Inscrição na biblioteca: Guest
Portal Digital Begell Biblioteca digital da Begell eBooks Diários Referências e Anais Coleções de pesquisa
International Journal for Multiscale Computational Engineering
Fator do impacto: 1.016 FI de cinco anos: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Imprimir: 1543-1649
ISSN On-line: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2018026804
pages 181-200

VIRTUAL ELEMENT FORMULATION FOR PHASE-FIELD MODELING OF DUCTILE FRACTURE

Fadi Aldakheel
Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstrasse 11, 30167 Hannover, Germany
Blaž Hudobivnik
Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstrasse 11, 30167 Hannover, Germany
Peter Wriggers
Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstrasse 11, 30167 Hannover, Germany

RESUMO

An efficient low-order virtual element method (VEM) for the phase-field modeling of ductile fracture is outlined within this work. The recently developed VEM is a competitive discretization scheme for meshes with highly irregular shaped elements. The phase-field approach is a very powerful technique to simulate complex crack phenomena in multi-physical environments. The formulation in this contribution is based on a minimization of a pseudo-potential density functional for the coupled problem undergoing large strains. The main aspect of development is the extension toward the virtual element formulation due to its flexibility in dealing with complex shapes and arbitrary number of nodes. Two numerical examples illustrate the efficiency, accuracy, and convergence properties of the proposed method.

Referências

  1. Aldakheel, F., Mechanics of Nonlocal Dissipative Solids: Gradient Plasticity and Phase Field Modeling of Ductile Fracture, PhD, Institute ofApplied Mechanics (CE), Chair I, University of Stuttgart, 2016. DOI: 10.18419/opus-8803.

  2. Aldakheel, F., Micromorphic Approach for Gradient-Extended Thermo-Elastic-Plastic Solids in the Logarithmic Strain Space, Continuum Mech. Thermodynam., vol. 29, no. 6, pp. 1207-1217, 2017.

  3. Aldakheel, F., Hudobivnik, B., and Wriggers, P., Virtual Elements for Finite Thermo-Plasticity Problems, Comput. Mech, 2019. DOI: 10.1007/s00466-019-01714-2.

  4. Aldakheel, F., Hudobivnik, B., Hussein, A., and Wriggers, P., Phase-Field Modeling of Brittle Fracture Using an Efficient Virtual Element Scheme, Comput. Methods Appl. Mech. Eng., vol. 341, pp. 443-466, 2018a.

  5. Aldakheel, F., Kienle, D., Keip, M.A., and Miehe, C., Phase Field Modeling of Ductile Fracture in Soil Mechanics, Proc. Appl. Math. Mech, vol. 17, no. 1, pp. 383-384, 2017.

  6. Aldakheel, F., Mauthe, S., and Miehe, C., Towards Phase Field Modeling of Ductile Fracture in Gradient-Extended Elastic-Plastic Solids, Proc. Appl. Math. Mech, vol. 14, pp. 411-412, 2014.

  7. Aldakheel, F. and Miehe, C., Coupled Thermomechanical Response of Gradient Plasticity, Int. J. Plasticity, vol. 91, pp. 1-24, 2017.

  8. Aldakheel, F., Wriggers, P., and Miehe, C., A Modified Gurson-Type Plasticity Model at Finite Strains: Formulation, Numerical Analysis and Phase-Field Coupling, Comput. Mech, vol. 62, no. 4, pp. 815-833,2018b.

  9. Alessi, R., Ambati, M., Gerasimov, T., Vidoli, S., and De Lorenzis, L., Comparison of Phase-Field Models of Fracture Coupled with Plasticity, in Advances in Computational Plasticity, Springer International Publishing, pp. 1-21, 2018a.

  10. Alessi, R., Vidoli, S., and De Lorenzis, L., A Phenomenological Approach to Fatigue with a Variational Phase-Field Model: The One-Dimensional Case, Eng. Fracture Mech., vol. 190, no. 1, pp. 53-73, 2018b.

  11. Ambati, M., Gerasimov, T., and De Lorenzis, L., Phase-Field Modeling of Ductile Fracture, Comput. Mech., vol. 55, pp. 1017-1040,2015.

  12. Amor, H., Marigo, J., and Maurini, C., Regularized Formulation of the Variational Brittle Fracture with Unilateral Contact: Numerical Experiments, J. Mech. Phys. Solids, vol. 57, pp. 1209-1229, 2009.

  13. Artioli, E., Beirao da Veiga, L., Lovadina, C., and Sacco, E., Arbitrary Order 2D Virtual Elements for Polygonal Meshes: Part I, Elastic Problem, Comput. Mech., vol. 60, no. 3, pp. 355-377, 2017.

  14. Bathe, K.J., Finite Element Procedures, Englewood Cliffs, NJ: Prentice-Hall, 1996.

  15. Beirao Da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., and Russo, A., Basic Principles of Virtual Element Methods, Math. Models Methods Appl. Sci., vol. 23, no. 1, pp. 199-214, 2013a.

  16. Beirao Da Veiga, L., Brezzi, F., and Marini, L.D., Virtual Elements for Linear Elasticity Problems, SIAMJ. Numer. Anal., vol. 51, no. 2, pp. 794-812, 2013b.

  17. Bellis, M.L.D., Wriggers, P., Hudobivnik, B., and Zavarise, G., Virtual Element Formulation for Isotropic Damage, Finite Elements Anal. Des., vol. 144, pp. 38-48,2018.

  18. Belytschko, T. and Bindeman, L.P., Assumed Strain Stabilization of the 4-Node Quadrilateral with 1-Point Quadrature for Nonlinear Problems, Comput. Methods Appl. Mech. Eng., vol. 88, no. 3, pp. 311-340,1991.

  19. Bleyer, J. and Alessi, R., Phase-Field Modeling of Anisotropic Brittle Fracture Including Several Damage Mechanisms, Comput. Methods Appl. Mech. Eng., vol. 336, no. 1, pp. 213-236,2018.

  20. Boerner, E., Loehnert, S., and Wriggers, P., A New Finite Element based on the Theory of a Cosserat Point-Extension to Initially Distorted Elements for 2D Plane Strain, Int. J. Numer. Methods Eng., vol. 71, pp. 454-472, 2007.

  21. Boger, L., Keip, M.A., and Miehe, C., Minimization and Saddle-Point Principles for the Phase-Field Modeling of Fracture in Hydrogels, Comput. Mater. Sci., vol. 138, pp. 474-485, 2017.

  22. Borden, M.J., Hughes, T. J., Landis, C.M., Anvari, A., and Lee, I.J., A Phase-Field Formulation for Fracture in Ductile Materials: Finite Deformation Balance Law Derivation, Plastic Degradation, and Stress Triaxiality Effects, Comput. Methods Appl. Mech. Eng., vol. 312, pp. 130-166, 2016.

  23. Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J.R., and Landis, C.M., A Phase-Field Description of Dynamic Brittle Fracture, Comput. Methods Appl. Mech. Eng., vols. 217-220, pp. 77-95, 2012.

  24. Brezzi, F., Buffa, A., and Lipnikov, K., Mimetic Finite Differences for Elliptic Problems, ESAIM: Math. Model. Numer. Anal., vol. 43, no. 2, pp. 277-295, 2009.

  25. Cangiani, A., Manzini, G., Russo, A., and Sukumar, N., Hourglass Stabilization and the Virtual Element Method, Int. J. Numer. Methods Eng., vol. 102, nos. 3-4, pp. 404-436, 2015.

  26. Chi, H., Beirao da Veiga, L., and Paulino, G., Some Basic Formulations of the Virtual Element Method (VEM) for Finite Deformations, Comput. Methods Appl. Mech. Eng., vol. 318, pp. 148-192, 2017.

  27. Choo, J. and Sun, W., Coupled Phase-Field and Plasticity Modeling of Geological Materials: From Brittle Fracture to Ductile Flow, Comput. Methods Appl. Mech. Eng., vol. 330, pp. 1-32, 2018.

  28. Dittmann, M., Aldakheel, F., Schulte, J., Wriggers, P., and Hesch, C., Variational Phase-Field Formulation of Non-Linear Ductile Fracture, Comput. Methods Appl. Mech. Eng., vol. 342, pp. 71-94, 2018.

  29. Dittmann, M., Hesch, C., Schulte, J., Aldakheel, F., and Franke, M., Multi-Field Modelling and Simulation of Large Deformation Ductile Fracture, Proc. of the XIV International Conference on Computational Plasticity. Fundamentals and Applications, Barcelona, Spain, pp. 556-567, 2017.

  30. Dittmann, M., Kriiger, M., Schmidt, F., SchuB, S., and Hesch, C., Variational Modeling of Thermomechanical Fracture and Anisotropic Frictional Mortar Contact Problems with Adhesion, Comput. Mech, vol. 63, no. 3, pp. 571-591, 2019.

  31. Duda, F.P., Ciarbonetti, A., Sanchez, P. J., and Huespe, A.E., A Phase-Field/Gradient Damage Model for Brittle Fracture in Elastic-Plastic Solids, Int. J. Plasticity, vol. 65, pp. 269-296,2014.

  32. Ehlers, W. and Luo, C., A Phase-Field Approach Embedded in the Theory of Porous Media for the Description of Dynamic Hydraulic Fracturing, Comput. Methods Appl. Mech. Eng., vol. 315, pp. 348-368,2017.

  33. Flanagan, D. and Belytschko, T., A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hour-Glass Control, Int. J. Numer. Methods Eng, vol. 17, pp. 679-706, 1981.

  34. Gain, A.L., Talischi, C., and Paulino, G.H., On the Virtual Element Method for Three-Dimensional Linear Elasticity Problems on Arbitrary Polyhedral Meshes, Comput. Methods Appl. Mech. Eng., vol. 282, pp. 132-160, 2014.

  35. Gultekin, O., Dal, H., and Holzapfel, G.A., A Phase-Field Approach to Model Fracture of Arterial Walls: Theory and Finite Element Analysis, Comput. Methods Appl. Mech. Eng, vol. 312, pp. 542-566,2016.

  36. Hackl, K., Generalized Standard Media and Variational Principles in Classical and Finite Strain Elastoplasticity, J. Mech. Phys. Solids, vol. 45, no. 5, pp. 667-688,1997.

  37. Hallquist, J.O., Nike 2D: An Implicit, Finite Deformation, Finite Element Code for Analyzing the Static and Dynamic Response of Two-Dimensional Solids, Lawrence Livermore National Laboratory, University of California, Livermore, CA, Rep. UCRL- 52678, 1984.

  38. Heider, Y. and Markert, B., A Phase-Field Modeling Approach of Hydraulic Fracture in Saturated Porous Media, Mech. Res. Commun., vol. 80, pp. 38-46, 2017.

  39. Hesch, C., Franke, M., Dittmann, M., and Temizer, I., Hierarchical NURBS and a Higher-Order Phase-Field Approach to Fracture for Finite-Deformation Contact Problems, Comput. Methods Appl. Mech. Eng., vol. 301, pp. 242-258, 2016.

  40. Hesch, C. and Weinberg, K., Thermodynamically Consistent Algorithms for a Finite-Deformation Phase-Field Approach to Fracture, Int. J. Numer. Methods Eng., vol. 99, pp. 906-924,2014.

  41. Hudobivnik, B., Aldakheel, F., and Wriggers, P., A Low Order 3D Virtual Element Formulation for Finite Elasto-Plastic Deformations, Comput. Mech., vol. 63, no. 2, pp. 253-269, 2019.

  42. Korelc, J., Solinc, U., and Wriggers, P., An Improved EAS Brick Element for Finite Deformation, Comput. Mech, vol. 46, pp. 641-659,2010.

  43. Korelc, J. and Stupkiewicz, S., Closed-Form Matrix Exponential and Its Application in Finite-Strain Plasticity, Int. J. Numer. Methods Eng., vol. 98, pp. 960-987, 2014.

  44. Korelc, J. and Wriggers, P., Automation of Finite Element Methods, Switzerland: Springer, 2016.

  45. Krysl, P., Mean-Strain Eight-Node Hexahedron with Optimized Energy-Sampling Stabilization for Large-Strain Deformation, Int. J. Numer. Methods Eng., vol. 103, pp. 650-670, 2015a.

  46. Krysl, P., Mean-Strain Eight-Node Hexahedron with Stabilization by Energy Sampling Stabilization, Int. J. Numer. Methods Eng., vol. 103, pp. 437-449, 2015b.

  47. Kuhn, C., Schluter, A., and Muller, R., On Degradation Functions in Phase Field Fracture Models, Comput. Mater. Sci., vol. 108, pp. 374-384,2015.

  48. Miehe, C., Aldakheel, F., and Raina, A., Phase Field Modeling of Ductile Fracture at Finite Strains. A Variational Gradient-Extended Plasticity-Damage Theory, Int. J. Plasticity, vol. 84, pp. 1-32, 2016a.

  49. Miehe, C., Aldakheel, F., and Teichtmeister, S., 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., vol. 111, no. 9, pp. 816-863, 2017.

  50. Miehe, C., Hofacker, M., Schanzel, L.M., and Aldakheel, F., Phase Field Modeling of Fracture in Multi-Physics Problems. Part II. Brittle-to-Ductile Failure Mode Transition and Crack Propagation in Thermo-Elastic-Plastic Solids, Comput. Methods Appl. Mech. Eng., vol. 294, pp. 486-522,2015a.

  51. Miehe, C., Kienle, D., Aldakheel, F., and Teichtmeister, S., 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., vol. 312, pp. 3-50, 2016b.

  52. Miehe, C., Schanzel, L., and Ulmer, H., Phase Field Modeling of Fracture in Multi-Physics Problems. Part I. Balance of Crack Surface and Failure Criteria for Brittle Crack Propagation in Thermo-Elastic Solids, Comput. Methods Appl. Mech. Eng., vol. 294, pp. 449-485, 2015b.

  53. Miehe, C., Teichtmeister, S., and Aldakheel, F., Phase-Field Modeling of Ductile Fracture: A Variational Gradient-Extended Plasticity-Damage Theory and Its Micromorphic Regularization, Philos. Trans. R. Soc. A: Math., Phys. Eng. Sci, vol. 374, no. 2066,2016c.

  54. Miehe, C., Welschinger, F., and Aldakheel, F., Variational Gradient Plasticity at Finite Strains. Part II: Local-Global Updates and Mixed Finite Elements for Additive Plasticity in the Logarithmic Strain Space, Comput. Methods Appl. Mech. Eng., vol. 268, pp. 704-734, 2014.

  55. Miehe, C., Welschinger, F., and Hofacker, M., Thermodynamically Consistent Phase-Field Models of Fracture: Variational Principles and Multi-Field FE Implementations, Int. J. Numer. Methods Eng., vol. 83, pp. 1273-1311,2010.

  56. Mueller-Hoeppe, D.S., Loehnert, S., and Wriggers, P., A Finite Deformation Brick Element with Inhomogeneous Mode Enhancement, Int. J. Numer. Methods Eng., vol. 78, pp. 1164-1187,2009.

  57. Nadler, B. and Rubin, M., A New 3D Finite Element for Nonlinear Elasticity Using the Theory of a Cosserat Point, Int. J. Solids Struct., vol. 40, pp. 4585-4614,2003.

  58. Reese, S., Kuessner, M., and Reddy, B.D., A New Stabilization Technique to Avoid Hourglassing in Finite Elasticity, Int. J. Numer. Methods Eng., vol. 44, pp. 1617-1652, 1999.

  59. Reese, S. and Wriggers, P., A New Stabilization Concept for Finite Elements in Large Deformation Problems, Int. J. Numer. Methods Eng., vol. 48, pp. 79-110, 2000.

  60. Reinoso, J., Paggi, M., and Linder, C., Phase Field Modeling of Brittle Fracture for Enhanced Assumed Strain Shells at Large Deformations: Formulation and Finite Element Implementation, Comput. Mech., vol. 59, no. 6, pp. 981-1001,2017.

  61. Simo, J.C., A Framework for Finite Strain Elastoplasticity based on Maximum Plastic Dissipation and the Multiplicative Decom-position. Part II: Computational Aspects, Comput. Methods Appl. Mech. Eng., vol. 68, pp. 1-31,1988.

  62. Simo, J.C. and Miehe, C., Associative Coupled Thermoplasticity at Finite Strains: Formulation, Numerical Analysis and Implementation, Comput. Methods Appl. Mech. Eng., vol. 98, pp. 41-104,1992.

  63. Taylor, R.L. and Artioli, E., VEM for Inelastic Solids, Adv. Comput. Plasticity. Comput. Methods Appl. Sci., vol. 46, pp. 381-394, 2018.

  64. Teichtmeister, S., Kienle, D., Aldakheel, F., and Keip, M.A., Phase Field Modeling of Fracture in Anisotropic Brittle Solids, Int. J. Non-Linear Mech, vol. 97, pp. 1-21, 2017.

  65. Verhoosel, C.V. and de Borst, R., A Phase-Field Model for Cohesive Fracture, Int. J. Numer. Methods Eng., vol. 96, pp. 43-62, 2013.

  66. Wriggers, P., Nonlinear Finite Elements, Berlin: Springer, 2008.

  67. Wriggers, P. and Hudobivnik, B., A Low Order Virtual Element Formulation for Finite Elasto-Plastic Deformations, Comput. Methods Appl. Mech. Eng., vol. 327, pp. 459-477,2017.

  68. Wriggers, P., Hudobivnik, B., and Korelc, J., Efficient Low Order Virtual Elements for Anisotropic Materials at Finite Strains, Cham, Switzerland: Springer International Publishing, pp. 417-434, 2018a.

  69. Wriggers, P., Hudobivnik, B., and Schroder, J., Finite and Virtual Element Formulations for Large Strain Anisotropic Material with Inextensive Fibers, Cham, Switzerland: Springer International Publishing, pp. 205-231, 2018b.

  70. Wriggers, P., Reddy, B.D., Rust, W., and Hudobivnik, B., Efficient Virtual Element Formulations for Compressible and Incompressible Finite Deformations, Comput. Mech, vol. 60, no. 2, pp. 253-268, 2017.

  71. Wriggers, P., Rust, W.T., and Reddy, B.D., A Virtual Element Method for Contact, Comput. Mech., vol. 58, no. 6, pp. 1039-1050, 2016.

  72. Zhang, X., Vignes, C., Sloan, S.W., and Sheng, D., Numerical Evaluation of the Phase-Field Model for Brittle Fracture with Emphasis on the Length Scale, Comput. Mech, vol. 59, no. 5, pp. 737-752, 2017.

  73. Zienkiewicz, O.C., Taylor, R., and Zhu, J.Z., The Finite Element Method: Its Basis and Fundamentals, Amsterdam: Elsevier, 2005.


Articles with similar content:

MOLECULAR DYNAMICS/XFEM COUPLING BY A THREE-DIMENSIONAL EXTENDED BRIDGING DOMAIN WITH APPLICATIONS TO DYNAMIC BRITTLE FRACTURE
International Journal for Multiscale Computational Engineering, Vol.11, 2013, issue 6
M. Silani, S. P. A. Bordas, Timon Rabczuk, Hossein Talebi, Pierre Kerfriden
MULTI-LEVEL K-d TREE-BASED DATA-DRIVEN COMPUTATIONAL METHOD FOR THE DYNAMIC ANALYSIS OF MULTI-MATERIAL STRUCTURES
International Journal for Multiscale Computational Engineering, Vol.18, 2020, issue 4
Hongwu Zhang, Zhangcheng Zheng, Yonggang Zheng, Zhen Chen, Hongfei Ye
METHOD FOR DETERMINING STRUCTURES OF NEW CARBON-BASED 2D MATERIALS WITH PREDEFINED MECHANICAL PROPERTIES
International Journal for Multiscale Computational Engineering, Vol.15, 2017, issue 5
Waclaw Kus, Adam Mrozek, Tadeusz Burczynski
IMPROVED CRACK TIP ENRICHMENT FUNCTIONS AND INTEGRATION FOR CRACK MODELING USING THE EXTENDED FINITE ELEMENT METHOD
International Journal for Multiscale Computational Engineering, Vol.11, 2013, issue 6
Hans Minnebo, Nicolas Chevaugeon, Nicolas Moes
Simulation of High-Speed Transonic and Supersonic Flows with Anisotropic Mesh Refinement and Coarsening Strategies
International Journal of Fluid Mechanics Research, Vol.43, 2016, issue 5-6
Renato V. Linn, Armando M. Awruch