Revision: Class 11 >> Mechanical Properties of Solids NEET (UG) Mechanical Properties of Solids

A body that regains its original shape and size after removal of the deforming force is called an elastic body, and the property is called elasticity.

OR

The property by which a body returns to its original shape after the removal of a deforming force is called elasticity.

When a solid is deformed and returns to its original shape upon removal of the force, the deformation is called elastic deformation.

If a body regains its original shape and size after removal of the deforming force, it is called an elastic body and the property is called elasticity.

The permanent deformation in substances (like putty and mud) that do not return to their original shape after the deforming force is removed is called plastic deformation (or plasticity).

A body that regains its original shape and size completely and instantaneously upon removal of the deforming force is said to be perfectly elastic.

A body that does not regain its original shape and size and retains its altered shape or size upon removal of the deforming force is called a plastic body, and the property is called plasticity.

The internal restoring force per unit area of a body is called stress.

OR

The internal restoring force acting per unit area of a deformed body is called stress.

• SI Unit: N/m² (pascal, Pa)
   Dimensions: [M¹L⁻¹T⁻²]

The ratio of change in length of the body to its initial length is called longitudinal strain: ε = ΔL/L.

The angular displacement of the surface in direct contact with the applied shear stress from its original position is called shear strain: τ = W/L = tan ⁡θ.

When there is an increase in the length or extension of the body in the direction of the applied force, the stress produced is called tensile stress.

When there is a decrease in the length or compression of the body due to the applied force, the stress produced is called compressive stress.

When equal normal forces are applied on every surface of a body causing a change in volume, the restoring force opposing this change per unit area is called hydraulic stress (also called volume stress).

Strain is defined as the ratio of the change in dimensions of the body to its original dimensions.

OR

The ratio of change in configuration to the original configuration is called strain.

• It has no unit and no dimensions (pure ratio).

The strain is defined as the ratio of change in dimensions of the body to its original dimensions.

Strain = `"change in dimensions"/"original dimensions"`

The ratio of change in volume of the body to its original volume is called volume strain: ΔV/V.

The modulus of elasticity of a material is the ratio of stress to the corresponding strain. It is defined as the slope of the stress-strain curve in the elastic deforming region and depends on the nature of the material.

\[\frac {stress}{strain}\] = Constant

The constant is called the modulus of elasticity.

OR

The constant ratio of stress to strain within the elastic limit is called the Modulus of Elasticity.

A graph drawn by taking tensile strain along the x-axis and tensile stress along the y-axis, obtained by gradually increasing the load on a metal wire suspended vertically from a rigid support until the wire breaks, and measuring the elongation produced during each step.

In case of some materials like vulcanized rubber, when the stress applied on a body decreases to zero, the strain does not return to zero immediately. The strain lags behind the stress. This lagging of strain behind the stress is called elastic hysteresis.

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

OR

The ratio of tensile (or compressive) stress to the longitudinal strain is called Young's Modulus of Elasticity, denoted by Y.

"Shear modulus or modulus of rigidity: It is defined as the ratio of shear stress to shear strain within elastic limits."

OR

The ratio of shearing stress to the corresponding shearing strain in a material is called the Shear Modulus or Modulus of Rigidity, denoted by G.

"Bulk modulus is defined as the ratio of volume stress to volume strain."

OR

The ratio of hydraulic stress to the corresponding hydraulic strain (change in volume) is called the Bulk Modulus, denoted by B.

Within elastic limit, the ratio of lateral strain to the linear strain is called the Poisson's ratio.

The reciprocal of the bulk modulus is called compressibility: k = \[\frac {1}{B}\].

The point on the stress-strain curve up to which Hooke's Law is valid is called the proportional limit (Point A).

The stress at the yield point (end of elastic behavior and start of plastic deformation) is called the yield strength.

The maximum stress that a material can withstand is called the Ultimate Tensile Strength (Point D).

The point at which the material breaks and failure of the material takes place is called the fracture point (Point E).

The mathematical expression for Young's modulus (Y) is:

Y = \[\frac{MgL}{\pi r^2l}\] or \[\frac {FL}{AΔL}\]

Where:

• Y = Young’s Modulus
• M = Mass of the load attached
• g = Acceleration due to gravity
• L = Original length of the wire
• r = Radius of the wire cross-section
• l = Extension or elongation produced in the wire

The formula for modulus of rigidity is:

η = \[\frac{\text{Shear Stress}}{\text{Shear Strain}}=\frac{F/A}{\theta}=\frac{F}{A\cdot\theta}\]

Where:

• η = Modulus of rigidity (Pa or N/m²)
• F = Tangential force applied (N)
• A = Cross-sectional area on which force acts (m²)
• θ = Shear strain = Δl/l (in radians)
• Δl = Displacement of the upper surface relative to the lower surface (m)
• l = Original height of the block (m)

SI Unit: Pascal (Pa) or N/m²​

Dimensional Formula: M¹L⁻¹T⁻²

The mathematical representation of Bulk Modulus (K) is:

K = \[\frac{\text{Volume Stress}}{\text{Volume Strain}}\]

K = \[\frac{dP}{\left(\frac{dV}{V}\right)}\] = V \[\frac {dP}{dV}\]

Where:

• K: Bulk Modulus
• dP: Change in pressure (Volume Stress)
• dV: Change in volume
• V: Original volume

\[\sigma=\frac{\text{Lateral strain}}{\text{Linear strain}}=\frac{\frac{d}{D}}{\frac{\Delta l}{l}}=\frac{d\cdot l}{D\cdot\Delta l}\]

Where:

• σ = Poisson's ratio
• l = original length of the wire
• ∆l = increase or decrease in length of the wire
• D = original diameter of the wire
• d = corresponding change in diameter of the wire

Important Note: Poisson's ratio has no unit. It is dimensionless.

Hooke's Law was discovered by English scientist Robert Hooke in 1660. He first stated it as a Latin anagram: "As the extension, so the force."

Statement: For small deformations, stress is directly proportional to strain, within the elastic limit.

Key Points:

• Hooke's Law is a measure of elasticity.
• It is valid only up to the elastic limit. Beyond this, the material does not return to its original shape and Hooke's Law no longer applies.
• In springs: The force needed to extend or compress a spring by distance x is proportional to that distance → F = −kx (where k is the spring constant).
• Hooke's Law is applicable only in the case of elastic deformation.

• OA (Proportional Region): Stress ∝ Strain; material behaves elastically; Hooke's Law valid.
• Point A: Proportional limit — end of Hooke's Law.
• Region AB: Non-linear elastic region; material still returns to original shape.
• Point B (Elastic/Yield Limit): End of elastic behavior; start of plastic deformation.
• Region BD: Permanent (plastic) deformation; material does not return to original shape.
• Point D: Ultimate Tensile Strength — maximum stress the material can bear.
• Point E: Fracture point — material breaks.