Explicit Modelling of Thermo-Plastic Fracture for Metals under Finite Deformations

In the present work, a numerical framework for predicting the forming behaviour and ductile fracture of low carbon steel under moderate temperatures is elaborated. Therefore, a thermodynamically consistent material model based on the concept of thermo-elastic isomorphism is developed for the macro scale and for the poly-crystalline microstructure as well. The constitutive equations are thermo-mechanically fully coupled and consist of a micro mechanically motivated damage model. The coupling of the damage model to the thermo-elastic regime is achieved by studying an artificial micro structure. The effective stiffness and the effective heat transfer are analysed for different void volume fractions. The integration of the constitutive evolution equations of the derived model needs some special treatment under finite deformation plasticity. Standard integration schemes fail because they do not fulfil the axiom of isochoric plastic deformation. In this work, the axiom is enforced by estimating and later cancelling out the volumetric error. For micromechanical analysis a thermo-mechanically coupled, visco-plastic regularised crystal-plasticity model is derived. The artificial microstructure is generated by taking into account experimentally observed statistical quantities. The resulting initial boundary value problem is then transformed to its weak form and solved by the finite element method (FEM). Softening materials tend to localise mesh dependently. Therefore, a strategy based on the fulfilment of an additional balance equation depending on the damage value and its gradient is applied. To avoid volumetric locking, a 7-field ansatz is derived by Hu-Washizus variational form. This leads to an element formulation strongly related to the well known Q1P0 element. To model cracks explicitly, the fractured domain is discretised by the extended finite element method (XFEM). This method allows a nearly mesh independent discretisation of the cracks. The classical displacement based extend finite element method ansatz is therefore extended to the temperature and damage field and some adequate enrichment functions are presented. Based on this numerical technique, a crack propagation criterion is defined and different strategies to evolve the implicitly modelled crack face are studied. Finally a combination of these different strategies is proposed and an algorithm for quasi static crack growth is elaborated. In every chapter numerical examples show the strengths and weaknesses of the presented methods.