EFFECT OF COMPRESSION ON THE SHEAR MODULUS OF RUBBER A.
R . PAYNE
Rubber and Plastics Research Association of Great Britain, Shawbury, Shrewsbury, Shropshire, England
When a cylinder of small diameter-height ratio is compressed, it buckles. A theory has been developed by Haringx to describe this, based on the Euler theory of struts. This paper shows how Haringx‘s theory is applicable to the compression of rubber and to cylinders of moderate shape factor, which do not visibly buckle, and relates the changes in the shear modulus of a rubber which is compressed, to the degree of compression and the shape factor of the cylinder.
a heavy machine such as a hammer is supported on rubber mountings, considerable compression of the rubber occurs and this alters the stiffness and stability in shear of the mounting. Shock and vibration isolation is achieved by using rubber mountings having a small ratio of diameter to height to obtain a sufficiently low natural frequency for the system. The instability in the shear plane to which this small diameter-height ratio leads has been corrected by the empirical insertion of stabilizing pads ( 7 ) . I t would be of value to designers of mountings for heavy machinery if data were available showing the effect of compression on the shear modulus for a range of shape factors. Recent reference to this phenomenon ( 3 ) emphasizes the need for data and demonstrates that the relevance of the theoretical work of Haringx (2) has not been appreciated (or, probably, is not known) in the rubber-engineering industry. This article draws attention to the analysis by Haringx of the behavior of highly compressible helical springs and rubber rods and, combined with the shape-factor investigations of the present author ( d ) , shows its application to the problem considered. HEN
Theory Following the procedure and symbolism adopted in the author’s textbook (5), the stress-strain relation of bonded rubber cylinders in compression is given by : f = -G(A-
A-)( 1
+ BP)
-GO,
- A--2)
=
-E ( A - A--2) 3
(2)
where E = Young’s modulus. Turning now to the analysis of “highly compressible helical springs and rubber rods, and their application for vibration 86
f = E(?)
(3)
Even though the bracketed terms which are functions of X in Equations 1, 2, and 3 are not formally identical, in practice they are nearly so. Consider the ratio of Equations 2 and 3, -l/3 (X-X-2)/(l-X)X-1; its value would be unity if the functions were identical. As is seen in Table I, its value varies with increasing compression (decreasing value of A), but even with 50% compression the departure from unity is less than 20%. For X 21.1)
0.103 0.225 0.056
0.080
0.063
0.056
0.175
0,140
0,120
0,044
0.035
0,030
1
JUNE 1962
VOL.
NO. 2
>300
87
normal shear mountings), then the resulting deformation is not simple shear but a combination of shearing and bending. This problem is similar to that encountered in the bending of a beam loaded a t one end, and has been solved by Rivlin and Saunders (7), who treated the deformation at any cross section as the superposition of the displacements due to simple shear and to bending under the action of the bending moment at that cross section. Figure 3 shows the condition found in the deformation of a cylindrical shear mounting of length l, and radius r . The displacement of the end of the sample perpendicular to its length is d,. This deflection results from the combination of bending and shearing where the bending component, X,, is given by (8;
Xb
=
SIo3/12E,FoR2
(7)
and the shearing component by
X, =
Sl,/F,G
(8)
where S is the shear load. K is the radius of gyration of the cross section about the neutral axis of bending. Thus d, =
X, + Xb
=
SI,( 1
+ Io2/36 K 2 )
(9)
The relation of the shearing stress, S/F,, to the apparent shear modulus is given by Equation 10 S/F,
=
G, X d a / l o
(10)
and, sincr K = d/4 for circular cross-sectioned rods, from Equations 9 and 10, the relation between the true arid the effective shear moduli is
G, = G/(1 Figure 2. Relation of shear modulus to height-radius ratio of cylinders 0
+ 4 12/9 d 2 )
(11)
Compariscn of this function with that of Equation 5b shows them to be thr same.
Experimental values of shear modulus a t X = 1
Conclusions
When a cylinder of small diameter-height iatio is compressed, it buckles. In classical metals theory this phenomenon is associated with Euler’s theory of struts and such a cylinder is known as an Euler strut. Haringx’s theory is based on the Euler strut, but is obviously applicable when a cylinder of rubber of moderate shape factor is greatly compressed, even though it does not visibly buckle. The agreement between experiment and theory which this paper shows enables the change in shear modules to be estimated for rubber mountings which are compressed by static load. The appropriate shape functions for mountings other than cylinders and the necessary values of constants are discussed above to enable actual design problems to be solved. Figure 3.
Rubber in shear
Showing bending and shearing action
The constants required in the shape functions vary with shear modulus and the appropriate values are given in Table 11. Alternative Equation
for G, at X
=1
Rivlin (6) has pointed out that a simple shear cannot be maintained by the application of constraints merely to the end faces of the rubber being deformed. If constraints are applied to the end faces of a rubber so that one face can move relative to the other, but only in a plane parallel to it (as in 88
l & E C P R O D U C T RESEARCH A N D DEVELOPMENT
literature Cited
(1) Crockett, J. H. A., O’Neill, D. B., Proc. Zirst. Civil Engrs. (London) 13, 133 (1959). (2) Haringx, J. A,, Philips Research Repts. 3, 401-99 (1948). (3) Hint, A. J., “Applied Science of Rubber,” i V . J. S. Naunton, p. 510, Arnold, London, 1961. 4) Payne, A. R., Engineer 207, 328, 368 (1959). 5) Payne, .4.R., Scott, J. R., “Engineering Design with Rubber,” p. 142, McLaren, London, Interscience, New York, 1960. (6) Rivlin, R. S., Proc. 2nd Rubber Technol. Conf., p. 204, Heffer, London, 1948. (7) Rivlin, R. S., Saunders, D. W., Trans. Inst. Rubber Ind. 24, 296 (1949). ( 8 ) Southwell, R. V., “Introduction to the Theory of Elasticity,” p. 161, Oxford University Press, London, 1944.
I
RECEIVED for review December 4, 1961 ACCEPI’ED February 16, 1962
