Profile
Settings
Refer your friends
Sign out
For a material that is placed under an external force, the shear modulus of rigidity of the material is defined as the ratio between the shear stress and shear strain experienced by the material due to the external force.
The modulus of rigidity gives the expression for the rigid property of a material. The rigidity can be explained as the property of a material to remain in a certain size and shape.
The value of the modulus of rigidity varies from material to material. For instance, the modulus of rigidity of steel is approximately 200 gigaPascals (GPa). This article will talk about the modulus of rigidity and its expression.
Stress and strain
A physical body is said to be rigid if it has a definite shape and size. When an external force is applied to a body’s surface, it undergoes a certain kind of deformation. However, the body itself produces a force against the external force to retain its original shape, called stress. The deformation that occurs in the body due to the influence of external force is called strain.
Modulus of rigidity
The modulus of rigidity or shear modulus of rigidity is defined as the ratio of shear stress to shear strain in a material that undergoes an external force.
Consider a rectangular box. Let the lower face of this box be fixed to the ground, and the upper face (A) undergoes a shear force (F). This will result in an equal and opposite force F applied to the lower fixed face. Since the lower face of the rectangular box is fixed, the upper face A will slide in the direction of force F, moving at a point A’.
The deformation in the box AA’ will be given by:
AA’ = ∆x
Shear stress is given by:
F/A,
Shear strain: Ө = tanӨ = AA’/AB = ∆x/x.
The shear modulus of rigidity is given by:
η = (shear stress) / (shear strain)
η = (F/A) / (∆x/x),
Or: η = (F.x)/(A.∆x).
The unit of the shear modulus of rigidity is N/m².
The above expression gives the formula for the shear modulus of rigidity for a body undergoing a shear force.
Relationship between shear modulus and elastic modulus
The elastic moduli of Hooke’s law are interrelated. These parameters of Hooke’s law include the shear modulus of rigidity, Young’s modulus, and bulk modulus. The following expression gives a relationship between the values of these parameters:
2G(1+u)=3K(1-2u)=E.
In the above expression:
G = shear modulus of rigidity,
u = ratio (Poisson’s ratio)
K = bulk modulus
E = Young’s modulus.
Solved examples
Example 1: A block is kept so that the bottom face of the box is placed on the table. Say the rectangular box is experiencing a shear force. Consider that the dimensions of the block = 80 mm x 80 mm x 20 mm, shearing force = 0.255 N, deformation = 10 mm. For the given values, find the shear modulus of rigidity of the material.
Solution: From the values above, we can calculate:
Shear stress:
Shear strain:
∆x/x,
= 10/80,
= 1/8.
Therefore the shear modulus of rigidity is given by:
G = (shear stress) / (shear strain),
G = 159.37 / (1/8),
G = 1,274.96 N/m-².
Example 2: A steel block is kept so that the bottom face of the box is placed on the table. Say the rectangular box is experiencing a shear force. Consider the dimensions of the block = 50 mm x 50 mm x 10 mm, shearing Force = 0.300 N, deformation = 8 mm. For the given values, find the shear modulus of rigidity of the steel.
Solution: From the values above, we can calculate:
Shear stress:
Shear strain:
∆x/x,
= 8 /80
= 1/10.
Therefore, the shear modulus of rigidity is given by:
G = (shear stress) / (shear strain),
G = 600 / (1/10),
G = 6,000 N/m-².
Example 3: A block of some material is kept so that the bottom face of the box is placed jointed. Say the rectangular box is experiencing a shear force. Consider the dimensions of the block = 30 mm x 30 mm x 20 mm, shearing force = 0.230 N, deformation = 9 mm. For the given values, find the shear modulus of rigidity of the material.
Solution: Let us calculate using the provided values:
Shear stress:
F/A,
Shear strain:
∆x/x,
= 9/30,
= 3/10.
Therefore, the shear modulus of rigidity is given by:
G = (shear stress) / (shear strain),
G = 383.33 / (3/10),G = 114.99 N/m-².
Conclusion
When a rigid body is placed under a compressive force, the loose side of the body experiences a shear force in the direction of the external force; the restraining force that the body applies in response is known as the shear force, and the deformation is known as the shear strain.
The modulus of rigidity or the shear modulus of rigidity is defined as the ratio of shear stress to shear strain in a rigid material experiencing an external force, shear or tangential. The value of modulus varies from different materials. In the above article, we learn about the modulus of rigidity and the modulus of rigidity of steel by solving various examples.
