# Why is the continuity equation hardly used in solid mechanics when it is essential in fluid mechanics?

For any continuum, fluid or solid, we can express mass conservation through the continuity equation

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 ,$$

where $$\rho$$ is density and $$\mathbf{v}$$ is velocity. We can also express momentum conservation through the Cauchy momentum equation

$$\frac{\partial \rho \mathbf{v}}{\partial t} + \nabla \cdot (\rho\mathbf{v}\mathbf{v}) = \nabla \cdot \mathbf{\sigma} + \mathbf{f} ,$$

where $$\mathbf{\sigma}$$ is the stress tensor and $$\mathbf{f}$$ is the body force density. Fluids and solids only differ in their stress tensor: In fluids, the stress is determined by the strain rate and pressure, giving the Navier-Stokes equation, while in solids (assuming linear elasticity), the stress is determined by the strain.

In fluid mechanics, both these equations are essential and very much used, often in an incompressible form where the density $$\rho$$ is assumed constant and the continuity equation hence becomes $$\nabla \cdot \mathbf{v} = 0$$. However, the continuity equation hardly seems to be used in solid mechanics. I have searched through a few books on elasticity, continuum mechanics, and waves in solids, and the continuity equation seems only to be mentioned in the context of fluid mechanics.

Why is it, then, that the continuity equation is essential in fluid mechanics but not in solid mechanics, when both solids and fluids are continua, governed by the same fundamental equations?

• As a consequence of Eulerian and Lagrangian frames being the same the density does not change and the continuity equation is satisfied trivially. Check this answer: scicomp.stackexchange.com/a/26324/9667 – nicoguaro Jun 11 '20 at 20:07