The response of different materials to shear deformation is often measured in terms of shear modulus. Shear stress is characterised as the probability of shear stress to shear strain. This ratio is useful in determining how resistant a material is to shearing deformation. If a substance is highly resistant to shearing, the energy produced will also be shared quickly. 

The shear modulus is often referred to as the modulus of rigidity. Occasionally, it is distinguished by the gigapascals (GPa) or thousands of pounds per square inch. In this article, we will learn about the shear modulus. 

Meaning of Shear Modulus

Shear modulus can be defined as one of the techniques by which we measure the mechanical properties of solids. It is frequently cited as the shear modulus of elasticity. Young’s modulus and bulk modulus are two other techniques to gauge elastic modulus. The ratio of shear stress to shear strain in a substance can be determined by finding out the material’s shear modulus.

Some of its characteristics are as follows:

• It is written by the SI unit of pascal or Pa.
• The dimensional formula can be expressed as M¹L-¹T-²
• Shear modulus is represented by G. 

A solid that possesses a high shear modulus value will be more rigid than other substances. In simple words, to change the shape of an object with more shear modulus, you will require a lot of force.  

A low shear modulus value will indicate that the material is quite soft with low flexibility. To deform such substances, you will need a small amount of force.  

Lastly, a fluid is defined as a substance with a zero shear modulus value. 

Representing Shear Modulus in the Form of an Equation 

You can find the value of shear modulus by calculating the deformation of a solid if any amount of force is applied to it. The applied force would be parallel to any surface, while the opposing force would keep the surface intact. For example, if a shear force helps in pushing a box, there would be a frictional force that will restrict the box from the other side.  

The equation would thus be expressed in the form of

G = τxy / γxy = F/A / Δx/l = Fl / AΔx

Here,

• G would be the modulus of rigidity
• τxy would represent shear stress
• γxy is the strain
• Δx would be the transverse replacement
• l would be the initial length

The shear strain would be written as Δx/l = tan θ, and often, the value becomes θ, which is for the angle formed by the deformation caused by the applied force.

Shear Modulus of Different Materials

Here is a list of different materials with different values of shear modulus. The values of young modulus are the same as of shear. The elastic limit would be defined by the ratio of stress (σ) concerning strain (€). 

E = σ / €

The shear modulus of mild steel is approximately 79 GPa or 11,460 ksi.

Solved Question Based on Shear Modulus

Q.A metal plate has a thickness of 0.3 inches and we want to drill a hole that has a radius of 0.6 inches. Find out the force required to drill a hole if the shear strength is FA = 4 x 10⁴ lb square inch.

Answer: The shear stress which will be applied to the cylindrical shape’s object would be expressed as

2πrh = 2×3.14x 0.06x 0.30

= 0.11304 square inch

Here, it is given that FA = 4 x 10⁴ lb square inch. Therefore, the force needed to make a hole in the object would be 

4×10⁴ x 0.11304

That is, the required force would be 4521.6 lb. 

Conclusion

In this article, we have learned that the elastic coefficient is often called the shear modulus of rigidity. Or, in simple words, it can be a shear force applied to a body that will lead to distortion. This technique is used to evaluate a surface’s rigidity. On a surface, it’s also the percentage of shear stress to shear strain. Along with that, we also learned  about the shear modulus of mild steel and the shear modulus of concrete.