What exactly is Hooke's law defined for?

Question

I used to think that Hooke's law was a relationship between how much a bar under uniaxial loading deformed and the internal force (per unit area) that developed within that bar. But this clearly isn't the case as I have recently seen that Hooke's law is used in analyzing the stress in pure bending of beams. So it seems that Hooke's law is a lot more general than I had thought. If Hooke's law isn't specifically defined for a bar under uniaxial loading, what physical object is it exactly defined for? What system/object is Hooke's law trying to describe? My guess is that Hooke's law is defined for an infinitesimal cubic element which feels a normal stress on its sides from neighboring elements (picture below). That is, Hooke's law relates the normal stress and normal strain on an infinitesimal cubic element. This might make Hooke's law general enough so that it could be applied to many loading situations since we can think of any object as being made of many of these infinitesimal elements and the total strain of the object is the sum of the strains of all of the elements. Is this correct? I'd appreciate it if someone could clear this up for me.

Answer 1

Hooke’s law describes the linear relationship between stress and strain as well as the linear relationship between force and displacement. It is not “defined “ for “things”. It is applied to things. One of those things is a spring.

In the case of pure bending of beams, the outer fibers are subjected to tension and the inner fibers subjected to compression resulting in stretching or shortening the fibers, respectively, and beam deflection. For small displacements the radius of curvature $r$ of the deflection of the beam is proportional to the modulus of elasticity $E$ according to

$$r=\frac{EI}{M}$$

Where $M$ = bending moment and $I$ = moment of inertia about the centroidal axis.

So yes, Hooke’s law can be applied to many loading situations.

Hope this helps

Answer 2

I'd argue that it's even more general than you are guessing.

Suppose you have an object that has some parameter of motion $\theta$, and that parameter has an equilibrium position $\theta_0$. Maybe $\theta$ represents stretching, maybe it represents bending, maybe it represents twisting, maybe it actually represents position in 3D space, who knows.

Lagrangian mechanics teaches us that we can define a generalized force for this parameter (like force, or torque, or bending moment) which tells us how the parameter evolves in time. And, importantly, the generalized force is determined by the potential energy function of the object, $V(\theta)$, which depends on the parameter. Specifically $F = -\frac{dV}{d\theta}$.

For the object to be in equilibrium the generalized force must be zero, so $\left.\frac{dV}{d\theta}\right|_{\theta_0} = 0$. Then, if we look at motions that are near the equilibrium we find $\frac{dp_\theta}{dt} = -\frac{dV}{d\theta} = -\left.\frac{dV}{d\theta}\right|_{\theta_0}-\left.\frac{d^2V}{d\theta^2}\right|_{\theta_0}(\theta-\theta_0)+\dots$ $$\frac{dp_\theta}{dt} = -\left.\frac{d^2V}{d\theta^2}\right|_{\theta_0}(\theta-\theta_0)+\dots$$ As long as $\theta$ is close to equilibrium, the force is basically just Hooke's law with "spring constant" $k = \left.\frac{d^2V}{d\theta^2}\right|_{\theta_0}$.

Whenever a combination of mostly conservative forces (like the forces between atoms in a solid object) combine to make an equilibrium, the motion near that equilibrium will be governed by Hooke's law. That's why you see it compression and twisting and bending and all sorts of things.

Answer 3

Here is an excerpt out of the Wiki article on Hooke's Law. It gives the relationship between the local stresses and local strains in 3D. It applies at any given location within the material.

$${\displaystyle {\begin{aligned}\varepsilon _{11}&={\frac {1}{E}}{\big (}\sigma _{11}-\nu (\sigma _{22}+\sigma _{33}){\big )}\\\varepsilon _{22}&={\frac {1}{E}}{\big (}\sigma _{22}-\nu (\sigma _{11}+\sigma _{33}){\big )}\\\varepsilon _{33}&={\frac {1}{E}}{\big (}\sigma _{33}-\nu (\sigma _{11}+\sigma _{22}){\big )}\\\varepsilon _{12}&={\frac {1}{2G}}\sigma _{12}\,;\qquad \varepsilon _{13}={\frac {1}{2G}}\sigma _{13}\,;\qquad \varepsilon _{23}={\frac {1}{2G}}\sigma _{23}\end{aligned}}}$$ In these equations, E is the Young's modulus and G is the shear modulus (which is related by a simple equation to the Young's modulus and the Poisson ratio).

Just Google Hooke's Law.