The phenomenology of rubber elasticity

Assuming that there is uniform stress throughout a semicrystalline pol5mier sample and that the average crystal and amorphous moduli, and are 5 X 10 Pa and 0.25 x 10 Pa, respectively, calculate the modulus of the sample if its volume crystallinity x = 0.6. 

Because the polymer is under uniform stress the modulus will be the same as the effective modulus of a rod of material of uniform unit cross-section consisting of lengths equal to the volume fractions of the crystalline and amorphous materials placed in series. Applying a stress a to such a rod results in a strain e equal to Xv lEc + (1 — XuV/ a and a modulus E equal to a/e = l/[x / c + (1 - Xv)/E ]. 

The problem with the amorphous material is that, even though it may be in a rubbery state, there are likely to be constraints on the chains due to the crystallisation process, which will give the material different properties from those of a purely amorphous rubbery polymer. The difficulty with the two averaging steps is that the states of stress and strain are not homogeneous in materials made up of components with different elastic properties. The simple assumption of uniform stress often gives results closer to experiment than does the assumption of uniform strain, but neither is physically realistic. For polyethylene, values of the average erystal modulus Ec and the average amorphous modulus are found to be about 5 x 10 Pa and 0.25 x 10 Pa, respectively. 

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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... 

In spite of its lack of success in this direction the Mooney equation has provided the basis for a number of phenomenological theories of rubber elasticity such as those by Rivlin and Saunders (1951). Discussion of such non-molecular theories is considered outside the scope of this book. 

Yeoh O, Fleming P (1997) A new attempt to reconcile the statistical and phenomenological theories of rubber elasticity. J Polym Sci Pt B Polym Phys 35 1919-1931... 

An expression of this form is often associated with the stretching of mbber elastic networks, having been derived on physical grounds (see Chapter 4). However, we can see here that it arises on purely phenomenological grounds once the assumptions of isotropy and incompressibility have been made, and does not imply that the material is rubber-like. These materials are sometimes referred to as neo-Hookean. 

An earlier and different approach from the molecular models of rubber elasticity presented so far is based on a phenomenological model and continuum mechanics. It considers the elastic energy stored in the system. The work done on the system must be stored as elastic energy W and given by... 

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>... 

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... 

The main phenomenological approach for rubber elasticity at small deformations is due to Langley The shear modulus is assumed to be the sum of two terms... 

A totally different approach to rubber elasticity has been developed by Stepto and co-workers [15, 16], which also accounts for the Mooney-Rivlin softening. Their approach is not phenomenological, but is based on structural considerations that give an accurate description of the moleeular eonformational states of the units in the polymer chains as the network is stretched. They have proposed a method for calculating the free energy of a stretched molecular network based on the rotational isomeric state of the network chains, with conformational energies determined from observations of conformational properties. 

The phenomenological approach to rubber-like elasticity is based on continuum mechanics and symmetry arguments rather than on molecular concepts [2, 17, 26, 27]. It attempts to fit stress-strain data with a minimum number of parameters, which are then used to predict other mechanical properties of the same material. Its best-known result is the Mooney-Rivlin equation, which states that the modulus of an elastomer should vary linearly with reciprocal elongation [2],... 

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