  Physical systems are modeled by field equations; these are coupled, partial
differential equations in space and time. Field equations are often given by
balance equations and constitutive equations, where the former are
axiomatically given and the latter are thermodynamically derived. This approach
is useful in thermomechanics and electromagnetism, yet challenges arise once we
apply it in damage mechanics for generalized continua. For deriving governing
equations, an alternative method is based on a variational framework known as
the extended Noether's formalism. Its formal introduction relies on
mathematical concepts limiting its use in applied mechanics as a field theory.
In this work, we demonstrate the power of extended Noether's formalism by using
tensor algebra and usual continuum mechanics nomenclature. We demonstrate
derivation of field equations in damage mechanics for generalized continua,
specifically in the case of strain gradient elasticity.
