Abstract
An augmented finite element method (“A-FEM”) is presented that is a variant of the method of Hansbo and Hansbo (Comput Methods Appl Mech Eng, 193: 3523–3540, 2004), which can fully account for arbitrary discontinuities that traverse the interior of elements. Like the method of Hansbo and Hansbo, the A-FEM preserves elemental locality, because element augmentation is implemented within single elements and involves nodal information from the modified element only. The A-FEM offers the additional convenience that the augmentation is implemented via separable mathematical elements that employ standard finite element nodal interpolation only. Thus, the formulation is fully compatible with standard commercial finite element packages and can be incorporated as a user element without access to the source code. Because possible discontinuities include both elastic heterogeneity and cracks, the A-FEM is ideally suited to modeling damage evolution in structural or biological materials with complex morphology. Elements of a multi-scale approach to analyzing damage mechanisms in laminated or woven textile composites are used to validate the A-FEM and illustrate its possible uses. Key capabilities of the formulation include the use of meshes that need not conform to the surfaces of heterogeneities; the ability to apply the augmented element recursively, enabling modeling of multiple discontinuities arising on different, possibly intersecting surfaces within an element; and the ease with which cohesive zone models of nonlinear fracture can be incorporated.
Similar content being viewed by others
References
Areias PMA, Belytschko T (2006) A comment on the article: a finite element method for simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 195: 1275–1276. doi:10.1016/j.cma.2005.03.006
Ashby MF (1992) Physical modelling or materials problems. Mater Sci Technol 8: 102–111
Barber JR (2002) Elasticity. Kluwer Academic Press, New York
Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45: 601–620. doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S
Belytschko T, Moes N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50: 993–1013. doi:10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M
Camanho PP, Dávila CG et al (2003) Numerical simulation of mixed-mode progressive delamination in composite materials. J Compos Mater 37: 1415–1438. doi:10.1177/0021998303034505
Carpinteri A, Ferro G (2003) Fracture assessment in concrete structures. In: Milne I, Ritchie RO, Karihaloo B (eds) Concrete structure integrity, vol 7. Elsevier Science, Amsterdam
Cavalli MN, Thouless MD (2001) The effect of damage nucleation on the toughness of an adhesive joint. J Adhes 76: 75–92. doi:10.1080/00218460108029618
Chang KY, Liu S et al (1991) Damage tolerance of laminated composites containing an open hole and subjected to tensile loadings. J Compos Mater 25: 274–301
Chessa J, Wang H et al (2003) On the construction of blending elements for local partitioning if unity enriched finite elements. Int J Numer Methods Eng 57: 1015–1038. doi:10.1002/nme.777
Cox BN, Dadkhah MS (1995) The macroscopic elasticity of 3D woven composites. J Compos Mater 29: 785–819
Cox BN, Flanagan G (1997) Handbook of analytical methods for textile composites. NASA Contractor Report 4750, NASA Langley Research Center, Hampton, Virginia
Cox BN, Yang QD (2006) In quest of virtual tests for structural composites. Science 314: 1102–1107. doi:10.1126/science.1131624
Cox BN, Dadkhah MS, Inman RV, Morris WL, Zupon J (1992) Mechanisms of compressive failure in 3D composites. Acta Metall Mater 40: 3285–3298. doi:10.1016/0956-7151(92)90042-D
Cox BN, Dadkhah MS, Morris WL, Flintoff JG (1994) Failure mechanisms of 3D woven composites in tension, compression, and bending. Acta Metall Mater 42: 3967–3984. doi:10.1016/0956-7151(94)90174-0
Cox BN, Dadkhah MS, Morris WL (1996) On the tensile failure of 3D woven composites. Compos A 27: 447–458. doi:10.1016/1359-835X(95)00053-5
Cox BN, Spearing SM, Mumm DR (2008). In: Camanho PP, Dávila CG, Pinho ST, Remmers JJC (eds) Mechanical Response of Composites. Springer Science and Business Media, Dordrecht, pp. 57–75
De Borst R (2003) Numerical aspects of cohesive-zone models. Eng Fract Mech 70: 1743–1752. doi:10.1016/S0013-7944(03)00122-X
Elices M, Guinea GV et al (2002) The cohesive zone model: advantages, limitations and challenges. Eng Fract Mech 69: 137–163. doi:10.1016/S0013-7944(01)00083-2
Fries TP, Belytschko T (2006) The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int J Numer Methods Eng 68: 1358–1385. doi:10.1002/nme.1761
Gonzalez C, Lorca JL (2006) Multiscale modeling of fracture in fiber-reinforced composites. Acta Mater 54: 4171–4181. doi:10.1016/j.actamat.2006.05.007
Gravouil A, Moes N et al (2002) Non-planar 3D crack growth by the extended finite element and level sets—part II: level set update. Int J Numer Methods Eng 53: 2569–2586. doi:10.1002/nme.430
Hallett S, Wisnom MR (2006) Numerical investigation of progressive damage and the effect of layup in notched tensile tests. J Compos Mater 40: 1229–1245. doi:10.1177/0021998305057432
Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Methods Appl Mech Eng 191: 5537–5552. doi:10.1016/S0045-7825(02)00524-8
Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193: 3523–3540. doi:10.1016/j.cma.2003.12.041
Iarve EV, Mollenhauer D et al (2005) Theoretical and experimental investigation of stress redistribution in open-hole composite laminates due to damage accumulation. Compos Part A 36: 163–171
Jager P, Steinmann P et al (2008) On local tracking algorithms for the simulation of three-dimensional discontinuities. Comput Mech 42: 395–406. doi:10.1007/s00466-008-0249-3
Karihaloo BL, Xiao QZ (2003) Modeling of stationary and growing cracks in FE framework without remeshing: a state-of-art review. Comput Struc 81: 119–129. doi:10.1016/S0045-7949(02)00431-5
Ko FK (1989) Preform architecture for ceramic-matrix composites. Ceram Bull 68: 401–414
McCartney LN (2003) Physically based damage models for laminated composites. J Mater Des Appl 217: 163–199
Melenk JM, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139: 289–314. doi:10.1016/S0045-7825(96)01087-0
Mergheim J, Huhl E et al (2005) A finite element method for the computational modeling of cohesive cracks. Int J Numer Methods Eng 63: 276–289. doi:10.1002/nme.1286
Mergheim J, Huhl E et al (2007) Towards the algorithmic treatment of 3D strong discontinuities. Commun Numer Methods Eng 23: 97–108. doi:10.1002/cnm.885
Moes N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69: 813–833. doi:10.1016/S0013-7944(01)00128-X
Moes N, Dolbow J et al (1999) Finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150. doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
Moes N, Gravouil A et al (2002) Non-planar 3D crack growth by the extended finite element and level sets—part I: mechanical model. Int J Numer Methods Eng 53: 2549–2568. doi:10.1002/nme.429
Pastore CM, Bogdanovich AE, Gowayed YA (1993) Applications of a meso-volume-based analysis for textile composite structures. Compos Eng 3: 181–194. doi:10.1016/0961-9526(93)90041-H
Remmers JC (2006) Discontinuities in materials and structures—a unifying computational approach. Ph.D Thesis, Delft University
Shahwan KW, Waas AM (1997) Non-self-similar decohesion along a finite interface of unilaterally constrained delaminations. Proc R Soc Lond A 453: 515–555. doi:10.1098/rspa.1997.0029
Talreja R (2006) Mulitscale modeling in damage mechanics of composite materials. J Mater Sci 41: 6800–6812. doi:10.1007/s10853-006-0210-9
Tay TE (2003) Characterization and analysis of delamination fracture in composites: an overiew of developments from 1990 to 2001. Appl Mech Rev 56: 1–32. doi:10.1115/1.1504848
Tvergaard V, Hutchinson JW (1996) On the toughness of ductile adhesive joints. J Mech Phys Solids 44: 789–800
Van de Meer FP, Sluys LJ (2008) Continuum models for the analysis of progressive failure in composite laminates. J Compos Mater (in press)
Wang JS, Suo Z (1990) Experimental determination of interfacial toughness using Brazil-nut-sandwich. Acta Metall 38: 1279–1290. doi:10.1016/0956-7151(90)90200-Z
Wells GN, Sluys LJ (2001) A new method for modeling cohesive cracks using finite elements. Int J Numer Methods Eng 50: 2667–2682. doi:10.1002/nme.143
Wells GN, De Borst R et al (2002) A consistent geometrically non-linear approach for delamination. Int J Numer Methods Eng 54: 1333–1355. doi:10.1002/nme.462
Wisnom MR, Chang F-K (2000) Modeling of splitting and delamination in notched cross-ply laminates. Compos Sci Technol 60: 2849–2856. doi:10.1016/S0266-3538(00)00170-6
Xie D, Amit G et al (2006) Discrete cohesive zone model to simulate static fracture in 2D tri-axially braided carbon fiber composites. J Compos Mater 40: 2025–2046. doi:10.1177/0021998306061320
Yang QD, Thouless MD (2001) Mixed mode fracture of plastically-deforming adhesive joints. Int J Fract 110: 175–187. doi:10.1023/A:1010869706996
Yang QD, Cox BN (2003) Spatially averaged local strains in textile composites via the binary model formulation. J Eng Mater Technol 125: 418–425. doi:10.1115/1.1605117
Yang QD, Cox BN (2005) Cohesive zone models for damage evolution in laminated composites. Int J Fract 133(2): 107–137. doi:10.1007/s10704-005-4729-6
Yang QD, Thouless MD et al (1999) Numerical simulations of adhesively-bonded beams failing with extensive plastic deformation. J Mech Phys Solids 47: 1337–1353. doi:10.1016/S0022-5096(98)00101-X
Yang QD, Rugg KL et al (2005) Evaluation of macroscopic and local strains in a 3D woven C/SiC composite. J Am Ceram Soc 88: 719–725. doi:10.1111/j.1551-2916.2005.00156.x
Zienkiewicz OC, Taylor RL (2005) The finite element method. Elsevier, Oxford
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ling, D., Yang, Q. & Cox, B. An augmented finite element method for modeling arbitrary discontinuities in composite materials. Int J Fract 156, 53–73 (2009). https://doi.org/10.1007/s10704-009-9347-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-009-9347-2