Hooke's law in component form

 We start with Hooke's law for the strain tensor, and then substitute the E and nu material properties for K and mu.

The result is

We see that there is still a compression term, now on the right, but not a pure shear term, and both terms contain both material constants.  

We mention here that  nu is between 0.2 and 0.3 for many materials.  Applying this range of values to the expressions for the bulk and shear moduli above, we see that K, the bulk modulus value is about 60 to 80 percent of the Young's modulus, and mu, the shear modulus is about is about 40 % of the Young's modulus.  The bulk modulus (of compression) is therefore about 1.5 to 2 times larger than the shear modulus.  

Since K and mu are in the denominators of the compression and shear terms of the strain tensor, respectively, and in the numerators of the corresponding terms of the stress tensor, we see that most materials are more resistant to compressive forces than to shear.  This effect is magnified further since the denominator for compression (expansion) is 9K, whereas the denominator for pure shear is only 2 mu.

As I may have mentioned earlier, this seems reasonable from a microscopic point of view, that it would be harder to push atoms or molecules closer together, or farther apart, than it would be to move them around without changing the volume of the body.

We can now write Hooke's law in a convenient component form:

The shear terms at the bottom are analogous to the shear terms with the K, G (or K, mu) formulation, with (1 + nu)/E replaced by 1/(2 mu) or 1/(2G).  Recall that the shear modulus is designated by either G or mu.  The numerical results should be the same in either formulation, of course.

You could say that with the appropriate use of Poisson's ratio (using the formulas given in the last blog), we can transform Young's modulus into either the bulk or shear modulus.  One nice thing about the Young's modulus/Poisson's ratio formulation of Hooke's law is that it makes the role of the material properties so clear:  Young's modulus for compression/expansion, and Poisson's ratio for transverse compression/longitudinal extension, or vice versa.

You can write these component equations with the bulk and shear moduli, but the coefficients of each term are combinations of the two constants, so you don't see their role so well.  Of course, if you keep the pure compression and shear terms separate, then the moduli are also separated, so that's another way to look at it, I guess.  Also, you can always write the 1/E and sigma/E coefficients in terms of K and mu.  Wikipedia has a  conversion table for elastic constants at http://en.wikipedia.org/wiki/Elastic_modulus.

The equations for the normal strains in the above set display clearly that there can be strain components perpendicular to the  stress components, even in the absence of  of the corresponding stress.  I think this is another indication of why you need tensors to describe elastic behavior.

It would be hard to imagine a vector formulation that could describe this behavior, because I think in the end one would have to match up x-components of a vector strain with x-components of vector stress, etc., and there would be no way to relate, say, x-components of one with y-components of the other.