Question regarding the conditions required for Hooke's law

Question

Given a material which obeys Hooke's law, and given a point on its force-extension graph that indicates its limit of proportionality, call the point $P(x_1,y_1)$, would the following be a correct way of demonstrating Hooke's law mathematically:

$$F=k\Delta L$$ $$\forall\lbrace{F:F\leq y_1\rbrace}\cup\lbrace{\Delta L:\Delta L \leq x_1\rbrace}$$

and if not, what should I do differently so that it if?

Answer 1

I think the notation $F(x) = \left\{ \begin{array}{ll} kx & x\leq x_1 \\ \text{something here} & x > x_1 \end{array}\right.$ is most clear.

You're right to observe that the graph of $F(x)$ is a set of points of the form $(x,F(x))$, and you might use set builder notation like $\{(x,kx):x \in [0,x_1]\}$.

You could also write $F : [0,x_1] \rightarrow \mathbb{R}$ to indicate if you only have a force law for positive $x \leq x_1$.

Answer 2

You cannot demonstrate this law. It is a law, in the sense that if you measure the strain of many materials under many different stresses, the expression $F \sim \Delta L$ seems to hold always for $F < y_1$. But that is an observation!