A Phase Transition Model Describing Auxetic Materials

  • Elena BonettiEmail author
  • Mauro Fabrizio
  • Michel Frémond
Part of the Springer INdAM Series book series (SINDAMS, volume 22)


In this paper we introduce a new model describing the behavior of auxetic materials in terms of a phase-field PDE system. More precisely, the evolution equations are recovered by a generalization of the principle of virtual power in which microscopic motions and forces, responsible for the phase transitions, are included. The momentum balance is written in the setting of a second gradient theory, and it presents nonlinear contributions depending on the phases. The evolution of the phases is governed by variational inclusions with non-linear coupling terms. By use of a fixed point theorem and monotonicity arguments, we are able to show that the resulting initial and boundary value problem admits a weak solution.


Auxetic materials Nonlinear PDE system Phase transitions Second gradient theory 

AMS (MOS) Subject Classification

35K87 74N25 



This paper is dedicated to Gianni Gilardi, with friendship and gratitude.


  1. 1.
    Auricchio, F., Bonetti, E., Scalet, G., Ubertini, F.: Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation. Int. J. Plast. 59, 30–54 (2014)CrossRefGoogle Scholar
  2. 2.
    Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden (1976)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bezazi, A., Scarpa, F.: Mechanical behaviour of conventional and negative Poisson’s ratio thermoplastic polyurethane foams under compressive cyclicloading. Int. J. Fatigue 29, 922–930 (2007)CrossRefGoogle Scholar
  4. 4.
    Bonetti, E.: Global solution to a Frémond model for shape memory alloys with thermal memory. Nonlinear Anal. 46, 535–565 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bonetti, E.: Global solvability of a dissipative Frémond model for shape memory alloys. Part II: existence. Q. Appl. Math. 62, 53–76 (2004)zbMATHGoogle Scholar
  6. 6.
    Bonetti, E., Frémond, M.: A phase transition model for the helium supercooling. Asymptot. Anal. 87, 29–42 (2014)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Bonetti, E., Colli, P., Fabrizio, M., Gilardi, G.: Existence of solutions for a mathematical model related to solid–solid phase transitions in shape memory alloys. Arch. Ration. Mech. Anal. 219, 203–254 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Ciarletta, M., Fabrizio, M., Tibullo, V.: Shape memory and phase transitions for auxetic materials. Science 37, 2864–2871 (2014)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Colli, P., Frémond, M., Visintin, A.: Thermo-mechanical evolution of shape memory alloys. Q. Appl. Math. 48, 31–47 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Evans, K.E., Alderson, A.: Auxetic materials: functional materials and structures from lateral thinking. Adv. Mater. 12, 617–628 (2000)CrossRefGoogle Scholar
  11. 11.
    Fabrizio, M.: Ginzburg-Landau equations and first and second order phase transitions. Int. J. Eng. Sci. 44, 529–539 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Frémond, M.: Phase Change in Mechanics. Springer, Berlin, Heidelberg (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Frémond, M.: Virtual Work and Shape Change in Solid Mechanics. Springer Series in Solid and Structural Mechanics, vol. 7. Springer, Berlin (2016)Google Scholar
  14. 14.
    Li, D., Ma, J., Dong, L., Lakes, R.S.: A bi-material structure with Poisson’s ratio tunable from positive to negative via temperature control. Mater. Lett. 181, 285–288 (2016)CrossRefGoogle Scholar
  15. 15.
    Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris (1969)zbMATHGoogle Scholar
  16. 16.
    Moreau, J.J.: Sur les lois de frottement, de viscosité et plasticité. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 271, 608–611 (1970)Google Scholar
  17. 17.
    Passarella, F., Tibullo, V., Zampoli, V.: Decay properties of solutions of a Mindlin-type plate model for rhombic systems. J. Mech. Mater. Struct. 5, 323–339 (2010)CrossRefGoogle Scholar
  18. 18.
    Passarella, F., Tibullo, V., Zampoli, V.: On the strong ellipticity for orthotropic micropolar elastic bodies in a plane strain state. Mech. Res. Commun. 38, 512–517 (2011)CrossRefzbMATHGoogle Scholar
  19. 19.
    Scalet, G., Auricchio, F., Bonetti, E., Castellani, L., Ferri, D., Pachera, M., Scavello, F.: An experimental, theoretical and numerical investigation of shape memory polymers. Int. J. Plast. 67, 127–147 (2015)CrossRefGoogle Scholar
  20. 20.
    Teik-Cheng, L.: Auxetic Materials and Structures. Springer, Berlin (2015)Google Scholar
  21. 21.
    Yang, W., Li, Z.M., Shi, W., Xie, B.H., Yang, M.B.: Review on auxetic materials. J. Mater. Sci. 39, 3269–3279 (2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Elena Bonetti
    • 1
    • 2
    Email author
  • Mauro Fabrizio
    • 3
  • Michel Frémond
    • 4
  1. 1.Dipartimento di Matematica F. EnriquesUniversità di MilanoMilanoItaly
  2. 2.IMATI-CNRPaviaItaly
  3. 3.Università di BolognaScuola di Ingegneria e ArchitetturaBolognaItaly
  4. 4.Dipartimento di Ingegneria Civile e InformaticaUniversità di Roma Tor VergataRomaItaly

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