In linear elastic shear, shear stress is proportional to shear strain:

$$\tau=G\gamma.$$

The constant $G$ is the shear modulus, also called the modulus of rigidity.

What it measures

A larger shear modulus means that more shear stress is required to produce the same angular distortion. Like Young's modulus, $G$ has units of stress.

Relation to Young's modulus

For an isotropic linear elastic material,

$$G=\frac{E}{2(1+\nu)},$$

where $E$ is Young's modulus and $\nu$ is Poisson's ratio.

This relation does not hold for arbitrary anisotropic materials, where different elastic constants can be independent.

Shear modulus becomes central in torsion because twisting a shaft produces shear strain and shear stress.