  The weak-field limit of Einstein--Cartan (EC) relativity is studied. The
equations of EC theory are rewritten such that they formally resemble those of
Einstein General Relativity (EGR); this allows ideas from post-Newtonian theory
to be imported without essential change. The equations of motion are then
written both at first post-Newtonian (1PN) order and at 1.5PN order. EC
theory's 1PN equations of motion are found to be those of a micropolar/Cosserat
elastic medium, along with a decoupled evolution equation for non-classical,
spin-related fields. It seems that a necessary condition for these results to
hold is that one chooses the non-classical fields to scale with the speed of
light in a certain empirically reasonable way. Finally, the 1.5PN equations
give greater insight into the coupling between energy-momentum and spin within
slowly moving, weakly gravitating matter. Specifically, the weakly relativistic
modifications to Cosserat theory involve a gravitational torque and an
augmentation of the gravitational force due to a `dynamic mass moment density'
with an accompanying `dynamic mass moment density flux', and new forms of
linear momentum density captured by a `dynamic mass density flux' and a
`dynamic momentum density'.
