When an external force is applied to a rigid body there is a change in its length, volume (or) shape. When external forces are removed the body tends to regain its original shape and size. Such a property of a body by virtue of which a body tends to regain its original shape (or) size when external forces are removed is called elasticity.

## Elastic Stress and Strain

### What is Stress?

When the body is deformed by the application of external forces, forces within the body are brought into play. Elastic bodies regain their original shape due to internal restoring forces. The internal forces and external forces are opposite in direction. If a force F is applied uniformly over a surface of area A then the stress is defined as the force per unit area.

Stress = Force/Area

S.I unit for stress is Nm^{-2}

### Types of Stress

There are three types of stress

- Longitudinal stress
- Volume stress or Bulk Stress
- Tangential stress (or) shear stress

#### Longitudinal Stress

When the stress is normal to the surface area of the body and there is a change in the length of the body it is known as longitudinal stress.

Again it is classified into two types

- Tensile stress
- Compressive stress.

**Tensile stress:** When longitudinal stress produced due to an increase in the length of the object is known as tensile stress.

**Compressive stress:**Longitudinal stress produced due to the decrease in length of the object is known as compressive stress.

#### Volume Stress or Bulk Stress

If equal normal forces applied to the body causes a change in volume of the body, the stress is called volume stress.

#### Tangential Stress

When the stress is tangential (or) parallel to the surface of the body is known as Tangential (or) Shear stress. Due to this shape of body changes (or) gets twisted.

### What is a Strain?

A body under stress gets deformed. The fractional change in the dimension of a body produced by the external stress acting on is called strain. The ratio of charge of any dimension to its original dimension is called strain. Since strain is the ratio of two identical physical quantities, it is just a number. It has no unit and dimension.

\(\begin{array}{l}Strain=\frac{Change\,in\,dimension}{initial\,dimension}\end{array} \)

**Strain also classified into three types**

- Longitudinal strain
- Volume strain
- Shearing strain or tangential strain

#### Longitudinal Strain

The strain under longitudinal stress is called longitudinal strain.

\(\begin{array}{l}Longitudinal\,strain=\frac{Change\,in\,length\,of\,the\,body}{initial\,length\,of\,the\,body}\end{array} \)

\(\begin{array}{l}=\frac{\Delta L}{L}\end{array} \)

#### Volume Strain

The strain caused by the volume stress is called volume strain.

\(\begin{array}{l}Volume\,strain=\frac{Change\,in\,volume\,of\,the\,body}{\,Original\,volume\,of\,the\,body}\end{array} \)

\(\begin{array}{l}=\frac{\Delta V}{V}\end{array} \)

#### Shearing Strain

When a deforming force is applied to a body parallel to its surface its shape (not size) changes this is known as shearing strain. The angle of shear Φ

\(\begin{array}{l}\tan \phi =\frac{\ell }{L}=\frac{displacement\,of\,upper\,face}{distance\,between\,two\,faces}\end{array} \)

### Stress- Strain Curve

**Proportion limit:**The limit in which Hooke’s law is valid and stress is directly proportional to strain.**Elastic limit:**That maximum stress which on removing the deforming force makes the body recoverits original state completely.**Lower Yield point:**The point beyond the elastic limit at which the length of the wire starts increasing with increasing stress. It is defined as the yield point.**Breaking point or Fracture Point:**The point when the strain becomes so large that the wire breaks is called the breaking point.

#### Elastic hysteresis

The strain persists even when the stress is removed. This lagging behind of strain is called elastic hysteresis. This is why the values of strain for the same stress are different while increasing the load and while decreasing the load.

## Hooke’s Law

If deformation is small, the stress in a body is proportional to the corresponding strain, this fact is known as Hooke’s law.

Within elastic limit, Stress & strain

\(\begin{array}{l}\Rightarrow \frac{Stress}{Strain}=Constant\end{array} \)

This constant is known as modulus of elasticity (or) coefficient of elasticity. The elastic modulus has the same physical unit as stress. It only depends on the type of material used. It is independent of stress and strain. The modulus of elasticity is of three types.

- Young’s modulus of elasticity “y”
- The bulk modulus of elasticity “B”
- Modulus of rigidity

### Young’s modulus of elasticity “y”

Within the elastic limit, the ratio of longitudinal stress and longitudinal strain is called Young’s modulus of elasticity (y).

\(\begin{array}{l}y=\frac{Longitudinal\,stress}{Longitudinal\,strain}=\frac{\frac{F}{A}}{\frac{\ell }{L}}=\frac{FL}{A\ell }\end{array} \)

Within the elastic limit, the force acting upon a unit area of a wire by which the length of wire becomes double is equivalent to Young’s modulus of elasticity of the material of the wire. If L is the length of wire, r – radius and is the increase in the length of wire by suspending a weight (mg) at its one end then young’s modulus of elasticity of the wire becomes,

\(\begin{array}{l}y=\frac{F/A}{\ell /L}=\frac{FL}{A\ell }=\frac{mgL}{\pi {{r}^{2}}\ell }\end{array} \)

**(a) The increment of the length of an object by its own weight:**

Let a rope of mass M and length (L) is hanged vertically. As the tension of different point on the rope is different, similarly stress as well as the strain will be different at different points.

- Maximum stress at hanging point
- Minimum stress at a lower point

Consider a dx element of rope at x distance from the lower end, then tension;

\(\begin{array}{l}T=\left( \frac{M}{L} \right)\times g\end{array} \)

So stress

\(\begin{array}{l}=\frac{T}{A}=\left( \frac{M}{L} \right)\frac{xg}{A}\end{array} \)

Let the increase in length of element dx is dy then

\(\begin{array}{l}Strain=\frac{Change\,in\,length}{Original\,length}=\frac{\Delta y}{\Delta x}=\frac{dy}{dx}\end{array} \)

Now we got stress and strain then young’s modulus of elasticity “y”

\(\begin{array}{l}y=\frac{Stress}{Strain}=\frac{\left( \frac{M}{L} \right)\frac{xg}{A}}{\frac{dy}{dx}}\Rightarrow \left( \frac{M}{L} \right)\frac{xg}{A}dx=\frac{1}{dy}\end{array} \)

The total change in length of the wire is

\(\begin{array}{l}\frac{Mg}{LA}\int\limits_{o}^{L}{x\,dx}=y\int\limits_{o}^{\Delta l}{dy}\end{array} \)

\(\begin{array}{l}\frac{Mg}{LA}\frac{{{L}^{2}}}{2}=y\Delta \ell\end{array} \)

\(\begin{array}{l}\frac{MgL}{2Ay}=\Delta \ell\end{array} \)

**(b) Work done in stretching a wire**

If we need to stretch a wire, we have to do work against its inter atomic forces, which is stored in the form of elastic potential energy.

For a wire of length (L_{0}) stretched by a distance (x) the restoring elastic force is

\(\begin{array}{l}F=(Stress)(Area)=y\left[ \frac{x}{{{L}_{0}}} \right]A\end{array} \)

Work required for increasing an element length

\(\begin{array}{l}dW=Fdx=\frac{{{y}_{A}}}{{{L}_{0}}}x\,dx\end{array} \)

Total work required in stretching the wire is

\(\begin{array}{l}W=\int\limits_{0}^{\Delta \ell }{Fdx=\frac{{{y}_{A}}}{{{L}_{0}}}}\int\limits_{0}^{\Delta \ell }{x\,dx}\end{array} \)

\(\begin{array}{l}=\frac{{{y}_{A}}}{{{L}_{0}}}\left[ \frac{{{x}^{2}}}{2} \right]_{0}^{\Delta \ell }\end{array} \)

\(\begin{array}{l}=\frac{{{y}_{A}}{{\left( \Delta \ell \right)}^{2}}}{2{{L}_{0}}}\end{array} \)

**(c) Analogy of rod as a spring**

From definition of young’s modulus

\(\begin{array}{l}y=\frac{Stress}{Strain}=\frac{FL}{A\,\Delta L}\end{array} \)

\(\begin{array}{l}F=\frac{{{y}_{A}}\Delta L}{L}\end{array} \)

**This expression is analogy of spring force**

\(\begin{array}{l}F=kx\end{array} \)

\(\begin{array}{l}k=\frac{yA}{L}=\text{constant}\end{array} \)

### Bulk Modulus (B)

Within elastic limit the ratio of the volume stress and the volume strain is called bulk modulus of elasticity.

\(\begin{array}{l}B=\frac{Volume\,stress}{Volume\,strain}=\frac{\frac{F}{A}}{-\frac{\Delta V}{V}}=\frac{\Delta P}{\frac{-\Delta V}{V}}\end{array} \)

### Rigidity Modulus

Within elastic limit, the ratio of shearing stress (or) tangential stress and shearing strain (or) tangential strain is called modulus of rigidity.

\(\begin{array}{l}\eta =\frac{Shearing\,stress}{Shearing\,strain}=\frac{\frac{{{F}_{tangential}}}{A}}{\phi }\end{array} \)

Φ – angle of shear.

### Poisson’s Ratio

Within the elastic limit, the ratio of lateral strain (or) transverse train and longitudinal strain is called Poisson’s ratio. In case of a circular bar of material, the change in the diameter of the circular bar material to its diameter due to deformation in the longitudinal direction.

\(\begin{array}{l}Poisson’s\,ratio(\sigma )=\frac{lateral\,strain}{longitudinal\,strain}=\frac{\beta }{\alpha }\end{array} \)

### You might also be interested in:

- HC Verma Solutions
- HC Verma Solutions Part 1
- HC Verma Solutions Part 2
- JEE Main Elasticity Previous year Questions

## Hooke’s Law

## Frequently Asked Questions on Elasticity

### What is elasticity?

The ability of the deformed objects to regain their actual shape and size when the force causing the deformation is removed.

### What are the types of modulus of elasticity?

There are three types of modulus of elasticity namely Young’s modulus, shear modulus and bulk modulus.

### What is the SI unit for modulus of elasticity?

The SI unit for modulus of elasticity is Pascal.

### Define Hooke’s law.

Within the elastic limit of the material, the strain caused in the material is proportional to the applied stress.

### Define Young’s Modulus of Elasticity.

Young’s modulus of elasticity is the ratio of the normal stress to the longitudinal strain.

### Is stress a vector quantity?

No. Stress is a scalar quantity.

### The Young’s modulus of steel is much more than that of rubber. For the same longitudinal strain, which one will have greater tensile stress?

Tensile stress = Young’s modulus x longitudinal strain.

Therefore, steel will have greater tensile stress.

### What are ductile materials?

The materials whose plastic range is comparatively large.

## FAQs

### What are the different types of modules of elasticity? ›

There are three types of modulus of elasticity namely **Young's modulus, shear modulus and bulk modulus**.

**What do you mean by modules of elasticity? ›**

: **the ratio of the stress in a body to the corresponding strain** (as in bulk modulus, shear modulus, and Young's modulus) called also coefficient of elasticity, elastic modulus.

**What is elasticity and modulus of elasticity? ›**

The Elastic Modulus is **the measure of the stiffness of a material**. In other words, it is a measure of how easily any material can be bend or stretch. It is the slope of stress and strain diagram up to the limit of proportionality.

**What are the three types of modulus of elasticity? ›**

**Three types of moduli of Elasticity.**

- 1∙ Youngs modulus − tensile elasticity.
- 2∙ Shear modulus − modulus of rigidity, shear elasticity.
- 3∙ Bulk modulus − volumetric elasticity.

**What are the 4 types of elasticity? ›**

What Are the 4 Types of Elasticity? Four types of elasticity are **demand elasticity, income elasticity, cross elasticity, and price elasticity**.

**What are the 5 types of elasticity? ›**

Elasticities can be usefully divided into five broad categories: **perfectly elastic, elastic, perfectly inelastic, inelastic, and unitary**.

**What is elasticity and its types? ›**

On the basis of different factors affecting the quantity demanded for a product, elasticity of demand is categorized into mainly three categories: **Price Elasticity of Demand (PED), Cross Elasticity of Demand (XED), and Income Elasticity of Demand (YED)**. Let us look at them in detail and their examples.

**Why modulus of elasticity is important? ›**

Modulus of elasticity is a measure of the stress–strain relationship and is **an important parameter in the evaluation of the deformation response of concrete under working loads**. The load-deformation behaviour of concrete is in fact non-linear, though generally in practice, an elastic modulus is adopted for convenience.

**How do you find the modulus of elasticity? ›**

**Modulus =(σ2 - σ1) / (ε2 - ε1)** where stress (σ) is force divided by the specimen's cross-sectional area and strain (ε) is the change in length of the material divided by the material's original gauge length.

**What is elasticity and its properties? ›**

Elasticity is defined as **a physical property of materials which return to their original shape after the stress that had caused the deformation is no longer applied**.

### What is the meaning of the term elasticity? ›

1. uncountable noun. The elasticity of a material or substance is **its ability to return to its original shape, size, and condition after it has been stretched**.

**What is elasticity and its causes? ›**

Elasticity is **the ability of a body to resist any permanent change to it when stress is applied**. Causes of elasticity. One of the main cause of elasticity is applied external force. Elasticity is also caused due to restoring force of the material.

**What are the 3 factors that determine elasticity? ›**

Factors that determine the elasticity of demand would be **the availability of substitutes, the share of the good's expense in individuals' income, and the passage of time**. More substitutes imply individuals have more choices and therefore consumers are more sensitive to price changes.

**What are the three examples of elastic materials? ›**

Some of the examples of elastic materials are: **Bungee Jumping**. Elastic Waistband. Rubber Bands.

**What are the 3 cases of demand elasticity? ›**

Demand can be classified as **elastic, inelastic or unitary**. An elastic demand is one in which the change in quantity demanded due to a change in price is large.

**What are the 4 Determinants of elasticity of supply? ›**

Determinants of Elasticity of Supply

**Effortlessness of switching**. Ease of storage. Length of the period of production. The time frame of training.

**What does a 5 elasticity of demand mean? ›**

As a rule of thumb, **if the quantity of a product demanded or purchased changes more than the price changes**, then the product is considered to be elastic (for example, the price goes up by 5%, but the demand falls by 10%).

**What is the best definition of elasticity in economics? ›**

Elastic is a term used in economics to describe **a change in the behavior of buyers and sellers in response to a change in price for a good or service**. In other words, demand elasticity or inelasticity for a product or good is determined by how much demand for the product changes as the price increases or decreases.

**What is elasticity give two examples? ›**

Elasticity is **the ability of an object or material to resume its normal shape after being stretched or compressed**. Example: A rubber regains its shape after long stretch because of its elastic property.

**What are the main types of elasticity of demand? ›**

The four main types of elasticity of demand are **price elasticity of demand, cross elasticity of demand, income elasticity of demand, and advertising elasticity of demand**.

### What increases elastic modulus? ›

The authors reported that **heat treatment** has positive effect on elastic modulus and increases it by 8.4%. The use of suitable heat treatment technique brings about a significant improvement to the compressive strength as well as elastic modulus, due to the enhanced adhesion between OPS and the cement matrix. Yew et al.

**How important is elasticity in our daily lives? ›**

**Elasticity plays a very important role in daily life in designing a structure which undergoes different types of stress**. Some of the examples are: In order to calculate the elastic limit of metallic machineries so they cannot be subjected above a particular level of stress.

**What does modulus tell us? ›**

The Young's modulus (E) is a property of the material that tells us **how easily it can stretch and deform** and is defined as the ratio of tensile stress (σ) to tensile strain (ε).

**What is the difference between yield stress and Young's modulus? ›**

Traditionally, **Young's modulus is used up to the material's yield stress**. (Yield stress is the stress at which a material begins to deform plastically. Prior to the yield point, the material deforms elastically and returns to its original shape when the applied stress is removed.)

**What is the basic formula of elasticity? ›**

...

Summary.

Base Formula | Mid-Point Formula | Point-Slope Formula |
---|---|---|

%ΔQuantity%ΔPrice | ΔQΔP⋅(P1+P2)(Q1+Q2) | ΔQΔP⋅PQ |

**How do you calculate elastic value? ›**

To calculate price elasticity, **divide the change in demand (or supply) for a product, service, resource, or commodity by its change in price**. That figure will tell you which bucket your product falls into.

**What is the law of elasticity? ›**

Hooke's law, law of elasticity discovered by the English scientist Robert Hooke in 1660, which states that, **for relatively small deformations of an object, the displacement or size of the deformation is directly proportional to the deforming force or load**.

**What is the unit for elasticity? ›**

Units of Modulus of Elasticity/Young's modulus are: **Nm ^{-}^{2} or Pa**. The practical units used in plastics are megapascals (MPa or N/mm

^{2}) or gigapascals (GPa or kN/mm

^{2}). In the United States customary units, it is often expressed as pounds (force) per square inch (psi).

**Which material is most elastic? ›**

The correct answer is Steel. Steel is the most elastic material. If the object is elastic, the body regains its original shape when the pressure is removed. Steel having the steepest linear stress-strain curve among all.

**What is meaning of elasticity of demand? ›**

An elastic demand is **one in which the change in quantity demanded due to a change in price is large**. An inelastic demand is one in which the change in quantity demanded due to a change in price is small. The formula used here for computing elasticity.

### What are the properties of elastic materials? ›

Properties of Elastic Material

Elastic materials are **always inversely proportional to the strain of the material**. In which the material remains the same before and after removing the externally applied load. Elastic material having a limit of loading is known as the elastic limit.

**How is elasticity made? ›**

Elastic is made from **a series of rubber (or stretchable synthetic, such as spandex) cores that are bound or wrapped in polyester, cotton, nylon or a blend of fiber threads**. The exterior threads are braided, woven or knit together to create the elastic.

**What are the 3 questions asked to determine demand elasticity? ›**

**Let's examine the three questions.**

- Can the purchase be delayed? The ability to delay or postpone the purchase of a product is one of the determinants of elasticity. ...
- Are adequate substitutes available? ...
- Does the purchase use a large portion of income?

**What are the two types of elasticity? ›**

As mentioned above in the blog, there are mainly two types of elasticity- **Elasticity of Demand and Elasticity of Supply**. Elasticity of demand is an economic measure of the sensitivity of demand relative to a change in another variable.

**What are the forms of modulus? ›**

**There are three different types of elastic modulus:**

- young's modulus.
- shear modulus.
- bulk modulus.

**What are the units of modulus of elasticity? ›**

Units of Modulus of Elasticity/Young's modulus are: **Nm ^{-}^{2} or Pa**. The practical units used in plastics are megapascals (MPa or N/mm

^{2}) or gigapascals (GPa or kN/mm

^{2}). In the United States customary units, it is often expressed as pounds (force) per square inch (psi).