Rubber phenomenology

Mars, W.V. and Fatemi, A., A phenomenological model for the effect of R ratio on fatigue of strain crystallizing rubbers. Rubber Chem. Tech., 76, 1241, 2003. 

With this condition, there are a great many possible choices for the form of W as a function of Our ultimate purpose in the phenomenologic study of rubber elasticity is to find out its form applicable for an accurate and coherent description of the elastic behavior of rubber-like materials under various modes of deformation. We may use /j, J2, and J3 for the set of /<, which are defined by... 

This consists of experimental measurements of stress-strain relations and analysis of the data in terms of the mathematical theory of elastic continua. Rivlin7-10 was the first to pply the finite (or large) deformation theory to the phenomenologic analysis of rubber elasticity. He correctly pointed out the above-mentioned restrictions on W, and proposed an empirical form... 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

It is well known that the equation of state of Eq. (28) based on the Gaussian statistics is only partially successful in representing experimental relationships tension-extension and fails to fit the experiments over a wide range of strain modes 29-33 34). The deviations from the Gaussian network behaviour may have various sources discussed by Dusek and Prins34). Therefore, phenomenological equations of state are often used. The most often used phenomenological equation of state for rubber elasticity is the Mooney-Rivlin equation 29 ,3-34>... 

A comprehensive consideration of new phenomenological equations of state for rubber elasticity have been carried out lately by Tschoegl et al.41 45One of their equations of state is given by... 

In a recent series of papers, Kilian 9,50 52) proposed a new phenomenological approach to rubber elasticity and suggested a molecular network might be considered as a formelastic fluid the conformational abilities of which were adequately characterized by the model of a van der Waals conformational gas with weak interaction. The ideal network is treated as an ideal conformational gas. According to... 

In this connection, it is very interesting that the volume and intrachain changes obtained by various experimental methods 24,29,85) [Eq. (101)] agree well with Eq. (56) following from the Tobolsky-Shen semiempirical equation of state or the related phenomenological Eq. (76). The values of y determined from the data are rather small (0.1-0.3). As has been mentioned above, according to the semiempirical approach by Tobolsky and Shen one can formally suggest that the front-factor in Eq. (28) is pressure dependent. If it is really so, then the parameter y for rubbers can be considered as an experimental coefficient similar to the coefficient of thermal expansion and compressibility 29). 

The relationships resulting from the statistical theory fall well short of fully describing the stress strain curves of filled rubbers. From the alternative phenomenological approach a general relation for W is given by ... 

Hedreul et al. (1998) examined a model of the cure kinetics of a thermally and microwave-cured rubber-modified epoxy-resin formulation. The phenomenological cure kinetic model used was... 

A further phenomenological theory, which uses the concept of strain-energy functions, deals with more general kinds of stress than uniaxial stress. When a rubber is strained work is done on it. The strain-energy function, U, is defined as the work done on unit volume of material. It is unfortunate that the symbol U is conventionally used for the strain-energy function and it will be important in a later section to distinguish it from the thermodynamic internal-energy function, for which the same symbol is also conventionally used, but which is not the same quantity. 

Section 6.3 deals with purely phenomenological theories. In this section the predictions of a theory based on the microstructure of a rubber are considered. By 1788 at the latest the term rubber was being applied to the material obtained from the latex of the tree Hevea braziliensis because of its ability to remove pencil marks from paper. The first printed account of this use for wiping off from paper the marks of black lead pencil was given by Joseph Priestley as early as 1770. This material is now called natural rubber and its chemical structure is shown in fig. 6.9. 

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