Shear modulus: description of the stiffness of a material

The transverse modulus of elasticity, also called shear modulus, shear modulus, or stiffness modulus, is an elastic constant that characterizes the change in shape that an elastic material undergoes when shear stresses are applied and is defined as the ratio between the shear stress and shear deformation. It is named as  G  or less commonly by  S  or μ . The unit as the transverse modulus of elasticity is expressed in the international system of units is the Pascal (Pa), but the values ​​are generally expressed in gigapascals (GPa). 

• A large shear modulus value indicates that a body is very stiff. In other words, a great force is required to produce the deformation.
• A small shear modulus value indicates that a solid is soft or flexible. Little force is needed to deform it.
• A 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 by applying a parallel force to one surface of the solid, while an opposite force acts on its opposite surface and holds the solid in place. Think of shear as pushing against the side of a block, with friction as the opposing force. Another example would be trying to cut wire or hair with dull scissors.

The equation for the shear modulus is:

G = τxy _/  γxy ₌  F / A / Δx / l = Fl / AΔx

Where:

• G is the shear modulus or stiffness modulus
• 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

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

Isotropic and Anisotropic Materials

There are basically two types of material responses, some are isotropic with respect to shear, which means that 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 grain of the wood compared to its response to a force applied perpendicular to the grain. Consider the way a diamond responds to an applied force. The ease with which the crystal is cut depends on the orientation of the force with respect to the crystal lattice.

Effect of temperature and pressure

As expected, the response of a material to an applied force changes with temperature and pressure. In metals, the shear modulus generally decreases with increasing temperature. Stiffness decreases with increasing pressure. Three models that are used to predict the effects of temperature and pressure on shear modulus are the plastic flow stress or mechanical threshold stress (MTS) model, the Nadal and LePoac (NP ) and the Steinberg-Cochran-Guinan (SCG) shear modulus model. For metals, there tends to be a region of temperature and pressures over which the change in shear modulus is linear. Outside of this range, the modeling behavior is more complicated.

Table of values ​​of the cutting module

This is a table of sample shear modulus values ​​at room temperature. Soft and flexible materials tend to have low shear modulus values. The alkaline earth and base metals have intermediate values. Transition metals and alloys have high values. For example, diamond is a hard and rigid substance, therefore it has an extremely high cutting modulus.

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

Sources

• Crandall, Dahl, Lardner. (1959). 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 LD, Pitaevskii, LP, Kosevich, AM, Lifshitz EM (1970). Theory of Elasticity, Vol. 7. (Theoretical Physics). 3rd Ed. Pergamum: Oxford. ISBN: 978-0750626330
• Varshni, Y. (1981). “Temperature dependence of elastic constants”. Physical Review B. 2(10):3952.