en>Citation bot: [442]Add: title, year, last1, first1, last2, first2, last3, first3, last4, first4, last5, first5, last6, first6, journal, pages. Tweak: last2, last3, last4, last5, last6. User-activated.

2013-10-23T18:54:22Z

[442]Add: title, year, last1, first1, last2, first2, last3, first3, last4, first4, last5, first5, last6, first6, journal, pages. Tweak: last2, last3, last4, last5, last6. User-activated.

New page

'''Wagner model''' is a [[rheology|rheological]] model developed for the prediction of the [[Viscoelasticity|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 process|isothermal]] 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]] 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 (physics)|relaxation]]:
:<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 of tensors|invariants]] of [[Finite deformation tensors#Finger tensor|Finger tensor]] <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]].

==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]]

Cyanimide - Revision history

en>Citation bot: [442]Add: title, year, last1, first1, last2, first2, last3, first3, last4, first4, last5, first5, last6, first6, journal, pages. Tweak: last2, last3, last4, last5, last6. User-activated.