressibility of rubber was mainly deter-mined by PR as we discussed before. The parametersof each constitutive model are shown in Table 1. Theparameter D1 can be adjusted by changing the value ofPR v, while the other parameters are specific and deter-mined by the experimental data. And these hyperelasticmodels were applied in finite element analysis (FEA) insection ‘‘FEA.’’

From equation (7) and (16), bulk modulus K can beexpressed in terms of PR v. For Mooney–Rivlin model

KMooney =4(v+ 1)(C10 +C01)

3(1� 2v)ð19Þ

For neo-Hookean and Yeoh model, the form of K isthe same

KYeoh =KNeo�Hookean =4(v+ 1)C10

3(1� 2v)ð20Þ

Table 1. Parameter values for constitutive models.

Model Parameters of the deviatoric term Values of D1

Mooney–Rivlin C10 = 0:8997 MPaC01 = � 0:0724 MPa

3(1� 2v)

2(v + 1)(C10 +C01)Neo-Hookean C10 = 0:8027 MPa 3(1� 2v)

2(v + 1)C10

Yeoh C10 = 0:7705 MPaC20 = � 0:0103 MPaC30 = 0:0041 MPa

3(1� 2v)

2(v + 1)C10

Ogden a1 = � 0:6911,a2 = 4:6046,a3 = � 1:3018m1 = 6:0558 MPa,m2 = 0:2162 MPa,m3 = � 4:6652 MPa

3(1� 2v)

(m1 +m2 +m3)(v + 1)

Figure 5. Fitted curves of different constitutive models. Figure 6. Relative error between experimental and theoreticalresults.

Shi et al. 5

For Ogden model

KOgden =2(m1 +m2 +m3)(1+ v)

3(1� 2v)ð21Þ

With the values of parameters in Table 1, the rela-tion between K and v is shown in Figure 7. For thevalue of v less than 0.49, there is an approximately lin-ear relationship with the bulk modulus. But when it islarger than 0.49, these curves show a very strong nonli-nearity, and the value of K increases rapidly as v

increases. For Mooney–Rivlin, neo-Hookean, andYeoh model, the curves are similar. But the curve ofOgden model is different, because its parameter andform are different from other models.

FEA

Structure of the rubber isolator

As shown in Figure 8(a), the rubber isolator consists ofnine rubber pads, each of which is made up with a rub-ber layer bonded to two steel plates. The rubber padswere bolted together. The whole height of the rubberisolator is 1.26m, and the diameter of the steel plate is1.6m. It can only bear vertical load. The rubber isolatoris a part of the leg mating unit as shown in Figure 8(b),which is used in the installation of oil field platforms toreduce the dynamic load caused by the waves. The sizeof the rubber isolators is very large, and it is differentfrom those commonly used in seismic bridge

Figure 7. Relation between the bulk modulus K and Poisson’s ratio v: (a) bulk modulus against PR and (b) PR against bulk modulus.

Figure 8. Structure of the rubber isolators: (a) rubber isolators and (b) leg mating unit.

6 Advances in Mechanical Engineering

applications. The thickness of rubber layers is largerthan bridge rubber bearings that are commonly used, asshown in Figures 12 and 18; the rubber layer couldbulge in the radial direction when the rubber pad iscompressed. And the lateral surfaces would touch thesteel surfaces, which increases the stiffness of the rubberisolator and makes it bear more loads in extreme situa-tion. So the diameter of the steel plates is larger thanthe diameter of the rubber layers.

The structure of a half rubber pad is shown inFigure 9 and the geometry is defined in Table 2. Theupper and lower surfaces of the rubber are bonded torigid steel plates and are highly constrained. Only thevertical lateral surfaces of the rubber can expand freelywhen the rubber pad is subjected to vertical loads.

Modeling of rubber pads

As we can see from Figure 8, the rubber pads werestacked together just like springs in series. So the char-acteristic of the rubber isolator can be obtained fromanalyzing only one rubber pad.

The two-dimensional (2D) plane FE model is shownin Figure 10. Only a half of the cross section was mod-eled, as the shape and load of rubber pads were axisym-metric. In numerical modeling, the axisymmetric modelis used to reduce the number of elements and improvethe calculation speed. The element size of the rubberpads is about 2mm. The rubber section of the pad was

modeled with plane strain four-node hybrid elements.The type of rubber and steel elements is CAX4H andCAX4R, respectively.5

The steel plates are made of low-carbon steel withmaterial properties: PR is 0.3, and elastic modulus is210GPa. Rubber’s material performance is determinedby strain energy functions as we discussed in section‘‘Constitutive models with compressibility.’’ Four typesof hyperelastic constitutive models were applied, andtheir parameters are defined in Table 1. To study theinfluence of rubber compressibility on the simulation, arange of values of PR from 0.48 to 0.49999 weredefined separately, and the simulation was calculatedrepeatedly.

Figure 11 shows the boundary conditions and con-straints of FE model. The displacement of the base steelplate was restrained in all directions. The rubber layerand steel plates were bonded together by ‘‘tie.’’ Whenthe rubber pad is compressed, the rubber layer couldbulge in the radial direction, and its lateral surfaces willbe in contact with the surface of the steel plates. So thefrictions of these contact surfaces were defined and thefriction coefficient was 0.3. Displacement load was puton a reference point, which was coupled with the topsteel plate. Finally, the displacement–force data of thereference point were obtained during the simulation.Although the FE model is planar, it can be displayed inthree dimensions by sweeping along the Y-axis. In thisway, it could be more visualized. In Figure 12, the pla-nar elements swept 90� along the Y-axis, showing thedeformation of a quarter rubber pad.

To verify that the 2D model equals to the three-dimensional (3D) model, the two types of the FE modelwere simulated, and the vertical displacement was

Figure 9. Half model of a rubber pad.

Table 2. Geometric details of rubber pad.

Variables Notation Values (mm)

Thickness of rubber layer H 100Thickness of steel plate Hs 20Outer radius of steel plate R0o 800Inner radius of steel sheet R0i 250Outer radius of rubber layer R 685Inner radius of rubber layer Ri 360Flank radius of rubber layer r 70

Figure 10. Axisymmetric FEA model: initial and deformedmeshes.

Figure 11. Boundary conditions and constraints of FE model.

Shi et al. 7

applied to the top of the pad. The displacement–forcecurves are shown in Figure 13. It is obvious that thetwo curves are almost the same, which verifies the accu-racy of the 2D model.

Effect of bulk compressibility

To compare the simulation result to the test data of thevertical stiffness, the compression test of the rubber iso-lator was performed on a universal testing machine.The displacement–force history was obtained in thetest. As shown in Figure 14, the universal testingmachine is driven by hydraulic pressure from accumu-lators. And its maximum vertical load is 30MN. In thecompression test, the rubber isolator was put on thelower plate of the test machine, and the vertical forcewas applied to the upper plate. Vertical displacementwas measured by four sensors in different locations,and the average was calculated as the measured valueof the displacement. The force was measured by loadcells inside the machine. And the air temperature isabout 25�C.

Because of rubber’s nonlinear behavior of hysteresis,creep, and stress softening phenomenon, the compres-sion speed is slow and is 6mm/min. After six repeatedload–unload cycles, the force–displacement data weremeasured. The compression displacement is 250mm

totally (the displacement of each rubber pad is about28mm).

The load–displacement history obtained from FEAis compared with the test data in Figure 15. It is clearthat the value of PR has a great influence on the simu-lation results. The greater the value of PR, the harder itis compressed. In addition, the linear elastic materialmodel was simulated. It is obvious that the hyperelasticmodel is more accurate than the linear elastic materialmodel, especially in the large displacement.

In all ranges of deformation, the results of Mooney–Rivlin and neo-Hookean models are similar, for thestrain energy function of neo-Hookean is simplifiedfrom Mooney–Rivlin model and the parameters C10 inboth models are very similar in Table 1. For these twomodels, the simulation stiffness of the rubber pad isnearly linear and seriously deviates from test results inlarge displacement, because they are one-order modelsand can only describe the rubber behavior accurately atlow strains.24,35

When the value of PR is greater than 0.49, Ogdenand Yeoh models capture the nonlinear behavior ofrubber pad in large deformation successfully. For Yeohmodel, the simulation results for PR=0.499 prove agood conformity with the test data. So the hyperelasticYeoh model is taken for further analysis, and the valueof PR is set to 0.499.

In this article, the static stiffness of the rubber pad isdefined as the ratio between the load and the verticaldisplacement of the top steel plate. When the verticaldisplacement is 27mm, the static stiffness of the rubberpad with different values of PR is shown in Figure 16.The static stiffness can be greatly influenced by PR.When the value of PR is less than 0.48, the stiffnessesof four hyperelastic constitutive models are similar, andthe relation between load and PR is linear. But whenthe value of PR is larger than 0.49, the stiffness variesin different constitutive models, and the stiffnessincreases rapidly as PR increases. It emphasizes theimportance of compressibility in FE simulation ofrubber-like materials. In addition, PR could be used tomodify the FE model to improve the accuracy.

Figure 12. FE model in 3D form (sweeping 90� along Y-axis).

Figure 13. Displacement–force curves of 2D and 3D model.

Figure 14. Compression test of the rubber isolator.

8 Advances in Mechanical Engineering

Structure improvement

Knowledge of the modified FE model by PR makes itpossible to improve the rubber pad structure to meetthe design request. According to the test and simulationresults, the initial structure of the rubber pad should beimproved for the reasons: (1) cracks could appear onthe lateral surfaces of rubber pads in compression and(2) the initial static stiffness is larger than the designvalue.

Analysis of crack

When the rubber pad is compressed in the verticaldirection, the vertical lateral surfaces of it could bulgehorizontally. But in some conditions, cracks wouldappear on the middle of the bulged lateral surface. Forinstance, the crack is shown in Figure 17. During thecompression test, the probability of crack occurrencevaries with the value of the flank radius r (Figure 9),and the cracks appear on the rubber pad with the larger more often than those with the small r. So it is

Figure 15. Comparison of measured and predicted force: (a) Mooney–Rivlin, (b) Neo, (c) Yeoh, and (d) Ogden.

Figure 16. Static stiffness for various Poisson’s ratio (verticaldisplacement is 27 mm).

Figure 17. Cracks on the rubber pad.

Shi et al. 9

important to analyze the influence of the flank radius ron the distribution of the stress field.

The stress field of the rubber pad’s cross section isshown in Figure 18. The red color means a large valueof stress while blue means a small value. The lateral sur-face is all in red especially in the middle of it. The stressalong the rubber’s lateral surface in the Y-direction isshown in Figure 19. It is clear that the stress is high inthe middle of the lateral surface, which is in agreementwith the location of the crack in Figure 17. The stressdecreases as the value of r decreases. The stress is sensi-tive to r when its value is less than 70mm. While thevalue of r is larger than 80mm, the change in the stressis neglectable as r increases. So the value of r has beenchanged to 50mm to reduce the stress and avoid thecracks. In Figure 20, the stiffness of rubber pads canhardly be influenced by the values of r.

Adjustment of stiffness

As shown in Figure 21, the design stiffness of the rubberpad is 1100 ton as the vertical displacement is 25mm,which means the static stiffness is 44 ton/mm and equalsto 440kN/mm. The maximum design displacement ofthe rubber pad in the vertical direction is 32mm, while

the thickness of the rubber pad is 100mm. Due to lim-itations of the structure and process, only the parameterouter radius R (Figure 9) could be changed to meet thedesired stiffness. By FEA, the stiffness of rubber padswith different radius is shown in Figure 21. As the outerradius increases, the vertical stiffness also increases. Inthe vertical direction, the force and displacement beha-vior is linear in the small range of load. But when thevertical displacement is beyond 20mm, the nonlinearityof the stiffness is obvious. When the value of the outerradius is 670mm, the stiffness approaches 1100 ton. Sothe outer radius of the rubber pad is changed to670mm.

Experimental verification

By FEA, the values of R and r have been changed to670 and 50mm. As described in section ‘‘Effect of bulkcompressibility,’’ the compression test of the improvedstructure was performed to get the displacement–forcehistory. The improved rubber isolator under compres-sion is shown in Figure 22. There is no crack appearingon the vertical lateral surfaces during the test. In Figure23, the vertical stiffness of the rubber pad under pure

Figure 18. Stress field in the rubber pad (initial and deformedmeshes, r = 70 mm, vertical displacement is 29 mm).

Figure 19. Stress on the vertical lateral surface in Y-direction.

Figure 20. The stiffness of rubber pads with different valuesof r.

Figure 21. Displacement–force curves of rubber pads withvarious R.

10 Advances in Mechanical Engineering

compression obtained by FEA is in good agreementwith the test data. The maximum deviation between thetest data and the simulation is only 5.1% at the verticaldisplacement 29mm. It proves the accuracy of the FEmodel modified by PR, and it could be used to provideguidance for the design and production of rubberisolator.

Conclusion

The conclusion is as follows:

1. For the structure of a rubber pad bonded torigid plates, the static stiffness is very sensitiveto bulk compressibility.

2. Stress on the lateral surfaces could be decreasedby adjusting the flank radius without changingthe stiffness of rubber pad of rubber isolators.

3. PR could be used to modify the FE model toimprove the simulation accuracy. And it couldprovide guidance for the design and productionof rubber isolator.

4. Yeoh and Ogden constitutive models are moreaccurate than Mooney–Rivlin and neo-Hookean models in the FE simulation of rubberisolators. Yeoh and Ogden constitutive modelsfit the stress–strain data of rubber specimensmore accurately than the other two models.Compared with the test results, the stiffness ofrubber pads simulated with the former twomodels is more accurate than with the lattertwo models.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) received no financial support for the research,authorship, and/or publication of this article.

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