Elastic Modulus – Ultrasonic Measurement

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Elastic modulus – Ultrasonic measurement

Fundamentals

In quality assurance and materials testing, it is often necessary to convert the characteristic values determined by non-destructive testing into elastic or mechanical characteristic values. This requires calibration measurements in order to establish clear correlations between the mechanical and acoustic material parameters. The correlation between elastic and acoustic characteristic values listed here refers specifically to the material parameters modulus of elasticity E_t (from the tensile test) and the sound velocity of the longitudinal waves c_L.

Acoustic-mechanical property correlations

The desired correlation can take different functional forms. In the simplest case, there is a linear relationship between parameter K₁ and parameter K₂ (see Eq. (1)).

$K_{1}\sim K_{2}\!$

(1)

The proportionality constant or calibration function is determined graphically or using a regression function, allowing each K₁ value to be clearly assigned a K₂ value.

Analogous to this procedure, the characteristic values from the mechanical and acoustic measurements are compared or correlated with each other. From elasticity theory, the relationship between the parameters modulus of elasticity E_t and sound velocity c_L is known, which is described by the following Eq. (2).

$c_{L}={\sqrt {{\frac {E_{t}}{\rho }}{\frac {\left(1-\mu \right)}{\left(1-2\mu \right)\left(1+\mu \right)}}}}$

(2)

To determine the modulus of elasticity from the known sound velocity, Eq. (2) is rearranged to give the calibration function for linear-elastic material behaviour (see Eq. (3)), provided that the density ρ of the material under investigation is known.

$E_{t}=c_{L}^{2}\cdot \rho {\frac {\left(1-2\mu \right)\left(1+\mu \right)}{1-\mu }}$

(3)

In practice, Eq. (3) applies to plastics only to a very limited extent, as it is only relevant for elastic material behaviour (see: elasticity) and thus for the area of validity of HOOKE's law. However, plastics have a very small elastic range and are characterised by viscoelastic properties at small deformations. Since these are not taken into account in Eq. (3), it is replaced by Eq. (4), in which the proportionality factor is described by a function that depends on density, Poisson's ratio and temperature. This proportionality function is considered a black box, as these influencing factors can only be represented separately.

$E_{t}=c_{L}^{2}\cdot f\left(\rho ,\mu ,T\right)$

(4)

Example of creating a calibration function for PP

As an example, a measurement on polypropylene (abbreviation: PP) is shown, which has a typical exponential dependence (Fig. 1). The mechanical and acoustic characteristic values were determined at defined temperatures in the range from –40 to +40 °C, entered into the diagram, and a quadratic regression function was then determined.

Fig. 1:

Dependence of the tensile modulus Et on the sound velocity cL of polypropylene (homopolymer)

For the entire curve, there is an exponential relationship between the two correlated parameters. Therefore, the corresponding modulus of elasticity can be determined from the diagram for each sound velocity. For example, for a sound velocity of 2,750 m/s, a modulus of elasticity of 2,600 MPa is determined (see Fig. 1). Using the deviations of the measurement points from the exponential compensation curve, the relative error for the determined modulus of elasticity can be estimated at approx. 10 %.

According to Eq. (3), the functionality between the modulus of elasticity and the sound velocity is defined based on elasticity theory. However, since the dependence of the modulus of elasticity on the sound velocity is also influenced by the temperature-dependent Poisson's ratio (transverse contraction ratio), the increase transitions from a quadratic curve to an exponential functionality.

See also

• Elastic modulus
• Elastic modulus – Examples and material values
• Energy elasticity
• HOOKE´s law

References

[1]	Berktay, H. O.: Possible Exploitation of Non-Linear Acoustics in Underwater Transmitting Applications. J. Sound and Vibration, 1965, 2(4) pp. 435–461; https://doi.org/10.1016/0022-460X(65)90122-7
[2]	Moussatov, A., Castagnede, B.: Ultrasonic Defectoscopy of Damaged Materials by Modulation Transfer Method: Nonlinear Pump-Probe Interaction. WCU 2003, Paris, September 7–10, 2003
[3]	Šutilov, V. A.: Physik des Ultraschalls. Akademie-Verlag, Berlin (1984) (ISBN 978-3-7091-8751-7)
[4]	Landau, L. D., Lifschitz, E. M.: Lehrbuch der theoretischen Physik – Elastzitätstheorie Vol. 7, Akademie-Verlag Berlin (2009) (ISBN 978-3-8085-5498-2)