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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


Contents

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

See also

 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Kelvin-Voigt_material". A list of authors is available in Wikipedia.
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