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Handbook of Damage Mechanics pp 1–32Cite as

Lattice and Particle Modeling of Damage Phenomena

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Abstract

Lattice (spring network) models offer a powerful way of simulating mechanics of materials as a coarse scale cousin to molecular dynamics and, hence, an alternative to finite element models. In general, lattice nodes are endowed with masses, thus resulting in a quasiparticle model. These models, having their origins in spatial trusses and frameworks, work best when the material may naturally be represented by a system of discrete units interacting via springs or, more generally, rheological elements. This chapter begins with basic concepts and applications of spring networks, in particular the anti-plane elasticity, planar classical elasticity, and planar nonclassical elasticity. One can easily map a specific morphology of a composite material onto a particle lattice and conduct a range of parametric studies; these result in the so-called damage maps. Considered next is a generalization from statics to dynamics, with nodes truly acting as quasiparticles, application being the comminution of minerals. The chapter closes with a discussion of scaling and stochastic evolution in damage phenomena as stepping-stone to stochastic continuum damage mechanics.

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References

  • M.J. Alava, P.K.V.V. Nukala, S. Zapperi, Statistical models of fracture. Adv. Phys. 55(3–4), 349–476 (2006)

    Article  Google Scholar 

  • A. Al-Ostaz, I. Jasiuk, Crack initiation and propagation in materials with randomly distributed holes. Eng. Fract. Mech. 58, 395–420 (1997)

    Article  Google Scholar 

  • K. Alzebdeh, A. Al-Ostaz, I. Jasiuk, M. Ostoja-Starzewski, Fracture of random matrix-inclusion composites: scale effects and statistics. Int. J. Solids Struct. 35(19), 2537–2566 (1998)

    Article  MATH  Google Scholar 

  • M.F. Ashby, D.R.H. Jones, Engineering Materials 1: An Introduction to their Properties and Applications (Pergamon Press, Oxford, 1980)

    Google Scholar 

  • T. Belytschko, Y.Y. Lu, L. Gu, Crack propagation by element-free Galerkin method. Eng. Fract. Mech. 51, 295–313 (1995)

    Article  Google Scholar 

  • M.D. Bird, C.R. Steele, A solution procedure for Laplace’s equation on multiply connected circular domains. J. Appl. Mech. 59(2), 398–404 (1992)

    Article  MATH  Google Scholar 

  • X. Blanc, C. LeBris, P.-L. Lions, From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164, 341–381 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • A. Bonamy, E. Bouchad, Failure of heterogeneous materials: a dynamic phase transition. Phys. Rep. 498, 1–44 (2011)

    Article  Google Scholar 

  • E. Bouchad, Scaling properties of cracks. J. Phys. Conden. Matter 9, 4319–4343 (1997)

    Article  Google Scholar 

  • G.A. Buxton, C.M. Care, D.J. Cleaver, A lattice spring model of heterogeneous materials with plasticity. Model. Simul. Mater. Sci. Eng. 9, 485–497 (2001)

    Article  Google Scholar 

  • D. Capecchi, G. Giuseppe, P. Trovalusci, From classical to Voigt’s molecular models in elasticity. Arch. Hist. Exact Sci. 64, 525–559 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • L. De Arcangelis, S. Redner, H.J. Hermann, A random fuse model for breaking processes. J. Phys. Lett. 46, 585–590 (1985)

    Article  Google Scholar 

  • E.J. Garboczi, M.F. Thorpe, M.S. DeVries, A.R. Day, Universal conductance curve for a plane containing random holes. Phys. Rev. A 43, 6473–6480 (1991)

    Article  MathSciNet  Google Scholar 

  • D. Greenspan, Particle Modeling (Birkhauser Publishing, Basel, 1997)

    Book  MATH  Google Scholar 

  • D. Greenspan, New approaches and new applications for computer simulation of N-body problems. Acta Appl. Math. 71, 279–313 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • A. Hansen, E.L. Hinrichsen, S. Roux, Scale invariant disorder in fracture and related breakdown phenomena. Phys. Rev. B 43(1), 665–678 (1991)

    Article  Google Scholar 

  • F.W. Hehl, Y. Itin, The Cauchy relations in linear elasticity. J. Elast. 66, 185–192 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • R. Hill, Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids. 11, 357–372 (1963)

    Google Scholar 

  • R.W. Hockney, J.W. Eastwood, Computer Simulation Using Particles (Institute of Physics Publishing, Bristol, 1999)

    Google Scholar 

  • C. Huet, Universal conditions for assimilation of a heterogeneous material to an effective medium, Mech. Res. Comm. 9(3), 165–170 (1982)

    Google Scholar 

  • C. Huet, Application of variational concepts to size effects in elastic heterogeneous bodies, J. Mech. Phys. Solids 38, 813–841 (1990)

    Google Scholar 

  • B. Kahng, G. Batrouni, S. Redner, Electrical breakdown in a fuse network with random, continuously distributed breaking strengths. Phys. Rev. B 37(13), 7625–7637 (1988)

    Article  Google Scholar 

  • P.N. Keating, Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure. Phys. Rev. 145, 637–645 (1966)

    Article  Google Scholar 

  • J.G. Kirkwood, The skeletal modes of vibration of long chain molecules. J. Chem. Phys. 7, 506–509 (1939)

    Article  Google Scholar 

  • D. Krajcinovic, Damage Mechanics (North-Holland, Amsterdam, 1996)

    Google Scholar 

  • G. Lapasset, J. Planes, Fractal dimension of fractured surfaces: a universal value? EuroPhys. Lett. 13(1), 73–79 (1990)

    Article  Google Scholar 

  • J. Lemaitre, J.-L. Chaboche, Mechanics of Solid Materials (Cambridge University Press, Cambridge, 1994)

    Google Scholar 

  • A.E.H. Love, The Mathematical Theory of Elasticity (Cambridge University Press, New York, 1934)

    Google Scholar 

  • M.B. Mandelbrot, A.J. Paullay, Fractal nature of fracture surfaces of metals. Nature 308(19), 721–722 (1984)

    Article  Google Scholar 

  • J. Mandel, P. Dantu, Contribution à l'étude théorique et expérimentale du coefficient délasticité d'un milieu hétérogénes mais statisquement homog ène, Annales des Ponts et Chaussées Paris 6, 115–145 (1963)

    Google Scholar 

  • T.J. Napier-Munn, S. Morrell, R.D. Morrison, T. Kojovic, Mineral Comminution Circuits – Their Operation and Optimisation (Julius Kruttschnitt Mineral Research, The University of Queensland, Indooroopilly, 1999)

    Google Scholar 

  • M. Ostoja-Starzewski, Lattice models in micromechanics. Appl. Mech. Rev. 55(1), 35–60 (2002a)

    Article  MathSciNet  Google Scholar 

  • M. Ostoja-Starzewski, Microstructural randomness versus representative volume element in thermomechanics. ASME J. Appl. Mech. 69, 25–35 (2002b)

    Article  MATH  Google Scholar 

  • M. Ostoja-Starzewski, Microstructural Randomness and Scaling in Mechanics of Materials (Chapman & Hall/CRC Modern Mechanics and Mathematics Series, Boca Raton, 2008)

    MATH  Google Scholar 

  • M. Ostoja-Starzewski, J.D. Lee, Damage maps of disordered composites: a spring network approach. Int. J. Fract. 75, R51–R57 (1996)

    Article  Google Scholar 

  • M. Ostoja-Starzewski, G. Wang, Particle modeling of random crack patterns in epoxy plates. Probab. Eng. Mech. 21(3), 267–275 (2006)

    Article  Google Scholar 

  • A. Rinaldi, Statistical model with two order parameters for ductile and soft fiber bundles on nanoscience and biomaterials. Phys. Rev. E 83, 046126-1-10 (2011)

    Article  Google Scholar 

  • A. Rinaldi, Bottom-up modeling of damage in heterogeneous quasi-brittle solids. Contin. Mech. Thermodyn. 25(2–4), 359–373 (2013)

    Article  Google Scholar 

  • A. Rinaldi, D. Krajcinovic, P. Peralta, Y.C. Lai, Lattice models of polycrystalline microstructures: a quantitative approach. Mech. Mater. 40, 17–36 (2008)

    Article  Google Scholar 

  • K.A. Snyder, E.J. Garboczi, A.R. Day, The elastic moduli of simple two-dimensional composites: Computer simulation and eective medium theory. J. Appl. Phys.72, 5948–5955 (1992)

    Article  Google Scholar 

  • K. Sab, Principe de Hill et homogénéisation des matériaux aléatoires, C.R. Acad. Sci. Paris II. 312, 1–5 (1991)

    Google Scholar 

  • K. Sab, On the homogenization and the simulation of random materials. Europ. J. Mech., A Solids 11, 585–607 (1992)

    Google Scholar 

  • P. Trovalusci, D. Capecchi, G. Ruta, Genesis of the multiscale approach for materials with microstructure. Arch. Appl. Mech. 79(11), 981–997 (2009)

    Article  MATH  Google Scholar 

  • O. Vinogradov, A static analog of molecular dynamics method for crystals. Int. J. Comput. Methods 3(2), 153–161 (2006)

    Article  MATH  Google Scholar 

  • O. Vinogradov, Vacancy diffusion and irreversibility of deformations in the Lennard–Jones crystal. Comput. Mater. Sci. 45, 849–854 (2009)

    Article  Google Scholar 

  • O. Vinogradov, On reliability of molecular statics simulations of plasticity in crystals. Comput. Mater. Sci. 50, 771–775 (2010)

    Article  Google Scholar 

  • G. Wang, M. Ostoja-Starzewski, Particle modeling of dynamic fragmentation – I: theoretical considerations. Comput. Mater. Sci. 33(4), 429–442 (2005)

    Article  Google Scholar 

  • G. Wang, M. Ostoja-Starzewski, P.M. Radziszewski, M. Ourriban, Particle modeling of dynamic fragmentation - II: Fracture in single- and multi-phase materials. Comp. Mat. Sci. 35(2), 116–133 (2006)

    Google Scholar 

  • M.P. Wnuk, Introducing Fractals to Mechanics of Fracture. Basic Concepts in Fractal Fracture Mechanics. Handbook of Damage Mechanics, Springer, New York (2014a)

    Google Scholar 

  • M.P. Wnuk, Introducing Fractals to Mechanics of Fracture. Toughening and Instability Phenomena in Fracture. Smooth and Rough Cracks. Handbook of Damage Mechanics, Springer, New York (2014b)

    Google Scholar 

  • H. Yserentant, A new class of particle methods. Numer. Math. 76, 87–109 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • P. Zhang, Y. Huang, H. Gao, K.C. Hwang, Fracture nucleation in single-wall carbon nanotubes under tension: a continuum analysis incorporating interatomic potentials. ASME J. Appl. Mech. 69, 454–458 (2002)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Science & Engineering, also Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Mechanical Engineering Building, 1206 W. Green Street, Urbana, IL, 61801, USA

    Sohan Kale & Martin Ostoja-Starzewski

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  1. Sohan Kale
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  2. Martin Ostoja-Starzewski
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Correspondence to Sohan Kale .

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Editors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, Louisiana, USA

    George Z. Voyiadjis

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Kale, S., Ostoja-Starzewski, M. (2013). Lattice and Particle Modeling of Damage Phenomena. In: Voyiadjis, G. (eds) Handbook of Damage Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8968-9_20-1

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  • DOI: https://doi.org/10.1007/978-1-4614-8968-9_20-1

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  • Online ISBN: 978-1-4614-8968-9

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