Primitive chain segmental equilibrium time

According to the Doi-Edwards theory, after time t = Teq following a step deformation at t = 0, the stress relaxation is described by Eqs. (8.52)-(8.56). In obtaining these equations, it is assumed that the primitive-chain contour length is fixed at its equilibrium value at all times. And the curvilinear diffusion of the primitive chain relaxes momentarily the orientational anisotropy (as expressed in terms of the unit vector u(s,t) = 5R(s,t)/9s), or the stress anisotropy, on the portion of the tube that is reached by either of the two chain ends. The theory based on these assumptions, namely, the Doi-Edwards theory, is called the pure reptational chain model. In reality, the primitive-chain contour length should not be fixed, but rather fluctuates (stretches and shrinks) because of thermal (Brownian) motions of the segments. 

First, consider a simpliflcation in which it is assumed that the contour length of the primitive chain remains at the equilibrium length L under the imposed deformation. This assumes an inextensible primitive chain and is seen as a reasonable approximation for slow flows or long times. Then, the deformation of the primitive chain is given by considering that the segment in the middle of the chain changes position affinely as... 

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