Wagner model

Template:For 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) = - p 𝐈 + \int_{- \infty}^{t} M (t - t^{'}) h (I_{1}, I_{2}) 𝐁 (t^{'}) d t^{'}

where:

$𝝈 (t)$ is the Cauchy stress tensor as function of time t,
p is the pressure
$𝐈$ 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}}{θ_{i}} \exp (\frac{- x}{θ_{i}})

, where for each mode of the relaxation,

g_{i}

is the relaxation modulus and

θ_{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 $𝐁$ .

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