Strain Energy

Strain energy is the energy stored in a body due to deformation. The strain energy stored per unit volume is known as the strain energy density. When strain energy is represented in a graph, the area under the stress-strain curve is called strain energy. It is denoted by a capital “U”.

Strain Energy

So, in simple terms, strain energy is the energy stored in a material when work is done on it. It is also called “resilience.” Strain energy is a kind of potential energy that is stockpiled in a body due to deformation in elasticity. For example, if force is applied to the bar and the bar is bent, then it is said that the bar is bent in an unstressed state. The quantity of total energy of strain stored in the bar is equal to the activity done on it. It is calculated as the area under the stress and strain curve towards the point of deformation. Only if the stress is beneath the elastic limit, can the energy of strain stored in the material be transferred to kinetic energy. When a material’s stress surpasses its elastic limit, plastic deformation occurs, and some energy is lost in the process. When a force is applied to an item formed of a deformable material, it will alter shape. If one applies greater force, the object will keep on stretching. Here, the stress is equal to the quantity of total applied force divided by the cross-sectional area of the object. So, the strain energy equals the total work done.

The strain energy formula 

U represents the strain energy.

U = Fδ / 2

Here,

F = applied force.

δ = compression

The formula for strain energy is 

U = 0.5 * V * strain * stress

V = volume of the body

The formula for strain is also derived from the young modulus.

U = = (σ2 / 2E )× V

σ = stress

E = Young’s modulus 

V = volume of the body

If the stress is proportional to strain, the formula for strain energy can be given as,

U = 1/2 * V * σ * ϵ

Where,

U = strain energy

V = volume of the body

 σ = stress

 ϵ = strain

The brittle materials are very strong as they can endure a lot of stress. For this, they don’t stretch easily and break easily if they try to be stretched. Ductile materials are, on the other hand, very elastic in nature. In the first peak, when the ductile materials are stretched, they break and can’t return to their original length. In the second turnover, it represents the maximum amount of tensile stress the object can endure. Plastic materials are not strong enough, but they can withstand a lot of stress. In the graph, the young modulus is represented by the gradient line of the stress and strain curve.

Solved problems on strain energy

Problem 1

What is the value of the strain energy if a force of 1000 N is applied to a body and it gets compressed by 1.2 mm?

Answer: The force given is F = 1,000 N.

Compressor = 1.2 mm

Strain energy formula = (1000 * 1.2 * 10 ^ -3) / 2

= 0.6 Joules 

So, the strain energy is 0.6 joules.

Problem 2

The area of a road is 90 sq. mm. and its length is 3 m. What is the stress energy when 300 MPa is applied and the given Young Modulus is 200 GPa?

Answer: The area is 90 sq. mm.

Length = 3m

Stress 300 MPa

Young’s modulus is 200 GPa.

The formula for volume is 

V = area * length

= 90 * 3 * 10 ^-6

= 270 * 10 ^ – 6 cubic meter

The value of strain energy = (300 * 10 ^ -6) ^ 2 / 2 * 200 * 10 ^ 9 * 270 * 10 ^ -6

So, U = 83.3 * 10 ^ 6 J

So, the value of strain energy is 83.3 * 10 ^ 6 Joules.

Problem 3.

Calculate the work done in stretching a wire with a length of 5 m and a cross-sectional area of 1 sq. mm that is deformed by a length of 1 mm if the wire’s Young’s Modulus is 21011N/m22.

Answer:

Given that the length of the wire is 5 meters,

The area of the cross section of the area is 1 sq. mm. = 10 –6 sq. m.

The quantity of  deformation due to stress is 1 mm, which is 10 ^ -3 m.

Young modulus of the wire is = 2 * 10 ^ 11 N / sq. m

The Young modulus of elasticity 

E = (F * l) * (A * deformation due to stress)

So, F is equal to (2 * 10 ^ 11 N/m2)(10 ^ – 6 sq. m)(10 ^ -3 m) / 5

= 40 N

Now the total work done is = (40 * 10 ^ -3) / 2 J

= 0.02 J

So, the total work done is 0.02 joules.

Conclusion

These concepts are very important for GATE exam aspirants. These concepts are from advanced physics but comparatively easy.