shearmodulus

The shear modulus, denoted G, is a material property that describes how resistant a material is to shear deformation in the linear elastic regime. It is also known as the modulus of rigidity and is defined as the ratio of shear stress to shear strain for small deformations.

In an isotropic, linear-elastic material, shear stress τ and shear strain γ are related by τ = G γ. For such materials, G is related to the more familiar Young’s modulus E and Poisson’s ratio ν through E = 2G(1+ν), or equivalently G = E/[2(1+ν)]. Because ν typically lies between 0 and 0.5 for most stable materials, G is generally less than E and, for incompressible materials with ν approaching 0.5, G ≈ E/3.

Measurement and frequency considerations: G can be determined from torsion tests on cylinders or from dynamic mechanical analysis that yields the storage modulus G’ and the loss modulus G’’, which describe elastic and viscous responses, respectively. In static, purely elastic solids, G is a constant value; in liquids or soft viscoelastic materials, the apparent shear modulus is frequency-dependent and can be described by a complex modulus G*(ω) with G’(ω) and G’’(ω).

Anisotropy and crystals: In isotropic materials a single shear modulus suffices, but many materials are anisotropic and exhibit direction-dependent shear responses. Such materials may have several independent shear moduli determined by their crystal symmetry or microstructure.

Typical values and notes: Metals commonly have high G, on the order of tens of gigapascals (for example, steel around 70–80 GPa, aluminum about 26 GPa). Polymers and rubbers span a wide range, from tens of megapascals to a few gigapascals; rubbers typically exhibit G in the MPa range. Temperature and strain rate can significantly affect G.