In strength of materials: Which named law states that, within the elastic limit of a material, the strain produced is directly proportional to the stress producing it (i.e., stress ∝ strain)?

Introduction / Context:
Hooke's law is a foundational concept in strength of materials and structural engineering. It links stress (internal force per unit area) to strain (deformation per unit length) for materials behaving elastically. Engineers use this relationship to size members, estimate deflections, and ensure serviceability before yielding occurs.

Given Data / Assumptions:

• Material behavior considered is within the elastic limit.
• Loading is such that linear elastic response is valid.
• Temperature and time-dependent (creep/relaxation) effects are neglected.

Concept / Approach:
Hooke's law states stress = E * strain for axial loading, where E is Young's modulus. In shear, tau = G * gamma, with G as shear modulus. The proportionality is valid only up to the proportional limit; beyond this, nonlinearities and plasticity occur.

Step-by-Step Solution:
Identify the statement: “within elastic limits, strain is proportional to stress.”Recall the named law matching this statement: Hooke's law.Therefore, select “Hooke's law.”

Verification / Alternative check:
Tensile test curves show a straight initial segment where stress/strain is constant (slope = E). This region corresponds to Hookean behavior.

Why Other Options Are Wrong:
Poisson's law: relates lateral and longitudinal strains via Poisson's ratio, not stress-proportionality.Bernoulli's law: fluid dynamics principle, not solids.“Stress law”/“None”: not standard names for this relationship.

Common Pitfalls:
Assuming proportionality holds beyond yield or during large deformations; ignoring effects like temperature, strain-rate, or viscoelasticity.

Final Answer:
Hooke's law.