Recently, different extensions of phase field fracture models to ductile fracture scenarios where the crack evolution is affected by plastic deformation have been proposed. In most of these phase field models for ductile fracture, the coupling of the elastic‐plastic material laws and the phase field variable is rather complicated and staggered finite element solution schemes are employed in order to avoid the computation of the complicated coupling terms in the tangent matrix. In this work, we propose an elastic‐plastic phase field fracture model, where the coupling is such that a monolithic solution is possible. Through the proposed coupling of the elastic‐plastic material law and the phase field order parameter, a reinterpretation of the meaning of the plastic material parameters is necessary in order to obtain reasonable effective material properties for the ductile phase field fracture model that can be related to measured data. The respective relations are derived from an analysis of the model in one dimension. Finite element simulations of plane strain fracture problems demonstrate the appropriateness of the derived effective material parameters and the applicability of the monolithic solution scheme. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

A phase field model for elastic-plastic fracture is presented, which is based on an energy functional composed of an elastic energy contribution, a plastic dissipation potential and a fracture energy. The coupling of the mechanical fields with the fracture field is modeled by a degradation function. Due to the proposed coupling it is possible to solve the global system of differential equations in a monolithic iterative solution scheme. Numerical simulations are presented, where the choice of the degradation function is investigated and a staggered is compared to monolithic iteration scheme. The proposed ductile phase field fracture model with linear isotropic hardening is an extension of an elastic phase field fracture model [1], which can be regarded as a regularized approximation of the variational formulation of brittle fracture by [2]. The crack field variable s ranging from 1 for undamaged to 0 for fully damaged material, together with the displacement vector represent the set of independent variables in the energy density formulation Ψ that consists of three parts: the Griffith-type fracture energy Ψ fr along with the contributions of elastic Ψ el and plastic deformation Ψ pl . Compared to the linear elastic model a dissipation potential Π pl accounting for the accumulated plastic strain, represented by the internal variable α, is added to the elastic energy contribution W el . The elastic strain is the difference between the total and the plastic strain ε p , the second internal variable. A small strain scenario is assumed. The fourth order stiffness tensor C is assumed to be isotropic. The plastic material behavior is characterized by the plastic variables, the linear hardening modulus H and the initial yield stress σ Y . The degradation function g(s) models the decrease in stiffness of broken material as well as the coupling between plastic deformation and the evolution of fracture. The dimensionless parameter η is introduced for numerical reasons. Parameter G c describes the fracture resistance and can be related to the fracture toughness in the elastic material. The width of the transition zone between broken and undamaged material is controlled by the length parameter .withandEquilibrium of stresses has to hold, wherein the stresses are computed from the energy density formulationThe Ginzburg-Landau equation describes the evolution of the fracture fieldThe parameter M in this viscous approximation has to be chosen high enough in order to approximate quasi-static conditions, where δΨ δs = 0. The hardening force q, which is driving the plasticity is given as