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Computational Mechanics
Published in: Computational Mechanics 3/2019

19-02-2019 | Original Paper

Adaptive consistent element-free Galerkin method for phase-field model of brittle fracture

Authors: Yulong Shao, Qinglin Duan, Shasha Qiu

Published in: Computational Mechanics | Issue 3/2019

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Abstract

Efficient implementation of the element-free Galerkin (EFG) method for a phase-field model of linear elastic fracture mechanics is presented, in which the convenience of the meshfree method to construct high order approximation functions and to implement h-adaptivity is fully exploited. A second-order moving-least squares approximation for both displacement and phase field is employed. Domain integration of the weak forms is evaluated by the quadratically consistent 3-point integration scheme. The refinement criterion using maximum residual strain energy history is proposed and the insertion of nodes is based on the background mesh. Numerical results show that the developed method is more efficient than the standard finite element method (3-node triangle element) due to the proposed h-adaptivity. In comparison with the standard EFG method, the proposed consistent EFG method significantly improves the computational efficiency and accuracy. The advantage of the quadratic approximation is also demonstrated. In addition, the feasibility of extending the proposed method to 3D is validated by the modeling of a twisting crack.
previous article Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels
next article A meshless method for the nonlinear von Kármán plate with multiple folds of complex shape

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Literature
1.
Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A 221:163–198CrossRef
2.
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150MathSciNetMATHCrossRef
3.
Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620MATHCrossRef
4.
Fleming M, Chu YA, Moran B, Belytschko T (1997) Enriched element-free galerkin methods for crack tip fields. Int J Numer Methods Eng 40(8):1483–1504MathSciNetCrossRef
5.
Wang YY (2017) Abaqus analysis users’ guide: analysis volume. China Machine Press, Beijing
6.
7.
8.
9.
Amor H, Marigo JJ, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57:1209–1229MATHCrossRef
10.
Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199:2765–2778MathSciNetMATHCrossRef
11.
Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83:1273–1311MathSciNetMATHCrossRef
12.
Borden MJ, Hughes TJR, Landis CM, Verhoosel CV (2014) A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework. Comput Methods Appl Mech Eng 273:100–118MathSciNetMATHCrossRef
13.
Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55:383–405MathSciNetMATHCrossRef
14.
Nguyen TT, Yvonnet J, Zhu QZ, Bornert M, Chateau C (2015) A phase field method to simulate crack nucleation and propagation in strongly heterogeneous materials from direct imaging of their microstructure. Eng Fract Mech 139:18–39CrossRef
15.
Wu J (2017) A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J Mech Phys Solids 103:72–99MathSciNetCrossRef
16.
Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220(1):77–95MathSciNetMATHCrossRef
17.
Bourdin B, Larsen CJ, Richardson CL (2010) A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 168(168):133–143MATH
18.
19.
Ambati M, Kruse R, Lorenzis LD (2016) A phase-field model for ductile fracture at finite strains and its experimental verification. Comput Mech 57(1):149–167MathSciNetMATHCrossRef
20.
Miehe C, Aldakheel F, Teichtmeister S (2017) Phase-field modeling of ductile fracture at finite strains: a robust variational-based numerical implementation of a gradient-extended theory by micromorphic regularization. Int J Numer Methods Eng 111:1–32MathSciNetCrossRef
21.
Crosby T, Ghoniem N (2012) Phase-field modeling of thermomechanical damage in tungsten under severe plasma transients. Comput Mech 50(2):159–168MathSciNetMATHCrossRef
22.
Miehe C, Dal H, Schänzel LM, Raina A (2016) A phase-field model for chemo-mechanical induced fracture in lithium-ion battery electrode particles. Int J Numer Methods Eng 106(9):683–711MathSciNetMATHCrossRef
23.
Zuo P, Zhao YP (2016) Phase field modeling of lithium diffusion, finite deformation, stress evolution and crack propagation in lithium ion battery. Extreme Mech Lett 9:467–479CrossRef
24.
Miehe C, Mauthe S, Teichtmeister S (2015) Minimization principles for the coupled problem of Darcy–Biot-type fluid transport in porous media linked to phase field modeling of fracture. J Mech Phys Solids 82:186–217MathSciNetCrossRef
25.
Miehe C, Kienle D, Aldakheel F, Teichtmeister S (2016) Phase field modeling of fracture in porous plasticity: a variational gradient-extended Eulerian framework for the macroscopic analysis of ductile failure. Comput Methods Appl Mech Eng 312:3–50MathSciNetCrossRef
26.
Heider Y, Markert B (2016) A phase-field modeling approach of hydraulic fracture in saturated porous media. Mech Res Commun 80:38–46CrossRef
27.
Abdollahi A, Arias I (2012) Phase-field modeling of crack propagation in piezoelectric and ferroelectric materials with different electromechanical crack conditions. J Mech Phys Solids 60(12):2100–2126MathSciNetCrossRef
28.
Wilson ZA, Borden MJ, Landis CM (2013) A phase-field model for fracture in piezoelectric ceramics. Int J Fract 183(2):135–153CrossRef
29.
Raina A, Miehe C (2016) A phase-field model for fracture in biological tissues. Biomech Model Mechanobiol 15(3):479–496CrossRef
30.
Gültekin O, Dal H, Holzapfel GA (2018) Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: a rate-dependent anisotropic crack phase-field model. Comput Methods Appl Mech Eng 331:23–52MathSciNetCrossRef
31.
Areias P, Rabczuk T, Msekh MA (2016) Phase-field analysis of finite-strain plates and shells including element subdivision. Comput Methods Appl Mech Eng 312:322–350MathSciNetCrossRef
32.
Artina M, Fornasier M, Micheletti S, Perotto S (2015) Anisotropic mesh adaptation for crack detection in brittle materials. SIAM J Sci Comput 37(4):B633–B659MathSciNetMATHCrossRef
33.
Radszuweit M, Kraus C (2017) Modeling and simulation of non-isothermal rate-dependent damage processes in inhomogeneous materials using the phase-field approach. Comput Mech 60(1):1–17MathSciNetMATHCrossRef
34.
Burke S, Ortner C, Süli E (2010) An adaptive finite element approximation of a variational model of brittle fracture. SIAM J Numer Anal 48(3):980–1012MathSciNetMATHCrossRef
35.
Heister T, Wheeler MF, Wick T (2015) A primal-dual active set method and predictor–corrector mesh adaptivity for computing fracture propagation using a phase-field approach. Comput Methods Appl Mech Eng 290:466–495MathSciNetMATHCrossRef
36.
Badnava H, Msekh MA, Etemadi E, Rabczuk T (2018) An h-adaptive thermo-mechanical phase field model for fracture. Finite Elem Anal Des 138(2018):31–47CrossRef
37.
Lee S, Wheeler MF, Wick T (2017) Iterative coupling of flow, geomechanics and adaptive phase-field fracture including level-set crack width approaches. Comput Appl Math 314(C):40–60MathSciNetMATHCrossRef
38.
Lee S, Wheeler MF, Wick T (2016) Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model. Comput Methods Appl Mech Eng 305:111–132MathSciNetMATHCrossRef
39.
Klinsmann M, Rosato D, Kamlah M, Mcmeeking RM (2015) An assessment of the phase field formulation for crack growth. Comput Methods Appl Mech Eng 294(2):313–330MathSciNetMATHCrossRef
40.
Li S, Liu WK (2004) Meshfree particle methods. Springer, BerlinMATH
41.
Liu GR (2009) Mesh free methods: moving beyond the finite element method, 2nd edn. CRC Press, Boca RatonCrossRef
42.
Belytschko T, Krongauz Y, Organ D (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47MATHCrossRef
43.
Nguyen VP, Rabczuk T, Bordas S (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79:763–813MathSciNetMATHCrossRef
44.
45.
Krysl P, Belytschko T (1997) Element-free Galerkin method convergence of the continuous and discontinuous shape functions. Comput Methods Appl Mech Eng 148:257–277MathSciNetMATHCrossRef
46.
Amiri F, Millán D, Arroyo M, Silani M, Rabczuk T (2016) Fourth order phase-field model for local max-ent approximants applied to crack propagation. Comput Methods Appl Mech Eng 312(2016):254–275MathSciNetCrossRef
47.
Duan QL, Li XK, Zhang HW, Belytschko T (2012) Second-order accurate derivatives and integration schemes for meshfree methods. Int J Numer Methods Eng 92:399–424MathSciNetMATHCrossRef
48.
49.
Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193(12):1257–1275MathSciNetMATHCrossRef
50.
Miehe C (1998) Comparison of two algorithms for the computation of fourth-order isotropic tensor functions. Comput Struct 66(1):37–43MathSciNetMATHCrossRef
51.
Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50:435–466MATHCrossRef
52.
Liu GR, Zhang J, Lam KY, Li H, Xu G, Zhong ZH, Li GY, Han X (2008) A gradient smoothing method (GSM) with directional correction for solid mechanics problems. Comput Mech 41:457–472MATHCrossRef
53.
Wang DD, Wu JC (2016) An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods. Comput Methods Appl Mech Eng 298:485–519MathSciNetMATHCrossRef
54.
Ortizbernardin A, Köbrich P, Hale JS, Olatesanzana E, Bordas SPA, Natarajan S (2018) A volume-averaged nodal projection method for the Reissner–Mindlin plate model. Comput Methods Appl Mech Eng 341:827–850MathSciNetCrossRef
55.
Duan QL, Gao X, Wang BB, Li XK, Zhang HW (2014) A four-point integration scheme with quadratic exactness for three-dimensional element-free Galerkin method based on variationally consistent formulation. Comput Methods Appl Mech Eng 280(10):84–116MathSciNetMATHCrossRef
56.
Goh CM, Nielsen PMF, Nash MP (2017) A stabilised mixed meshfree method for incompressible media: application to linear elasticity and stokes flow. Comput Methods Appl Mech Eng 329:575–598MathSciNetCrossRef
57.
Ortiz-Bernardin A, Hale JS, Cyron CJ (2015) Volume-averaged nodal projection method for nearly-incompressible elasticity using meshfree and bubble basis functions. Comput Methods Appl Mech Eng 285:427–451MathSciNetMATHCrossRef
58.
Ortiz-Bernardin A, Puso MA, Sukumar N (2015) Improved robustness for nearly-incompressible large deformation meshfree simulations on Delaunay tessellations. Comput Methods Appl Mech Eng 293:348–374MathSciNetMATHCrossRef
59.
Kästner M, Hennig P, Linse T, Ulbricht V (2016) Phase-field modelling of damage and fracture—convergence and local mesh refinement. In: Naumenko K, Aßmus M (eds) Advanced methods of continuum mechanics for materials and structures. Advanced structured materials, vol 60. Springer, Singapore
60.
Zhang X, Vignes C, Sloan SW, Sheng D (2017) Numerical evaluation of the phase-field model for brittle fracture with emphasis on the length scale. Comput Mech 59(5):1–16MathSciNetCrossRef
61.
Winkler B (2001) Traglastuntersuchungen von unbewehrten undbewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes fürBeton. Dissertation, University of Innsbruck, Austria
62.
Liu GW, Li QB, Zuo Z (2016) Implementation of a staggered algorithm for a phase field model in ABAQUS. Chin J Rock Mech Eng 35(5):1019–1030
63.
Citarella R, Buchholz F-G (2008) Comparison of crack growth simulation by DBEM and FEM for SEN-specimens undergoing torsion or bending loading. Eng Fract Mech 75(3):489–509CrossRef
64.
Cervera M, Barbat GB, Chiumenti M (2017) Finite element modeling of quasi-brittle cracks in 2D and 3D with enhanced strain accuracy. Comput Mech 4:1–30MathSciNetMATH
65.
Benedetti L, Cervera M, Chiumenti M (2017) 3D numerical modeling of twisting cracks under bending and torsion of skew notched beams. Eng Fract Mech 176:235–256CrossRef
Metadata
Title
Adaptive consistent element-free Galerkin method for phase-field model of brittle fracture
Authors
Yulong Shao
Qinglin Duan
Shasha Qiu
Publication date
19-02-2019
Publisher
Springer Berlin Heidelberg
Published in
Computational Mechanics / Issue 3/2019
Print ISSN: 0178-7675
Electronic ISSN: 1432-0924
DOI
https://doi.org/10.1007/s00466-019-01679-2

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