# Wagner model

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Wagner model

Wagner model is a rheological model developed for the prediction of the viscoelastic properties of polymers. It might be considered as a simplified practical form of the Bernstein-Kearsley-Zapas model. The model was developed by German rheologist Manfred Wagner.

For the isothermal conditions the model can be written as::$mathbf\left\{sigma\right\}\left(t\right) = -p mathbf\left\{I\right\} + int_\left\{-infty\right\}^\left\{t\right\} M\left(t-t\text{'}\right)h\left(I_1,I_2\right)mathbf\left\{B\right\}\left(t\text{'}\right), dt\text{'}$

where:
*$mathbf\left\{sigma\right\}\left(t\right)$ is the stress tensor as function of time "t",
*"p" is the pressure
*$mathbf\left\{I\right\}$ is the unity tensor
*"M" is the memory function showing, usually expressed as a sum of exponential terms for each mode of relaxation::$M\left(x\right)=sum_\left\{k=1\right\}^m frac\left\{g_i\right\}\left\{ heta_i\right\}exp\left(frac\left\{-x\right\}\left\{ heta_i\right\}\right)$, where for each mode of the relaxation, $g_i$ is the relaxation modulus and $heta_i$ is the relaxation time;
*$h\left(I_1,I_2\right)$ is the "strain damping" function that depends upon the first and second invariants of Finger tensor $mathbf\left\{B\right\}$.

The "strain damping function" is usually written as::$h\left(I_1,I_2\right)=m^*exp\left(-n_1 sqrt\left\{I_1-3\right\}\right)+\left(1-m^*\right)exp\left(-n_2 sqrt\left\{I_2-3\right\}\right)$,The strain hardening function equal to one, then the deformation is small and approaching zero, then the deformations are large.

The Wagner equation can be used in the non-isothermal cases by applying time-temperature shift factor.

References

*M.H. Wagner "Rheologica Acta", v.15, 136 (1976)
*M.H. Wagner "Rheologica Acta", v.16, 43, (1977)
*B. Fan, D. Kazmer, W. Bushko, "Polymer Engineering and Science", v44, N4 (2004)

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