# Hooke's Law and stretched fabric

• verdigris

#### verdigris

A piece of fabric obeys Hooke's Law - force is proportional to extension - when stretched in one direction.Is it possible for the fabric to continue to obey Hooke's Law if it is simultaneously stretched in another direction at right angles to the first direction?

Basically yes (to some extent), since every minimum looks like a parabola on some scale.

It depends on the material. If the material is isotropic the direction in which you stretch doesn't matter. For some materials, it will stretch more easily in one direction than in another (and different for the 3rd independent direction still). You'd have 3 spring constants (one for each direction).

Linear or nonlinear material behaviour is indepedent of whether the material is isotropic or anisotropic.

For an anisotropic materal the stiffness is different in different directions, and if you pull in one direction you might get stretching or bending in a different direction!. That doesn't contradict Hooke's law, provided twice the force gives twice the stretch.

The general answer to the OP is "yes, provided the forces and stretches are small enough so the material remains linear".

If by "fabric" you mean something like a piece of woven cloth, then it is highly anisotropic, and also highly nonlinear even if you pull in just one direction. The initial pull tends to straighten out the fibers, so the stiffness is low and the stretch is quite large. As the fibers become straight the stiffness becomes much higher because you have to stretch the fibers themselves.

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