A user material approach for the solution of multi-field problems in Abaqus: Theoretical foundations, gradient-enhanced damage mechanics and thermo-mechanical coupling

The solution of multi-field problems and the numerical implementation by means of the finite element method constitute a sophisticated part of the characterisation of industrial processes. A comprehensive implementation framework for such a system of coupled field equations into a non-linear large strain finite element formulation is provided. The procedure is derived for a micromorphic approach in a thermo-mechanical setting. Although the provided framework contributes to a particular three-field problem it is not limited to a specific application or a specific number of coupled field equations from a conceptual point of view. The solution of the considered system of equations is separated into two coupled domains, with the balance of linear momentum and a balance equation of heat equation-type being solved on each of them. Since both, the balance of micromorphic momentum and the heat balance equation, are partial differential equations of Laplace-type, the resulting two-instance problem can be solved in the framework of commercial finite element software, such as Abaqus, based on a thermo-mechanical user material. To assess the framework for a particular constitutive model, a gradient-enhanced damage model in a thermo-mechanical setting is applied and representative simulation results are discussed. The Abaqus framework is made available as an open-source code on GitHub (https://github.com/InstituteOfMechanics/Thermomechanical_ Gradient_Enhanced_Damage_UMAT).