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')h(I_1,I_2)\mathbf{B}(t'), dt'

where:

• \mathbf{\sigma}(t) is the Cauchy 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}{\theta_i}\exp(\frac{-x}{\theta_i}), where for each mode of the relaxation, g_i is the relaxation modulus and \theta_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)