An enhanced beam-theory model of the mixed-mode bending (MMB) test. I: Literature review and mechanical model. (English) Zbl 1293.74256

Meccanica 48, No. 2, 443-462 (2013); erratum ibid. 48, No. 2, 463 (2013).
Summary: The paper presents a mechanical model of the mixed-mode bending (MMB) test used to assess the mixed-mode interlaminar fracture toughness of composite laminates. The laminated specimen is considered as an assemblage of two sublaminates partly connected by an elastic-brittle interface. The problem is formulated through a set of 36 differential equations, accompanied by suitable boundary conditions. Solution of the problem is achieved by separately considering the two subproblems related to the symmetric and antisymmetric parts of the loads, which for symmetric specimens correspond to fracture modes I and II, respectively. Explicit expressions are determined for the interfacial stresses, internal forces, and displacements.


74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74-05 Experimental work for problems pertaining to mechanics of deformable solids
74E30 Composite and mixture properties
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
Full Text: DOI


[1] Garg AC (1988) Delamination–a damage mode in composite structures. Eng Fract Mech 29(5):557–584. doi: 10.1016/0013-7944(88)90181-6
[2] Sela N, Ishai O (1989) Interlaminar fracture toughness and toughening of laminated composite materials: a review. Composites 20(5):423–435. doi: 10.1016/0010-4361(89)90211-5
[3] Tay TE (2003) Characterization and analysis of delamination fracture in composites: an overview of developments from 1990 to 2001. Appl Mech Rev 56(1):1–31. doi: 10.1115/1.1504848
[4] Friedrich K (ed) (1989) Application of fracture mechanics to composite materials. Elsevier, Amsterdam
[5] Adams DF, Carlsson LA, Pipes RB (2003) Experimental characterization of advanced composite materials, 3rd edn. CRC Press, Boca Raton · Zbl 1067.74003
[6] Brunner AJ, Blackman BRK, Davies P (2008) A status report on delamination resistance testing of polymer–matrix composites. Eng Fract Mech 75(9):2779–2794. doi: 10.1016/j.engfracmech.2007.03.012
[7] Crews JH Jr, Reeder JR (1988) A mixed-mode bending apparatus for delamination testing. NASA TM-100662. http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19890001574_1989001574.pdf
[8] Reeder JR, Crews JH Jr (1990) Mixed-mode bending method for delamination testing. AIAA J 28(7):1270–1276. doi: 10.2514/3.25204
[9] Reeder JR, Crews JH Jr (1991) Nonlinear analysis and redesign of the mixed-mode bending delamination test. NASA TM-102777. http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19910010169_1991010169.pdf
[10] Reeder JR, Crews JH Jr (1992) Redesign of the mixed-mode bending delamination test to reduce nonlinear effects. J Compos Technol Res 14(1):12–19. doi: 10.1520/CTR10078J
[11] Reeder JR (1992) An evaluation of mixed-mode delamination failure criteria. NASA TM-104210. http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19920009705_1992009705.pdf
[12] Reeder JR (2003) Refinements to the mixed-mode bending test for delamination toughness. J Compos Technol Res 25(4):191–195. doi: 10.1520/CTR10961J
[13] ASTM (2006) Standard test method for mixed mode I–mode II interlaminar fracture toughness of unidirectional fiber reinforced polymer matrix composites. D6671/D6671M-06. American Society for Testing and Materials, West Conshohocken. doi: 10.1520/D6671_D6671M-06
[14] Allix O, Ladevèze P (1992) Interlaminar interface modelling for the prediction of delamination. Compos Struct 22(4):235–242. doi: 10.1016/0263-8223(92)90060-P
[15] Corigliano A (1993) Formulation, identification and use of interface models in the numerical analysis of composite delamination. Int J Solids Struct 30(20):2779–2811. doi: 10.1016/0020-7683(93)90154-Y · Zbl 0782.73055
[16] Bennati S, Colleluori M, Corigliano D, Valvo PS (2009) An enhanced beam-theory model of the asymmetric double cantilever beam (ADCB) test for composite laminates. Compos Sci Technol 69(11–12):1735–1745. doi: 10.1016/j.compscitech.2009.01.019
[17] Bennati S, Fisicaro P, Valvo PS (2013) An enhanced beam-theory model of the mixed-mode bending (MMB) test–Part II: Applications and results. Meccanica. doi: 10.1007/s11012-012-9682-7 · Zbl 1293.74258
[18] Chen JH, Sernow R, Schultz E, Hinrichsen G (1999) A modification of the mixed-mode bending test apparatus. Composites, Part A, Appl Sci Manuf 30(7):871–877. doi: 10.1016/S1359-835X(98)00193-6
[19] Kinloch AJ, Wang Y, Williams JG, Yayla P (1993) The mixed-mode delamination of fibre composite materials. Compos Sci Technol 47(3):225–237. doi: 10.1016/0266-3538(93)90031-B
[20] Kenane M, Benzeggagh ML (1997) Mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites under fatigue loading. Compos Sci Technol 57(5):597–605. doi: 10.1016/S0266-3538(97)00021-3
[21] Yum Y-J, You H (2001) Pure mode I, II and mixed mode interlaminar fracture of graphite/epoxy composite materials. J Reinf Plast Compos 20(9):794–808. doi: 10.1177/073168401772678571
[22] Soboyejo WO, Lu G-Y, Chengalva S, Zhang J, Kenner V (1999) A modified mixed-mode bending specimen for the interfacial fracture testing of dissimilar materials. Fatigue Fract Eng Mater Struct 22(9):799–810. doi: 10.1046/j.1460-2695.1999.00203.x
[23] Marannano GV, Pasta A (2007) An analysis of interface delamination mechanisms in orthotropic and hybrid fiber-metal composite laminates. Eng Fract Mech 74(4):612–626. doi: 10.1016/j.engfracmech.2006.09.004
[24] Suárez JC, López F, Miguel S, Pinilla P, Herreros MA (2009) Determination of the mixed-mode fracture energy of elastomeric structural adhesives: evaluation of debonding buckling in fibre-metal hybrid laminates. Fatigue Fract Eng Mater Struct 32(2):127–140. doi: 10.1111/j.1460-2695.2008.01317.x
[25] Bhashyan S, Davidson BD (1997) Evaluation of data reduction methods for the mixed mode bending test. AIAA J 35(3):546–552. doi: 10.2514/2.129
[26] Kanninen MF (1973) An augmented double cantilever beam model for studying crack propagation and arrest. Int J Fract 9(1):83–92. doi: 10.1007/BF00035958
[27] Carlsson LA, Gillespie JW, Pipes RB (1986) On the analysis and design of the end notched flexure (ENF) specimen for mode II testing. J Compos Mater 20(6):594–604. doi: 10.1177/002199838602000606
[28] Fan C, Ben Jar P-Y, Cheng J-JR (2006) Revisit the analysis of end-notched-flexure (ENF) specimen. Compos Sci Technol 66(10):1497–1498. doi: 10.1016/j.compscitech.2006.01.016
[29] Valvo PS (2008) Does shear deformability influence the mode II delamination of laminated beams? In: ECF 17–17th European Conference on Fracture, 2–5 September 2008, Brno, Czech Republic
[30] Williams JG (1989) End corrections for orthotropic DCB specimens. Compos Sci Technol 35(4):367–376. doi: 10.1016/0266-3538(89)90058-4
[31] Hashemi S, Kinloch AJ, Williams JG (1990) The analysis of interlaminar fracture in uniaxial fibre-polymer composites. Proc R Soc Lond. Ser A 427(1872):173–199. doi: 10.1098/rspa.1990.0007
[32] Wang Y, Williams JG (1992) Corrections for mode II fracture toughness specimens of composites materials. Compos Sci Technol 43(3):251–256. doi: 10.1016/0266-3538(92)90096-L
[33] Wang JL, Qiao PZ (2004) Novel beam analysis of end notched flexure specimen for mode-II fracture. Eng Fract Mech 71(2):219–231. doi: 10.1016/S0013-7944(03)00096-1
[34] de Morais AB (2011) Novel cohesive beam model for the end-notched flexure (ENF) specimen. Eng Fract Mech 78(17):3017–3029. doi: 10.1016/j.engfracmech.2011.08.019
[35] Jumel J, Budzik MK, Ben Salem N, Shanahan MER (2013) Instrumented end notched flexure–crack propagation and process zone monitoring. Part I: Modelling and analysis. Int J Solids Struct 50(2):310–319. doi: 10.1016/j.ijsolstr.2012.08.028
[36] de Morais AB, Pereira AB (2006) Mixed mode I + II interlaminar fracture of glass/epoxy multidirectional laminates–Part 1: Analysis. Compos Sci Technol 66(13):1889–1895. doi: 10.1016/j.compscitech.2006.04.006
[37] Pereira AB, de Morais AB (2006) Mixed mode I + II interlaminar fracture of glass/epoxy multidirectional laminates–Part 2: Experiments. Compos Sci Technol 66(13):1896–1902. doi: 10.1016/j.compscitech.2006.04.008
[38] de Morais AB, Pereira AB (2007) Interlaminar fracture of multidirectional glass/epoxy laminates under mixed-mode I + II loading. Mech Compos Mater 43(3):233–244. doi: 10.1007/s11029-007-0023-1
[39] Pereira AB, de Morais AB (2008) Mixed mode I + II interlaminar fracture of carbon/epoxy laminates. Composites, Part A 39(2):322–333. doi: 10.1016/j.compositesa.2007.10.013
[40] Ducept F, Davies P, Gamby D (1997) An experimental study to validate tests used to determine mixed mode failure criteria of glass/epoxy composites. Composites, Part A 28(8):719–729. doi: 10.1016/S1359-835X(97)00012-2
[41] Ducept F, Gamby D, Davies P (1999) A mixed-mode failure criterion derived from tests on symmetric and asymmetric specimens. Compos Sci Technol 59(4):609–619. doi: 10.1016/S0266-3538(98)00105-5
[42] Ducept F, Davies P, Gamby D (2000) Mixed mode failure criteria for a glass/epoxy composite and an adhesively bonded composite/composite joint. Int J Adhes Adhes 20(3):233–244. doi: 10.1016/S0143-7496(99)00048-2
[43] Benzeggagh ML, Kenane M (1996) Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos Sci Technol 56(4):439–449. doi: 10.1016/0266-3538(96)00005-X
[44] Martin RH, Hansen PL (1997) Experimental compliance calibration for the MMB specimen. In: Armanios EA (ed) Composite materials: fatigue and fracture, vol 6. ASTM STP, vol 1285, pp 305–323. doi: 10.1520/STP19934S
[45] Ozdil F, Carlsson LA (1999) Beam analysis of angle-ply laminate mixed-mode bending specimens. Compos Sci Technol 59(6):937–945. doi: 10.1016/S0266-3538(98)00128-6
[46] Kim BW, Mayer AH (2003) Influence of fiber direction and mixed-mode ratio on delamination fracture toughness of carbon/epoxy laminates. Compos Sci Technol 63(5):695–713. doi: 10.1016/S0266-3538(02)00258-0
[47] Yokozeki T, Ogasawara T, Aoki T (2008) Correction method for evaluation of interfacial fracture toughness of DCB, ENF and MMB specimens with residual thermal stresses. Compos Sci Technol 68(3–4):760–767. doi: 10.1016/j.compscitech.2007.08.025
[48] Jagan U, Chauhan PS, Parameswaran V (2008) Energy release rate for interlaminar cracks in graded laminates. Compos Sci Technol 68(6):1480–1488. doi: 10.1016/j.compscitech.2007.10.027
[49] Quispitupa A, Berggreen C, Carlsson LA (2009) On the analysis of a mixed mode bending sandwich specimen for debond fracture characterization. Eng Fract Mech 76(4):594–613. doi: 10.1016/j.engfracmech.2008.12.008
[50] Allix O, Corigliano A (1996) Modeling and simulation of crack propagation in mixed-modes interlaminar fracture specimens. Int J Fract 77(2):111–140. doi: 10.1007/BF00037233
[51] Miravete A, Jiménez MA (2002) Application of the finite element method to prediction of onset of delamination growth. Appl Mech Rev 55(2):89–105. doi: 10.1115/1.1450763
[52] Jiménez MA, Miravete A (2004) Application of the finite-element method to predict the onset of delamination growth. J Compos Mater 38(15):1309–1335. doi: 10.1177/0021998304042734
[53] Camanho PP, Dávila CG, de Moura NF (2003) Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater 37(16):1415–1438. doi: 10.1177/0021998303034505
[54] Turon A, Camanho PP, Costa J, Dávila CG (2006) A damage model for the simulation of delamination in advanced composites under variable-mode loading. Mech Mater 38(11):1072–1089. doi: 10.1016/j.mechmat.2005.10.003
[55] Tumino D, Cappello F (2007) Simulation of fatigue delamination growth in composites with different mode mixtures. J Compos Mater 41(20):2415–2441. doi: 10.1177/0021998307075439
[56] Oliveira JMQ, de Moura MFSF, Silva MAL, Morais JJL (2007) Numerical analysis of the MMB test for mixed-mode I/II wood fracture. Compos Sci Technol 67(2):1764–1771. doi: 10.1016/j.compscitech.2006.11.007
[57] de Moura MFSF, Oliveira JMQ, Morais JJL, Xavier J (2010) Mixed-mode I/II wood fracture characterization using the mixed-mode bending test. Eng Fract Mech 77(1):144–152. doi: 10.1016/j.engfracmech.2009.09.014
[58] Warrior NA, Pickett AK, Lourenço NSF (2003) Mixed-mode delamination–experimental and numerical studies. Strain 39(4):153–159. doi: 10.1046/j.1475-1305.2003.00088.x
[59] Iannucci L (2006) Dynamic delamination modelling using interface elements. Comput Struct 84(15–16):1029–1048. doi: 10.1016/j.compstruc.2006.02.002
[60] Borg R, Nilsson L, Simonsson K (2004) Simulating DCB, ENF and MMB experiments using shell elements and a cohesive zone model. Compos Sci Technol 64(2):269–278. doi: 10.1016/S0266-3538(03)00255-0
[61] Aymerich F, Lecca G, Priolo P (2007) Modelling of delamination growth in composite laminates by the virtual internal bond method. Composites, Part A 39(2):145–153. doi: 10.1016/j.compositesa.2007.11.012
[62] van der Meer FP, Sluys LJ (2009) A phantom node formulation with mixed mode cohesive law for splitting in laminates. Int J Fract 158(2):107–124. doi: 10.1007/s10704-009-9344-5 · Zbl 1400.74106
[63] Blanco N, Turon A, Costa J (2006) An exact solution for the determination of the mode mixture in the mixed-mode bending delamination test. Compos Sci Technol 66(10):1256–1258. doi: 10.1016/j.compscitech.2005.10.028
[64] Tenchev RT, Falzon BG (2007) A correction to the analytical solution of the mixed-mode bending (MMB) problem. Compos Sci Technol 67(3–4):662–668. doi: 10.1016/j.compscitech.2006.05.007
[65] Massabò R, Cox BN (2001) Unusual characteristics of mixed-mode delamination fracture in the presence of large-scale bridging. Mech Compos Mater Struct 8(1):61–80. doi: 10.1080/107594101459833
[66] Szekrényes A, Uj J (2006) Comparison of some improved solutions for mixed-mode composite delamination coupons. Compos Struct 72(3):321–329. doi: 10.1016/j.compstruct.2005.01.002
[67] Szekrényes A (2007) Improved analysis of unidirectional composite delamination specimens. Mech Mater 39(10):953–974. doi: 10.1016/j.mechmat.2007.04.002 · Zbl 1121.74468
[68] Jones RM (1999) Mechanics of composite materials, 2nd edn. Taylor & Francis, Philadelphia
[69] Vannucci P, Verchery G (2001) A special class of uncoupled and quasi-homogeneous laminates. Compos Sci Technol 61(10):1465–1473. doi: 10.1016/S0266-3538(01)00039-2
[70] Timoshenko SP (1984) Strength of materials: elementary theory and problems, vol 1. Krieger, Melbourne
[71] Cotterell B, Hbaieb K, Williams JG, Hadavinia H, Tropsa V (2006) The root rotation in double cantilever beam and peel tests. Mech Mater 38(7):571–584. doi: 10.1016/j.mechmat.2005.11.001
[72] Andrews MG, Massabò R (2007) The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers. Eng Fract Mech 74(17):2700–2720. doi: 10.1016/j.engfracmech.2007.01.013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.