Intermediate

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Stress, Strain, Hooke's Law & Stress-Strain Curve

Master Stress and Strain for JEE & NEET. Covers stress types, strain types, Hooke's law, stress-strain curve all 5 points, ductile vs brittle materials, elastomers.

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Every solid object deforms when a force is applied — and understanding how it deforms, how much, and whether it recovers, is the subject of the mechanical properties of solids. The concepts of stress and strain provide the quantitative language for deformation: stress captures the internal restoring force per unit area, while strain measures the fractional change in shape or size. Hooke's Law — stress is proportional to strain within the elastic limit — is the simplest and most powerful statement in elasticity. The stress-strain curve narrates the full story of a material from initial elastic behaviour through yielding to fracture. For JEE and NEET, this topic yields direct numerical questions on stress, strain, and extension calculations, as well as conceptual questions on the stress-strain curve.

1. Elasticity and Plasticity

Elasticity: The property of a body by virtue of which it tends to regain its original shape and size after the deforming force is removed.

Plasticity: The property by which a body does not regain its original shape/size after the deforming force is removed — the deformation is permanent.

Perfectly elastic body: Regains original shape completely — ideal, no real body is perfectly elastic. Quartz fibre comes closest.
Perfectly plastic body: Retains deformed shape completely — putty, clay.
Real materials lie between these extremes — steel is highly elastic; lead is highly plastic.

Elastic limit: The maximum stress up to which a body shows elastic behaviour. Beyond this, permanent deformation sets in.

2. Stress

When a deforming force $F$ is applied to a body, an internal restoring force develops. Stress is defined as the restoring force per unit area of cross-section:

$Stress = \frac{F}{A}$
${SI unit: N m}^{- 2} = Pa (pascal)$

Types of Stress

Type	Direction of force relative to area	Effect
Tensile stress	Perpendicular (outward) — stretching	Elongation
Compressive stress	Perpendicular (inward) — compression	Shortening
Shear (tangential) stress	Parallel to the surface — tangential	Change in shape (no volume change)
Hydraulic (bulk) stress	Normal from all sides — uniform pressure	Change in volume (no shape change)

Tensile and compressive stresses are together called longitudinal stress or normal stress.

3. Strain

Strain is the ratio of the change in configuration to the original configuration. It is dimensionless — no units.

Type	Formula	Caused by
Longitudinal (linear) strain	$ε = \frac{Δ L}{L_{0}}$	Tensile or compressive stress
Shear strain	$ϕ = \frac{x}{L} = \tan ϕ \approx ϕ$ (for small angles)	Tangential (shear) stress
Volumetric (bulk) strain	$= \frac{Δ V}{V_{0}}$	Hydraulic (uniform) stress

Worked Example

A wire of length $2 m$ extends by $0.5 mm$ when loaded. Longitudinal strain:

$ε = \frac{Δ L}{L_{0}} = \frac{0.5 \times 10^{- 3}}{2.0} = 2.5 \times 10^{- 4}$

4. Hooke's Law

$Stress \propto Strain$
$\Rightarrow Stress = E \times Strain$

where $E$ is the modulus of elasticity (a material constant with units of Pa).

Valid only within the proportionality limit — below the elastic limit.
The proportionality constant $E$ depends on the material, not on the dimensions of the body.
Higher $E$ → stiffer material (less strain for same stress).

5. Stress-Strain Curve

When a wire is gradually loaded, the stress-strain graph passes through several distinct regions:

Point/Region	Name	Behaviour
O → A	Proportionality limit (A)	Linear; stress ∝ strain exactly; Hooke's law strictly obeyed
A → B	Elastic limit (B)	Non-linear but still elastic; body returns to original shape if force removed; Hooke's law no longer exact
B → C	Yield point (C) / Upper yield point	Beyond elastic limit; permanent deformation begins; strain increases without much increase in stress
C → D	Plastic region	Plastic flow (necking begins); material elongates significantly with little increase in load
D	Ultimate tensile stress (UTS)	Maximum stress the material can withstand; cross-sectional area decreases (necking)
E	Fracture point / Breaking point	Wire breaks; stress at fracture < UTS (due to necking)

Ductile vs Brittle Materials

Property	Ductile (e.g., steel, copper)	Brittle (e.g., glass, cast iron)
Plastic region	Large — significant plastic deformation before fracture	Very small — fractures very soon after elastic limit
UTS vs fracture stress	Clear gap — necking visible	Fracture ≈ UTS (almost coincide)
Warning before fracture	Visible (elongation)	None — sudden failure
Area under stress-strain	Large (absorbs more energy)	Small

Elastomers (Rubber-like materials)

Elastomers (e.g., rubber, tissue) show large strains for small stresses — they do not obey Hooke's law over a wide range. The stress-strain curve is nonlinear and shows hysteresis. The elastic limit is very high, and no well-defined plastic region exists.

6. Worked Example — Stress on a Wire

A steel wire of diameter $2 mm$ supports a load of $10 N$ . Find the stress.

$r = 1 mm = 10^{- 3} m$

$A = π r^{2} = π \times (10^{- 3})^{2} = 3.14 \times 10^{- 6} m^{2}$

$Stress = \frac{F}{A} = \frac{10}{3.14 \times 10^{- 6}} = 3.18 \times 10^{6} Pa$

Stress-Strain Curve — All 5 Points in One Table

Point	Name	Key fact
A	Proportionality limit	Hooke's law holds exactly; graph is a straight line up to A
B	Elastic limit	Still elastic (recovers fully); Hooke's law not obeyed between A and B
C	Yield point	Permanent deformation begins; large strain with small stress increase
D	Ultimate tensile stress	Maximum stress; necking begins here
E	Fracture point	Wire breaks; fracture stress < UTS for ductile; ≈ UTS for brittle

JEE / NEET Power Tips — Stress, Strain & Hooke's Law

Strain is dimensionless — it has no units. Stress has units of Pa. This difference is tested directly. The ratio stress/strain = modulus has units of Pa.
Proportionality limit ≠ elastic limit. Proportionality limit (A) is where Hooke's law stops being exact. Elastic limit (B) is where elastic behaviour stops. A comes before B — elastic limit is higher than proportionality limit. Many students conflate these two.
For brittle materials, fracture point ≈ UTS. For ductile materials, fracture stress < UTS because of necking. This is a common MCQ distinction — if a question asks which material fractures suddenly without warning, the answer is a brittle material.
Shear strain = $x / L$ = $\tan ϕ \approx ϕ$ for small angles. The shear strain is the angular deformation in radians, not a ratio of lengths in the usual sense. For solids, shear strain is always small so $\tan ϕ \approx ϕ$ .
Restoring force opposes deformation. The internal force that develops inside a body when deformed is the restoring force — it acts opposite to the applied deforming force, trying to restore the original configuration. Stress is defined in terms of this restoring force, not the applied force (though they are equal and opposite at equilibrium).

Practice Questions

Q1 (JEE Main / NEET): A copper wire of length 2.4 m and cross-sectional area $1.5 \times 10^{- 6}$ m² is stretched by 3.0 mm when a force of 18 N is applied. Find the stress and strain in the wire.

Explanation:

$Stress = \frac{F}{A} = \frac{18}{1.5 \times 10^{- 6}} = 1.2 \times 10^{7}$ Pa

$Strain = \frac{Δ L}{L} = \frac{3.0 \times 10^{- 3}}{2.4} = 1.25 \times 10^{- 3}$ (dimensionless)

Q2 (NEET MCQ): The stress-strain graph for a material shows a long, extended plastic region before fracture. This material is:

A) Brittle
B) Ductile
C) Elastic
D) Hard

Answer: B) Ductile.

Explanation: A large plastic region (meaning it can undergo large strain/deformation before fracturing) is the defining feature of ductile materials (e.g., steel, copper). Brittle materials (like glass or cast iron) fracture with almost no plastic deformation — their fracture point nearly coincides with their ultimate tensile strength (UTS).

Q3 (Board): A rubber cord has a cross-sectional area of 2 mm² and a natural length of 10 cm. It is stretched to 12 cm by a force of 4 N. Find the stress and strain. Does it obey Hooke's law?

Explanation:

$A = 2$ mm² $= 2 \times 10^{- 6}$ m²

$Stress = \frac{4}{2 \times 10^{- 6}} = 2 \times 10^{6}$ Pa

$Strain = \frac{12 - 10}{10} = 0.2$ (or 20%)

A strain of 20% is very large. While rubber (an elastomer) can sustain massive strains without breaking, it does not obey Hooke's law over such a range. Hooke's law is valid only for small deformations within the strict proportionality limit (where stress $\propto$ strain).

Q4 (JEE Main): Within the elastic limit, the stress-strain ratio for a material is $4 \times 10^{10}$ Pa. A wire of this material has a length of 2 m and a diameter of 2 mm. What force will extend it by 0.08 mm?

Explanation:

The stress-strain ratio is Young's modulus ( $E$ ):
$E = \frac{Stress}{Strain} = 4 \times 10^{10}$ Pa

Radius $r = 1$ mm $= 10^{- 3}$ m.
$A = π r^{2} = π (10^{- 3})^{2} = π \times 10^{- 6}$ m²

$Strain = \frac{Δ L}{L} = \frac{0.08 \times 10^{- 3}}{2} = 4 \times 10^{- 5}$

$Stress = E \times Strain$
$= (4 \times 10^{10}) \times (4 \times 10^{- 5}) = 1.6 \times 10^{6}$ Pa

$F = Stress \times A$
$= (1.6 \times 10^{6}) \times (π \times 10^{- 6}) = 1.6 π \approx 5.03 N$

Q5 (NEET / Board): Which of the following correctly distinguishes between the proportionality limit and the elastic limit?

A) They are the same point on the stress-strain curve.
B) The elastic limit is below the proportionality limit.
C) The proportionality limit is below the elastic limit; Hooke's law holds up to the proportionality limit.
D) Beyond the elastic limit, the material returns to its original shape.

Answer: C) The proportionality limit is below the elastic limit; Hooke's law holds up to the proportionality limit.

Explanation: The proportionality limit (Point A) is the exact point where stress stops being directly proportional to strain — meaning Hooke's law ends here. The elastic limit (Point B) is slightly higher on the curve. Between A and B, the material is still perfectly elastic (it will return to its original shape if the load is removed), but it no longer obeys Hooke's law. Permanent plastic deformation only occurs if the material is stretched beyond the elastic limit (Point B).

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DifficultyIntermediate

Est. Read Time24 minutes

CategoryMechanical Properties Of Solids

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Stress, Strain, Hooke's Law & Stress-Strain Curve

1. Elasticity and Plasticity

2. Stress

Types of Stress

3. Strain

Worked Example

4. Hooke's Law

5. Stress-Strain Curve

Ductile vs Brittle Materials

Elastomers (Rubber-like materials)

6. Worked Example — Stress on a Wire

Stress-Strain Curve — All 5 Points in One Table

Practice Questions

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