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

I had a PhD interview yesterday. One of the faculty asked me if it is possible to use a modified version of the Navier-Stokes equation for solid simulation, knowing that both are derived from Newton's law of motion.
I could just say that even though solid simulation can be done in Eulerian description for example in the Hyperelastic models, there are fundamental differences in fluid and solids and it might be really challenging to modify the Navier-Stokes equation for solid simulation.

Implicitly, I said yes and no. Does anyone have their views on this?

• How would you define the difference between a solid and a fluid? (Two-word answer: "shear stress.") Jun 2 at 22:38
• I mean, solid and fluids have completely different constitutive relationship. Solid yields, they depict elasticity, plasticity. In solids, the density changes with deformation but it is considered only in molecular dynamics. Jun 4 at 17:55

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.

$$\partial_i\sigma^{ij}=-F^j$$

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.