summaryrefslogtreecommitdiff
path: root/source/know/concept/navier-cauchy-equation
diff options
context:
space:
mode:
authorPrefetch2022-10-14 23:25:28 +0200
committerPrefetch2022-10-14 23:25:28 +0200
commit6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae (patch)
treea0abb6b22f77c0e84ed38277d14662412ce14f39 /source/know/concept/navier-cauchy-equation
Initial commit after migration from Hugo
Diffstat (limited to 'source/know/concept/navier-cauchy-equation')
-rw-r--r--source/know/concept/navier-cauchy-equation/index.md108
1 files changed, 108 insertions, 0 deletions
diff --git a/source/know/concept/navier-cauchy-equation/index.md b/source/know/concept/navier-cauchy-equation/index.md
new file mode 100644
index 0000000..b9db548
--- /dev/null
+++ b/source/know/concept/navier-cauchy-equation/index.md
@@ -0,0 +1,108 @@
+---
+title: "Navier-Cauchy equation"
+date: 2021-04-02
+categories:
+- Physics
+- Continuum physics
+layout: "concept"
+---
+
+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:
+
+$$\begin{aligned}
+ \va{f^*} \dd{V}
+ = \pdvn{2}{\va{u}}{t} \dd{m}
+ = \rho \pdvn{2}{\va{u}}{t} \dd{V}
+\end{aligned}$$
+
+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 $\nabla_j = \ipdv{}{x_j}$ is the partial derivative
+with respect to the $j$th coordinate:
+
+$$\begin{aligned}
+ f_i + \sum_{j} \nabla_j \sigma_{ij}
+ = \rho \pdvn{2}{u_i}{t}
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+ \rho \pdvn{2}{u_i}{t}
+ &= f_i + 2 \mu \sum_{j} \nabla_j u_{ij} + \lambda \sum_{j} \nabla_j \bigg( \delta_{ij} \sum_{k} u_{kk} \bigg)
+ \\
+ &= f_i + 2 \mu \sum_{j} \nabla_j u_{ij} + \lambda \nabla_i \sum_{j} u_{jj}
+\end{aligned}$$
+
+And then into this we insert the definition of the strain components $u_{ij}$, yielding:
+
+$$\begin{aligned}
+ \rho \pdvn{2}{u_i}{t}
+ &= f_i + \mu \sum_{j} \nabla_j \big( \nabla_i u_j + \nabla_j u_i \big) + \lambda \nabla_i \sum_{j} \nabla_j u_{j}
+\end{aligned}$$
+
+Rearranging this a bit leads us to the Navier-Cauchy equation written in index notation:
+
+$$\begin{aligned}
+ \boxed{
+ \rho \pdvn{2}{u_i}{t}
+ = f_i + \mu \sum_{j} \nabla_j^2 u_i + (\mu + \lambda) \nabla_i \sum_{j} \nabla_j u_j
+ }
+\end{aligned}$$
+
+Traditionally, it is written in vector notation instead,
+in which case it looks like this:
+
+$$\begin{aligned}
+ \boxed{
+ \rho \pdvn{2}{\va{u}}{t}
+ = \va{f} + \mu \nabla^2 \va{u} + (\mu + \lambda) \nabla (\nabla \cdot \va{u})
+ }
+\end{aligned}$$
+
+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.