Abstract
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.
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Abeyaratne, R., Knowles, J.K.: Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114(2), 119–154 (1991)
Abeyaratne, R., Knowles, J.K.: Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge (2006)
Agrawal, A., Steigmann, D.J.: Coexistent fluid-phase equilibria in biomembranes with bending elasticity. J. Elast. 93(1), 63–80 (2008)
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)
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)
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)
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)
Berdichevsky, V.L.: Variational Principles of Continuum Mechanics. I. Fundamentals. Springer, Heidelberg (2009)
Berezovski, A., Engelbrecht, J., Maugin, G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, New Jersey (2008)
Berezovski, A., Maugin, G.A.: On the velocity of a moving phase boundary in solids. Acta Mech. 179(3–4), 187–196 (2005)
Bhattacharya, K.: Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect?. Oxford University Press, Oxford (2003)
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)
Bhattacharya, K., James, R.D.: The material is the machine. Science 307(5706), 53–54 (2005)
Bose, D.K., Kienzler, R.: On material conservation laws for a consistent plate theory. Arch. Appl. Mech. 75(10–12), 607–617 (2006)
Boulbitch, A.A.: Equations of heterophase equilibrium of a biomembrane. Arch. Appl. Mech. 69(2), 83–93 (1999)
Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i Dynamika Powłok Wielopłatowych. Nieliniowa Teoria i Metoda Elementów Skończonych. Wydawnictwo IPPT PAN, Warszawa (2004)
Ciarlet, P.G.: Mathematical elasticity:theory of shells, 2018. North-Holland, Amsterdam (2000)
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)
Epstein, M., De León, M.: On uniformity of shells. Int. J. Solids Struct. 35(17), 2173–2182 (1998)
Epstein, M., Roychowdhury, A.: On the notion of embedded homogeneity of thin structures. Math. Mech. Solids 21(6), 657–666 (2016)
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)
Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, New Jersey (2018)
Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2013)
Eremeyev, V.A., Pietraszkiewicz, W.: The non-linear theory of elastic shells with phase transitions. J. Elast. 74(1), 67–86 (2004)
Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech. 61(1), 41–67 (2009)
Eremeyev, V.A., Pietraszkiewicz, W.: Thermomechanics of shells undergoing phase transition. J. Mech. Phys. Solids 59(7), 1395–1412 (2011)
Eshelby, J.D.: The force on an elastic singularity. Phil. Trans. R. Soc. Lond. A 244(877), 87–112 (1951)
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)
Eshelby, J.D.: The elastic energy–momentum tensor. J. Elast. 5(3–4), 321–335 (1975)
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)
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)
Freidin, A.B.: On the chemical affinity tensor for chemical reactions in deformable materials. Mech. Solids 50(3), 260–285 (2015)
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)
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)
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)
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)
Gol’denveizer, A.L.: Theory of Elastic Thin Shells. Pergamon Press, Oxford (1961)
Grinfeld, M.: Thermodynamics Methods in the Theory of Heterogeneous Systems. Longman, Harlow (1991)
Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin (2000)
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
Kienzler, R., Herrmann, G.: Mechanics in Material Space with Applications to Defect and Fracture Mechanics. Springer, Berlin (2000)
Knowles, J.K.: On the dissipation associated with equilibrium shocks in finite elasticity. J. Elast. 9(2), 131–158 (1979)
Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010)
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)
Libai, A., Simmonds, J.G.: Nonlinear elastic shell theory. Adv. Appl. Mech. 23, 271–371 (1983)
Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)
Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman Hall, London (1993)
Maugin, G.A.: Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. CRC Press, Boca Raton (2011)
Maugin, G.A., Berezovski, A.: On the propagation of singular surfaces in thermoelasticity. J. Therm. Stress. 32(6–7), 557–592 (2009)
Miyazaki, S., Fu, Y.Q., Huang, W.M. (eds.): Thin Film Shape Memory Alloys: Fundamentals and Device Applications. Cambridge University Press, Cambridge (2009)
Naghdi, P.: The theory of plates and shells. In: S. Flügge (ed.) Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Heidelberg (1972)
Naumenko, K., Altenbach, H.: Modeling of Creep for Structural Analysis. Springer, Berlin (2007)
Nguyen, T.T., Bruinsma, R.F., Gelbart, W.M.: Elasticity theory and shape transitions of viral shells. Phys. Rev. E 72(5), 051923 (2005)
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)
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)
Pietraszkiewicz, W., Eremeyev, V.A., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM 87(2), 150–159 (2007)
Pietraszkiewicz, W., Konopińska, V.: Singular curve in the resultant thermomechanics of shells. Int. J. Eng. Sci. 80, 21–31 (2014)
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)
Roytburd, A., Slutsker, J.: Coherent phase equilibria in a bending film. Acta Mater. 50(7), 1809–1824 (2002)
Roytburd, A.L., Slutsker, J.: Theory of multilayer SMA actuators. Mater. Trans. 43(5), 1023–1029 (2002)
Stupkiewicz, S.: Micromechanics of Contact and Interface Layers, Lecture Notes in Applied and Computational Mechanics, vol. 30. Springer, Berlin (2007)
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin (2004)
Wilson, E.B.: Vector Analysis, Founded Upon the Lectures of G. W. Gibbs. Yale University Press, New Haven (1901)
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)
Acknowledgements
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.
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Communicated by Holm Altenbach.
In occasion of the 80th birthday of Prof. Wojciech Pietraszkiewicz
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Eremeyev, V.A., Konopińska-Zmysłowska, V. On the correspondence between two- and three-dimensional Eshelby tensors. Continuum Mech. Thermodyn. 31, 1615–1625 (2019). https://doi.org/10.1007/s00161-019-00754-6
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DOI: https://doi.org/10.1007/s00161-019-00754-6