We show a gradient-regularised damage formulation for the implementation in commercial finite element codes. The novelty is that the numerical implementation is established within a thermo-mechanically coupled finite element formulation, where the heat equation solution capabilities are utilised for the damage regularisation.
The non-local, gradient-extended and geometrically non-linear damage formulation is based on an overall free energy function, where the standard local free energy contribution is additively extended by two non-local terms. The first additional term basically contains the referential gradient of the non-local damage variable. Secondly, a penalty term is added to enforce equivalence between the local damage variable -- whose evolution is governed by an ordinary differential equation -- and the non-local damage field variable that is governed by an additional balance equation of elliptic type.
Since the additional elliptic balance equation is formally equivalent to the steady-state heat equation, the framework at hand allows for the regularisation of damage using the heat equation. In other words, existing finite element codes that provide thermo-mechanically coupled finite elements can be utilised to efficiently regularise the damage formulation. To this end, we show representative three-dimensional boundary value problems, the solution of which can take advantage of the features of existing, sophisticated finite element codes without the need for the implementation of user element routines.