Can Hooke's law be derived?

Question

Can we derive Hooke's law from the theory of elasticity? I know it is not a fundamental law and therefore can be derived from more basic considerations.

Answer 1

Yes, we can derive Hooke's Law from more basic continuum conditions, provided that the material be stable and at equilibrium, so that the strain energy is smoothly minimized with respect to the distance between atoms. (This energy is sometimes called the pair potential and is modeled using functions such as the Lennard-Jones potential.)

Consider just a pair of atoms. For slight positional deviations $\delta$ (corresponding to the small-strain assumption of linear elasticity) around the smooth energy minimum $E(0)$, the energy $E$, regardless of its true functional form, can be expanded using a Taylor series:

$$E(\delta)=E(0)+E^{\prime}(0)\delta+\frac{E^{\prime\prime}(0)\delta^2}{2!}+\frac{E^{\prime\prime\prime}(0)\delta^3}{3!}+\cdots,$$

where the prime notation denotes derivatives with respect to position. Now let's set our energy reference to $E(0)=0$, note that $E^\prime(0)=0$ because we're at an energy minimum, and drop all but the very next term. This gives

$$E(\delta)\approx\frac{1}{2}E^{\prime\prime}(0)\delta^2,$$

which describes the energy of an idealized spring with spring constant $E^{\prime\prime}$ and displacement $\delta$; the derivative of this equation with respect to position provides the restoring force, which is

$$F(\delta)\approx E^{\prime\prime}(0)\delta.$$

Now define stress as $\sigma=F/A$ and engineering strain as $\epsilon=\delta/L$ and we have $$\sigma=\frac{LE^{\prime\prime}(0)}{A}\epsilon$$ or $$\sigma=Y\epsilon,$$

which is simple Hooke's Law for a corresponding elastic modulus $Y$.

Answer 2

I like this particular derivation because it avoids the requirement to actually specify inter-atomic forces, such as the coulomb force and the Pauli force. Technically, however, you will have one relationship for compression and another for tension and therefore two elastic constants. On the macro-scale this method will yield the classic work and virtual work functions structural engineers are so familiar with. This is an excellent derivation.