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.

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