NDTnet - March 1996, Vol.1 No.03
Chapter II
By virtue of symmetry, the relationship of the ultrasonic velocities to the tangent modulus is found for isotropic homogeneous materials from a simplification of the 36 possible elastic constants derived for anisotropic materials. The formulae for isotropic homogeneous materials is shown below. The longitudinal velocity is given by
when the wavelength of the propagating ultrasound is greater than five times the diameter of the rod or bar. The effective stiffness modulus YL of such a constrained ¹ section will be higher than the Young's modulus Yo for a free bar. where the formula for Yo takes into account the restraining forces for CL : ,with the limit of b = 1 when n = 0.5. The stiffness modulus is a complex number with the imaginary part corresponding to the loss factor. When the loss factor is small, it is usually ignored and the stiffness modulus is reported as a scaler number.
Eq.(4)
where K = bulk modulus, G = shear modulus, and
Eq.(5)
¹ The propagating ultrasound through the length of a bar will cause the lateral dimensions to change, dynamically, with respect to the position of the pressure wave within the bar. A material with infinite lateral dimension does not constrain the lateral edges of the longitudinal stress wave.
where n = Poisson's Ratio. When n is 1/2, CL is indeterminate.
However, Young's modulus is equivalent to
Now, by substituting Eq. (6) into Eq. (5), a condition may be established that will hold for an ideal, simple fluid, i.e., a fluid with no viscosity. First let
3(1 - 2n)(1 - n)/[(1 + n)(1 - 2n)] = 1 Eq.(7)
The solution of Eq. (7) yields
Therefore, since n = 1/2, the material meets the fluid requirement, and the substitution for velocity can be completed for the perfect fluid condition
CF = ( K / )¹/² Eq.(8)
which describes the velocity of sound through many liquids and gases. The shear velocity is found in materials that sustain shearing forces, such as most solids and many viscous fluids. The shear velocity of sound is given by
Cs = ( G / )¹/² Eq.(9)
Now, one may mistakenly combine Eq. (8) and (9) and substitute the result back into Eq. (4), which would then yield
Eq.(10)
However, the earlier requirement placed upon the validity of Eq. (8) will be violated if the validity requirement for Eq. (9) remains extant, that is, a material may not be a perfect fluid and sustain shearing forces. It is left to the reader to conjecture whether Eq. (10) could prevail immediately after the detonation of a high explosive into a solid material, causing that material to become totally fluid for an instant. It is not likely, however, that the material could sustain shearing forces during that instant in time. There is another longitudinal sonic velocity that is encountered in materials capable of sustaining a shearing force. This longitudinal velocity is called the extensional, bar, or thin rod velocity. Rods, bars, or wires that are subjected to a compressional wave with a wavelength greater than five times the cross sectional dimension of the rod permits the wave to propagate at a reduced velocity compared with the bulk specimen longitudinal wave. This velocity, Co , is related to Young's modulus by Eq. (11).
Co = ( E / )¹/² Eq.(11)
and is either calculated from CL and Cs or measured from specially designed specimens and transducers. By solving Eq. (9) for the shear modulus,
G = C²s Eq.(12)
and by substituting this equation into Eq. (4), the bulk modulus is calculated:
[ K + (4/3) C²s ]/ Eq.(13)
and
K =[ C²L - (4/3) C²s ]
The relationship of Young's modulus to the bulk and shear modulus is
E = 9 KG/(G + 3 K) Eq.(14)
If Eq. (13) is substituted for K, and Eq. (12) is substituted for G in Eq. (14), Young's modulus becomes
E = C²s (3 C²L - 4 C²s )/( C²L - C²s ) Eq.(15)
and by substituting Eq. (15) into Eq. (11), the extensional velocity becomes
The dependence of the extensional velocity on a shear velocity is apparent and is consistant with the concept that materials that cannot support shearing forces, such as fluids with zero viscosity, cannot propagate an extensional velocity. The extensional velocity may be visualized as a compressional pressure wave that exhibits a maniscus effect on the specimen walls as the wave propagates along the rod axis.
It is axiomatic that the elastic constants are all interrelated. Therefore, once the isotropic moduli are established, Poisson's Ratio may be calculated. Poisson's Ratio is also a function of the ratio of the longitudinal and shear velocities,
v = [ 1 - 2 ( Cs/CL)²]/[ 2- 2 ( Cs/CL)²] Eq.(17)
and a set of curves is shown in the Nondestructive Testing Handbook ²¹ , depicting the ratio of any two of four ultrasonic velocities as a function of Poisson's Ratio.
There are two Lame elastic constants calculated from the ultrasonic-velocities. One of theLame constants has been given as the shear modulus. The other is sometimes listed as the Lame' modulus, . The Lame' modulus ¹° is a governing factor for both the longitudinal and shear velocities:
C²L =( + 2 G )/ Eq.(18)
and by rearranging the formula, the Lame' modulus is
=(C²L - 2 C²s ) Eq.(19)
The Lame modulus is perhaps of interest to those who work with plastic materials. As Poisson's Ratio increases, the Lame modulus will numerically approach the bulk modulus. The shear modulus will disappear as the viscosity of the fluid approaches zero. This may be observed from the recognition of the equation for bulk modulus: K = + [2 ยต/3 ], where shear approaches zero for fluids and, simultaneously, Poisson's ratio approaches 0.5. The compressional velocity for fluids is proportional to the bulk modulus as shown in equation (8)
The thermal loss mechanisms (temperature dependence) of materials is most suitably described in terms of the Debye temperature, . The Debye temperature may be calculated with the help of the infinite medium, shear, and longitudinal velocities. The mean integrated velocity 5 is calculated for the isotropic material and then fed into an equation containing Planck's constant, Boltzmann's constant, Avogadro's number, the mean atomic weight, and the density of the material. The mean integrated velocity, CM, is given by
CM = [ 3 ( Cs CL)³ / ( 2 C³L + C³s)]¹/³ Eq.(20)
and the Debye temperature, , is
(Degrees Kelvin)
Where:
h/k = Planck's constant / Boltzmann's constant = 47.99216
°K s [1/10¹²],
P = Number of atoms per molecule,
N = Avogadro's number (602.2167 x 10 ²¹ x [1/kmol])
M = Mean atomic weight.
Ultrasonic velocity measurements in material testing do not allow sufficient time for thermal diffusion. Therefore, Young's modulus, as calculated from the ultrasonic velocities, is the adiabatic modulus, which is slightly higher than the isothermal modulus. The relationship between the adiabatic and isothermal Young's modulus is given by:
EA/Ei = 1 + a²T(EA/ CP) Eq.(22)
Ei/EA/ [ 1 + {a²T(EA/ CP)}]
EA = Ei CP/ ( CP - Ei a² T )
Where:
a = coefficient of thermal expansion, m/m/ °K
Cp = specific heat at constant stress, J/kg °K
T = absolute temperature
r = density
EA = adiabatic Young's modulus
Ei = isothermal Young's modulus
Rolf Diederichs 22.Nov.1995, info@ndt.net