Relation Between Shear Modulus and Young's Modulus with Formulas and Examples
Shear modulus, also known as modulus of rigidity, is a key concept in understanding the mechanical properties of solids. It describes how a material responds when forces are applied tangentially (parallel to a surface), causing a change in the shape of the object without altering its volume. Along with Young’s modulus and bulk modulus, shear modulus forms one of the three main elastic moduli used for analyzing elasticity in physics.
A truly rigid body is rare. Any object we see—even those that appear hard—can bend, stretch, or compress slightly if subjected to an external force. The measure of a material's tendency to return to its original shape after being deformed is defined by its modulus of elasticity.
Definition and Types of Elastic Modulus
The modulus of elasticity is the relationship between stress (force per unit area) and strain (relative deformation) in a material under load. The three main types are:
- Young’s Modulus (E): Measures resistance to changes in length under tensile or compressive load.
- Bulk Modulus (K): Measures resistance to a change in volume under uniform pressure.
- Shear Modulus (G): Measures resistance to shape change under tangential force.
For any small deformation, the relations remain linear, helping analyze real-world engineering and physics applications.
Shear Modulus: Explanation and Formula
Shear modulus (G) is expressed as the ratio of shear stress to shear strain. Shear stress is the force per unit area applied parallel to a surface, while shear strain is the relative displacement between layers within a material.
The formula for shear modulus is:
Breaking this down:
- Shear Stress (τ) = F / A, where F is the tangential force and A is the area of the surface.
- Shear Strain (γ) = Δx / l, where Δx is the transverse displacement and l is the original length from the fixed layer.
Young's Modulus and Relationship With Shear Modulus
Young’s modulus (E) is defined as the ratio of longitudinal stress to longitudinal strain. It shows how materials stretch or compress in response to force applied perpendicularly:
Where:
- F = Force applied
- A = Cross-sectional area
- ΔL = Change in length
- L = Original length
There is a fundamental relationship between Young's modulus (E), shear modulus (G), and Poisson’s ratio (μ):
This allows calculation of one modulus if the other two are known.
Units and Dimensions
| Parameter | SI Unit | CGS Unit | Dimensions |
|---|---|---|---|
| Shear Modulus (G) | Pascal (Pa) or N/m2 | dyne/cm2 | [ML-1T-2] |
A higher shear modulus indicates a material is more rigid against shape deformation (for example, steel vs. wood).
Comparison of Elastic Moduli
| Property | Young’s Modulus (E) | Shear Modulus (G) | Bulk Modulus (K) |
|---|---|---|---|
| Type of Stress | Tensile/Compressive | Shearing | Volume |
| Physical Change | Length | Shape | Volume |
| Unit | Pa | Pa | Pa |
| Steel (Example Value) | 2.1 × 1011 Pa | 7.2 × 1010 Pa | 1.6 × 1011 Pa |
| Wood (Example Value) | — | 6.2 × 108 Pa | — |
How to Solve Shear Modulus Problems: Step-by-Step
- Identify type of force and direction (tensile, compressive, or shear).
- Calculate the appropriate stress using the force (F) and area (A): for shear, τ = F/A.
- Find the corresponding strain: for shear, γ = Δx/l.
- Use the formula: G = τ/γ.
- Interpret the result for rigidity or flexibility of the material.
Remember, all deformations for these formulas must be small and within the elastic limit of the material.
Example Problem
A steel rod has a shear modulus (G) of 7.2 × 1010 Pa. If a tangential force of 1000 N acts on a square face of side 2 cm, calculate the shear strain when the displacement at the top is 0.2 mm.
- Area, A = (0.02 m) × (0.02 m) = 0.0004 m2
- Length, l = 0.02 m
- Displacement, Δx = 0.2 mm = 0.0002 m
- Shear Stress, τ = 1000 N / 0.0004 m2 = 2.5 × 106 Pa
- Shear Strain, γ = Δx / l = 0.0002 / 0.02 = 0.01
- Shear Modulus, G = τ / γ = (2.5 × 106 Pa) / (0.01) = 2.5 × 108 Pa
Notice that in practice, you would compare the calculated modulus with known values to check the consistency of material properties.
Physical Meaning and Applications
A high shear modulus means the material strongly resists shape deformation under parallel forces—critical for beams, bridges, and machine components. Steel’s high shear modulus makes it much more rigid than wood, hence used where structural stability is vital.
Fluids (liquids and gases) have a shear modulus of zero; they cannot resist shear force and will flow when such force is applied.
Key Formulas at a Glance
| Modulus | Symbol | Formula | Physical Meaning |
|---|---|---|---|
| Young’s Modulus | E | E = Stress/Strain (tensile) | Length change under tension/compression |
| Shear Modulus | G | G = Shear Stress/Shear Strain | Shape change under parallel force |
| Bulk Modulus | K | K = Volume Stress/Volume Strain | Volume change under pressure |
Next Steps: Vedantu Resources for Elasticity and Beyond
To master elasticity, practice more solved examples and explore related material properties:
- Elastic Behaviour of Materials
- Young's Modulus
- Stress and Strain
- Bulk Modulus of Elasticity
- Elastic Potential Energy
- Elasticity
Continuous practice using Vedantu quizzes, revision notes, and previous year questions will strengthen your concepts and problem solving for all Physics branches.
FAQs on Shear Modulus and Elastic Moduli Explained for Physics Students
1. What are the three main types of modulus of elasticity as per the Class 11 Physics syllabus?
The three main types of modulus of elasticity describe a material's response to different kinds of deforming forces:
1. Young's Modulus (E): Measures resistance to length change under tensile or compressive stress.
2. Shear Modulus (G or Modulus of Rigidity): Measures resistance to shape change under shear stress.
3. Bulk Modulus (K): Measures resistance to volume change under uniform pressure.
These moduli help in understanding how materials behave under mechanical forces.
2. What is the formula for calculating Shear Modulus, and what do its components represent?
The formula for Shear Modulus (G) is:
G = Shear Stress (τ) / Shear Strain (γ)
Where:
Shear Stress (τ) = Force applied tangentially per unit area (F/A).
Shear Strain (γ) = Ratio of transverse displacement to original length (Δx/L).
SI unit: Pascal (Pa) or N/m².
This formula describes a material's rigidity when a shape-altering force is applied.
3. What is the fundamental relationship between Young's Modulus (E) and Shear Modulus (G)?
Young's Modulus (E), Shear Modulus (G), and Poisson's Ratio (μ) are related by the following formula:
E = 2G(1 + μ)
This relationship shows that for isotropic materials, knowing any two of these properties allows you to calculate the third, helping to understand the elastic behaviour of solids.
4. Why is Shear Modulus also referred to as the 'Modulus of Rigidity'?
Shear Modulus is called the Modulus of Rigidity because:
- It quantifies a solid’s resistance to shape change (shearing force).
- A high shear modulus means the material is very rigid and difficult to distort.
This term directly indicates the ability of a material to maintain its original shape when subjected to tangential forces.
5. How does the value of Shear Modulus explain the difference in properties between steel and wood?
The shear modulus reflects a material’s rigidity:
- Steel has a high shear modulus (about 72 GPa), so it is very rigid and resists shape deformation.
- Wood has a much lower shear modulus (about 0.6 GPa), so it is less rigid and easier to bend or twist.
This difference explains why steel is used for structural applications while wood is easier to shape and flex.
6. Do liquids and gases have a Shear Modulus? Explain why or why not.
No, liquids and gases do not have a shear modulus (G = 0).
- Liquids and gases cannot sustain shear stress; when a tangential force is applied, they flow rather than resisting deformation.
- Only solids offer resistance to shape change, which is measured by a non-zero shear modulus.
7. What is the key difference in how shear stress and tensile stress affect a solid body?
The key difference lies in direction and type of deformation:
- Tensile Stress causes elongation or compression along the line of force.
- Shear Stress causes layers of the material to slide over each other, changing shape without significantly altering length or volume.
This leads to different physical changes in the material.
8. What are the SI unit and dimensional formula for Shear Modulus?
The SI unit of Shear Modulus is Pascal (Pa) or N/m².
Dimensional formula: [M L-1 T-2]
These units are the same as other types of stress and elasticity moduli.
9. How can Shear Modulus be calculated from Young's Modulus and Poisson's Ratio?
The formula connecting Shear Modulus (G), Young's Modulus (E), and Poisson's Ratio (ν) is:
G = E / [2(1 + ν)]
Use this formula to find G when E and ν are given for a material, as required in competitive exam problems.
10. What is the physical significance of a high Shear Modulus value for a material?
A high value of Shear Modulus indicates:
- The material is very rigid and strongly resists shape deformation under tangential/shearing forces.
- Such materials are ideal for structural, load-bearing, and engineering applications where minimal distortion is required.
11. What are some practical applications of Shear Modulus in engineering?
Shear Modulus is crucial in:
- Designing beams, shafts, and mechanical components subjected to torsion or shear forces
- Selecting suitable materials for bridges, machinery, and construction
- Assessing material performance for gears, springs, and structural supports
Engineers use Shear Modulus to ensure safety and functionality of devices under load.
12. How does Shear Modulus differ from Bulk Modulus and Young's Modulus?
Shear Modulus, Bulk Modulus, and Young's Modulus differ in the type of deformation measured:
- Shear Modulus (G): Resistance to shape change (shearing)
- Bulk Modulus (K): Resistance to volume change under uniform pressure
- Young's Modulus (E): Resistance to length change under tensile or compressive force
Each modulus quantifies a different aspect of elastic behaviour in materials.
