Relation between Y, K, η and σ

Sections 8.5-8.8 introduced four elastic quantities for a solid material -- Young's modulus $Y$, the bulk modulus $K$, the shear modulus of rigidity $\eta$, and Poisson's ratio $\sigma$ -- treating each, so far, as though it were an independent property that has to be separately measured. For a homogeneous, isotropic elastic solid (a material whose elastic properties are the same in every direction, a good approximation for most ordinary metals, glasses, and many other engineering materials), this is not actually the case: the four quantities are linked together by the relations

WBCHSE's own syllabus explicitly states that these relations are to be used, not derived, so no derivation is given here -- what matters for this chapter is knowing the relations exist and being able to apply them.

The direct, practical consequence of these relations is that an isotropic elastic solid is fully characterised by only two independent elastic constants, not four: given any two of $Y$, $K$, $\eta$, $\sigma$ for a particular material, the remaining two can always be calculated from the relations above, without any further experiment. For instance, if $K$ and $\eta$ are both known, $Y$ follows directly from $Y = 9K\eta/(3K+\eta)$, and $\sigma$ then follows from $Y = 2\eta(1+\sigma)$, i.e. $\sigma = Y/(2\eta) - 1$ (equivalently, directly in terms of $K$ and $\eta$, $\sigma = (3K - 2\eta)/(2(3K+\eta))$ -- both forms give the same value and either can be used, whichever is more convenient for the numbers given in a particular problem).

These relations also explain a pattern already noted in Section 8.6: for ordinary metals, with $\sigma$ typically around $0.3$, the relation $Y = 2\eta(1+\sigma)$ gives $\eta \approx Y/2.6 \approx 0.38\,Y$ -- i.e. $\eta$ comes out to be somewhere around a third to two-fifths of $Y$, exactly the rough rule of thumb mentioned earlier, and now seen to follow directly from the interlinking relation rather than being a separate coincidence.

The table below gives approximate values of $Y$, $K$, and $\eta$ for a few common materials, useful as a quick reference and for sanity-checking a computed value in a numerical problem against the real material it is supposed to represent: