Determination elastic constants wit ultrasonic in bulk metallic glasses

[noreau]

HAllo everyone,
I had a problem with the measurement of young modulus and shear modulus with ultrasonic measurements( pulse-echo overlap technique)

For my metallic glasses rods (amorphous material) I used these formula for the calculation:

1. determination Shear modulus: G= ro * v trans^2
where ro= density of material and v trans = transversale vawe packet velocity= 2*D * nu trans
where D= height of the rods and nu trans= the found transversale frequency.

2 for Young modulus the formula that I found was quite complex and say:
first I calculated the longitudinale velocity with the same formula as for the transversale:
v long = 2*D*nu long --> and here I'm not sure of this formula !

Then I used the following two equations:
nu (= poisson ratio) = E/2*G -1 and v long = E(1- nu)/ro(1-nu-2*nu^2)
to finally come to the following formula for the young modulus:
E= ((2-3*v long ^2 * ro)/2*G)/((-v long ^2*ro/2*G^2) + 1/(2*G)

Is it this formula correct? or maybe have you another for amorphous rods?!

Thank in advance

Noreau

 

[Enthalpy]

Longitudinal waves, still use
V²=E'/ρ
but if the rod's radius is small versus the (quarter) wavelength, E'=E
andif the rod is broad versus the wavelength, E'=E/[1-2μ²]
where I misuse µ as Poisson't coefficient.