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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?

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  • $\begingroup$ 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 $\endgroup$ – nicoguaro Jun 11 '20 at 20:07
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The simplest formulation for almost all solid mechanics modelling is Lagrangian, not Eulerian. You are considering the motion of an element of the structure which has constant mass, and therefore mass conservation is "baked in" to the model. You don't need an equation to represent it.

The continuity equation would determine the local density distribution of the deformed solid, but since plastic deformation takes place at constant volume, unless the material can have large elastic strains the density changes will usually be small and are not interesting from a practical point of view. If you don't care about density changes, you don't need the continuity equation to calculate them.

There are some situations where an Eulerian formulation is useful to avoid numerical problems with large plastic deformations, but they usually involve some sort of "re-meshing" algorithm, not the fluid mechanics type of Eulerian formulation.

The shape of boundary of a solid is determined by the fact that the material is a solid, unlike a fluid whose the shape is constrained by the boundary conditions imposed on the model. Trying to use a Eulerian type of discretization that is "fixed in space" isn't very useful for most solid modeling applications, because it doesn't accurately represent boundary of a solid object when it deforms.

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  • $\begingroup$ Very good point! However, with a phenomenon like linear wave propagation, the displacement is small enough that the difference between the Langrangian and Eulerian formulations is negligible. If you represent the medium in Eulerian coordinates, which is most common by far in linear acoustics in fluids and solids, a passing pressure wave causes a density variation that you would think would need to be taken into account. I may come back with another comment when I figure out why the continuity equation is only used for linear acoustics in fluid and not in solids. $\endgroup$ – Erlend Magnus Viggen Jun 10 '20 at 12:40

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