# Poisson's Ratio and Ultrasonic Velocity for Isotropic Material

1. Dec 4, 2006

### dsanyal

For an isotropic material, the relation between the longitudinal ultrasonic
velocity(VL), the transverse (shear) ultrasonic velocity(VT) and the Poisson's ratio (nu) is given by

(VT/VL)^2 = (1-2*nu)/(2*(1-nu))

From the above relation, one gets that VL=0 when nu=1 which is
not physically acceptable as nu varies between -1 and 0.5 for an
isotropic material. On the contrary when nu=0, VT/VL=sqrt(0.5).
However, when nu=0, what happens to the longitudinal ultrasonic
velocity. Does VL becomes zero when nu beccomes zero ?

2. Dec 4, 2006

### AlephZero

"Does VL becomes zero when nu beccomes zero?"

No. Why did you think it might be zero?

3. Dec 5, 2006

### dsanyal

Poisson's ratio (nu) zero means that there will be no lateral deformation of
the material. In that case, if VL exists, the material will also have a
longitudinal elastic modulus. Under extension, the volume of the material
will tend to increase without any lateral contraction. This can only happen
if simultaneously density change occurs ? Is it feasible.

4. Dec 5, 2006

### AlephZero

The transverse wave is a shear wave. The volume of material doesn't change in shear, for any value of nu. The shear modulus G = E/2(1+nu) is not zero when nu is zero.

Velocity of VL = sqrt(E/rho), velocity of VT = sqrt(G/rho).

There are materials where (in some specific coordinate system) E is non zero and G is zero, but they are not isotropic. A piece of woven cloth is one example. G = 0 in a coordinate system lined up with the fibres of the cloth.

If you think of these waves as stress waves, not "displacement" waves, the names make more sense. For the VL wave, there is a lateral displacement (expansion and contraction) when nu is non-zero, but there is no lateral stress. For the VT wave there is transverse shear stress, but no longtitudinal stress.

5. Dec 5, 2006

### dsanyal

Thanks. I am still confused.
For an isotropic material, when nu=0,
if E and G both are non-zero, then under
and the material volume should increase
as it cannot have lateral contraction.
This can happen only when if the density
decreases. Is this physcially tenable ?
If yes, then decreasing density should
imply increase in VL, otherwise elastic
instability will occur.

6. Dec 5, 2006

### AlephZero

The conventional way to define density for solids undergoing small strains is relative to the unstrained condition of the material - i.e. the density is (by definition) always constant, even for problems involving temperature change and thermal expansion.

The relevance of this definition is that when you set up the equations of motion you are considering the motion of a fixed piece of material (e.g a small box size dx.dy.dz) and the mass of that material is constant. This method is called the Lagrangian formulation of the equations.

You can indeed formulate the equations of motion by considering a fixed region in space instead of a fixed piece of material. Then, material with varying density moves in and out of that region during the motion. This is called the Euleran formulation and it's often used in fluid mechanics, but rarely in solid mechanics.

The results of both formulations are identical because they both use the same Newtonian mechanics, the same constitutive equations for the material, etc. There is no advantage in using an Euleran formulation for solid mechanics unless you have a problem involving large strains and/or displacements (e.g. plastic deformation of ductile materials).

To summarize, it's possible to formulate the equations of motion correctly with your "variable density" definition, but you would then have a different form of wave equation, and its physical solution would still be the same as the Lagrangian equations.

Finally, there are things called ALE formulations (Adaptive Lagrangian-Euleran) which combine both approaches - try a literature search for keywords ALE, DYNA3D (a computer program), and John Hallquist (its author) for more on those.

Hope this helps.

Last edited: Dec 5, 2006