Hooke's law states that, within certain limits, stress is directly proportional to strain. For a typical ductile material under uniaxial loading, Hooke's law is valid over which portion of the stress strain curve?

Introduction / Context:

In strength of materials and solid mechanics, Hooke's law is fundamental for describing elastic deformation. It states that, up to a certain limit, the extension or strain is proportional to the applied load or stress. This question asks you to identify the part of the stress strain curve for which this straight line proportionality strictly holds for a typical ductile material, such as mild steel.

Given Data / Assumptions:

• The material is ductile and follows a typical stress strain curve with a proportional limit, an elastic limit, yield point and plastic region.
• Hooke's law is written as stress proportional to strain or stress = E * strain, where E is Young modulus.
• We are interested in the regime where stress strain relationship is exactly linear.

Concept / Approach:

The stress strain curve of a ductile material initially rises as a straight line from the origin. In this region, stress is directly proportional to strain and Young modulus is constant. The upper limit of this perfectly linear portion is called the proportional limit. Beyond this point, the curve may continue to be elastic for a short range, meaning that the material will still return to its original shape when unloaded, but the relationship between stress and strain is no longer strictly linear. Therefore, Hooke's law in its original form applies only to the proportional region, not to the full elastic region and certainly not to the plastic region.

Step-by-Step Solution:

Verification / Alternative check:

Experimental stress strain diagrams for mild steel show a straight line from the origin up to a certain stress, followed by a curved region that is still recoverable on unloading, and then a yield plateau and strain hardening region. Textbooks define the proportional limit as the end of the region where Hooke's law is valid. This directly confirms that the law applies only in the initial proportional region, not over the entire elastic span or plastic portion.

Why Other Options Are Wrong:

Option B: The entire stress strain curve includes large plastic deformations and fracture, where the relationship is highly non linear and irreversible, so Hooke's law does not apply.

Option C: The entire elastic region may include slight non linearity beyond the proportional limit, so strict proportionality is not valid throughout.

Option D: The plastic region by definition involves permanent deformation and clearly does not satisfy Hooke's law.

Common Pitfalls:

Students sometimes equate elastic region with proportional region and assume Hooke's law applies up to the elastic limit. In reality, the proportional limit is usually slightly below the elastic limit. For accurate design, we use the strictly linear portion associated with Hooke's law, not the entire elastic span.

Final Answer:

Hooke's law is valid for only the initial proportional region of the stress strain curve.