# 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:

Where, $\rho $ is the material density.

The Cauchy true stress matrix at a point of $\Gamma $ is defined by:

Where, $n$ is the outward normal to $\Gamma $ at that point. Using this definition, Equation 1 is written:

Gauss' theorem allows the rewrite of the surface integral as a volume integral so that:

As the volume is arbitrary, the expression can be applied at any point in the body providing the differential equation of translation equilibrium:

Use of Gauss' theorem with this equation leads to the result that the true Cauchy stress matrix must be symmetric:

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.