Approaches to Fracture Mechanics based on Local and Global Energy Minimization

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The energetic framework of brittle fracture proposed by Griffith is unable to predict crack initiation in a body free of defects. This problem can be overcome by a global minimization of suitably defined incremental energy functionals having both volume and surface contributions. Problems governed by such functionals involving volume and surface energies appear in a variety of areas in applied sciences, ranging from image and signal processing to fracture mechanics. In order to deal numerically with these kind of problems, many approximate functional representations have been proposed, using high-order singular perturbations, finite differences or non-local energies. However, comprehensive theoretical treatments of incremental minimization methods for fracture in an engineering language and their associated computational implementations suitable for large-scale engineering applications are missing in the literature. The purpose of the project is to investigate theoretical and computational frameworks of brittle and cohesive fracture based on global energy minimization that a priori circumvent the main drawback of Griffith’s theory with regard to crack initiation. In the first period of the research project the following results have been obtained: Local energy minimization concepts of fracture based on configurational–force–driven sharp crack propagation were extended to the three–dimensional setting. Results obtained from this work provide reference solutions for the subsequent developments on global methods. Global energy minimization concepts were developed for a diffusive crack propagation, were a set of regularized crack discontinuities is described by a phase field which is driven by a gradient–type balance equation. The proposed framework results in a smooth continuum–damage–type theory of fracture with specific constitutive functions. An extended three–field model that consists of the displacement field, the fracture phase field and the dual dissipative force field was developed, whose viscous over–force structure provides a very robust computational setting of diffusive crack propagation with an enormous potential with regard to the modeling of complex three–dimensional crack topologies. In the subsequent period of the research project, we will improve the phase field model of fracture with regard to the modeling of more complex crack topologies as well as its extension to multi–field environments. As it turned out in the first research period, a very important aspect is its embedding into adaptive mesh refinement procedures which resolve diffusive crack zones. To this end, new configurational–force–driven mesh refinement indicators will be developed for the gradient–type phase field model of fracture. Furthermore, we plan to investigate problems of dynamic fracture including complex crack branching. Finally, we will underline the advantage of bulk constitutive modeling inherent in the phase field modeling of fracture with regard to its embedding into more complex multi–field problems such as coupled thermo–mechanical and electro–magneto–mechanical problems.