In the last blog, we saw that two material constants are required to describe elastic behavior in solids - the shear modulus G and the modulus of compression K, which is also called the bulk modulus.  (Timoshenko and Goodier use the letter G for the shear modulus, while Landau and Lifshitz use the Greek letter mu).

Apparently, materials may have a different response to compressive forces  than they do to shear forces.  This seems plausible from a microscopic viewpoint, that squeezing atoms or molecules closer together would require more force than moving them around without changing the volume.  However, we will see later that for many materials, K and G have about the same value.

Continuing from the previous blog, we find an expression for the strain tensor in terms of stress, following Landau and Lifshitz.  First we find the  trace of the stress tensor in terms of strain from the formula for Hooke's law obtained last time:

Setting i = j in this formula, we find the parenthetical expression vanishes (the trace of the Kronecker delta is equal to 3), and the trace of the stress tensor is 3K times the trace of the strain tensor.  Inverting this and substituting in Hooke's law,

Solving for the strain tensor gives the converse form of Hooke's law:

We observe that the trace term contains only the bulk modulus, and the shear term contains only the shear modulus, in both versions of Hooke's law, as they should.  There is another way to look at Hooke's law, though, and to find it we'll use this last formulation to revisit the stretched rubber tube problem, discussed earlier (in the March 13 blog), which can also be found in Chapter 1, Section 5 in Landau and Lifshitz, Theory of Elasticity.