Rubbers fixed junction model

The elastic modulus of a rubber according to the phantom network theory is much lower than the modulus of the same network with all junction fluctuations suppressed. If the fluctuations are partially suppressed, the calculated modulus lies between these limits. In fact, in many cases, the measured modulus is many times greater than predicted by fixed junction models (8,9). 

The model so set up is the simplest one to treat and known as the fixed junction model with reference to the last point. Let us refer again to a prismatic piece of rubber, with edges Lx, Ly and Lz, and consider a homogeneous orthogonal deformation, which changes the lengths of the edges as follows... 

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... 

FIGURE 7.1 Statistical model of the network structure of rubber (after Grassley, 1975) O, mobile junction X, peripheral fixed point A, junction under investigation. 

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