Advances in Materials Science and Engineering

Volume 2015 (2015), Article ID 159256, 8 pages

http://dx.doi.org/10.1155/2015/159256

## Stress-Softening Formulae of Polymer Bearings

Department of Applied Mathematics and Science, Khalifa University, P.O. Box 127788, Abu Dhabi, UAE

Received 1 July 2015; Accepted 28 September 2015

Academic Editor: Luigi Nicolais

Copyright © 2015 M. H. B. M. Shariff et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The motivation for this work was the absence of closed form solutions that can reasonably describe the axial deformation behaviour of stress-softening polymer bearings. In this paper, new closed form solutions that exhibit Mullins phenomenon are developed. We show that the apparent Young modulus depends on the shape factor and the minimal infinitesimal strain. We furthermore found that, in a nonlinear deformation, the shape factor plays an important role in stress softening. The solutions are design friendly and are consistent with expected results.

#### 1. Introduction

When subjected to cyclic loadings, many polymers exhibit an anisotropic stress-softening phenomenon widely known as the Mullins effect (Mullins [1]). Recently, several researchers (Shariff [2–4]; Itskov et al. [5]; Dorfmann and Pancheri [6]; Merckel et al. [7]) developed anisotropic constitutive equations for stress-softening polymers. Shariff’s 2014 model (Shariff [4]) compares well with different experimental data available in the literature for different types of rubberlike materials. In his 2014 model, Shariff [4] provides explicit theoretical results for homogeneous deformations. In this paper, however, we develop explicit closed form solutions for axial inhomogeneous deformation of polymer block bonded to two rigid plates. Polymer mounts have several applications and are, for instance, used in earthquake isolation bearings, in vehicle suspensions, and in bearings for bridges. The most important characteristic of a polymer mount is its load-deflection relation (stiffness). This property can be obtained experimentally or mathematically. However, for an axial compression (or tension) mounting, the deformation is significantly inhomogeneous. This results in great difficulties in mathematical treatment to obtain explicit closed form solutions. We note that explicit solutions have an advantage over numerical solutions, such as finite element solutions, in the sense that a numerical approach requires considerable computation effort to obtain specific values for design purposes. Since a numerical approach is not explicit in nature, it is not very convenient to use. In spite of these difficulties, approximate explicit closed form solutions have been developed in the past (Shariff [8–10]; Petrie and Shariff [11]; Haddow and Ogden [12]) by treating rubber as purely elastic. These approximate solutions predict fairly well the elastic mechanical behaviour of certain types of polymer (rubberlike) mounts. However, rubberlike materials are generally not purely elastic since they exhibit anisotropic stress-softening behaviour. Hence, the aim of this paper is to develop approximate explicit closed forms of solution for axial deformation of polymer mounts with Mullins’s behaviour. The solutions are obtained without resorting to simplifying physical assumptions. Indeed, they are rather sought via a damage function and a variational principle. This leads to relatively simple theoretical solutions which are easy to handle and can be used as initial estimates for design purposes.

This paper is divided into seven sections. In Section 2, we define a useful concept of damage function. An average constitutive equation is given in Section 3 and it is used in a variational principle in Section 4. Approximate solutions are constructed in Section 5 and their results are discussed in Section 6. Section 7 concludes this paper.

#### 2. Isotropic Damage Function

Before introducing the idea of a damage function, it is useful to present a few preliminaries needed in this paper. Throughout this document, all subscripts and take the values 1, 2, and 3, unless otherwise stated. An eigenvalue (principal stretch) of the right stretch tensor will be denoted by . The right Cauchy-Green tensor is given by , where is the deformation tensor. In our case, incompressible polymers are considered; hence the relation must be satisfied.

A measure of damage caused by strain is important in analysing stress-softening materials. Shariff [4] defined an anisotropic damage function to tackle anisotropic stress-softening behaviour. However, in the current work, we are only dealing with average stress. Moreover, due to intractable mathematical analysis, isotropic damage is studied here. In view of this, we propose a damage function such that where . The function has also the properties that where The first derivative may or may not exist at . If it exists, then . In view of our definition, increases monotonically as moves away in a straight line from the point . It can be easily shown that (tr denotes the trace of a second-order tensor) is a damage function since it has only one extremum (global minimum) at the point for and . In this paper, in order to obtain closed form solutions, we use the strain invariant as our damage function with ; that is, We note that, throughout the history of the deformation, there exist and (not necessarily unique) such that where the material is subjected to a deformation history up to the current time and denotes a running time variable.

#### 3. Isotropic Model for the Mullins Effect

In this study, we are only concerned with average stress softening as a first-order approximation. Moreover, we limit our study to isotropic stress softening in order to obtain explicit closed form expressions for the force-deflection relationships. Following the spirit of the work of Shariff [4] and Ogden and Roxburgh [14], we propose a “free” energy function of the formwhere is a constant parameter and is a function that behaves similarly to a strain energy function of a purely elastic isotropic polymer. In this paper, we focus on moderate strain deformation and, in order to obtain closed form solutions and following the work of Shariff [9], we only consider the neo-Hookean form where is the ground-state shear modulus of the virgin material.

The softening function is introduced to take into account the softening of the material under stress. We require (Ogden and Roxburgh [14]) expressing the softened Biot stress , say, in the form where is the Lagrange multiplier associated with the incompressible constraint; then we must impose the condition In view of (10), depends on (damage function) and and has the softening property (Ogden and Roxburgh [14]) If we impose the condition that in the stress-free reference state, then we must have , where . To satisfy the Clausius-Duhem inequality, we impose the conditions and . As we are only interested in evaluating the force-deflection expression and since the function does not appear in the stress-strain relationship such as (9), we will not elaborate on its properties here.

Figure 1 depicts nominal stress-strain paths for a simple tension deformation. The nominal stress on any path can be obtained by differentiating the area under an “elastic curve” path. On an elastic curve path, the value of is fixed. As a consequence, the “free” energy function for an inelastic solid can be portrayed by an infinite family of elastic strain energy functions parameterized by the internal variable .