Application:
Measurement on Young's Modulus and Shear Modulus of Elasticity, and Poisson's ratio, in nondispersive isotropic engineering materials.
Problem:
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 quickly and easily through computations based on sound velocities. Sound velocity can be easily measured using ultrasonic pulse-echo techniques. The procedure outlined below is valid for any nondispersive material and 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. Plastics and composites are generally dispersive, however, useful information may be obtained by ultrasonic measurements as long as the dispersive nature of the material is recognized when interpreting the results.
Equipment:
The technique requires an ultrasonic pulser-receiver such as a 5072PR, 5077PR, or 5800PR, an ultrasonic thickness gage such as a Model 35DL, 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. Generally, a V109 or M109 5MHz broadband longitudinal wave transducer and a V154 2.25MHz normal incidence shear wave
transducer will work for typical metal and fired ceramic samples. Different transducers will be required for very thick, thin, or highly attenuating samples.
The test samples may be of any geometry that permits clean pulse-echo measurement of sound transit time through a section of known thickness. Ideally this would be a slug or bar at least 0.5" (12.5 mm) thick with smooth parallel surfaces. The diameter or cross-section must exceed about five longitudinal wavelengths. In steel this means D>.25" (6mm) at 5MHz.
Procedure:
Measure the longitudinal and shear wave sound velocity of the test piece. If using a thickness gage or flaw detector, follow the instrument's recommended procedure for velocity measurement. 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:
Velocity = Thickness ÷ 1/2 Round-Trip
Transit Time
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.
Poisson's Ratio
= 1 - 2(V
T ÷ V
L)
2 ÷ 2 - 2(V
T ÷ V
L)
2
Where
VT = Shear (transverse) Velocity
VL = Longitudinal Velocity
Example for typical 304 stainless steel:
VT = 1.2 x 10
5 in/S or 3.1 x 10
5 cm/S
VL = 2.3 x 10
5 in/S or 5.4 x 10
5 cm/S
VT / VL = 0.522
Poisson's Ratio
= 1 - 2(.522)
2 ÷ 2 - 2(.522)
2
= .456 ÷ 1.456
= .313
Young's Modulus
= V
L
2r (1 + d) (1 - 2 d) ÷ 1 - d
Where
VL = Longitudinal Velocity
r = Density
d = Poisson's Ratio
If 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 a modulus expressed 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, it is necessary to multiply the solution of the above equation by a mass/force conversion constant.
1 ÷ Acceleration of Gravity
= 1 ÷ 32.174 ft/S
2
= 1 ÷ 386 in/S
2
or if the initial calculation is done in metric units, use the conversion factor.
1 psi = 6.89 x 104 dynes/cm
2
or as another alternative, 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.
Examples for typical 304 stainless steel:
Young's Modulus
= (5.9 x l0
5cm/S)
2 (7.9 g/cm
3)
[(1 + .313) (1 - .616) ÷ (1 - .313)]
= (3.48 x 1011 x cm
2 ÷ S2) (7.9 g/cm
3) (.733)
= 2.0 x 10
12 dynes/cm2
Then converting to PSI (2.0 x 10
12 dynes/cm2) x (1 psi ÷ 6.89 x 10
4 dynes/cm2)
= 29 x l0
6 psi
or, alternately doing the computation in English units:
Young's Modulus
= (2.3 x l05in/S)
2 (.29 lb/in
3) [(1 + .313) (1 - .616) ÷ (1 - .313)] [1 ÷ 386 in/S]
= (5.29 x 1010 in2/S
2) (.29 lb/in
3) (.733) [1 ÷ 386 in/S]
= 29 x l06 psi
Shear Modulus = V
T
2r
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.
Example for 304 stainless steel:
Shear Modulus
= (3.1 x 10
5 cm/S)2 (7.9 g/cm3)
= 7.6 x 10
11 dynes/cm
2
or
(1.2 x 10
5 in/S)
2 (.29 lb/in
3)
[1 ÷ 386 in/S
2]
= 10.8 x 10
6 psi
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.
3. Lynnworth, L. C., E. P. Papadakis, K. A. Fowler, "Ultrasound Propagation Measurements and Applications," International Advances in Nondestructive Testing, 1977, Vol. 5, pp.71-115.
4. Lynnworth, L. C., "Acoustical Nomograms for the Elastic Properties of Engineering Materials, " Ultrasonics, October 1969, pp. 254-256.