Let
be a volume occupied by a part of the body in the current
configuration, and
the boundary of the body. In the Lagrangian formulation,
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.
is the traction surface on
and
are the body
forces.
Force equilibrium for the volume is then:
(1)
Where,
-
- Material density
The Cauchy true stress matrix at a point of
is defined by:
(2)
Where,
is the outward normal to
at that point. Using this definition,
式 1 is
written:
(3)
Gauss' theorem allows the rewrite of the surface integral as a volume integral so
that:
(4)
As the volume is arbitrary, the expression can be applied at any point in the body providing the
differential equation of translation equilibrium:
(5)
Use of Gauss' theorem with this equation leads to the result that the true Cauchy stress matrix
must be symmetric:
(6)
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.