Abstract
Many materials exhibit a discontinuous and inhomogeneous nature on various spatial scales that can lead to complex mechanical behaviors difficult to reproduce with continuum-based models. “Among these complex phenomena is damage evolution with nucleation, propagation, interaction, and coalescence of cracks that can result in a plethora of macroscale deformation forms.” Discontinua-based models are computational methods that represent material as an assemblage of distinct elements interacting with one another. The mesoscale methods of computational mechanics of discontinua presented in this our two essays can be, arguably, divided into three broad and intervening categories: spring network (lattice) models, discrete/distinct-element methods (DEM), and particle models. The distinct-element computational methods such as molecular dynamics and smoothed-particle hydrodynamics are outside the scope of the present overview. The objective of this chapter is to briefly survey the spring network models and their main applications. The discrete-based models have been extensively applied in the last decade to three-dimensional configurations. However, since the scope of this article is limited to two-dimensional (2D) models for practical purposes, these important advances are ignored. Likewise, one-dimensional (1D) fiber bundle models are also excluded from this account.
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References
J. Alibert, P. Seppecher, F. Dell’Isola, Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Sol. 8, 51–73 (2003)
A. Arslan, R. Ince, B.L. Karihaloo, Improved lattice model for concrete fracture. J. Eng. Mech. 128(1), 57–65 (2002)
Z.P. Bažant, B.H. Oh, Crack band theory for fracture of concrete. Mater. Struct. (RILEM) 16(93), 155–177 (1983)
Z.P. Bažant, M.R. Tabbara, M.T. Kazemi, G. Pyaudier-Cabot, Random particle model for fracture of aggregate and fiber composites. J. Eng. Mech. 116(8), 1686–1705 (1990)
J.E. Bolander Jr., S. Saito, Fracture analyses using spring networks with random geometry. Eng. Fract. Mech. 61, 569–591 (1998)
J.E. Bolander, N. Sukumar, Irregular lattice model for quasistatic crack propagation. Phys. Rev. B 71, 094106 (2005)
J.E. Bolander Jr., G.S. Hong, K. Yoshitake, Structural concrete analysis using rigid-body-spring networks. Comput. Aided Civ. Infrast. Eng. 15, 120 (2000)
G. Cusatis, Z.P. Bažant, L. Cedolin, Confinement-shear lattice model for concrete damage in tension and compression I. Theory. J. Eng. Mech. 129(12), 1439–1448 (2003)
G. Cusatis, Z.P. Bazant, L. Cedolin, Confinement-shear lattice CSL model for fracture propagation in concrete. Comput. Methods Appl. Mech. Eng. 195, 7154–7171 (2006)
M. Grah, K. Alzebdeh, P.Y. Sheng, M.D. Vaudin, K.J. Bowman, M. Ostoja-Starzewski, Brittle intergranular failure in 2D microstructures: experiments and computer simulations. Acta Mater. 44(10), 4003–4018 (1996)
P. Hou, Lattice model applied to the fracture of large strain composite. Theory Appl. Fract. Mech. 47, 233–243 (2007)
A. Hrennikoff, Solution of problems of elasticity by the framework method. J. Appl. Mech. 8, A619–A715 (1941)
A. Jagota, S.J. Bennison, Spring-network and finite element models for elasticity and fracture, in Proceedings of a Workshop on Breakdown and Non-linearity in Soft Condensed Matter, ed. by K.K. Bardhan, B.K. Chakrabarti, A. Hansen (Springer, Berlin/Heidelberg/New York, 1994), pp. 186–201
M. Jirásek, Z.P. Bažant, Microscopic fracture characteristics of random particle system. Int. J. Fract. 69, 201–228 (1995)
B.L. Karihaloo, P.F. Shao, Q.Z. Xiao, Lattice modelling of the failure of particle composites. Eng. Fract. Mech. 70, 2385–2406 (2003)
T. Kawai, New discrete models and their application to seismic response analysis of structures. Nucl. Eng. Des. 48, 207–229 (1978)
A.R. Khoei, M.H. Pourmatin, A dynamic lattice model for heterogeneous materials. Comput. Methods Civ. Eng. 2, 1–20 (2011)
J. Kozicki, J. Tejchman, Modelling of fracture process in concrete using a novel lattice model. Granul. Matter 10, 377–388 (2008)
G. Lilliu, J.G.M. van Mier, 3D lattice type fracture model for concrete. Eng. Fract. Mech. 70, 927–941 (2003)
J.X. Liu, S.C. Deng, J. Zhang, N.G. Liang, Lattice type of fracture model for concrete. Theory Appl. Fract. Mech. 48, 269–284 (2007)
J.X. Liu, S.C. Deng, N.G. Liang, Comparison of the quasi-static method and the dynamic method for simulating fracture processes in concrete. Comput. Mech. 41, 647–660 (2008)
J.X. Liu, Z.T. Chen, K.C. Li, A 2-D lattice model for simulating the failure of paper. Theory Appl. Fract. Mech. 54, 1–10 (2010)
A. Misra, C.S. Chang, Effective elastic moduli of heterogeneous granular solids. Int. J. Sol. Struct. 30(18), 2547–2566 (1993)
M. Ostoja-Starzewski, Lattice models in micromechanics. Appl. Mech. Rev. 55(1), 35–60 (2002)
M. Ostoja-Starzewski, Microstructural Randomness and Scaling in Mechanics of Materials (Taylor & Francis Group, Boca Raton, 2007)
S.L. Phoenix, I.J. Beyerlein, Statistical strength theory for fibrous composite materials, in Comprehensive Composite Materials, ed. by A. Kelly, vol. 1 (Pergamon, Oxford, 2000), pp. 559–639
A. Rinaldi, A rational model for 2D disordered lattices under uniaxial loading. Int. J. Damage Mech. 18, 233–257 (2009)
A. Rinaldi, Statistical model with two order parameters for ductile and soft fiber bundles in nanoscience and biomaterials. Phys. Rev. E 83(4–2), 046126 (2011a)
A. Rinaldi, Advances in statistical damage mechanics: new modelling strategies, in Damage Mechanics and Micromechanics of Localized Fracture Phenomena in Inelastic Solids, ed. by G. Voyiadjis. CISM Course Series, vol. 525 (Springer, Berlin/Heidelberg/New York, 2011b)
A. Rinaldi, Bottom-up modeling of damage in heterogeneous quasi-brittle solids. Continuum Mech. Thermodyn. 25(2–4), 359–373 (2013)
A. Rinaldi, Y.C. Lai, Damage theory of 2D disordered lattices: energetics and physical foundations of damage parameter. Int. J. Plast. 23, 1796–1825 (2007)
A. Rinaldi, L. Placidi, A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices. Z. Angew. Math. Mech. (ZAMM) (2013). doi:10.1002/zamm.201300028
A. Rinaldi, S. Mastilovic, D. Krajcinovic, Statistical damage mechanics – 2. Constitutive relations. J. Theory Appl. Mech. 44(3), 585–602 (2006)
A. Rinaldi, D. Krajcinovic, S. Mastilovic, Statistical damage mechanics and extreme value theory. Int. J. Damage Mech. 16(1), 57–76 (2007)
E. Schlangen, E.J. Garboczi, New method for simulating fracture using an elastically uniform random geometry lattice. Int. J. Eng. Sci. 34, 1131–1144 (1996)
E. Schlangen, E.J. Garboczi, Fracture simulations of concrete using lattice models: computational aspects. Eng. Fract. Mech. 57, 319–332 (1997)
E. Schlangen, J.G.M. Van Mier, Micromechanical analysis of fracture of concrete. Int. J. Damage Mech. 1, 435–454 (1992)
V. Topin, J.-Y. Delenne, F. Radjaï, L. Brendel, F. Mabille, Strength and failure of cemented granular matter. Eur. Phys. J. E 23, 413–429 (2007)
K. Tsubota, S. Wada, Elastic force of red blood cell membrane during tank-treading motion: consideration of the membrane’s natural state. Int. J. Mech. Sci. 52, 356–364 (2010)
K. Tsubota, S. Wada, T. Yamaguchi, Simulation study on effects of hematocrit on blood flow properties using particle method. J. Biomech. Sci. Eng. 1(1), 159–170 (2006)
J.G.M. Van Mier, Fracture Processes of Concrete (CRC Press, New York, 1997)
J.G.M. Van Mier, M.B. Nooru-Mohamed, Geometrical and structural aspects of concrete fracture. Eng. Fract. Mech. 35(4/5), 617–628 (1990)
J.G.M. Van Mier, M.R.A. van Vliet, T.K. Wang, Fracture mechanisms in particle composites: statistical aspects in lattice type analysis. Mech. Mater. 34, 705–724 (2002)
R. Vidya Sagar, Size effect in tensile fracture of concrete – a study based on lattice model applied to CT-specimen, in Proceedings of the 21th International Congress of Theoretical and Applied Mechanics ICTAM04, ed. by W. Gutkowski, T.A. Kowaleski (Springer, Berlin/Heidelberg/New York, 2004), pp. 1–3
H.-J. Vogel, H. Hoffmann, A. Leopold, K. Roth, Studies of crack dynamics in clay soil II. A physically based model for crack formation. Geoderma 125, 213–223 (2005)
G. Wang, A. Al-Ostaz, A.H.-D. Cheng, P.R. Mantena, Hybrid lattice particle modeling: theoretical considerations for a 2D elastic spring network for dynamic fracture simulations. Comput. Mater. Sci. 44, 1126–1134 (2009a)
T. Wang, T.-W. Pan, Z.W. Xing, R. Glowinski, Numerical simulation of rheology of red blood cell rouleaux in microchannels. Phys. Rev. E 79, 041916 (2009b)
A. Zubelewicz, Z.P. Bažant, Interface element modeling of fracture in aggregate composites. J. Eng. Mech. 113(11), 1619–1630 (1987)
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Rinaldi, A., Mastilovic, S. (2015). Two-Dimensional Discrete Damage Models: Lattice and Rational Models. In: Voyiadjis, G. (eds) Handbook of Damage Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5589-9_22
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