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 throughthe-thickness procedure applied to a 3D shell-like body.
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- Category:
- Articles
- Type:
- artykuł w czasopiśmie wyróżnionym w JCR
- Published in:
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CONTINUUM MECHANICS AND THERMODYNAMICS
no. 31,
pages 1615 - 1625,
ISSN: 0935-1175 - Language:
- English
- Publication year:
- 2019
- Bibliographic description:
- Eremeev V., Konopińska-Zmysłowska V.: On the correspondence between two- and three-dimensional Eshelby tensors// CONTINUUM MECHANICS AND THERMODYNAMICS. -Vol. 31, (2019), s.1615-1625
- DOI:
- Digital Object Identifier (open in new tab) 10.1007/s00161-019-00754-6
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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) open in new tab
- Abeyaratne, R., Knowles, J.K.: Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge (2006) open in new tab
- Agrawal, A., Steigmann, D.J.: Coexistent fluid-phase equilibria in biomembranes with bending elasticity. J. Elast. 93(1), 63-80 (2008) open in new tab
- Altenbach, H., Eremeyev, V.: Thin-walled structural elements: classification, classical and advanced theories, new applica- tions. In: Altenbach, H., Eremeyev, V. (eds.) Shell-like Structures: Advanced Theories and Applications, pp. 1-62. Springer, Cham (2017) open in new tab
- 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) open in new tab
- 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) open in new tab
- 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) open in new tab
- Berdichevsky, V.L.: Variational Principles of Continuum Mechanics. I. Fundamentals. Springer, Heidelberg (2009) open in new tab
- Berezovski, A., Engelbrecht, J., Maugin, G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, New Jersey (2008) open in new tab
- Berezovski, A., Maugin, G.A.: On the velocity of a moving phase boundary in solids. Acta Mech. 179(3-4), 187-196 (2005) open in new tab
- Bhattacharya, K.: Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect?. Oxford University Press, Oxford (2003) open in new tab
- 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) open in new tab
- Bhattacharya, K., James, R.D.: The material is the machine. Science 307(5706), 53-54 (2005) open in new tab
- Bose, D.K., Kienzler, R.: On material conservation laws for a consistent plate theory. Arch. Appl. Mech. 75(10-12), 607-617 (2006) open in new tab
- Boulbitch, A.A.: Equations of heterophase equilibrium of a biomembrane. Arch. Appl. Mech. 69(2), 83-93 (1999) open in new tab
- 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) open in new tab
- 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) open in new tab
- Epstein, M., De León, M.: On uniformity of shells. Int. J. Solids Struct. 35(17), 2173-2182 (1998) open in new tab
- Epstein, M., Roychowdhury, A.: On the notion of embedded homogeneity of thin structures. Math. Mech. Solids 21(6), 657-666 (2016) open in new tab
- 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) open in new tab
- Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, New Jersey (2018) open in new tab
- Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2013) open in new tab
- Eremeyev, V.A., Pietraszkiewicz, W.: The non-linear theory of elastic shells with phase transitions. J. Elast. 74(1), 67-86 (2004) open in new tab
- Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech. 61(1), 41-67 (2009) open in new tab
- Eremeyev, V.A., Pietraszkiewicz, W.: Thermomechanics of shells undergoing phase transition. J. Mech. Phys. Solids 59(7), 1395-1412 (2011) open in new tab
- Eshelby, J.D.: The force on an elastic singularity. Phil. Trans. R. Soc. Lond. A 244(877), 87-112 (1951) open in new tab
- 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) open in new tab
- Eshelby, J.D.: The elastic energy-momentum tensor. J. Elast. 5(3-4), 321-335 (1975) open in new tab
- 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) open in new tab
- 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) open in new tab
- Freidin, A.B.: On the chemical affinity tensor for chemical reactions in deformable materials. Mech. Solids 50(3), 260-285 (2015) open in new tab
- 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) open in new tab
- 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) open in new tab
- 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) open in new tab
- 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) open in new tab
- Gol'denveizer, A.L.: Theory of Elastic Thin Shells. Pergamon Press, Oxford (1961) open in new tab
- Grinfeld, M.: Thermodynamics Methods in the Theory of Heterogeneous Systems. Longman, Harlow (1991) open in new tab
- Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin (2000) open in new tab
- Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291-323 (1975) open in new tab
- Kienzler, R., Herrmann, G.: Mechanics in Material Space with Applications to Defect and Fracture Mechanics. Springer, Berlin (2000) open in new tab
- Knowles, J.K.: On the dissipation associated with equilibrium shocks in finite elasticity. J. Elast. 9(2), 131-158 (1979) open in new tab
- Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010) open in new tab
- 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) open in new tab
- Libai, A., Simmonds, J.G.: Nonlinear elastic shell theory. Adv. Appl. Mech. 23, 271-371 (1983) open in new tab
- Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998) open in new tab
- Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman Hall, London (1993) open in new tab
- Maugin, G.A.: Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. CRC Press, Boca Raton (2011) open in new tab
- Maugin, G.A., Berezovski, A.: On the propagation of singular surfaces in thermoelasticity. J. Therm. Stress. 32(6-7), 557-592 (2009) open in new tab
- Miyazaki, S., Fu, Y.Q., Huang, W.M. (eds.): Thin Film Shape Memory Alloys: Fundamentals and Device Applications. Cambridge University Press, Cambridge (2009) open in new tab
- Naghdi, P.: The theory of plates and shells. In: S. Flügge (ed.) Handbuch der Physik, vol. VIa/2, pp. 425-640. Springer, Heidelberg (1972) open in new tab
- Naumenko, K., Altenbach, H.: Modeling of Creep for Structural Analysis. Springer, Berlin (2007) open in new tab
- Nguyen, T.T., Bruinsma, R.F., Gelbart, W.M.: Elasticity theory and shape transitions of viral shells. Phys. Rev. E 72(5), 051923 (2005) open in new tab
- 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) open in new tab
- 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) open in new tab
- Pietraszkiewicz, W., Eremeyev, V.A., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM 87(2), 150-159 (2007) open in new tab
- Pietraszkiewicz, W., Konopińska, V.: Singular curve in the resultant thermomechanics of shells. Int. J. Eng. Sci. 80, 21-31 (2014) open in new tab
- 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) open in new tab
- Roytburd, A., Slutsker, J.: Coherent phase equilibria in a bending film. Acta Mater. 50(7), 1809-1824 (2002) open in new tab
- Roytburd, A.L., Slutsker, J.: Theory of multilayer SMA actuators. Mater. Trans. 43(5), 1023-1029 (2002) open in new tab
- Stupkiewicz, S.: Micromechanics of Contact and Interface Layers, Lecture Notes in Applied and Computational Mechanics, vol. 30. Springer, Berlin (2007) open in new tab
- Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin (2004) open in new tab
- Wilson, E.B.: Vector Analysis, Founded Upon the Lectures of G. W. Gibbs. Yale University Press, New Haven (1901) open in new tab
- 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) open in new tab
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