Review of Damage Tolerant Analysis of Laminated Composites
محورهای موضوعی : EngineeringX.L Fan 1 , Q Sun 2 , M Kikuchi 3
1 - State Key Laboratory for Mechanical Structural Strength and Vibration, School of Aerospace, Xi’an Jiao Tong University
2 - School of Aeronautics, Northwestern Polytechnical University
3 - Faculty of Science and Engineering, Tokyo University of Science
کلید واژه: Delamination, Damage tolerant analysis, Virtual crack closure-integral technique, Cohesive zone model, Progressive failure analysis,
چکیده مقاله :
With advanced composites increasing replacing traditional metallic materials, the material inhomogeneity and inherent anisotropy of such materials lead to not only new attributes for aerospace structures, but also introduce new technology to damage tolerant design and analysis. The deleterious effects of changes in material properties and initiation and growth of structural damage must be addressed. The anisotropic and brittle properties make this requirement a challenging to composite structural designers. Accurate, reliable and user-friendly computational methods, design and analysis methods are vital for more damage tolerant composite structures. Both durability and damage tolerant methodologies must address the possible changes in mechanical properties and the evolving damage accumulations that may occur during the vehicle’s service lifetime. Delamination is a major failure mode in laminated composites and has received much research attention. It may arise out of manufacturing defects, free edge effects, structural discontinuities, low and high velocity impact damage, and even bird strikes. Early pioneering work established that the reduction in strength following delamination damages placed severe limits on the design allowable for highly loaded components such as aircraft wing and fuselage structure. In the present article, we provide a state-of-art survey on damage tolerant design correlated failure behavior and analysis methodologies of laminated composites. Particular emphasis is placed on some advanced formulations and numerical approaches for efficient computational modeling and damage tolerant analysis of laminated composites.
[1] Harris C.E., Starnes J.H., Shuart M.J., 2003, Advanced Durability and Damage Tolerance Design and Analysis Methods for Composite Structures, NASA/TM-2003-212420.
[2] Schmidt H.J., Brandecker B.S., 2003, Damage tolerance design and analysis of current and future aircraft structure, in: AIAA/ICAS International Air and Space Symposium and Exposition: the next 100 years, Dayton, Ohio, Paper No. 2003-2784.
[3] Sierakowski R.L., 2005, Damage tolerance: a status report, in: 46th AIAA/ ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 2005, Austin, Texas.
[4] Ransom J.B., Glaessgen E.H., Raju I.S., Harris, C.E., 2007, Recent advances in durability and damage tolerance methodology at NASA Langley Research Center, in: Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Paper No.AIAA 2007-2377-CP, April 23-26, Honolulu, HI, USA.
[5] Tomblin J., Lacy T., Smith B., Hooper S., Vizzini A., Lee S., 1999, Review of damage tolerance for composite sandwich airframe structures, FAA, Report No. DOT/FAA/AR-99/49.
[6] Tomblin J.S., Raju K.S., Walker T., Acosta J.F., 2005, Damage tolerance of composite sandwich airframe structures-additional results, FAA, Report No. DOT/FAA/AR-05/33.
[7] Tomblin J.S., Raju K.S., Liew J., Smith, B.L., 2001, Impact damage characterization and damage tolerance of composite sandwich airframe structures-final report, FAA, Report No. DOT/FAA/AR-00/44.
[8] McGowan D.M., Ambur D.R., 1997, Damage-Tolerance Characteristics of Composite Fuselage Sandwich Structures with Thick Facesheets, NASA TM 110303.
[9] Moody R.C., Vizzini A.J., 1999, Damage tolerance of composite sandwich structures, FAA, Report No. DOT/FAA/AR-99/91.
[10] Williams J.G., 2005, NASA research in composite structure damage tolerance and composite applications in the oil industry, in: 46th AIAA Structures, Structural Dynamics & Materials Conference, Austin, Texas.
[11] Echaabi J.F., Trochu F., 1996, Review of failure criteria of fibrous composite materials, Polymer Composites 17: 786-798.
[12] Hinton M.J., Soden P.D., 1998, Predicting failure in composite laminated composites: the background to the exercise, Composite Science and Technology 58: 1001-1010.
[13] Soden P.D., Hinton M.J., Kaddour A.S., 1998, A comparison of the redictive capabilities of current failure theories for composite laminated composites, Composite Science and Technology 58: 1225-1254.
[14] Hinton M.J., Kaddour A.S., Soden P.D., 2002, A comparison of the predictive capabilities of current failure theories for composite laminated composites: Judged against experimental evidence, Composite Science and Technology 62: 1725-1797.
[15] Paris F., 2001, A Study of Failure Criteria of Fibrous Composite Materials, NASA/CR-2001-210661.
[16] Icardi U., Locatto S., Longo A., 2007, Assessment of recent theories for predicting failure of composite laminated composites, Applied Mechanics Review 60(3): 76-86.
[17] Miller A.G., Lowell D.T., Seferis J.C., 1994, The evolution of an aerospace material: influence of design, manufacturing and in-service performance, Composite Structures 27: 193-206.
[18] Bolotin V.V., 1996, Delaminations in composite structures: its origin, buckling, growth and stability, Composites: Part B 27A: 129-145.
[19] Miravete A., Jimenez M. A., 2002, Application of the finite element method to prediction of onset of delamination growth, Applied Mechanics Review 55(2): 89-106.
[20] Tay T.E., 2003, Characterization and analysis of delamination fracture in composites – an overview of developments from 1990 to 2001, Applied Mechanics Review 56(1): 1-32.
[21] Harris B., 2003, Fatigue in Composites, edited by Bryan Harris, CRC Press, New York.
[22] Reeder J.R., Crews J.H.Jr., 1990, Mixed-mode bending method for delamination testing, AIAA Journal 28: 1270-1276.
[23] Test Method D6671-01, Standard test method for mixed mode I-mode II interlaminar fracture toughness of unidirectional fiber reinforced polymer matrix composites, American Society for Testing and Materials (ASTM), West Conshohocken, PA, USA.
[24] Ramkumar R.L., Whitcomb J.D., 1985, Characterization of mode I and mixed mode delamination growth in T300/5028 graphite epoxy, ASTM STP 876: 315-335.
[25] O’Brien T.K., 1984, Mixed-mode strain energy release rate effects on the edge delamination of composites, ASTM STP 836: 125-142.
[26] Arcan M., Hashin Z., Voloshin A., 1978, A method to produce uniform plane-stress states with application to fiber-reinforced materials, Experimental Mechanics 18(4): 141-146.
[27] Bradley W.L., Cohen R.N., 1985, Matrix deformation and fracture in graphite reinforced epoxies, Delamination and Debonding of Materials, ASTM STP 876: 389-410.
[28] Russell A.I., Street K.N., 1987, The effect of matrix toughness on delamination: static and fatigue fracture under mode II shear loading of graphite fiber composites, ASTM STP 937: 275-294.
[29] Hashemi S., Kinloch A.I., Williams J.G., 1987, Interlaminar fracture of composite materials, in: 6th ICCM & 2nd ECCM Conference Proceedings, London, 3: 254-264.
[30] Whitcomb J.D., 1984, Analysis of instability-related growth of a through-width delamination, NASA TM-86301.
[31] Goyal V.K., Johnson E.R., Davila C.G., 2004, Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities, Computers and Structures 65(3-4): 289-305.
[32] Benzeggagh M.L., Kenane M., 1996, Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus, Composite Science and Technology 56(4): 439-449.
[33] Gong X.J., Benszeggagh M., 1995, Mixed mode interlaminar fracture toughness of unidirectional glass/epoxy composite, Composite Materials: Fatigue and Fracture, ASTM STP 1230, 3: 100-123.
[34] Reeder J., Kyongchan S., Chunchu P.B., Ambur D.R., 2002, Postbuckling and growth of delaminations in composite plates subjected to axial compression, in: Proceeding of the 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, Colorado, Paper No. AIAA-2002-1746.
[35] Reeder J.R., 2006, 3D mixed-mode delamination fracture criteria—an experimentalist's perspective, in: Proceedings of the 21st Annual Technical Conference of the American Society for Composites, Lancaster, PA, 17-20.
[36] Davies P., Blackman B.R.K., Brunner A.J., 1998, Standard test methods for delamination resistance of composite materials: current status, Applied Composite Materials 5: 345-364.
[37] O’Brien T.K., 1998, Interlaminar fracture toughness: the long and winding road to standardization, Composites: Part B 29B: 57-62.
[38] Brunner A.J., 2000, Experimental aspects of mode I and mode II fracture toughness testing of fiber-reinforced polymer-matrix composites, Computer methods in applied mechanics and engineering 185(2-4): 161-172.
[39] Abrate S., 1998, Impact on Composite Structures, Cambridge University Press, Cambridge, UK.
[40] Schoeppner G.A., Abrate S., 2000, Delamination threshold loads for low velocity impact on composite laminated composites, Composites: Part A: Applied Science and Manufacturing 31: 903-915.
[41] Reid S.R., Zhou G., 2000, Impact Behavior of Fiber-Reinforced Composite Materials and Structures, edited by S.R. Reid, and G. Zhou, Loughborough University, UK.
[42] Poe C.C., Illg W., 1987, Strength of a Thick Graphite/Epoxy Rocket Motor Case after Impact by a Blunt Object, NASA TM-89099.
[43] Christoforou A.P., Swanson S. R., 1988, Strength loss in composite cylinders under impact, ASME Journal of Engineering Materials and Technology 110(2): 180-184.
[44] ASTM D6264/D6264M, 2007, Standard test method for measuring the damage resistance of a fibre reinforced polymer matrix composite to a concentrated quasi-static indentation force, American Society for Testing and Materials.
[45] ASTM D 7136/D 7136M, 2007, Standard test method for measuring the damage resistance of a fibre reinforced polymer matrix composite to a drop-weight impact event, American Society for Testing and Materials.
[46] Lopes C.S., Seresta O., Coquet Y., Gurdal Z., Camanho P.P., Thuis B., 2009, Low-velocity impact damage on dispersed stacking sequence laminated composites: Part I experiments, Composites Science and Technology 69(7-8): 926-936.
[47] Belingardi G., Vadori R., 2002, Low velocity impact tests of laminated glass-fiber-epoxy matrix composite material plates, International Journal of Impact Engineering27: 213-229.
[48] Shyr T.W., Pan Y.H., 2003, Impact resistance and damage characteristics of composite laminated composites, Composite Structures 62:193-203.
[49] Gao D., Zhang X., 1994, Impact damage prediction in carbon composite structure, International Journal of Impact Engineering 16(1): 149-170.
[50] Shen Z., Zhang Z.L., Wang J., Yang S.C., Ye L., 2004, Characterization of damage resistance and damage tolerance behaviour of composite laminates, Acta Materiae Composite Sinica 21(5): 140-145.
[51] Yew C.H., Kendrick R.B.,1987, A study of damage in composite panels produced by hypervelocity impact, International Journal of Impact Engineering5: 729-738.
[52] Lamontagne C.G., Manuelpillai G.N., Taylor E.A., Tennyson R.C., 1999, Normal and oblique hypervelocity impacts on carbon fibre/PEEK composites, International Journal of Impact Engineering23(1): 519-532.
[53] Christiansen E.L., 1990, Investigation of hypervelocity impact damage to space station truss tubes, International Journal of Impact Engineering 10: 125-133.
[54] Shortliffe G.D., Tennyson R.C., 1997, Hypervelocity impact tests on composite boom structures for space robot applications, Canadian Aeronautics and Space Journal 43(3): 195-202.
[55] Taylor E.A., Herbert M.K., Gardner D.J., Thomson R., Burchell M.J., 1998, Hypervelocity impact on spacecraft carbon fibre-reinforced plastic/aluminum honeycomb, Journal of Aerospace Engineering 221: 355-366.
[56] Vaidya U.K., Nelson S., Sinn B., Mathew B., 2001, Processing and high strain rate impact response of multi-functional sandwich composites, Composite Structures 52: 429-440.
[57] HICAS: High Velocity Impact of Composite Aircraft Structures, 1998-2000, CEC DG XII BRITE-EURAM Project BE 96-4238.
[58] Jiang F.C., Vecchio K.S., 2009, Hopkinson bar loaded fracture experimental technique: a critical review of dynamic fracture toughness tests, Applied Mechanics Review 62(6): 060802-060839.
[59] Elder D.J., Thomson R.S., Nguyen M.Q., Scott M.L., 2004, Review of delamination predictive methods for low speed impact of composite laminated composites, Composite Structures 66: 677-683.
[60] Rybichi E.F., Kanninen M.F., 1977, A finite element calculation of stress intensity factors by a modified crack closure integral, Engineering Fracture Mechanics 9: 931-938.
[61] Raju I.S., Shivakumar K.N., 1988, Three-dimensional elastic analysis of a composite double cantilever beam specimen, AIAA Journal 26(12): 1493-1498.
[62] Dugdale D.S.,1960, Yielding of steel sheets containing slits, Journal of the Mechanics and Physics of Solids 8: 100-104
[63] Barenblatt G.I., 1962, The mathematical theory of equilibrium cracks in brittle fracture, Advances in Applied Mechanics 7: 55-129.
[64] Murthy P.L.N., Chamis C.C., 1986, Integrated Composite Analyzer (ICAN): Users and Programmers Manual, NASA Technical Paper 2515.
[65] Allen D.H., Groves S.E., Harris C.E., 1988, A cumulative damage model for continuous fiber composite laminated composites with matrix cracking and interply delamination, ASTM STP 972: 57-80.
[66] Krueger R., 2004, Virtual crack closure technique: history, approach, and application, Applied Mechanics Review 57(2): 109-143.
[67] Irwin G.R.,1958, Fracture I, Handbuch der Physik VI, edited by S. Flügge, Springer Verlag, Berlin, Germany, 558-590.
[68] Krueger R., Minguet P.J., 2005, Skin-Stiffener Debond Prediction Based on Computational Fracture Analysis, NASA/CR-2005-213915.
[69] ABAQUS Analysis User’s Manual - Version 6.5, Volume I, 2005, ABAQUS Inc.
[70] Mabson G., Doper B., Deobald L., 2004, User Manual for Fracture and Traction Interface Elements - Version 1.3, The Boeing Company.
[71] Williams M.L., 1963, The Fracture of Visco-Elastic Material, Wiley Interscience Publishers, Drucker.
[72] Schapery R.A., 1975, A theory of crack initiation and growth in visco-elastic media, International Journal of Fracture 11(1): 141-159.
[73] Hillerborg A., Modeer M., Petersson P.E., 1976, Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement Concrete Research 6: 773-782.
[74] Unguwarungasri T., Knauss W.G., 1987, The role of damage-softened material behavior in the fracture of composites and adhesive, International Journal of Fracture 35:221-241.
[75] Needleman A., 1987, A continuum model for void nucleation by inclusion debonding, Journal of Applied Mechanics 54: 525-531.
[76] Needleman A., 1990, An analysis of decohesion along an imperfect interface, International Journal of Fracture 42: 21-40.
[77] Shahwan K.L., Wass A. M., 1997, Non-self-similar decohesion along a finite interface of unilaterally constrained delaminations, in: Proceeding of the Royal Society of London 453: 515-550.
[78] Ortiz M., Pandolfi A., 1999, Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis, International Journal for Numerical Methods in Engineering 44(9): 1267-1282.
[79] Yu C., 2001, Three Dimensional Cohesive Modeling of Impact Damage of Composites, PhD Thesis, Pasadena, CA.
[80] Corigliano A., Mariani S., Pandolfi A., 2006, Numerical analysis of rate-dependent dynamic composite delamination, Composites Science and Technology 66: 766-775.
[81] Reddy Jr.E.D., Mello F.J., Guess T.R.,1997, Modeling the initiation and growth of delaminations in composite structures, Journal of Composite Materials 31: 812-831.
[82] Petrossian Z., Wisnom M.R., 1998, Prediction of delamination initiation and growth from discontinuous plies using interface elements, Composites: Part A 29: 503-515.
[83] Mi Y., Crisfield M.A., Davies G.A.O., Hellweg H.B.,1998, Progressive delamination using interface elements, Journal of Composite Materials 32: 1246-1273.
[84] Turon A., Camanho P.P., Costa J., Davila C.G., 2006, A damage model for the simulation of delamination in advanced composites under variable-mode loading, Mechanics of Materials 38(11): 1072-1089.
[85] Rice J.R., 1968, A path independent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics 31: 379-386.
[86] Crisfield M.A., Hellweg H.B., Davies G.A.O., 1997, Failure analysis of composite structures using interface elements, in: Proceedings of the NAFEMS Conference on Application of Finite Elements to Composite Materials, London, U.K.
[87] Camanho P.P., Dávila C.G., Ambur D.R., 2001, Numerical Simulation of Delamination Growth in Composite Materials, NASA TP-2001-211041.
[88] Pandey A.K., Reddy J.N., 1987, A post first-ply failure analysis of composite laminated composites, in: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 28th Structures, Structural Dynamics, and Materials Conference, 788-797.
[89] Ochoa O.O., Engblom J.J., 1987, Analysis of failure in composites, Composites Science and Technology 28: 87-102.
[90] Chang F.K., Chang K.Y., 1987, A progressive damage model for laminated composites containing stress concentrations, Journal of Composite Materials 21: 834-855.
[91] Chang F.K., Chang K.Y., 1987, Post-failure analysis of bolted composite joints in tension or shear-out Mode failure, Journal of Composite Materials 21: 809-833.
[92] Chang F.K., Lessard L.,1989, Modeling compression failure in laminated composite plates containing an open hole, in: Proceedings of the AIAA/ASME/ASCE/AHS/ASC 30th Structures, Structural Dynamics and Materials Conference, 979-988.
[93] Reddy Y.S., Reddy J.N., 1992, Linear and non-linear failure analysis of composite laminated composites with transverse shear, Composites Science and Technology 44: 227-255.
[94] Reddy Y.S., Reddy J.N., 1993, Three-dimensional finite element progressive failure analysis of composite laminated composites under axial extension, Journal of Composites Technology and Research 15(2): 73-87.
[95] Engelstad S.P., Reddy J.N., Knight N.F., 1992, Postbuckling response and failure prediction of graphite-epoxy plates loaded in compression, AIAA Journal 30(8):2106-2113.
[96] Coats T.W., 1996, A Progressive Damage Methodology for Residual Strength Predictions of Center-Crack Tension Composite Panels, PhD Dissertation, Old Dominion University.
[97] Irvin F.B., Ginty C.A., 1986, Progressive fracture of fiber composites, Journal of Composite Materials 20: 166-184.
[98] Huang D., Minnetyan L., 1998, Damage progression in carbon-fiber reinforced I-beams, ASCE Journal of Composites for Construction 2: 38-45.
[99] Sleight D.W., 1999, Progressive Failure Analysis Methodology for Laminated Composite Structures, NASA/TP-209107.
[100] Knight Jr.N.F., 2006, User-Defined Material Model for Progressive Failure Analysis, NASA/CR-214526.
[101] Gotsis P.K., Chamis C.C., Minnetyan L., 1995, Effect of combined loads in the durability of a stiffened adhesively bonded composite structure, in: Proceedings of the 36th AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics, and Material Conference, AIAA-95-1283-CP 2: 1083-1092.
[102] Gotsis P.K., Chamis C.C., David K., Abdi F., 2007, Progressive Fracture of Laminated Composite Stiffened Plate, NASA/TM-2007-214927.
[103] Chamis C.C., Gotsis P.K., Minnetyan L., 1996, Damage progression in bolted composite structures, in: Proceedings of the 1995 USAF Structural Integrity Program Conference ASIP II: 663-679.
[104] Coats T.W., Harris C.E, 1998, A Progressive Damage Methodology for Residual Strength Predictions of Notched Composite Panels, NASA TM-1998-207646.
[105] Gotsis P.K., Chamis C.C., Minnetyan L., 1996, Progressive Fracture of Fiber Composite Shell Structures Under Internal Pressure and Axial Loads, NASA TM-07234.
[106] Ochoa O., Reddy J.N., 1992, Finite Element Analysis of Composite Lamaintes, Kluwer Academic Publishers, Dordrecht, Netherlands.
[107] Garnich M.R., Akula M.K., 2009, Review of degradation models for progressive failure analysis of fiber reinforced polymer composites, Applied Mechanics Review 62: 010801.
[108] Nahas M.N., 1986, Survey of failure and post-failure theories of laminated fiber-reinforced composites, Journal of Composites Technology and Research 8(4): 138-153.
[109] Tsai S.W., 1984, A survey of macroscopic failure criteria for composite materials, Journal of Reinforced Plastics and Composites 3: 40-63.
[110] Icardi U., Locatto S., Longo A., 2007, Assessment of recent theories for predicting failures of composite laminated composites, Applied Mechanics Review 60(2): 76-86.
[111] Tsai S. W., 1965, Strength Characteristics of Composite Materials, NASA CR-224.
[112] Hill R., 1948, A theory of the yielding and plastic flow of anisotropic metals, in: Proceedings of the Royal Society of London, Series A, 193: 281-297.
[113] Tsai S.W., Wu E.M., 1971, A general theory of strength for anisotropic materials, Journal of Composite Materials 5: 58-80.
[114] Hashin Z., 1980, Failure criteria for unidirectional fiber composites, ASME Journal of Applied Mechanics 47(2): 329-334.
[115] Hoffman O., 1967, The brittle strength of orthotropic materials, Journal of Composite Materials 1: 200-206.
[116] Chamis C.C., 1969, Failure criteria for filamentary composites, Composite Materials: Testing and Design, ASTM STP 460: 336-351.
[117] Azzi V.D., Tsai S.W., 1965, Anisotropic Strength of Composites, Experimental Mechanics, 283-288.
[118] Hashin Z., Rotem A., 1973, A fatigue failure criterion for fiber reinforced materials, Journal of Composite Materials 7(4): 448-464.
[119] Christensen R.M., 1997, Stress based yield/failure criteria for fiber composites, International Journal of Solids and Structures 34(5): 529-543.
[120] Mayes J.S., Hansen A.C., 2001, Multicontinuum failure analysis of composite structural laminated composites, Mechanics of Composite Materials and Structures 8(4): 249-262.
[121] Murray Y., Schwer L., 1990, Implementation and verification of fiber-composite damage models, Failure Criteria and Analysis in Dynamic Response, ASME AMD, 107: 21-30.
[122] Lee J.D., 1982, Three dimensional finite element analysis of damage accumulation in composite laminated, Computers and Structures 15(3): 335-350.
[123] Engblom J.J., Ochoa O.O., 1986, Finite element formulation including interlaminar stress calculations, Computers and Structures 23(2): 241-249.
[124] Hwang W.C., Sun C.T., 1989, Failure analysis of laminated composites by using iterative three-dimensional finite element method, Computers and Structures 33(1): 41-47.
[125] Kweon J.H., 2002, Crippling analysis of composite stringers based on complete unloading method, Computers and Structures 80(27–30): 2167-2175.
[126] Huang Z.M., 2004, A bridging model prediction of the ultimate strength of composite laminated composites subjected to biaxial loads, Composite Science and Technology 64(3–4): 395-448.
[127] Sandhu R.S., 1974, Nonlinear behavior of unidirectional and angle ply laminated composites, AIAA Journal of Aircraft 13: 104-111.
[128] Kwon Y.W., Berner J.M., 1994, Analysis of matrix damage evolution in laminated composite plates, Engineering Fracture Mechanics 48(6): 811-817.
[129] Reddy Y.S.N., Moorthy C.M.D., Reddy J.N., 1995, Non-linear progressive failure analysis of laminated composite plates, International Journal of Non-Linear Mechanics 30(5): 629-649.
[130] Puck A., Schürmann H., 1998, Failure analysis of FRP laminated composites by means of physically based phenomenological models, Composite Science and Technology 58(7): 1045-1067.
[131] Joo S.G., Hong C.S., 2000, Progressive failure analysis of composite laminated composites using 3-D finite element method, Key Engineering Materials 183-187: 535-540.
[132] Puck A., Schürmann H., 2002, Failure analysis of FRP laminated composites by means of physically based phenomenological models—Part B, Composite Science and Technology 62(12–13): 1633-1662.
[133] Voyiadjis G.Z., Kattan P.I., 2005, Damage Mechanics, CRC Press, New York.
[134] Knops M., Bogle C., 2006, Gradual failure in fibre/polymer laminated composites, Composite Science and Technology 66(5): 616-625.
[135] Hahn H.T., Tsai S.W., 1983, On the behavior of composite laminated composites after initial failures, Astronautics and Aeronautics 21: 58-62.
[136] Belytschko T., Black T., 1999, Elastic crack growth in finite elements with minimal remeshing, International Journal of Fracture Mechanics 45: 601-620.
[137] Moës N., Dolbow J., Belytschko T.,1999, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46: 131-150.
[138] Dolbow J.E., 1999, An Extended Finite Element Method with Discontinuous Enrichment for Applied Mechanics, PhD dissertation, Theoretical and Applied Mechanics, Northwestern University, USA.
