,

(1 Offshore Heavy Industries Design&Research Institute,Shanghai Zhenhua Heavy Industry Co.,Ltd.,Shanghai 200125, China;2 School of Naval Architecture,Ocean and Civil Engineering,Shanghai Jiao Tong University,Shanghai 200240, China)

Load-Compression Relationships of Bonded Rubber Ring

ZHENG Yi-kan1,ZHANG Shi-lian2

(1 Offshore Heavy Industries Design&Research Institute,Shanghai Zhenhua Heavy Industry Co.,Ltd.,Shanghai 200125, China;2 School of Naval Architecture,Ocean and Civil Engineering,Shanghai Jiao Tong University,Shanghai 200240, China)

Bonded rubber rings are widely used in many engineering domains to buffer the impact.In general,accurate load-deformation relationships are required in these applications.However,previous researches merely discussed the cylindrical rubber pad rather than the rubber ring.Besides,few of them are based on continuum mechanics theory.In this paper,the load-compression relationships of the bonded incompressible rubber ring are derived for three boundary conditions.The Mooney-Rivlin material is considered and the derivation is based on continuum mechanics theory.The results calculated by the derived formulae are compared with the FEM solutions and proved to have adequate accuracy for various shape factors and materials,even in the finite strain.The typical load-compression curves of the rubber rings are also presented and the characteristic of the compression stiffness in different boundaries are discussed.

load-compression relationship;rubber ring;mooney-rivlin;boundary condition; finite strain

0 Introduction

Bonded rubber rings are widely used in many engineering domains to buffer the impact. LMU(Leg Mating Units)is a good example of its application,which is used in the float-over installations on the sea.As shown in Fig.1,the rubber rings are bonded between metal plates to provide higher compression stiffness.The rigid center cylinder,such as a cast tubular member,is often set inside the rubber ring to prevent the instability of stacked elastomeric pads and spacer plates[1].On this occasion,rubber rings are preferred than circular rubber pads.

In general,accurate load-deformation relationships are required in these applications.The in-depth theoretical mechanical analysis of the compression performance will provide an important reference for the engineers,especially in the early stages of the design.Many researchers have put in the effort to this issue.Gent,Lindley and Meinecke[2-3]first derived the compressive stiffness of the incompressible elastic layer bonded between rigid plates for infinite-strip,circular pads and other shapes.Kelly et al[4-7]developed a theoretical approach andtook the effect of bulk compressibility into consideration.Koh and Kelly[8]abandoned the stress assumption and derived the compression stiffness of the bonded square layer using only the kinematic assumptions.Tsai and Lee[9]derived the compression stiffness of va-rious shaped rubber layers without limitation on the values of Poisson’s ratio.Whereafter,simplified forms of those formulae are given[10]. Furthermore,Lindley[11]derived a load-compression relationship for the circular rubber pad. Hill[12]derived partial solutions of finite elasticity for various situations based on the incompressible Mooney-Rivlin material with certain limiting condition.

However,none of above researches has given a discussion about the rubber ring and most of them are based on linear elastic theory.In this paper,the load-compression relationships of the bonded incompressible rubber ring are derived for three boundary conditions,i.e.,when the inner surface is restrained,when the outer surface is restrained and when both surfaces are free.The kinematic assumptions mentioned above are adopted:(i)Planes parallel to the rigid bounding plates remain plane and parallel;(ii)Lines normal to the rigid bounding plates before deformation become parabolic after loading[8].The Mooney-Rivlin material is considered and the derivation is based on continuum mechanics theory.The results calculated by the derived formulae are compared with the FEM solutions.The typical load-compression curves of the rubber rings are also presented and the characteristic of the compression stiffness in different boundaries are discussed.

Fig.1 Rubber ring bonded between rigid plates

1 Inner surface restrained

Fig.2 Undeformed and deformed configurations of the rubber ring when the inner surface is restrained

The first situation in consideration is when the inner surface of the rubber ring is restrained in both radial direction and circumferential direction.This is an extreme case when arigid column with a same diameter as the inner diameter of the rubber ring is set up.The rubber ring is assumed as homogeneous,isotropic and incompressible.The undeformed and deformed configurations of the rubber ring under compressive load in this situation are shown in Fig.2,as well as the material coordinates(R,Θ,Z)and spatial coordinates(r,h,z).The inner and outer radii are RB0and RA0,respectively.The origin points are located in the center and the mid-height of the rubber ring.The transformation relations between these coordinates are as follows considering two kinematic assumptions:planes parallel to the rigid plates remain plane and parallel and vertical lines become parabolic.

The inverse transformation is:

where λ is the length ratio in z direction and α0is the relative extension in radial direction at the mid-height of the rubber ring,regarded as the first order small quantity.

From the incompressible condition,ignoring the second order small quantities,α0is:

where η is defined as RB0/R.

The covariant and contravariant components of the metric tensor of the material and spatial cylindrical coordinates,denoted as GAB,GAB,gij,gij,respectively,are as follows:

Denoting the material and spatial coordinates bythe deformation gradient F is written as:

where giand GAare covariant base vectors ofand reciprocal base vectors ofrespectively.The left Cauchy-Green deformation tensor is[13]:

The inverse of B is:

where cijcan be calculated by the following equation:

Now define three parameters:

Then equations(1)and(4)turn to be:

Utilizing Eqn(8)through Eqn(19),we can get the contravariant components of B and B-1:

The Cauchy stress tensor of the incompressible hyperelastic material can be expressed by[13]:

where I is the metric tensor of{xi};p is the unknown hydrostatic pressure;ψ1and ψ2are the partial derivatives of the potential function with respect to the first invariant I1and second invariant I2of B,respectively.That is,

For the incompressible Mooney-Rivlin material,

where C1and C2are two material parameters.

Utilizing Eqn(20)through Eqn(25)and ignoring the second order small quantities,the nonzero physical components of the Cauchy stress tensor are:

Consider the equilibrium equation in r direction and the boundary condition:

where f1is equal to zero;r0is the outer radius of the rubber ring after deformation,which is a function of z.

From equations(26)through(29),we obtain:Since α,β and γ are all first order small quantities,by neglecting the second order small quantities,equations(32)and(33)become:

Substituting equations(34),(35)and(17)to Eqn(30),we have:

Integrating Eqn(36)from r to r0and utilizing the boundary condition Eqn(31),we have:

In the above integration,the itemsare treated as constant.Although in fact they are functions of r,this treatment will not induce obvious error because α and γ are the first order small quantities.

From equations(26)and(28),

It will become the following equation by neglecting the second order small quantities:

The effective compression modulus is,on average of the volume,defined as:

where the integral variable z has been transformed to Z to simplify the form of the integral and d is the compression displacement,which equalshere.

Substituting equations(37)and(39)into Eqn(40),Ec1becomes:

Using Taylor’s series,it can be proved readily that the mean value of the polynomial ofin Z direction equals the polynomial of the mean value ofand the error is the second order small quantity,namely:

where k is an integer.This derivation also holds forAs a consequence,the integral in Eqn(41)can be calculated approximately as:

Therefore,

Utilizing Eqn(7)and Eqn(14),the mean valuein the whole volume can be derived:

Similarly,

Substituting Eqn(44)through Eqn(47)into Eqn(43),the effective compression modulusis obtained.

2 Outer surface restrained

The second situation is when the outer surface of the rubber ring is restrained.This is an extreme case when a rigid sleeve with the same diameter as the outer diameter of the rubber ring exists.The undeformed and deformed configurations of the rubber ring under compressive load in this situation are shown in Fig.3.The inner and outer radii are RC0and RB0,respectively.It is similar with the situation in the previous section except the outer and inner surface switch roles.

Fig.3 Undeformed and deformed configurations of the rubber ring when the outer surface is restrained

From Fig.3,the transformation relation between r and R changes to:

Meanwhile,from the incompressible condition,α0changes to:

where η is defined as RB0/R.Take advantage of these equations,we can derive the transformation relation with the same form as in Chapter 1:

Further,it is apparently all other transformation relations are the same as those in section 2.Using the same method in Chapter 1,identical equations from Eqn(8)through Eqn(36) can be gotten.The only difference is that r0represents the inner radius of the rubber ring after deformation.Then,similar with equations(37)and(43),the stress component in the radial direction is:and the effective compression modulus is:

3 Outer and inner surfaces free

Fig.4 Undeformed and deformed configurations of the rubber ring when the outer and inner surfaces are free

The last situation is when the outer and inner surfaces of the rubber ring are free.The undeformed and deformed configurations of the rubber ring under compressive load in this situation are shown in Fig.4.The extensions in radial direction at the mid-height of the outer surface and inner surface are RA0α0′and RC0α0″,respectively,where RA0is the outer radius and RB0is the inner radius.

This issue can be solved using the results achieved already.As shown in Fig.4,the rubber ring can be divided into two parts,separating by an imaginary neutral cylinder surface.This surface is assumed to keep unchanged in radial direction and circumferential direction during the deformation.In this way,the outer and inner parts become rubber rings as described in Chapter 1 and Chapter 2,whose effective compression modulus are already known.The only problem is the radius of the neutral surface RB0is not known yet.To determine RB0,let Eqn (37)equals Eqn(51)on average of the height and neglect the small quantities in the equation for simplification,i.e.,letand λ equal 1;let rA0equal RA0,rC0equal RC0and r equal RB0.Then we can get the following formula for RB0:

Substituting Eqn(57)into equations(43)and(52),we can get the effective compression modulus of the outer and inner parts of the rubber ring,i.e.,EC1and EC2.These two parts are parallelly connected.The total effect of the whole rubber ring is:

4 FEM solutions and discussion

The load-compression curves in the three situations from equations(43),(52)and(58) were compared with the solutions of the nonlinear FEM program Abaqus.The FEM analysis used axisymmetric models and implicit algorithm.The hybrid stress element CAX4RH was adopted to avoid volumetric locking.The materials are all incompressible Mooney-Rivlin types, including four sets of representative material parameters as shown in Tab.1,indicated by Mat-1 through Mat-4.To evaluate these formulae as thoroughly as possible,a series of rubber rings with different geometric dimensioning are checked.These rubber rings contain five shape factors and three diameter ratios,as in Tab.2,and with the same outer radius RA0=200 mm.Here, the shape factor S is defined traditionally as RA0/2h,and η0is the ratio of the inner radius to the outer radius.Taking the three boundary conditions into consideration,180 models were calculated in total.The rubber rings are compressed until the free surface is about to contact with the rigid plates.The maximal compression strain is 0.18.

Tab.1 Material parameters used in the calculation

Tab.2 Geometric dimensioning of the rubber rings

Fig.5 through Fig.7 plot three typical load-compression curves for Mat-1.Similar results are found in other cases and the charts are omitted.It is seen from these figures that:(i)The results of all the three formulae derived in this paper fit very well with the FEM calculation; (ii)The effective compression modulus Ecincreases obviously with compression;(iii)Ecvaries quite considerably in different boundary conditions.When the outer surface is restrained,the value is much higher than the one when the inner surface is restrained,and it reaches the minimum when both the surfaces are free.As a consequence,for the rubber rings with a rigid center cylinder,the vertical stiffness may have significant change during compression,depending on the diameters of the rubber inner surface and the center cylinder.If the diameters are close,at first the compression modulus can be gained by Eqn(58).With the increasing of compression,the inner surface could contact with the center cylinder,which makes the situation more like that described in Chapter 1 and leads a much higher vertical stiffness.This phenomenon deserves to be noticed in the design stage.

Fig.5 Load-deformation curves for Mat-1,S=1,RB0/RA0=1/2,inner surface restrained

Fig.6 Load-deformation curves for Mat-1,S=1,RB0/RA0=1/2,outer surface restrained

Fig.7 Load-deformation curves for Mat-1,S=2,RB0/RA0=1/2,outer and inner surfaces free

For the better discussion of the results,a new shape factor which describes the aspect ratio of the rubber ring’s cross section is defined as:

Fig.8 Absolute value error of Ecfor Mat-1 at maximum compression,inner surface restrained

Fig.9 Absolute value error of Ecfor Mat-1 at maximum compression,outer surface restrained

Fig.10 Absolute value error of Ecfor Mat-1 at maximum compression,outer and inner surfaces free

The absolute value errors of Ecat the maximum compression of Mat-1 are plotted in Fig.8 trough Fig.10.The other materials have the similar results,which are tabulated in Tab.3 through Tab.5.From these figures and tables,we can see that equations(43),(52)and(58)show very good accuracy in most cases.In general,when S′is larger than 1.0 for Eqn(43),and larger than 2.0 for equations(52)and(58),the error is less than 3 percent in most cases.There is one exception yet.It is shown the error of Eqn(52)is sensitive to the ratio of the inner radius to the outer radius,which becomes apparent when η equals 1/4.However,the error drops quickly with the decrease of λ,i.e.,the initial stiffness obtained by Eqn(52)is in fact very accurate.The main reason is that the average processing is applied for some terms to get EC. When the outer surface is restrained,the rubber ring is harder to be compressed than in other situations and small change of λ will bring a relative large change ofIn this way,the error is induced.In the termof Eqn(52),this error is further magnified by the square and the coefficientwhich is much larger thanin Eqn(43). As a consequence,the error may reach 10 percent when η is small.It is suggested Eqn(52) be modified to consider this error source as follows,i.e.,use the mean value in the whole volume rather than the mean value in z for1+（ ）α in the above term.

Using Eqn(60)instead of Eqn(52),the absolute value errors of Ecat the maximum compression are tabulated in Tab.6.The accuracy is very good for all cases as long as S′≥2.0.

Tab.3 Absolute value errors of Ecfor Mat-2 at maximum compression

Tab.4 Absolute value errors of Ecfor Mat-3 at maximum compression

Tab.5 Absolute value errors of Ecfor Mat-4 at maximum compression

Tab.6 Absolute value errors of Ecfor Eqn(60)at maximum compression

From the above discussing,it is recommended in general that Eqn(43)be used for S′≥1.0,and equations(60)and(58)be used for S′≥2.0.This range is sufficient for most engineering applications and the formulae will provide a good approximation to Ecof the rubber ring.

5 Conclusions

The load-compression relationships of the incompressible rubber ring bonded between rigid plates are derived in this paper.The relationships are based on two kinematics assumptions.The hyper-elastic Mooney-Rivlin type material is considered and the derivation complies with the theory of continuum mechanics.Three boundary conditions are considered in the deriva-tion,i.e.,when the inner surface is restrained,when the outer surface is restrained and when both surfaces are free.

The theoretical solutions are obtained.The comparison with the FEM results shows these proposed formulae has a very good accuracy in predicting the behavior of the bonded rubber ring with various shape factors,even in the finite strain.The typical load-compression curves of the rubber rings are also presented and the characteristics of the compression stiffness in different boundaries are discussed.

[1]Yuan R H,Wang A M,et al.Design considerations of leg mating units for floatover installations[C]//ISOPE,Proceedings of the Twenty-second(2012)International Offshore and Polar Engineering Conference.Greece,ISOPE,2012: 1091-1098.

[2]Gent A N,Lindley P B.The compression of bonded rubber blocks[J].Proceeding of the Institution of Mechanical Engineers,1959,173:111-117.

[3]Gent A N,Meinecke E A.Compression,bending and shear of bonded rubber blocks[J].Polymer Engineering and Sciences,1970,10:48-53.

[4]Kelly J M.Earthquake-resistant design with rubber[M].London:Springer,1993.

[5]Chalhoub M S,Kelly J M.Analysis of infinite-strip-shaped base isolator with elastomer bulk compression[J].Journal of Engineering Mechanics ASCE,1991,117:1791-1805.

[6]Chalhoub M S,Kelly J M.Effect of bulk compressibility on the stiffness of cylindrical base isolation bearings[J].International Journal of Solids and Structures,1990,26:734-760.

[7]Koh C G,Kelly J M.Effects of axial load on elastomeric isolation bearings[R].Report no.UCB/EERC-86/12,Berkeley: Earthquake Engineering Research Center,University of California,1987.

[8]Koh C G,Kelly J M.Compression stiffness of bonded square layers of nearly incompressible material[J].Engineering Structures,1989,11:9-15.

[9]Tsai H C,Lee C C.Compressive stiffness of elastic layers bonded between rigid plates[J].International Journal of Solids and Structures,1998,35:3053-3069.

[10]Tsai H C,Pai W J.Simplified stiffness formulae for elastic layers bonded between rigid plates[J].Engineering Structures, 2003,25:1443-1454.

[11]Lindley P B.Load-compression relationships of rubber units[J].J Strain Anal.,1966,1:190-195.

[12]Hill J M.A review of partial solutions of finite elasticity and their applications[J].International Journal of Non-Linear Mechanics,2001,36:447-463.

[13]Huang Z P.Fundamentals of continuum mechanics[M].Beijing:Higher Education Press,2011.

粘結(jié)橡膠環(huán)的載荷—壓縮關(guān)系

鄭軼刊1，張世聯(lián)2

（1上海振華重工（集團(tuán)）股份有限公司，海上重工設(shè)計(jì)研究院，上海200125；2上海交通大學(xué)船舶海洋與建筑工程學(xué)院，上海 200240）

粘結(jié)橡膠環(huán)在很多工程領(lǐng)域被廣泛應(yīng)用于緩沖沖擊。這些應(yīng)用一般都要求掌握橡膠墊準(zhǔn)確的載荷壓縮關(guān)系。但以往的研究僅討論了橡膠柱體而忽略了橡膠環(huán)，而且僅少數(shù)研究是基于連續(xù)介質(zhì)力學(xué)理論的。文章推導(dǎo)了剛性板間不可壓縮橡膠環(huán)在三種邊界條件下的載荷—壓縮關(guān)系，考慮了Mooney-Rivlin材料，并基于連續(xù)介質(zhì)力學(xué)理論進(jìn)行了推導(dǎo)。文中將推導(dǎo)公式的計(jì)算結(jié)果與有限元計(jì)算結(jié)果進(jìn)行了比較，證明了這些公式有足夠的精度，適用于各種形狀系數(shù)和材料參數(shù)，在有限應(yīng)變下仍然適用。最后，文中還給出了典型橡膠環(huán)的載荷—壓縮曲線，并對不同邊界條件下的壓縮剛度特性進(jìn)行了討論。

載荷—壓縮關(guān)系；橡膠環(huán)；Mooney-Rivlin；邊界條件；有限應(yīng)變

O342

:A

鄭軼刊（1983－），男，上海振華重工（集團(tuán)）股份有限公司，海上重工設(shè)計(jì)研究院博士，通訊作者；

O342

A

10.3969/j.issn.1007-7294.2016.03.012

1007-7294（2016）03-0363-17

張世聯(lián)（1952-），男，上海交通大學(xué)教授，博士生導(dǎo)師。

Received date：2015-07-18

Biography：ZHENG Yi-kan(1983-),male,Ph.D.,E-mail:zykorzht@sjtu.edu.cn;

ZHANG Shi-lian(1952-),male,professor/tutor.