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Advances in Cohesive Zone Modeling of Dynamic Fracture

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Dynamic Failure of Materials and Structures

Abstract

In this chapter, we review the state of the-art in computational methods for modeling dynamic fracture of brittle solids based on the popular cohesive element approach. The discussion includes a detailed review of the underlying theory, its implementation via interface elements in its two different flavors: the intrinsic and extrinsic approach, as well as the application of the method to different concrete problems in dynamic fracture. Limitations and numerical issues are discussed in detail. As a means to address some of these issues, we describe an alternative approach based on a discontinuous Galerkin (DG) reformulation of the continuum problem that exploits the virtues of the existing cohesive element methods. The scalability and accuracy of the DG method for fracture mechanics is demonstrated through wave propagation and spall tests in ceramics. Lastly, some unresolved open problems and numerical issues pertaining to cohesive zone modeling of fracture are briefly discussed.

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References

  • Anvari M, Liu J, Thaulow C (2007) Dynamic ductile fracture in aluminum round bars: experiments and simulations. Int J Fracture 143:317–332

    Article  Google Scholar 

  • Anvari M, Scheider I, Thaulow C (2006) Simulation of dynamic ductile crack growth using strain-rate and triaxiality-dependent cohesive elements. Eng Fract Mech 73:2210–2228

    Article  Google Scholar 

  • Areias PMA, Belytschko T (2005) Analysis of three-dimensional crack intitiation and propagation using the extended finite element method. Int J Numer Meth Eng 63:760–788

    Article  MATH  Google Scholar 

  • Arias I, Knap J, Chalivendra VB, Hong S, Ortiz M, Rosakis AJ (2007) Modeling and experimental validation of dynamic fracture events along weak planes. Comput Methods Appl Mech Eng 196:3833–3840

    Article  MATH  Google Scholar 

  • Arnold DN (1982) An interior penalty finite element method with discontinuous elements. SIAM J Numer Anal 19:742–760

    Article  MATH  MathSciNet  Google Scholar 

  • Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J Numer Anal 39(5):1749–1779

    Article  MATH  MathSciNet  Google Scholar 

  • Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129

    Article  MathSciNet  Google Scholar 

  • Bassi F, Rebay S (1997) A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier-stokes equations. J Comput Phys 131:267–279

    Article  MATH  MathSciNet  Google Scholar 

  • Beltz G, Rice JR (1991) Dislocation nucleation versus cleavage decohesion at crack tips. In: Lowe TC, Rollett AD, Follansbee PS, Daehn (eds) Modeling and deformation of crystalline solids, TMS, Warrendale, PA, p 457

    Google Scholar 

  • Bozzolo G, Ferrante J, Smith JR (1991) Universal behavior in ideal slip. Scr Metal Mater 25:1927–1931

    Article  Google Scholar 

  • Brezzi F, Manzini M, Marini D, Pietra P, Russo A (2000) Discontinuous galerkin approximations for elliptic problems. Numer Methods Partial Differ Equ 16:47–58

    Article  MathSciNet  Google Scholar 

  • Camacho GT (1996) Computational modeling of impact damage and penetration of brittle and ductile solids. PhD thesis, Brown University

    Google Scholar 

  • Camacho GT, Ortiz M (1996) Computational modeling of impact damage in brittle materials. Int J Solids Struct 33(20–22):2899–2983

    Article  MATH  Google Scholar 

  • Chiluveru S (2007) Computational modeling of crack initiation in cross-roll piercing. Master’s thesis, Massachusetts Institute of Technology

    Google Scholar 

  • Cirak F, Ortiz M, Pandolfi A (2005) A cohesive approach to thin-shell fracture and fragmentation. Comput Methods Appl Mech Eng 194:2604–2618

    Article  MATH  Google Scholar 

  • Cockburn B, Shu CW (1998) The local discontinuous galerkin method for time-dependent convection diffusion problems. SIAM J Numer Anal 35:2440–2463

    Article  MATH  MathSciNet  Google Scholar 

  • Dolbow J, Moes N, Belytschko T (2001) An extended finite element method for modeling crack growth with frictional contact. Comput Methods Appl Mech Eng, 190:6825–6846

    Article  MATH  MathSciNet  Google Scholar 

  • Dugdale DS (1960) Yielding of steel sheets containing clits. J Mech Phys Solids 8:100–104

    Article  Google Scholar 

  • Dvorkin EN, Assanelli AP (1991) 2d Finite elements with displacement interpolated embedded localization lines: the analysis of fracture in frictional materials. Comput Methods Appl Mech Eng 90:829–844

    Article  Google Scholar 

  • Dvorkin EN, Cuitino AM, Gioia G (1990) Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. Int J Numer Methods Eng 30:541–564

    Article  MATH  Google Scholar 

  • Espinosa HD, Dwivedi S, Lu H-C (2000) Modeling impact induced delamination of woven fiber reinforced composites with contact/cohesive laws. Comput Methods Appl Mech Eng 183:259–290

    Article  MATH  Google Scholar 

  • Espinosa HD, Zavattieri PD (2003)1 A grain level model for the study of failure initiation and evolution in polycrystalline brittle materials, part i: theory and numerical implementation. Mech Mater 35: 333–364

    Google Scholar 

  • Espinosa HD, Zavattieri PD (2003)2 A grain level model for the study of failure initiation and evolution in polycrystalline brittle materials, part ii: numer examples. Mech Mater, 35: 365–394

    Google Scholar 

  • Espinosa HD, Zavattieri PD, Dwivedi SK (1998) A finite deformation continuum discrete model for the description of fragmentation and damage in brittle materials. J Mech Phys Solids 46(10):1909–1942

    Article  MATH  MathSciNet  Google Scholar 

  • Falk ML, Needleman A, Rice JR (2001) A critical evaluation of cohesive zone models of dynamic fracture. J Phys IV 11:Pr5–43–51

    Google Scholar 

  • Ferrante J, Smith JR (1985) Theory of the bimetallic interface. Phys Rev B 31(6):3427–3434

    Article  Google Scholar 

  • Field JE (1988) Investigation of the impact performance of various glass and ceramic systems. Technical report, Cambridge University, Cambridge

    Google Scholar 

  • Freund LB (1989) Dynamic fracture mechanics. Cambridge University Press, Cambridge

    Google Scholar 

  • Geubelle PH, Baylor JS (1998) Impact-induced delamination of composites: a 2d simulation. Compos B 29B:589–602

    Article  Google Scholar 

  • Grady DE, Benson DA (1983) Fragmentation of metal rings by electromagnetic loading. Exp Mech 12:393–400

    Article  Google Scholar 

  • Griffith AA (1920) The phenomena of rupture and flow in solids. In: Royal society (GB) (ed) Philosophical transactions of the Royal society of London, vol A221: Mathematical and physical sciences Cambridge University Press, Cambridge, pp 163–198

    Google Scholar 

  • Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J Eng Mater Technol 99:2–15

    Article  Google Scholar 

  • Klein PA, Foulk JW, Chen EP, Wimmer SA, Gao HJ (2001) Physics-based modeling of brittle fracture, cohesive formulations and the applications of meshfree methods. Theor Appl Fracture Mech 37:99–166

    Article  Google Scholar 

  • Kubair DV, Geubelle PH (2003) A comparative analysis of intrinsic and extrinsic cohesive models of dynamic fracture. Int J Solids Struct 40:3853–3868

    Article  MATH  Google Scholar 

  • Lew A, Neff P, Sulsky D, Ortiz M (2004) Optimal bv estimates for a discontinuous galerkin method for linear elasticity. Appl Math Res eXpress 3:73–106

    Article  MathSciNet  Google Scholar 

  • Lubliner J (1972) On the thermodynamic foundations of non-linear solid mechanics. Int J Non-Linear Mech 7:237–254

    Article  MATH  Google Scholar 

  • Lubliner J (1973) On the structure of rate equations of materials with internal variables. Int J Non-Linear Mech, 17:109–119

    MATH  MathSciNet  Google Scholar 

  • Maiti S, Geubelle PH (2004) Mesoscale modeling of dynamic fracture of ceramic materials. Comput Methods Eng Sci 5(2):2618–2641

    Google Scholar 

  • Maiti S, Rangaswamy K, Geubelle PH (2005) Mesoscale analysis of dynamic fragmentation of ceramics under tension. Acta Mater 53:823–834

    Article  Google Scholar 

  • Mergheim J, Kuhl E, Steinmann P (2004) A hybrid discontinuous Galerkin/interface method for the computational modelling of failure. Commun Numer Methods Eng 20:511–519

    Article  MATH  MathSciNet  Google Scholar 

  • Miller O, Freund LB, Needleman A (1999)1 Energy dissipation in dynamic fracture of brittle materials. Modeling Simul Mater Sci Eng 7:573–586

    Google Scholar 

  • Miller O, Freund LB, Needleman A (1999)2 Modeling and simulation of dynamic fragmentation in brittle materials. Int J Fracture 96(2):101–125

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Mota A, Klug WS, Ortiz M, Pandolfi A (2003) Finite-element simulation of firearm damage to the human cranium. Comput Mech 31:115–121

    Article  MATH  Google Scholar 

  • Mota A, Knap J, Ortiz M (2008) Fracture and fragmentation of simplicial finite element meshes using graphs. Int J Numer Methods Eng 73:1547–1570

    Article  MATH  MathSciNet  Google Scholar 

  • Needleman A (1987) A continuum model for void nucleation by inclusion debonding. J Appl Mech, 54:525–531

    Article  MATH  Google Scholar 

  • Needleman A (1990) An analysis of decohesion along an imperfect interface. Int J Fract, 42:21–40

    Article  Google Scholar 

  • Needleman A (1990) An analysis of tensile decohesion along an interface. J Mech Phys Solids 38(3):289–324

    Article  Google Scholar 

  • Needleman A (1997) Numer modeling of crack growth under dynamic loading conditions. Comput Mech 19:463–469

    Article  MATH  Google Scholar 

  • Nguyen O, Repetto EA, Ortiz M, Radovitzky R (2001) A cohesive model of fatigue crack growth. Int J Fract, 110:351–369

    Article  Google Scholar 

  • Nittur PG, Maitit S, Geubelle PH (2008) Grain-level analysis of dynamic fragmentation of ceramics under multi-axial compression. J Mech Phys Solids, 56:993–1017

    Article  MATH  Google Scholar 

  • Noels L, Radovitzky R (2006) A general discontinuous Galerkin method for finite hyperelasticity. formulation and numerical applications. Int J Numer Methods Eng 68:64–97

    Article  MATH  MathSciNet  Google Scholar 

  • Noels L, Radovitzky R (2008) An explicit discontinuous galerkin method for non-linear solid dynamics: formulation, parallel implementation and scalability properties. Int J Numer Methods Eng 74:1393–1420

    Article  MATH  MathSciNet  Google Scholar 

  • Ortiz M, Leroy Y, Needleman, A (1987) A finite element method for localized failure analysis. Comput Meth Appl M 61(2):189–214

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Ortiz M, Suresh S (1993) Statistical properties residual stresses and intergranular fracture in ceramic materials. J Appl Mech 60:77–84

    Article  Google Scholar 

  • Pandolfi A, Guduru PR, Ortiz M, Rosakis AJ (2000) Three dimensional cohesive-element analysis and experiments of dynamic fracture in c300 steel. Int J Solids Struct 37:3733–3760

    Article  Google Scholar 

  • Pandolfi A, Krysl P, Ortiz M (1999) Finite element simulation ring expansion experiments and fragmentation: The capturing of length and time scales through cohesive models of fracture. Int J Fract 95:279–297

    Article  Google Scholar 

  • Pandolfi A, Ortiz M (2002) An efficient adaptive procedure for three-dimensional fragmentation simulations. Eng Comput 18:148–159

    Article  Google Scholar 

  • Panfolfi A, Ortiz M (1998) Solid modeling aspects of three-dimensional fragmentation. Eng Comput 14:287–308

    Article  Google Scholar 

  • Papoulia KD, Sam C-H, Vavasis SA (2003) Time continuity in cohesive finite element modeling. Int J Numer Methods Eng 58:679–701

    Article  MATH  Google Scholar 

  • Papoulia KD, Vavasis SA, Ganguly P (2006) Spatial convergence of crack nucleation using a cohesive finite-element model on a pinwheel-based mesh. Int J Numer Methods Eng 67:1–16

    Article  MATH  MathSciNet  Google Scholar 

  • Paulino GH, Celes W, Espinha R, Zhang Z (2008) A general topology-based framework for adaptive insertion of cohesive elements in finite element meshes. Eng Comput 24:59–78

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Remmers JJC, de Borst R, Needleman A (2008) The simulation of dynamic crack propagation using the cohesive segments method. J Mech Phys Solids 56:70–92

    Article  MATH  MathSciNet  Google Scholar 

  • Repetto EA, Radovitzky R, Ortiz M (2000) Finite element simulation of dynamic fracture and fragmentation of glass rods. Comput Methods Appl Mech Eng 183(1-2):3–14

    Article  MATH  Google Scholar 

  • Rice JR (1968) Mathematical analysis in the mechanics of fracture. In: H Liebowitz, (ed) Fracture: an advanced treatise, vol 2. Academic, New York, pp 191–311

    Google Scholar 

  • Rice JR (1967) A path-independent integral and approximate analysis of strain concentrations by notches and cracks. J Appl Mech 35:379–386

    Google Scholar 

  • Rose JH, Ferrante J, Smith JR (1981) Universal binding energy curves for metals and bimetallic interfaces. Phys Rev Lett 47(9):675–678

    Article  Google Scholar 

  • Rose JH, Smith JR, Ferrante J (1983) Universal features of bonding in metals. Phys Rev B 28(4):1835–1845

    Article  Google Scholar 

  • Ruiz G, Ortiz M, Pandolfi A (2000) Three-dimensional finite element simulation of the dynamic brazilian tests on concrete cylinders. Int J Numer Methods Eng 48:963–994

    Article  MATH  Google Scholar 

  • Ruiz G, Pandolfi A, Ortiz M (2001) Three-dimensional cohesive modeling of dynamic mixed-mode fracture. Int J Numer Methods Eng 52(1–2):97–120

    Article  Google Scholar 

  • Sam C-H, Papoulia KD, Vavasis SA (2005) Obtaining initially rigid cohesive finite element models that are temporally convergent. Eng Fract Mech 72:2247–2267

    Article  Google Scholar 

  • Scheider I, Brocks W (2003) Simulation of cup-cone fracture using the cohesive model. Eng Fract Mech 70:1943–1961

    Article  Google Scholar 

  • Seagraves A, Jérusalem A, Radovitzky R, Noels L (in preparation) A hybrid DG/cohesive method for modeling dynamic fracture brittle solids

    Google Scholar 

  • Siegmund T, Brocks W (1999) Prediction of the work of separation and implications to modeling. Int J Fract 99:97–116

    Article  Google Scholar 

  • Siegmund T, Brocks W (2000) A numerical study on the correlation between the work of separation and the dissipation rate in ductile fracture. Eng Fract Mech 67:139–154

    Article  Google Scholar 

  • Siegmund T, Needleman A (1997) A numerical study of dynamic crack growth in elastic-viscoplastic solids. Int J Solids Struct 34(7):769–787

    Article  MATH  Google Scholar 

  • Ten Eyck A, Lew A (2006) Discontinuous galerkin methods for nonlinear elasticity. Int J Numer Methods Eng 00:1–6

    MathSciNet  Google Scholar 

  • Tijssens MGA, Sluys LJ, van der Giessen E (2001) Simulation of fracture of cementitious composites with explicit modeling of microstructural features. Eng Fract Mech 68:1245–1263

    Article  Google Scholar 

  • Tvergaard V (1990) Effect of fibre debonding in a whisker-reinforced metal. Mater Sci Eng A125:203–213

    Google Scholar 

  • Tvergaard V (2001) Crack growth prediction by cohesive zone model for ductile fracture. J Mech Phys Solids 49:2191–2207

    Article  MATH  Google Scholar 

  • Tvergaard V, Hutchinson JW (1992) The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J Mech Phys Solids 40(6):1377–1397

    Article  MATH  Google Scholar 

  • Tvergaard V, Hutchinson JW (1993) The influence of plasticity on mixed mode interface toughness. J Mech Phys Solids 41(6):1119–1135

    Article  MATH  Google Scholar 

  • Tvergaard V, Hutchinson JW (1996) Effect of strain dependent cohesive model on predictions of interface crack growth. J Phys IV 6:C6–165–C6–172

    Google Scholar 

  • Tvergaard V, Needleman A (1993) An analysis of the brittle-ductile transition in dynamic crack growth. Int J Fract 59:53–67

    Google Scholar 

  • Xu X-P, Needleman A (1993) Void nucleation by inclusion debonding in a crystal matrix. Model Simul Mater Sci Eng 1:111–132

    Article  Google Scholar 

  • Xu X-P, Needleman A (1995) Numerical simulations of dynamic interfacial crack growth allowing for crack growth away from the bond line. Int J Fract 74:253–275

    Google Scholar 

  • Xu XP, Needleman A (1994) Numerical simulation of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397–1434

    Article  MATH  Google Scholar 

  • Xu XP, Needleman A (1996) Numerical simulations of dynamic crack growth along an interface. Int J Fract 74:289–324

    Article  Google Scholar 

  • Xu XP, Needleman A, Abraham FF (1997) Effect of inhomogeneities on dynamic crack growth in an elastic solid. Model Simul Mater Sci Eng 5:489–516

    Article  Google Scholar 

  • Yang Z, Xu XF (2008) A heterogeneous cohesive model for quasi-brittle materials considering spatially varying random fracture properties. Comput Methods Appl Mech Eng 197:4027–4039

    Article  MATH  Google Scholar 

  • Yu C, Ortiz M, Rosakis AJ (2003) 3d Modeling of impact failure in sandwich structures. In Blackman BRK, Pavan A, Williams JG, (ed) Fract polymers, compos adhesives II Elsevier Science, Amsterdam, pp 527–537

    Google Scholar 

  • Yu C, Pandolfi A, Ortiz M, Coker D, Rosakis AJ (2002) Three-dimensional modeling of intersonic shear-crack growth in asymmetrically loaded unidirectional composite plates. Int J Solids Struct 39:6135–6157

    Article  MATH  Google Scholar 

  • Yu RC, Ruiz G, Pandolfi A (2004) Numerical investigation on the dynamic behavior of advanced ceramics. Eng Fract Mech 71:897–911

    Article  Google Scholar 

  • Zavattieri PD, Espinosa HD (2001) Grain level analysis of crack initiation and propagation in brittle materials. Acta Mater 49:4291–4311

    Article  Google Scholar 

  • Zavattieri PD, Hector LG, Bower AF (2008) Cohesive zone simulations of crack growth along a rough interface between two elastic-plastic solids. Eng Fract Mech 75:4309–4332

    Article  Google Scholar 

  • Zhang Z, Paulino GH (2005) Cohesive zone modeling of dynamic failure in homogeneous and functionally graded materials. Int J Plasticity 21:1195–1254

    Article  MATH  Google Scholar 

  • Zhang Z, G.H. Paulino, Celes W (2007) Extrinsic cohesive modeling of dynamic fracture and microbranching instability in brittle materials. Int J Numer Methods Eng 72:893–923

    Article  MATH  Google Scholar 

  • Zhou F, Molinari JF (2004) Dynamic crack propagation with cohesive elements: a methodology to address mesh dependency. Int J Numer Methods Eng 59:1–24

    Article  MATH  Google Scholar 

  • Zhou F, Molinari JF (2004) Stochastic fracture of ceramics under dynamic tensile loading. Int J Solids Struct 41:6573–6596

    Article  MATH  Google Scholar 

  • Zhou F, Molinari JF, Li Y (2004) Three-dimensional numerical simulations of dynamic fracture in silicon carbide reinforced aluminum. Eng Fract Mech 71:1357–1378

    Article  Google Scholar 

  • Zhou F, Molinari JF, Ramesh KT (2005) A cohesive model based fragmentation analysis: effects of strain rate and initial defects distribution. Int J Solids Struct 42:5181–5207

    Article  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by the U.S. Army through the MIT Institute for Soldier Nanotechnologies, under Contract DAAD-19-02-D-0002 with the U.S. Army Research Office. The content does not necessarily reflect the position of the Government, and no official endorsement should be inferred.

Partial support from the office of Naval Research through a Multidisciplinary University Research Initiative program on “Cellular material concepts for force protection”, Prime Award no. N00014-07-1-0764 is also gratefully acknowledged.

The authors would like to acknowledge the contributions of Ludovic Noels who has kindly read the manuscript and suggested many useful modifications.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

    Andrew Seagraves

  2. Institute for Soldier Nanotechnologies, Massachusetts Institute of Technology, Cambridge, MA, USA

    Andrew Seagraves & Raúl Radovitzky

  3. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Raúl Radovitzky

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  1. Andrew Seagraves
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Editors and Affiliations

  1. Dept. Mechanical Engineering &, University of Rhode Island, Upper College Rd. 92, Kingston, 02881, U.S.A.

    Arun Shukla

  2. Graduate Aeronautical Laboratories, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, U.S.A.

    Guruswami Ravichandran

  3. Solid Mechanics Div., Office of Naval Research, Arlington, 22203-1995, U.S.A.

    Yapa D.S. Rajapakse

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Seagraves, A., Radovitzky, R. (2009). Advances in Cohesive Zone Modeling of Dynamic Fracture. In: Shukla, A., Ravichandran, G., Rajapakse, Y. (eds) Dynamic Failure of Materials and Structures. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0446-1_12

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