Shear Modulus of Rigidity (η)

The shear modulus of rigidity, denoted $\eta$ (also written $G$ or $\mu$ in some textbooks), is the modulus of elasticity that applies to shearing stress and the corresponding shearing strain -- the elastic constant governing how much a body's shape changes (with no change in volume) when a tangential force is applied to one of its faces while the opposite face is held fixed. By Hooke's law,

where $F$ is the tangential force, $A$ is the area of the face it acts on, $\Delta x$ is the resulting sideways displacement of that face relative to the fixed opposite face, $L$ is the perpendicular distance between the two faces, and $\theta$ (in radians) is the small shear angle. Like $Y$, the shear modulus has SI unit pascal ($\text{Pa}$), since shearing strain, like longitudinal strain, is a dimensionless ratio.

For essentially every common solid material, the shear modulus $\eta$ is noticeably smaller than the same material's Young's modulus $Y$ -- typically somewhere around one-third of $Y$ for ordinary metals (this ratio is not a coincidence: Section 8.9 shows it follows directly from the relation connecting $Y$, $K$, $\eta$, and $\sigma$, once a typical value of Poisson's ratio $\sigma$ is used). Physically, this reflects the fact that it generally takes noticeably less stress to change a solid's shape (shear it) than to change its length by the same fractional amount (stretch or compress it) -- shape-changing deformation involves comparatively weaker interatomic bond distortions than length-changing (bond-stretching) deformation. …