Ideal rubber constitutive equation

However, specific new theories, called the equation of state theories, have been worked out in an effort to understand the special conditions of mixing long chains (29-34). At equilibrium, an equation of state is a constitutive equation that relates the thermodynamic variables of pressure, volume, and temperature. A simple example is the ideal gas equation, PV = nRT.The van der Waals equation provides a fundamental correction for molecular volume and attractive forces. Equations of state may also include mechanical terms. Thus rubber elasticity phenomena are also described by equations of state see Chapter 9. 

Whether to use the first or the second form of Finger s constitutive equa tion is just a matter of convenience, depending on the expression obtained for the free energy density in terms of the one or the other set of invariants. For the system under discussion, a body of rubbery material, the choice is clear The free energy density of an ideal rubber is most simply expressed when using the invariants of the Finger strain tensor. Equation (7.22), giving the result of the statistical mechanical treatment of the fixed junction model, exactly corresponds to... 

Knowing /, we can formulate the constitutive equation of an ideal rubber. Since only bi gives a contribution, we simply obtain... 

There are various suggestions for a better choice, however, they are all rather complicated and their discussion is outside our scope. It appears today that a short analytical expression for the free energy density of a real rubber in the form of a simple extension of the free energy density of an ideal rubber does not exist. Even so, the general constitutive equation, Eq. (7.75), certainly provides us with a sound basis for treatments. Once the functional dependence of the free energy density, /(/b,// ), has been mapped by a suitable set of experiments, and one succeeds in representing the data by an empirical expression, one can predict the stresses for any kind of deformation. 

Equation (10.7.8) is called a stress constitutive equation, and it relates a three-dimensional measure of strain to the three-dimensional stress. For rubbers, Eq. (10.7.8) obviously holds for the specialized case of imiaxial extension. By similar reasoning, it can be shown to hold for other idealized deformations such as biaxial extension and shear. Indeed, Eq. (10.7.8) is valid for all volumepreserving deformations [10]. The only material quantity appearing in this... 

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