Skip to main content
SpringerLink
Log in

On the Optimal Prediction of the Stress Field Associated with Discrete Element Models

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This work presents an optimized and convergent regularization procedure for the computation of the stress field exhibited by particle systems subject to self-equilibrated short-range interactions. A regularized definition of the stress field associated with arbitrary force networks is given, and its convergence behavior in the continuum limit is demonstrated analytically, for the first time in the literature. The analyzed systems of forces describe pair interactions between lumped masses in ‘atomistic’ models of 2D elastic bodies and 3D membrane shells based on non-conforming finite element methods. We derive such force networks from polyhedral stress functions defined over arbitrary triangulations of 2D domains. The stress function associated with an unstructured force network is projected onto a structured triangulation, producing a new force network with ordered structure. The latter is employed to formulate a ‘microscopic’ definition of the Cauchy stress of the system in the continuum limit. The convergence order of such a stress measure to its continuum limit is given, as the mesh size approaches zero. Benchmark examples illustrate the application of the proposed regularization procedure to the prediction of the stress field exhibited by a variety of 2D and 3D membrane networks.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Phillips, R., Ortiz, M.: An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method. J. Mech. Phys. Solids 47, 611–642 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Knap, J., Ortiz, M.: An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49, 1899–1923 (2001)

    MATH  Google Scholar 

  3. Belytschko, T., Xiao, S.P.: Coupling methods for continuum model with molecular model. Int. J. Multiscale Comput. Eng. 1(1), 115–126 (2003)

    Google Scholar 

  4. Liu, B., Huang, Y., Jiang, H., Qu, S., Hwang, K.C.: The atomic-scale finite element method. Comput. Methods Appl. Mech. Eng. 193, 1849–1864 (2004)

    MATH  Google Scholar 

  5. Badia, S., Parks, M., Bochev, P., Gunzburger, M., Lehoucq, R.: On atomistic-to-continuum coupling by blending. Multiscale Model. Simul. 7(1), 381–406 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Yang, Q., Biyikli, E., Zhang, P., Tian, R., To, A.C.: Atom collocation method. Comput. Methods Appl. Mech. Eng. 237–240, 67–77 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Yang, Q., Biyikli, E., To, A.C.: Multiresolution molecular mechanics: statics. Comput. Methods Appl. Mech. Eng. 258, 26–38 (2013)

    MATH  Google Scholar 

  8. Miller, R.E., Tadmor, E.B.: A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. J. Model. Simul. Mater. Sci. Eng. 17, 053001 (2009)

    Google Scholar 

  9. Tu, Z.C., Ou-Yang, Z.C.: Elastic theory of low-dimensional continua and its application in bio- and nano-structures. J. Comput. Theor. Nanosci. 5, 422–448 (2008)

    Google Scholar 

  10. Fraternali, F., Blegsen, M., Amendola, A., Daraio, C.: Multiscale mass-spring models of carbon nanotube foams. J. Mech. Phys. Solids 59(1), 89–102 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Raney, J.R., Fraternali, F., Amendola, A., Daraio, C.: Modeling and in situ identification of material parameters for layered structures based on carbon nanotube arrays. Compos. Struct. 93, 3013–3018 (2011)

    Google Scholar 

  12. Blesgen, T., Fraternali, F., Raney, J.R., Amendola, A., Daraio, C.: Continuum limits of bistable spring models of carbon nanotube arrays accounting for material damage. Mech. Res. Commun. 45, 58–63 (2012)

    Google Scholar 

  13. Trovalusci, P., Augusti, G.: A continuum model with microstructure for materials with flaws and inclusions. J. Phys. IV 8(8), 383–390 (1998)

    Google Scholar 

  14. Trovalusci, P., De Bellis, M.L., Masiani, R.: A multiscale description of particle composites: from lattice microstructures to micropolar continua. Compos. Part B Eng. 128, 164–173 (2017)

    Google Scholar 

  15. Reccia, E., De Bellis, M.L., Trovalusci, P., Masiani, R.: Sensitivity to material contrast in homogenization of random particle composites as micropolar continua. Compos. Part B Eng. 136, 39–45 (2018)

    Google Scholar 

  16. Zhou, Z., Joós, B.: Mechanisms of membrane rupture: from cracks to pores. Phys. Rev. B 56, 2997–3009 (1997)

    Google Scholar 

  17. Nelson, D., Piran, T., Weinberg, S. (eds.): Statistical Mechanics of Membranes and Surfaces, 2nd edn. World Scientific, Singapore (2004)

    Google Scholar 

  18. Muller, M., Katsov, K., Schick, M.: Biological and synthetic membranes: what can be learned from a coarse-grained description? Phys. Rep. 434, 113–176 (2006)

    Google Scholar 

  19. Dao, M., Li, J., Suresh, S.: Molecular based analysis of deformation of spectrin network and human erythrocyte. Mater. Sci. Eng. 26, 1232–1244 (2006)

    Google Scholar 

  20. Fraternali, F., Lorenz, C., Marcelli, G.: On the estimation of the curvatures and bending rigidity of membrane networks via a local maximum-entropy approach. J. Comput. Phys. 231, 528–540 (2012)

    MATH  Google Scholar 

  21. Schmidt, B., Fraternali, F.: Universal formulae for the limiting elastic energy of membrane networks. J. Mech. Phys. Solids 60, 172–180 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Fraternali, F., Marcelli, G.: A multiscale approach to the elastic moduli of biomembrane networks. Biomech. Model. Mechanobiol. 11, 1097–1108 (2012)

    Google Scholar 

  23. Kohlhoff, S., Gumbsch, P., Fischmeister, F.: Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model. Philos. Mag. A 64(4), 851–878 (1991)

    Google Scholar 

  24. Jones, R.E., Zimmerman, J.A.: The construction and application of an atomistic J-integral via Hardy estimates of continuum fields. J. Mech. Phys. Solids 58, 1318–1337 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Davini, C., Pitacco, I.: Relaxed notions of curvature and a lumped strain method for elastic plates. SIAM J. Numer. Anal. 35, 677–691 (2000)

    MathSciNet  MATH  Google Scholar 

  26. Fraternali, F., Angelillo, M., Fortunato, A.: A lumped stress method for plane elastic problems and the discrete-continuum approximation. Int. J. Solids Struct. 39, 6211–6240 (2002)

    MATH  Google Scholar 

  27. Heyman, J.: The Stone Skeleton. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  28. O‘Dwyer, D.: Funicular analysis of masonry vaults. Int. J. Solids Struct. 73, 187–197 (1999)

    MATH  Google Scholar 

  29. Kilian, A., Ochsendorf, J.: Particle-spring systems for structural form finding. IASS J. 46(2), 77–85 (2005)

    Google Scholar 

  30. Block, P., Ochsendorf, J.: Thrust network analysis: a new methodology for three-dimensional equilibrium. J. IASS 48(3), 167–173 (2007)

    Google Scholar 

  31. Block, P.: Thrust network analysis: exploring Three-dimensional equilibrium. Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, USA (2009)

  32. Fraternali, F.: A thrust network approach to the equilibrium problem of unreinforced masonry vaults via polyhedral stress functions. Mech. Res. Commun. 37, 198–204 (2010)

    MATH  Google Scholar 

  33. Fraternali, F.: A mixed lumped stress–displacement approach to the elastic problem of masonry walls. Mech. Res. Commun. 38, 176–180 (2011)

    MATH  Google Scholar 

  34. Greco, F., Leonetti, L., Luciano, R., Trovalusci, P.: Multiscale failure analysis of periodic masonry structures with traditional and fiber-reinforced mortar joints. Compos. Part B Eng. 118, 75–95 (2017)

    Google Scholar 

  35. Skelton, R.E.: Structural systems: a marriage of structural engineering ans system science. J. Struct. Control 9, 113–133 (2002)

    Google Scholar 

  36. Vera, C., Skelton, R.E., Bosscns, F., Sung, L.A.: 3-D nanomechanics of an erythrocyte junctional complex in equibiaxial and anisotropic deformations. Ann. Biomed. Eng. 33(10), 1387–1404 (2005)

    Google Scholar 

  37. Mofrad, M.R.K., Kamm, R.D. (eds.): Cytoskeletal Mechanics: Models and Measurements. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  38. Skelton, R.E., de Oliveira, M.C.: Tensegrity Systems. Springer, Berlin (2010)

    MATH  Google Scholar 

  39. Fraternali, F., Senatore, L., Daraio, C.: Solitary waves on tensegrity lattices. J. Mech. Phys. Solids 60, 1137–1144 (2012)

    Google Scholar 

  40. Zhu, H.P., Zhou, Z.Y., Yang, R.Y., Yu, A.B.: Discrete particle simulation of particulate systems: a review of mayor applications and findings. Chem. Eng. Sci. 63, 5728–5770 (2008)

    Google Scholar 

  41. Daraio, C., Ngo, D., Nesterenko, V.F., Fraternali, F.: Highly nonlinear pulse splitting and recombination in a two-dimensional granular network. Phys. Rev. E 82, 036603 (2010)

    Google Scholar 

  42. Leonard, A., Fraternali, F., Daraio, C.: Directional wave propagation in a highly nonlinear square packing of spheres. Exp. Mech. 53, 327–337 (2013)

    Google Scholar 

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

    MATH  Google Scholar 

  44. Capecchi, D., Ruta, G., Trovalusci, P.: Voigt and Poincaré’s mechanistic-energetic approaches to linear elasticity and suggestions for multiscale modelling. Arch. Appl. Mech. 81(11), 1573–1584 (2011)

    MATH  Google Scholar 

  45. Trovalusci, P.: Molecular approaches for multifield continua: origins and current developments. In: Sadowski, T., Trovalusci, P. (eds.) Multiscale Modeling of Complex Materials: Phenomenological, Theoretical and Computational Aspects. CISM (International Centre for Mechanical Sciences) Series, vol. 556, pp. 211–278. Springer, Berlin (2014)

    Google Scholar 

  46. Schlaich, J., Schäfer, K., Jennewein, M.: Toward a consistent design of structural concrete. J. Prestress. Concr. Inst. 32, 74–150 (1987)

    Google Scholar 

  47. Admal, N.C., Tadmor, E.B.: A unified interpretation of stress in molecular systems. J. Elast. 100, 63–143 (2010)

    MathSciNet  MATH  Google Scholar 

  48. Morante, S., Rossi, G., Testa, M.: The stress tensor of an atomistic system. Cent. Eur. J. Phys. 10(3), 552–559 (2012)

    Google Scholar 

  49. Alicandro, R., Cicalese, M.: A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36, 1–37 (2005)

    MathSciNet  MATH  Google Scholar 

  50. Braides, A., Gelli, M.S.: From discrete systems to continuous variational problems: an introduction. In: Braides, A., Chiadò, V. (eds.) Topics on Concentration Phenomena and Problems with Multiple Scales. Lecture Notes of the Unione Matematica Italiana, vol. 2, pp. 3–77. Springer, Berlin (2006)

    MATH  Google Scholar 

  51. Schmidt, B.: A derivation of continuum nonlinear plate theory from atomistic models. SIAM Multiscale Model. Simul. 5, 664–694 (2006)

    MathSciNet  MATH  Google Scholar 

  52. Braides, A., Solci, M., Vitali, E.: A derivation of linear elastic energies from pair-interaction atomistic systems. Netw. Heterog. Media 2, 551–567 (2007)

    MathSciNet  MATH  Google Scholar 

  53. Rodin, G.J.: Higher-order macroscopic measures. J. Mech. Phys. Solids 55, 1103–1119 (2007)

    MathSciNet  MATH  Google Scholar 

  54. Schmidt, B.: On the passage from atomic to continuum theory of thin films. Arch. Ration. Mech. Anal. 190, 1–55 (2008)

    MathSciNet  MATH  Google Scholar 

  55. Fraternali, F.: Complementary energy variational approach for plane elastic problems with singularities. Theor. Appl. Fract. Mech. 35, 129–135 (2001)

    Google Scholar 

  56. Fraternali, F.: Error estimates for a lumped stress method for plane elastic problems. Mech. Adv. Mater. Struct. 14(4), 309–320 (2007)

    Google Scholar 

  57. Espriu, D.: Triangulated random surfaces. Phys. Lett. B 194, 271–276 (1987)

    MathSciNet  Google Scholar 

  58. Bailiie, C.F., Johnston, D.A., Williams, R.D.: Nonuniversality in dynamically triangulated random surfaces with extrinsic curvature. Mod. Phys. Lett. A 5, 1671–1683 (1990)

    Google Scholar 

  59. Gompper, G., Kroll, D.M.: Random surface discretization and the renormalization of the bending rigidity. J. Phys. I Fr. 6, 1305–1320 (1996)

    Google Scholar 

  60. Discher, D.E., Boal, D.H., Boey, H.S.: Phase transitions and anisotropic responses of planar triangular nets under large deformation. Phys. Rev. E 55(4), 4762–4772 (1997)

    Google Scholar 

  61. Fraternali, F., Carpentieri, G.: On the correspondence between 2D force networks and polyhedral stress functions. Int. J. Space Struct. 29(3), 145–159 (2014)

    Google Scholar 

  62. Baratta, A., Corbi, O.: On the equilibrium and admissibility coupling in NT vaults of general shape. Int. J. Solids Struct. 47, 2276–2284 (2010)

    MATH  Google Scholar 

  63. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Boston (1984)

    MATH  Google Scholar 

  64. Ericksen, J.L.: On the Cauchy–Born rule. Math. Mech. Solids 13, 199–220 (2008)

    MathSciNet  MATH  Google Scholar 

  65. Gurtin, M.E.: The linear theory of elasticity. In: Handbuch der Physik (Encyclopedia of Physics), VIa/2, pp. 1–295 Springer-Verlag (1972)

  66. Adams, R.A.: Sobolev Spaces. Academic Press, Cambridge (1975)

    MATH  Google Scholar 

  67. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM Editions, Philadelphia (2002)

    MATH  Google Scholar 

  68. Babuska, I., Podnos, G., Rodin, G.J.: New fictitious domain methods: formulation and analysis. Math. Models Methods Appl. Sci. 15, 1575–1594 (2005)

    MathSciNet  MATH  Google Scholar 

  69. SURF, Visual Numerics, Inc.: Smooth bivariate interpolant to scattered data that is locally a quintic polynomial in two variables. IMSL\(^{\textregistered }\) Fortran Numerical Math Library, Chap. 3. http://www.vni.com (2007)

  70. Fraternali, F., Marino, A., El Sayed, T., Della Cioppa, A.: On the structural shape optimization through variational methods and evolutionary algorithms. Mech. Adv. Mater. Struct. 18, 225–243 (2011)

    Google Scholar 

  71. Novotny, A.A., Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals. Part I: theory in singularly perturbed geometrical domains. J. Optim. Theory Appl. 180(2), 341–373 (2019)

    MathSciNet  MATH  Google Scholar 

  72. Novotny, A.A., Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals. Part II: first-order method and applications. J. Optim. Theory Appl. 180(3), 683–710 (2019)

    MathSciNet  MATH  Google Scholar 

  73. Fraternali, F., Carpentieri, G., Montuori, R., Amendola, A., Benzoni, G.: On the use of mechanical metamaterials for innovative seismic isolations systems. In: Proceedings of the 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), pp. 349–358 (2015)

  74. Amendola, A., Fabbrocino, F., Feo, L., Auricchio, F., Fraternali, F.: Dependence of the mechanical properties of pentamode materials on the lattice microstructure. In: Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2016), vol. 1, pp. 2134–2150 (2016). https://doi.org/10.7712/100016.1947.6004

  75. Saleem, W., Khan, M.A., Ch, S.R.: Formulation and execution of structural topology optimization for practical design solutions. J. Optim. Theory Appl. 152(2), 517–536 (2012)

    MathSciNet  MATH  Google Scholar 

  76. Achtziger, W.: Multiple-load truss topology and sizing optimization: some properties of minimax compliance. J. Optim. Theory Appl. 98(2), 255–280 (1998)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author acknowledges financial support from the Italian Ministry of Education, University and Research (MIUR) under the ‘Departments of Excellence’ grant L.232/2016.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, University of Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano, SA, Italy

    Ada Amendola

Authors
  1. Ada Amendola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ada Amendola.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amendola, A. On the Optimal Prediction of the Stress Field Associated with Discrete Element Models. J Optim Theory Appl 187, 613–629 (2020). https://doi.org/10.1007/s10957-019-01572-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01572-1

Keywords

Mathematics Subject Classification

Search

Navigation