Closure to “Discussion of ‘Axial Loading of Bonded Rubber Blocks’ ” (2005, ASME J. Appl. Mech., 72, p. 453)

Abstract

Our immediate reaction to the comments of Pence was one of amazement that such a supposedly prestigious journal would consider it appropriate to devote valuable space to what, we feel, should be obvious to a reasonably well-informed reader who has delved into the related literature.In his first paragraph, Pence appears to review a few random elements of the very well-known basics of classical elasticity theory (even to the extent of giving the absolutely fundamental relationships between the displacement and strain components). Several of these were, in fact, willingly deleted from our original manuscript, upon receiving the observation of a reviewer that, “It is not necessary to remind readers of Journal of Applied Mechanics … for a linear elastic model.”He then, seemingly unnecessarily, simply reproduces a few of our expressions, before making the observation that the equilibrium equation in the z direction is not satisfied. Actually this would clearly not be expected with the basic objectives and assumptions of the analyses presented. However, he naively appears to regard this as a hugely serious flaw that casts doubt on the worth of the realistic approximate expressions derived for the apparent Young’s modulus, offering improvements on those previously available.The approximations developed by Gent and Lindley (1) and Gent (2) have been widely quoted and used in the engineering industry for assessing axial stiffness. As we hoped was clearly explained in our introduction in Sec. 1, they were derived on the basis of two fundamental assumptions: (i) that planes initially normal to the direction of loading remain normal after loading, and (ii) that the deformed shapes of the free lateral surfaces are parabolic. Subsequently, as we pointed out in Secs. 1 and 5.2, the validity of the assumption (ii) has been questioned by several authors in interpreting their experimental results—with comments including “the assumption of a parabolic profile is erroneous” and that the next step “would be an improved method of estimating the ‘bulgeability.’ ” It was our aim therefore to provide such estimates with this assumption of parabolic profiles removed, while maintaining the more reasonable first assumption.However, it is a natural consequence of this simplifying assumption (i) that, for an incompressible material, it becomes unrealistic to satisfy exactly the equilibrium equation in the z direction (except on the central plane, z=h∕2) on an infinitesimal volume. Further, this same assumption leads to the prediction of the existence of a shear stress on the unloaded boundary which, as observed in the paper, cannot actually physically exist.If Pence were to refer to the already cited references of Gent and Lindley (1) and Gent (2), he would deduce that the theories therein lead to solutions for the stress components that also do not satisfy the equilibrium equation in the direction of loading, nor do the expressions derived in the later considerations of, for example, Constantinuo, Kartoum, and Kelly (3), Chalhoub and Kelly (4), and Tsai and Lee (5).Additionally, it should perhaps be pointed out that all the above papers, and others, have assumed the rubber block to have a small thickness and have either predicted or assumed parabolic deformed profiles. In contrast, our analysis applies to a block of any thickness, and predicts that, especially for blocks of small shape factor, the profile is noticeably not parabolic. This is reassuringly in agreement with the experimental findings of Mott and Roland (6) and others.In conclusion, we would suggest that, contrary to Pence’s concerns in his final sentence, methodologies similar to that adopted in our paper have proved invaluable and extremely useful in related analyses. Particularly worthy of note are the expressions that Horton, Gover, and Tupholme (78) presented for the radial stiffness and tilting stiffness of a rubber bush mounting of finite length. Not only were there no useful estimates available previously, but moreover they yield numerical values that agree well with the available experimental data.We are grateful to the Editor of the Journal of Applied Mechanics for giving us this opportunity to respond. We hope that our comments will enhance the appreciation of the potential importance and value of our results, for those readers who have not worked directly in this area of rubber technology and are therefore less familiar with the relevant literature.