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 nondispersive material and sample geometry (i.e., velocity does not change with frequency). This includes most metals, ceramics, and glasses as long as cross sectional dimensions are not close to the test frequency wavelength.

Rubber usually cannot be characterized ultrasonically because of its high dispersion and nonlinear elastic properties. In the case of anisotropic materials, elastic properties vary with direction, and so does 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.

Equipment:

The technique requires ultrasonic pulser-receiver such as a 5072PR or 5077PR, an ultrasonic thickness gage such as a Model 35DL or 38DL PLUS, or a flaw detector with velocity measurement capability such as the EPOCH series instruments. It 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, thin, or highly attenuating samples.

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.

Procedure:

Measure the longitudinal and shear wave sound velocity of the test piece using the appropriate transducers. A Model 35DL or 38DL PLUS 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 106 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/cm³, then Young's modulus will be expressed in units of dynes/cm². If English units of in/S and lbs/in³ 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 104 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 10⁴to 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 ³to obtain modulus in dynes/cm ²or English units of in/S and lbs/in ³and 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.