LARGE ELASTIC COMPRESSION OF FINITE RECTANGULAR BLOCKS OF RUBBER · 2016. 10. 18. · LARGE ELASTIC...

LARGE ELASTIC COMPRESSIONOF FINITE RECTANGULAR

BLOCKS OF RUBBERBy JAMES M. HILL and ALEXANDER I. LEE

{Department of Mathematics, University of Wollongong, Wollongong,New South Wales, Australia)

[Received 16 March 1988]

SUMMARYFor the problem of large elastic compression of a finite rectangular rubber block

which has bonded metal plates to its upper and lower surfaces, a new load-deflectionrelation is derived assuming an isotropic incompressible Mooney material and thatthe pointwise vanishing of the stress on the free faces can be replaced by thevanishing of the force resultant on these faces. The load-deflection relation soobtained is derived from a fully three-dimensional deformation which is a generali-zation of a plane-strain deformation given previously for infinitely long rectangularrubber pads. The 'finite' and 'infinite' load-deflection relations are compared withexperimental results and with that predicted by the conventional engineeringapproximation deduced from the so-called 'shape-factor' method.

1. Introduction

THE problems arising from large elastic deformations of finite rectangularrubber blocks are still important and relevant both to the constructionengineer, who utilizes such devices in bridges for example, and to themanufacturer of rubber engineering components, both of whom need topredict accurately their in situ mechanical performance. Typically suchdevices have bonded metal plates on their upper and lower surfaces and thethree principal deformation modes of most concern are compression,shearing and tilting (rotational) which are effected by fixing the lower metalplate and applying forces or moments to the upper plate as indicated in Fig.1. In this paper we consider the problem of large elastic compression ofbonded finite rectangular blocks of rubber which is assumed to be theisotropic incompressible perfectly elastic Mooney material.

Since the problem involves finite deformations there is no exact solutionand much of the engineering literature is based on load-deflection relationsof the type emanating from the linear theory of elasticity but couched interms of the 'shape factor' 5 which is defined by

one loaded areaforce-free area

and incorporating an 'apparent Young's modulus' Ea which varies with 5.

[Q. Jl Meeh. ippi. Matfa., Vol. 42, Pt. 2, 1989] © Oxfonl LWrtntty Prtu 1989

268 JAMES M. HILL AND ALEXANDER I. LEE

(i) Compression (ii) Shearing (iii) Tilting or rotational

FIG. 1

For compression the resulting force-deflection relation arising from in-tegrating the equation

dF (apparent Young's modulus) x (one loaded area)

db (thickness after a compression of magnitude 6)

where F is the applied force, frequently provides a surprisingly accuratedescription of the actual behaviour of bonded rubber blocks. Such analysishas no rational or continuum mechanical basis and its apparent successderives from the fact that the assumed relations are fudged to fitexperimental results. For many engineering purposes such a superficialanalysis may be perfectly adequate. However, it only conveys reasonablyaccurate information about the force-deflection relation which is a macro-scopic property of the block. It provides no information whatsoever on thepointwise variation of stresses and deformation throughout the block.Further, the extension to slightly more sophisticated problems is by nomeans obvious and depends upon whether there already exists a substantialbody of experimental results on the problem. Thus although the shape-factor approach by no means describes the whole story, it must neverthelessbe compared and considered in the context of any new proposed force-deflection relation. For the shape-factor approach and the engineeringliterature relating to this method we refer the reader to Gent andLindley (1).

In order to provide a rational framework for the derivation of load-deflection relations, KJingbeil and Shield (2) produced an approximatesolution for infinitely long rectangular blocks assuming firstly that materialplanes Y = constant are mapped into planes y = constant (see Fig. 2) andsecondly that the pointwise requirement of zero stress on the free surfaces isreplaced by the vanishing of resultant forces on these surfaces. Accordinglyfor the isotropic incompressible perfectly elastic Mooney material, Klingbeiland Shield (2) exploited solutions of the form

x=f(Y)X, y=g(Y), z = Z, (1.3)

FINITELY DEFORMED RUBBER BLOCKS 269

Y

FIG. 2

where (X, Y, Z) and (x, y, z) denote material and spatial rectangularCartesian coordinates respectively. Further, f(Y) and g(Y) denote functionsof Y only which are determined completely from the constraint ofincompressibility, the equilibrium equations and appropriate displacementboundary conditions. The resulting load-deflection relation provides areasonably accurate representation of experimental behaviour. In practicehowever, rectangular rubber blocks tend not to be particularly long in onedirection and the purpose of this paper is to investigate a possiblethree-dimensional extension of the solution of Klingbeil and Shield (2).

Accordingly for the Mooney material we adopt a deformation of the form

x=f(Y)X, y=g(X), z=h{Y)Z, (1.4)

and now, based on the assumption that the vanishing of pointwise stresseson four free faces can be replaced by the vanishing of two resultant forces,we may deduce yet another force-deflection relation. In this paper weexamine the three force-deflection relations arising from the shape-factormethod, (2) and the deformation (1.4) and we make a comparison withexperimental results. In the next three sections of the paper we giverespectively the basic equations for finite deformations of a Mooney


material, the solutions for the deformation (1.4) and the approximateanalysis leading to the load-deflection relation for compression of rectangu-lar rubber blocks of finite length. In section 5 we give the correspondinglinear estimate which we relate to that arising from the shape-factor methodand in section 6 we compare the various load-deflection relations withexperimental results.

2. Bask equations for finite elastic deformations of a Mooney materialFor fully three-dimensional deformations of the isotropic incompressible

perfectly elastic Mooney material the calculations associated with theequilibrium equations are frequently awkward and involved. One approachwhich minimizes the labour is to exploit the 'reciprocal equations' given byHill (3,4). Such equations utilize both the deformation and its inverse. Thuswith material and spatial rectangular cartesian coordinates ZK (K = 1, 2, 3)and z1 (J = 1, 2, 3) respectively, the deformation and its inverse take theform

z> = z'(ZK), ZK = ZK{zj), (2.1)

with strain tensors

azKdzK'( 2 2 )( }

where as usual the repeated index implies summation. The Mooneystrain-energy function is given by

S = C 1 ( / 1 -3 ) + C 2 ( / 2 -3 ) , (2.3)

where C, and C2 are material constants and /, and I2 are the traces of thematrices c"1 and c respectively. From either of (3,4) we may readily deducethe reciprocal equations for the deformation (2.1) of a Mooney material,

—t (p + C212) = 2C, W - 2C2 - ^ V\ZK, (2.4)

where V2 and V? are the Laplacians with respect to material and spatialvariables respectively, namely

V2 = V? = (2 5)dZKdzK' l az'dz'' K ^

and p is the pressure function occurring in Finger's stress-strain relations forthe Cauchy stress tensor /*, thus

-1" - 2C2c", (2.6)

and 6U is the Kronecker delta. Although, of course, (2.4) involves both thedeformation and its inverse, these reciprocal equations are preferable for


complicated calculations involving fully three-dimensional deformationsbecause they preserve the inherent symmetry of the equations of finiteelasticity under interchanges of material and spatial coordinates. Thestrategy in utilizing reciprocal equations is to initially exploit both forms ofthe deformation (2.1) and then, in the final stages of the calculation, useonly one of these forms. This process is demonstrated in the followingsection. We remark that for the neo-Hookean material (C2 = 0) or for theso-called extreme Mooney material (Cx = 0) we need only use one form ofthe deformation at the outset.

3. Solutions for compressive deformations

Together with (1.4) we utilize the inverse deformation which takes theform

= F(y)x, Y=G(y), (3.1)

where F, G and H denote functions of y only such that Ff = 1 and Hh = 1.For the deformation (1.4) the incompressibility condition \dz'/9ZK\ = 1becomes

fhg' = 1, (3.2)

where here and subsequently primes denote differentiation with respect toY. On introducing q defined by

q=p + C2I2, (3.3)

it is not difficult to show that the reciprocal equations (2.4) become

ldY2

dy ~ I x dY2 2L dy dy2 dy dy2 dy dy2 \Y(3.4)

Clearly for the second-order partial derivatives we have qxz = qzx = 0 and onequating expressions for qv and qzy we obtain two equations which can beimmediately integrated, thus

(3.5)d2h dH\


where ax and a2 denote integration constants. On using the incompress-ibility condition we have

dF -hf dH -fh1

dy f dy hand (3.5) becomes

( d + C-zh2)/" + C2hh'f = or,/, (C, + C2f2)h" + C-df h' = a2h, (3.7)

which can be readily integrated again to give2 2 + ^2, (3.8)

where /3, and /32 are further integration constants.From the above equations it can be shown that the appropriate function

q(x, y, z) is given by

(3.9)

where p0 is a constant. Altogether, for the pressure function p(x, y, z) wefind that

p{x, y, z) = aJ'X2 + a2h2Z2 + Cig'

2 - C2[-2 + -2j

— ZL2{Jn) \X —j + Z T 7 + 1 )+/><>• (3.10)

We observe that from (3.8) we have

which although yielding the separable equation

df dh=C2/2)(o-1/2 + /5,)]i = [(C, + C2h

2)(a2h2 + fc)]l' ( 3 ' 1 2 )

in general immediately gives rise to elliptic functions (except in the specialcase when aJPi = 0-2//32 = C2ICX). We further note that if / = h (and«i = 2̂> /?i = Pi) then (1.4) is essentially the axially symmetric deformationstudied by Klingbeil and Shield (2) in the context of squashing circular padsof rubber and these authors also noted that the particular deformationstudied resulted in elliptic functions for the Mooney material. Finally in thissection, for the neo-Hookean material (C2 = 0) we have immediately from(3.11) that

C\df C\dhdY ( 3 1 3 )


which readily yield simple trigonometric functions as solutions for / and h,while for the extreme Mooney material ( d = 0), (3.6) and (3.8) give

which again results in simple trigonometric functions, the explicit nature ofwhich depends on the signs of the various integration constants. In thefollowing section we detail such solutions appropriate to the compressing ofrectangular pads.

4. Approximate analysis for compression of finite rectangular blocks

We consider a rectangular rubber block with undeformed dimensions oflength 2L, breadth 2B and thickness 2T as indicated in Fig. 2. We supposethat in the undeformed state the rectangular Cartesian coordinate system(A', Y, Z) has its origin located at the central point of the block and that thefaces Y= ±T have bonded rigid metal plates over which there is an appliednormal force F resulting in a final thickness 2t. Assuming a deformation ofthe form (1.4), we require that / and h are even while g is an odd function.In addition we require that

/(±r) = l, g(±T)=±t, h(±T) = l. (4.1)

We assume that the pointwise zero-stress conditions on the free surfacesZ=±L and X=±B can be approximated by the vanishing of forceresultants on these surfaces, and because of the symmetries of /, g and hthese become

fL rT fB ,T

TR dZdY = i), TR dXdY = 0, (4.2)J-LJ-T J-BJ-T

on surfaces X=±B and Z = ±L respectively. In addition the appliednormal force on the metal end-plates is given by

F=-\ \ TjfdXdZ, (4.3)J—BJ—L

where T$ denotes the first Piola-Kirchhoff stress tensor with components

TH = -pg'h + 2CJ - 2C2g'h3(g'2 +f'2X2),

TR = -Pfh + 2Clg' - 2C2fh[(f2 +f'2X2)h2 + <Jh')2Z2],

T% = -pg'f + 2Cxh - 2C2g'f3(g'2 + h>2Z2),

rj? ={P + 2C2[(J2 + g'2 +f'2X2)h2 + (Jh')2Z2))f'hX, y (4.4)

rj? ={p+ 2C2[(h2 + g'2 + h'2Z2)f2 + (f'h)2X2]}fh'Z,

Tl1 = 2(C, + Ch^tf'X, tf = 2(C, + C2f2)h 'Z,

TR3 = -2C2ff'h 'XZ, TR

X = -2C2hf'h 'XZ.


From the above equations is it not difficult to deduce the expression for theapplied force F, namely

{ ^ 2 ^ } (4.5)

If we introduce the mid-plane stretches At and A2 such that

Ai=/(0), A2 = *(0), (4.6)

then using the fact that / ' (0) and h'(0) are zero we see that /3, = -axk\,02= - t t2A| a n d equation (3.8) becomes

(C, + C2A2)f2= *,( /*-A?) , (C1 + C2 /2)/ l '2 = or2(/I

2-Al). (4.7)

In principle for a given applied force F the five conditions (4.1)!, (4.1)3,(4.2)i, (4.2)2 and (4.3) constitute five equations for the determination of thefive unknown constants a^, a2, A^ A2 and p0 from which a theoretical valuefor the new thickness 2/ can be predicted. In the remainder of this sectionwe consider the neo-Hookean material for which solutions of (4.7) canbe given explicitly, and numerical results for this material are given insection 6.

If C2 = 0 then it can be shown that the appropriate solutions of (4.7) are

A1cos)t1y( h(Y) = ^ cos k2Y, (4.8)

where the constants kt and k2 are related to at and a2 as follows:

k^-aJCu k\ = -oc2ICx. (4.9)

Further, from (3.2) we may deduce that

• sec k2ldl, (4.10)

while (4.1) gives

. = secJfc2T,

Further for the neo-Hookean material the conditions (4.2) and (4.5)become

ER (T(^M)^+ fl^MV/^M)2^Cik VsecJfc,77 Jo Vsec^r/ \scck2T/


2 D 22 tanJt2r _ k\B

'~k~2

k\B2

Jo §>

| k2L p0

(4.12)

In principle these three equations consitute three equations for the threeunknowns kt, k2 and p0 which in general need to be solved numerically.However in the special case when L = Bwe have the following simplifica-tions resulting form (4.11) and (4.12):

kx = k2 = k, Aj = A2 = sec kT, t =

{^r + - cos4 kT\ log [sec kT + tan kT]ICj 8 J

sin2A:r

l + - - cos2 kT - - cos4

4 8 itan kT sec kT,

(4.13)

These results are displayed numerically in section 6. Finally in this sectionwe note that similar formulae can also be deduced for the extreme Mooneymaterial (Ct = 0). However since this strain-energy function has no realphysical basis we do not give these results here.

5. linear estimate and the shape-factor methodFor small compression we define the applied strain e = (T — t)/T and we

suppose that the deformation and stresses can be expanded in powers of eso that consistently neglecting e2 and higher powers gives the usual lineartheory of elasticity. Thus for the deformation we have

/(Y) = 1 + eu(Y) + ... = 1 + eYl(T2 - Y2) + ....

ev(Y) (5.1)

h(Y) = 1 + e»v(Y) + ... = 1 + ey2(T2 - Y2) + ...,

while for the pressure function and diagonal stress components we have

p(X, Y, Z) = 2(C, - C2) + eP(X, Y, Z) + ...

= 2(Ci - C2) + e/i[y,( Y2 - A-2) + y2(Y2 - Z2) + y3] + ..., (5.2)


and

TiR

1 = e[-P + 2nu] + ..., Tf = e[-P + 2pv'] + ...A

Jwhere y1; y2 and y3 denote arbitrary constants and /z = 2(C, + C2) is thelinear shear modulus. Now from the above, (4.2) and (4.3) we may deducethe two equations,

3y i(T2 + B2) - Y2(T2 - L2) = 3y3> 3y2(7"2 + L2) - Yl(T

2 - B2) = 3y3,(5.4)

which have solutions3y3(L2 + 2T2)

15T\L*. . . .,

72 [5T2(L2 + B2) + 4(7* + L2B2)]"

But from the definition of the applied strain e we may deduce thatv(T) = — T and thus we have

2 7 % , + y2) = 3, (5.6)

which together with (5.5) is the determining equation for y3, thus

_ [57-2(L2 + B2) + 4(7* + L2g2)]Y3~ -iT2/r2 i D2 , >T2\ " {?•')

Now, from (4.3) and (5.3)2) we have

F = ̂ | ^ {(3r2 - B2)y, + (3T2 - L2)y2 + 3y3}, (5.8)

which, on using the above equations, simplifies to give

F 2 + 2T2)(L2

+ 2T2)}B2 + L2 + 4T2)T2 VABL 3 I ( )

where E is the linear Young's modulus, that is

£ = 6(Ci + C2) = 3/i. (5.10)

We observe that in the limit as L tends to infinity (5.9) agrees with theexpression given by Klingbeil and Shield (2) resulting from their plane-strain analysis for long rubber blocks.

Lindley (5) and Gent and Meinecke (6) propose shape-factor methods forthe compression of rectangular rubber blocks which utilize equations (1.1)and (1.2). For a compression of magnitude 6 we have

6 = 27 - 2t = 2Te, (5.11)

where e = (T - t)/T is the applied strain. Now the shape factor as defined


by (1.1) at a compression of magnitude 6 is 'approximately'

S = 2{4Lt + 4Bt) = (l=e)' ( 5 1 2 )

where SQ is the original shape factor of the rectangular rubber block. Nowthe shape-factor methods for rectangular blocks with B =£ L are based onassuming that the apparent Young's modulus Ea in (1.2) takes the form

Ea = Eh + ECS2, (5.13)

where E is the linear Young's modulus, C is a constant given by_ 4 B

and Eh is known as the 'homogeneous' compression modulus. Lindley (5)proposes the expression

while Gent and Meinecke (6) suggest an alternative expression:

( 5 1 6 )

We observe that both expressions yield the linear Young's modulus if B = Lwhile both predict the plane strain limit 4E/3 for L tending to infinity. Nowfrom (1.2) we have

dF _ ABLE.dd~(2T-6)' ( 5 1 7 )

and thus from (5.11) to (5.13) we may deduce that

Hence, on integration with respect to e and using the fact that the appliedstrain e is zero when F is zero, we obtain

£ f 1 } (519)

For small applied strains this gives

F = 4BLe{Eh + ECSl). (5.20)

Equation (5.19) with Lindley's expression (5.15) for Eh is used for thenumerical work in the following section.

Finally in this section we simply note that if we also allow a variation withcompression of the homogeneous compression modulus (5.16), thus


then (5.18) can still be integrated, with Et in place of Eh, with the resultthat.

F/"~*C2 J7/ 7 D\2 r 1 _l_ (f ^ i f>2\ /O'T12/i ^\2~i

< T ^ + W ^ ) l o g L i + (L2 + B2)/8r2 J

-E log ( 1 - 0 ^- 2 | . ( 5 2 2 )

which actually gives almost identical results to (5.19) with Lindley'sexpression (5.15) for the homogeneous compression modulus.

6. Experimental and numerical results

We first compare numerical values of the applied force for the finite modelfor the neo-Hookean material with the plane-strain approximation given byKlingbeil and Shield (2). Table 1 gives numerical values of F for six blockseach of thickness 10 cm, breadth 20 cm and lengths ranging from 30 cm to960 cm and a shear modulus /i = 0-064 kN/cm2. Interestingly enough thethree-dimensional deformation produces results which tend to those ofKlingbeil and Shield (2) as the length of the block tends to infinity.Although this is a highly desirable feature it is not altogether obvious whythis should happen because the finite model has an additional constraint(namely (4.2)2) over the end-faces of the block. No such constraint exists forthe Klingbeil and Shield (2) model and the authors are not entirely clearwhy (4.2)2 becomes redundant as 2L tends to infinity. For the neo-Hookeanmaterial, the length L tending to infinity corresponds to the constant k2

tending to zero and formally the two conditions (4.12) are compatiblebecause the product k2L remains finite. In the limit as L tends to infinityand k2 tends to zero, (4.12) gives

(k2L)2

= l^" l D '~2A?/~*7~~2^'(6.1)

C, 3 H2 + ^B)-^\ kxt 2X2'

and these two equations together fix the product k2L. However, (6.1)i is thecondition used by Klingbeil and Shield (2) and since the quantity on theleft-hand side of this equation also appears in the expression (4.12)3 for theapplied force F, we have at least some explanation why the numericalresults of Table 1 coincide for long blocks.

In order to make an objective assessment of the various force-deflectionrelations for the compression of rectangular blocks we need to prescribe thelinear shear modulus fi or Young's modulus E (E = 3fi). In this regard, wecan either adopt the strategy that for each force-deflection relation £ is avariable parameter of the model determined from linear experimental data,

Z mT

AB

LE 1

. C

ompa

riso

n of

app

lied

forc

es fo

r th

e fi

nite

-blo

ck m

odel

for

the

neo-

Hoo

kean

mat

eria

l an

d (2

) for

six

blo

cks

£ea

ch o

f th

ickn

ess

10 c

m,

brea

dth

20 c

m

and

leng

ths

incr

easi

ng fr

om

30 c

m

to

960

cm f

or

a sh

ear

mod

ulus

of

0-06

4 kN

/cm

2 g

.train

004

008

012

016

0-20

0-24

0-28

0-32

2L =

(4.12) 8 17 27 39 52 68 86 108

30 cm (2) 12 26 42 62 86 115

150

194

2L =

(4.12)

20 44 72 104

142

187

240

304

60 cm (2)

23 51 84 124

172

230

301

388

2L =

(4.12)

45 98 161

235

324

432

561

720

Force (k

N)120 cm (2)

47 102

168

248

343

459

601

777

2L =

(4.12)

93 202

333

489

676

904

1181

1522

240 cm (2) 94 204

337

495

686

919

1203

1554

2L =

(4.12)

187

408

671

987

1368

1830

2396

3093

480 cm (2)

187

409

674

991

1373

1837

2406

3108

2L =

(4.12)

375

818

1347

1980

2744

3672

4808

6210

960 cm (2)

374

818

1348

1981

2746

3675

4812

6215

w o c ts tn 7>

toLOCKS


TABLE 2. Values of shear modulus for varying hardness from Lindley (7),Payne and Scott (8), Gobel (9) and British Standard data (BS5400)

Shear modulus (kN/cm2)

Hardness(IRHD)

3040506070

Lindley

00300045006401060173

Payne andScott (8)

003300530074011001740-310

G6bel (9)

0047007100960-1340191

BritishStandard

006000900120

in which case we obtain a different value for each theory, or we canindependently measure the hardness of the rubber block and obtain theshear or Young's modulus from tables. For our own experimental results weadopt both points of view and give two sets of graphs. In our analysis of theexperimental results of Gent and Lindley (1) we adopt the former approach.The shear modulus of rubber is related to its hardness and there are varioushardness systems including International Rubber Hardness Degrees(IRHD), Shore and British Standard Hardness. In our experiments thehardness of the rubber blocks was measured using a Shore durometer(model A) which actually gives values which are almost the same as thosefrom the IRHD system. Various authors provide information which relateshardness to shear modulus (see for example Lindley (7), Payne and Scott(8), Gdbel (9) and Table 2).

The data provided by Lindley (7) results from experiments on naturalrubber springs with hardness above 48 IRHD and measured at low shearstrain. Lindley (7) maintains that hardness measurements are subject tosome uncertainty and states that there should be ±2 degrees tolerance inhardness in Table 2. Payne and Scott (8) claim that their data form an idealrelationship and that a difference of 1 degree in hardness corresponds toapproximately 4-5 per cent difference in modulus. G6bel (9) does not giveany details about his data and the British Standard data (BS5400) is basedon shear strains ranging from 5 to 25 per cent.

Before proceeding to our own experimental work we first compare ourresults with existing experimental results of Gent and Lindley (1). Table 3gives experimental values of F/4BL for four square blocks of varying shapefactor SQ. The experimental values are obtained from (magnified) graphicalresults and accordingly while they are not completely precise they arenevertheless reasonably accurate. The values of the Young's modulus usedin the table are obtained from the initial linear portion of the curves given in(1) and the formulae

(6.2)


TABLE 3. Experimental results of Gent and Lindley (1) compared withKUngbeil and Shield (2) and the present model for the neo-Hookean material

Force per unit area (F/4flL)(kgf/cm2)Shape Shape Finitefactor Percentage Experimental factor neo-Hookean

So strain (2) (1) (5.20) (4.12)

018

0-39

0.51

0.85

102030102030102030102030

2-66-2

11-3

3-48-5

16-6

4-210-921-8

6-517-335-6

2-35-39-42-86-4

1203-47-6

14-2

5011-820-8

2-45-28-7306-7

11-2

3-78-3

14-2

5-613022-9

2-45-28-73-17-2

12-6

3-99-2

16-6

6 014-727-8

being the appropriate linear approximations for the KUngbeil and Shield (2)estimate, the shape factor estimate (see (5.20), (5.14) and (5.15)) and thepresent work (see (5.9)) respectively where for the square block SQ = B/4T.The actual values of E used in Table 3 are given in Table 4 and result fromthe given experimental results in Table 3 at 10 per cent strain. For the finitemodel, results for only the neo-Hookean material are given. It is apparentfrom the table that the full three-dimensional approximation presented hereis considerably superior to the KUngbeil and Shield (2) model but bear inmind the latter model is at its worst for the square block. We also observefrom the table that the shape-factor estimate appears to provide aconsistently good approximation.

In order to obtain a more objective assessment of the situation we

TABLE 4

Shapefactor

So

018

0-39

0-51

0-85

Young's modulus E (kgf/cm2)

(6.2),

19-61

15-46

14-24

10-30

Shape factor(6.2)2

21-45

20-90

21-51

1913

Finite(6.2),

30-54

2611

2500

19-28


TABLE 5

Figure

3(a)

4(a)

5(a)

6(a)

Young's modulus £ (kN/cm2)

(2)

01767

0-5897

01613

0-2005

Shapefactor

0-2355

0-7263

0-2463

0-2517

Finite

0-2533

0-9573

0-2603

0-2875

Figure

3(b)

4(b)

5(b)

6{b)

Young's modulus E (kN/cm2)

(2) andfinite

01863

0-5793

01920

01983

Shapefactor

01710

0-3870

01800

01845

conducted experiments on rubber blocks using a hydraulic compressiontesting machine. For forces in the range 0 to 2000 kN the machine measureshydraulic pressure. However, for forces in the range 0 to 200 kN a load cellcan be inserted to obtain more accurate results. The hydraulic compressorhas an accuracy of ± 10 to 15 per cent in the range 0 to 2000 kN while the200 kN load cell has an accuracy of ±5 per cent. Sandpaper was used tosimulate a bonded rubber block and the area of contact between the rubberand the compressing metal plates remained substantially constant. The forcemeasurements were displayed digitally and the deflection was read fromfour dial gauges situated at the corners of the compressing metal plates. Theaverage of these four readings was taken as the deflection which took place.The Young's modulus used in the theoretical curves of Figs 3(a), 4(a), 5(a)and 6(a) are obtained from the initial linear portion of the experimentalcurves. Table 5 gives the values of the Young's modulus obtained in thismanner and also the value of the Young's modulus obtained by measuringthe hardness of the rubber and using Table 2. For the Klingbeil and Shieldapproximation and the present estimate the Young's modulus in Table 5arising from the hardness test is that obtained from Lindley (7) while thevalue given in Table 5 for the shape-factor approximation is that obtainedusing British Standard data which is generally used in conjunction with thisestimate. From Figs 3(a), 4(a), 5(a) and 6(a) it is apparent that although thefinite model is an improvement on the Klingbeil and Shield estimate, theshape-factor approximation appears to be superior to both. However, if weadopt the alternative strategy and assign each theory the value of E,obtained from the hardness test, then Figs 3(b), 4(b), 5(b) and 6(b) appearto indicate that the finite model is superior to both the Klingbeil and Shieldestimate and the shape-factor approximation.


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7. Conclusion

In modelling the compression of rectangular rubber blocks, we haveutilized a three-dimensional deformation for the isotropic incompressibleperfectly elastic Mooney material, to generate an approximate solution ofthe problem, for which the pointwise vanishing of the stress on free surfacesis approximated by the vanishing of force resultants over the surface. Thisnew approximation appears to be a substantial improvement on a similarapproximation due to Klingbeil and Shield (2) which is strictly only valid forextremely long rubber blocks. Moreover we have compared the* newload-deflection relation with new experimental data and with results arisingfrom the shape-factor approximation which is commonly used in engineer-ing. Our work tends to indicate that if the shape-factor approximation isused in conjunction with a value of the Young's modulus obtained from thelinear experimental data then this generates a consistently good estimate.However, our work also indicates that the finite model presented here alsoprovides a reasonably accurate approximation irrespective of whether theYoung's modulus is determined from the linear experimental data or from ahardness test. The related problem of combined compression and torsion ofcircular cylindrical bonded rubber mounts is examined in a companionpaper (10).

Acknowledgements

The second author gratefully acknowledges the financial support of anAustralian National Research Fellowship. This work has been undertakenas a part of a collaborative research programme with the Sydney manufac-turer of rubber engineering components P. J. O'Connor. The authors areextremely grateful for the use of their testing facilities and particularly tothe Manager, Mr Peter Snowball, who has enthusiastically assisted theprogramme.

REFERENCES1. A. N. GENT and P. B. LINDLEY, Proc. Instn mech. Engng 173 (1959) 111-117.2. W. W. KUNOBEIL and R. T. SHIELD, Z. angew. Math. Phys. 17 (1966) 281-305.3. J. M. HILL, ibid. 24 (1973) 609-618.4. , Ph.D. thesis, Queensland University, Australia 1972.5. P. B. LINDLEY, Plastics and Rubber Processing and Applications 1 (1981)

331-337.6. A. N. GENT and E. A. MEINECKE, Poly. Eng. Sci. 10 (1970) 48-53.7. P. B. LINDLEY, Engineering Design with Natural Rubber. Natural Rubber

Technical Bulletin (1966).8. A. R. PAYNE and J. R. SCOTT, Engineering Design with Rubber (Maclaren,

London 1960).9. E. F. GOBEL, Rubber Springs Design (Newnes-Butterworths, London 1974).

10. J. M. HILL and A. I. LEE, / . Mech. Phys. Solids to appear.

LARGE ELASTIC COMPRESSION OF FINITE RECTANGULAR BLOCKS OF RUBBER · 2016. 10. 18. · LARGE ELASTIC...

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