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Two-phase equilibrium microstructures against optimal composite microstructures

  • Alexander B. FreidinEmail author
  • Leah L. Sharipova
Original
  • 23 Downloads

Abstract

We relate two problems which arise from different branches of mechanics of materials: construction of limiting phase transformation surfaces in strain space and stress–strain diagrams for stress-induced phase transitions and optimal design of two-phase 3D composites in the sense of minimizing its energy. In Antimonov et al. (Int J Eng Sci 98:153–182, 2015) for the case of isotropic phases, it was shown that given a new phase volume fraction and depending on average strain, the strain energy of a two-phase linear-elastic composite is minimized by either direct or inclined simple laminates, direct or skew second-rank laminates or third-rank laminates. Then these results were applied for the construction of direct and reverse transformations limiting surfaces in strain space for elastic solids undergoing phase transformations by additional minimization with respect to the new phase volume fraction and finding the strains at which minimizing volume fraction equals zero or one. In the present paper we construct stress–strain diagrams on various straining paths at which a material undergoes the phase transformation. We demonstrate that an additional degree of freedom—new phase volume fraction—may crucially result in instability of two-phase microstructures even if the microstructures are energy minimizers for composites with given volume fractions of phases. This in turn may lead to incompleteness of monotonic phase transformations and broken stress–strain diagrams. We study how such a behavior depends on a loading path and chemical energies of the phases.

Keywords

Stress-induced phase transformations Optimal composite microstructure Limit transformation surfaces Stress–strain behavior 

Notes

Acknowledgements

This paper became possible thanks to fruitful collaboration with Andrej Cherkaev who gave Alexander Freidin inspiring lessons on optimal microstructures, which, in turn, arise to Prof. Konstantin Lurie, and for invaluable comments on the draft of the paper. The authors also greatly appreciate financial support of Russian Foundation for Basic Research (Grant No. 16-01-00815)

References

  1. 1.
    Albin, N., Cherkaev, A., Nesi, V.: Multiphase laminates of extremal effective conductivity in two dimensions. J. Mech. Phys. Solids 55(7), 1513–1553 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allaire, G., Aubry, S.: On optimal microstructures for a plane shape optimization problem. Struct. Optim. 17, 86–94 (1999)CrossRefGoogle Scholar
  3. 3.
    Antimonov, M.A., Cherkaev, A.V., Freidin, A.B.: On transformation surfaces construction for phase transitions in deformable solids. In: Proceedings of XXXVIII International Summer School—Conference Advanced Problems in Mechanics (APM-2010). IPME, St. PetersburgRAS, pp. 23–29, 1–5 July 2010Google Scholar
  4. 4.
    Antimonov, M.A., Cherkaev, A.V., Freidin, A.B.: Optimal microstructures and exact lower bound of energy of elastic composites comprised of two isotropic phases. St. Petersb. State Polytech. Univ. J. Phys. Math. 3, 113–122 (2010). (in Russian) Google Scholar
  5. 5.
    Antimonov, M.A., Freidin, A.B.: Equilibrium cylindrical anisotropic phase inclusion in isotropic elastic solid. St. Petersb. State Polytech. Univ. J. Phys. Math. 4(4), 37–44 (2010). (in Russian) Google Scholar
  6. 6.
    Antimonov, M.A., Cherkaev, A., Freidin, A.B.: Phase transformations surfaces and exact energy lower bounds. Int. J. Eng. Sci. 98, 153–182 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Avellaneda, M., Cherkaev, A., Gibiansky, L., Milton, G., Rudelson, M.: A complete characterization of the possible bulk and shear moduli of planar polycrystals. J. Mech. Phys. Solids 44(7), 1179–1218 (1996)CrossRefGoogle Scholar
  8. 8.
    Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100, 13–52 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berdichevsky, V.L.: Seed of a melt in a solid. Dokl. Acad. Nauk SSSR 27, 80–84 (1983)Google Scholar
  10. 10.
    Briane, M.: Correctors for the homogenization of a laminate. Adv. Math. Sci. Appl. 2, 357–379 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chenchiah, I.V., Bhattacharya, K.: The relaxation of two-well energies with possibility unequal moduli. Arch. Rat. Mech. Anal. 187(3), 409–479 (2008)CrossRefzbMATHGoogle Scholar
  12. 12.
    Cherkaev, A., Gibiansky, L.: Extremal structures of multiphase heat conducting composites. Int. J. Solids Struct. 33(18), 2609–2623 (1996)CrossRefzbMATHGoogle Scholar
  13. 13.
    Cherkaev, A.V.: Variational Methods for Structural Optimization. Springer, Berlin (2002)zbMATHGoogle Scholar
  14. 14.
    Cherkaev, A., Zhang, Y.: Optimal anisotropic three-phase conducting composites: plane problem. Int. J. Solids Struct. 48(20), 2800–2813 (2011)CrossRefGoogle Scholar
  15. 15.
    Freidin, A.: Crazes and shear bands in glassy polymer as layers of a new phase. Mech. Compos. Mater. 1, 1–7 (1989)CrossRefGoogle Scholar
  16. 16.
    Freidin, A.B.: Small strains approach in the theory of strain induced phase transformations. In: Morozov, N.F. (ed.) Strength and Fracture of Materials, Studies on Elasticity and Plasticity, vol. 18, pp. 266–290. St. Petersburg University, St. Petersburg (1999). (in Russian) Google Scholar
  17. 17.
    Freidin, A., Sharipova, L.: On a model of heterogenous deformation of elastic bodies by the mechanism of multiple appearance of new phase layers. Meccanica 41(3), 321–339 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Freidin, A.B.: On new phase inclusions in elastic solids. ZAMM 87(2), 102–116 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Freidin, A.B., Vilchevskaya, E.N.: Multiple development of new phase inclusions in elastic solids. Int. J. Eng. Sci. 47(2), 240–260 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gibiansky, L., Cherkaev, A.: Microstructures of composites of extremal rigidity and exact bounds on the associated energy density. In: Cherkaev, A., Kohn, R. (eds.) Topics in the Mathematical Modelling of Composite Materials. Vol. 31 of Progress in Nonlinear Differential Equations and Their Applications, pp. 273–317. Birkhäuser, Boston (1997)CrossRefGoogle Scholar
  21. 21.
    Grinfeld M.A.: On heterogeneous equilibrium of non-linear elastic phases and chemical potential tensors. In: Zvolinsky N.V. et al. (eds.) Problems of the Nonlinear Mechanics of a Continuous Medium, pp. 33–47. Valgus, Tallin (1985) (in Russian) Google Scholar
  22. 22.
    Grinfeld, M.A.: Thermodynamic Methods in the Theory of Heterogeneous Systems. Longman, New York (1991)Google Scholar
  23. 23.
    Kaganova, I., Roitburd, A.: Equilibrium of elastically interacting phases. Sov. Phys JETP 67, 1174–1186 (1988)Google Scholar
  24. 24.
    Kohn, R.: The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3(3), 193–236 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kublanov, L., Freidin, A.: Solid phase seeds in a deformable material. J. Appl. Math. Mech. 52(3), 382–389 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lurie, K., Cherkaev, A.: Optimal structural design and relaxed controls. Optim. Control Appl. Methods 4(4), 387–392 (1983)CrossRefzbMATHGoogle Scholar
  27. 27.
    Lurie, K., Cherkaev, A.: On a certain variational problem of phase equilibrium. Material instabilities in continuum mechanics. In: Proceedings of the Symposium Year Held at Heriot-Watt University, Edinburgh, 1985–1986, pp. 257–268 (1988)Google Scholar
  28. 28.
    Milton, G.W.: Modelling the properties of composites by laminates. In: Ericksen, J., Kinderlehrer, D., Kohn, R., Lions, J.-L. (eds.) Homogenization and Effective Moduli of Materials and Media. The IMA Volumes in Mathematics and its Applications, vol. 1, pp. 150–174. Springer, New York (1986)CrossRefGoogle Scholar
  29. 29.
    Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  30. 30.
    Morozov, N.F., Freidin, A.B.: Phase transition zones and phase transformations of elastic solids under different stress states. In: Proceedings of the Steklov Institute of Mathematics, vol. 223, pp. 220–232 (1998)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering of Russian Academy of SciencesSaint PetersburgRussia
  2. 2.Peter the Great St. Petersburg Polytechnic UniversitySaint PetersburgRussia

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