Bundesanstalt für Materialforschung und -prüfung

International Symposium (NDT-CE 2003)

Non-Destructive Testing in Civil Engineering 2003
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Determination of Third Order Elastic Constants Using a Simple Ultrasonic Apparatus

Abdelmalek Bouhadjera, NDT Lab, Institute of Technology, Jijel University, Algeria


The difficulty in measuring third order elastic constants has been always in the design of the complex apparatus needed to generate both compressional and shear waves under uniaxial and hydrostatic pressures. Parameters, such as the angle of incidence have to be determined with accuracy using expensive devices. The apparatus described in this paper leaves just one variable to be determined with accuracy, namely, the time of flight of ultrasonic waves within the specimen under test. A variety of materials could be tested in this way, including rock specimens and concrete, thus enabling the evaluation of stresses in-situ within buildings and structures.


Non-linear elastic moduli are important physical properties in conventional materials. They provide information about the inter-atomic bonding forces in crystalline solids.

Non-linear properties are also important in the non-destructive determination of applied and residual stress (strain) and several investigations have also established a possible relationship between non-linear elastic properties and ultimate strength in materials [1,2,3,4,5,6,7,8].

Changes in the velocity of ultrasonic waves brought about by the application of relatively high stresses are usually of the order of a few parts in ten thousand for most materials. Direct measurement of the transit-time of ultrasonic pulses may be made to about this level of accuracy with conventional techniques, but it is often more convenient and precise to use interferometric methods, or modern digital processing techniques.

In this research, an apparatus has been developed to determine the non-linear elastic moduli in a variety of engineering and building materials. This was accomplished by first determining the linear elastic moduli using a pulse-echo method. A special formula which relates wave velocities to transit-time measurements was derived.

The stress and temperature dependence of the ultrasonic wave velocity can also be measured from the same formula, however digital signal processing techniques are needed for the determination of transit-times in order to increase the sensitivity of the system.

2- Acousto-elastic theory

Murnaghan has shown that an elastic material initially in any state of stress other than a simple hydrostatic or compression cannot be elastically isotropic. Thus a body acted upon or containing a general stress system will exhibit anisotropic characteristics when ultrasonic waves are propagated through it [9].

By using the finite strain formulation of Murnaghan, Hughes and Kelly obtained five expressions for the stress dependence of the velocities of principal ultrasonic waves in initially isotropic materials related to simple cases of an applied uniaxial stress. The results are shown in the following equations:

In these relationships, r0 is the density of the material in the initial unstrained state, v is the velocity of the ultrasonic wave, l and m are the familiar second-order Lamé's elastic constants for an isotropic material; l, m, n are the so-called Murnaghan third-order elastic constants for an isotropic material, C is the uniaxial compression acting on the material, and K0 is the bulk modulus for an isotropic material in the unstrained state.

The co-ordinate system relevant to the propagation direction, wave polarizations, and applied stress directions is indicated in figure1. The first subscript to the velocity-squared terms indicates the nature of the wave motion (longitudinal or shear); the second subscript indicates the direction of applied stress, in y-direction or z-direction.

Fig 1: Co-ordinate system for elastic wave propagation in stressed solids.The stress is applied in either the y or the z direction.

It is readily seen that when the uniaxial compression is zero, the expressions reduce to the well known forms:

From which Lamé's Constants could be computed:

The bulk modulus could also be derived from these constants:

3- The apparatus

An immersion technique based on a pulse-echo method was used to design an ultrasonic apparatus for measuring second order elastic constants in a variety of materials. Both compressional and shear waves are generated by mode conversion in a prism-shaped specimen. The latter is fixed on a circular platform, whereas the transducer is made to rotate around the specimen. The whole set is put in a water tank (Figure2).

Fig 2: The immersion unit. T1: Immersion transducer1; T2: Immersion transducer2; SUT: specimen under test; C: Compressive load.

The angle arrangement allows for just one variable to be left in the equation that enables the computation of the velocity of ultrasonic waves:

t1: Transit-time in water with specimen in place.
t2 : Transit-time in water without specimen.
Tt : Total transit-time with the presence of specimen.
R: Radius of circle made by the transducer around the specimen.

Thus, a considerable improvement could be made on an earlier method designed by the author and based on the same principle [10], thereby improving the accuracy and reliability of measurements.

The previous equation used with a similar apparatus is:

Notice the presence of the angle , which requires a relatively complex system for its measurement.

At the start of the experiment, the incident waves from the immersion transducer are made to impinge normally on the largest face of the prism (Figure4).

Fig 4: Trajectory of ultrasonic beam within the immersion unit.

The cylindrically focused transducer helps to concentrate the energy of ultrasonic waves at the centre of the prism. Most of the energy is reflected from the face and the related echo gives the time-of-flight of the waves within the water interface, and also helps in measuring the thickness of the cube.

Next, the angle of incidence is increased slowly until the disappearance of the first echo and the emergence of a second echo related to compressional waves.

After that, the angle is increased further, a third echo appears, and this time it is related to shear waves within the specimen.

Finally, the velocity of compressional waves is cross-checked by using a contact-transducer put on top of the specimen (on one of the parallel faces).

Special care must be undertaken while preparing the materials samples (Figure5).

The starting point should be from a perfectly made cube, which has a side of about 50 mm. It is then divided through its diagonal into two equal prisms. The prism could also be made from a cylindrically shaped specimen. The critical points here are the right angle, and the two parallel faces.

Fig 5: The prism-shaped specimens. Fig 6: General view of the immersion unit.

There would be no influence on the quality of measurements if the prism were more or less chipped on the edges. Nevertheless, the parallel faces are important for measuring the velocity of compressional waves by means of a contact transducer, whereas the right angle is vital for computing the velocity of shear waves using the immersion transducer.

Unlike cylindrically shaped specimens, the prism-shaped ones could be tested in three different directions, which would give a fairly good idea about the anisotropy of the material. There is also the possibility of cross-checking the velocity of compressional waves using a contact transducer. Therefore, the drawback related to the difficulty of preparing prism-shaped specimens compared to cylindrically shaped ones is more than compensated by the wealth of information yielded from the former.

The accuracy of measurements depends only on one parameter, namely the time-of-flight of ultrasonic waves. It could be determined with high precision using well-known signal processing techniques, such as the overlap, cross-correlation or cepstrum algorithms.

The nearly finished immersion unit is shown in Figure6 above.

Table1 shows some preliminary results related to second order elastic constants for two specimens of Perspex and aluminum.

Measurements are being carried out in order to determine higher order elastic constants in a variety of materials, before investigating stresses in-situ, which would require, in its simplest form, only the measurement of the velocity of compressional waves from a hammer impact on the part of the structure to be tested.

      r0(Kg/m3) Vl0 (m/s) Vs0 (m/s) l (GPa) m (GPa) K0 (GPa)
Perspex 1179 2730 1430 3.96 2.41 5.57
Aluminum 2699 6320 3130 54.92 26.44 72.55
Table 1: Second-Order elastic constants.

4- Conclusion

This research provides initial measurements related to the characterization of the non-linear elastic properties of materials using a simple ultrasonic apparatus. These will provide the basis for studying the relationships between non-linear properties and the more important engineering properties such as ultimate strength, residual strength after impact, and fatigue loading.

The results related to second order elastic constants were obtained from the computation of both longitudinal and shear wave velocities using a single formula with just one variable: the transit-time of ultrasonic pulses. There is the potential of making high precision transit-time measurements using digital signal processing techniques, such as cross-correlation.

There is also the exciting prospect of exploiting shear wave birefringence present within the specimen, which would provide additional information about internal fractures and pore structure of materials.

5- References

  1. William, H., et al, "Characterization of the Non-linear Elastic Constants of Graphite/Epoxy Composites Using Ultrasound", Journal of Reinforced Plastics and Composites, pp.162-173, Vol.9, March 1990.
  2. Bray, D. E., "Current Directions of Ultrasonic Stress Measurement techniques", 15th WCNDT, Roma 2000.
  3. Landa, M., "Ultrasonic Techniques for Non-destructive Evaluation of Internal Stresses", 15th WCNDT, Roma 2000.
  4. Popovics, J.S., "Determination of Elastic Constants of a Concrete Specimen Using Transient Elastic Waves", J.Acoust.Soc.Am., Vol.98, pp.2142-2148, 1995.
  5. Hernandez, M. G., et al, "Porosity Estimation of Concrete by Ultrasonic NDE", Ultrasonics, vol.38, pp.531-533, 2000.
  6. McCann, D. M., "Review of NDT Methods in the Assessment of Concrete and Masonry Structures", NDT&E International, Vol. 34, pp.71-84, 2001.
  7. Martin, L. P., "Ultrasonic Determination of Elastic Moduli in Cement During Hydrostatic Loading to 1 GPa". Materials Science and Engineering, A279, pp87-94, 2000.
  8. Hwa, L. G., Pressure and Temperature Dependence of Elastic Properties of a ZBLAN Glass", Materials Chemistry and Physics, Vol. 74, pp.160-166, 2002.
  9. Smith, R.T., "Stress-Induced Anisotropy in Solids: The Acousto-Elastic Effect", Ultrasonics, pp. 135-147, July-September 1963.
  10. Bouhadjera, A., "A mode Conversion Method for Evaluating Elastic Properties of Materials", INSIGHT, (The Journal of the British Institute of Non-Destructive Testing), Vol.38, N°10, October 1996.
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