Wagner model 3404702 150138834 2007-08-09T07:40:56Z 218.185.0.106 '''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 [[Stress (physics)|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]]: :<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]] [[es:Modelo de Wagner]]