Elasticity

Elasticity is a fundamental property of solid materials that enables them to regain their original shape and size after the removal of an external force. When a deforming force is applied to a body, it causes a change in its shape or size. If the body returns to its original form after the force is removed, the material is said to be elastic. This behavior is commonly observed in materials like rubber, metal wires, and springs.The study of elasticity involves understanding the concepts of stress and strain.

1.0Definition Of Elasticity

Elasticity is that property of a material of a body by virtue of which it opposes any change in its shape or size when deforming forces are applied on it, and recover its original state as soon as the deforming force is removed.

2.0Terms Associated with Elasticity

1. Deforming Forces: An external force that causes a change in the length, volume, or shape of a body is known as a deforming force.
2. Restoring Forces: When an external force acts on any object then an internal resistance is produced in the material due to the intermolecular forces which are called restoring force.
3. Rigid Body: A body is said to be rigid if the relative positions of its constituent particles remain unchanged when external deforming forces are applied to it. The nearest approach to a rigid body is diamond or carborundum.
4. Perfectly Elastic Body: A body which perfectly regains its original form on removing the external deforming force, is defined as a perfectly elastic body. Example : quartz.It is quite close to a perfect elastic body.
5. Plastic Body: A body which does not have the property of opposing the deforming forces, is known as a plastic body. All bodies which remain in the deformed state even after the removal of the deforming forces are known as plastic bodies. Example : clay, wax, putty.

3.0Stress and Its Types

Stress: The restoring force acting per unit area of the deformed body is called stress.

$Stress=\frac{InternalRestoringForce}{AreaofCross-Section}$  

$Stress=\frac{F_{external}(AtEquilibrium)}{A}$

$S.IUnit:N/m^2$

$Dimensions:[M1L^{-1}{T}^{-2}]$

Types of stress

1. Longitudinal Stress: When the stress is normal to the cross sectional area, then it is known as longitudinal stress.

$LongitudinalStress=\frac{F_{\perp }}{A}$

There are two types of longitudinal stress:

Tensile Stress	Compressive Stress
Tensile stress is the longitudinal stress caused by an increase in a body's length.	The longitudinal stress, produced due to the decrease in the length of a body, is defined as compressive stress.

2. Volume Stress/Hydraulic Stress: If equal normal forces are applied over every unit surface of a body, then it undergoes a certain change in volume. The force opposing this change in volume per unit area is defined as volume stress.

$VolumeStress/HydraulicStress=\frac{F_{\perp }}{A}$

3. Shear Stress/Tangential Stress: Shear stress acts parallel to a surface, causing a change in shape or twist without altering the body's volume.

$Shearstress=\frac{F_t}{A}=\frac{F_{Tangential}}{A}$

4. Breaking Stress :The minimum stress required to cause the actual fracture of a material is called the breaking stress or ultimate strength.                           

$Breakingstress=\frac{F_{Max}}{A}$

$F_{Max}$= force required to break the body

Dependence of breaking stress:

(1) Nature of material 

(2) Temperature

(3) Impurities.

Independence of breaking stress :

(1) Cross sectional area or thickness 

(2) Applied force

Maximum load (force) which can applied on the wire depends on

(1) Cross sectional area or thickness 

(2) Nature of material

(3) Temperature

(4) Impurities

4.0Strain And Its Types

Strain: It is defined as the fraction of the change in length to the material's original length.

$Strain=\frac{Changeinthedimensionofthebody}{OriginalDimensionofthebody}$

• It is a unitless and dimensionless quantity.

1. Longitudinal Strain: When applied force is ⊥ to cross-section, length changes.

$LongitudinalStrain=\frac{changeinlengthofthebody}{initiallengthofthebody}=\frac{\Delta L}{L}$         

2. Volumetric strain: When pressure is applied on the body, volume changes.

$VolumeStrain=\frac{changeinvolumeofthebody}{originalvolumeofthebody}=\frac{\Delta V}{V}$       

3. Shear strain: When applied force is parallel (∥) to cross-section, shape changes.

$tan\varphi =\frac{x}{L}$(Here $\varphi$  is very small)

$\varphi =\frac{x}{L}=\frac{Displacementofupperfacerelativetothelowerface}{distancebetweentwofaces}$

$\varphi$=shear strain OR angle of shear

Relation between angle of twist and angle of shear

When a cylinder of length L and radius r is fixed at one end and a tangential force is applied at the other end, then the cylinder gets twisted. Figure shows the angle of shear Φ  and angle of twist Ө .

$r\theta =L\varphi \Rightarrow \varphi =\frac{r\theta }{L}$

5.0Hooke’s Law

Statement of Hooke’s Law: According to this law within the elastic limit the stress produced in a body is directly proportional to the corresponding strain.

Stress ∝ Strain Stress = E ✕ Strain

E=Coefficient of Elasticity or Modulus of Elasticity

$E=\frac{Stress}{Strain}$

6.0Graph of Hookes’ Law

The slope of the stress & strain graph gives a coefficient of elasticity.

(A) E depends on :

1. Nature of material

2. Impurities

3. Temperature

(B) E independent from:-

1. Stress

2. Strain

7.0Modulus of Elasticity And Its Significance

• The modulus of elasticity depends on the material's nature and not on its dimensions.

$ModulusofElasticity=\frac{Stress}{Strain}$

• Unit :  $\frac{N}{m^2}$ or Pascal (Pa).
• Dimension:$[M^1{L}^{-1}{T}^{-2}]$
• More is the value of Modulus of Elasticity, more is the Elasticity of material.It means more elastic material will have more tendency to regain its shape under elastic limit deformation (not permanent deformation).

Types of Modulus of Elasticity

1. Young's modulus of elasticity

$Youn{g}^{\prime }smodulus(Y)=\frac{Longitudinalstress}{Longitudinalstrain}$

$Youn{g}^{\prime }smodulus(Y)=\frac{\frac{F}{A}}{\frac{\Delta L}{L}}=\frac{FL}{A\Delta L}$

2. Bulk Modulus:  It is defined as the ratio of the volume stress to the volume strain

$B=\frac{Pressure}{VolumetricStrain}=\frac{p}{-\frac{\Delta V}{V}}=-\frac{pV}{\Delta V}$

Negative sign shows that increase in pressure (p) causes decrease in volume ($\Delta V$).

$B=\frac{p}{-\frac{\Delta V}{V}}=-V\frac{\Delta P}{\Delta V}=-V\frac{dP}{dV}$

Compressibility : Compressibility is the inverse of the bulk modulus of elasticity.

 SI Unit-

$N^{-1}{m}^{2}orPasca{l}^{-1}$

$B_{Solids}>B_{Liquids}>B_{Gases}$

Isothermal bulk modulus of elasticity of gas B = P (pressure of gas)

Adiabatic bulk modulus of elasticity of gas $B=\gamma \times P$ Where $\gamma =\frac{C_p}{C_v}$

3. Modulus of Rigidity: Within the elastic limit it is the fraction of shearing stress to shearing strain.

$\eta =\frac{shearingstress}{shearingstrain}$$=\frac{\frac{F_{Tangential}}{A}}{\varphi }$$=\frac{F_{Tangential}}{A\varphi }$

$\eta =\frac{F}{bc.\varphi }$

Note: Angle of shear 'Ф' is always taken in radians

8.0Potential Energy Of a Stretched Wire

When a wire of length 'L' is extended by an external force F , then work has to be done against the restoring force. This work is stored as potential energy of wire.

For spring : W.D.= Elastic Potential energy =$\frac{1}{2}k{x}^2$

For wire:

$W=PE=\frac{1}{2}\left(\frac{YA}{L}\right)(\Delta L{)}^2$

$EPE=\frac{1}{2}\left(\frac{YA}{L}\Delta L\right)(\Delta L)=\frac{1}{2}F(\Delta L)$

$EPE=\frac{1}{2}\left(\frac{F}{A}\cdot \frac{\Delta L}{L}\right)(AL)$

$EPE=\frac{1}{2}(\text{stress})(\text{strain})(\text{volume})$

$EPE=\frac{1}{2}Y(\text{strain}{)}^2(\text{volume})$

$EPE=\frac{1}{2}\left(\frac{(\text{stress}{)}^2}{Y}\right)(\text{volume})$

9.0Potential Energy Density

Potential Energy per unit volume stored in the wire

$PED=\frac{PE}{\text{Vol}}$

$SIUnit-J/m^{3}orN/m^2$

$PED=\frac{1}{2}(\text{stress})(\text{strain})=\frac{1}{2}Y(\text{strain}{)}^2=\frac{1}{2}\left(\frac{(\text{stress}{)}^2}{Y}\right)$

$\text{Note:}slope=\tan \theta =E=\text{Coefficient of Elasticity}$

$\text{Area}=\frac{1}{2}\times \text{stress}\times \text{strain}=\text{Potential Energy Density}$

10.0Poisson’s Ratio$(\sigma )$

longitudinal strain=$\frac{\Delta L}{L}$ ; lateral strain=$\frac{\Delta r}{r}$

$\sigma =\frac{\text{lateral strain}}{\text{longitudinal strain}}\Rightarrow \sigma =-\frac{\Delta r/r}{\Delta L/L}$

-1 ≤ ≤ 0.5(theoretical limit)

=0.2-0.4(experimental limit)

11.0Stress-Strain Relationship And Graphical Analysis

1. O to A: Linear region where stress is proportional to strain — Hooke’s Law is obeyed; material is perfectly elastic.
2. A to B: Non-linear but still elastic — material returns to original shape; B is the elastic limit.
3. Beyond B: Enters plastic region — permanent deformation occurs; strain remains even after removing stress.
4. Point C: Ultimate tensile strength — maximum stress the material can bear before weakening.
5. Point D: Fracture point — material breaks; large plastic region = ductile, small = brittle.

Note: For some materials the elastic region is very large and the material does not obey Hooke's law over most of the region. These are called elastomers e.g. Tissue of Aorta, rubber, etc.