Shear Modulus Calculator

The Shear Modulus is defined as the ratio of shear stress to shear strain within the elastic limit:

$$G = \frac{\tau}{\gamma}$$

where:

• G is the Shear Modulus (Pa or GPa)
• τ (tau) is the shear stress — tangential force per unit area (Pa)
• γ (gamma) is the shear strain — the angular deformation in radians

Shear stress arises when a force acts parallel (tangential) to a surface rather than perpendicular to it:

$$\tau = \frac{F_{\text{tangential}}}{A}$$

Shear strain represents the angular distortion of the material, measured as the tangent of the deformation angle. For small angles, shear strain approximately equals the angle itself in radians:

$$\gamma = \tan(\theta) \approx \theta$$

The shear modulus relates to other elastic constants for isotropic materials. The connection to Young's Modulus (E) and Poisson's ratio (ν) is:

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

Since Poisson's ratio for most engineering materials ranges from 0.2 to 0.5, the shear modulus is typically 33–42% of Young's Modulus. For steel with E = 200 GPa and ν = 0.3, G ≈ 77 GPa. For rubber with ν ≈ 0.5 (nearly incompressible), G ≈ E/3.

Shear modulus is critical in designing shafts under torsion, bolted and riveted joints, adhesive bonds, beam webs resisting transverse loads, and seismic isolation bearings. It also determines the speed of shear waves (S-waves) in seismology:

$$v_s = \sqrt{\frac{G}{\rho}}$$

where ρ is the material density. This relationship is used in geophysics to study Earth's interior structure and in non-destructive testing of materials.

A higher shear modulus indicates greater resistance to shape change under shear loading. Steel (G ≈ 80 GPa) resists shearing far more than aluminum (G ≈ 26 GPa) or lead (G ≈ 5.6 GPa). If your result seems unusually high or low for the expected material, verify that your strain measurement represents pure shear deformation without any normal stress contribution.