Abstract
Phase field models have been successfully applied in recent years to a variety of fracture mechanics problems, such as quasi-brittle materials, dynamic fracture mechanics, fatigue cracks in brittle materials, as well as ductile materials. The basic idea of the method is to introduce an additional term in the energy functional describing the state of material bodies. A new state variable is included in this term, the so-called phase field, and enables to determine the surface energy of the crack. This approach allows to model phenomena such as crack initiation, crack branching and buckling of cracks, as well as the modelling of the crack front in three-dimensional geometries, without further assumptions. There is yet no systematic investigation of the influence of strain hardening on crack development within the phase field method. Thus, the aim of the paper is to provide an analysis of the effect of kinematic and isotropic hardening on the evolution of the phase field variable.
Highlights
In fracture mechanics, the description of complex crack phenomena such as initiation, propagation, kinking and branching of cracks is a very demanding task
A different approach is given by phase field theories, which have been successfully introduced in fracture mechanics to capture complex phenomena in a unified framework
The motivation for the introduction of such part in the state functional arises from a regularisation of the crack geometry in the context of the classical Griffith theory for brittle materials and enables the calculation of the surface energy during crack propagation
Summary
The description of complex crack phenomena such as initiation, propagation, kinking and branching of cracks is a very demanding task. A different approach is given by phase field theories, which have been successfully introduced in fracture mechanics to capture complex phenomena in a unified framework. The applications include both elastic and plastic material behaviour, as well as extensions to fatigue crack propagation. A phase field model in common use is introduced by extension of a von Mises-plasticity model with the aid of the concept of energy equivalence known from continuum damage mechanics. In Hofacker et al [2], e.g., conventional thermodynamics is adopted while the gradient of the damage variable is incorporated in the postulated damage criterion and the associated damage dissipation function This implies that crack propagation is modelled as a completely dissipative process. Model responses for pure isotropic and pure kinematic hardening are examined for one- and two-dimensional problems
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