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# Hooke’s Law

Hooke’s law is a principle of physics that states that the force needed to extend or compress a spring by some distance is proportional to that distance. That is: where is a constant factor characteristic of the spring, its stiffness.

$F = -kX,$ where $k$ is a constant factor characteristic of the spring, its stiffness.

Hooke’s law for a spring is often stated under the convention that $F$ is the restoring (reaction) force exerted by the spring on whatever is pulling its free end. In that case the equation becomes

$F= -k X\,$

since the direction of the restoring force is opposite to that of the displacement.

In SI units, displacements are measured in metres (m), and forces in newtons (N or kg·m/s2). Therefore the spring constant $k$, and each element of the tensor $\kappa$, is measured in newtons per metre (N/m), or kilograms per second squared (kg/s2).

For continuous media, each element of the stress tensor $\sigma$ is a force divided by an area; it is therefore measured in units of pressure, namely pascals (Pa, or N/m2, or kg/(m·s2). The elements of the strain tensor $\epsilon$ are dimensionless (displacements divided by distances). Therefore the entries of $c_{ijk\ell}$ are also expressed in units of pressure.