Equation of state, elastic

Other forms of the elastic equation of state will appear below. 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Brotzman, R. W. Eichinger, B. E., "Volume Dependence of the Elastic Equation of State. 3. Bulk-Cured Poly(dimethylsiloxane)," Macromolecules, 15, 531 (1982). 

Mark, J. E., The Volume Dependence of the C2 Correction in the Mooney-Rivlin Elastic Equation of State. J. Polym. Sci., Part C Polym. Symp. 1970,31,97-106. 

Expression 38 is one of the forms of the thermodynamic elastic equation of state. Measurements of stress at constant length as a function of temperature have been... 

The early versions of the statistical theory of rubber elasticity assumed an affine displacement of the average positions of the network junctiOQs with the macroscopic strain This is tantamount to the assertion that the network Junctions are firmly embedded in the medium of which they are part. The elastic equation of state derived on this basis for simple extension at constant volume takes the familiar neo-Hookean form, i.e. Eq. (7), with... 

Polymer networks at first sight appear to present insurmountable complexities that would seem to preclude rational analysis of their properties in molecular terms. The basic premise that underlies the theory of rubber elasticity, a premise that has been fully validated, permits circumvention of most of these complexities. Recent advances of theory in conjunction with a wealth of empirical evidence gained from well chosen, carefully executed experiments offer the prospect of a comprehensive understanding of the elastic equation of state and associated properties of elastomeric materials in the foreseeable future. 

Equations (1.9) and (1.10) are basic to the molecular theories of rubber-Uke elasticity and can be used to obtain the elastic equations of state for any type of deformation [1-3], i.e. the equations interrelating the stress, strain, temperature, and number or number density of network chains. Their appUcation is best illustrated... 

The elastic equation of state in the form given in Eq. (1.14) is strikingly similar to the molecular form of the equation of state for an ideal gas ... 

Some of the elastic equations of state resulting from these various approaches are discussed further in subsequent sections. 

The thermodynamic behavior of rubber shows, therefore, a close analogy with that of an ideal gas, the entropy of which decreases during compression. In fact, the elastic equation of state for an ideal rubber is similar to the molecular form of the equation of state for an ideal gas. The stress replaces the pressure, and the number of network chains the number of gas molecules. 

As illustrated in Figure 2, elastomeric networks consist of chains joined by multifunctional junctions. As early as 1934, it was suggested by Guth and Mark and by Kuhn that the elastic retractive force exhibited by rubber upon deformation arises from the entropy decrease associated with the diminished number of conformations available to deformed polymer chains. It is, therefore, of primary interest to study the statistics of a polymer chain and to establish the elastic equation of state for a single chain. 

Transformation of equation (83) to a form suitable for analysis of such isobaric data requires recourse to an appropriate elastic equation of state. Such an equation for simple elongation, valid for both swollen and unswollen samples, is obtained through the statistical mechanical treatment of phantom Gaussian networks. It is given in equation (84). The extension a is measured relative to the... 

The elastic equation of state for simple elongation along the axis denoted 1 takes the familiar neo-Hookean form shown in equation (93), where Lq is the length of the specimen in the isotropic state of volume Vq. Because of the symmetry of the elongation, 2 3 equal and may be calculated... 

The elastic free energy of a phantom network of Gaussian chains was obtained rigorously by Flory and is valid for networks of any functionality, irrespective of their structural imperfections. It is given in equation (101). The elastic equation of state for phantom networks may then be expressed by equation (102). Equation (84) is then recovered, as expected, because of the relationship shown in equation (103). 

In view of the expressions for the elastic equation of state for affine networks (equation 98) and phantom networks (equation 102), it is customary to plot the reduced nominal stress [/ ] measured in uniaxial extension (or compression) against reciprocal extension The reduced nominal... 

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