Keywords: bonded-mounts, stress-softening, explicit-formulae, Mullins, nonlinear.

Bonded rubber mounts are used, for example, in bridge bearings, base isolation for the earthquake protection of buildings and suspensions of vehicles. It is well known that exact explicit closed form solutions for linear and nonlinear elastic deformations of bonded mounts do not exist because the mathematical analysis proves intractable. However, approximate explicit closed form solutions have been developed in the past [1,2,3,4,5]. These approximate solutions predict fairly well the elastic mechanical behaviour of certain types of bonded mounts. Explicit solutions have an advantage over numerical solutions, such as finite element solutions, in the sense that a numerical approach requires considerable computation to obtain specific values for design purposes and, not being explicit in nature, it is not very convenient to use. However, rubberlike materials are generally not purely elastic; when rubber is subjected to cyclic loading it exhibits a stress-softening phenomenon widely known as the Mullins effect [6]. Explicit closed forms of solution, for axial deformation of bonded rubber mounts that exhibit stress softening behaviour, currently do not exist in the literature. Hence, the aim of this paper is to develop approximate explicit closed form solutions for stress-softening bonded rubber mounts. The constitutive equation used to describe the Mullins effect is based on the work of Shariff [7] and Ogden and Roxburgh [8].

Quite often, approximate solutions of physical problems are obtained by making simplifying physical assumptions. These solutions generally do not satisfy all the governing equations. To avoid this, we use a variational method to obtain approximate solutions without resorting to simplifying physical assumptions. To facilitate our analysis a concept of damage function is introduced. The relatively simple forms of explicit formula are easy to use for design purposes.

The stiffness of the bonded mount is measured in terms of its shape factor. We show that the apparent Young's modulus depends on the shape factor and the history of previous axial loadings. Theoretical results are given and analysed.

1

M.H.B.M. Shariff, "An approximate analysis of infinitesimal deformations of bonded elastic mounts", J. Strain Anal., 23, 115-120, 1988. doi:10.1243/03093247V233115

2

M.H.B.M. Shariff, "An analysis of non-linear deformation of bonded rubber mounts", Proc. Inst. Mech. Eng., 203, 113-119, 1989. doi:10.1243/PIME_PROC_1989_203_094_02

3

C.J.S Petrie, M.H.B.M. Shariff, "Finite deformation of bonded compressible elastic mounts", J. Strain Anal., 27, 157-169, 1992. doi:10.1243/03093247V273157

4

M.H.B.M. Shariff, "A general approach to axial deformation of bonded elastic mounts of various cross-sectional shapes", Appl. Math. Modelling, 17, 430-436, 1993. doi:10.1016/0307-904X(93)90118-Z

5

J.B. Haddow, R.W. Ogden, "Compression of bonded elastic bodies", J. Mech. Phys. Solids, 36, 551-579, 1988. doi:10.1016/0022-5096(88)90010-5

6

L. Mullins, "Effect of stretching on the properties of rubber", J. Rubber Res., 16, 275-289, 1947.

7

M.H.B.M. Shariff, "An Anisotropic model of the Mullins effect", J. Eng. Math., 56, 415-435, 2006. doi:10.1007/s10665-006-9051-4

8

R.W. Ogden, D.G. Roxburgh, "A pseudo-elastic model for the Mullins effect in filled rubber", Proc. R. Soc. London A, 455, 2861-2877, 1999. doi:10.1098/rspa.1999.0431

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