the shear modulus

Push the top of a thick deck of cards sideways while the bottom stays put and the deck tilts into a slanted parallelogram — the layers slide over one another without the deck getting longer. That sideways, shape-twisting kind of deformation is shear, and the shear modulus measures a material's resistance to it: how stiff it is against being skewed or twisted rather than stretched.

It is defined as G = shear stress / shear strain, where shear stress tau is a force acting parallel to a surface divided by that area, and shear strain is the tilt angle (in radians). For an isotropic material it is tied to Young's modulus and Poisson's ratio by G = E / (2 x (1 + nu)); with nu about 0.3, that makes G roughly 0.38 x E. For steel with E = 200 GPa, G is about 80 GPa. Like Young's modulus it is set by bond stiffness and quoted in GPa.

The shear modulus is the property that governs twisting, so it runs the design of anything that carries a torque: drive shafts, torsion bars, springs, and bolts. A coil spring, for instance, works almost entirely by twisting its wire, so its stiffness depends on G, not on E. Ceramics and metals have high shear moduli and resist twisting stiffly; polymers and especially rubbers have low ones, which is exactly why a rubber bushing can soak up twisting vibration.