I've recently been re-learning some physics, and a question came to me when looking over Hooke's law:
In the following I am always assuming that the force required for permanent deformation is sufficiently large and thus irrelevant for my purposes.
Hooke's Law gives that the force applied to a spring is linearly proportional to the displacement, i.e. $F \propto x$. I am wondering how one might accomplish a system where the proportionality is quadratic ($F \propto x^2$). What about cubic? ... What about for any arbitrary degree? ($F \propto x^n$) by using only linear components such as ideal springs that obey Hooke's law.
One thought that came to my mind was that this might be possibly by using some combination of multiple springs, some in series and some in parallel. But I am not certain if this would work.
Alternatively, in my research prior to posting this I discovered other types of springs such as a torsion spring, possibly something like this could be used in combination in some way?
The primary focus of this question is to understand whether a system built out of ideal spring-like components that react with forces linear to their displacement, can as a whole, exhibit a force that reacts nonlinearly to its displacement.