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.

When subjected to cyclic loadings, many polymers exhibit an anisotropic stress-softening phenomenon widely known as the Mullins effect (Mullins [

This paper is divided into seven sections. In Section

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

A measure of damage caused by strain is important in analysing stress-softening materials. Shariff [

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 [

The softening function

Figure

Schematic loading-unloading curves in simple tension (Mullins effect).

In this paper, for simplicity, we consider

The Clausius-Duhem inequality is given by the relation

The equation of equilibrium of an incompressible material with negligible body force is

The essential boundary condition is

When

In the past, several researchers used simplifying physical assumptions in order to obtain approximate closed form solutions (Shariff [

Bonded mounts.

In the present study, we are mainly interested in calculating the force-deflection curve. To facilitate the evaluation of explicit closed form solutions, we use the average damage function

The geometry of the disc body is defined by

It can be easily shown from (

The physical components of the Cauchy stress relative to the

Let

The results for tension are obtained from those for compression simply by replacing

The geometry of the “infinite” strip in the

Following the work of Shariff [

In this section, we only discuss our results for the bonded disc. Results for disc tension and axial deformations of rectangular strip are similar, and we, hence, omit their discussions. It is clear from (

Load-compression curves of bonded disc.

Comparison of theory with Mullins and Tobin [

For an infinitesimal deformation, the apparent Young modulus

This work has demonstrated that explicit nonlinear (finite deformation) formulae may be obtained for axial deformations of bonded polymer (rubberlike) mounts with Mullins behaviour. The proposed formulae can be used as an initial approximation to facilitate design procedures since the forms of solution are relatively simple. Furthermore, the solutions satisfy all the governing equations exactly with the exception of the traction free surface where the governing equation is approximated in a weighted sense. The theoretical results obtained are consistent with expected behaviour. From our analysis, we found that the shape factor plays an important role in stress softening. This result is new and can not be found in the previous literature. In addition to the specific formulae developed and presented here, our work furthermore demonstrates an approach which may be adapted to other free energy functions. There is clearly a potential to apply our method to a wide variety of nonlinear polymeric materials that exhibit stress softening.

The authors declare that there is no conflict of interests regarding the publication of this paper.