Introduction
Shear Modulus of elastic Moduli is the measure of mechanical properties present in solids. Other types of elastic moduli are Young’s modulus and bulk modulus. Let us help you understand the shear modulus elastic moduli. Shear modulus of elasticity is also known as modulus of rigidity. It is used to measure the rigidity of a body. Shear modulus is the ratio of shear stress to shear strain. This property is denoted using G or S or μ.
Any element that has a definite shape and shape is a rigid body. All the things and shapes that we see around us cannot be said to be completely rigid. When an external force is applied, these shapes can be bent, stretched or even compressed. On the application of an external force, these shapes tend to deform.
Stress is defined as the internal force that acts per unit area and is applied to regain the original shape.
A strain is defined as the ratio of change in the dimension produced and the body's original dimension.
The relationship so established between stress and strain is called the modulus of elasticity.
E=Stress/Strain
SI unit= Nm-2
Dimensions= [M L-1 T -2]
We can use this to explain how transformation is resisted from transverse deformations. However, this is practically possible just for small deformations.
Corresponding to the three types of strain namely longitudinal, volumetric and shear, there are mainly three types of modulus of elasticity. These are:
Young’s modulus
Bulk modulus
Shear modulus
Young’s Modulus of Elasticity
Young’s modulus is defined as the material’s ability to withstand the compression or expansion in accordance with its length.
It is represented using E or Y.
It is used to define the relationship between the longitudinal stress vs the longitudinal strain of an object. Whenever a load is applied to an object, its deformation takes place. The object can regain its original shape by removing the pressure and if it is elastic.
Young’s modulus formula is given by:
E= σ/ε
σ= F/A
ε= ΔL/L
Therefore,
\[E= \frac{\sigma (\epsilon )}{\epsilon }=\frac{\frac{F}{A}}{\frac{\Delta L}{L}}=\frac{FL}{A\Delta L}\]
Here,
E= Young’s modulus in Pascal
σ= Stress in Pascal
ε= Strain/ Proportional deformation
F= Force exerted on the object placed under tension
A= Cross-sectional area of the object
ΔL= Change in the length of the object
L= Original length of the object
The Formula For Modulus of Rigidity:
\[G=\frac{\tau _{xy}}{\gamma _{xy}}=\frac{Fl}{A\Delta x}\]
\[\tau _{xy}=\frac{F}{A}\] which is the shear stress.
F= Force acting on the object
A= Area on which the force acts
Δx= Transverse displacement
l= initial length
Difference Between Modulus of Rigidity and Modulus of Elasticity
Modulus of rigidity helps in calculating the deformation of an object when the deforming force is applied at right angles to the surface.
While modulus of elasticity helps in calculating the deformations of an object when the deforming force is applied parallel to the surface.
Modulus of Elasticity and Shear Modulus Relationship
Modulus of rigidity= G
Modulus of elasticity= E
The relationship between modulus of elasticity and shear modulus is given by:
E= 2G(1+μ)
The SI unit of the relation is Pascal (Pa).
Examples For Shear Modulus of Rigidity
For some material, the shear modulus of rigidity is given as follows:
Wood= 6.2 x 108 Pa
Steel= 7.2 x 1010 Pa
This shows that steel is highly rigid in comparison to wood.
FAQs on Shear Modulus - Elastic Moduli
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. They are:
- Young's Modulus (E): This measures a solid's resistance to a change in its length when a tensile or compressive force is applied perpendicular to its cross-sectional area.
- Shear Modulus (G): Also known as the Modulus of Rigidity, this measures a solid's resistance to a change in its shape when a tangential force is applied.
- Bulk Modulus (K): This measures a material's resistance to a change in its volume when subjected to uniform pressure from all sides.
2. What is the formula for calculating Shear Modulus, and what do its components represent?
The formula for Shear Modulus (G) is the ratio of shear stress to shear strain. It is expressed as:
G = Shear Stress (τ) / Shear Strain (γ)
Here's what each component means:
- Shear Stress (τ) is the deforming force acting parallel to the surface per unit area (F/A).
- Shear Strain (γ) is the ratio of the transverse displacement of a layer to its distance from the fixed layer (Δx/L).
The SI unit for shear modulus is Pascals (Pa) or N/m².
3. What is the fundamental relationship between Young's Modulus (E) and Shear Modulus (G)?
Young's Modulus (E), Shear Modulus (G), and a material's Poisson's Ratio (μ) are interconnected. The relationship is given by the formula: E = 2G(1 + μ). This equation highlights that the elastic properties of an isotropic material are not independent of each other; if you know any two, you can determine the third.
4. Why is Shear Modulus also referred to as the 'Modulus of Rigidity'?
Shear Modulus is called the Modulus of Rigidity because it directly quantifies a solid's ability to maintain its shape, or its structural stiffness. A high shear modulus indicates a very rigid material that strongly resists twisting and shearing forces. This property is crucial for understanding how a material will behave under forces that try to distort its shape rather than stretch or compress it.
5. How does the value of Shear Modulus explain the difference in properties between steel and wood?
The Shear Modulus value is a direct indicator of a material's rigidity. For example:
- Steel has a very high shear modulus (approximately 72 GPa). This means it is extremely rigid and requires a massive force to deform its shape, making it ideal for structural components like beams and gears.
- Wood has a much lower shear modulus (around 0.6 GPa). This indicates it is far less rigid and can be bent or twisted much more easily than steel. This difference in rigidity is a key reason for their different applications in construction and engineering.
6. Do liquids and gases have a Shear Modulus? Explain why or why not.
No, liquids and gases do not have a shear modulus; their shear modulus is considered to be zero. The reason is that fluids, by definition, cannot sustain a shear stress. When a tangential force is applied, the layers of a fluid simply slide past one another and do not return to their original position. They flow rather than deforming elastically in shape. Only solids, which resist changes in shape, possess 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 the direction of the applied force and the resulting deformation:
- Tensile Stress results from a force applied perpendicular to the cross-sectional area, causing the body to elongate or stretch along the line of force.
- Shear Stress results from a force applied parallel or tangential to a surface, causing the layers of the body to slide relative to one another, resulting in a change of shape without a significant change in volume.
