Lattice and Particle Modeling of Damage Phenomena

Living reference work entry


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.


Triangular Lattice Stiffness Tensor Diffusive Fracture Brittle Transition Markov Jump Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. M.J. Alava, P.K.V.V. Nukala, S. Zapperi, Statistical models of fracture. Adv. Phys. 55(3–4), 349–476 (2006)CrossRefGoogle Scholar
  2. A. Al-Ostaz, I. Jasiuk, Crack initiation and propagation in materials with randomly distributed holes. Eng. Fract. Mech. 58, 395–420 (1997)CrossRefGoogle Scholar
  3. 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)CrossRefMATHGoogle Scholar
  4. M.F. Ashby, D.R.H. Jones, Engineering Materials 1: An Introduction to their Properties and Applications (Pergamon Press, Oxford, 1980)Google Scholar
  5. T. Belytschko, Y.Y. Lu, L. Gu, Crack propagation by element-free Galerkin method. Eng. Fract. Mech. 51, 295–313 (1995)CrossRefGoogle Scholar
  6. 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)CrossRefMATHGoogle Scholar
  7. X. Blanc, C. LeBris, P.-L. Lions, From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164, 341–381 (2002)CrossRefMATHMathSciNetGoogle Scholar
  8. A. Bonamy, E. Bouchad, Failure of heterogeneous materials: a dynamic phase transition. Phys. Rep. 498, 1–44 (2011)CrossRefGoogle Scholar
  9. E. Bouchad, Scaling properties of cracks. J. Phys. Conden. Matter 9, 4319–4343 (1997)CrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. D. Capecchi, G. Giuseppe, P. Trovalusci, From classical to Voigt’s molecular models in elasticity. Arch. Hist. Exact Sci. 64, 525–559 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. L. De Arcangelis, S. Redner, H.J. Hermann, A random fuse model for breaking processes. J. Phys. Lett. 46, 585–590 (1985)CrossRefGoogle Scholar
  13. 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)CrossRefMathSciNetGoogle Scholar
  14. D. Greenspan, Particle Modeling (Birkhauser Publishing, Basel, 1997)CrossRefMATHGoogle Scholar
  15. D. Greenspan, New approaches and new applications for computer simulation of N-body problems. Acta Appl. Math. 71, 279–313 (2002)CrossRefMATHMathSciNetGoogle Scholar
  16. A. Hansen, E.L. Hinrichsen, S. Roux, Scale invariant disorder in fracture and related breakdown phenomena. Phys. Rev. B 43(1), 665–678 (1991)CrossRefGoogle Scholar
  17. F.W. Hehl, Y. Itin, The Cauchy relations in linear elasticity. J. Elast. 66, 185–192 (2002)CrossRefMATHMathSciNetGoogle Scholar
  18. R. Hill, Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids. 11, 357–372 (1963)Google Scholar
  19. R.W. Hockney, J.W. Eastwood, Computer Simulation Using Particles (Institute of Physics Publishing, Bristol, 1999)Google Scholar
  20. C. Huet, Universal conditions for assimilation of a heterogeneous material to an effective medium, Mech. Res. Comm. 9(3), 165–170 (1982)Google Scholar
  21. C. Huet, Application of variational concepts to size effects in elastic heterogeneous bodies, J. Mech. Phys. Solids 38, 813–841 (1990)Google Scholar
  22. 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)CrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. J.G. Kirkwood, The skeletal modes of vibration of long chain molecules. J. Chem. Phys. 7, 506–509 (1939)CrossRefGoogle Scholar
  25. D. Krajcinovic, Damage Mechanics (North-Holland, Amsterdam, 1996)Google Scholar
  26. G. Lapasset, J. Planes, Fractal dimension of fractured surfaces: a universal value? EuroPhys. Lett. 13(1), 73–79 (1990)CrossRefGoogle Scholar
  27. J. Lemaitre, J.-L. Chaboche, Mechanics of Solid Materials (Cambridge University Press, Cambridge, 1994)Google Scholar
  28. A.E.H. Love, The Mathematical Theory of Elasticity (Cambridge University Press, New York, 1934)Google Scholar
  29. M.B. Mandelbrot, A.J. Paullay, Fractal nature of fracture surfaces of metals. Nature 308(19), 721–722 (1984)CrossRefGoogle Scholar
  30. 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
  31. 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
  32. M. Ostoja-Starzewski, Lattice models in micromechanics. Appl. Mech. Rev. 55(1), 35–60 (2002a)CrossRefMathSciNetGoogle Scholar
  33. M. Ostoja-Starzewski, Microstructural randomness versus representative volume element in thermomechanics. ASME J. Appl. Mech. 69, 25–35 (2002b)CrossRefMATHGoogle Scholar
  34. M. Ostoja-Starzewski, Microstructural Randomness and Scaling in Mechanics of Materials (Chapman & Hall/CRC Modern Mechanics and Mathematics Series, Boca Raton, 2008)MATHGoogle Scholar
  35. M. Ostoja-Starzewski, J.D. Lee, Damage maps of disordered composites: a spring network approach. Int. J. Fract. 75, R51–R57 (1996)CrossRefGoogle Scholar
  36. M. Ostoja-Starzewski, G. Wang, Particle modeling of random crack patterns in epoxy plates. Probab. Eng. Mech. 21(3), 267–275 (2006)CrossRefGoogle Scholar
  37. 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)CrossRefGoogle Scholar
  38. A. Rinaldi, Bottom-up modeling of damage in heterogeneous quasi-brittle solids. Contin. Mech. Thermodyn. 25(2–4), 359–373 (2013)CrossRefGoogle Scholar
  39. A. Rinaldi, D. Krajcinovic, P. Peralta, Y.C. Lai, Lattice models of polycrystalline microstructures: a quantitative approach. Mech. Mater. 40, 17–36 (2008)CrossRefGoogle Scholar
  40. 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)CrossRefGoogle Scholar
  41. K. Sab, Principe de Hill et homogénéisation des matériaux aléatoires, C.R. Acad. Sci. Paris II. 312, 1–5 (1991)Google Scholar
  42. K. Sab, On the homogenization and the simulation of random materials. Europ. J. Mech., A Solids 11, 585–607 (1992)Google Scholar
  43. P. Trovalusci, D. Capecchi, G. Ruta, Genesis of the multiscale approach for materials with microstructure. Arch. Appl. Mech. 79(11), 981–997 (2009)CrossRefMATHGoogle Scholar
  44. O. Vinogradov, A static analog of molecular dynamics method for crystals. Int. J. Comput. Methods 3(2), 153–161 (2006)CrossRefMATHGoogle Scholar
  45. O. Vinogradov, Vacancy diffusion and irreversibility of deformations in the Lennard–Jones crystal. Comput. Mater. Sci. 45, 849–854 (2009)CrossRefGoogle Scholar
  46. O. Vinogradov, On reliability of molecular statics simulations of plasticity in crystals. Comput. Mater. Sci. 50, 771–775 (2010)CrossRefGoogle Scholar
  47. G. Wang, M. Ostoja-Starzewski, Particle modeling of dynamic fragmentation – I: theoretical considerations. Comput. Mater. Sci. 33(4), 429–442 (2005)CrossRefGoogle Scholar
  48. 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
  49. 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
  50. 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
  51. H. Yserentant, A new class of particle methods. Numer. Math. 76, 87–109 (1997)CrossRefMATHMathSciNetGoogle Scholar
  52. 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)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mechanical Science & Engineering, also Institute for Condensed Matter Theory and Beckman InstituteUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations