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 .
In this paper, for simplicity, we considerwhere is a material constant. We could, of course, propose a more elaborate function for , but it is not the purpose of this paper. In view of (10) to (12) and the condition , we have
3.1. Dissipation
The Clausius-Duhem inequality is given by the relationwhere the superposed dot represents, for example, the time derivative. Note that In view of (det denotes the determinant of a tensor), we have Hence, in view of (9) and (14), we have It is clear from the property of that . The functional properties of and ensure that which is consistent with the Clausius-Duhem inequality that indicates energy dissipation.
4. A Variational Principle
The equation of equilibrium of an incompressible material with negligible body force is where Div denotes the divergence operator relative to , and is the first Piola-Kirchhoff stress tensor.
The essential boundary condition is and the natural (traction) boundary condition on the stress-free surface is where and is the unit outward normal vector to .
When is fixed, can be treated as an isotropic elastic strain energy function. Equilibrium equation (18), traction boundary condition (21), and the constraint can be obtained via the first variation of the functional for an incompressible material where is any deformation field satisfying essential boundary condition (20) and is the volume element appropriate to . We note that , where is a class of kinematically admissible deformation field defined by where is a class of twice continuously differentiable functions on . The Lagrange multiplier belongs to the admissible class where is a class of differentiable functions on . Indeed, vanishing of the first variation (23) yields where is the surface element appropriate to and on . Since and are arbitrary, then (18), (21), and follow by the calculus of variation, and being interpreted as a variation in and , respectively.
5. Approximate Solution
In the past, several researchers used simplifying physical assumptions in order to obtain approximate closed form solutions (Shariff [9], Gent and Lindley [15], and Klingbeil and Shield [16]). However, in this paper, we use the variational principle given in the previous section to obtain approximate explicit closed form solutions without resorting to facilitating physical assumptions. In this section, approximate solutions for axial deformations of bonded disc and rectangular strip are given. Figure 2 illustrates geometries of rectangular strip and bonded disc deformed under axial load. The approximate deformation solutions belong to the set , where . Since is not arbitrary on , traction-free condition (21) will not be satisfied exactly. However, it will be satisfied in a weighted sense. For instance, if is constant, then the zero traction on the free surface is approximated by zero average traction (see, e.g., Klingbeil and Shield [16]). Nevertheless, our solutions satisfy in an exact manner all three of the following: equilibrium equation (18), essential boundary condition (20), and incompressibility equation .
5.1. Average Stress Softening
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 (instead of ) in our calculation, where where the parameter depends on the axial strain and where the strip problem is considered as a plane strain problem. We note that is a damage function with properties given in Section 3. Instead of given in (7), we will use the free energy in our calculation, where . The function in (29) is independent of the principal stretches and .
5.2. Circular Cylindrical Disc
The geometry of the disc body is defined by in the reference configuration, where are the cylindrical polar coordinates. Let denote corresponding coordinates in the deformed configuration. The material is bonded to rigid plates at . The essential boundary condition then takes the form Following the work of Shariff [9], we use the following deformation field in the cylindrical problem:where for compression and for tension. The term is related to the axial strain via the equation where and . Up to moderate strain (e.g., 30%) and the right-hand side of (35) can be approximated using Taylor series up to the 4th order and we can write explicitly in terms of ; that is,
5.2.1. Compression
It can be easily shown from (35) and (36) that is a monotonically decreasing function of . The damage function invariant takes the form and in view of (27), we have where is the shape factor (Shariff [9]) of the cylinder. From (38) it is clear that the shape factor plays a role in the average stress softening. With some algebra, we can show that for the relevant range of . This implies that is a monotonically increasing function of and hence it is a damage function with respect to the single variable .
The physical components of the Cauchy stress relative to the system are The constant in (39) can be obtained using (26); that is, since the left term of (26) vanishes. Note that our approximate solution , and hence on the traction free surface. A different functional form of will give a different value of . As mentioned earlier in this section, if is a constant, then our average traction on the free surface is zero. Alternatively, if we choose a value for (via some physical means), then seeking a specific functional form of (so that (40) is satisfied) is not so important. In the current study, we choose a value of based on the work of Shariff [9]; that is, where is an even function of the deformation . For moderate strain, using Taylor’s expansion, we have Note that neo-Hookean form (8) may not represent accurately some actual rubberlike materials. Hence, can be used to compensate both the error of representing some rubberlike materials via the neo-Hookean form and error due to the approximations used in our model. Following the work of Shariff [9], we let and .
Let be the area of the bonded surface and let be the normal force acting on the rigid plates. We then have where is the ground-state Young modulus.
5.2.2. Tension
The results for tension are obtained from those for compression simply by replacing by .
5.3. Plane Strain Rectangular Strip
The geometry of the “infinite” strip in the direction is defined by where are the Cartesian coordinates in the reference configuration. If denote corresponding coordinates in the deformed configuration and the material is bonded to rigid plates at and , then the essential boundary condition takes the form Following the work of Shariff [9], we use the following deformation field in the strip problem:By symmetry is an odd function in and an even function in . In view of incompressibility condition, the essential boundary condition, and the relation we have where .
5.3.1. Compression
Following the work of Shariff [9], we have where The damage functions take the form where the shape factor . The Cartesian components of the Cauchy stress are given by where (Shariff [9])
6. Results
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 (38) that the softening function depends on the shape factor . Figure 3 shows the load-deformation curves of the primary loading and “softened” elastic loading for and which behave as expected. Unfortunately, no stress-softening experiment for bonded polymers exists in the literature. Hence, we are not able to compare our theoretical results for various values of . However, as approaches zero, the load-deflection behaviour (in tension) is equivalent to stretching a string (simple tension) and simple tension experiments do exist in the literature. In Figure 4, we compare our force-tension curves for with the experimental data of Mullins and Tobin [13]. The material used in the experiment was filler-reinforced GR-S tread vulcanizate rubber. It is clear in the figure that our theory compares well with the uniaxial experiment. We note that our theory is only developed for moderate strain.
For an infinitesimal deformation, the apparent Young modulus takes the simple form where () is the minimal infinitesimal strain. We note that is softened by the softening parameter which is independent of the shape factor . “Purely elastic” can be easily recovered by letting . Similarly, for the nonlinear case, the load-deflection curves of a purely elastic material can be easily recovered by letting . Moreover, it is shown in Shariff [9] that these elastic curves compare well with experimental data of elastic rubberlike materials.
7. Conclusion
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.