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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:

σ(t)=−pI+∫−∞tM(t−t′)h(I1,I2)B(t′)dt′displaystyle mathbf sigma (t)=-pmathbf I +int _-infty ^tM(t-t')h(I_1,I_2)mathbf B (t'),dt'

where:

• 
  σ(t)displaystyle mathbf sigma (t) is the Cauchy stress tensor as function of time t,


• 
  p is the pressure


• 
  Idisplaystyle 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)=∑k=1mgiθiexp⁡(−xθi)displaystyle M(x)=sum _k=1^mfrac g_itheta _iexp(frac -xtheta _i), where for each mode of the relaxation, gidisplaystyle g_i is the relaxation modulus and θidisplaystyle theta _i is the relaxation time;

• 
  h(I1,I2)displaystyle h(I_1,I_2) is the strain damping function that depends upon the first and second invariants of Finger tensor Bdisplaystyle mathbf B .

The strain damping function is usually written as:

h(I1,I2)=m∗exp⁡(−n1I1−3)+(1−m∗)exp⁡(−n2I2−3)displaystyle h(I_1,I_2)=m^*exp(-n_1sqrt I_1-3)+(1-m^*)exp(-n_2sqrt 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)