NOTICE: this is the author’s version of a work that was accepted for publication in Engineering Fracture Mechanics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Engineering Fracture Mechanics, [VOL#, ISSUE#, (DATE)] DOI 10.1016/j.engfracmech.2013.03.018
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[en] The recently developed hybrid discontinuous Galerkin/extrinsic cohesive law framework is extended to the study of intra{laminar fracture of composite materials. Toward this end, micro-volumes of di erent sizes are studied.
The method captures the debonding process, which is herein proposed to be assimilated to a damaging process, before the strain softening onset, and the density of dissipated energy resulting from the damage (debonding) remains the same for the di erent studied cell sizes. Finally, during the strain softening phase a micro{crack initiates and propagates in agreement with experimental observations. We thus extract a resulting mesoscale cohesive law, which is independent on the cell sizes, using literature methods.
FP7 - 235303 - MATERA+ - ERA-NET Plus on Materials Research
Name of the research project :
SIMUCOMP
Funders :
The authors acknowledge funding through SIMUCOMP, an ERA-NET +, MATERA+ project nanced by Consejer a de Educaci on y Empleo of Comunidad de Madrid, the Walloon region (agreement no 1017232, CT-EUC 2010-10-12), the Luxembourg region and by the European Union s Seventh Framework Programme (FP7/2007-2013). Computational resources have been provided by the supercomputing facilities of the Consortium des Equipements de Calcul Intensif en F ed eration Wallonie Bruxelles (C ECI) funded by the Fond de la Recherche Scienti que de Belgique (FRS-FNRS). CÉCI - Consortium des Équipements de Calcul Intensif [BE] CE - Commission Européenne [BE]
Ladevèze P., Lubineau G., Marsal D. Towards a bridge between the micro- and mesomechanics of delamination for laminated composites. Compos Sci Technol 2006, 66(6):698-712. ISSN: 0266-3538. http://www.sciencedirect.com/science/article/pii/S0266353804003665. 10.1016/j.compscitech.2004.12.026.
LLorca J., González C., Molina-Aldareguía J.M., Segurado J., Seltzer R., Sket F., et al. Multiscale modeling of composite materials: a Roadmap towards virtual testing. Adv Mater 2011, 23(44):5130-5147. ISSN: 1521-4095. 10.1002/adma.201101683.
Barenblatt G. The mathematical theory of equilibrium cracks in brittle fracture 1962, vol. 7. Elsevier, doi:10.1016/S0065-2156(08)70121-2. http://www.sciencedirect.com/science/article/B7RNC-4S867RJ-6/2/24d2fbaac93457d135f935d6cfa35836.
Hillerborg A., Modéer M., Petersson P.-E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 1976, 6(6):773-781. ISSN: 0008-8846. http://www.sciencedirect.com/science/article/pii/0008884676900077. 10.1016/0008-8846(76)90007-7.
Needleman A. A continuum model for void nucleation by inclusion debonding. J Appl Mech 1987, 54(3):525-531. http://link.aip.org/link/?AMJ/54/525/1. 10.1115/1.3173064.
Tvergaard V. Effect of fibre debonding in a whisker-reinforced metal. Mater Sci Engng: A 1990, 125(2):203-213. ISSN: 0921-5093. http://www.sciencedirect.com/science/article/pii/0921509390901708. 10.1016/0921-5093(90)90170-8.
Inglis H.M., Geubelle P.H., Matouš K., Tan H., Huang Y. Cohesive modeling of dewetting in particulate composites: micromechanics vs. multiscale finite element analysis. Mech Mater 2007, 39(6):580-595. ISSN: 0167-6636. http://www.sciencedirect.com/science/article/pii/S0167663606001141. 10.1016/j.mechmat.2006.08.008.
Tvergaard V., Hutchinson J.W. The influence of plasticity on mixed mode interface toughness. J Mech Phys Solids 1993, 41(6):1119-1135. ISSN: 0022-5096. http://www.sciencedirect.com/science/article/pii/002250969390057M. 10.1016/0022-5096(93)90057-M.
Tvergaard V., Hutchinson J.W. Toughness of an interface along a thin ductile layer joining elastic solids. Philos Mag A 1994, 70(4):641-656. http://www.tandfonline.com/doi/abs/10.1080/01418619408242253. 10.1080/01418619408242253.
Needleman A. An analysis of decohesion along an imperfect interface. Int J Fract 1990, 42:21-40. ISSN: 0376-9429. http://dx.doi.org/10.1007/BF00018611.
Camanho P.P., Dávila C.G., Pinho S.T. Fracture analysis of composite co-cured structural joints using decohesion elements. Fatigue Fract Engng Mater Struct 2004, 27(9):745-757. ISSN 1460-2695. 10.1111/j.1460-2695.2004.00695.x.
Miron M.C., Constantinescu D.M. Strain fields at an interface crack in a sandwich composite. Mech Mater 2011, 43(12):870-884. ISSN: 0167-663. http://www.sciencedirect.com/science/article/pii/S0167663611001839. 10.1016/j.mechmat.2011.10.002.
Geubelle P.H., Baylor J.S. Impact-induced delamination of composites: a 2D simulation. Compos Part B: Engng 1998, 29(5):589-602. ISSN: 1359-8368. http://www.sciencedirect.com/science/article/pii/S1359836898000134. 10.1016/S1359-8368(98)00013-4.
Pantano A., Averill R.C. A mesh-independent interface technology for simulation of mixed-mode delamination growth. Int J Solids Struct 2004, 41(14):3809-3831. ISSN: 0020-7683. http://www.sciencedirect.com/science/article/pii/S0020768304000745. 10.1016/j.ijsolstr.2004.02.018.
Turon A., Dávila C.G., Camanho P.P., Costa J. An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models. Engng Fract Mech 2007, 74(10):1665-1682. ISSN: 0013-7944. http://www.sciencedirect.com/science/article/pii/S0013794406003808. 10.1016/j.engfracmech.2006.08.025.
Tvergaard V. Crack growth predictions by cohesive zone model for ductile fracture. J Mech Phys Solids 2001, 49(9):2191-2207. ISSN: 0022-5096. http://www.sciencedirect.com/science/article/B6TXB-43RTTVV-P/2/f3f973c370359693df9c4261c78d5525. 10.1016/S0022-509(01)00030-8.
Xu X.-P., Needleman A. Void nucleation by inclusion debonding in a crystal matrix. Model Simul Mater Sci Engng 1993, 1:111-132. 10.1088/0965-0393/1/2/001.
Tvergaard V., Hutchinson J.W. The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J Mech Phys Solids 1992, 40(6):1377-1397. ISSN: 0022-5096. http://www.sciencedirect.com/science/article/pii/0022509692900203. 10.1016/0022-509(92)90020-3.
Nguyen O., Ortiz M. Coarse-graining and renormalization of atomistic binding relations and universal macroscopic cohesive behavior. J Mech Phys Solids 2002, 50(8):1727-1741. ISSN: 0022-5096. http://www.sciencedirect.com/science/article/pii/S0022509601001338. 10.1016/S0022-509(01)00133-8.
Xu X.-P., Needleman A. Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 1994, 42(9):1397-1434. ISSN: 0022-5096. http://www.sciencedirect.com/science/article/pii/0022509694900035. 10.1016/0022-509(94)90003-5.
Song S.H., Paulino G.H., Buttlar W.G. A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material. Engng Fract Mech 2006, 73(18):2829-2848. ISSN: 0013-7944. http://www.sciencedirect.com/science/article/pii/S0013794406001755. 10.1016/j.engfracmech.2006.04.030.
Blal N., Daridon L., Monerie Y., Pagano S. Criteria on the artificial compliance inherent to the intrinsic cohesive zone. CR Méc 2011, 339(12):789-795. ISSN: 1631-0721. http://www.sciencedirect.com/science/article/pii/S1631072111001677. 10.1016/j.crme.2011.10.001.
Zavattieri P.D., Espinosa H.D. Grain level analysis of crack initiation and propagation in brittle materials. Acta Mater 2001, 49(20):4291-4311. ISSN: 1359-6454. http://www.sciencedirect.com/science/article/pii/S1359645401002920. 10.1016/S1359-645(01)00292-0.
Camacho G.T., Ortiz M. Computational modelling of impact damage in brittle materials. Int J Solids Struct 1996, 33(20-22):2899-2938. ISSN: 0020-7683. http://www.sciencedirect.com/science/article/B6VJS-3VTSPVJ-3/2/98fe3c31d0cee8e029d78d1631c6bbdd. 10.1016/0020-768(95)00255-3.
Ortiz M., Pandolfi A. Finite-deformation irreversible cohesive elements for three-dimensional crack propagation analysis. Int J Numer Methods Engng 1999, 44(9):1267-1282. ISSN: 1097-0207. http://dx.doi.org/10.1002/(SICI)1097-0207(19990330)44:91267::AID-NME4863.0.CO;2-7.10.1002/(SICI)1097-0207(19990330)44:91267::AID-NME4863.0.CO;2-7.
Pandolfi A., Guduru P.R., Ortiz M., Rosakis A.J. Three dimensional cohesive-element analysis and experiments of dynamic fracture in C300 steel. Int J Solids Struct 2000, 37(27):3733-3760. ISSN: 0020-7683. http://www.sciencedirect.com/science/article/pii/S0020768399001559. 10.1016/S0020-768(99)00155-9.
Arias I., Knap J., Chalivendra V.B., Hong S., Ortiz M., Rosakis A.J. Numerical modelling and experimental validation of dynamic fracture events along weak planes. Comput Methods Appl Mech Engng 2007, 196(37-40):3833-3840. ISSN: 0045-7825. http://www.sciencedirect.com/science/article/pii/S0045782507001107. 10.1016/j.cma.2006.10.052.
Molinari J.F., Gazonas G., Raghupathy R., Rusinek A., Zhou F. The cohesive element approach to dynamic fragmentation: the question of energy convergence. Int J Numer Methods Engng 2007, 69(3):484-503. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.177.
Pandolfi A., Ortiz M. An efficient adaptive procedure for three-dimensional fragmentation simulations. Engng Comput 2002, 18(2):148-159. ISSN: 0177-0667. http://dx.doi.org/10.1007/s003660200013. 10.1007/s003660200013.
Mota A., Knap J., Ortiz M. Fracture and fragmentation of simplicial finite element meshes using graphs. Int J Numer Methods Engng 2008, 73(11):1547-1570. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.2135. 10.1002/nme.2135.
Paulino G., Celes W., Espinha R., Zhang Z. A general topology-based framework for adaptive insertion of cohesive elements in finite element meshes. Engng Comput 2008, 24:59-78. ISSN: 0177-0667. http://dx.doi.org/10.1007/s00366-007-0069-7. 10.1007/s00366-007-0069-7.
Moës N., Dolbow J., Belytschko T. A finite element method for crack growth without remeshing. Int J Numer Methods Engng 1999, 46(1):131-150. ISSN: 1097-0207. http://dx.doi.org/10.1002/(SICI)1097-0207(19990910)46:1131::AID-NME7263.0.CO;2-J.10.1002/(SICI)1097-0207(19990910)46:1131:: AID-NME7263.0.CO;2-J.
Moës N., Belytschko T. Extended finite element method for cohesive crack growth. Engng Fract Mech 2002, 69(7):813-833. ISSN: 0013-7944. http://www.sciencedirect.com/science/article/B6V2R-457VNPR-3/2/6a76fcf28df8e401ac5588bb6c48b3cd.10.1016/S0013-794(01)00128-X.
Armero F., Linder C. Numerical simulation of dynamic fracture using finite elements with embedded discontinuities. Int J Fract 2009, 160:119-141. ISSN: 0376-9429. http://dx.doi.org/10.1007/s10704-009-9413-9. 10.1007/s10704-009-9413-9.
de Borst R., Gutiérrez M.A., Wells G.N., Remmers J.J.C., Askes H. Cohesive-zone models, higher-order continuum theories and reliability methods for computational failure analysis. Int J Numer Methods Engng 2004, 60(1):289-315. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.963. 10.1002/nme.963.
Mergheim J., Kuhl E., Steinmann P. A hybrid discontinuous Galerkin/interface method for the computational modelling of failure. Commun Numer Methods Engng 2004, 20(7):511-519. http://dx.doi.org/10.1002/cnm.689.
Radovitzky R., Seagraves A., Tupek M., Noels L. A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method. Comput Methods Appl Mech Engng 2011, 200(14):326-344. ISSN: 0045-7825. http://www.sciencedirect.com/science/article/pii/S0045782510002471. 10.1016/j.cma.2010.08.014.
Prechtel M., Leugering G., Steinmann P., Stingl M. Towards optimization of crack resistance of composite materials by adjustment of fiber shapes. Engng Fract Mech 2011, 78(6):944-960. ISSN: 0013-7944. http://www.sciencedirect.com/science/article/pii/S0013794411000117. 10.1016/j.engfracmech.2011.01.007.
Noels L., Radovitzky R. A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications. Int J Numer Methods Engng 2006, 68(1):64-97. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.1699. 10.1002/nme.1699.
Ten Eyck A., Lew A. Discontinuous Galerkin methods for non-linear elasticity. Int J Numer Methods Engng 2006, 67(9):1204-1243. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.1667. 10.1002/nme.1667.
Noels L., Radovitzky R. An explicit discontinuous Galerkin method for non-linear solid dynamics: formulation, parallel implementation and scalability properties. Int J Numer Methods Engng 2008, 74(9):1393-1420. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.2213. 10.1002/nme.2213.
Lew A., Ten Eyck A., Rangarajan R. Some applications of discontinuous Galerkin methods in solid mechanics, IUTAM symposium on theoretical. Comput Model Aspects Inelastic Media 2008, 227-236. http://dx.doi.org/10.1007/978-1-4020-9090-5_21. 10.1007/978-1-4020-9090-5_21.
Ten Eyck A., Celiker F., Lew A. Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: motivation, formulation, and numerical examples. Comput Methods Appl Mech Engng 2008, 197(45-48):3605-3622. ISSN: 0045-7825. http://www.sciencedirect.com/science/article/B6V29-4RY6WKK-1/2/c484a571e4c532aa6448233fe12e43a6. 10.1016/j.cma.2008.02.020.
Becker G., Noels L. A fracture framework for Euler-Bernoulli beams based on a full discontinuous Galerkin formulation/extrinsic cohesive law combination. Int J Numer Methods Engng 2011, 85(10):1227-1251. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.3008. 10.1002/nme.3008.
Becker G., Geuzaine C., Noels L. A one field full discontinuous Galerkin method for Kirchhoff-Love shells applied to fracture mechanics. Comput Methods Appl Mech Engng 2011, 200(4546):3223-3241. ISSN: 0045-7825. http://www.sciencedirect.com/science/article/pii/S0045782511002490. 10.1016/j.cma.2011.07.008.
Becker G., Noels L. A full discontinuous Galerkin formulation of non-linear Kirchhoff-Love shells: elasto-plastic finite deformations, parallel computation & fracture applications. Int J Numer Methods Engng 2013, 93(1):80-117. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.4381. 10.1002/nme.4381.
Gitman I., Askes H., Sluys L. Representative volume: existence and size determination. Engng Fract Mech 2007, 74(16):2518-2534. ISSN: 0013-7944. http://www.sciencedirect.com/science/article/pii/S0013794406004772. 10.1016/j.engfracmech.2006.12.021.
Pelissou C., Baccou J., Monerie Y., Perales F. Determination of the size of the representative volume element for random quasi-brittle composites. Int J Solids Struct 2009, 46(14-15):2842-2855. ISSN: 0020-7683. http://www.sciencedirect.com/science/article/pii/S0020768309001334. 10.1016/j.ijsolstr.2009.03.015.
Nguyen V.P., Lloberas-Valls O., Stroeven M., Sluys L.J. On the existence of representative volumes for softening quasi-brittle materials - a failure zone averaging scheme. Comput Methods Appl Mech Engng 2010, 199(45-48):3028-3038. ISSN: 0045-7825. http://www.sciencedirect.com/science/article/pii/S0045782510001854. 10.1016/j.cma.2010.06.018.
Coenen E.W.C., Kouznetsova V.G., Geers M.G.D. Novel boundary conditions for strain localization analyses in microstructural volume elements. Int J Numer Methods Engng 2012, 90(1):1-21. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.3298. 10.1002/nme.3298.
Jérusalem A. Continuum modeling of the reverse HallPetch effect in nanocrystalline metals under uniaxial tension: how many grains in a finite element model?. Philos Mag Lett 2011, 91(9):599-609. ISSN 0950-0839. 10.1080/09500839.2011.598478.
Verhoosel C.V., Remmers J.J.C., Gutiérrez M.A., de Borst R. Computational homogenization for adhesive and cohesive failure in quasi-brittle solids. Int J Numer Methods Engng 2010, 83(8-9):1155-1179. ISSN: 1097-0207. http://dx.doi.org/10.1002/nme.2854. 10.1002/nme.2854.
Nguyen V.P., Stroeven M., Sluys L.J. An enhanced continuous-discontinuous multiscale method for modeling mode-I cohesive failure in random heterogeneous quasi-brittle materials. Engng Fract Mech 2012, 79(0):78-102. ISSN: 0013-7944. http://www.sciencedirect.com/science/article/pii/S0013794411003845. 10.1016/j.engfracmech.2011.10.005.
Leugering G., Prechtel M., Steinmann P., Stingl M. A cohesive crack propagation model: mathematical theory and numerical solution. Commun Pure Appl Anal 2013, 12:1705-1729. http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=7926. 10.3934/cpaa.2013.12.1705.
Seagraves A., Radovitzky R. Advances in cohesive zone modeling of dynamic fracture. Dynamic failure of materials and structures 2010, 349-405. Springer, US, ISBN 978-1-4419-0446-1,10.1007/978-1-4419-0446-1_12. A. Shukla, G. Ravichandran, Y.D. Rajapakse (Eds.).
Geuzaine C., Remacle J.-F. Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Methods Engng 2009, 79(11):1309-1331. http://dx.doi.org/10.1002/nme.2579. 10.1002/nme.2579.
Hulbert G.M., Chung J. Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Comput Methods Appl Mech Engng 1996, 137(2):175-188. ISSN: 0045-7825. http://www.sciencedirect.com/science/article/B6V29-3WFNRDY-5/2/2ec79b924e3dd1b9231ffe40c6a40306. 10.1016/S0045-782(96)01036-5.
Karypis G., Kumar V. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 1998, 20(1):359-392.
Dooley I., Mangala S., Kale L., Geubelle P.H. Parallel simulations of dynamic fracture using extrinsic cohesive elements. J Sci Comput 2009, 39:144-165. ISSN: 0885-7474. http://link.springer.com/article/10.1007%2Fs10915-008-9254-0?LI=true. 10.1007/s10915-008-9254-0.
Byström J. Optimal design of a long and slender compressive strut. Int J Multiphys 2009, 3:235-257. http://dx.doi.org/10.1260/175095409788922275. 10.1260/175095409788922275.
Sato N., Kurauchi T., Kamigaito O. Fracture mechanism of unidirectional carbon-fibre reinforced epoxy resin composite. J Mater Sci 1986, 21:1005-1010. ISSN: 0022-2461. http://dx.doi.org/10.1007/BF01117386. 10.1007/BF01117386.
Moloney A.C., Kausch H.H., Kaiser T., Beer H.R. Parameters determining the strength and toughness of particulate filled epoxide resins. J Mater Sci 1987, 22:381-393. ISSN: 0022-2461. http://dx.doi.org/10.1007/BF01160743. 10.1007/BF01160743.
Jia Y., Yan W., Hong-Yuan L. Carbon fibre pullout under the influence of residual thermal stresses in polymer matrix composites. Comput Mater Sci 2012, 62(0):79-86. ISSN: 0927-0256. http://www.sciencedirect.com/science/article/pii/S092702561200290X. 10.1016/j.commatsci.2012.05.019.
Zhou F., Molinari J.-F.c. Stochastic fracture of ceramics under dynamic tensile loading. Int J Solids Struct 2004, 41(22-23):6573-6596. ISSN: 0020-7683. http://www.sciencedirect.com/science/article/pii/S0020768304002562. 10.1016/j.ijsolstr.2004.05.029.
Bonet J., Burton A.J. A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations. Comput Methods Appl Mech Engng 1998, 162(1-4):151-164. ISSN: 0045-7825. http://www.sciencedirect.com/science/article/pii/S0045782597003393. 10.1016/S0045-7825(97)00339-3.
Cuitiño A., Ortiz M. A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics. Engng Comput 1992, 9:437-451. ISSN: 0264-4401. http://dx.doi.org/10.1108/eb023876. 10.1108/eb023876.
