On the (a)symmetry of the stress tensor in continuum mechanics

It is erroneous to believe that, in continuum mechanics, there exists a first principle about the angular momentum budget of a material volume. The latter must be established from the momentum equation (i.e., Newton's second law) with the help of common vector calculus tools (divergence theorem, Reynolds' transport theorem, etc.). If there are no body moments (i.e., no local production/destruction of angular momentum), which is often but not always the case, then the stress tensor must be symmetric. Various forms of the angular momentum budget of a material volume are derived, including that in which the angular momentum is evaluated with respect to the centre of mass