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 viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linear ...
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
Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
and German physicist
Woldemar Voigt.
Definition
The Kelvin-Voigt model, also called the Voigt model, is represented by a purely
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
damper and purely
elastic 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.
Since the two components of the model are arranged in parallel, the strains in each component are identical:
:
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:
:
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:
:
or, in dot notation:
:
where ''E'' is a modulus of elasticity and
is the
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
. The equation can be applied either to the
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
or
normal stress of a material.
Effect of a sudden stress
If we suddenly apply some constant stress
to Kelvin-Voigt material, then the deformations would approach the deformation for the pure elastic material
with the difference decaying exponentially:
:
where ''t'' is time and
is the
retardation time Retardation is the delayed response to an applied force or stress and can be described as "delay of the elasticity".
Ideal elastic materials show an immediate deformation after applying a jump-like stress, and an immediate reformation after removin ...
.
If we would free the material at time
, then the elastic element would retard the material back until the deformation becomes zero. The retardation obeys the following equation:
:
The picture shows the dependence of the dimensionless deformation
on dimensionless time
. In the picture the stress on the material is loaded at time
, and released at the later dimensionless time
.
Since all the deformation is reversible (though not suddenly) the Kelvin–Voigt material is a
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structura ...
.
The Voigt model predicts creep more realistically than the Maxwell model, because in the infinite time limit the strain approaches a constant:
:
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 c ...
of the Kelvin-Voigt material is given by:
:
Thus, the real and imaginary components of the dynamic modulus are:
:
:
Note that
is constant, while
is directly proportional to frequency (where the apparent viscosity,
, is the constant of proportionality).
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
*
Burgers material
*
Generalized Maxwell model
*
Maxwell material
*
Standard linear solid model
{{DEFAULTSORT:Kelvin-Voigt Material
Non-Newtonian fluids
Materials science
William Thomson, 1st Baron Kelvin