Linear elastic restoring force

The Gaussian approximation can therefore be seen to be equivalent to a quadratic potential or linear elastic restoring force. Deviations from the Gaussian distribution will correspondingly yield nonlinear force terms in the dynamics. The Gaussian approximation should therefore be an appropriate simplification for describing systems close to equilibrium or at most linearly displaced from the equilibrium state. 

It is important to realize that this type of behavior is not just a simple addition of linear elastic and viscous responses. An ideal elastic solid would display an instantaneous elastic response to an applied (non-destructive) stress (top of Figure 13-74). The strain would then stay constant until the stress was removed. On the other hand, if we place a Newtonian viscous fluid between two plates and apply a shear stress, then the strain increases continuously and linearly with time (bottom of Figure 13-74). After the stress is removed the plates stay where they are, there is no elastic force to restore them to their original position, as all the energy imparted to the liquid has been dissipated in flow. 

Thus, the restoring force is proportional to the extension and the onedimensional chain behaves as a Hookean spring. This important result simplifies the analysis of the normal modes of motion of a polymer. Polymer chain models can be treated mathematically by the much simpler linear differential equations because second order effects are absent. (It should be noted diat, while the elastic equation for a polymer chain is identical in form with Hooke s law, the molecular origin of the restoring force is very different). 

For small stresses, we can use the approximation sinha w a in equation (8.3) so that the strain rate is proportional to the applied stress. In this case, the behaviour is linear and viscous. As stresses are small, the deformation is not plastic, but elastic, for there is a restoring force corresponding to the spring element in figure 8.7(a), whereas equation (8.3) describes the dashpot element of the Kelvin model. The behaviour is thus linear viscoelastic. At larger stresses, deviations from linearity occur, although the behaviour is still viscoelastic. 

In the amorphous state the linear macromolecule is more or less randomly coiled with the root mean square end-to-end distance proportional to the square root or a slightly higher power of molecular weight. Any deviation from this average value yields a mechanical force by which the sample tends to restore the average value of chain conformation. This entropic force is the basis of rubber elasticity. 

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