Kelvin

Kelvin, William Thomson, 1st Baron, 1824-1907, British mathematician and physicist, b. Belfast. He was professor of natural philosophy at the Univ. of Glasgow (1846-99). He is known especially for his work on heat and electricity. In thermodynamics his work of coordinating the theories of heat held by various leading scientists of his time established firmly the law of the conservation of energy as proposed by Joule. He introduced the Kelvin temperature scale, or absolute scale, of temperature. He also discovered the Thomson effect in thermoelectricity. The importance of the discoveries and improvements that he made in connection with the transmission of messages by submarine cables led to his establishment as a leading authority in this field. He invented the reflecting galvanometer and the siphon recorder, an instrument by which telegraphic messages are recorded in ink fed from a siphon.

His brother, James Thomson, 1822-92, an engineer, was professor at Queen's College, Belfast, from 1857 to 1873. He is known for his studies of the variation in melting point with pressure as well as for his research in hydraulics.

kelvin, abbr. K, official name in the International System of Units (SI) for the degree of temperature as measured on the Kelvin temperature scale.

A Kelvin-Voigt material, also called a Voigt material, is a viscoelastic material having the properties both of elasticity and viscosity. It is named after the British physicist and engineer William Thomson, 1st Baron Kelvin and after German physicist Woldemar Voigt

Definition

The Kelvin-Voigt model, also called the Voigt model, can be represented by a purely viscous damper and purely elastic spring connected in parallel as shown in the picture:

If 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:

$\{epsilon\_\{Total\}\}=\{epsilon\_\{D\}\}=\{epsilon\_\{S\}\}$

Similarly, the total stress will be the sum of the stress in each component:

$\{sigma\_\{Total\}\}=\{sigma\_\{D\}\}+\{sigma\_\{S\}\}$

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:

$sigma\; (t)\; =\; E\; epsilon(t)\; +\; eta\; frac\; \{depsilon(t)\}\; \{dt\}$

where E is a modulus of elasticity and $eta$ is the viscosity. The equation can be applied either to the shear stress or normal stress 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:

$varepsilon(t)=frac\; \{sigma\_0\}\{E\}\; (1-e^\{-lambda\; t\})$,

where t is time and the rate of relaxation $lambda=frac\; \{E\}\{eta\}$

$lambda$ is also the inverse of the relaxation time.

The picture shows dependence of dimensionless deformation $frac\; \{Eepsilon(t)\}\; \{sigma\_0\}$ upon dimensionless time $lambda\; t$. The material is loaded by the stress at time $t=0$ that is released at different dimensionless times $t\_1^*=lambda\; t\_1$

If we would free the material at time $t\_1$, then the elastic element would retard the material back until the deformation become zero. The retardation obeys the following equation:

$varepsilon(t>t\_1)=varepsilon(t\_1)e^\{-lambda\; t\}$.

Since all the deformation is reversible (though not suddenly) the Kelvin-Voigt material is a solid.

The Voigt model predicts creep more realistically than the Maxwell model, since for

$lim\_\{ttoinfty\}varepsilon\; =\; frac\{sigma\_0\}\{E\}$

while a Maxwell model predicts a linear relationship between strain and time, which is most often not the case. Alternatively, 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 of the Kelvin-Voigt material would be:

$E^star\; (omega\; )\; =\; E\; +\; i\; eta\; omega$

Thus, the real and imaginary components of the dynamic modulus are:

$E\_1\; =\; Re\; [E(omega\; )]\; =\; E$

$E\_2\; =\; Im\; [E(omega\; )]\; =\; eta\; omega$

Note that $E\_1$ is constant, while $E\_2$ is directly proportional to frequency (where the apparent viscosity, $eta$, is the constant of proportionality).

References

• Meyers and Chawla (1999): Section 13.10 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

• Maxwell material
• Standard Linear Solid Material
• Generalized Maxwell material