nach oben

Erschienen in:

21.01.2020 | Original Paper

An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

verfasst von: Karsten Paul, Christopher Zimmermann, Kranthi K. Mandadapu, Thomas J. R. Hughes, Chad M. Landis, Roger A. Sauer

Erschienen in: Computational Mechanics | Ausgabe 4/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires \(C^1\)-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-\(\alpha \) method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching.

Vorheriger Artikel Modal analysis of rotating pre-twisted viscoelastic sandwich beams

Nächster Artikel A poro-viscoelastic substitute model of fine-scale poroelasticity obtained from homogenization and numerical model reduction

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

This is a part of the Saint Venant–Kirchhoff model, see Duong et al. [19].

To avoid confusion, we write discrete arrays, such as the shape function array \({\mathbf {N}}\), in roman font, whereas continuous tensors, such as the normal vector \({\varvec{N}}\), are written in italic font.

The free nodes refer to the degrees of freedom, which are not given by boundary conditions.

\(T_0\) refers to a reference time used to obtain a dimensionless formulation, see Sect. 5.7

Note that \(\rho _0\) is the surface density and has units \([\mathrm {kg}/\mathrm {m}^2]\).

Also see the remark on stress waves at the beginning of this section.

We can compute the shear wave speed based on \(c_\mathrm {s}=\sqrt{G/\rho }\approx 6.2\,L_0/T_0\). An approximate value for the Rayleigh wave speed is then obtained as \(c_\mathrm {R}\approx 0.9162\cdot c_\mathrm {s}\approx 5.7\,L_0/T_0\). Based on the experiments by Ravi-Chandar and Knauss [58], the crack tip velocity stays below \(60\%\) of the Rayleigh wave speed. We can thus formulate a condition for the minimum time step, i.e. \(\Delta t\le \Delta t_\mathrm {max}<\Delta x_\mathrm {min}/(0.6\cdot c_\mathrm {R})\approx 1.1\cdot 10^{-3}\,T_0\), where the minimum element size is \(\Delta x_\mathrm {min}=1/256\,L_0\).

Ambati M, De Lorenzis L (2016) Phase-field modeling of brittle and ductile fracture in shells with isogeometric NURBS-based solid-shell elements. Comput Methods Appl Mech Eng 312:351–373MathSciNet

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(2):383–405MathSciNetMATH

Ambati M, Kruse R, De Lorenzis L (2016) A phase-field model for ductile fracture at finite strains and its experimental verification. Comput Mech 57(1):149–167MathSciNetMATH

Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M (2014) Phase-field modeling of fracture in linear thin shells. Introducing the new features of Theoretical and Applied Fracture Mechanics through the scientific expertise of the Editorial Board. Theor Appl Fract Mech 69:102–109

Amor H, Marigo J-J, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229MATH

Areias P, Rabczuk T, Msekh M (2016) Phase-field analysis of finite-strain plates and shells including element subdivision. Comput Methods Appl Mech Eng 312:322–350MathSciNet

Badnava H, Msekh MA, Etemadi E, Rabczuk T (2018) An h-adaptive thermo–mechanical phase field model for fracture. Finite Elem Anal Des 138:31–47

Benson DJ, Hartmann S, Bazilevs Y, Hsu M-C, Hughes TJR (2013) Blended isogeometric shells. Comput Methods Appl Mech Eng 255:133–146MathSciNetMATH

Borden MJ, Hughes TJR, Landis CM, Anvari A, Lee IJ (2016) 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 312:130–166MathSciNet

10.

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 273:100–118MathSciNetMATH

11.

Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95MathSciNetMATH

12.

Bourdin B, Francfort G, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826MathSciNetMATH

13.

Bourdin B, Larsen CJ, Richardson CL (2011) A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 168(2):133–143MATH

14.

Chen L, de Borst R (2018) Locally refined T-splines. Int J Numer Methods Eng 114(6):637–659MathSciNet

15.

Chen L, Verhoosel CV, de Borst R (2018) Discrete fracture analysis using locally refined T-splines. Int J Numer Methods Eng 116(2):117–140MathSciNet

16.

Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-alpha method. J Appl Mech 60(2):371–375MathSciNetMATH

17.

Ciarlet PG (1993) Mathematical elasticity: three dimensional elasticity. Elsevier, North-HollandMATH

18.

Dokken T, Lyche T, Pettersen KF (2013) Polynomial splines over locally refined box-partitions. Comput Aided Geom Des 30(3):331–356MathSciNetMATH

19.

Duong TX, Roohbakhshan F, Sauer RA (2017) A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries. Comput Methods Appl Mech Eng 316:43–83MathSciNet

20.

Echter R, Oesterle B, Bischoff M (2013) A hierarchic family of isogeometric shell finite elements. Comput Methods Appl Mech Eng 254:170–180MathSciNetMATH

21.

Forsey DR, Bartels RH (1988) Hierarchical B-spline refinement. SIGGRAPH Comput Graph 22(4):205–212

22.

Francfort G, Marigo J-J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342MathSciNetMATH

23.

Geelen RJ, Liu Y, Hu T, Tupek MR, Dolbow JE (2019) A phase-field formulation for dynamic cohesive fracture. Comput Methods Appl Mech Eng 348:680–711MathSciNet

24.

Geelen RJM, Liu Y, Dolbow JE, Rodríguez-Ferran A (2018) An optimization-based phase-field method for continuous-discontinuous crack propagation. Int J Numer Methods Eng 116(1):1–20MathSciNet

25.

Gerasimov T, Lorenzis LD (2016) A line search assisted monolithic approach for phase-field computing of brittle fracture. Comput Methods Appl Mech Eng 312:276–303MathSciNet

26.

Gerasimov T, Lorenzis LD (2019) On penalization in variational phase-field models of brittle fracture. Comput Methods Appl Mech Eng 354:990–1026MathSciNet

27.

Gerasimov T, Noii N, Allix O, De Lorenzis L (2018) A non-intrusive global/local approach applied to phase-field modeling of brittle fracture. Adv Model Simul Eng Sci 5:14

28.

Gomez H, Reali A, Sangalli G (2014) Accurate, efficient, and (iso)geometrically flexible collocation methods for phase-field models. J Comput Phys 262:153–171MathSciNetMATH

29.

Griffith AA (1921) VI. The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A 221:163–198

30.

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 290:466–495MathSciNetMATH

31.

Hesch C, Franke M, Dittmann M, Temizer İ (2016a) Hierarchical NURBS and a higher-order phase-field approach to fracture for finite-deformation contact problems. Comput Methods Appl Mech Eng 301:242–58MathSciNetMATH

32.

Hesch C, Schuß S, Dittmann M, Franke M, Weinberg K (2016b) Isogeometric analysis and hierarchical refinement for higher-order phase-field models. Comput Methods Appl Mech Eng 303:185–207MathSciNetMATH

33.

Hirmand MR, Papoulia KD (2018) A continuation method for rigid-cohesive fracture in a discontinuous Galerkin finite element setting. Int J Numer Methods Eng 115(5):627–650

34.

Hirmand MR, Papoulia KD (2019) Block coordinate descent energy minimization for dynamic cohesive fracture. Comput Methods Appl Mech Eng 354:663–688MathSciNet

35.

Hofacker M, Miehe C (2013) A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. Int J Numer Methods Eng 93(3):276–301MathSciNetMATH

36.

Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39–41):4135–4195MathSciNetMATH

37.

Johannessen KA, Kvamsdal T, Dokken T (2014) Isogeometric analysis using LR B-splines. Comput Methods Appl Mech Eng 269:471–514MathSciNetMATH

38.

Karma A, Kessler D, Levine H (2001) Phase-field model of mode III dynamic fracture. Phys Rev Lett 75:045501

39.

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. Springer, Singapore, pp 307–324

40.

Kiendl J, Ambati M, De Lorenzis L, Gomez H, Reali A (2016) Phase-field description of brittle fracture in plates and shells. Comput Methods Appl Mech Eng 312:374–394MathSciNet

41.

Kiendl J, Hsu M-C, Wu MC, Reali A (2015) Isogeometric Kirchhoff–Love shell formulations for general hyperelastic materials. Comput Methods Appl Mech Eng 291:280–303MathSciNetMATH

42.

Krueger R (2004) Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 57(2):109–143

43.

Kuhn C, Müller R (2010) A continuum phase field model for fracture. Computational mechanics in fracture and damage: a special issue in Honor of Prof. Gross. Eng Fract Mech 77(18):3625–3634

44.

Kuhn C, Schlüter A, Müller R (2015) On degradation functions in phase field fracture models. Selected articles from phase-field method 2014 international seminar. Comput Mater Sci 108:374–384

45.

Larsen C, Ortner C, Süli E (2010) Existence of solutions to a regularized model of dynamic fracture. Math Model Methods Appl Sci 20:1021–1048MathSciNetMATH

46.

Larsen CJ (2010) Models for dynamic fracture based on Griffith’s criterion. In: Hackl K (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials. Springer, Dordrecht, pp 131–140

47.

Linse T, Hennig P, Kästner M, de Borst R (2017) A convergence study of phase-field models for brittle fracture. Eng Fract Mech 184:307–318

48.

Miehe C, Hofacker M, Welschinger F (2010a) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45):2765–2778MathSciNetMATH

49.

Miehe C, Welschinger F, Hofacker M (2010b) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83(10):1273–1311MathSciNetMATH

50.

Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150MathSciNetMATH

51.

Molinari JF, Gazonas G, Raghupathy R, Rusinek A, Zhou F (2007) The cohesive element approach to dynamic fragmentation: the question of energy convergence. Int J Numer Methods Eng 69(3):484–503MATH

52.

Nagaraja S, Elhaddad M, Ambati M, Kollmannsberger S, De Lorenzis L, Rank E (2018) Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method. Comput Mech 63:1283–1300MathSciNetMATH

53.

Naghdi PM (1973) The theory of shells and plates. In: Truesdell C (ed) Linear theories of elasticity and thermoelasticity: linear and nonlinear theories of rods, plates, and shells. Springer, Berlin, pp 425–640

54.

Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Methods Eng 44(9):1267–1282MATH

55.

Papoulia KD (2017) Non-differentiable energy minimization for cohesive fracture. Int J Fract 204(2):143–158

56.

Parvizian J, Düster A, Rank E (2007) Finite cell method. Comput Mech 41(1):121–133MathSciNetMATH

57.

Radovitzky R, Seagraves A, Tupek M, Noels L (2011) A scalable 3d fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method. Comput Methods Appl Mech Eng 200(1):326–344MathSciNetMATH

58.

Ravi-Chandar K, Knauss W G (1984) An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching. Int J Fract 26(2):141–154

59.

Reali A, Hughes TJ R (2015) An introduction to isogeometric collocation methods. Springer, Vienna, pp 173–204MATH

60.

Reinoso J, Paggi M, Linder C (2017) Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: formulation and finite element implementation. Comput Mech 59(6):981–1001MathSciNetMATH

61.

Remmers JJC, de Borst R, Needleman A (2003) A cohesive segments method for the simulation of crack growth. Comput Mech 31(1):69–77MATH

62.

Sahu A, Sauer RA, Mandadapu KK (2017) Irreversible thermodynamics of curved lipid membranes. Phys Rev E 96:042409

63.

Sargado JM, Keilegavlen E, Berre I, Nordbotten JM (2018) High-accuracy phase-field models for brittle fracture based on a new family of degradation functions. J Mech Phys Solids 111:458–489MathSciNet

64.

Sauer RA (2018) On the computational modeling of lipid bilayers using thin-shell theory. In: Steigmann DJ (ed) The role of mechanics in the study of lipid bilayers. Springer, Cham, pp 221–286

65.

Sauer RA, Duong TX (2017) On the theoretical foundations of thin solid and liquid shells. Math Mech Solids 22(3):343–371MathSciNetMATH

66.

Sauer RA, Duong TX, Corbett CJ (2014) A computational formulation for constrained solid and liquid membranes considering isogeometric finite elements. Comput Methods Appl Mech Eng 271:48–68MathSciNetMATH

67.

Sauer RA, Duong TX, Mandadapu KK, Steigmann DJ (2017) A stabilized finite element formulation for liquid shells and its application to lipid bilayers. J Comput Phys 330:436–466MathSciNetMATH

68.

Schillinger D, Borden MJ, Stolarski HK (2015) Isogeometric collocation for phase-field fracture models. Isogeometric analysis special issue. Comput Methods Appl Mech Eng 284:583–610MATH

69.

Schlüter A, Willenbücher A, Kuhn C, Müller R (2014) Phase field approximation of dynamic brittle fracture. Comput Mech 54(5):1141–1161MathSciNetMATH

70.

Sederberg TW, Zheng J, Bakenov A, Nasri A (2003) T-splines and T-NURCCs. ACM Trans Graph 22(3):477–484

71.

Steigmann DJ (1999) Fluid films with curvature elasticity. Arch Ration Mech Anal 150:127–152MathSciNetMATH

72.

Ulmer H, Hofacker M, Miehe C (2012) Phase field modeling of fracture in plates and shells. PAMM 12(1):171–172

73.

Vavasis SA, Papoulia KD, Hirmand MR (2020) Second-order cone interior-point method for quasistatic and moderate dynamic cohesive fracture. Comput Methods Appl Mech Eng 358:112633MathSciNet

74.

Zhou S, Zhuang X (2018) Adaptive phase field simulation of quasi-static crack propagation in rocks. Computational modeling of fracture in geotechnical engineering part I. Undergr Sp 3(3):190–205

75.

Zimmermann C, Sauer RA (2017) Adaptive local surface refinement based on LR NURBS and its application to contact. Comput Mech 60:1011–1031MathSciNetMATH

76.

Zimmermann C, Toshniwal D, Landis CM, Hughes TJR, Mandadapu KK, Sauer RA (2019) An isogeometric finite element formulation for phase transitions on deforming surfaces. Comput Methods Appl Mech Eng 351:441–477MathSciNet

Titel: An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS
verfasst von: Karsten Paul
Christopher Zimmermann
Kranthi K. Mandadapu
Thomas J. R. Hughes
Chad M. Landis
Roger A. Sauer
Publikationsdatum: 21.01.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Computational Mechanics / Ausgabe 4/2020
Print ISSN: 0178-7675
Elektronische ISSN: 1432-0924
DOI: https://doi.org/10.1007/s00466-019-01807-y

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 4/2020

Heart valve isogeometric sequentially-coupled FSI analysis with the space–time topology change method

Computational shape optimisation for a gradient-enhanced continuum damage model

Thermodynamic properties and thermoelastic constitutive relation for cubic crystal structures based on improved free energy

An FFT-based spectral solver for interface decohesion modelling using a gradient damage approach

A phase-field model for fracture of unidirectional fiber-reinforced polymer matrix composites

Element length calculation in B-spline meshes for complex geometries