Categories: Continuum physics, Physics.

# Navier-Cauchy equation

The Navier-Cauchy equation describes elastodynamics: the movements inside an elastic solid in response to external forces and/or internal stresses.

For a particle of the solid, whose position is given by the displacement field $$\va{u}$$, Newton’s second law is as follows, where $$\dd{m}$$ and $$\dd{V}$$ are the particle’s mass and volume, respectively:

\begin{aligned} \va{f^*} \dd{V} = \pdv[2]{\va{u}}{t} \dd{m} = \rho \pdv[2]{\va{u}}{t} \dd{V} \end{aligned}

Where $$\rho$$ is the mass density, and $$\va{f^*}$$ is the effective force density, defined from the Cauchy stress tensor $$\hat{\sigma}$$ like so, with $$\va{f}$$ being an external body force, e.g. from gravity:

\begin{aligned} \va{f^*} = \va{f} + \nabla \cdot \hat{\sigma}^\top \end{aligned}

We can therefore write Newton’s second law as follows, while switching to index notation, where $$\nabla_j = \pdv*{x_j}$$ is the partial derivative with respect to the $$j$$th coordinate:

\begin{aligned} f_i + \sum_{j} \nabla_j \sigma_{ij} = \rho \pdv[2]{u_i}{t} \end{aligned}

The components $$\sigma_{ij}$$ of the Cauchy stress tensor are given by Hooke’s law, where $$\mu$$ and $$\lambda$$ are the Lamé coefficients, which describe the material:

\begin{aligned} \sigma_{ij} = 2 \mu u_{ij} + \lambda \delta_{ij} \sum_{k} u_{kk} \end{aligned}

In turn, the components $$u_{ij}$$ of the Cauchy strain tensor are defined as follows, where $$u_i$$ are once again the components of the displacement vector $$\va{u}$$:

\begin{aligned} u_{ij} = \frac{1}{2} \big( \nabla_i u_j + \nabla_j u_i \big) \end{aligned}

To derive the Navier-Cauchy equation, we start by inserting Hooke’s law into Newton’s law:

\begin{aligned} \rho \pdv[2]{u_i}{t} %= f_i + \sum_{j} \nabla_j \sigma_{ij} &= f_i + 2 \mu \sum_{j} \nabla_j u_{ij} + \lambda \sum_{j} \nabla_j \bigg( \delta_{ij} \sum_{k} u_{kk} \bigg) \\ &= f_i + 2 \mu \sum_{j} \nabla_j u_{ij} + \lambda \nabla_i \sum_{j} u_{jj} \end{aligned}

And then into this we insert the definition of the strain components $$u_{ij}$$, yielding:

\begin{aligned} \rho \pdv[2]{u_i}{t} &= f_i + \mu \sum_{j} \nabla_j \big( \nabla_i u_j + \nabla_j u_i \big) + \lambda \nabla_i \sum_{j} \nabla_j u_{j} \end{aligned}

Rearranging this a bit leads us to the Navier-Cauchy equation written in index notation:

\begin{aligned} \boxed{ \rho \pdv[2]{u_i}{t} = f_i + \mu \sum_{j} \nabla_j^2 u_i + (\mu + \lambda) \nabla_i \sum_{j} \nabla_j u_j } \end{aligned}

Traditionally, it is written in vector notation instead, in which case it looks like this:

\begin{aligned} \boxed{ \rho \pdv[2]{\va{u}}{t} = \va{f} + \mu \nabla^2 \va{u} + (\mu + \lambda) \nabla (\nabla \cdot \va{u}) } \end{aligned}

A special case is the Navier-Cauchy equilibrium equation, where the left-hand side is just zero. That version describes elastostatics: the deformation of a solid at rest.

## References

1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.