According to Hooke's Law, when is stress in an elastic body proportional to strain?

Hooke's Law states that the stress applied to an elastic body is directly proportional to the strain it produces, provided that the material remains within its elastic limit. In this context, the elastic limit is the maximum stress that a material can withstand without undergoing permanent deformation. Therefore, when the elastic limit is not exceeded, the material will return to its original shape after the load is removed, and the relationship between stress and strain remains linear, characterized by Young's Modulus.

When the elastic limit is exceeded, a material will no longer behave elastically, meaning it may deform permanently or take on a different structure, causing the proportional relationship between stress and strain to no longer hold. Similarly, while exceeding the yield point means the material has passed the point where it can no longer return to its original state, which disrupts the linear relationship defined by Hooke's Law.

In scenarios where Young's Modulus is constant, the relationship described by Hooke's Law can be true during linear elasticity; however, this does not address whether the elastic limit is exceeded or not. The definitive condition for stress to be proportional to strain according to Hooke's Law is specifically tied to remaining within the elastic limit of the material, confirming that the relationship exists under those circumstances