## Application:

Measurement on Young's Modulus and Shear Modulus of Elasticity, and Poisson's ratio, in nondispersive isotropic engineering materials.

## Background:

**Young's Modulus of Elasticity** is defined as the ratio of stress (force per unit area) to corresponding strain (deformation) in a material under tension or compression.
**Shear Modulus of Elasticity** is similar to the ratio of stress to strain in a material subjected to shear stress.
**Poisson's Ratio** is the ratio of transverse strain to corresponding axial strain on a material stressed along one axis.

These basic material properties, which are of interest in many manufacturing and research applications, can be determined through computations based on measured sound velocities and material density. Sound velocity can be easily measured using ultrasonic pulse-echo techniques with appropriate equipment. The general procedure outlined below is valid for any homogeneous, isotropic, nondispersive material (velocity does not change with frequency). This includes most common metals, industrial ceramics, and glasses as long as cross sectional dimensions are not close to the test frequency wavelength. Rigid plastics such as polystyrene and acrylic can also be measured, although they are more challenging due to higher sound attenuation.

Rubber cannot be characterized ultrasonically because of its high dispersion and nonlinear elastic properties. Soft plastics similarly exhibit very high attenuation in shear mode and as a practical matter usually cannot be tested. In the case of anisotropic materials, elastic properties vary with direction, and so do longitudinal and/or shear wave sound velocity. Generation of a full matrix of elastic moduli in anisotropic specimens typically requires six different sets of ultrasonic measurements. Porosity or coarse granularity in a material can affect the accuracy of ultrasonic modulus measurement since these conditions can cause variations in sound velocity based on grain size and orientation or porosity size and distribution, independent of material elasticity.

## Equipment:

The velocity measurements for modulus calculation are most commonly made with precision thickness gages such as models 38DL PLUS and 45MG with Single Element software, or a flaw detector with velocity measurement capability such as the EPOCH series instruments. Pulser/receivers such as the Model 5072PR or 5077PR can also be used with an oscilloscope or waveform digitizer for transit time measurements. This test also requires two transducers appropriate to the material being tested, for pulse-echo sound velocity measurement in longitudinal and shear modes. Commonly used transducers include an M112 or V112 broadband longitudinal wave transducer (10 MHz) and a V156 normal incidence shear wave transducer (5 MHz). These work well for many common metal and fired ceramic samples. Different transducers will be required for very thick, very thin, or highly attenuating samples. Some cases may also require use of through transmission techniques, with pairs of transducers positioned on opposite sides of the part. It is recommended that in all cases the user consult Olympus for specific transducer recommendations and assistance with instrument setup.

The test sample may be of any geometry that permits clean pulse/echo measurement of sound transit time through a section on thickness. Ideally this would be a sample at least 0.5 in. (12.5 mm) thick, with smooth parallel surfaces and a width or diameter greater than the diameter of the transducer being used. Caution must be used when testing narrow specimens due to possible edge effects that can affect measured pulse transit time. Resolution will be limited when very thin samples are used due to the small changes in pulse transit time across short sound paths. For that reason we recommend that samples should be at least 0.2 in. (5 mm) thick, preferably thicker. In all cases the thickness of the test sample must be precisely known.

## Procedure:

Measure the longitudinal and shear wave sound velocity of the test piece using the appropriate transducers and instrument setup. The shear wave measurement will require use of a specialized high viscosity couplant such as our SWC. A Model 38DL PLUS a 45MG thickness gage can provide a direct readout of material velocity based on an entered sample thickness, and an EPOCH series flaw detector can measure velocity through a velocity calibration procedure. In either case, follow the recommended procedure for velocity measurement as described in the instrument's operating manual. If using a pulser/receiver, simply record the round-trip transit time through an area of known thickness with both longitudinal and shear wave transducers, and compute:

Convert units as necessary to obtain velocities expressed as inches per second or centimeters per second. (Time will usually have been measured in microseconds, so multiply in/uS or cm/uS by 10^{6} to obtain in/S or cm/S.) The velocities thus obtained may be inserted into the following equations.

Note on units: If sound velocity is expressed in cm/S and density in g/cm^{3}, then Young's modulus will be expressed in units of dynes/cm^{2}. If English units of in/S and lbs/in^{3} are used to compute modulus in pounds per square inch (PSI), remember the distinction between "pound" as a unit of force versus a unit of mass. Since modulus is expressed as a force per unit area, when calculating in English units it is necessary to multiply the solution of
the above equation by a mass/force conversion constant of (1 / Acceleration of Gravity) to obtain modulus in PSI. Alternately, if the initial calculation is done in metric units, use the conversion factor 1 psi = 6.89 x 10^{4} dynes/cm ^{2}. Another alternative is to enter velocity in in/S, density in g/cm ^{3}, and divide by a conversion constant of 1.07 x 10^{4} to obtain modulus in PSI.

For shear modulus simply multiply the square of the shear wave velocity by the density.

Again, use units of cm/S and g/cm ^{3}to obtain modulus in dynes/cm ^{2}or English units of in/S and lbs/in ^{3}and multiply the result by the mass/force conversion constant.
**Bibliography**

For further information on ultrasonic measurement of elastic modulus, consult the following:

1. Moore, P. (ed.), *Nondestructive Testing Handbook,* Volume 7, American Society for Nondestructive Testing, 2007, pp. 319-321.

2. Krautkramer, J., H. Krautkramer, *Ultrasonic Testing of Materials*, Berlin, Heidelberg, New York 1990 (Fourth Edition), pp. 13-14, 533-534.