According to Hooke's Law, what relationship exists between stress and strain?

Hooke's Law states that the force or stress applied to an elastic material produces a proportional amount of deformation or strain, provided that the material has not exceeded its elastic limit. This relationship can be summarized by the equation ( \sigma = E \epsilon ), where ( \sigma ) represents stress, ( E ) is the modulus of elasticity (a constant specific to the material), and ( \epsilon ) is the strain.

When the material is within its elastic limit, stress increases in direct proportion to strain, meaning that if you double the applied stress, the strain will also double, assuming constant material properties. This linear relationship holds true only up to the brink of the material's elastic limit. Beyond this point, the material may become permanently deformed, making the relationship no longer linear.

The other answer options do not accurately reflect the principles of Hooke's Law. For instance, stating that stress is inversely proportional to strain directly contradicts Hooke’s Law. Similarly, the assertions that strain varies with temperature or that stress equals constant strain are not aligned with the fundamental understanding of stress-strain relationships in materials under mechanical loads.