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 u⃗\va{u}, Newton’s second law is as follows, where dm\dd{m} and dV\dd{V} are the particle’s mass and volume, respectively:

f∗⃗dV=∂2u⃗∂t2dm=ρ∂2u⃗∂t2dV\begin{aligned} \va{f^*} \dd{V} = \pdvn{2}{\va{u}}{t} \dd{m} = \rho \pdvn{2}{\va{u}}{t} \dd{V} \end{aligned}

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

f∗⃗=f⃗+∇⋅σ^⊤\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 ∇j=∂/∂xj\nabla_j = \ipdv{}{x_j} is the partial derivative with respect to the jjth coordinate:

fi+∑j∇jσij=ρ∂2ui∂t2\begin{aligned} f_i + \sum_{j} \nabla_j \sigma_{ij} = \rho \pdvn{2}{u_i}{t} \end{aligned}

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

σij=2μuij+λδij∑kukk\begin{aligned} \sigma_{ij} = 2 \mu u_{ij} + \lambda \delta_{ij} \sum_{k} u_{kk} \end{aligned}

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

uij=12(∇iuj+∇jui)\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:

ρ∂2ui∂t2=fi+2μ∑j∇juij+λ∑j∇j(δij∑kukk)=fi+2μ∑j∇juij+λ∇i∑jujj\begin{aligned} \rho \pdvn{2}{u_i}{t} &= 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 uiju_{ij}, yielding:

ρ∂2ui∂t2=fi+μ∑j∇j(∇iuj+∇jui)+λ∇i∑j∇juj\begin{aligned} \rho \pdvn{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:

ρ∂2ui∂t2=fi+μ∑j∇j2ui+(μ+λ)∇i∑j∇juj\begin{aligned} \boxed{ \rho \pdvn{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:

ρ∂2u⃗∂t2=f⃗+μ∇2u⃗+(μ+λ)∇(∇⋅u⃗)\begin{aligned} \boxed{ \rho \pdvn{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.