Elasticity with Hierarchical Disarrangements: A Field Theory That Admits Slips and Separations at Multiple Submacroscopic Levels

  • Luca Deseri
  • David R. OwenEmail author


The complexity and variety of geometrical changes in physical systems at submacroscopic levels has led to various approaches to the broadening of the classical theory of finite elasticity. One approach, the field theory “elasticity with disarrangements”, employed the multiscale geometry of structured deformations in order to incorporate the effects of disarrangements such as slips and separations at a single submacroscopic level on the macroscopic response of a continuous body. This article extends that field theory by enriching the underlying geometry so as to include the effects of disarrangements at more than one submacroscopic level. The resulting field theory broadens the scope of this approach, sharpens the description of the physical nature of dissipative mechanisms that can arise, and increases the variety of systems of contact forces that can serve as boundary loadings for a body that evolves via multiscale geometrical processes.


Structured deformations Hierarchies Multiscale geometry Elasticity Disarrangements Field theory 

Mathematics Subject Classification

74A 74B20 74C15 74M25 



The partial support from the grant ERC-2013-ADG-340561-INSTABILITIES is gratefully acknowledged.


  1. 1.
    Baia, M., Matias, J., Santos, P.M.: A relaxation result in the framework of structured deformation in a bounded variation setting. Proc. R. Soc. Edinb. A 142(2), 239–271 (2012) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barroso, A.C., Matias, J., Morandotti, M., Owen, D.R.: Second-order structured deformations: relaxation, integral representation and applications. Arch. Ration. Mech. Anal. 225, 1025–1072 (2017) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barroso, A.C., Matias, J., Morandotti, M., Owen, D.R.: Explicit formulas for relaxed disarrangement densities arising from structured deformations. Math. Mech. Complex Syst. 5, 163–189 (2017) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bertoldi, K., Bigoni, D., Drugan, W.J.: Nacre: an orthotropic and bimodular elastic material. Compos. Sci. Technol. 68, 1363–1375 (2008) CrossRefGoogle Scholar
  5. 5.
    Carita, G., Matias, J., Morandotti, M., Owen, D.R.: Dimension reduction in the context of structured deformations. J. Elast. (2018). MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, Q., Pugno, N.M.: Bio-mimetic mechanisms of natural hierarchical materials: a review. J. Mech. Behav. Biomed. Mater. 19, 3–33 (2013) CrossRefGoogle Scholar
  7. 7.
    Choksi, R., Fonseca, I.: Bulk and interfacial energy densities for structured deformations of continua. Arch. Ration. Mech. Anal. 138, 37–103 (1997) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Choksi, R., Del Piero, G., Fonseca, I., Owen, D.R.: Structured deformations as energy minimizers in some models of fracture and hysteresis. Math. Mech. Solids 4, 321–356 (1999) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Daniels, H.E.: The statistical theory of the strength of bundles of threads I. Proc. R. Soc. A 183(995) (1945).
  11. 11.
    Del Piero, G.: Limit analysis and no-tension materials. Int. J. Plast. 14, 259–271 (1998) CrossRefGoogle Scholar
  12. 12.
    Del Piero, G., Owen, D.R.: Structured deformations of continua. Arch. Ration. Mech. Anal. 124, 99–155 (1993) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Del Piero, G., Owen, D.R.: Integral-gradient formulae for structured deformations. Arch. Ration. Mech. Anal. 131, 121–138 (1995) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Del Piero, G., Owen, D.R.: Structured Deformations, XXII Scuola Estiva di Fisica Matematica, Ravello, Settembre 1997. Quaderni dell’ Istituto Nazionale di Alta Matematica. Gruppo Nazionale di Fisica Matematica, Firenze (2000) zbMATHGoogle Scholar
  15. 15.
    Deseri, L., Owen, D.R.: Active slip-band separation and the energetics of slip in single crystals. Int. J. Plast. 16, 1411–1418 (2000) CrossRefGoogle Scholar
  16. 16.
    Deseri, L., Owen, D.R.: Energetics of two-level shears and hardening of single crystals. Math. Mech. Solids 7, 113–147 (2002) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Deseri, L., Owen, D.R.: Invertible structured deformations and the geometry of multiple slip in single crystals. Int. J. Plast. 18, 833–849 (2002) CrossRefGoogle Scholar
  18. 18.
    Deseri, L., Owen, D.R.: Toward a field theory for elastic bodies undergoing disarrangements. J. Elast. 70, 197–236 (2003) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Deseri, L., Owen, D.R.: Submacroscopically stable equilibria of elastic bodies undergoing dissipation and disarrangements. Math. Mech. Solids 15, 611–638 (2010) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Deseri, L., Owen, D.R.: Moving interfaces that separate loose and compact phases of elastic aggregates: a mechanism for drastic reduction or increase in macroscopic deformation. Contin. Mech. Thermodyn. 25, 311–341 (2013) ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Deseri, L., Owen, D.R.: Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with non-linear response in compression. Meccanica 49, 2907–2932 (2014) CrossRefGoogle Scholar
  22. 22.
    Deseri, L., Owen, D.R.: Stable disarrangement phases arising from expansion/contraction or from simple shearing of a model granular medium. Int. J. Eng. Sci. 96, 111–130 (2015) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Deseri, L., Owen, D.R.: Submacroscopic disarrangements induce a unique, additive and universal decomposition of continuum fluxes. J. Elast. 122, 223–230 (2016) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gibson, L.: The hierarchical structure and mechanics of plant materials. J. R. Soc. Interface 9(76), 2749–2766 (2012) CrossRefGoogle Scholar
  25. 25.
    Gu, G.X., Takaffoli, M., Buehler, M.J.: Hierarchically enhanced impact resistance of bioinspired composites. Adv. Mater. 29(28), 1700060 (2017) CrossRefGoogle Scholar
  26. 26.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, New York (1981) zbMATHGoogle Scholar
  27. 27.
    Habibi, M.K., Lu, Y.: Crack propagation in bamboo′s hierarchical cellular structure. Sci. Rep. 4, 5598 (2014). ADSCrossRefGoogle Scholar
  28. 28.
    Lakes, R.: Materials with structural hierarchy. Nature 361, 511–515 (1993) ADSCrossRefGoogle Scholar
  29. 29.
    Launey, M.E., Buehler, M.J., Ritchie, R.O.: On the mechanistic origins of toughness in bone. Annu. Rev. Mater. Res. 40, 25–53 (2010) ADSCrossRefGoogle Scholar
  30. 30.
    Matias, J., Morandotti, M., Zappale, E.: Optimal design of fractured media with prescribed macroscopic strain. J. Math. Anal. Appl. 449, 1094–1132 (2017) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Newman, W.I., Gabrielov, A.M.: Failure of hierarchical distributions of fibre bundles I. Int. J. Fract. 50(1), 1–14 (1991) Google Scholar
  32. 32.
    Oliver, K., Seddon, A., Trask, R.S.: Morphing in nature and beyond: a review of natural and synthetic shape-changing materials and mechanisms. J. Mater. Sci. 51, 10663–10689 (2016) ADSCrossRefGoogle Scholar
  33. 33.
    Owen, D.R.: Elasticity with disarrangements. In: Multiscale Modeling in Continuum Mechanics and Structured Deformations. CISM Courses and Lectures, vol. 447. Springer, Berlin (2004) Google Scholar
  34. 34.
    Owen, D.R.: Field equations for elastic constituents undergoing disarrangements and mixing. In: Šilhavý, M. (ed.) Mathematical Modelling of Bodies with Complicated Bulk and Boundary Behavior. Quaderni di Matematica, vol. 20. Aracne, Rome (2007) Google Scholar
  35. 35.
    Owen, D.R.: Elasticity with gradient disarrangments: a multiscale perspective for strain-gradient theories of elasticity and of plasticity. J. Elast. 127, 115–150 (2017) CrossRefGoogle Scholar
  36. 36.
    Owen, D.R., Paroni, R.: Second-order structured deformations. Arch. Ration. Mech. Anal. 155, 215–235 (2000) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Owen, D.R., Paroni, R.: Optimal flux densities for linear mappings and the multiscale geometry of structured deformations. Arch. Ration. Mech. Anal. 218, 1633–1652 (2015) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Ozcoban, H., Yilmaz, E.D., Schneider, G.A.: Hierarchical microcrack model for materials exemplified at enamel. Dent. Mater. 34(1), 69–77 (2018) CrossRefGoogle Scholar
  39. 39.
    Pugno, N., Bosia, F., Abdalrahman, T.: Hierarchical fiber bundle model to investigate the complex architectures of biological materials. Phys. Rev. E 85, 011903 (2012) ADSCrossRefGoogle Scholar
  40. 40.
    Šilhavý, M.: On the approximation theorem for structured deformations from BV(\(\Omega \)). Math. Mech. Complex Syst. 3, 83–100 (2015) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Šilhavý, M.: The general form of the relaxation of a purely interfacial energy for structured deformations. Math. Mech. Complex Syst. 5, 191–215 (2017) MathSciNetCrossRefGoogle Scholar
  42. 42.
    Simonini, I., Pandolfi, A.: Customized finite element modelling of the human cornea. PLoS ONE 10(6), e0130426 (2015) CrossRefGoogle Scholar
  43. 43.
    Wang, R., Gupta, H.S.: Deformation and fracture mechanisms of bone and nacre. Annu. Rev. Mater. Res. 41, 41–73 (2011) ADSCrossRefGoogle Scholar
  44. 44.
    Wegst, U.G.K., Bai, H., Saiz, E., Tomsia, A.P., Ritchie, R.O.: Bioinspired structural materials. Nat. Mater. 14, 23–36 (2015) ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.DICAM-Mechanical, Civil and Environmental EngineeringUniversity of TrentoTrentoItaly
  2. 2.MEMS-Mechanical Engineering and Materials Sciences, Swanson School of EngineeringUniversity of PittsburghPittsburghUSA
  3. 3.Department of Civil and Environmental EngineeringCarnegie Mellon UniversityPittsburghUSA
  4. 4.Department Mechanical EngineeringCarnegie Mellon UniversityPittsburghUSA
  5. 5.THMR-Department of Nanomedicine-Houston Methodist HospitalHoustonUSA
  6. 6.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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