Variationally consistent eXtended FE model for 3D planar and curved imperfect interfaces. (English) Zbl 1286.74095

Summary: We propose an eXtended Finite Element Method convergent to the asymptotic solution of a thin interface problem for both planar and curved imperfect interfaces in three dimensions. The main advantage over standard cohesive-zone models is the bulk-mesh size independence. With respect to standard eXtended Finite Element Method, in the proposed procedure, blending and quadrature sub-domains are not required. The focus is on the evaluation of the accuracy of the proposed approach in solving three-dimensional benchmark tests. The numerical results are compared with those available from analytical solutions and spring-like interface models.


74S05 Finite element methods applied to problems in solid mechanics
74A50 Structured surfaces and interfaces, coexistent phases


Full Text: DOI


[1] Eshelby, J., The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. A: Math. Phys. Eng. Sci., 241, 376-396, (1957) · Zbl 0079.39606
[2] Eshelby, J., The elastic field outside an ellipsoidal inclusion, Proc. Roy. Soc. A: Math. Phys. Eng. Sci., 252, 561-569, (1959) · Zbl 0092.42001
[3] Mori, T.; Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall., 21, 571-574, (1973)
[4] Mura, T.; Furuhashi, R., The elastic inclusion with a sliding interface, J. Appl. Mech., 51, 308-310, (1984) · Zbl 0544.73018
[5] Hashin, Z.; Shtrikman, S., A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Sol., 11, 127-140, (1990) · Zbl 0108.36902
[6] Hashin, Z., The spherical inclusion with imperfect interface, J. Appl. Mech., 58, 444-449, (1991)
[7] Klarbring, A., Derivation of a model of adhesively bonded joints by the asymptotic expansion method, Int. J. Eng. Sci., 29, 493-512, (1991) · Zbl 0762.73069
[8] Bigoni, D.; Serkov, S.; Movchan, A.; Valentini, M., Asymptotic models of dilute composites with imperfectly bonded inclusions, Int. J. Sol. Struct., 35, 3239-3258, (1998) · Zbl 0918.73042
[9] Geymonat, G.; Krasucki, F.; Lenci, S., Mathematical analysis of a bonded joint with a soft thin adhesive, Math. Mech. Solids, 4, 201-225, (1999) · Zbl 1001.74591
[10] Benveniste, Y.; Miloh, T., The effective mechanical behavior of composite materials with imperfect contact between the constituents, Mech. Mat., 33, 309-323, (2001)
[11] Caillerie, D., The effect of a thin inclusion of high rigidity in an elastic body, Math. Methods Appl. Sci., 2, 251-270, (1980) · Zbl 0446.73014
[12] E. Acerbi, G. Buttazzo, Reinforcement problems in the calculus of variations, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986) 273-284. · Zbl 0607.73018
[13] Suquet, P., Discontinuities and plasticity, (1988), Springer Verlag Wien New York · Zbl 0667.73025
[14] Lebon, F.; Rizzoni, R., Asymptotic analysis of a thin interface: the case involving similar rigidity, Int. J. Engrg. Sci., 48, 473-486, (2010) · Zbl 1213.74249
[15] Tomar, V.; Zhai, J.; Zhou, M., Bounds for element size in a variable stiffnesss cohesive finite element model, Int. J. Numer. Methods Engrg., 61, 1894-1920, (2004) · Zbl 1075.74683
[16] Turon, A.; Dàvila, C. G.; Camanho, P.; Costa, J., An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models, Engrg. Fract. Mech., 74, 1665-1682, (2007)
[17] Blal, N.; Daridon, L.; Monerie, Y.; Pagano, S., Artificial compliance inherent to the intrinsic cohesive zone models: criteria and application to planar meshes, Int. J. Fract., 178, 71-83, (2012)
[18] Melenk, J.; Babuska, I., The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139, 289-314, (1996) · Zbl 0881.65099
[19] Benvenuti, E., A regularized XFEM framework for embedded cohesive interfaces, Comput. Methods Appl. Mech. Engrg., 197, 4367-4378, (2008) · Zbl 1194.74364
[20] Benvenuti, E.; Tralli, A.; Ventura, G., A regularized XFEM framework for embedded cohesive interfaces, Int. J. Numer. Methods Engrg., 197, 4367-4378, (2008) · Zbl 1194.74364
[21] Benvenuti, E., Mesh-size-objective XFEM for regularized continuousdiscontinuous transition, Finite Elem. Anal. Des., 47, 1326-1336, (2011)
[22] Benvenuti, E.; Tralli, A., Simulation of finite-width process zone for concrete-like materials, Comput. Mech., 50, 479-497, (2012) · Zbl 1398.74302
[23] E. Benvenuti, O. Vitarelli, A. Tralli, Delamination of FRP-reinforced concrete by means of an extended finite element formulation, Composites Part B 43 (2012) 3258-3269.
[24] Benvenuti, E.; Ventura, G.; Ponara, N., Finite element quadrature of regularized discontinuous and singular level set functions in 3D problems, Algorithms, 5, 529-544, (2012) · Zbl 1461.65027
[25] Yvonnet, J.; Quang, H. L.; He, Q.-C., An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites, Comput. Mech., 42, 119-131, (2008) · Zbl 1188.74076
[26] Löhnert, S.; Müller-Hoeppe, D.; Wriggers, P., 3D corrected XFEM approach and extension to finite deformation theory, Int. J. Numer. Methods Engrg., 86, 431-452, (2011) · Zbl 1216.74026
[27] Zhu, Q. Z.; Gu, S.; Yvonnet, J.; Shao, J.; He, Q., Three-dimensional numerical modelling by XFEM of spring-layer imperfect curved interfaces with applications to linearly elastic composite materials, Int. J. Numer. Methods Engrg., 88, 307-328, (2011) · Zbl 1242.74181
[28] Sukumar, N.; Chopp, D.; Moës, N.; Belytschko, T., Modeling holes and inclusions by level sets in the extended finite-element method, Comput. Methods Appl. Mech. Engrg., 190, 6183-6200, (2001) · Zbl 1029.74049
[29] T. Belytschko, R. Gracie, G. Ventura, A review of the extended/generalized finite element methods for material modelling, Model. Simul. Mater. Sci. Engrg. 17, doi:10.1088/0965-0393/17/4/043001. · Zbl 1195.74201
[30] Fries, T.; Belytschko, T., The extended/generalized finite element method: an overview of the method and its applications, Int. J. Numer. Methods Engrg., 84, 253-304, (2010) · Zbl 1202.74169
[31] Wells, G.; Sluys, L., A new method for modelling cohesive cracks using finite elements, Int. J. Numer. Methods Engrg., 50, 2667-2682, (2001) · Zbl 1013.74074
[32] Moës, N.; Belytschko, T., Extended finite element method for cohesive crack growth, Engrg. Fract. Mech., 69, 813-833, (2002)
[33] Fries, T., A corrected XFEM approximation without problems in blending elements, Int. J. Numer. Methods Engrg., 75, 503-532, (2007) · Zbl 1195.74173
[34] Ventura, G.; Gracie, R.; Belytschko, T., Fast integration and weight function blending in the extended finite element method, Int. J. Numer. Methods Engrg., 77, 1-29, (2009) · Zbl 1195.74201
[35] Abbas, S.; Alizada, A.; Fries, T., The XFEM for high-gradient solutions in convection-dominated problems, Int. J. Numer. Methods Engrg., 82, 1044-1072, (2010) · Zbl 1188.76224
[36] Vandoren, B.; Proft, K. D.; Simone, A.; Sluys, L., Mesoscopic modelling of masonry using weak and strong discontinuities, Comput. Methods Appl. Mech. Engrg., 255, 167-182, (2013) · Zbl 1297.74100
[37] Iarve, E.; Gurvich, M.; Mollenhauer, D.; Rose, C.; Dávila, C., Mesh-independent matrix cracking and delamination modeling in laminated composites, Int. J. Numer. Methods Engrg., 88, 749-773, (2011) · Zbl 1242.74126
[38] Iarve, E.; Swindeman, M.; Hoos, K., Mesh dependence of the discrete damage modeling in laminated composites, in: 53rd AIAA/ASME/ASCE/AHS/ASC structures 20th, structural dynamics and materials conference, 23-26 April 2012, (2012), Honolulu Hawaii
[39] Osher, S.; Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12-49, (1988) · Zbl 0659.65132
[40] Patzák, B.; Jirásek, M., Process zone resolution by extended finite elements, Engrg. Fract. Mech., 70, 957-977, (2003)
[41] Ventura, G., On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method, Int. J. Numer. Methods Engrg., 66, 761-795, (2006) · Zbl 1110.74858
[42] Chessa, J.; Wang, H.; Belytschko, T., On the construction of blending elements for local partition of unity enriched finite elements, Int. J. Numer. Methods Engrg., 57, 1015-1038, (2003) · Zbl 1035.65122
[43] Gracie, R.; Wang, H.; Belytschko, T., Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods, Int. J. Numer. Methods Engrg., 74, 1645-1669, (2008) · Zbl 1195.74175
[44] Moës, N.; Cloirec, M.; Cartraud, P.; Remacle, J., A computational approach to handle complex microsctructures geometries, Comput. Methods Appl. Mech. Engrg., 192, 3163-3177, (2003) · Zbl 1054.74056
[45] Lian, W.; Legrain, G.; Cartraud, P., Image-based computational homogenization and localization: comparison between X-FEM/levelset and voxel-based approaches, Comput. Mech., 51, 279-293, (2013) · Zbl 06149122
[46] Fries, T.; Belytschko, T., The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns, Int. J. Numer. Methods Engrg., 68, 1358-1385, (2006) · Zbl 1129.74045
[47] Moës, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. Numer. Methods Engrg., 46, 131-150, (1999) · Zbl 0955.74066
[48] Mousavi, S.; Sukumar, N., Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method, Comput. Methods Appl. Mech. Engrg., 199, 3237-3249, (2012) · Zbl 1225.74099
[49] Müller, B.; Kummer, F.; Oberlack, M., Simple multidimensional integration of discontinuous functions with application to level set methods, Int. J. Numer. Methods Engrg., 92, 637-651, (2012) · Zbl 1352.65084
[50] Sukumar, N.; Chopp, D.; Moran, B., Extended finite element method and fast marching method for three-dimensional fatigue crack propagation, Engrg. Fract. Mech., 70, 29-48, (2003)
[51] Cools, R.; Haegemans, A., Algorithm 824: CUBPACK: A package for automatic cubature; framework description, ACM Trans. Math. Software, 29, 287-296, (2003) · Zbl 1070.65517
[52] Tornberg, A.; Engquist, B., Regularization techniques for numerical approximation of PDEs with singularities, J. Scient. Comput., 19, 527-552, (2003) · Zbl 1035.65085
[53] Benveniste, Y., The effective mechanical behavior of composite materials with imperfect contact between the constituents, Mech. Mater., 4, 197-208, (1985)
[54] Hashin, Z., Thermoelastic properties of fiber composites with imperfect interface, Mech. Mater., 8, 333-348, (1990)
[55] Zhong, Z.; Meguid, S., On the elastic field of a spherical inhomogeneity with an imperfectly bonded interface, J. Elast., 46, 91-113, (1997) · Zbl 0889.73010
[56] Hashin, Z., Thin interphase/imperfect interface in elasticity with application to coated fiber composites, J. Mech. Phys. Sol., 50, 2509-2537, (2002) · Zbl 1080.74006
[57] Zhu, Q. Z., On enrichment functions in the extended finite element method, Int. J. Numer. Methods Engrg., 91, 186-217, (2012) · Zbl 1246.74066
[58] Kolmogorov, A.; Fomin, S., Introductory real analysis, (1975), Dover
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