Wagner model explained

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:

\sigma (t) = - p I + \int ^{t} _{-inf ty} M(t - t')h(I _{1,I} _{2)B(t')} dt'

where:

\sigma (t) is the Cauchy stress tensor as function of time t,

• p is the pressure

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 ^{m} _{k =1} (g _{i})/(\exp( \theta _{i}) (- 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 B .

The strain damping function is usually written as:

h(I _{1,I} ^{*\exp( - n} ^{1} \sqrt{I ^{*)\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)