Main Constitutive Equation of the Viscoelastic Behavior of Unixially Co-Oriented Polymers

The kinetic-equations method is used to obtain a constitutive equation of viscoelasticity based on representations of the existence of different conformational states of the macromolecules of oriented polymers. The equation makes it possible to describe and predict the stress-strain state of synthetic fibers and other polymeric materials under non-destructive loads.

Appendix 1.

$$ \begin{array}{c}\hfill m{}_{0 n}{\tilde{W}}_{12 n}-\left({\tilde{W}}_{21 n}+{\tilde{W}}_{12 n}\right){m}_{2 n}^P={m}_{0 n}{\tilde{W}}_{12 n}-\frac{\left({\tilde{W}}_{12 n}+{\tilde{W}}_{21 n}\right){m}_{0 n}{W}_{12}}{W_{12}+{W}_{21}}={m}_{0 n}\frac{{\tilde{W}}_{12}{W}_{21}-{\tilde{W}}_{21}{W}_{12}}{W_{12}+{W}_{21}}=\hfill \\ {}\hfill ={m}_{0 n}\frac{\upnu_{0 n}^2 \exp \left(-2{H}^{*}+{U}^{*}\right)\left[ \exp \left({\upgamma}^{*}{X}^2\right)- \exp \left(-{\upgamma}^{*}{X}^2\right)\right]}{W_{12}+{W}_{21}}=\frac{2{m}_{0 n}{\upnu}_{0 n} \exp \left[-{H}_n^{*}\right]}{ \exp \left[-{U}_n^{*}\right]} sh\left({\upgamma}_n^{*}{X}^2\right).\hfill \end{array} $$

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Appendix 2.

$$ \begin{array}{c}\hfill \upvarepsilon (t)=\frac{\sigma (t)}{E_0}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{t}{\int }}\left\{ q(H) sh\left[{\upgamma}^{*}(H)\frac{\sigma^2}{E_0^2}\right]-{\upvarepsilon}_0(H)\left( \exp \left[{\upgamma}^{*}(H)\frac{\sigma^2}{E_0^2}\right]+ A(H) \exp \left[\right]\right)\right\}\times \hfill \\ {}\hfill \times \exp \left\{-\underset{\varTheta}{\overset{t}{\int }}\left(\left[{\upgamma}^{*}(H)\frac{\sigma^2}{E_0^2}\right]+ A(H) \exp \left[-{\upgamma}^{*}(H)\frac{\sigma^2}{E_0^2}\right]\right){\upnu}_0(H) \exp \left(-\frac{H}{T}\right) d\uptau \right\}{\upnu}_0(H) \exp \left(\frac{- H}{T}\right) d Hd\varTheta, \hfill \end{array} $$

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where we have introduced the following notation: γ*(H) is a structure-sensitive function;

$$ \begin{array}{c}\hfill A(H)== \exp \left[ U*(H)\right]; q(H)=\frac{2\xi (H) A(H)}{1+ A(H)}; B\left( H,\sigma \right)= \exp \left[{\upgamma}^{*}(H)\frac{\sigma^2}{E_0^2}\right].\hfill \\ {}\hfill R\left( t,\varTheta \right)=\underset{0}{\overset{\infty }{\int }}\left\{ q(H) sh\left[{\upgamma}^{*}(H)\frac{\sigma^2}{E_0^2}\right]-{\upvarepsilon}_0(H) B\left[ H,\sigma \left(\varTheta \right)\right]\right\} \exp \left[-\underset{\varTheta}{\overset{t}{\int }} B\Big( H,\sigma \left(\uptau \right){\upnu}_0(H) \exp \left(-\frac{H}{T}\right) d\uptau \right]{\upnu}_0(H) \exp \left(-\frac{H}{T}\right) d H,\hfill \end{array} $$

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