Hooke’s law in mechanics of materials: Within the elastic limit of a homogeneous material, the normal stress is ______ to the corresponding normal strain.

Introduction / Context:
Hooke’s law is the foundational linear relationship used in elasticity and structural analysis. It links stress and strain for materials behaving elastically and underlies the use of Young’s modulus in design calculations.

Given Data / Assumptions:

• Material behaves linearly within its elastic (proportional) limit.
• Small deformations; unloading follows loading without permanent set.
• Uniaxial normal stress–strain state is considered.

Concept / Approach:
Hooke’s law states: σ = E * ε, where σ is normal stress, ε is normal strain, and E is Young’s modulus (a constant for a given material and temperature, within the elastic range). This is a direct proportionality: doubling strain doubles stress.

Step-by-Step Solution:
Identify regime: elastic and proportional region.Apply linear law: σ = E * ε.Conclude: stress is directly proportional to strain.

Verification / Alternative check:
Tensile test curves show an initial straight line whose slope is E. The relationship deviates after the proportional limit as plasticity begins.

Why Other Options Are Wrong:
Inversely proportional/Curvilinear: contradict the linear segment of a stress–strain curve in the elastic region.Not related/Depends on time: ignores elastic material behavior (viscoelastic exceptions are beyond the simple Hooke’s law).

Common Pitfalls:
Extending Hooke’s law beyond the elastic limit or confusing elastic modulus with yield strength.

Final Answer:
Directly proportional.