Strain Energy: Formula, Examples & Important Questions

Strain Energy is the energy stored in any substance when it undergoes deformation. This energy can be reversed back into kinetic energy. A perfect example is the stretching of a rubber band. The elongation causes a deformation which results in storing of energy in the rubber band. This is Strain Energy. Once the applied force causing the deformation is released, the rubber band comes back to its original shape. 

This article aims to explain strain energy caused by stress or strain, derive its formula, and discuss some important examples. 

Read More: Solid Deformation

Key Terms: Stress, Strain, Strain Energy, Strain Energy Formula, Elastic, Deformation, Hooke’s Law, Strain Energy Density, Strain Energy per Unit Volume

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What is Strain Energy?

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Strain energy is the potential energy that gets stored in a substance when it experiences a change in its structure in the form of elastic deformation. The work done on the substance to change its initial shape gets stored as strain energy after deformation. 

Let us understand strain energy with an example:

• A perfect example is a spring attached to a block. 
• When the spring gets compressed, the deformation causes strain energy to be stored in the spring. 
• When the spring returns to its original shape, the strain energy gets transferred as kinetic energy causing the block to move.

Strain Energy Stored in a Spring

Read More: Yield Strength

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Stress and Strain

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Strain Energy forms as a result of the deformation of an object. Hence, it is important to understand the stress-strain relation and the significance of the stress-strain curve. 

Stress

Stress is the resistive force acting on the object when an equal force is applied to it. Stress is hence the resistive force inside the object per unit area. The formula for stress is given as:

Stress (σ)= \(\frac{F}{A}\)

Where,

F= Force Applied on the object

A= Cross-sectional Area

Strain

The linear deformation suffered by an object per unit length is longitudinal strain. It is given by:

Strain (ε) = \(\frac{Change in Length}{Original Length}\) = \(\frac{ \Delta L}{L}\)

Where, 

L= Initial Length

ΔL= Change in the length after deformation

Stress and Strain Schematic

Read More: Shear Modulus

Relation Between Stress and Strain

The relation between stress and strain is given by the famous Hooke’s Law. The law states that the strain is directly proportional to the stress for a small deformation of an elastic material. 

Hence mathematically, the relation becomes,

σ ∝ ε

Replacing proportionality with a constant,

Stress = E x Strain

E here is the proportionality constant also known as the elastic strain energy.

Stress-Strain Graph

The stress-strain graph is an easy representation of the stress-strain response of an object on which a force is applied. Following is the stress-strain graph and its important characteristics:

Stress-Strain Graph

• Proportional Limit: The point where the linear deformation changes to the non-linear region.
• Plastic Region: The point at which complete deformation of the material occurs.
• Yield Strength: Separates the elastic region from the plastic region.
• Ultimate Strength: The maximum stress the material can withstand.
• Fracture: Point at which the material separates out owing to the maximum strain taken by it.
• Young’s Modulus: The property of a material to withstand compression or elongation when the length changes. It is denoted by the letter Y and can be represented mathematically as:

Y = \(\frac{Stress}{Strain}\)

The unit of Young’s Modulus is N/m² or Pa.

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Strain Energy Formula

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The strain energy (U) stored in an object that undergoes deformation is given by the formula:

U = \(\frac{F\delta}{2}\)

Where, 

U= Strain Energy

F= force applied 

δ= Compression

• When stress is proportional to the strain, strain energy is given by:

U =  \(\frac{1}{2}\) V σ ε

Where, 

U = Strain Energy

σ = Stress on the body

V = Volume of the object

ε = Compression

• For Young’s Modulus E, the strain energy formula is given by:

U =  \(\frac{\sigma^2}{2E}\) x V

U = Strain Energy

σ = Stress on the body

V = Volume of the object

E = Young’s Modulus

Read More: Shearing Stress

Strain Energy Density

Strain energy density is the stress stored in the material when it undergoes deformation uniformly. Hence, the strain energy stored per unit volume of the material is given by:

U = \(\frac{Total Strain Energy}{Volume of the Material}\)

= 12 × stress × strain

= 12 × (Stress )²/E

Strain Energy Formula: Derivation

Let us derive the strain energy formula by making certain assumptions.

Assumptions

The assumptions to arrive at the strain energy formula are:

1. The material is elastic in nature.
2. The stress developed as part of deformation is within the proportional limit.
3. Load application happens gradually.

Let the original length of the material be L and A be the cross-sectional area. Let x be the elongation limit of the material when force is applied to it. 

We know,

Stress = E X Strain

\(\frac{F}{A}\) = E\(\frac{x}{L}\)

⇒ F = \(\frac{EAx}{L}\)

The force is not a constant and increases as the deformation increases. Consider a small deformation dx. The differential work dW is given as per the following equations:

dW = \(\overrightarrow{F}.\overrightarrow{ds} = F dx\)

W = ∫₀^ΔL Fdx

W = ∫₀^ΔL\(\frac{EAx}{L}\)dx

Note: The displacement and the force are in the same direction.

Therefore, 

W = [\(\frac{EAx^2}{2L}\)]₀^ΔL = \(\frac{EA(\Delta L^2)}{2L}\)

Thus, the total strain energy, U for this small deformation is

U = \(\frac{EA(\Delta L^2)}{2L}\)

= \(\frac{1}{2}\) x (E x \(\frac{ \Delta L}{L}\)) x AL x \(\frac{ \Delta L}{L}\)

= \(\frac{1}{2}\) x (E x strain) x AL x strain

= \(\frac{1}{2}\) x Stress x strain x volume of material

This is hence equal to the formula for the Strain energy density: \(\frac{1}{2}\) × (Stress )² / E

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Solved Examples on Strain Energy Formula

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Some important examples to understand the concept of strain energy are discussed below:

Example:1 

If a force of 1500 N is applied on a body, the object gets compressed by 1.5 mm. What is the Strain Energy?

Solution: We know,

Force, F = 1500N

Compression δ = 1.5 mm

Strain energy formula is given by,

U = \(\frac{F\delta}{2}\)

Applying the values in the above formula, we get

= (1500 ×1.5×10^-3 )/ 2

Therefore, U = 1.125 J.

Example:2

A rod has a cross-sectional area of 90 mm² and a length of 3 m. What is the strain energy if 300MPa of stress is applied to stretching the rod. E = 200 GPa. 

Solution:

From the question we know, 

Cross-sectional Area, A = 90 mm²

Length, L = 3m

Stress, σ = 300 MPa

Young’s modulus, E = 200 GPa

Volume, V = area X length

= (90 × 10⁻⁶) × 3

V = 270 x 10⁻⁶ m³

Strain energy formula is given as,

U= \(\frac{\sigma^2}{2E}\)× V

={(300×10⁶)² / [2 x 200×10⁹ ]}x 270 x 10⁻⁶

Thus, strain energy

U = 83.3 x 10⁶ J

= 83.3 x 10⁶ J

Read More: Tensile Stress

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Things to Remember

• Deformation causes an alteration in the structure of the object.
• The alteration from the initial shape causes the object to store energy in the form of strain energy.
• Stress and strain are characteristic of the elastic limits of materials.
• Stress-Strain graph is a characteristic of a material that provides information on the elastic strength and fracture point.
• Strain energy formula is based on the stress applied, the compression suffered and the volume over which the deformation occurred.
• The strain energy stored per unit volume is called the strain energy density.
• The Young’s Modulus is different for different materials and hence the strain energy varies.

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