= Solution
An isotropic linear elastic solid has two independent elastic constants. For infinitesimal <strain>,
$$
\sigma_{ij}=2G\epsilon_{ij}+\lambda\epsilon_{kk}\delta_{ij},
\qquad K=\lambda+\frac23G.
$$
The <shear modulus> $G$ governs shape change and the <bulk modulus> $K$ governs volume change. <Young's modulus> $E$ and <Poisson's ratio> $\nu$ are equivalent parameters:
$$
G=\frac{E}{2(1+\nu)},\qquad K=\frac{E}{3(1-2\nu)}.
$$
Positive <strain> <energy> requires $G,K>0$, hence $-1<\nu<1/2$; $\nu\to1/2$ is the incompressible limit. Small-amplitude bulk waves have speeds $c_P=\sqrt{(K+4G/3)/\rho}$ and $c_S=\sqrt{G/\rho}$. \b[Isotropy removes directional dependence, not the distinction between compression and shear.] These laws apply within the elastic small-strain regime, not after yielding or fracture.
Back to article page
