Wagner model is a rheological model developed for the prediction of the

viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linear ...

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 In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...

conditions the model can be written as: :

\mathbf(t) = -p \mathbf + \int_^ M(t-t')h(I_1,I_2)\mathbf(t')\, dt'

where: *

\mathbf(t)

is the

Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...

as function of time ''t'', *''p'' is the pressure *

\mathbf

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_^m \frac\exp(\frac)

, where for each mode of the relaxation,

g_i

is the relaxation modulus and

\theta_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 In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...

\mathbf

. The ''strain damping function'' is usually written as: :

h(I_1,I_2)=m^*\exp(-n_1 \sqrt)+(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) Non-Newtonian fluids