Wagner model


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 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}{ heta_i}exp(frac{-x}{ heta_i}), where for each mode of the relaxation, g_i is the relaxation modulus and heta_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.

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)


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