Abstract
This paper presents the lattice element models, as a class of discrete models, in which the structural solid is represented as an assembly of one-dimensional elements. This idea allows one to provide robust models for propagation of discontinuities, multiple cracks interaction or cracks coalescence. Many procedures for computation of lattice element parameters for representing linear elastic continuum have been developed, with the most often used ones discussed herein. Special attention is dedicated to presenting the ability of this kind of models to consider material disorder, heterogeneities and multi-phase materials, which makes lattice models attractive for meso- or micro-scale simulations of failure phenomena in quasi-brittle materials, such as concrete or rocks. Common difficulties encountered in material failure and a way of dealing with them in the lattice models framework are explained in detail. Namely, the size of the localized fracture process zone around the propagating crack plays a key role in failure mechanism, which is observed in various models of linear elastic fracture mechanics, multi-scale theories, homogenization techniques, finite element models, molecular dynamics. An efficient way of dealing with this kind of phenomena is by introducing the embedded strong discontinuity into lattice elements, resulting with mesh-independent computations of failure response. Moreover, mechanical lattice can be coupled with mass transfer problems, such as moisture, heat or chloride ions transfer which affect the material durability. Any close interaction with a fluid can lead to additional time dependent degradation. For illustration, the lattice approach to porous media coupling is given here as well. Thus, the lattice element models can serve for efficient simulations of material failure mechanisms, even when considering multi-physics coupling. The main peculiarities of such an approach have been presented and discussed in this work.
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References
Hrennikoff A (1941) Solution of problems of elasticity by the framework method. ASME J Appl Mech 8:A619–A715
Schlangen E, Garboczi EJ (1996) New method for simulating fracture using an elastically uniform random geometry lattice. Int J Eng Sci 34:1131–1144
Ostoja-Starzewski M (2002) Lattice models in micromechanics. Appl Mech Rev 55(1):35–60
Schlangen E, Van Mier JGM (1992) Simple lattice model for numerical simulation of fracture of concrete materials and structures. Mater Struct 25:534–542
Schlangen E, Van Mier JGM (1992) Experimental and numerical analysis of micromechanisms of fracture of cement-based composites. Cem Concr Compos 14:105–118
Benkemoun N, Hautefeuille M, Colliat JB, Ibrahimbegovic A (2010) Failure of heterogeneous materials: 3D meso-scale FE models with embedded discontinuities. Int J Numer Methods Eng 82:1671–1688
Benkemoun N, Ibrahimbegovic A, Colliat JB (2012) Anisotropic constitutive model of plasticity capable of accounting for details of meso-structure of two-phase composite material. Comput Struct 90–91:153–162
Nikolic M, Ibrahimbegovic A, Miscevic P (2015) Brittle and ductile failure of rocks: embedded discontinuity approach for representing mode I and mode II failure mechanisms. Int J Numer Methods Eng 102:1507–1526
Nikolic M, Ibrahimbegovic A (2015) Rock mechanics model capable of representing initial heterogeneities and full set of 3D failure mechanisms. Comput Methods Appl Mech Eng 290:209–227
Vassaux M, Richard B, Ragueneau F, Millard A, Delaplace A (2015) Lattice models applied to cyclic behavior description of quasi-brittle materials: advantages of implicit integration. Int J Numer Anal Meth Geomech 39:775–798
Vassaux M, Oliver-Leblond C, Richard B, Ragueneau F (2016) Beam-particle approach to model cracking and energy dissipation in concrete: identification strategy and validation. Cem Concr Compos 70:1–14
Cusatis G, Pelessone D, Mencarelli A (2011) Lattice discrete particle model (LDPM) for failure behavior of concrete. I: theory. Cem Concr Compos 33:881–890
Cusatis G, Mencarelli A, Pelessone D, Baylot J (2011) Lattice discrete particle model (LDPM) for failure behavior of concrete. I: calibration and validation. Cem Concr Compos 33:891–905
Nikolic M, Ibrahimbegovic A, Miscevic P (2016) Discrete element model for the analysis of fluid-saturated fractured poro-plastic medium based on sharp crack representation with embedded strong discontinuities. Comput Methods Appl Mech Eng 298:407–427
Nikolic M, Ibrahimbegovic A, Miscevic P (2016) Modelling of internal fluid flow in cracks with embedded strong discontinuities. In: Ibrahimbegovic A (ed) Computational methods for solids and fluids—multiscale analysis, probability aspects and model reduction. Springer, Switzerland, pp 315–341
Grassl P (2009) A lattice approach to model flow in cracked concrete. Cem Concr Compos 31:454–460
Grassl P, Fahy C, Gallipoli D, Wheeler SJ (2015) On a 2D hydro-mechanical lattice approach for modelling hydraulic fracture. J Mech Phys Solids 75:104–118
Kirkwood JG (1939) The skeletal modes of vibration of long chain molecules. J Chem Phys 7:506–509
Keating PN (1966) Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure. Phys Rev 145:637–645
Hassold GN, Srolovitz DJ (1989) Brittle fracture in materials with random defects. Phys Rev 39:9273–9281
Cusatis G, Bazant Z, Cedolin L (2006) Confinement-shear lattice CSL model for fracture propagation in concrete. Comput Methods Appl Mech Eng 195:7154–7171
Chang CS, Wang TK, Sluys LJ, van Mier JGM (2002) Fracture modeling using a micro-structural mechanics approach I. Theory and formulation. Eng Fract Mech 69:1941–1958
Karihaloo BL, Shao PF, Xiao QZ (2003) Lattice modelling of the failure of particle composites. Eng Fract Mech 70:2385–2406
Bolander J, Saito S (1998) Fracture analyses using spring networks with random geometry. Eng Fract Mech 61:569–591
Green PJ, Sibson R (1978) Computing Dirichlet tessellations in the plane. Comput J 21:168–173
Bolander J, Sukumar N (2005) Irregular lattice model for quasistatic crack propagation. Phys Rev 71:094106-1–12
Berton S, Bolander J (2006) Crack band model of fracture in irregular lattices. Comput Methods Appl Mech Eng 195:7172–7181
Grassl P, Jirasek M (2010) Meso-scale approach to modelling the fracture process zone of concrete subjected to uniaxial tension. Int J Solids Struct 47:957–968
Grassl P, Gregoire D, Solano LR, Pijaudier-Cabot G (2012) Meso-scale modelling of the size effect on the fracture process zone of concrete. Int J Solids Struct 49:1818–1827
Gregoire D, Verdon L, Lefort V, Grassl P, Saliba J, Regoin J-P, Loukili A, Pijaudier-Cabot G (2015) Mesoscale analysis of failure in quasi-brittle materials: comparison between lattice model and acoustic emission data. Int J Numer Anal Meth Geomech 39:1639–1664
Griffith A (1921) The phenomena of rupture and flow in solids. Phil Trans R Soc A 221:163–198
Irwin G (1957) Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech 24:361–364
Orowan E (1948) Fracture and strength of solids. Rep Prog Phys 12:185
Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35:379–386
Herrmann HJ, Roux S (1990) Modelization of fracture in disordered systems. In: Herrmann HJ, Roux S (eds) Statistical models for the fracture of disordered media. Elsevier, North Holland, pp 159–188
Ibrahimbegovic A (2009) Nonlinear solid mechanics: theoretical formulations and finite element solution methods. Springer, London
Bazant ZP, Lin FB (1988) Non-local yield limit degradation. Int J Numer Methods Eng 26:1805–1823
Bazant ZP, Pijaudier-Cabot G (1988) Non linear continuous damage, localization instability and convergence. J Appl Mech 55:287–293
Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys LJ (2011) Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks. Comput Methods Appl Mech Engrg 200:1220–1236
Contrafatto L, Cuomo M, Gazzo S (2016) A concrete homogenisation technique at meso-scale level accounting for damaging behaviour of cement paste and aggregates. Comput Struct 173:1–18
Toro S, Sanchez PJ, Blanco PJ, de Souza Neto EA, Huespe AE, Feijoo RA (2016) Multiscale formulation for material failure accounting for cohesive cracks at the macro and micro scales. Int J Plast 76:75–110
Oliver J, Caicedo M, Roubin E, Huespe AE, Hernandez JA (2015) Continuum approach to computational multiscale modeling of propagating fracture. Comput Methods Appl Mech Engrg 294:384–427
Fish J (2006) Bridging the scales in nano engineering and science. J Nanopart Res 8:577–594
Ibrahimbegovic A, Niekamp R, Kassiotis C, Markovic D, Matthies H (2014) Code-coupling strategy for efficient development of computer software in multiscale and multiphysics nonlinear evolution problems in computational mechanics. Adv Eng Softw 72:8–17
Rountree CL, Kalia RK, Lidorikis E, Nakano A, Van Brutzel L, Vashishta P (2002) Atomistic aspects of crack propagation in brittle materials: multimillion atom molecular dynamics simulations. Ann Rev Mater Res 32:377–400
Bonamy D, Bouchaud E (2011) Failure of heterogeneous materials: a dynamic phase transition? Phys Rep 498:1–44
Kalia RK, Nakano A, Vashishta P, Rountree CL, Van Brutzel L, Ogata S (2003) Multiresolution atomistic simulations of dynamic fracture in nanostructured ceramics and glasses. Int J Fract 121:71–79
Barenblatt GI (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129
Dugdale DS (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8:100–104
Lilliu G, van Mier JGM (2003) 3D lattice type fracture model for concrete. Eng Fract Mech 70:927941
Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150
Fries TP, Belytschko T (2006) The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int J Numer Methods Eng 68:1358–1385
Fries TP, Belytschko T (2010) The generalized/extended finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84:253–304
Jirasek M (2000) Comparative study on finite elements with embedded discontinuities. Comput Methods Appl Mech Engrg 188:307–330
Oliver J, Huespe AE, Sanchez PJ (2006) A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM. Comput Methods Appl Mech Engrg 195:4732–4752
Linder C, Armero F (2007) Finite elements with embedded strong discontinuities for the modeling of failure in solids. Int J Numer Meth Eng 72:1391–1433
Brancherie D, Ibrahimbegovic A (2009) Novel anisotropic continuum-discrete damage model capable of representing localized failure of massive structures, Part I: theoretical formulation and numerical implementation. Eng Comput 26:100–127
Dujc J, Brank B, Ibrahimbegovic A (2013) Stress-hybrid quadrilateral finite element with embedded strong discontinuity for failure analysis of plane stress solids. Int J Numer Meth Eng 94:1075–1098
Gedik YH, Nakamura H, Yamamoto Y, Kunieda M (2011) Evaluation of three-dimensional effects in short deep beams using a rigid-body-spring-model. Cem Concr Compos 33:978–991
Yamamoto Y, Nakamura H, Kuroda I, Furuya N (2014) Crack propagation analysis of reinforced concrete wall under cyclic loading using RBSM. Eur J Environ Civ Eng 18:780–792
Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Géotechnique 29:47–65
Obermayr M, Dressler K, Vrettos C, Eberhard P (2013) A bonded-particle model for cemented sand. Comput Geotech 49:299–313
Camborde F, Mariotti C, Donzé FV (2000) Numerical study of rock and concrete behaviour by discrete element modelling. Comput Geotech 27:225–247
Ergenzinger C, Seifried R, Eberhard P (2010) A discrete element model to describe failure of strong rock in uniaxial compression. Granul Matter 13:1–24
Utili S, Nova R (2008) Dem analysis of bonded granular geomaterials. Int J Numer Anal Methods Geomech 32:1997–2031
Obermayr M, Dressler K, Vrettos C, Eberhard P (2011) Prediction of draft forces in cohesionless soil with the discrete element method. J Terramechanics 48:347–358
D’Addetta GA, Kun F, Ramm E, Herrmann HJ (2001) From solids to granulates - Discrete element simulations of fracture and fragmentation processes in geomaterials. In: Vermeer PA, Diebels S, Ehlers W, Herrmann HJ, Luding S, Ramm E (eds) Continuous and discontinuous modelling of cohesive frictional materials. Springer, Berlin, pp 231–258
D’Addetta GA, Kun F, Ramm E (2002) On the application of a discrete model to the fracture process of cohesive granular materials. Granul Matter 4:77–90
Ibrahimbegovic A, Delaplace A (2003) Microscale and mesoscale discrete models for dynamic fracture of structures built of brittle material. Comput Struct 81:1255–1265
Delaplace A, Ibrahimbegovic A (2006) Performance of time-stepping schemes for discrete models in fracture dynamic analysis. Int J Numer Meth Engng 65:1527–1544
Rots JG, Invernizzi S (2004) Regularized sequentially linear saw-tooth softening model. Int J Numer Anal Meth Geomech 28:821–856
Simo JC, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12:277–296
Ortiz M, Leroy Y, Needleman A (1987) A finite element method for localized failure analysis. Comput Methods Appl Mech Eng 61:189–214
Simo J, Rifai M (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29:1595–1638
Ibrahimbegovic A, Wilson E (1991) A modified method of incompatible modes. Commun Appl Numer Methods 7:187–194
Ibrahimbegovic A, Melnyk S (2007) Embedded discontinuity finite element method for modeling of localized failure in heterogeneous materials with structured mesh: an alternative to extended finite element method. Comput Mech 40:149–155
Pham BM, Brancherie D, Davenne L, Ibrahimegovic A (2013) Stress resultant models for ultimate load design of reinforced concrete frames and multi-scale parameter estimates. Comput Mech 51:347–360
Bui NN, Ngo M, Nikolic M, Brancherie D, Ibrahimbegovic A (2014) Enriched Timoshenko beam finite element for modeling bending and shear failure of reinforced concrete frames. Comput Struct 143:9–18
Stambuk Cvitanovic N, Nikolic M, Ibrahimbegovic A (2015) Influence of specimen shape deviations on uniaxial compressive strength of limestone and similar rocks. Int J Rock Mech Min Sci 80:357–372
Bolander J, Berton S (2004) Simulation of shrinkage induced cracking in cement composite overlays. Cem Concr Compos 26:861–871
Nakamura H, Srisoros W, Yashiro R, Kunieda M (2006) Time-dependent structural analysis considering mass transfer to evaluate deterioration process of RC structures. J Adv Concr Technol 4:147–158
Wang L, Soda M, Ueda T (2008) Simulation of chloride diffusivity for cracked concrete based on RBSM and truss network model. J Adv Concr Technol 6:143–155
Wang L, Ueda T (2011) Mesoscale modelling of the chloride diffusion in cracks and cracked concrete. J Adv Concr Technol 9:241–249
Savija B, Pacheco J, Schlangen E (2013) Lattice modeling of chloride diffusion in sound and cracked concrete. Cem Concr Compos 42:30–40
Asahina D, Houseworth JE, Birkholzer JT, Rutqvist J, Bolander J (2014) Hydro-mechanical model for wetting/drying and fracture development in geomaterials. Comput Geosci 65:13–23
Damjanac B, Detournay C, Cundall PA (2016) Application of particle and lattice codes to simulation of hydraulic fracturing. Comput Part Mech 3:249–261
Biot MA (1965) Mechanics of incremental deformations. Wiley, Chichester
Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York
Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media, 2nd edn. Wiley, Chichester
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Nikolić, M., Karavelić, E., Ibrahimbegovic, A. et al. Lattice Element Models and Their Peculiarities. Arch Computat Methods Eng 25, 753–784 (2018). https://doi.org/10.1007/s11831-017-9210-y
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DOI: https://doi.org/10.1007/s11831-017-9210-y