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Mechanical Properties of Solids

Master Mechanical Properties of Solids for JEE Main, JEE Advanced, NEET & Board Exams. Covers stress, strain, Hooke's law, Young's modulus, bulk modulus, shear modulus, stress-strain curve, Poisson's ratio and thermal stress with solved examples.

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When external forces are applied to a solid body, it deforms — and if the forces are removed, it may or may not return to its original shape. The study of how solids respond to applied forces is called Mechanical Properties of Solids. Key concepts include stress, strain, elastic moduli, and the limits of elasticity. This topic is important for JEE (Main & Advanced), NEET, and board exams, with frequent questions on Young's modulus, stress-strain graphs, and energy stored in stretched wires.

1. Elasticity and Plasticity

Elasticity is the property of a solid by which it regains its original shape and size after the removal of deforming forces. Example: rubber band, spring steel.
Plasticity is the property by which a body does not regain its original shape after removal of deforming forces and remains permanently deformed. Example: clay, putty, lead.
A perfectly elastic body regains its shape completely; a perfectly plastic body does not regain it at all. Real materials lie between these two extremes.
Elastic limit: The maximum stress up to which a body behaves elastically. Beyond this, permanent deformation sets in.

2. Stress

Stress is the internal restoring force developed per unit area when a body is deformed by an external force:

$Stress = \frac{Restoring Force}{Area} = \frac{F}{A}$

SI unit: Pascal (Pa) = N/m². Stress is a scalar for normal stress.

Type of Stress	Direction of Force	Effect on Body
Tensile Stress	Perpendicular to area, outward (stretching)	Elongation
Compressive Stress	Perpendicular to area, inward (compressing)	Compression
Shear (Tangential) Stress	Parallel (tangential) to area	Change in shape (no volume change)
Hydraulic (Bulk) Stress	Normal to all surfaces equally (pressure)	Change in volume (no shape change)

3. Strain

Strain is the ratio of the change in dimension to the original dimension. It is a dimensionless quantity (no unit).

Type of Strain	Formula	What Changes
Longitudinal (Tensile/Compressive)	$ε = \frac{Δ L}{L}$	Length
Shear Strain	$ϕ = \frac{x}{h} = \tan ϕ \approx ϕ$ (small angles)	Shape (angle of shear)
Volumetric Strain	$ε_{V} = \frac{Δ V}{V}$	Volume

4. Hooke's Law and Elastic Moduli

Hooke's Law

Within the elastic limit, stress is directly proportional to strain:

$Stress \propto Strain$

$\Rightarrow \frac{Stress}{Strain} = Elastic Modulus (constant)$

The elastic modulus (modulus of elasticity) is a measure of the stiffness of a material — it does not depend on the dimensions of the body, only on the material.

(i) Young's Modulus ( $Y$ )

Ratio of longitudinal (tensile or compressive) stress to longitudinal strain:

$Y = \frac{Longitudinal Stress}{Longitudinal Strain}$ $= \frac{F / A}{Δ L / L}$ $= \frac{F L}{A Δ L}$

SI unit: Pa (N/m²). Young's modulus is defined only for solids (not liquids or gases, which cannot sustain tensile stress).

Greater $Y$ means the material is stiffer (harder to stretch). Steel has higher $Y$ than rubber.
For a wire of length $L$ , cross-section $A$ , stretched by $Δ L$ under force $F$ :
$Δ L = \frac{F L}{A Y}$

(ii) Bulk Modulus ( $B$ or $K$ )

Ratio of hydraulic (volume) stress to volumetric strain. Defined for solids, liquids, and gases:

$B = - \frac{Δ P}{Δ V / V} = - \frac{V \cdot Δ P}{Δ V}$

The negative sign ensures $B$ is positive (volume decreases when pressure increases). SI unit: Pa.

Compressibility $= \frac{1}{B}$ . Liquids and solids have very high $B$ (nearly incompressible). Gases have very low $B$ (highly compressible).
For gases: Isothermal bulk modulus $= P$ ; Adiabatic bulk modulus $= γ P$ .

(iii) Modulus of Rigidity / Shear Modulus ( $G$ or $η$ )

Ratio of shear stress to shear strain. Defined only for solids:

$G = \frac{Shear Stress}{Shear Strain} = \frac{F / A}{ϕ}$

SI unit: Pa. Liquids and gases have $G = 0$ (they cannot sustain shear stress).

Summary of Elastic Moduli

Modulus	Symbol	Stress Type	Applicable To
Young's Modulus	$Y$	Tensile / Compressive	Solids only
Bulk Modulus	$B$	Hydraulic (volume)	Solids, liquids, gases
Shear Modulus	$G$	Shear (tangential)	Solids only

5. Stress-Strain Curve

The stress-strain graph for a ductile material (e.g., mild steel) is one of the most important diagrams in this chapter. It reveals different regimes of material behaviour:

O to A — Proportional Limit: Stress $\propto$ Strain (Hooke's Law holds). The graph is a straight line. The slope equals Young's modulus $Y$ .
A to B — Elastic Limit: Material still behaves elastically (returns to original shape on unloading) but Hooke's Law is no longer obeyed. Beyond B, permanent deformation begins.
B to C — Yield Point (C is lower yield point): Strain increases rapidly with little or no increase in stress. The material begins to "flow." Yield stress is the stress at this point.
C to D — Plastic Region: Material deforms permanently. Stress increases slowly (strain hardening). D is the Ultimate Tensile Strength (UTS) — the maximum stress the material can withstand.
D to E — Necking and Fracture: The material narrows (necks) and eventually breaks at E, the fracture point.

Region	Key Point	Material Behaviour
O → A	Proportional limit	Elastic + Hooke's Law
A → B	Elastic limit	Elastic, non-linear
B → C	Yield point	Onset of plasticity
C → D	UTS (point D)	Plastic deformation (strain hardening)
D → E	Fracture point	Necking and fracture

Ductile vs. Brittle Materials

Ductile materials (e.g., steel, copper, aluminium): Large plastic region before fracture. Can be drawn into wires.
Brittle materials (e.g., glass, cast iron, ceramics): Very small or no plastic region — fracture occurs suddenly near the elastic limit without warning.
Elastomers (e.g., rubber): Very large strain for small stress; elastic limit and fracture point nearly coincide. Not ductile.

6. Elastic Potential Energy Stored in a Stretched Wire

When a wire is stretched, work is done against the internal restoring forces. This work is stored as elastic potential energy:

$U = \frac{1}{2} \times Stress \times Strain \times Volume$

$U = \frac{1}{2} \cdot \frac{F}{A} \cdot \frac{Δ L}{L} \cdot (A L)$ $= \frac{1}{2} F \cdot Δ L$

Energy density (energy per unit volume):

$u = \frac{U}{V} = \frac{1}{2} \times Stress \times Strain$ $= \frac{{Stress}^{2}}{2 Y}$ $= \frac{1}{2} Y \times {Strain}^{2}$

This is analogous to the elastic potential energy $\frac{1}{2} k x^{2}$ stored in a spring. In fact, a wire under tension behaves like a spring with:

$k_{w i r e} = \frac{Y A}{L}$

where $Y$ is Young's modulus, $A$ is cross-sectional area, and $L$ is the natural length.

7. Poisson's Ratio

When a wire is stretched longitudinally, it contracts laterally (gets thinner). The ratio of lateral strain to longitudinal strain is called Poisson's ratio ( $σ$ or $ν$ ):

$σ = - \frac{Lateral Strain}{Longitudinal Strain}$ $= - \frac{Δ D / D}{Δ L / L}$

The negative sign accounts for the fact that lateral and longitudinal strains are always opposite in sign (one increases while the other decreases).

Poisson's ratio is dimensionless.
Theoretical limits: $- 1 \leq σ \leq 0.5$ . For most real materials: $0 < σ < 0.5$ .
For rubber: $σ \approx 0.5$ (nearly incompressible). For steel: $σ \approx 0.3$ . For cork: $σ \approx 0$ .
Relation between elastic moduli:
$Y = 2 G (1 + σ)$
$Y = 3 B (1 - 2 σ)$

8. Thermal Stress

When a rod fixed at both ends is subjected to a temperature change $Δ T$ , it cannot expand freely. This sets up an internal thermal stress:

$Thermal Strain = α Δ T$

$Thermal Stress = Y α Δ T$

$Force developed = Y A α Δ T$

where $α$ is the coefficient of linear thermal expansion and $Y$ is Young's modulus of the material. This concept is crucial in engineering design — bridges and rail tracks have expansion gaps to prevent thermal stress buildup.

Wires in Series and Parallel — Spring Analogy

Since a wire behaves like a spring with $k = \frac{Y A}{L}$ , multiple wires can be analysed using the series/parallel spring formulae:

Wires in Series (same force, extensions add):
$\frac{1}{k_{e f f}} = \frac{1}{k_{1}} + \frac{1}{k_{2}}$ $\Rightarrow$ $Δ L_{t o t a l} = Δ L_{1} + Δ L_{2}$
Wires in Parallel (same extension, forces add):
$k_{e f f} = k_{1} + k_{2}$ $\Rightarrow$ $F_{t o t a l} = F_{1} + F_{2}$
For a wire of uniform cross-section and material but two different segments of lengths $L_{1}$ and $L_{2}$ : these are in series.
For two wires of different materials supporting the same load from a rigid rod: these are in parallel.

JEE/NEET Power Tips — Avoid These Common Mistakes

Strain has no unit — it is a pure ratio. Never write Pa or N/m² as the unit of strain.
Young's modulus is a property of the material, not the wire. It does not change if the length or diameter of the wire changes. However, the spring constant $k = Y A / L$ does change with dimensions.
Elastic limit ≠ Proportional limit. The proportional limit (where Hooke's law stops) is always less than or equal to the elastic limit (where permanent deformation begins).
Liquids and gases have Bulk modulus but NOT Young's modulus or Shear modulus — they cannot sustain tensile or shear stresses.
For a soap film (two surfaces), surface energy = $2 T L^{2}$ . Do not confuse with a single liquid surface.
Poisson's ratio for cork ≈ 0 — this is why wine bottle corks can be pushed in without bulging sideways, and why they are used as stoppers.
Energy stored in a wire = $\frac{1}{2} F Δ L$ , not $F Δ L$ . The factor of $\frac{1}{2}$ arises because force increases linearly from 0 to $F$ as the wire stretches.

Practice Questions (JEE / NEET / Board Level)

Q1: A steel wire of length $2$ m and cross-sectional area $2 \times 10^{- 6}$ m² is stretched by $1$ mm under a force of $200$ N. The Young's modulus of steel is:

A) $2 \times 10^{11}$ Pa
B) $1 \times 10^{11}$ Pa
C) $4 \times 10^{11}$ Pa
D) $0.5 \times 10^{11}$ Pa

Answer: A) $2 \times 10^{11}$ Pa.

Solution:
$Y = \frac{F L}{A Δ L}$
$= \frac{200 \times 2}{2 \times 10^{- 6} \times 1 \times 10^{- 3}}$
$= \frac{400}{2 \times 10^{- 9}}$
$= 2 \times 10^{11}$ Pa.

Q2: A wire stretches by $Δ L$ under a load $F$ . If both the length and diameter of the wire are doubled and the same load is applied, the new extension will be:

A) $Δ L$
B) $Δ L / 2$
C) $2 Δ L$
D) $Δ L / 4$

Answer: B) $Δ L / 2$ .

Solution:
$Δ L = \frac{F L}{A Y} = \frac{F L}{π r^{2} Y}$
When length is doubled: $L^{'} = 2 L$ , and diameter doubled means $r^{'} = 2 r$ , so $A^{'} = π (2 r)^{2} = 4 A$ .
$Δ L^{'} = \frac{F \cdot 2 L}{4 A \cdot Y} = \frac{2}{4} \cdot \frac{F L}{A Y} = \frac{Δ L}{2}$

Q3: The bulk modulus of a liquid is $3 \times 10^{10}$ Pa. The pressure required to reduce its volume by $2 %$ is:

A) $3 \times 10^{8}$ Pa
B) $6 \times 10^{8}$ Pa
C) $1.5 \times 10^{10}$ Pa
D) $6 \times 10^{10}$ Pa

Answer: B) $6 \times 10^{8}$ Pa.

Solution:
Volumetric strain $= \frac{Δ V}{V} = 2 % = 0.02$
$B = \frac{Δ P}{Δ V / V}$
$\Rightarrow Δ P = B \times \frac{Δ V}{V}$
$= 3 \times 10^{10} \times 0.02$
$= 6 \times 10^{8}$ Pa.

Q4: A wire of length $L$ and radius $r$ has Young's modulus $Y$ . It is used as a spring. Its spring constant is:

A) $\frac{Y r^{2}}{π L}$
B) $\frac{π Y r^{2}}{L}$
C) $\frac{Y L}{π r^{2}}$
D) $\frac{π r^{2} L}{Y}$

Answer: B) $\frac{π Y r^{2}}{L}$ .

Solution:
The spring constant of a wire is:
$k = \frac{Y A}{L} = \frac{Y \cdot π r^{2}}{L} = \frac{π Y r^{2}}{L}$

Q5: A metal rod of length $1$ m, Young's modulus $Y = 2 \times 10^{11}$ Pa, and coefficient of linear expansion $α = 1.2 \times 10^{- 5}$ /°C is fixed between two rigid walls. If the temperature rises by $20 °$ C, the thermal stress developed is:

A) $4.8 \times 10^{7}$ Pa
B) $2.4 \times 10^{7}$ Pa
C) $9.6 \times 10^{7}$ Pa
D) $1.2 \times 10^{7}$ Pa

Answer: A) $4.8 \times 10^{7}$ Pa.

Solution:
Thermal Stress $= Y α Δ T$
$= 2 \times 10^{11} \times 1.2 \times 10^{- 5} \times 20$
$= 2 \times 10^{11} \times 2.4 \times 10^{- 4}$
$= 4.8 \times 10^{7}$ Pa.

Q6: The elastic potential energy stored per unit volume in a stretched wire with stress $σ$ and Young's modulus $Y$ is:

A) $\frac{σ^{2}}{Y}$
B) $\frac{σ^{2}}{2 Y}$
C) $\frac{2 σ^{2}}{Y}$
D) $\frac{σ}{2 Y}$

Answer: B) $\frac{σ^{2}}{2 Y}$ .

Solution:
Energy density $= \frac{1}{2} \times Stress \times Strain$
$= \frac{1}{2} \times σ \times \frac{σ}{Y}$
$= \frac{σ^{2}}{2 Y}$

Q7 (JEE Advanced type): Two wires A and B of the same material and length are connected in series and support a mass $M$ . Wire A has radius $r$ and Wire B has radius $2 r$ . The ratio of extensions $Δ L_{A} : Δ L_{B}$ is:

A) $1 : 1$
B) $2 : 1$
C) $4 : 1$
D) $1 : 4$

Answer: C) $4 : 1$ .

Solution:
In series, both wires carry the same tension $F = M g$ .
$Δ L = \frac{F L}{A Y} = \frac{F L}{π r^{2} Y}$
Since $F$ , $L$ , $Y$ are the same for both:
$Δ L \propto \frac{1}{r^{2}}$
$\frac{Δ L_{A}}{Δ L_{B}} = \frac{r_{B}^{2}}{r_{A}^{2}} = \frac{(2 r)^{2}}{r^{2}} = \frac{4 r^{2}}{r^{2}} = 4 : 1$

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DifficultyIntermediate

Est. Read Time22 minutes

CategoryProperties Of Solids

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Mechanical Properties of Solids

1. Elasticity and Plasticity

2. Stress

3. Strain

4. Hooke's Law and Elastic Moduli

Hooke's Law

(i) Young's Modulus ( $Y$ )

(ii) Bulk Modulus ( $B$ or $K$ )

(iii) Modulus of Rigidity / Shear Modulus ( $G$ or $η$ )

Summary of Elastic Moduli

5. Stress-Strain Curve

Ductile vs. Brittle Materials

6. Elastic Potential Energy Stored in a Stretched Wire

7. Poisson's Ratio

8. Thermal Stress

Wires in Series and Parallel — Spring Analogy

Practice Questions (JEE / NEET / Board Level)

Related Study Material

Practice Questions

Topic Information

Quick Actions

Track Your Learning

Study Tips

We Value Your Privacy

Edvaya Target

Edvaya Aspire

Take a Free Mock Test

Target Subjects

Aspire Subjects

Subject Mastery

Mechanical Properties of Solids

1. Elasticity and Plasticity

2. Stress

3. Strain

4. Hooke's Law and Elastic Moduli

Hooke's Law

(i) Young's Modulus (Y)

(ii) Bulk Modulus (B or K)

(iii) Modulus of Rigidity / Shear Modulus (G or η)

Summary of Elastic Moduli

5. Stress-Strain Curve

Ductile vs. Brittle Materials

6. Elastic Potential Energy Stored in a Stretched Wire

7. Poisson's Ratio

8. Thermal Stress

Wires in Series and Parallel — Spring Analogy

Practice Questions (JEE / NEET / Board Level)

Related Study Material

Practice Questions

Topic Information

Quick Actions

Track Your Learning

Study Tips

We Value Your Privacy

(i) Young's Modulus ( $Y$ )

(ii) Bulk Modulus ( $B$ or $K$ )

(iii) Modulus of Rigidity / Shear Modulus ( $G$ or $η$ )