Notes : Hooke's Law and Stress-Strain Curve | Class 11 Physics Chapter 8 | IIT JEE, NEET , NCERT

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Hooke's Law and Stress-Strain Curve Notes | Class 11 Physics Chapter 8 | IIT JEE, NEET & NCERT - Physics kund

When a force is applied to a body, it undergoes deformation. The relationship between stress and strain is explained by Hooke's Law, while the complete behaviour of a material under load is represented by the Stress–Strain Curve.

Hooke's Law

Definition: Hooke's Law states that within the elastic limit, stress is directly proportional to strain.

Mathematical Expression:

$\text{Stress} \propto \text{Strain}$

$\text{Stress} = k \times \text{Strain}$

where:

Stress = Restoring force per unit area
Strain = Fractional change in dimension
$k$ = Modulus of Elasticity (proportionality constant)

Important Points of Hooke's Law

• Valid only for small deformations.
• Applicable only within the elastic limit.
• It is an empirical law based on experiments.
• Most solids obey Hooke's law up to a certain limit.
• Rubber and biological tissues do not obey Hooke's law over most of their elastic region.

Stress–Strain Curve

A stress-strain curve is a graph obtained by plotting stress against strain for a material under tensile loading. It helps us understand the elastic and plastic behaviour of materials.

O–A Region (Linear Region)

• Stress is directly proportional to strain.
• Hooke's Law is obeyed.
• Graph is a straight line.
• Material behaves as an elastic body.
• Original dimensions are completely recovered when load is removed.

Point A (Proportional Limit)

• Highest point up to which stress remains proportional to strain.
• End of Hooke's Law region.

A–B Region (Elastic Region)

• Stress and strain are no longer proportional.
• Hooke's Law is not obeyed.
• Material still regains its original dimensions when load is removed.

Point B (Yield Point / Elastic Limit)

• Maximum stress up to which the material behaves elastically.
• Corresponding stress is called Yield Strength.

$\sigma_y$

B–D Region (Plastic Region)

• Plastic deformation occurs.
• Small increase in stress causes large increase in strain.
• Material does not regain original dimensions after load removal.
• Permanent deformation remains.

This permanent deformation is called Permanent Set.

Point D (Ultimate Tensile Strength)

• Maximum stress sustained by the material.
• Corresponding stress is called Ultimate Tensile Strength.

$\sigma_u$

D–E Region (Necking Region)

• Material starts weakening.
• Stress decreases while strain increases.

Point E (Fracture Point)

• Material breaks completely.
• End point of the stress-strain curve.

Ductile and Brittle Materials

Ductile Materials

If points D and E are far apart:

• Large plastic deformation occurs.
• Material can be drawn into wires.

Examples: Copper, Aluminium, Mild Steel

Brittle Materials

If points D and E are close together:

• Very little plastic deformation occurs.
• Material breaks suddenly.

Examples: Glass, Cast Iron

Stress–Strain Curve of Elastomers

Materials such as rubber and aorta tissue show a different stress-strain behaviour.

Characteristics

• Large elastic region.
• Non-linear stress-strain curve.
• Hooke's Law is not obeyed over most of the elastic region.
• No clearly defined plastic region.
• Can undergo large strains and still recover their original shape.

Elastomers

Substances that can be stretched to produce large strains and still return to their original shape are called Elastomers.

Examples: Rubber, Aorta Tissue, Elastic Polymers

Difference Between Hooke's Law and Stress–Strain Curve

Hooke's Law Stress–Strain Curve
Describes proportionality between stress and strain. Describes complete deformation behaviour of a material.
Valid only within elastic limit. Includes elastic, plastic and fracture regions.
Represented by an equation. Represented by a graph.
Applicable only in linear region. Applicable to entire loading process.

Frequently Asked Questions (FAQs)

Q1. What is Hooke's Law?

Answer: Hooke's Law states that within the elastic limit, stress is directly proportional to strain.

Q2. Write the mathematical expression of Hooke's Law.

Answer: $ \text{Stress} = k \times \text{Strain} $

Q3. What is the proportionality constant in Hooke's Law?

Answer: Modulus of Elasticity.

Q4. What is yield point?

Answer: The point beyond which permanent deformation begins.

Q5. What is yield strength?

Answer: Stress corresponding to the yield point.

Q6. What is ultimate tensile strength?

Answer: Maximum stress that a material can withstand.

Q7. What is permanent set?

Answer: Permanent deformation left after removal of load.

Q8. What are elastomers?

Answer: Materials that can sustain large strains and recover their original shape.

Multiple Choice Questions (MCQs)

1. Hooke's Law is valid for:
A) Large deformations
B) Small deformations
C) Plastic region
D) Fracture region
Answer: B) Small deformations

2. The proportionality constant in Hooke's Law is:
A) Force
B) Strain
C) Modulus of Elasticity
D) Stress
Answer: C) Modulus of Elasticity

3. Point B on the stress-strain curve is:
A) Ultimate strength
B) Fracture point
C) Yield point
D) Proportional limit
Answer: C) Yield point

4. Point D represents:
A) Yield strength
B) Ultimate tensile strength
C) Fracture point
D) Elastic limit
Answer: B) Ultimate tensile strength

5. Point E represents:
A) Yield point
B) Elastic limit
C) Fracture point
D) Permanent set
Answer: C) Fracture point

True / False

1. Hooke's Law is valid only within elastic limit. Answer: True

2. Stress is directly proportional to strain within elastic limit. Answer: True

3. Point B is known as yield point. Answer: True

4. Plastic deformation is reversible. Answer: False

5. Point D represents ultimate tensile strength. Answer: True

6. Point E is the fracture point. Answer: True

Very Short Answer Questions (1 Mark)

1. State Hooke's Law.
Answer: Within elastic limit, stress is directly proportional to strain.

2. What is stress?
Answer: Restoring force per unit area.

3. What is strain?
Answer: Fractional change in dimension.

4. What is elastic limit?
Answer: Maximum stress up to which a body regains its original shape.

5. What is yield strength?
Answer: Stress corresponding to the yield point.

Short Answer Questions (2–3 Marks)

1. State Hooke's Law and write its mathematical expression.

Answer:

Within elastic limit, stress is directly proportional to strain.

$\text{Stress} \propto \text{Strain}$

$\text{Stress} = k \times \text{Strain}$

2. What is the significance of stress-strain curve?

Answer:

The stress-strain curve helps in understanding:
• Elastic behaviour
• Yielding
• Plastic deformation
• Ultimate strength
• Fracture behaviour

3. Differentiate between elastic and plastic deformation.

Elastic Deformation:
• Temporary deformation.
• Original shape is regained.

Plastic Deformation:
• Permanent deformation.
• Original shape is not regained.

Long Answer Questions (5 Marks)

1. Explain Hooke's Law and its limitations.

Answer:

Hooke's Law states that within the elastic limit, stress is directly proportional to strain.

$\text{Stress} \propto \text{Strain}$

$\text{Stress} = k \times \text{Strain}$

Limitations:
• Valid only for small deformations.
• Applicable only within elastic limit.
• Not obeyed by all materials.
• Rubber and biological tissues exhibit non-linear behaviour.

2. Explain the stress-strain curve for a metallic wire.

Answer:

The stress-strain curve consists of:
• O–A: Hooke's Law region.
• A–B: Elastic region.
• B: Yield point.
• B–D: Plastic region.
• D: Ultimate tensile strength.
• D–E: Necking region.
• E: Fracture point.

3. What are elastomers? Explain their stress-strain behaviour.

Answer:

Elastomers are materials that can undergo large strains and still return to their original shape after removal of stress.

Examples: Rubber, Aorta Tissue

Characteristics:
• Large elastic region.
• Non-linear stress-strain curve.
• No well-defined plastic region.
• Hooke's Law is not obeyed over most of the elastic range.

Key Formulae

$\text{Stress} \propto \text{Strain}$

$\text{Stress} = k \times \text{Strain}$

$k = \frac{\text{Stress}}{\text{Strain}}$

where $k$ is the Modulus of Elasticity.

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