On the Nonlinear Theory of Two-Phase Shells

  • Victor A. EremeyevEmail author
  • Wojciech Pietraszkiewicz
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)


We discuss the nonlinear theory of shells made of material undergoing phase transitions (PT). The interest to such thin-walled structures is motivated by applications of thin films made of martensitic materials and needs of modeling biological membranes. Here we present the resultant, two-dimensional thermodynamics of non-linear theory of shells undergoing PT. The global and local formulations of the balances of momentum, moment of momentum, energy and entropy are given. Two temperature fields on the shell base surface are introduced: the referential mean temperature and its deviation, as well as two corresponding dual fields: the referential entropy and its deviation. Additional surface heat flux and the extra heat flux vector fields appear as a result of through-the-thickness integration procedure. Within the framework of the resultant shell thermodynamics we derive the continuity conditions along the curvilinear phase interface which separates two material phases. These conditions allow us to formulate the kinetic equation describing the quasistatic motion of the interface relative to the shell base surface. The kinetic equation is expressed by the jump of the Eshelby tensor across the phase interface. In the case of thermodynamic equilibrium the variational statement of the static problem of two-phase shell is presented.


Non-linear shell Shell thermodynamics Phase transition Cosserat shell Micropolar shell Kinetic equation Singular curve 


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  1. 1.
    Abeyaratne, R., Knowles, J.K.: Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  2. 2.
    Agrawal, A., Steigmann, D.J.: Coexistent fluid-phase equilibria in biomembranes with bending elasticity. Journal of Elasticity 93(1), 63–80 (2008)CrossRefGoogle Scholar
  3. 3.
    Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Archive for Rational Mechanics and Analysis 100(1), 13–52 (1987)CrossRefGoogle Scholar
  4. 4.
    Berezovski, A., Engelbrecht, J., Maugin, G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, New Jersey (2008)CrossRefGoogle Scholar
  5. 5.
    Bhattacharya, K.: Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press, Oxford (2003)Google Scholar
  6. 6.
    Bhattacharya, K., DeSimone, A., Hane, K.F., James, R.D., Palmstrøm, C.J.: Tents and tunnels on martensitic films. Materials Science and Engineering A 273(Sp. Iss. SI), 685–689 1999CrossRefGoogle Scholar
  7. 7.
    Bhattacharya, K., James, R.D.: A theory of thin films of martensitic materials with applications to microactuators. Journal of the Mechanics and Physics of Solids 47(3), 531–576 (1999)CrossRefGoogle Scholar
  8. 8.
    Bhattacharya, K., James, R.D.: The material is the machine. Science 307(5706), 53–54 (2005)CrossRefGoogle Scholar
  9. 9.
    Bhattacharya, K., Kohn, R.V.: Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials. Archive for Rational Mechanics and Analysis 139(2), 99–180 (1997)CrossRefGoogle Scholar
  10. 10.
    Boulbitch, A.A.: Equations of heterophase equilibrium of a biomembrane. Archive of Applied Mechanics 69(2), 83–93 (1999)CrossRefGoogle Scholar
  11. 11.
    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statics and Dynamics of Multifold Shells: Nonlinear Theory and Finite Element Method (in Polish). Wydawnictwo IPPT PAN, Warszawa (2004)Google Scholar
  12. 12.
    Elliott, C.M., Stinner, B.: A surface phase field model for two-phase biological membranes. SIAM J. Appl. Math. 70(8), 2904–2928 (2010)CrossRefGoogle Scholar
  13. 13.
    Eremeyev, V.A., Pietraszkiewicz, W.: The non-linear theory of elastic shells with phase transitions. Journal of Elasticity 74(1), 67–86 (2004)CrossRefGoogle Scholar
  14. 14.
    Eremeyev, V.A., Pietraszkiewicz, W.: Local symmetry group in the general theory of elastic shells. Journal of Elasticity 85(2), 125–152 (2006)CrossRefGoogle Scholar
  15. 15.
    Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Archives of Mechanics 61(1), 41–67 (2009)Google Scholar
  16. 16.
    Eremeyev, V.A., Pietraszkiewicz, W.: On tension of a two-phase elastic tube. In: Pietraszkiewicz, W., Kreja, I., (eds.) Shell Structures: Theory and Applications, vol. 2, pp. 63–66. CRC Press, Boca Raton (2010)Google Scholar
  17. 17.
    Eremeyev, V.A., Pietraszkiewicz, W.: Thermomechanics of shells undergoing phase transition. Journal of the Mechanics and Physics of Solids doi: 10.1016/j.jmps.2011.04.005 (2011)
  18. 18.
    Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells. Nauka, Moscow (2008) (in Russian)Google Scholar
  19. 19.
    Gibbs, J.W.: On the equilibrium of heterogeneous substances. In: The Collected Works of Willard Gibbs, J. pp. 55–353. Longmans, Green & Co, New York (1928)Google Scholar
  20. 20.
    Green, A.E., Naghdi, P.M.: Non-isothermal theory of rods, plates and shells. International Journals of Solids and Structures 6(2), 635–648 (1970)Google Scholar
  21. 21.
    Green, A.E., Naghdi, P.M.: On thermal effects in the theory of shells. Proceedings of the Royal Society of London Series A 365(1721), 161–190 (1979)CrossRefGoogle Scholar
  22. 22.
    Grinfeld, M.: Thermodynamics Methods in the Theory of Heterogeneous Systems. Longman, Harlow (1991)Google Scholar
  23. 23.
    Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin (2000)Google Scholar
  24. 24.
    Hane, K.F.: Bulk and thin film microstructures in untwinned martensites. Journal of the Mechanics and Physics of Solids 47, 1917–1939 (1999)CrossRefGoogle Scholar
  25. 25.
    He, Y.J., Sun, Q.P.: Effects of structural and material length scales on stress-induced martensite macro-domain patterns in tube configurations. International Journal of Solids and Structures 46(16), 3045–3060 (2009)CrossRefGoogle Scholar
  26. 26.
    He, Y.J., Sun, Q.P.: Scaling relationship on macroscopic helical domains in NiTi tubes. International Journal of Solids and Structures 46(24), 4242–4251 (2009)CrossRefGoogle Scholar
  27. 27.
    He, Y.J., Sun, Q.P.: Macroscopic equilibrium domain structure and geometric compatibility in elastic phase transition of thin plates. International Journal of Mechanical Sciences 52(2), 198–211 (2010)CrossRefGoogle Scholar
  28. 28.
    He, Y.J., Sun, Q.P.: Rate-dependent domain spacing in a stretched NiTi strip. International Journal of Solids and Structures 47(20), 2775–2783 (2010)CrossRefGoogle Scholar
  29. 29.
    James, R.D., Hane, K.F.: Martensitic transformations and shape-memory materials. Acta Materialia 48(1), 197–222 (2000)CrossRefGoogle Scholar
  30. 30.
    James, R.D., Rizzoni, R.: Pressurized shape memory thin films. Journal of Elasticity 59(1–3), 399–436 (2000)CrossRefGoogle Scholar
  31. 31.
    Kienzler, R., Herrman, G.: Mechanics in Material Space with Applications to Defect and Fracture Mechanics. Springer, Berlin (2000)Google Scholar
  32. 32.
    Lagoudas, D.C. (ed.): Shape Memory Alloys. Modeling and Engineering Applications. Springer, Berlin (2008)Google Scholar
  33. 33.
    Li, Z.Q., Sun, Q.P.: The initiation and growth of macroscopic martensite band in nano-grained NiTi microtube under tension. International Journal of Plasticity 18(11), 1481–1498 (2002)CrossRefGoogle Scholar
  34. 34.
    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  35. 35.
    Makowski, J., Pietraszkiewicz, W.: Thermomechanics of shells with singular curves. Zesz. Nauk. No 528/1487/2002, IMP PAN, Gdańsk (2002)Google Scholar
  36. 36.
    Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman Hall, London (1993)Google Scholar
  37. 37.
    Mielke, A., Theil, F., Levitas, V.I.: A variational formulation of rate-independent phase transformations using an extremum principle. Archive for Rational Mechanics and Analysis 162(2), 137–177 (2002)CrossRefGoogle Scholar
  38. 38.
    Miyazaki, S., Fu, Y.Q., Huang, W.M. (eds.): Thin Film Shape Memory Alloys: Fundamentals and Device Applications. Cambridge University Press, Cambridge (2009)Google Scholar
  39. 39.
    Murdoch, A.I.: On the entropy inequality for material interfaces. ZAMP 27(5), 599–605 (1976)CrossRefGoogle Scholar
  40. 40.
    Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. The Quarterly Journal of Mechanics and Applied Mathematics 29(3), 245–274 (1976)CrossRefGoogle Scholar
  41. 41.
    Pieczyska, E.: Activity of stress-induced martensite transformation in TiNi shape memory alloy studied by infrared technique. Journal of Modern Optics 57(18, Sp. Iss. SI), 1700–1707 (2010)CrossRefGoogle Scholar
  42. 42.
    Pietraszkiewicz, W.: On non-linear shell thermodynamics with interstitial working. In: Wilmański, K., Jędrysiak, J., Michalak, B. (eds.) Mathematical Methods in Continuum Mechanics, Chapter 11 (in print). Politechnika \(\L\)ódzka, \(\L\)ódź (2011)Google Scholar
  43. 43.
    Pietraszkiewicz, W., Chróścielewski, J., Makowski, J.: On dynamically and kinematically exact theory of shells. In: Pietraszkiewicz, W., Szymczak, C. (eds.) Shell Structures: Theory and Applications, pp. 163–167. Taylor & Francis, London (2005)Google Scholar
  44. 44.
    Pietraszkiewicz, W., Eremeyev, V.A., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM 87(2), 150–159 (2007)CrossRefGoogle Scholar
  45. 45.
    Rubin, M.B.: Restrictions on linear constitutive equations for a rigid heat conducting Cosserat shell. International Journal of Solids and Structures 41(24–25), 7009–7033 (2004)CrossRefGoogle Scholar
  46. 46.
    Rubin, M.B.: Heat conduction between confocal elliptical surfaces using the theory of a Cosserat shell. International Journal of Solids and Structures 43(2), 295–306 (2006)CrossRefGoogle Scholar
  47. 47.
    Shu, Y.C.: Heterogeneous thin films of martensitic materials. Archive of Rational Mechanics and Analysis 153(1), 39–90 (2000)CrossRefGoogle Scholar
  48. 48.
    Shu, Y.C.: Shape-memory micropumps. Materials Transactions 43(5, Sp. Iss. SI), 1037–1044 (2002)CrossRefGoogle Scholar
  49. 49.
    Simmonds, J.G.: The thermodynamical theory of shells: Descent from 3-dimensions without thickness expansions. In: Axelrad, E.K., Emmerling, F.A. (eds.) Flexible Shells, Theory and Applications, pp. 1–11. Springer, Berlin (1984)Google Scholar
  50. 50.
    Simmonds, J.G.: A simple nonlinear thermodynamic theory of arbitrary elastic beams. Journal of Elasticity 81(1), 51–62 (2005)CrossRefGoogle Scholar
  51. 51.
    Simmonds, J.G.: A classical, nonlinear thermodynamic theory of elastic shells based on a single constitutive assumption. Journal of Elasticity, doi: 10.1007/s10659-010-9293-2 (2011)
  52. 52.
    Sun, Q.P. (ed.): Mechanics of Martensitic Phase Transformation in Solids. Kluwer, Dordrecht (2002)Google Scholar
  53. 53.
    Tobushi, H., Pieczyska, E.A., Nowacki, W.K., Sakuragi, T., Sugimoto, Y.: Torsional deformation and rotary driving characteristics of SMA thin strip. Archives of Mechanics 61(3–4), 241–257 (2009)Google Scholar
  54. 54.
    Truesdell, C.: Rational Thermodynamics, 2nd edn. Springer, New York (1984)Google Scholar
  55. 55.
    Truesdell, C.A.: The Elements of Continuum Mechanics. Springer, Berlin (1966)Google Scholar
  56. 56.
    Zhang, X.H., Feng, P., He, Y.J., Yu, T.X., Sun, Q.P.: Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips. International Journal of Mechanical Sciences 52(12), 1660–1670 (2010)CrossRefGoogle Scholar
  57. 57.
    Zhilin, P.A.: Mechanics of deformable directed surfaces. International Journals of Solids and Structures 12(9–10), 635–648 (1976)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Victor A. Eremeyev
    • 1
    • 2
    Email author
  • Wojciech Pietraszkiewicz
    • 3
  1. 1.Martin-Luther-Universität Halle-WittenbergHalleGermany
  2. 2. South Scientific Center of RASci and South Federal UniversityRostov on DonRussia
  3. 3.Institute of Fluid-Flow Machinery of the Polish Academy of SciencesGdańskPoland

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