Q.State Hooke's law of elasticity in words and as an equation. Explain, in terms of a stress-strain graph, exactly what the "proportional limit" represents, and why Hooke's law is said to hold only up to this limit and not beyond it.

Hooke's law states that, within a certain limiting value of stress, the strain produced in an elastic body is directly proportional to the stress applied to it:

where $k$ is a constant (the modulus of elasticity) for the given material and type of deformation.

The proportional limit is the specific value of stress up to which this proportionality genuinely holds -- on a stress-strain graph, it is the point where the initial straight line (through the origin) stops being straight. Below the proportional limit, doubling the stress exactly doubles the strain, tripling the stress exactly triples the strain, and so on; beyond it, the strain starts to grow at a different (and generally increasing) rate relative to the stress, so the simple proportionality of Hooke's law no longer applies, even though -- for a small additional range of stress beyond the proportional limit, up to the (separately defined) elastic limit -- the material may still fully recover its original shape once unloaded. …

State the law as a direct proportionality between stress and strain; identify the proportional limit as the boundary stress value that marks the en …

• Treating Hooke's law as holding for all values of stress, rather than only up to the proportional limit.
• Conflating the proportional limit with the fracture point -- the material does not break at the proportional limit; it simply stops obeying the …