Approaches to Fracture Mechanics based on Local and Global Energy Minimization
Final Report Abstract
Within this research project, we have developed a consistent framework for the phase field modeling of fracture. This framework is rooted in variational principles for the evolution problems of gradient-type damage mechanics, but incoporates characteristic feature of fracture mechanics. This concerns a clear geometric concept for the regularization of sharp crack discontinuities based on the formulation of a minimization principle for a regularized crack surface. This regularized crack surface plays the key role in the phase field modeling of fracture. Focusing on irreversible crack process, its evolution is constraint to be positive. Evolution equations that are variationally consistent follow from mixed variational principles which contain threshold functions. Here, crack driving forces are dual to the phase field and derived from stored energy functions. In brittle fracture, a key aspect is the decomposition of the free energy in terms of positive and negative parts, related to tension and compression in isotropic solids. This allowed a consistent formulation of mode-I fracture in the phase field modeling. From the viewpoint of the numerical implementation, we have developed fully monolithic solution procedures for the coupled problem, as well as efficient operator splitting techniques. The latter provide an extremely robust and easy-to-implement scheme for the analysis of regularized fracture. The formulation was developed in the first part of the project for brittle fracture in elastic solids under quasi-static conditions. In a second step, we have extended the framework to dynamic brittle fracture, and demonstrated the superior performance of the phase field modeling by means of simulations of complex crack patterns including branching. It was shown, that these complex patterns appear in the phase field modeling of fracture in a straightforward manner without any additional branching criterion. The last part of the research project was related to the extension of ductile fracture in elastic-plastic solids, and in particular the modeling of brittle-to-ductile failure mode transition in dynamic impact problems to metals. Here, we developed a full variational setting of elasto-plasticity coupled with phase field fracture, and an associated numerical implementation. This formulation accounts for adiabatic temperature raise in plastic shear bands. The driving force decomposes into a brittle elastic and a ductile plastic contribution, where the latter is characterized by a locking energy related to a plastic strain. It was shown, that this formulation is able to reproduce experimental tests for failure mechanisms in impact problems. In summary, the key results of the projects are • A variational concept for the regularization of sharp crack discontinuities by a fracture phase field, based on assumed profiles of the regularization. • A consistent variational framework for the phase field modeling of fracture in brittle elastic solids under quasi-static conditions, including its effective numerical implementation by operator splits. • The extension of this variational framework to dynamic fracture in brittle solids and the development of a dynamic analysis tool. • The extension to a variational setting for brittle-to-ductile mode-transition under dynamic conditions, including thermomechanical coupling phenomena that allow the analysis of both brittle and ductile failure. The framework for phase field modeling of fracture is considered to provide a fundamental new aspect in fracture mechanics, allowing the modeling of complex phenomena by a pure continuum modeling. In contrast to computational methods which resolve discontinuities, such as the strong discontinuity method or the extended finite element method, this regularized continuum setting can be easily implemented in standard finite element programs. This makes the developed framework very attractive for future extension, such as the modeling of complex multi-physics problems at fracture.
Publications
- [2010]: A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 199: 2765– 2778
Miehe, C.; Hofacker, M.; Welschinger, F.
- [2010]: A phasefield model of electromechanical fracture. Journal of the Mechanics and Physics of Solids, 58: 1716– 1740
Miehe, C.; Welschinger, F.; Hofacker, M.
- [2010]: Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations. International Journal of Numerical Methods in Engineering, 83: 1273– 1311
Miehe, C.; Welschinger, F.; Hofacker, M.
- [2012]: A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns. International Journal for Numerical Methods in Engineering, 93: 276–301
Hofacker, M.; Miehe, C.
(See online at https://doi.org/10.1002/nme.4387) - [2012]: Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation. International Journal of Fracture, 178: 113–129
Hofacker, M.; Miehe, C.
(See online at https://doi.org/10.1007/s10704-012-9753-8)
