Kelvin–Voigt Material
   HOME

TheInfoList



OR:

A Kelvin–Voigt material, also called a Voigt material, is the most simple model
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both Viscosity, viscous and Elasticity (physics), elastic characteristics when undergoing deformation (engineering), deformation. Viscous mate ...
material showing typical rubbery properties. It is purely elastic on long timescales (slow deformation), but shows additional resistance to fast deformation. The model was developed independently by the British physicist
Lord Kelvin William Thomson, 1st Baron Kelvin (26 June 182417 December 1907), was a British mathematician, Mathematical physics, mathematical physicist and engineer. Born in Belfast, he was the Professor of Natural Philosophy (Glasgow), professor of Natur ...
in 1865 and by the German physicist
Woldemar Voigt Woldemar Voigt (; 2 September 1850 – 13 December 1919) was a German mathematician and physicist. Biography Voigt was born in Leipzig, and died in Göttingen. He was a student of Franz Ernst Neumann. Voigt taught at the Georg August Universi ...
in 1890.


Definition

The Kelvin–Voigt model, also called the Voigt model, is represented by a purely
viscous Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...
damper and purely
elastic Elastic is a word often used to describe or identify certain types of elastomer, Elastic (notion), elastic used in garments or stretch fabric, stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rub ...
spring connected in parallel as shown in the picture. If, instead, we connect these two elements in series we get a model of a
Maxwell material A Maxwell model is the most simple model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. It is named for James Clerk Maxwell who p ...
. Since the two components of the model are arranged in parallel, the strains in each component are identical: : \varepsilon_\text = \varepsilon_ = \varepsilon_. 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: : \sigma_\text = \sigma_ + \sigma_. From these equations we get that in a Kelvin–Voigt material, stress ''σ'', strain ''ε'' and their rates of change with respect to time ''t'' are governed by equations of the form: :\sigma (t) = E \varepsilon(t) + \eta \frac , or, in dot notation: :\sigma = E \varepsilon + \eta \dot , where ''E'' is a modulus of elasticity and \eta is the
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
. The equation can be applied either to the
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
or
normal stress In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to ''tensile'' stress and may undergo elongati ...
of a material.


Effect of a sudden stress

If we suddenly apply some constant stress \sigma_0 to Kelvin–Voigt material, then the deformations would approach the deformation for the pure elastic material \sigma_0/E with the difference decaying exponentially: :\varepsilon(t)=\frac (1-e^), where ''t'' is time and \tau_R=\frac is the retardation time. If we would free the material at time t_1, then the elastic element would retard the material back until the deformation becomes zero. The retardation obeys the following equation: :\varepsilon(t>t_1)=\varepsilon(t_1)e^. The picture shows the dependence of the dimensionless deformation \frac on dimensionless time t/\tau_R. In the picture the stress on the material is loaded at time t=0, and released at the later dimensionless time t_1^*=t_1/\tau_R. Since all the deformation is reversible (though not suddenly) the Kelvin–Voigt material is a
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
. The Voigt model predicts creep more realistically than the Maxwell model, because in the infinite time limit the strain approaches a constant: :\lim_\varepsilon = \frac, 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
dynamic modulus Dynamic modulus (sometimes complex modulusThe Open University (UK), 2000. ''T838 Design and Manufacture with Polymers: Solid properties and design'', page 30. Milton Keynes: The Open University.) is the ratio of stress to strain under ''vibratory ...
of the Kelvin–Voigt material is given by: :E^\star ( \omega ) = E_0 (1 + i \omega \tau). Thus, the real and imaginary components of the dynamic modulus are referred to as storage modulus E^ and E^ respectively: :E^ = \Re ^\star( \omega ) :E^ = \Im ^\star( \omega )= E_0 \omega \tau. Note that E^ is constant, while E^ is directly proportional to frequency (where time-scale \tau is the constant of proportionality). Often, this constant \tau multiplied with angular frequency \omega is called the loss modulus \eta=\omega\tau.


References


See also

* Burgers material * Generalized Maxwell model *
Maxwell material A Maxwell model is the most simple model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. It is named for James Clerk Maxwell who p ...
*
Standard linear solid model The standard linear solid (SLS), also known as the Zener model after Clarence Zener, is a method of modeling the behavior of a viscoelastic material using a linear combination of springs and dashpots to represent elastic and viscous components, r ...
{{DEFAULTSORT:Kelvin-Voigt Material Non-Newtonian fluids Materials science William Thomson, 1st Baron Kelvin