I'm trying to model isotropic linear elastic deformation in two dimensions. In one dimension, I know that a linear elastic material can be thought of as a spring which obeys Hooke's law $F=-k\Delta x$. In two dimensions, I want to describe a material that obeys Hooke's law in both directions (for simplicity, let's say the x and y directions). I want to say that the deformation in one direction does not influence the deformation in the other directions (i.e. it is isotropic).
Because it is in two dimensions, I don't think the 'spring' analogy applies. Is there an analogous object similar to a spring but which obeys Hooke's law in a two dimensional isotropic sense? Also, since it is two dimensional, can I write Hooke's law as $\vec{F}=k\Delta \vec{x}$? Is there another way to describe hooke's law in higher dimensions?
Best Answer
Hooke's Law is frequently used to model multi-dimensional materials because the stress tensor is simple (linear). The full expression can be found on Wikipedia. The simplification for 2D is straight forward (drop any terms with a 3 in the subscript). Note that whether deformation in one dimension affects the others is a property of the material and shows up through Poisson's Ratio ($\nu$). Independence between deformations in X and Y imply $\nu = 0$
If you imagine two perpendicular springs only, then the terms with $\gamma$ (or different subscripts like 12, 23, 31, depending on the form of the equation) drop out of the expression as those are shear terms. The shear terms can be thought of as a spring across the diagonal. The stress tensor $\sigma$ is defined as the force per unit area.