# Equilibrium Equations

Let $\Omega$ be a volume occupied by a part of the body in the current configuration, and $\Gamma$ the boundary of the body. In the Lagrangian formulation, $\Omega$ is the volume of space occupied by the material at the current time, which is different from the Eulerian approach where a volume of space through which the material passes is examined. $\tau$ is the traction surface on $\Gamma$ and $b$ are the body forces.

Force equilibrium for the volume is then:(1)
$\underset{\Gamma }{\overset{}{\int }}{\tau }_{i}d\Gamma +\underset{\Omega }{\int }\rho {b}_{i}d\Omega =\underset{\Omega }{\int }\rho \frac{\partial {v}_{i}}{\partial t}d\Omega$
Where,
$\rho$
Material density
The Cauchy true stress matrix at a point of $\Gamma$ is defined by:(2)
${\tau }_{i}={n}_{j}{\sigma }_{ji}$
Where, $n$ is the outward normal to $\Gamma$ at that point. Using this definition, Equation 1 is written:(3)
$\underset{\Gamma }{\int }{n}_{j}{\sigma }_{ji}d\Gamma +\underset{\Omega }{\int }\rho {b}_{i}d\Omega =\underset{\Omega }{\int }\rho \frac{\partial {v}_{i}}{\partial t}d\Omega$
Gauss' theorem allows the rewrite of the surface integral as a volume integral so that:(4)
$\underset{\Gamma }{\int }{n}_{j}{\sigma }_{ji}d\Gamma =\underset{\Omega }{\int }\frac{\partial {\sigma }_{ij}}{\partial {x}_{j}}d\Omega$
As the volume is arbitrary, the expression can be applied at any point in the body providing the differential equation of translation equilibrium:(5)
$\frac{\partial {\sigma }_{ij}}{\partial {x}_{j}}+\rho {b}_{i}=\rho \frac{\partial {v}_{i}}{\partial t}$
Use of Gauss' theorem with this equation leads to the result that the true Cauchy stress matrix must be symmetric:(6)
$\sigma ={\sigma }^{T}$

So that at each point there are only six independent components of stress. As a result, moment equilibrium equations are automatically satisfied, thus only the translational equations of equilibrium need to be considered.