On the wave dispersion in microstructured solids


In this paper, elastic wave propagation in a one-dimensional micromorphic medium characterized by two internal variables is investigated. The evolution equations are deduced following two different approaches, namely using: (i) the balance of linear momentum and the Clausius–Duhem inequality, and (ii) an assumed Lagrangian functional (including a gyroscopic coupling) together with a variational principle. The dispersion relation is obtained and the possibility of the emerging band gaps is shown in such microstructured materials. Some numerical simulations are also performed in order to highlight the dispersive nature of the material under study.


  1. 1.

    Alessandroni, S., dell’Isola, F., Porfiri, M.: A revival of electric analogs for vibrating mechanical systems aimed to their efficient control by PZT actuators. Int. J. Solids Struct. 39(20), 5295–5324 (2002)

    MATH  Article  Google Scholar 

  2. 2.

    Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Altenbach, H., Eremeyev, V.A.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Complex Syst. 3(3), 273–283 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Andrianov, I.V., Bolshakov, V.I., Danishevs’kyy, V.V., Weichert, D.: Higher order asymptotic homogenization and wave propagation in periodic composite materials. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 464(2093), 1181–1201 (2008)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Arnol’d, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin (2013)

    Google Scholar 

  6. 6.

    Askes, H., Metrikine, A.V., Pichugin, A.V., Bennett, T.: Four simplified gradient elasticity models for the simulation of dispersive wave propagation. Philos. Mag. 88(28–29), 3415–3443 (2008)

    ADS  Article  Google Scholar 

  7. 7.

    Auffray, N., dell’Isola, F., Eremeyev, V.A., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Berezovski, A., Engelbrecht, J., Berezovski, M.: Waves in microstructured solids: a unified viewpoint of modeling. Acta Mech. 220(1–4), 349–363 (2011)

    MATH  Article  Google Scholar 

  9. 9.

    Berezovski, A., Engelbrecht, J., Maugin, G.A.: Generalized thermomechanics with dual internal variables. Arch. Appl. Mech. 81(2), 229–240 (2011)

    ADS  MATH  Article  Google Scholar 

  10. 10.

    Bertram, A., Glüge, R.: Gradient materials with internal constraints. Math. Mech. Complex Syst. 4(1), 1–15 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Biswas, R., Poh, L.H.: A micromorphic computational homogenization framework for heterogeneous materials. J. Mech. Phys. Solids 102, 187–208 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Bloch, A.: XXXVIII: A new approach to the dynamics of systems with gyroscopic coupling terms. Lond. Edinb. Dublin Philos. Mag. J. Sci. 35(244), 315–334 (1944)

    MATH  Article  Google Scholar 

  13. 13.

    Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford University Press, Oxford (1954)

    Google Scholar 

  14. 14.

    Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Brillouin, L.: Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices. Dover Publications, Mineola (1946)

    Google Scholar 

  16. 16.

    Capriz, G.: Continua with Microstructure. Springer, Berlin (1989)

    Google Scholar 

  17. 17.

    Chen, W., Fish, J.: A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales. Trans. ASME J. Appl. Mech. 68(2), 153–161 (2001)

    ADS  MATH  Article  Google Scholar 

  18. 18.

    Crandall, S.H.: Dynamics of Mechanical and Electromechanical Systems. McGraw-Hill, New York (1968)

    Google Scholar 

  19. 19.

    De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles. J. Stat. Phys. 133(2), 281–345 (2008)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids 20(8), 887–928 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    dell’Isola, F., Corte, A.D., Giorgio, I.: Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math. Mech. Solids 22(4), 852–872 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    dell’Isola, F., Cuomo, M., Greco, L., Della Corte, A.: Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. 103(1), 127–157 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    dell’Isola, F., Della Corte, A., Esposito, R., Russo, L.: Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola’s peridynamics and generalized continuum theories. In: Generalized Continua as Models for Classical and Advanced Materials, pp. 77–128. Springer (2016)

  24. 24.

    dell’Isola, F., Seppecher, P., Della Corte, A.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. A 471(2183), 20150,415 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060,804 (2015)

    Article  Google Scholar 

  26. 26.

    Duhem, P.: Sauver les phénomènes. Essai sur la notion de théorie physique de Platon à Galilée. Sozein ta phainomena (2005)

  27. 27.

    Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mech. Solids 21(2), 210–221 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple micro-elastic solids I. Int. J. Eng. Sci. 2(2), 189–203 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Eugster, S.R., dell’Isola, F.: An ignored source in the foundations of continuum physics “Die Allgemeinen Ansätze der Mechanik der Kontinua” by E. Hellinger. In: Proceedings in Applied Mathematics and Mechanics (2017)

    Article  Google Scholar 

  30. 30.

    Eugster, S.R., et al.: Exegesis of the introduction and sect. I from “Fundamentals of the mechanics of continua” by E. Hellinger. ZAMM J. Appl. Math. Mech. (Z. Angew. Math. Mech.) 97(4), 477–506 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Eugster, S.R., et al.: Exegesis of sect. II and III. A from “Fundamentals of the mechanics of continua” by E. Hellinger. ZAMM J. Appl. Math. Mech. (Z. Angew. Math. Mech.) 98(1), 31–68 (2018)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Eugster, S.R., et al.: Exegesis of sect. III. B from “Fundamentals of the mechanics of continua” by E. Hellinger. ZAMM J. Appl. Math. Mech. (Z. Angew. Math. Mech.) 98(1), 69–105 (2018)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Fish, J., Chen, W.: Higher-order homogenization of initial/boundary-value problem. J. Eng. Mech. 127(12), 1223–1230 (2001)

    Article  Google Scholar 

  34. 34.

    Fish, J., Kuznetsov, S.: From homogenization to generalized continua. Int. J. Comput. Methods Eng. Sci. Mech. 13(2), 77–87 (2012)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Forest, S., Sab, K.: Cosserat overall modeling of heterogeneous materials. Mech. Res. Commun. 25(4), 449–454 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Geers, M.G., Kouznetsova, V.G., Brekelmans, W.: Multi-scale computational homogenization: trends and challenges. J. Comput. Appl. Math. 234(7), 2175–2182 (2010)

    MATH  Article  Google Scholar 

  37. 37.

    Giorgio, I., Culla, A., Del Vescovo, D.: Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network. Arch. Appl. Mech. 79(9), 859 (2009)

    ADS  MATH  Article  Google Scholar 

  38. 38.

    Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)

    Article  Google Scholar 

  39. 39.

    Goldstein, H.: Classical Mechanics. Addison-Wesley, Boston, United States (1965)

    Google Scholar 

  40. 40.

    Grimmett, G.R.: Correlation inequalities for the potts model. Math. Mech. Complex Syst. 4(3–4), 327–334 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Harrison, P.: Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh. Compos. A Appl. Sci. Manuf. 81, 145–157 (2016)

    Article  Google Scholar 

  42. 42.

    Lagrange, J.L.: Mécanique Analytique, vol. 1. Mallet-Bachelier, Paris (1853)

    Google Scholar 

  43. 43.

    Mariano, P.M.: Multifield theories in mechanics of solids. Adv. Appl. Mech. 38, 1–93 (2001)

    Google Scholar 

  44. 44.

    Maugin, G.A.: Infernal variables and dissipative structures. J. Non-equilib. Thermodyn. 15(2), 173–192 (1990)

    ADS  Article  Google Scholar 

  45. 45.

    Maugin, G.A.: Material Inhomogeneities in Elasticity. CRC Press, Boca Roton (1993)

    Google Scholar 

  46. 46.

    Maugin, G.A.: On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Arch. Appl. Mech. 75(10–12), 723–738 (2006)

    ADS  MATH  Article  Google Scholar 

  47. 47.

    Maugin, G.A.: The saga of internal variables of state in continuum thermo-mechanics (1893–2013). Mech. Res. Commun. 69, 79–86 (2015)

    Article  Google Scholar 

  48. 48.

    Maugin, G.A., Muschik, W.: Thermodynamics with internal variables. Part I. General concepts. J. Non-equilib. Thermodyn. 19, 217–249 (1994)

    ADS  MATH  Google Scholar 

  49. 49.

    Millet, O., Hamdouni, A., Cimetière, A.: A classification of thin plate models by asymptotic expansion of non-linear three-dimensional equilibrium equations. Int. J. Non-linear Mech. 36(1), 165–186 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Milton, G.W., Briane, M., Harutyunyan, D.: On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials. Math. Mech. Complex Syst. 5(1), 41–94 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Misra, A., Lekszycki, T., Giorgio, I., Ganzosch, G., Müller, W.H., dell’Isola, F.: Pantographic metamaterials show atypical Poynting effect reversal. Mech. Res. Commun. 89, 6–10 (2018)

    Article  Google Scholar 

  53. 53.

    Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst. 3(3), 285–308 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Mura, T.: Micromechanics of Defects in Solids. Springer, Berlin (1987)

    Google Scholar 

  55. 55.

    Neff, P., Ghiba, I.D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech. Thermodyn. 26(5), 639–681 (2014)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier, New York (1993)

    Google Scholar 

  57. 57.

    Pfeuty, P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57(1), 79–90 (1970)

    ADS  Article  Google Scholar 

  58. 58.

    Porfiri, M., dell’Isola, F., Santini, E.: Modeling and design of passive electric networks interconnecting piezoelectric transducers for distributed vibration control. Int. J. Appl. Electromagn. Mech. 21(2), 69–87 (2005)

    Article  Google Scholar 

  59. 59.

    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z. Angew. Math. Phys. 67(4), 85 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Turco, E., Giorgio, I., Misra, A., dell’Isola, F.: King post truss as a motif for internal structure of (meta) material with controlled elastic properties. R. Soc. Open Sci. 4(10), 171,153 (2017)

    Article  Google Scholar 

  61. 61.

    Ván, P., Berezovski, A., Engelbrecht, J.: Internal variables and dynamic degrees of freedom. J. Non-equilib. Thermodyn. 33(3), 235–254 (2008)

    ADS  MATH  Article  Google Scholar 

  62. 62.

    Vinogradov, A.M., Kupershmidt, B.A.: The structures of Hamiltonian mechanics. Russ. Math. Surv. 32(4), 177 (1977)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Daria Scerrato.

Additional information

Publisher's Note

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

Communicated by Francesco dell’Isola.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Berezovski, A., Yildizdag, M.E. & Scerrato, D. On the wave dispersion in microstructured solids. Continuum Mech. Thermodyn. 32, 569–588 (2020). https://doi.org/10.1007/s00161-018-0683-1

Download citation


  • Micromorphic media
  • Wave propagation
  • Internal variables