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# Kelvin-Voigt material

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:

εTotal = εD = εS

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

σTotal = σD + σS

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:

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

where E is a modulus of elasticity and η 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 σ0 to Kelvin-Voigt material, then the deformations would approach the deformation for the pure elastic material σ0 / 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 $\lambda=\frac {E}{\eta}$

λ is also the inverse of the relaxation time.

The picture shows dependence of dimensionless deformation $\frac {E\epsilon(t)} {\sigma_0}$ upon dimensionless time λt. 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 t1, 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

$\lim_{t\to\infty}\varepsilon = \frac{\sigma_0}{E}$

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 E1 is constant, while E2 is directly proportional to frequency (where the apparent viscosity, η, 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