Is it possible to use the Navier-Stokes equation for solid simulation?

I would say the following. The whole point of the Navier-Stokes equations is to determine the local distributions, e.g. pressure, flow-velocity, density, throughout the fluid, as a function of time.

In a rigid body, the flow-velocity is constant (in the possibly non-inertial frame fixed to center-of-mass and fixed in orientation). In this case the Navier-Stokes equations simplify tremendously to give a simple equilibrium equation: the divergence of the stress-tensor must balance external (and fictitious) forces.

$$partial_isigma^{ij}=-F^j$$

where $sigma^{ij}$ is the Cauchy stress-tensor and $F^j$ are the external & fictitious forces.

So Navier-Stokes for a solid would simply turn into two separate equations:

1. An equilibrium condition relating the external force-fields and the divergence of the stress-tensor, which when solved would give the local tensorial stress distribution throughout the body.

2. The ordinary Newtonian equations of motion for a rigid-body, which when solved would give the center-of-mass position (or velocity) and orientation (or angular velocity) as a function of time.

If you're not interested in the first item though, then all you're left with is good old Newton's equations for a rigid body.

For an elastic body, I see no reason why one couldn't use Navier-Stokes, but I am totally ignorant on this subject. I would guess one would introduce local restorative "elastic" forces.