https://www.crm.sns.it/media/course/3060/miehe+hofacker+welschinger10.pdf

A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits

A phase-field description of dynamic brittle fracture

In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities. Within the formulations stemming from Griffith's theory, we focus on quasi-static models featuring a tension-compression split, which prevent cracking in compression and interpenetration of the crack faces upon closure, and on the staggered algorithmic implementation due to its proved robustness. In this paper, we establish an appropriate stopping criterion for the staggered scheme. Moreover, we propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem. This enables a significant reduction of computational cost--about one order of magnitude--with respect to the available (non-linear) models. The conceptual and structural similarities of the hybrid formulation to gradient-enhanced continuum damage mechanics are outlined as well. Several benchmark problems are solved, including one with own experimental verification.

A review on phase-field models of brittle fracture and a new fast hybrid formulation

An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications

Phase-field modeling of brittle fracture in elastic solids is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths including branching and merging. We propose a novel phase-field model for ductile fracture of elasto-plastic solids in the quasi-static kinematically linear regime. The formulation is shown to capture the entire range of behavior of a ductile material exhibiting $$J_2$$J2-plasticity, encompassing plasticization, crack initiation, propagation and failure. Several examples demonstrate the ability of the model to reproduce some important phenomenological features of ductile fracture as reported in the experimental literature.

http://congress.cimne.com/coupled2015/admin/files/fileabstract/a482.pdf

Phase-field modeling of ductile fracture

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that Γ-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. In this paper, we outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi- field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that -converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions which govern the evolution of the crack phase field. Here, we propose alternative rate-independent and viscous over-force models which ensure the local growth of the phase field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase field formulations of fracture by means of representative numerical examples.

Thermodynamically-Consistent Phase Field Models of Fracture: Variational Principles and Multi-Field

Structural reliability analyses of piezoelectric solids need the modeling of failure under coupled electromechanical actions. However, the numerical simulation of failure due to fracture based on sharp crack discontinuities may suffer in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. In this work, we develop a framework of diffusive fracture in piezoelectric solids. We start our investigation with the definition of a crack surface functional of the phase field that Γ -converges for vanishing length-scale parameter to a sharp crack topology. This functional provides the basis for the definition of suitable dissipation functions which govern the evolution of the crack phase field. Based on experimental results available in the literature, we suggest a non-associative dissipative framework where the fracture phase field is driven by the mechanical part of the coupled electromechanical driving force. This accounts for a hierarchical view that considers (i) the decrease of stiffness due to mechanical rupture as the primary action that is followed by (ii) the decrease of electric permittivity due to the generated free space. The proposed definition of mechanical and electrical parts of the fracture driving force follows in a natural format from a kinematic assumption, that decomposes the total strains and the total electric field into energy–enthalpy-producing parts and fracture parts, respectively. Such an approach allows the insertion of well-known anisotropic piezoelectric storage functions without change. We end up with a three-field-problem that couples the displacement with the electric potential and the fracture phase field. The latter is governed by a micro-balance equation, which appears in a very transparent form in terms of a history field containing a maximum fracture source obtained in the time history of the electromechanical process. This representation allows the construction of a very robust algorithmic treatment based on a operator split scheme, which successively updates in a typical time step the history field, the crack phase field and finally the two piezoelectric fields. The proposed model is considered to be the canonically simple scheme for the simulation of diffusive electromechanical crack propagation in solids. We demonstrate its modeling capacity by means of representative numerical examples.

Fabian Welschinger

Papers

A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits

Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations

Thermodynamically-Consistent Phase Field Models of Fracture: Variational Principles and Multi-Field

A phase field model of electromechanical fracture

Microstructural analysis of short glass fiber reinforced thermoplastics based on x-ray micro-computed tomography