In this contribution a phase field model for ductile fracture with linear isotropic hardening is presented. An energy functional consisting of an elastic energy, a plastic dissipation potential and a Griffith type fracture energy constitutes the model.The application of an unaltered radial return algorithm on element level is possible due to the choice of an appropriate coupling between the nodal degrees of freedom, namely the displacement and the crack/fracture fields. The degradation function models the mentioned coupling by reducing the stiffness of the material and the plastic contribution of the energy density in broken material. Furthermore, to solve the global system of differential equations comprising the balance of linear momentum and the quasi-static Ginzburg-Landau type evolution equation, the application of a monolithic iterative solution scheme becomes feasible. The compact model is used to perform 3D simulations of fracture in tension. The computed plastic zones are compared to the dog-bone model that is used to derive validity criteria for K IC measurements. K E Y W O R D Sdog-bone model, ductile fracture, fine element, phase field model INTRODUCTIONBefore a material fails it can sustain external loads by deforming itself in such a way, that once the external loads are withdrawn, it does not return to its initial shape, that is, the material performs a plastic transformation. Dependent on how strong the capability of a material to perform a plastic transformation is developed, different theories have to be employed to describe the failure process of the respective material. From an engineering point of view however, generalized failure parameters are often of greater interest than the actual failure mechanism. Thus, for example, validity criteria for CT-specimen that have to be met in tensile testing exist to measure a valid fracture toughness K IC . These criteria shall guarantee that assumptions of small scale yielding and linear fracture mechanics are fulfilled as well as a state of plane strain dominates at the crack tip. The latter assumption is based on the so called "dog-bone" model, predicting a state of plane strain in the center of the specimen and a plane stress state at the surface. However, the validity of the criteria is frequently questioned. [1,2] Assessing failure of ductile materials by means of numerical simulations has been proved to be difficult so far. The concept of J-integrals, where, analogously to the K-concept in linear elastic fracture mechanics, the parameter J is a measure for This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

One technique to describe the failure of mechanical structures is a phase field model for fracture. Phase field models for fracture consider an independent scalar field variable in addition to the mechanical displacement [1]. The phase field ansatz approximates crack surfaces as a continuous transition zone in which the phase field variable varies from a value that indicates intact material to another value that represents cracks. For a good approximation of cracks, these transition zones are required to be narrow, which leads to steep gradients in the fracture field. As a consequence, the required mesh density in a finite element simulation and thus the computational effort increases. In order to circumvent this efficiency problem, exponential shape functions were introduced in the discretization of the phase field variable, see [2]. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the steep transition with less elements. Unfortunately, the exponential shape functions are not symmetric, which requires a certain orientation of elements relative to the crack surfaces. This adaptation is not uniquely determined and needs to be set up in the correct way in order to improve the approximation of smooth cracks. The issue is solved in this work by reorientating the exponential shape functions according to the nodal value of phase field gradient in a particular element. To be precise, this work discusses an adaptive algorithm that implements such a reorientation for 2d and 3d situations.

Phase field models for fracture are energy-based and employ a continuous field variable, the phase field, to indicate cracks. The width of the transition zone of this field variable between damaged and intact regions is controlled by a regularization parameter. Narrow transition zones are required for a good approximation of the fracture energy which involves steep gradients of the phase field. This demands a high mesh density in finite element simulations if 4-node elements with standard bilinear shape functions are used. In order to improve the quality of the results with coarser meshes, exponential shape functions derived from the analytic solution of the 1D model are introduced for the discretization of the phase field variable. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the fracture field. Unfortunately, lower-order Gauss-Legendre quadrature schemes, which are sufficiently accurate for the integration of bilinear shape functions, are not sufficient for an accurate integration of the exponential shape functions. Therefore in this work, the numerical accuracy of higher-order Gauss-Legendre formulas and a double exponential formula for numerical integration is analyzed.

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