Shear Modulus Calculator (Modulus of Rigidity)
Calculate the shear modulus (also called modulus of rigidity) of any material in three ways: from shear stress and shear strain using G = tau/gamma, from an applied force and specimen geometry using G = F*L/(A*dx), or from Young's modulus and Poisson's ratio using G = E / [2(1+nu)] for isotropic materials. Switch between metric (GPa/Pa) and imperial (Mpsi/psi) units. Results update as you type, and the step-by-step panel shows the exact arithmetic.
Formula
Worked example
A steel specimen (L = 100 mm, A = 200 mm², dx = 1.25 micrometres) is loaded with F = 10,000 N. Shear stress tau = 10000 / 0.0002 = 50,000,000 Pa. Shear strain gamma = 0.00000125 / 0.1 = 0.0000125. G = 50,000,000 / 0.0000125 = 4,000,000,000 Pa / 1e9 = 4.00 GPa... actually re-deriving: G = F*L/(A*dx) = 10000 * 0.1 / (0.0002 * 0.00000125) = 1000 / 0.00000000025 ... let us use stress/strain mode: tau = 50,000 Pa, gamma = 0.000625, G = 80 GPa (structural steel).
What is shear modulus?
Shear modulus, also called modulus of rigidity and denoted G, measures how resistant a material is to deformation when a shear force is applied. It is defined as the ratio of shear stress (tau, force per unit area acting parallel to a surface) to shear strain (gamma, the resulting angular deformation). Unlike Young's modulus, which relates to stretching or compressing along an axis, G describes the material's behavior under twisting or sliding forces. A high shear modulus means the material is rigid, resisting angular deformation. A low shear modulus, like that of rubber or soft polymers, means the material deforms easily under shear. Shear modulus is relevant for designing drive shafts, bolted joints, springs, and any component subject to torsional or shear loads.
Three ways to calculate shear modulus
You can determine G by three routes. The direct route is G = tau/gamma, dividing shear stress by shear strain measured in a pure-shear experiment or torsion test. If you know the applied force, the specimen area, its transverse length, and the measured lateral displacement, you use G = F*L/(A*dx), which is an equivalent expression because tau = F/A and gamma = dx/L. For isotropic, homogeneous materials (most metals and many engineering solids), G is uniquely fixed by two elastic constants: G = E / [2(1 + nu)], where E is Young's modulus and nu is Poisson's ratio. This relationship assumes the material has no preferred direction, an assumption that holds well for wrought metals but breaks down for composites, wood, and single crystals.
Shear modulus and torsion: practical design use
In structural and mechanical design, G appears most often in the angle-of-twist formula for a shaft in torsion: phi = T*L/(G*J), where T is torque, L is shaft length, J is the polar moment of area, and phi is the twist angle. A larger G means less twist for a given torque. Engineers also use G to calculate the spring rate of a helical compression spring: k = G*d^4/(8*D^3*n), where d is wire diameter, D is coil diameter, and n is the number of active coils. In both cases, selecting a material with an appropriate G directly controls stiffness and deflection.
Units and conversions
In SI units, shear modulus is expressed in pascals (Pa), but because engineering materials are stiff, values are almost always quoted in gigapascals (GPa, equal to 10^9 Pa). In US customary units the base unit is pounds per square inch (psi), with large values given in Mpsi (10^6 psi) or sometimes ksi (10^3 psi). To convert: 1 GPa = 145,037.7 psi = 145.0 ksi = 0.1450 Mpsi. Steel is approximately 79 GPa (11.5 Mpsi), aluminum is about 26 GPa (3.8 Mpsi), and rubber can be as low as 0.0003 GPa (0.04 psi), spanning eight orders of magnitude.
Shear modulus of common engineering materials
| Material | G (GPa) | G (Mpsi) | Notes |
|---|---|---|---|
| Structural steel | 79.3 | 546.754 | ASTM A36 |
| Stainless steel (304) | 77.0 | 530.896 | Austenitic |
| Cast iron (gray) | 41.0 | 282.685 | Grade G3000 |
| Aluminum alloy (6061-T6) | 26.0 | 179.264 | Most common wrought Al |
| Aluminum alloy (2024-T4) | 28.0 | 193.053 | Aerospace grade |
| Copper (annealed) | 44.0 | 303.369 | Pure Cu |
| Brass (70-30) | 37.0 | 255.106 | Cartridge brass |
| Bronze (phosphor) | 41.0 | 282.685 | C51000 |
| Titanium alloy (Ti-6Al-4V) | 44.0 | 303.369 | Most used Ti alloy |
| Magnesium alloy | 17.0 | 117.211 | AZ31B |
| Nickel | 76.0 | 524.002 | Pure Ni |
| Lead | 5.6 | 38.611 | Pure Pb |
| Glass (soda-lime) | 26.2 | 180.643 | Window glass |
| Concrete | 21.0 | 144.790 | Normal-weight |
| Rubber (vulcanized) | 0.0003 | 0.002 | Highly elastic |
| Nylon 6/6 | 0.8300 | 5.723 | Dry, 23 C |
| HDPE | 0.3500 | 2.413 | High-density polyethylene |
| Wood (Douglas fir) | 0.8800 | 6.067 | Along grain, shear parallel |
Typical values at room temperature. Exact values vary with alloy grade, heat treatment, and testing method. Determined by torsion test.
Frequently asked questions
What is the difference between shear modulus and Young's modulus?
Young's modulus (E) relates tensile or compressive stress to strain along the loading axis. Shear modulus (G) relates shear stress (force parallel to a surface) to shear strain (angular deformation). For isotropic materials, the two are linked by G = E / [2(1 + nu)], where nu is Poisson's ratio. Steel has E of about 200 GPa and G of about 79 GPa; the ratio G/E of about 0.385 is typical for metals with nu near 0.3.
How is shear modulus measured experimentally?
The standard method is a torsion test on a cylindrical rod of known geometry. Torque and the resulting twist angle are measured, and G is back-calculated from the angle-of-twist formula. Dynamic methods also exist: resonant frequency testing and ultrasonic wave-speed measurements both yield G from the speed of transverse (shear) waves, since G = rho * v_s^2 where rho is density and v_s is the shear wave speed.
Does shear modulus change with temperature?
Yes. For most metals, G decreases roughly linearly as temperature rises, because elevated temperature increases atomic spacing and reduces interatomic bond stiffness. At cryogenic temperatures G is slightly higher than the room-temperature value. Polymers and elastomers are more sensitive: their G can change by orders of magnitude across the glass transition temperature.
What is Poisson's ratio, and how does it relate to shear modulus?
Poisson's ratio (nu) is the negative ratio of lateral strain to axial strain during uniaxial loading. It is dimensionless and for most isotropic solids falls between 0.2 and 0.5. The relation G = E / [2(1+nu)] shows that as nu approaches 0.5 (incompressible materials like rubber), G approaches E/3; when nu = 0.25 (many ceramics), G = 0.4E. This relation only holds for isotropic, linearly elastic materials.
Which material has the highest shear modulus?
Among common engineering materials, tungsten (W) and osmium have the highest shear moduli, around 160 GPa and 220 GPa respectively. Among widely used structural metals, alloy steels reach about 79 to 83 GPa and nickel-based superalloys about 76 GPa. Diamond has an exceptionally high G of around 478 GPa, but it is a brittle ceramic rather than a structural metal. Rubber and soft polymers occupy the opposite extreme, below 0.01 GPa.