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::$mathbf\{sigma\}(t)\; =\; -p\; mathbf\{I\}\; +\; int\_\{-infty\}^\{t\}\; M(t-t\text{'})h(I\_1,I\_2)mathbf\{B\}(t\text{'}),\; dt\text{'}$

where:
*$mathbf\{sigma\}(t)$ is the stress tensor as function of time "t",
*"p" is the pressure
*$mathbf\{I\}$ is the unity tensor
*"M" is the memory function showing, usually expressed as a sum of exponential terms for each mode of relaxation::$M(x)=sum\_\{k=1\}^m\; frac\{g\_i\}\{\; heta\_i\}exp(frac\{-x\}\{\; heta\_i\})$, where for each mode of the relaxation, $g\_i$ is the relaxation modulus and $heta\_i$ is the relaxation time;
*$h(I\_1,I\_2)$ is the "strain damping" function that depends upon the first and second invariants of Finger tensor $mathbf\{B\}$.

The "strain damping function" is usually written as::$h(I\_1,I\_2)=m^*exp(-n\_1\; sqrt\{I\_1-3\})+(1-m^*)exp(-n\_2\; sqrt\{I\_2-3\})$,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)