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.