What Is the Shear Modulus?

The shear modulus is defined as the ratio of shear stress to shear strain. It is also known as the modulus of rigidity and may be denoted by G or less commonly by S or μ. The SI unit of shear modulus is the Pascal (Pa), but values are usually expressed in gigapascals (GPa). In English units, shear modulus is given in terms of pounds per square inch (PSI) or kilo (thousands) pounds per square in (ksi).

• A large shear modulus value indicates a solid is highly rigid. In other words, a large force is required to produce deformation.
• A small shear modulus value indicates a solid is soft or flexible. Little force is needed to deform it.
• One definition of a fluid is a substance with a shear modulus of zero. Any force deforms its surface.

Shear Modulus Equation

The shear modulus is determined by measuring the deformation of a solid from applying a force parallel to one surface of a solid, while an opposing force acts on its opposite surface and holds the solid in place. Think of shear as pushing against one side of a block, with friction as the opposing force. Another example would be attempting to cut wire or hair with dull scissors.

The equation for the shear modulus is:

G = τ_xy / γ_xy = F/A / Δx/l = Fl / AΔx

Where:

• G is the shear modulus or modulus of rigidity
• τ_xy is the shear stress
• γ_xy is the shear strain
• A is the area over which the force acts
• Δx is the transverse displacement
• l is the initial length

Shear strain is Δx/l = tan θ or sometimes = θ, where θ is the angle formed by the deformation produced by the applied force.

Example Calculation

For example, find the shear modulus of a sample under a stress of 4x10⁴ N/m² experiencing a strain of 5x10^-2.

G = τ / γ = (4x10⁴ N/m²) / (5x10^-2) = 8x10⁵ N/m² or 8x10⁵ Pa = 800 KPa

Isotropic and Anisotropic Materials

Some materials are isotropic with respect to shear, meaning the deformation in response to a force is the same regardless of orientation. Other materials are anisotropic and respond differently to stress or strain depending on orientation. Anisotropic materials are much more susceptible to shear along one axis than another. For example, consider the behavior of a block of wood and how it might respond to a force applied parallel to the wood grain compared to its response to a force applied perpendicular to the grain. Consider the way a diamond responds to an applied force. How readily the crystal shears depends on the orientation of the force with respect to the crystal lattice.

Effect of Temperature and Pressure

As you might expect, a material's response to an applied force changes with temperature and pressure. In metals, shear modulus typically decreases with increasing temperature. Rigidity decreases with increasing pressure. Three models used to predict the effects of temperature and pressure on shear modulus are the Mechanical Threshold Stress (MTS) plastic flow stress model, the Nadal and LePoac (NP) shear modulus model, and the Steinberg-Cochran-Guinan (SCG) shear modulus model. For metals, there tends to be a region of temperature and pressures over which change in shear modulus is linear. Outside of this range, modeling behavior is trickier.

Table of Shear Modulus Values

This is a table of sample shear modulus values at room temperature. Soft, flexible materials tend to have low shear modulus values. Alkaline earth and basic metals have intermediate values. Transition metals and alloys have high values. Diamond, a hard and stiff substance, has an extremely high shear modulus.

Note that the values for Young's modulus follow a similar trend. Young's modulus is a measure of a solid's stiffness or linear resistance to deformation. Shear modulus, Young's modulus, and bulk modulus are modulii of elasticity, all based on Hooke's law and connected to each other via equations.

Sources

• Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. Boston: McGraw-Hill. ISBN 0-07-013441-3.
• 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, vol. 7. (Theoretical Physics). 3rd Ed. Pergamon: Oxford. ISBN:978-0750626330
• Varshni, Y. (1981). "Temperature Dependence of the Elastic Constants". Physical Review B. 2 (10): 3952.