Entangled system normal stresses

The shear stress o 2 and the differences between the normal stresses o —0x2 and 022 — 033 are usually measured in the experiment (Meissner et al. 1989). The results of calculation of the stresses up to the third-order terms with respect to the velocity gradient will be demonstrated further on. For simplicity, we shall neglect the effect of anisotropy of the environment when the case of strongly entangled systems will be considered. 

For the weakly entangled system, the steady-state modulus depends on the molecular weight of polymer as M 1, while for strongly entangled system, the steady-state modulus does not depend on the molecular weight of polymer, which is consistent with typical experimental data for concentrated polymer systems (Graessley 1974). The expression for the modulus is exactly the same as for the plateau value of the dynamic modulus (equations (6.52) and (6.58)) Expressions (9.42) lead to the following relation for the ratio of the normal stresses differences... 

There are different approaches to the calculation of stresses in the system of entangled macromolecules [9, 34, 93, 94]. We consider the system to be a suspension of Brownian particles and start here to determine a expression for the stress tensor through correlation functions of the normal co-ordinates of the macromolecule, while we shall follow a method developed in the theory of liquids [95, 96] which was used by Pokrovskii and Volkov [55] in this case. 

In the figure below a series of terms is defined in order to help describe and define viscosity and viscoelasticity. The diagram represents a simple three - dimensional volume of fluid [or solid] and shear stress strain and shear rate are described as well as the parameters for viscosity and elasticity [shear modulus]. The origin of normal forces is also shown, but this is not common to all fluids and is likely to occur in certain cases where polymers are exhibiting entanglement coupling or filled systems are colliding with each other. 

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