Interpretation of Hooke's Law I often see people interpreting Hooke's Law $σ=Eε$ as,

"The deformation $ε$ that occurs when you subject a material to a stress $σ$."

This makes it sound like stress is an external stimulus that causes the material to deform. But from what I know, stress is an internal phenomenon, not an external one.
So technically, isn't it more correct to interpret Hooke's law as

"The stress σ that develops in a material given a deformation of $ε$"?

I would greatly appreciate it if someone could clear this up for me.
 A: Unfortunately, both descriptions are misleading because both describe the relationship as a one-way cause-and-effect relationship. The correct way to describe it is

the stress is proportional to the strain

It doesn’t matter if the stress is given and the strain is obtained by Hooke’s law or if the strain is given and the stress is obtained by Hooke’s law. Either way they are proportional to each other.
Unfortunately, whether it is Newton’s laws, Ohm’s law, Hooke’s law, or Maxwell’s equations, the tendency to verbally express such equations in cause-and-effect language is fairly strong and common in many introductory physics courses. It is almost universally inappropriate. A real cause-and-effect relationship is given by an equation of the form $$f(t)=g(t-\Delta t)$$ In this equation $g$ is the cause and $f$ is the effect and $0<\Delta t$ so that the cause always precedes the effect. In an expression like Hooke’s law both stress and strain are happening at the same time and causes happen before effects. It is a simple proportionality, not a cause-and-effect relationship.
A: Hooke's Law—in the standard form as you've written it—says that the normal stress $\sigma$ and normal strain $\varepsilon$ are linearly coupled by a constant of proportionality $E$ (termed Young's modulus). This is a good approximation for a long rod of a stable, constant-temperature solid for small axial deformations over moderate time scales. (All these qualifiers are needed to eliminate the effects of stresses in other directions, temperature dependence, creep, etc.) Note that no specific cause or effect regarding the stress or strain is implied.
