Hooke’s law — the correct validity range Every material obeys Hooke’s law within which portion of its stress–strain response?

Introduction / Context:
Hooke’s law states that stress is proportional to strain. While widely used, it only holds over a specific range of behavior. Distinguishing between the proportional limit and the elastic limit helps avoid design errors, especially for metals that show slight nonlinearity before yield.

Given Data / Assumptions:

• Tension test on a typical metal specimen.
• Small strains; room temperature.
• Standard engineering definitions for limits.

Concept / Approach:
The limit of proportionality is the largest stress at which stress is strictly proportional to strain (linear relation). The elastic limit is the maximum stress at which the material will still fully recover its original dimensions on unloading. Some materials exhibit a small nonlinear elastic region between these two limits. Therefore, the strict validity of Hooke’s law is only up to the proportional limit.

Step-by-Step Solution:

Verification / Alternative check:
Compare unloading paths: up to the elastic limit, unloading returns to zero strain; however, strict linear proportionality ends earlier at the proportional limit.

Why Other Options Are Wrong:

• Elastic limit (option a): elastic recovery can include slight nonlinearity; not strictly Hookean.
• Plastic point and upper yield (options b and e): by then, proportionality is lost.
• None of these (option d): incorrect because the proportional limit is the accepted answer.

Common Pitfalls:
Using the terms proportional limit and elastic limit interchangeably; they are close but not identical for many materials.

Final Answer:
Limit of proportionality