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. This property, by which a body tends to return to its original shape or size when external forces are removed, is called elasticity.

Elastic Stress and Strain

What is Stress?

Stress is a feeling of emotional or physical tension. It can come from any event or thought that makes you feel frustrated, angry, or nervous. Stress is your body’s reaction to a challenge or demand. In short, it is the body’s way of rising to a challenge and preparing to meet a tough situation with focus, strength, stamina, and heightened alertness.

When the body is deformed by the application of external forces, internal forces within the body are activated to restore its original shape. These internal forces and external forces act in opposite directions. The stress is defined as the force per unit area, when a force F is uniformly distributed over an area A.

Stress = $\frac{Force}{Area}$

The S.I. unit for stress is N/m²

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 parallel 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 a force is applied to an object, causing it to elongate, the resulting longitudinal stress is known as tensile stress.

Compressive Stress: Longitudinal stress caused by a decrease in length of an object is referred to as compressive stress.

Volume Stress or Bulk Stress

If equal normal forces applied to a body cause a change in its volume, the stress is referred to as volume stress.

Tangential Stress

The stress that is tangential or parallel to the surface of a body is known as Tangential or Shear Stress. This type of stress can cause the shape of the body to change or become twisted.

What is a Strain?

A strain is a biological variety of a species that has distinct characteristics and is usually identified by its geographic origin or the source from which it was derived.

The fractional change in the dimension of a body produced by the external stress acting on it is known as strain. It is a ratio of the change of any dimension to its original dimension and hence, is simply a number with no unit or dimension.

$$\begin{array}{l}Strain=\frac{Changeinlength}{Initiallength}\end{array}$$

• Strain is classified into three types
  • Longitudinal Strain
  • Volume Strain
  • Shearing Strain or Tangential Strain

Longitudinal Strain

The strain under longitudinal stress is called **longitudinal strain.

$$\begin{array}{l}Longitudinalstrain=\frac{Changeinlengthofthebody}{Initiallengthofthebody}\end{array}$$

$\frac{\mathrm{\Delta }L}{L}$

Volume Strain

The strain caused by the volume stress is referred to as volume strain.

$$\begin{array}{l}VolumeStrain=\frac{ChangeinVolumeoftheBody}{OriginalVolumeoftheBody}\end{array}$$

$\frac{\mathrm{\Delta }V}{V}$

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 is denoted by Phi.

$$\begin{array}{l}\tan \varphi =\frac{\text{displacement of upper face}}{\text{distance between two faces}}=\frac{\ell }{L}\end{array}$$

Stress-Strain Curve

1. Proportion Limit: Hooke’s Law is valid up to a certain limit, where stress is directly proportional to strain.

2. Elastic Limit: The maximum stress at which a body can be subjected to before it no longer returns to its original state when the deforming force is removed.

3. Lower Yield Point:** The point at which stress applied to a wire exceeds its elastic limit, resulting in an increase in the wire’s length. This point is defined as the yield point.

4. Fracture Point: The point at which the strain becomes so large that the wire breaks is referred to as the fracture 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

This fact, known as Hooke’s law, states that if the deformation is small, the stress in a body is proportional to the corresponding strain.

Within elastic limit, Stress and Strain $\Rightarrow \frac{\mathrm{Stress}}{\mathrm{Strain}}=\mathrm{Constant}$

The modulus of elasticity (or coefficient of elasticity) is a constant that is used to describe the elasticity of a material. It has the same physical unit as stress and is only dependent on the type of material used, being 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)

The ratio of longitudinal stress to longitudinal strain within the elastic limit is known as Young’s Modulus of Elasticity (Y).

$\begin{array}{l}y=\frac{\mathrm{Longitudinal}\mathrm{Stress}}{\mathrm{Longitudinal}\mathrm{Strain}}=\frac{F}{A}\cdot \frac{L}{\ell }=\frac{FL}{A\ell }\end{array}$

The Young’s modulus of elasticity of the wire can be calculated using the formula $$E=\frac{mgL}{\pi r^2\mathrm{\Delta }L}$$ where L is the initial length of the wire, r is the radius of the wire, ΔL is the increase in the length of the wire due to the force of gravity (mg).

$\frac{mgL}{\pi r^2\ell }=\frac{F}{A}\cdot \frac{L}{\ell }$

(a) The increase in the length of an object due to its own weight:

The rope of mass M and length (L) that is hanged vertically will experience different tensions, stresses, and strains at different points.

• Maximum Stress at Hanging Point
• Minimum stress at a lower level

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

$$T=\left(\frac{M}{L}\right)\times g$$

So stress $$=\frac{T}{A}=\left(\frac{M}{L}\right)\frac{xg}{A}$$

If 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 have stress and strain, then we can calculate Young’s Modulus of Elasticity (y).

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

The total change in length of the wire is

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

$$\frac{Mg}{LA}\frac{{{L}^{2}}}{2}=y\Delta \ell$$

$$\frac{MgL}{2Ay}=\Delta \ell$$

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 then stored as elastic potential energy.

For a wire of length (L_0) stretched by a distance (x), the restoring elastic force is

$F=(Stress)(Area)=\frac{y[x/L_0]A}{}$

Increasing the Length of an Element

‘\(dW=Fdx=\frac{{{y}_{A}}}{{{L}_{0}}}x,dx\)’

Total work required in stretching the wire is

$W={\int }_0^{\mathrm{\Delta }\ell }Fdx=\frac{y_A}{L_0}{\int }_0^{\mathrm{\Delta }\ell }xdx$

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

$\frac{y_A{\left(\mathrm{\Delta }\ell \right)}^2}{2L_0}$

Analogy of a Rod as a Spring

From Definition of Young’s Modulus

Young’s modulus is a measure of the stiffness of a material and is used to describe the elastic properties of objects like wires, rods, and columns. It is defined as the ratio of the stress applied to the material to the resulting strain within the material.

\begin{array}{l}y=\frac{Force}{Area \times Change \ in \ Length}=\frac{FL}{A,\Delta L}\end{array}

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

An analogy of spring force can be seen in this expression.

begin{array}{l}F=k\cdot x\end{array}

\begin{array}{l}k=\frac{yA}{L} \text{ is a constant}\end{array}

Bulk Modulus (B)

The ratio of the volume stress to the volume strain within the elastic limit is known as the Bulk Modulus of Elasticity.

\begin{array}{l}B=\frac{\Delta P \cdot V}{-\Delta V} \end{array}

Rigidity Modulus

The ratio of shearing stress (or tangential stress) to shearing strain (or tangential strain) within the elastic limit is known as the Modulus of Rigidity.

$\eta =\frac{ShearingStress}{ShearingStrain}=\frac{\frac{F_{Tangential}}{A}}{\varphi }$

Φ - Angle of Shear.

Poisson’s Ratio

The Poisson’s ratio is the ratio of lateral strain (or transverse strain) to longitudinal strain within the elastic limit. When a circular bar of material is deformed in the longitudinal direction, the change in its diameter is proportional to its diameter.

$$Poisson’s,ratio(\sigma )=\frac{lateral\ strain}{longitudinal\ strain}=\frac{\beta }{\alpha }$$

Frequently Asked Questions on Elasticity

What is Elasticity?

Elasticity is a measure of the responsiveness of a given variable to a change in another variable. It measures the degree to which a change in one variable affects a change in another variable.

The ability of 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?

• Young’s Modulus
• Bulk Modulus
• Shear Modulus

There are three types of modulus of elasticity:

1. Young’s Modulus
2. Shear Modulus
3. Bulk Modulus

The SI unit for modulus of elasticity is Pa (Pascal).

The SI unit for modulus of elasticity is Pascal).

Define Hooke’s law.

• Hooke’s law states that the force (F) needed to extend or compress a spring by some distance x scales linearly with respect to that distance. Mathematically, it can be written as F = -kx, where k is a constant factor characteristic of the spring, called its stiffness.
• Within the elastic limit of the material, the strain caused is directly proportional to the applied stress.

Define Young’s Modulus of Elasticity.

• Young’s Modulus of Elasticity (also known as the Elastic Modulus or the Young Modulus) is a measure of the stiffness of a material and is calculated as the ratio of stress to strain in a material when it is subjected to a tension or compression.
• Young’s modulus of elasticity** is the ratio of normal stress to 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?

Steel will have greater tensile stress than rubber for the same longitudinal strain due to its higher Young’s modulus.

Tensile stress = (Young’s modulus) × (longitudinal strain)

Therefore, steel will have higher tensile stress.

What are Ductile Materials?

Ductile materials are materials that can be deformed plastically without breaking. They are malleable and can be stretched or drawn into various shapes without fracturing. Examples of ductile materials include metals such as gold, silver, copper, and aluminum.

The materials with a wide range of plastics.