Damping Uniaxial extension

They assumed that the parameter j8 is constant and found that a value of -0.27 fitted their planar extension data. While it is now understood that the normal stress ratio is a function of strain, it is shown in Section 10.4.5 that it does approach a specific limiting value as y 0. Based on data for one melt in step shear and start-up of steady uniaxial extension, Wagner and Demarmels proposed the following empirical relationship for the damping function with... 

Uniaxial extension is an axi-symmetric deformation in which a tensile stress is appHed in one direction, we will call it the z-direction, while the free surfaces of the sample are under a uniform normal stress, usually one atmosphere of compression. The quantity measured is the net tensile stress t7g defined as (- a ), which is the applied axial stress minus that acting on the free surface. One could, in principle, carry out step-strain (stress relaxation) in extension, and if the tensile relaxation modulus (t,e) can be separated into time and strain-dependent contributions, a damping function could be determined as a function of strain. 

Kasehagen and Macosko [34] used Eq. 10.32 to fit their data for a linear polybutadiene and for several randomly branched samples derived from it. For the linear sample they reported that a value of a = 0.26 fitted their data, while for the branched samples, the value of a decreased, reaching a = 0.07 for a sample with 39 wt.% branched molecules. Damping functions for uniaxial and biaxial extension are discussed in Section 10.9. 

Figure 10.18 Doi-Edwards damping functions for three types of extensional flow.The iA approximation has little effect, and the predictions for uniaxial and biaxial extension are similar to each other. From Urakawa et al. [24].

Urakawa, O., Takahashi, M., Masuda, T., Ebrahimi, N. G. Damping functions and chain relaxation in uniaxial and biaxial extensions Comparison with the Doi-Edwards theory. Macromol (1995) 28, pp. 7196-7201... 

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