On the correspondence between two- and three-dimensional Eshelby tensors

  • Victor A. Eremeyev
  • Violetta Konopińska-ZmysłowskaEmail author
Open Access
Original Article


We consider both three-dimensional (3D) and two-dimensional (2D) Eshelby tensors known also as energy–momentum tensors or chemical potential tensors, which are introduced within the nonlinear elasticity and the resultant nonlinear shell theory, respectively. We demonstrate that 2D Eshelby tensor is introduced earlier directly using 2D constitutive equations of nonlinear shells and can be derived also using the through-the-thickness procedure applied to a 3D shell-like body.


Eshelby tensor Nonlinear elasticity Nonlinear shell Phase transformations Through-the-thickness integration 



Authors are grateful to Prof. Arkadi Berezovski for the fruitful discussions.

V. Konopińska-Zmysłowska acknowledges the support from the National Centre of Science of Poland with the grant DEC-2012/05/D/ST8/02298.


  1. 1.
    Abeyaratne, R., Knowles, J.K.: Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114(2), 119–154 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abeyaratne, R., Knowles, J.K.: Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  3. 3.
    Agrawal, A., Steigmann, D.J.: Coexistent fluid-phase equilibria in biomembranes with bending elasticity. J. Elast. 93(1), 63–80 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Altenbach, H., Eremeyev, V.: Thin-walled structural elements: classification, classical and advanced theories, new applications. In: Altenbach, H., Eremeyev, V. (eds.) Shell-like Structures: Advanced Theories and Applications, pp. 1–62. Springer, Cham (2017)CrossRefGoogle Scholar
  5. 5.
    Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech. 80(1), 73–92 (2010)ADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Contin. Mech. Thermodyn. 30(5), 1–21 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.H.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Contin. Mech. Thermodyn., pp. 1–13 (2018)Google Scholar
  8. 8.
    Berdichevsky, V.L.: Variational Principles of Continuum Mechanics. I. Fundamentals. Springer, Heidelberg (2009)zbMATHCrossRefGoogle Scholar
  9. 9.
    Berezovski, A., Engelbrecht, J., Maugin, G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, New Jersey (2008)zbMATHCrossRefGoogle Scholar
  10. 10.
    Berezovski, A., Maugin, G.A.: On the velocity of a moving phase boundary in solids. Acta Mech. 179(3–4), 187–196 (2005)zbMATHCrossRefGoogle Scholar
  11. 11.
    Bhattacharya, K.: Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect?. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  12. 12.
    Bhattacharya, K., James, R.D.: A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47(3), 531–576 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bhattacharya, K., James, R.D.: The material is the machine. Science 307(5706), 53–54 (2005)CrossRefGoogle Scholar
  14. 14.
    Bose, D.K., Kienzler, R.: On material conservation laws for a consistent plate theory. Arch. Appl. Mech. 75(10–12), 607–617 (2006)ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Boulbitch, A.A.: Equations of heterophase equilibrium of a biomembrane. Arch. Appl. Mech. 69(2), 83–93 (1999)ADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i Dynamika Powłok Wielopłatowych. Nieliniowa Teoria i Metoda Elementów Skończonych. Wydawnictwo IPPT PAN, Warszawa (2004)Google Scholar
  17. 17.
    Ciarlet, P.G.: Mathematical elasticity:theory of shells, 2018. North-Holland, Amsterdam (2000)Google Scholar
  18. 18.
    De Angelo, M., Barchiesi, E., Giorgio, I., Abali, B.E.: Numerical identification of constitutive parameters in reduced-order bi-dimensional models for pantographic structures: application to out-of-plane buckling. Arch. Appl. Mech., pp. 1–26 (2019)Google Scholar
  19. 19.
    Epstein, M., De León, M.: On uniformity of shells. Int. J. Solids Struct. 35(17), 2173–2182 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Epstein, M., Roychowdhury, A.: On the notion of embedded homogeneity of thin structures. Math. Mech. Solids 21(6), 657–666 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Eremeyev, V., Altenbach, H.: Basics of mechanics of micropolar shells. In: Altenbach, H., Eremeyev, V. (eds.) Shell-like Structures: Advanced Theories and Applications, pp. 63–111. Springer, Cham (2017)CrossRefGoogle Scholar
  22. 22.
    Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, New Jersey (2018)zbMATHCrossRefGoogle Scholar
  23. 23.
    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2013)zbMATHCrossRefGoogle Scholar
  24. 24.
    Eremeyev, V.A., Pietraszkiewicz, W.: The non-linear theory of elastic shells with phase transitions. J. Elast. 74(1), 67–86 (2004)zbMATHCrossRefGoogle Scholar
  25. 25.
    Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech. 61(1), 41–67 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Eremeyev, V.A., Pietraszkiewicz, W.: Thermomechanics of shells undergoing phase transition. J. Mech. Phys. Solids 59(7), 1395–1412 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Eshelby, J.D.: The force on an elastic singularity. Phil. Trans. R. Soc. Lond. A 244(877), 87–112 (1951)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 241(1226), 376–396 (1957)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Eshelby, J.D.: The elastic energy–momentum tensor. J. Elast. 5(3–4), 321–335 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Feng, P., Sun, Q.P.: Experimental investigation on macroscopic domain formation and evolution in polycrystalline NiTi microtubing under mechanical force. J. Mech. Phys. Solids 54(8), 1568–1603 (2006)ADSCrossRefGoogle Scholar
  31. 31.
    Franciosi, P., Spagnuolo, M., Salman, O.U.: Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Contin. Mech. Thermodyn., pp. 1–32 (2018)Google Scholar
  32. 32.
    Freidin, A.B.: On the chemical affinity tensor for chemical reactions in deformable materials. Mech. Solids 50(3), 260–285 (2015)ADSCrossRefGoogle Scholar
  33. 33.
    Freidin, A.B., Fu, Y.B., Sharipova, L.L., Vilchevskaya, E.N.: Spherically symmetric two-phase deformations and phase transition zones. Int. J. Solids Struct. 43(14–15), 4484–4508 (2006)zbMATHCrossRefGoogle Scholar
  34. 34.
    Freidin, A.B., Vilchevskaya, E.N., Korolev, I.K.: Stress-assist chemical reactions front propagation in deformable solids. Int. J. Eng. Sci. 83, 57–75 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Fu, Y.B., Freidin, A.B.: Characterization and stability of two-phase piecewise-homogeneous deformations. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 460(2051), 3065–3094 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Giorgio, I., Harrison, P., dell’Isola, F., Alsayednoor, J., Turco, E.: Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches. Proc. R. Soc. A Math. Phys. Eng. Sci. 474(2216), 20180,063 (2018)CrossRefGoogle Scholar
  37. 37.
    Gol’denveizer, A.L.: Theory of Elastic Thin Shells. Pergamon Press, Oxford (1961)Google Scholar
  38. 38.
    Grinfeld, M.: Thermodynamics Methods in the Theory of Heterogeneous Systems. Longman, Harlow (1991)Google Scholar
  39. 39.
    Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin (2000)zbMATHGoogle Scholar
  40. 40.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Kienzler, R., Herrmann, G.: Mechanics in Material Space with Applications to Defect and Fracture Mechanics. Springer, Berlin (2000)zbMATHGoogle Scholar
  42. 42.
    Knowles, J.K.: On the dissipation associated with equilibrium shocks in finite elasticity. J. Elast. 9(2), 131–158 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010)zbMATHCrossRefGoogle Scholar
  44. 44.
    Li, Z.Q., Sun, Q.P.: The initiation and growth of macroscopic martensite band in nano-grained NiTi microtube under tension. Int. J. Plast. 18(11), 1481–1498 (2002)CrossRefGoogle Scholar
  45. 45.
    Libai, A., Simmonds, J.G.: Nonlinear elastic shell theory. Adv. Appl. Mech. 23, 271–371 (1983)zbMATHCrossRefGoogle Scholar
  46. 46.
    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)zbMATHCrossRefGoogle Scholar
  47. 47.
    Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman Hall, London (1993)zbMATHCrossRefGoogle Scholar
  48. 48.
    Maugin, G.A.: Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. CRC Press, Boca Raton (2011)zbMATHGoogle Scholar
  49. 49.
    Maugin, G.A., Berezovski, A.: On the propagation of singular surfaces in thermoelasticity. J. Therm. Stress. 32(6–7), 557–592 (2009)CrossRefGoogle Scholar
  50. 50.
    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
  51. 51.
    Naghdi, P.: The theory of plates and shells. In: S. Flügge (ed.) Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Heidelberg (1972)Google Scholar
  52. 52.
    Naumenko, K., Altenbach, H.: Modeling of Creep for Structural Analysis. Springer, Berlin (2007)zbMATHCrossRefGoogle Scholar
  53. 53.
    Nguyen, T.T., Bruinsma, R.F., Gelbart, W.M.: Elasticity theory and shape transitions of viral shells. Phys. Rev. E 72(5), 051923 (2005)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Nicholson, J.W., Simmonds, J.G.: Sanders’ energy-release rate integral for arbitrarily loaded shallow shells and its asymptotic evaluation for a cracked cylinder. J. Appl. Mech. 47(2), 363–369 (1980)ADSzbMATHCrossRefGoogle Scholar
  55. 55.
    Pietraszkiewicz, W.: The resultant linear six-field theory of elastic shells: What it brings to the classical linear shell models? ZAMM 96(8), 899–915 (2016)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    Pietraszkiewicz, W., Eremeyev, V.A., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM 87(2), 150–159 (2007)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Pietraszkiewicz, W., Konopińska, V.: Singular curve in the resultant thermomechanics of shells. Int. J. Eng. Sci. 80, 21–31 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Poluektov, M., Freidin, A.B., Figiel, Ł.: Modelling stress-affected chemical reactions in non-linear viscoelastic solids with application to lithiation reaction in spherical si particles. Int. J. Eng. Sci. 128, 44–62 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Roytburd, A., Slutsker, J.: Coherent phase equilibria in a bending film. Acta Mater. 50(7), 1809–1824 (2002)CrossRefGoogle Scholar
  60. 60.
    Roytburd, A.L., Slutsker, J.: Theory of multilayer SMA actuators. Mater. Trans. 43(5), 1023–1029 (2002)CrossRefGoogle Scholar
  61. 61.
    Stupkiewicz, S.: Micromechanics of Contact and Interface Layers, Lecture Notes in Applied and Computational Mechanics, vol. 30. Springer, Berlin (2007)zbMATHGoogle Scholar
  62. 62.
    Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin (2004)zbMATHCrossRefGoogle Scholar
  63. 63.
    Wilson, E.B.: Vector Analysis, Founded Upon the Lectures of G. W. Gibbs. Yale University Press, New Haven (1901)Google Scholar
  64. 64.
    Yeremeyev, V.A., Freidin, A.B., Sharipova, L.L.: The stability of the equilibrium of two-phase elastic solids. J. Appl. Math. Mech. 71(1), 61–84 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Gdańsk University of TechnologyGdańskPoland

Personalised recommendations