# Wagner model

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{{for|the American labour relations model|Wagner Act}}
'''Wagner model''' is a [rheological](/source/rheology) model developed for the prediction of the [viscoelastic](/source/Viscoelasticity) properties of polymers. It might be considered as a simplified practical form of the [Bernstein-Kearsley-Zapas](/source/Bernstein-Kearsley-Zapas) model. The model was developed by German rheologist [Manfred Wagner](/source/Manfred_Wagner).

For the [isothermal](/source/Isothermal_process) conditions the model can be written as:
:<math>\mathbf{\sigma}(t) = -p \mathbf{I} + \int_{-\infty}^{t} M(t-t')h(I_1,I_2)\mathbf{B}(t')\, dt'</math>

where:
*<math>\mathbf{\sigma}(t)</math> is the [Cauchy stress tensor](/source/Cauchy_stress_tensor) as function of time ''t'',
*''p'' is the pressure
*<math>\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](/source/Relaxation_(physics)):
:<math>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>g_i</math> is the relaxation modulus and <math>\theta_i</math> is the relaxation time;
*<math>h(I_1,I_2)</math> is the ''strain damping'' function that depends upon the first and second [invariants](/source/Invariants_of_tensors) of [Finger tensor](/source/Finite_strain_theory) <math>\mathbf{B}</math>.

The ''strain damping function'' is usually written as:
:<math>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](/source/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)

Category:Non-Newtonian fluids

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