## What is the meaning of strain?

The term strain indicates the variation in an object's length, area, and volume when the object is subjected to an external force. The external force may be of normal, shear, moment, torque, etc. The strain in an object depends on the types of external force and cross-section of the object. Mathematically, the evaluation of strain can be obtained with the help of elastic constants.

## What is the meaning of strain energy?

Whenever an object is subjected to different load types (gradual, sudden, or impact load), the object deforms, and work is done upon it. If the elastic limit of the object does not exceed, then the work would be stored in the object as potential energy. This amount of energy stored in the object throughout the volume internally refers to strain energy.

## Strain energy due to different types of loading

Whenever different types of load act on an object, the value of strain energy depends on the types of loading on the object. The work done by the load on the object referred to as strain energy is stored in the object. The strain energy due to different types of loadings are as follows:

- Strain energy due to gradual loading
- Strain energy due to sudden loading
- Strain energy due to shock/impact loading
- Strain energy due to shear loading
- Strain energy due to pure torsion
- Strain energy due to bending (flexure)

### Strain energy due to gradual loading

When a specific gradual force acts on one end of a rectangular object with the other end fixed, a strain is generated in the object due to the applied force. The area of the curve between the force and extension in the object represents the work done by the force or strain energy. The expression of the strain energy due to gradual force can be represented as:

$U=\left(\frac{{\sigma}^{2}}{2E}\right)V$

Here, $U$ represents the strain energy due to gradual load, $\sigma $ represents the stress in the object due to gradual load, $E$ represents the Young modulus, $V$ represents the volume of the object, $\u2206L$ represents the extension in object's length and $A$ represents the cross-sectional area of the object.

### Strain energy due to sudden loading

Whenever a specific load is applied suddenly on an object, the load is constant throughout the object's deformation process. The area of the curve between the load and extension in the object represents the strain energy. The relation between the stress due to gradual load and the stress due to suddenly applied load is given below:

${\left(\sigma \right)}_{S}=2{\left(\sigma \right)}_{G}$

Here, ${\left(\sigma \right)}_{S}$ represents the stress in the object due to sudden applied load and ${\left(\sigma \right)}_{G}$ represents the stress in the object due to gradual load.

The expression of the strain energy due to the suddenly applied load can be represented below:

${U}_{s}=4\left(\frac{{\sigma}^{2}}{2E}\right)V$

Here, ${U}_{s}$ represents the strain energy due to the sudden loading.

### Strain energy due to impact/shock loading

Whenever a specific load is dropped from a certain height on the surface of a collar and bar system fixed at one end, it refers to the impact loading on the bar and collar system. Due to the impact of load on the collar, there would be a strain generated in the bar. The expression of the strain in the bar can be represented in the step below:

$U=\left(\frac{{\sigma}^{2}}{2E}\right)\left(A\times L\right)$

Here, $U$ represents the strain energy in the bar due to impact load, $A$ represents the cross-sectional area of the bar, and $L$ represents the length of the bar.

The expression of stress in the bar can be represented as,

$\sigma =\frac{P}{A}\left[1+\sqrt{1+\frac{2hAE}{PL}}\right]$

Here, $\sigma $ represents the stress in the bar, $h$ represents the height of the load above the collar, $P$ represents the load that acts on the bar, $L$ represents the length of the bar, and $A$ represents the cross-sectional area of the bar.

### Strain energy due to shear loading

When a shear load/force acts on a rectangular object that is fixed on a plane surface, then due to the shear load on the object's top surface, an angular strain is developed in the object. The expression of the strain energy due to shear loading within elastic limit can be represented as:

$U=\left(\frac{{\tau}^{2}}{2G}\right)V$

Here, $U$ represents the strain energy due to shear loading, $\tau $ represents the shear stress generated in the rectangular object, $G$ represents the modulus of rigidity of object material, and $V$ represents the volume of the object.

### Strain energy due to pure torsion

Whenever a pure torsional load acts on one end of a circular rod which is fixed at the other end, there would be shear stress generated in the rod, and due to this shear stress, an angular strain will be developed. Under the elastic limit, due to the induced strain, there would be strain energy stored by the rod that can be represented by using a mathematical expression as follows:

$U=\left(\frac{{\tau}^{2}}{4G}\right)V$

Here, $U$ represents strain energy stored due to torsion, $\tau $ represents the shear stress in the rod and $V$ represents the volume of the rod.

### Strain energy due to bending (flexure)

Whenever a specific uniformly distributed load acts on the entire length of a simply supported beam, then there is a bending moment that works on the beam and results in a displacement in the beam. Due to this, there would be strain energy stored in the beam within the elastic limit that can be represented by using a mathematical expression as,

$U={\int}_{0}^{L}\left(\frac{{M}^{2}}{2EI}\right)dx$

Here, $U$ represents the strain energy in the beam, $M$ represents the bending moment in the beam, $L$ represents the length of the beam, $E$ represents the Young modulus and $I$ represents the moment of inertia.

## Strain energy due to elastic property of material

The strain energy due to the elastic property of the material consists of the terms, resilience, proof resilience, modulus of resilience, and modulus of toughness. The details about these three terminologies are given as follows:

- Resilience
- Proof resilience
- Modulus of resilience

### Resilience

The term resilience indicates the ability of a material to regain its original shape before the loading of the material. After loading the material, there are changes in the shape within the elastic limit. In other words, the term resilience refers to the strain energy stored in material within the elastic limit.

### Proof resilience

The term proof resilience refers to the value of the maximum amount of strain energy stored in a material up to its elastic limit. The mathematical expression of proof resilience can be represented as follows:

${U}_{p}=\left(\frac{{{\sigma}_{e}}^{2}}{2E}\right)V$

Here, ${U}_{p}$ represents the proof resilience and ${\sigma}_{e}$ represents the stress induced up to elastic limit.

### Modulus of resilience

The term modulus of resilience of an object under loading refers to the ratio of proof resilience to object volume. The mathematical expression of the modulus of resilience can be represented as follows:

$Modulusofresilience=\frac{Proofresilience}{Volumeofobject}\phantom{\rule{0ex}{0ex}}=\frac{\left({\displaystyle \frac{{\sigma}^{2}}{2E}}\right)V}{V}\phantom{\rule{0ex}{0ex}}=\frac{{\sigma}^{2}}{2E}$

### Modulus of toughness

The term modulus of toughness represents the amount of energy stored by an object under specific loading before it fractures. The area between the graph of stress and strain before the fracture point of material represents the modulus of toughness. The graph of modulus of toughness can be represented as,

### Common mistakes

- Students sometimes get confused about the difference between the effect of stress and strain developed in an object due to gradual and sudden applied loads. However, due to gradually applied load, there would be a linear variation of load throughout the object's deformation, whereas due to sudden load, there would be a constant load throughout the object's deformation.
- Sometimes, students also get confused about why the formula of Poisson's ratio has a negative sign in it. However, there would be a positive expansion (positive strain) along the longitudinal direction under tensile loading, whereas a contraction (negative strain) along the lateral direction.
- Students also get confused about the difference between resilience and proof resilience. However, resilience represents the amount of energy stored within the elastic limit, whereas proof resilience represents the amount of energy stored in an object up to the elastic limit of the material.

## Context and Applications

Strain energy is very significant in several professional exams and courses for undergraduate, Diploma level, graduate, postgraduate. For example:

- Bachelor of Technology in Mechanical Engineering
- Bachelor of Technology in Civil Engineering
- Master of Technology in Mechanical and Civil Engineering
- Doctor of Philosophy in Mechanical Engineering
- Diploma in Mechanical Engineering
- Diploma in Civil Engineering

## Related concepts

- Hooke's law
- Strain energy density
- Fracture point and stress
- Poisson's ratio
- Lateral strain
- Longitudinal strain
- Strength of material
- Failure criterion of materials
- Elasticity and plasticity

## Practice problems

**Q 1**. The other equivalent name of the term resilience is known as:

a. Strain energy

b. Modulus of energy

c. Tenacity

d. None of these

Correct option:** (a)**

**Q 2**. Strain energy can be represented in the form of a mathematical expression which is equal to:

a. Young modulus

b. Work done

c. Power

d. Shear modulus

Correct option:** (b)**

**Q 3**. The expression of strain energy stored in an object under a gradual load is:

a. $U=\left(\frac{{\sigma}^{2}}{2E}\right)\left(A\times L\right)$

b. $U=\left(\frac{{\sigma}^{2}}{E}\right)\left(A\times L\right)$

c. $U=\left(\frac{\sigma}{2E}\right)\left(A\times L\right)$

d. $U=\left(\frac{{\sigma}^{2}}{4E}\right)\left(A\times L\right)$

Correct option:** (a)**

**Q 4. **The relation of stress developed in an object due to sudden load would be equal to:

a. Stress developed when the load applied gradually.

b. half of the stress developed when the load is applied gradually.

c. four times of the stress developed when the load is applied gradually.

d. twice of the stress developed when the load is applied gradually.

Correct option:** (d)**

**Q 5**. Mathematically, the modulus of resilience can be represented as:** **

a. The ratio of maximum strain and unit volume of the object.

b. The ratio of minimum strain and unit volume of the object.

c. The ratio of proof resilience and unit volume of the object.

d. None of these

Correct option:** (c)**

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