What is Shear Modulus?
Shear modulus is also known as the modulus of rigidity, it is a constant number that describes the elastic properties of a solid, under the use of transverse internal forces such as arise. For example, in torsion, twisting of the metal about its own axis is known as the shear modulus. In these types of material, any of the small volumes of the material is distorted, in a way that both faces of the material slide parallel to each other for a small distance and the other two faces change from the square to the diamond shape.
Rigidity Modulus Definition - It is the measure of the rigidity of the body, represented as the ratio of shear stress to the shear strain.
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The shear modulus measures the strength of an object to withstand flexible deformities and is a valid indicator of behaviour that extends only to a small deformity, after which the object is able to return to its original configuration. Excessive shear force leads to the permanent deformation and rotation or breakage of the material.
Mathematically
In physics materials science, shear modulus or modulus of rigidity is denoted by the letter ‘G’, or sometimes it is also denoted as ‘S’ or ‘μ’. Shear modulus is the measure of elastic shear stiffness of a material and the shear modulus definition - is the ratio of shear stress to the shear strain:
G = \[\frac{\tau_{xy}}{\gamma_{xy}}\] = \[\frac{F/A}{\frac{\Delta x}{l}}\] = \[\frac{Fl}{A\Delta x}\]
where,
\[\tau\]\[_{xy}\] = F/A = shear stress.
‘F’ is the force that acts.
‘A’ is the area of the material, on which force will be acting.
\[\gamma\]\[_{xy}\] = shear strain. In engineering \[\frac{\Delta x}{l}\] = tanθ,
elsewhere it will be ‘θ’
\[\Delta\]x’ is the transverse displacement.
‘l’ is the initial length of the area.
The SI unit of the shear modulus is the pascal (pa), but it is usually expressed in the gigapascal (GPa) or some of the time it is also expressed in thousand per square inch (KSI). And the dimensional formula of the shear modulus is given as M1L−1T−2, replacing force by mass times acceleration.
Explanation
Shear modulus is one among the various quantities used for measuring the stiffness of the materials, all of them arising keeping Hooke’s law in general.
Young's modulus denoted by ‘E’, describes the strain of the material (change in length to the original length) response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire, with the wire getting longer).
The Poisson's ratio denoted by ‘ν’, is the ratio that describes the response in the orthogonal directions to this uniaxial stress (the wire getting thinner compared to the previous thickness).
The bulk modulus ‘k’ of the material describes the response to hydrostatic (uniform) pressure (like the pressure at the bottom of the swimming pool).
The shear modulus G of a material describes the response of the shear stress, these moduli are not independent, and for isotropic materials, they are connected through the equations 2G(1 + v) = E = 3K(1 - 2v)
The shear modulus is concerned with the conversion of a solid-state when it encounters the same force as one of its surfaces. While its opposing face encounters opposing forces (such as friction). In the case of a rectangular prism object, it will be disabled as a parallelepiped. Anisotropic materials such as wood, paper and basically all single crystals show a different reaction of substances to stress or difficulty when tested in different directions. In this case, one may need to use a full-strength accent of the elastic constant, rather than a single scalar value of the material.
Shear Modulus Value of Different Materials
From the above-given table, we get to know the value of the shear modulus of the different material for example shear modulus of steel/modulus of rigidity of steel is 79.3 GPa.
Example Problem
1. Consider, a block of unknown material kept on a table (The square face of the material is placed on the table), is under a shearing force. Some of the data are given below to calculate the shear modulus of the material, dimensions = 60 mm x 60 mm x 20 mm, Shearing Force = 0.245 N, displacement = 5 mm.
Solution:
Substituting the values in the formula we get-
Shear stress = \[\frac{F}{A}\] = \[\frac{0.245}{60 \times 20 \times 10^{-6}}\] = \[\frac{2450}{50}\] N/m2
Shear strain = \[\frac{\Delta x}{l}\] = \[\frac{5}{60}\] = \[\frac{1}{12}\]
Thus,
Shear modulus, G = \[\frac{\text{shear stress}}{\text{shear strain}}\] = \[\frac{2450 \times 12}{12}\]
= 2450 N/m2.
Do you know?
What is the relation between Elastic Constants?
The elastic moduli of a material, like Young’s Modulus, Bulk Modulus, Shear Modulus are specific forms of Hooke’s law, which states that, for an elastic material, the strain experienced by the corresponding stress applied is proportional to that stress. Thus, we can write the relation between elastic constants by the following equation: 2G(1 + v) = E = 3K(1 - 2v)
Where,
G is the Shear Modulus
E is the Young’s Modulus
K is the Bulk Modulus elasticity
υ is Poisson’s Ratio
FAQs on Shear Modulus
1. What is Shear Modulus?
Shear Modulus, also known as the Modulus of Rigidity (G), is a measure of a material's resistance to shearing deformation. It is defined as the ratio of shear stress (force per unit area parallel to the surface) to shear strain (the resulting deformation angle). Essentially, it tells you how stiff a material is when you try to slide its layers past each other.
2. How is Shear Modulus calculated? What is its formula?
The formula for Shear Modulus (G) is derived from its definition as the ratio of shear stress to shear strain. As per the CBSE 2025-26 syllabus, the formula is:
G = (F/A) / (Δx/l)
Where:
- F is the tangential force applied.
- A is the area over which the force acts.
- Δx is the transverse displacement or the distance the sheared face moves.
- l is the initial length or height of the object.
3. What is the SI unit and dimensional formula for shear modulus?
The SI unit for shear modulus is the Pascal (Pa), which is equivalent to newtons per square meter (N/m²). Due to the large values for most solids, it is often expressed in Gigapascals (GPa). The dimensional formula for shear modulus is [M¹L⁻¹T⁻²].
4. How does Shear Modulus differ from Young's Modulus?
The primary difference lies in the type of stress and strain they measure:
- Shear Modulus (G) measures a material's resistance to a shearing or twisting force. It describes how a material deforms when its layers slide past each other.
- Young's Modulus (E) measures a material's resistance to a tensile or compressive force along its length. It describes how a material stretches or compresses.
5. Why is the shear modulus for liquids and gases considered to be zero?
Liquids and gases are considered fluids, which cannot sustain a shear stress. When a tangential force is applied, their layers do not resist but instead flow freely past one another. Since they offer no lasting resistance to a change in shape, their shear strain can be considered infinite for any sustained shear stress, making their Shear Modulus effectively zero.
6. What does a high shear modulus value indicate about a material?
A high shear modulus value signifies that a material is very rigid and highly resistant to deformation by shearing forces. For example, steel (Shear Modulus ≈ 79 GPa) and diamond have high shear moduli, meaning a large force is required to cause a small change in their shape. In contrast, a material with a low shear modulus, like rubber, is much more flexible and deforms easily.
7. Does the shear modulus of a material change if you increase the applied shearing force?
No, the shear modulus is an intrinsic property of a material and does not change with the amount of applied force, as long as the material remains within its elastic limit. Shear modulus defines the ratio of stress to strain. While increasing the force (stress) will cause a proportional increase in deformation (strain), the ratio between them—the shear modulus—remains constant for that material.
8. What is the relationship between Shear Modulus (G), Young's Modulus (E), and Bulk Modulus (K)?
For isotropic materials (materials with uniform properties in all directions), the elastic constants are related through equations involving Poisson's ratio (ν). The primary relationships taught in the NCERT syllabus are:
- E = 2G(1 + ν)
- E = 3K(1 - 2ν)
