Stress, Strain, and Hooke's Law

10.8. Stress, Strain, and Hooke's Law

The force per unit area required to produce any elastic deformation is referred to as the stress. The fractional change in a quantity (DL/L₀, DX/L₀, or DV/V₀) that results when a stress is applied is referred to as the strain. In each situation described in section 10.8, the stress was found to be directly proportional to the strain, the constants of proportionality being either Young's modulus, the shear modulus, or the bulk modulus. The relationship between stress and strain is known as Hooke's Law. The SI unit of stress is the pascal (Pa) where 1 Pa = 1 N/m², while the strain is a dimensionless quantity. Materials obey Hooke's law up to a certain limit. If too much stress is applied to a material, it becomes unable to "spring back" to its original size. We would say that the material has exceeded its "elastic limit". Beyond this limit the material can no longer obey Hooke's law.

Example 8

A steel cable is able to withstand a stress of 8.0 × 10⁷ Pa up to its elastic limit. If Young's modulus for steel is Y = 2.0 × 10¹¹ N/m², what is the strain?

Hooke's law states that the stress is proportional to the strain. In this case, since the wire is being stretched, the constant of proportionality is Young's modulus. We have, therefore,