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:

${\displaystyle \mathbf {\sigma } (t)=-p\mathbf {I} +\int _{-\infty }^{t}M(t-t')h(I_{1},I_{2})\mathbf {B} (t')\,dt'}$

where:

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

${\displaystyle M(x)=\sum _{k=1}^{m}{\frac {g_{i}}{\theta _{i}}}\exp({\frac {-x}{\theta _{i}}})}$, where for each mode of the relaxation, ${\displaystyle g_{i}}$ is the relaxation modulus and ${\displaystyle \theta _{i}}$ is the relaxation time;

• ${\displaystyle h(I_{1},I_{2})}$ is the strain damping function that depends upon the first and second invariants of Finger tensor ${\displaystyle \mathbf {B} }$.

The strain damping function is usually written as:

${\displaystyle h(I_{1},I_{2})=m^{*}\exp(-n_{1}{\sqrt {I_{1}-3}})+(1-m^{*})\exp(-n_{2}{\sqrt {I_{2}-3}})}$,

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)