Wagner model

Last updated April 22, 2019

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.

Rheology is the study of the flow of matter, primarily in a liquid state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force. It is a branch of physics which deals with the deformation and flow of materials, both solids and liquids.

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 linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

Manfred Hermann Wagner is the author of Wagner model and the molecular stress function theory for polymer rheology. He is a Professor for Polymer engineering and Polymer physics at the Technical University of Berlin.

For the isothermal conditions the model can be written as:

An isothermal process is a change of a system, in which the temperature remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and the change in the system will occur slowly enough to allow the system to continue to adjust to the temperature of the reservoir through heat exchange. In contrast, an adiabatic process is where a system exchanges no heat with its surroundings (Q = 0). In other words, in an isothermal process, the value ΔT = 0 and therefore the change in internal energy ΔU = 0 but Q ≠ 0, while in an adiabatic process, ΔT ≠ 0 but Q = 0.

\mathbf {\sigma } (t)=-p\mathbf {I} +\int _{-\infty }^{t}M(t-t')h(I_{1},I_{2})\mathbf {B} (t')\,dt'

where:

$\mathbf {\sigma } (t)$ is the Cauchy stress tensor as function of time t,
p is the pressure
$\mathbf {I}$ is the unity tensor
M is the memory function showing, usually expressed as a sum of exponential terms for each mode of relaxation:

In continuum mechanics, the Cauchy stress tensor $, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the stress vector T (n) across an imaginary surface perpendicular to n :$

In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp(-t/τ).

M(x)=\sum _{k=1}^{m}{\frac {g_{i}}{\theta _{i}}}\exp({\frac {-x}{\theta _{i}}})

, 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 $\mathbf {B}$ .

In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor $are the coefficients of the characteristic polynomial$

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.

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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)

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