Hooke's Law

In order to combine the shear and compression terms into one expression for Hooke's law, Landau and Lifshitz make use of the Kronecker delta symbol.  For the compression term, this is straightforward:

If i = j, the delta symbol is equal to one, and we have the expression given for compression in the previous blog. For the shear term, we need to subtract the trace components from the strain tensor, leaving only the off-diagonal shear components:

Note the reason for the factor 1/3:  When i = j, the terms with repeated indices are summed, so that the delta symbol is equal to 3.  As a result, the trace of the stress tensor vanishes in the shear term.

Finally, the formula for the stress tensor in terms of the strain, for Hooke's law (in Cartesian coordinates) is written

This covers all the possibilities.  However, before we write this out in component form, we will obtain the converse formula, for the strain tensor in terms of the stress.  The latter expression is more easily understood, and allows us to define a different pair of material constants which are often used in practice.