Shear Modulus of Rigidity

What is Shear Modulus

Shear modulus or modulus of rigidity is a measure of a material’s resistance to elastic (non-permanent) lateral deformation when a shearing force is acting on it. In other words, it measures the shear stiffness of a material when it’s under shear stress.

Diagram showing a rectangular sample under a shering force

The shear modulus value of a material indicates its resistance to deformation under shear stress. A high shear modulus value means a material is pretty rigid, and it would take quite a large shearing force to cause a small deformation. On the other hand, a low shear modulus value means a material is soft solid or flexible, and it can be deformed easily with little force.

For instance, the shear modulus of steel is approximately 79 GPa, whereas, for copper, it’s 45 GPa, which indicates that a higher shearing force is required to cause the same amount of lateral deformation in steel as in copper.

The shear modulus value of a material is an important factor to consider in materials selection and engineering design. It affects the behavior of the material under different types of loads, such as shear, torsion, and bending. In addition, it can also influence the material’s durability, fatigue resistance, and overall performance in various applications.

Definitions:

Shear modulus, denoted by the letter (G), (S), or (μ) "Mu", is defined as the ratio between the shear stress and the shear strain.

Shear force is a force parallel to the cross-sectional area of an object that causes the two sides of its cross-section to move in opposite directions.

Shear stress is the shear force divided by the cross-sectional area parallel to the applied force.

Shear strain is the lateral displacement per unit length.

The shear modulus, same as all the other elastic moduli, is only practical while the deformation is elastic, meaning that deformation is recoverable once the load / force is removed, and it is not permanent.

Formula & Units

Diagram showing a shear force acting on a rectangular object and the resulting shear strain — Deformation of an object due to a shear force acting on it and the resulting shear strain

Shear Modulus (G) = \frac{𝜏_{x y}}{𝛾_{x y}}

Where:

$G$ is the shear modulus or modulus of rigidity
$𝜏_{x y}$ is the shear stress (F / A)
$𝛾_{x y}$ is the shear strain (Δx / L)

Shear strain (Δx/L) is equal to (tan θ), where θ is the angle formed by the lateral deformation produced by the applied force.

The SI unit of shear modulus is pascal (Pa), which is equal to 1 Newton per square meter (N/m²).

The US customary unit of shear modulus is pounds per square inch (psi).

Shear Modulus Calculation

Find the shear modulus of a rectangular sample under a shearing force of 6 N and a lateral displacement of 5 mm. The dimensions of the sample are given as: width = 200 mm, depth 200 mm, and length = 500 mm. The shearing force is acting on the material parallel to the square face as shown in the diagram below:

Solution

Step 1) We write down the given parameters:

F = 6 N Δx = 5 mm w = 200 mm L = 500 mm

Step 2) We find the shear stress caused by the force:

𝜏 = \frac{F}{A} = \frac{6}{0.2 \times 0.2} = 150 N / m^{2}

Step 3) We find the shear strain:

𝛾 = \frac{Δ x}{L} = \frac{5}{500} = 0.01

Step 4) Now, we can find the shear modulus of the sample:

G = \frac{𝜏}{𝛾} = \frac{150}{0.01} = 15, 000 N / m^{2}

Factors Affecting Shear Modulus

There are several factors that can affect the shear modulus of a material, including:

Temperature: As temperature increases, the shear modulus of most materials tends to decrease. This is due to the increase in thermal energy and the resulting increase in molecular motion, which can lead to greater deformation and reduced stiffness.
Microstructure: The microstructure of a material, including its grain size, crystallographic orientation, and defects such as dislocations, can significantly affect its shear modulus.
Chemical composition: The chemical composition of a material can also impact its shear modulus. For example, the addition of alloying elements can improve the strength and stiffness of a material, increasing its shear modulus.
Strain rate: The rate at which a material is deformed can affect its shear modulus. In general, materials tend to have higher shear moduli at lower strain rates.
Pressure: High pressure can alter the interatomic or intermolecular bonding within a material, affecting its resistance to deformation under shear stress and thus its shear modulus.

Applications of Shear Modulus

The shear modulus is a fundamental material property that has many applications in various fields of science and engineering. Some of the most common applications of shear modulus are:

Structural engineering: The shear modulus is used to design and analyze structures such as bridges, buildings, and other infrastructure. It is an important parameter for determining the stability and strength of a structure under lateral or shear loads.
Geology: The shear modulus is an important parameter in geotechnical engineering and seismology. It is used to determine the mechanical properties of soils and rocks, and to understand the behavior of seismic waves as they propagate through the earth.
Materials research: The shear modulus is used to study the mechanical properties of materials at the atomic and molecular level. It can help to understand the deformation and failure mechanisms of materials under different loading conditions and to develop new materials with specific mechanical properties.

Shear Modulus Summary
Definition	The material’s resistance to lateral - elastic deformation when under a shear stress
Symbol	G, S, or μ "Mu"
Formula	$G = \frac{𝜏_{x y}}{𝛾_{x y}}$
Units	Si unit (Pa)	US unit (psi)

What does shear modulus tell you?: Shear modulus describes a material’s lateral stiffness or resistance to lateral-elastic deformation when under a shearing force.

Why shear modulus of liquids is zero?: Because liquids cannot withstand shearing stress, their shear modulus is zero.

How does shear modulus relate to the strength of a material?: Materials with a high shear modulus are typically stronger and more rigid, while those with a low shear modulus are generally weaker and more deformable. However, the relationship between shear modulus and strength depends on other factors, such as the material’s composition, microstructure, and other mechanical properties.