View source for Physics:Kelvin–Voigt material

A '''Kelvin-Voigt material''', also called a '''Voigt material''', is the most simple model viscoelastic material showing typical rubbery properties. It is purely elastic on long timescales (slow deformation), but shows additional resistance to fast deformation. It is named after the British physicist and engineer [[Biography:William Thomson, 1st Baron Kelvin|Lord Kelvin]] and German physicist [[Biography:Woldemar Voigt|Woldemar Voigt]].
== Definition ==

The Kelvin-Voigt model, also called the Voigt model, is represented by a purely [[Physics:Viscosity|viscous]] damper and purely [[Physics:Elasticity|elastic]] spring connected in parallel as shown in the picture.

[[Image:Kelvin Voigt diagram.svg|right|frame| Schematic representation of Kelvin–Voigt model.]]

If, instead, we connect these two elements in series we get a model of a [[Physics:Maxwell material|Maxwell material]].

Since the two components of the model are arranged in parallel, the strains in each component are identical:

:<math> \varepsilon_\text{Total} = \varepsilon_S = \varepsilon_D. </math>

where the subscript D indicates the stress-strain in the damper and the subscript S indicates the stress-strain in the spring. Similarly, the total stress will be the sum of the stress in each component:

:<math> \sigma_\text{Total} = \sigma_S + \sigma_D. </math>

From these equations we get that in a Kelvin-Voigt material, [[Physics:Stress|stress]] σ, [[Physics:Strain (materials science)|strain]] ε and their rates of change with respect to time ''t'' are governed by equations of the form:

:<math>\sigma (t) = E \varepsilon(t) + \eta \frac {d\varepsilon(t)} {dt},</math>

or, in dot notation:

:<math>\sigma  = E \varepsilon + \eta \dot {\varepsilon},</math>

where ''E'' is a modulus of elasticity and <math>\eta</math> is the [[Physics:Viscosity|viscosity]]. The equation can be applied either to the [[Physics:Shear stress|shear stress]] or normal stress of a material.

== Effect of a sudden stress ==

If we suddenly apply some constant stress <math>\sigma_0</math> to Kelvin-Voigt material, then the deformations would approach the deformation for the pure elastic material <math> \sigma_0/E</math> with the difference decaying exponentially:

:<math>\varepsilon(t)=\frac {\sigma_0}{E} (1-e^{-t/\tau_R }), </math>

where ''t'' is time and <math> \tau_R=\frac {\eta}{E} </math> is the [[Physics:Retardation time|retardation time]].

If we would free the material at time <math>t_1</math>, then the elastic element would retard the material back until the deformation becomes zero. The retardation obeys the following equation:

:<math>\varepsilon(t>t_1)=\varepsilon(t_1)e^{-(t-t_1)/\tau_R}. </math>

The picture shows the dependence of the dimensionless deformation <math>\frac {E\varepsilon(t)} {\sigma_0} </math> 
on dimensionless time <math>t/\tau_R</math>. In the picture the stress on the material is loaded at time <math>t=0</math>, and released at the later dimensionless time <math>t_1^*=t_1/\tau_R</math>.
[[Image:Kelvin deformation 2.png|right|frame|Dependence of dimensionless deformation
upon dimensionless time under constant stress]]

Since all the deformation is reversible (though not suddenly) the Kelvin–Voigt material is a [[Physics:Solid|solid]].

The Voigt model predicts creep more realistically than the Maxwell model, because in the infinite time limit the strain approaches a constant:

:<math>\lim_{t\to\infty}\varepsilon = \frac{\sigma_0}{E},</math>

while a Maxwell model predicts a linear relationship between strain and time, which is most often not the case. Although the Kelvin-Voigt model is effective for predicting creep, it is not good at describing the relaxation behavior after the stress load is removed.

== Dynamic modulus ==

The complex [[Physics:Dynamic modulus|dynamic modulus]] of the Kelvin-Voigt material is given by:

:<math>E^\star ( \omega ) = E_0 (1 + i \omega \tau). </math>

Thus, the real and imaginary components of the dynamic modulus are referred to as storage modulus <math>E^{\prime}</math> and <math>E^{\prime\prime}</math> respectively:

:<math>E^{\prime} = \Re [E^\star( \omega )], </math>
:<math>E^{\prime\prime} = \Im [E^\star( \omega )] = E_0 \omega \tau. </math>

Note that <math>E^{\prime}</math> is constant, while <math>E^{\prime\prime}</math> is directly proportional to frequency (where time-scale <math>\tau</math> is the constant of proportionality). Often, this constant <math>\tau</math> multiplied with angular frequency <math>\omega</math> is called the loss modulus <math>\eta=\omega\tau</math>.

== References ==

* Meyers and Chawla (1999): Section 13.11 of Mechanical Behaviors of Materials, ''Mechanical behavior of Materials'', 570–580. Prentice Hall, Inc.
* http://stellar.mit.edu/S/course/3/fa06/3.032/index.html

== See also ==
*[[Physics:Burgers material|Burgers material]]
*[[Physics:Generalized Maxwell model|Generalized Maxwell model]]
*[[Physics:Maxwell material|Maxwell material]]
*[[Physics:Standard linear solid model|Standard linear solid model]]

{{DEFAULTSORT:Kelvin-Voigt Material}}
[[Category:Non-Newtonian fluids]]
[[Category:Materials science]]

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