Physics: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:

[math]\displaystyle{ \mathbf{\sigma}(t) = -p \mathbf{I} + \int_{-\infty}^{t} M(t-t')h(I_1,I_2)\mathbf{B}(t')\, dt' }[/math]

where:

• [math]\displaystyle{ \mathbf{\sigma}(t) }[/math] is the Cauchy stress tensor as function of time t,
• p is the pressure
• [math]\displaystyle{ \mathbf{I} }[/math] is the unity tensor
• M is the memory function showing, usually expressed as a sum of exponential terms for each mode of relaxation:

[math]\displaystyle{ M(x)=\sum_{k=1}^m \frac{g_i}{\theta_i}\exp(\frac{-x}{\theta_i}) }[/math], where for each mode of the relaxation, [math]\displaystyle{ g_i }[/math] is the relaxation modulus and [math]\displaystyle{ \theta_i }[/math] is the relaxation time;

• [math]\displaystyle{ h(I_1,I_2) }[/math] is the strain damping function that depends upon the first and second invariants of Finger tensor [math]\displaystyle{ \mathbf{B} }[/math].

The strain damping function is usually written as:

[math]\displaystyle{ h(I_1,I_2)=m^*\exp(-n_1 \sqrt{I_1-3})+(1-m^*)\exp(-n_2 \sqrt{I_2-3}) }[/math],

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)