Continuous and discontinuous modelling of ductile fracture

Abstract

In many metal forming processes (e.g. blanking, trimming, clipping, machining, cutting) fracture is triggered in order to induce material separation along with a desired product geometry. This type of fracture is preceded by a certain amount of plastic deformation and requires additional energy to be supplied in order for the crack to propagate. It is known as ductile fracture, as opposed to brittle fracture (as in ceramics, concrete, etc). Ductile fracture originates at a microscopic level, as the result of voids initiated at inclusions in the material matrix. These microscopic degradation processes lead to the degradation of the macroscopic mechanical properties, causing softening, strain localisation and finally the formation of macroscopic cracks. The initiation and propagation of cracks has traditionally been studied by fracture mechanics. Yet, the application of this theory to ductile fracture, where highly nonlinear degradation processes (material and geometrical) take place in the fracture process zone, raises many questions. To model these processes, continuum models can be used, either in the form of softening plasticity or continuum damage mechanics. Yet, continuous models can not be applied to model crack propagation, because displacements are no longer continuous across the crack. Hence, a proper model for ductile fracture requires a different approach, one that combines a continuous softening model with a strategy to represent cracks, i.e. displacement discontinuities. This has been the main goal of the present work. In a combined approach, the direction of crack propagation is automatically determined by the localisation pattern, and its rate strongly depends on the evolution of damage (or other internal variables responsible for the strain softening). This contrasts with fracture mechanics, where the material behaviour is not directly linked to the crack propagation criteria. Softening materials have to be supplied with an internal length, which acts as a localisation limiter, thereby ensuring the well-posedness of the governing partial differential equations and mesh independent results. For this purpose, a nonlocal gradient enhancement has been used in this work, which gives similar results to nonlocal models of an integral form. A number of numerical methods are available to model displacement discontinuities in a continuum. In the present context, we have used a remeshing strategy, since it has additional advantages when used with large strain localising material models: it prevents excessive element distortions and allows to optimise the element distriviii bution through mesh adaptivity. As a first step towards a continuum-discontinuum approach, an uncoupled damage model is used first, in which damage merely acts as a crack initiation-propagation indicator, without causing material softening. Since uncoupled models do not lead to material localisation, no regularisation is needed. Yet, uncoupled approaches can not capture the actual failure mechanisms and therefore, in general, can give reliable results only when the size of the fracture process zone is so small that its effect can be neglected. When the size of the fracture process zone is large enough, a truly combined model must be used, which is developed in the second part of this study. Due to softening, the transition from the continuous damage material to the discrete crack occurs gradually, with little stress redistribution, in contrast with the previous uncoupled approach. The gradient regularised softening behaviour is introduced in the yield behaviour of an elastoplastic material. The combined model has been applied satisfactorily to the prediction of ductile failure under shear loading conditions. Third, to be able to apply the model to more general loading conditions, the material description has been improved by introducing the influence of stress triaxiality in the damage evolution of a gradient regularised elastoplastic damage model. The model has been obtained using the continuum damage mechanics concept of effective stress. Results show how compressive (tensile) states of triaxiality may increase (decrease) the material ductility. Finally, the combined approach is applied to the modelling of actual metal forming processes, e.g. blanking, fine blanking, score forming. The gradient regularisation has been implemented in an operator-split manner, which can be very appealing for engineering purposes. To capture the large strain gradients in the localisation zones, a new mesh adaptivity criterion has been proposed. The results of the simulations are in good agreement with experimental data from literature.