---
title: "Navier-Cauchy equation"
firstLetter: "N"
publishDate: 2021-04-02
categories:
- Physics
- Continuum physics

date: 2021-04-02T15:04:55+02:00
draft: false
markup: pandoc
---

# 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](/know/concept/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](/know/concept/hookes-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](/know/concept/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.