# Shear Modulus Formula and Definition Shear modulus is the shear stiffness of a material, which is the ratio of shear stress to shear strain.

By definition, the shear modulus is shear stiffness of a material, which is the ratio of shear stress to shear strain. Another name for the shear modulus is the modulus of rigidity. The most common symbol for the shear modulus is the capital letter G. Other symbols are S or μ.

• A material with a high shear modulus is a rigid solid. It takes a large force to cause deformation.
• A material with a low shear modulus is a soft solid. It deforms with very little force.
• One definition of a fluid is that it is a substance with a shear modulus of zero. Any force causes deformation. So, the shear modulus of a liquid or a gas is zero.

### Shear Modulus Units

The SI unit of the shear modulus is the pressure unit pascal (Pa). However, the pascal is newtons per square meter (N/m2), so this unit is also in use. Other common units are the gigapascal (GPa), pounds per square inch (psi), and kilopounds per square inch (ksi).

### Shear Modulus Formula

The shear modulus formula takes different forms:

G = τxy / γxy = F/A / Δx/l = Fl / AΔx

• G is the shear modulus or modulus of rigidity
• τxy or F/A is the shear stress
• γxy is the shear strain
• Shear strain is Δx/l = tan θ or sometimes = θ
• θ is the angle formed by the deformation from the applied force
• A is the area over which the force acts
• Δx is the transverse displacement
• l is the initial length

### Example Shear Stress Calculation

For example, find the shear modulus of a sample that is under a stress of 4×104 N/m2 and experiencing a strain of 5×10-2.

G = τ / γ = (4×104 N/m2) / (5×10-2) = 8×105 N/m2 or 8×105 Pa = 800 KPa

### Isotropic and Anisotropic Materials

Materials are either isotropic or anisotropic with respect to shear. The deformation of an isotropic material is the same no matter what its orientation is with respect to an applied force. In contrast, the stress or strain of an anisotropic material depends on its orientation.

Many common materials are anisotropic. For example, a diamond crystal (which has a cubic crystal) shears much more readily when the force aligns with the crystal lattice. A square block of wood responds to a force differently, depending on whether you apply the force parallel to the wood grain or perpendicular to it. Examples of isotropic materials include glass and metals.

### Dependence on Temperature and Pressure

Temperature and pressure affect the way a material responds to an applied force. Usually, increasing the temperature or decreasing the pressure lowers rigidity and the shear modulus. For example, heating most metals makes them easier to work, while chilling them increases brittleness.

Other factors that influence the shear modulus include melting point and vacancy formation energy.

The Mechanical Threshold Stress (MTS) plastic flow model, Nadal and LePoac (NP) shear stress model, and Steinberg-Cochran, Guinan (SCG) shear stress model all predict the effects of temperature and pressure on shear stress. These models help scientists and engineers predict the temperature and pressure range over which the change in shear stress is linear.

### Table of Shear Modulus Values

The shear modulus value for a material depends on its temperature and pressure. Here is a table of shear modulus values for representative substances at room temperature. Note low shear modulus values describe soft and flexible materials, while hard, stiff substances have high shear modulus values. For example, transition metals, their alloys, and diamond have high shear modulus values. Rubber and some plastics have low values.

### Shear Modulus, Young’s Modulus, and Bulk Modulus

The shear modulus, Young’s modulus, and the bulk modulus each describe the elasticity or rigidity of a material, according to Hooke’s law. Young’s modulus measures the stiffness or linear resistance of a solid to deformation. The bulk modulus is a measure of a material’s resistance to compression. Each elasticity modulus relates to the other via equations:

2G(1+υ) = E = 3K(1−2υ)

• G is the shear modulus
• E is the Young’s Modulus
• K is the Bulk Modulus
• υ is Poisson’s Ratio

### References

• Crandall, Stephen; Lardner, Thomas (1999). Introduction to the Mechanics of Solids (2nd ed). McGraw-Hill. ISBN: 978-0072380415.
• Guinan, M.; Steinberg, D. (1974). “Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements”. Journal of Physics and Chemistry of Solids. 35 (11): 1501. doi:10.1016/S0022-3697(74)80278-7
• Landau, L.D.; Pitaevskii, L.P.; Kosevich, A.M.; Lifshitz, E.M. (1970). Theory of Elasticity (3rd ed.). vol. 7. Oxford:Pergamon. ISBN: 978-0750626330.
• Nadal, Marie-Hélène; Le Poac, Philippe (2003). “Continuous model for the shear modulus as a function of pressure and temperature up to the melting point: Analysis and ultrasonic validation”. Journal of Applied Physics. 93 (5): 2472. doi:10.1063/1.1539913
• Varshni, Y. (1981). “Temperature Dependence of the Elastic Constants”. Physical Review B. 2 (10): 3952.