Treatment of Brittle Fracture in Solids with the Virtual Element Method

Computational Mechanics has many applications in engineering. Its range of application has been enlarged widely in the last decades. Still new developments are made to which a new discretization scheme belongs: the virtual element method (VEM). Despite being only few years under development the application range of VEM in engineering includes formulations for linear and nonlinear material responses. In this contribution the focus is on fracture mechanics. Especially the treatment of crack propagation will be discussed where VEM has some advantages. The performance of the formulation is underlined by means of representative examples.

[1]  Christian Miehe,et al.  Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations , 2010 .

[2]  Peter Wriggers,et al.  A Virtual Element Method for 2D linear elastic fracture analysis , 2018, Computer Methods in Applied Mechanics and Engineering.

[3]  Peter Wriggers,et al.  Efficient virtual element formulations for compressible and incompressible finite deformations , 2017 .

[4]  S. Chan,et al.  On the Finite Element Method in Linear Fracture Mechanics , 1970 .

[5]  Wolfgang Ehlers,et al.  A phase-field approach embedded in the Theory of Porous Media for the description of dynamic hydraulic fracturing , 2017 .

[6]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[7]  Shuodao Wang,et al.  A Mixed-Mode Crack Analysis of Isotropic Solids Using Conservation Laws of Elasticity , 1980 .

[8]  Gianmarco Manzini,et al.  Hourglass stabilization and the virtual element method , 2015 .

[9]  Stefano Berrone,et al.  A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method , 2016 .

[10]  Peter Wriggers,et al.  Phase-field modeling of brittle fracture using an efficient virtual element scheme , 2018, Computer Methods in Applied Mechanics and Engineering.

[11]  Christian Hesch,et al.  Thermodynamically consistent algorithms for a finite‐deformation phase‐field approach to fracture , 2014 .

[12]  Peter Wriggers,et al.  A virtual element method for contact , 2016 .

[13]  Ercan Gürses,et al.  A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization , 2007 .

[14]  Ralf Müller,et al.  On degradation functions in phase field fracture models , 2015 .

[15]  Y. Heider,et al.  A phase-field modeling approach of hydraulic fracture in saturated porous media , 2017 .

[16]  Peter Wriggers,et al.  A stabilization technique to avoid hourglassing in finite elasticity , 2000 .

[17]  Christian Miehe,et al.  Phase Field Modeling of Fracture in Multi-Physics Problems. Part II. Coupled Brittle-to-Ductile Failure Criteria and Crack Propagation in Thermo-Elastic-Plastic Solids , 2015 .

[18]  J. Rice.  A path-independent integral and the approximate analysis of strain , 1968 .

[19]  Cv Clemens Verhoosel,et al.  A phase‐field model for cohesive fracture , 2013 .

[20]  Peter Wriggers,et al.  A finite deformation brick element with inhomogeneous mode enhancement , 2009 .

[21]  R. D. Henshell,et al.  CRACK TIP FINITE ELEMENTS ARE UNNECESSARY , 1975 .

[22]  F. Erdogan,et al.  On the Crack Extension in Plates Under Plane Loading and Transverse Shear , 1963 .

[23]  Stefano Vidoli,et al.  A phenomenological approach to fatigue with a variational phase-field model: The one-dimensional case , 2017 .

[24]  M. Williams,et al.  On the Stress Distribution at the Base of a Stationary Crack , 1956 .

[25]  A. Ingraffea,et al.  Automatic Modelling of Mixed-Mode Fatigue and Quasi-Static Crack Propagation Using the Boundary Element Method , 1983 .

[26]  T. Belytschko,et al.  A uniform strain hexahedron and quadrilateral with orthogonal hourglass control , 1981 .

[27]  Peter Wriggers,et al.  A computational framework for brittle crack-propagation based on efficient virtual element method , 2019, Finite Elements in Analysis and Design.

[28]  Theodore H. H. Pian,et al.  A hybrid‐element approach to crack problems in plane elasticity , 1973 .

[29]  Cv Clemens Verhoosel,et al.  A phase-field description of dynamic brittle fracture , 2012 .

[30]  R. E. Jones,et al.  Nonlinear finite elements , 1978 .

[31]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[32]  T. K. Hellen.  On the method of virtual crack extensions , 1975 .

[33]  Bhushan Lal Karihaloo,et al.  Accurate determination of the coefficients of elastic crack tip asymptotic field , 2001 .

[34]  R. Nuismer.  An energy release rate criterion for mixed mode fracture , 1975 .

[35]  Mark A Fleming,et al.  ENRICHED ELEMENT-FREE GALERKIN METHODS FOR CRACK TIP FIELDS , 1997 .

[36]  J. Prévost,et al.  Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation , 2003 .

[37]  F. Brezzi,et al.  Basic principles of Virtual Element Methods , 2013 .

[38]  P. C. Paris,et al.  Stress Analysis of Cracks , 1965 .

[39]  Michael Schönlein.  Ensemble reachability of parametric harmonic oscillators via mixed open‐loop and feedback control , 2018, PAMM.

[40]  Marco Paggi,et al.  Revisiting the problem of a crack impinging on an interface: A modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model , 2017, 1702.01102.

[41]  Peter Wriggers,et al.  Virtual element formulation for isotropic damage , 2018 .

[42]  Stefanie Reese,et al.  A new stabilization technique for finite elements in non-linear elasticity , 1999 .

[43]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[44]  Petr Krysl,et al.  Mean‐strain eight‐node hexahedron with stabilization by energy sampling , 2015 .

[45]  Peter Wriggers,et al.  An improved EAS brick element for finite deformation , 2010 .

[46]  D. Rooke,et al.  The dual boundary element method: Effective implementation for crack problems , 1992 .

[47]  Laura De Lorenzis,et al.  A review on phase-field models of brittle fracture and a new fast hybrid formulation , 2015 .

[48]  Peter Wriggers,et al.  A low order 3D virtual element formulation for finite elasto–plastic deformations , 2017, Computational Mechanics.

[49]  J. Rice,et al.  Plane strain deformation near a crack tip in a power-law hardening material , 1967 .

[50]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[51]  Glaucio H. Paulino,et al.  On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes , 2014 .

[52]  Xue Zhang,et al.  Numerical evaluation of the phase-field model for brittle fracture with emphasis on the length scale , 2017 .

[53]  R. Barsoum.  Application of quadratic isoparametric finite elements in linear fracture mechanics , 1974 .

[54]  Peter Wriggers,et al.  A Virtual Element Method for Crack Propagation , 2018, PAMM.

[55]  Ted Belytschko,et al.  Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems , 1991 .

[56]  Peter Wriggers,et al.  Automation of Finite Element Methods , 2016 .

[57]  Gilles A. Francfort,et al.  Revisiting brittle fracture as an energy minimization problem , 1998 .

[58]  Eirik Keilegavlen,et al.  High-accuracy phase-field models for brittle fracture based on a new family of degradation functions , 2017, 1705.04046.

[59]  Annalisa Buffa,et al.  Mimetic finite differences for elliptic problems , 2009 .

[60]  J. D. Eshelby.  The Calculation of Energy Release Rates , 1974 .

[61]  O. C. Zienkiewicz,et al.  The Finite Element Method: Its Basis and Fundamentals , 2005 .

[62]  Emmanouil Kakouris,et al.  Phase‐field material point method for brittle fracture , 2017 .