Measurement of Elastic Coefficients

For single crystals, the elastic coefficients may be determined by acoustic velocity measurements. For example, it may be shown that the velocity of a longitudinal wave propagating in the [100] direction is given by 

The velocity of a longitudinal wave moving in the [110] direction can be shown to be given by 

If the material is isotropic, the velocity in the [100] direction will be the same as in the [110] direction hence 

This is not true for single crystals because they are generally anisotropic, i.e., the velocity of propagation is not the same in the [100] as in the [110] directions. However, polycrystalline materials and glasses are isotropic because of the random nature (or absence) of their grain structure. Therefore, the above relationship holds for bulk polycrystalline solids. This allows a number of relationships between B, E, G, and v to be derived. Using the relation between the Poisson ratio and Cn and C12 to eliminate C12, the bulk modulus may be related to Young s modulus by 

Using Equation 7.12 to relate C44 to Cn and C12 for isotropic materials, we can write 

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