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Singular Surface Curves in the Resultant Thermodynamics of Shells

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Recent Developments in the Theory of Shells

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 110))

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Abstract

Within six-parameter shells theory we discuss the governing equations of shells with material or non-material singular curves. By singular curve we mean a surface curve where are discontinuities in some surface fields. As an example we consider shells with junctions and shells undergoing stress-induced phase transitions.

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References

  1. Abeyaratne, R., Knowles, J.K.: Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  2. Altenbach, H., Eremeyev, V.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Complex Syst. 3(3), 273–283 (2015)

    Article  Google Scholar 

  3. Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. J. Appl. Math. Mech. ZAMM 89(4), 242–256 (2009)

    Article  Google Scholar 

  4. Altenbach, H., Eremeyev, V.A.: Actual developments in the nonlinear shell theorystate of the art and new applications of the six-parameter shell theory. In: Pietraszkiewicz, W., Górski, J. (Eds.) Shell Structures: Theory and Applications, Taylor & Francis Group, London, vol. 3, pp. 3–12 (2014)

    Google Scholar 

  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)

    Article  Google Scholar 

  6. Berezovski, A., Engelbrecht, J., Maugin, G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, New Jersey (2008)

    Book  Google Scholar 

  7. 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 

  8. 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)

    Article  CAS  Google Scholar 

  9. Bhattacharya, K., James, R.D.: The material is the machine. Science 307(5706), 53–54 (2005)

    Article  CAS  Google Scholar 

  10. Burzyński, S., Chróścielewski, J., Witkowski, W.: Elastoplastic law of Cosserat type in shell theory with drilling rotation. Math. Mech. Solids 20(7), 790–805 (2015)

    Article  Google Scholar 

  11. Burzyński, S., Chróścielewski, J., Daszkiewicz, K., Witkowski, W.: Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory. Compos. Part B: Eng. 107, 203–213 (2016)

    Article  Google Scholar 

  12. Burzyński, S., Chróścielewski, J., Witkowski, W.: Geometrically nonlinear FEM analysis of 6-parameter resultant shell theory based on 2-D Cosserat constitutive model. J. Appl. Math. Mech. ZAMM 96(2), 191–204 (2016)

    Article  Google Scholar 

  13. Chernykh, K.F.: Linear Theory of Shells (in Russian). University Press, 1968 English Translation by NASA-TT-F-II (1964)

    Google Scholar 

  14. Chróścielewski, J., Witkowski, W.: On some constitutive equations for micropolar plates. ZAMM 90(1), 53–64 (2010)

    Article  Google Scholar 

  15. Chróścielewski, J., Makowski, J., Stumpf, H.: Finite element analysis of smooth, folder and multi-shells. Comp. Meth. Appl. Mech. Engrg. 141, 1–46 (1997)

    Article  Google Scholar 

  16. Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i dynamika powłok wielopłatowych: Nieliniowa teoria i metoda elementów skończonych (in Polish). Biblioteka Mechaniki Stosowanej, Wydawnictwo IPPT PAN (2004)

    Google Scholar 

  17. Chróścielewski, J., Pietraszkiewicz, W., Witkowski, W.: On shear correction factors in the non-linear theory of elastic shells. Int. J. Solids Struct. 47(25), 3537–3545 (2010)

    Article  Google Scholar 

  18. Chróścielewski, J., Konopińska, V., Pietraszkiewicz, W.: On elasto-plastic analysis of thin shells with deformable junctions. In: Altenbach, H., Eremeyev, V.A. (eds.) Shell-like Structures—Nonclassical Theories and Applications, pp. 441–452. Springer, Berlin (2011)

    Chapter  Google Scholar 

  19. Chróścielewski, J., Konopińska, V., Pietraszkiewicz, W.: On modelling and non-linear elasto-plastic analysis of thin shells with deformable junctions. J. Appl. Math. Mech. ZAMM 91(6), 477–484 (2011)

    Article  Google Scholar 

  20. Chróścielewski, J., Sabik, A., Sobczyk, B., Witkowski, W.: Nonlinear FEM 2D failure onset prediction of composite shells based on 6-parameter shell theory. Thin-Walled Struct. 105, 207–219 (2016)

    Article  Google Scholar 

  21. Cosserat, E., Cosserat, F.: Theorie des corps deformables. Herman et Fils, Paris (1909)

    Google Scholar 

  22. Cuomo, M., dell’Isola, F., Greco, L., Rizzi, N.L.: First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Compos. Part B: Eng. 115, 423–448 (2017)

    Article  Google Scholar 

  23. Dong, L., Sun, Q.P.: On equilibrium domains in superelastic NiTi tubes - helix versus cylinder. Int. J. Solids Struct. 49, 1063–1076 (2012)

    Article  CAS  Google Scholar 

  24. 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)

    Chapter  Google Scholar 

  25. Eremeyev, V.A., Konopińska-Zmysłowska, V.: On phase equilibrium of an elastic liquid shell with wedge disclination. In: Pietraszkiewicz W, Witkowski W (eds) Shell Structures: Theory and Applications, vol. 4, Taylor & Francis Group, London, pp. 89–92 (2018)

    Google Scholar 

  26. Eremeyev, V.A., Konopińska-Zmysłowska, V.: On the correspondence between two- and three-dimensional Eshelby tensors. Continuum Mechanics and Thermodynamics https://doi.org/10.1007/s00161-019-00754-6 (2019)

  27. Eremeyev, V.A., Pietraszkiewicz, W.: The non-linear theory of elastic shells with phase transitions. J. Elast. 74(1), 67–86 (2004)

    Article  Google Scholar 

  28. Eremeyev, V.A., Pietraszkiewicz, W.: Local symmetry group in the general theory of elastic shells. J. Elast. 85(2), 125–152 (2006)

    Article  Google Scholar 

  29. Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech. 61(1), 41–67 (2009)

    Google Scholar 

  30. Eremeyev, V.A., Pietraszkiewicz, W.: Thermomechanics of shells undergoing phase transition. J. Mech. Phys. Solids 59(7), 1395–1412 (2011)

    Article  CAS  Google Scholar 

  31. Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49(14), 1993–2005 (2012)

    Article  CAS  Google Scholar 

  32. Eremeyev, V.A., Pietraszkiewicz, W.: Phase transitions in thermoviscoelastic shells. In: Hetnarski, R. (ed.) Encyclopedia of Thermal Stresses, pp. 3667–3673. Springer, Berlin (2015)

    Google Scholar 

  33. Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mech. Solids 21(2), 210–221 (2016)

    Article  Google Scholar 

  34. Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2013)

    Book  Google Scholar 

  35. Eremeyev, V.A., Lebedev, L.P., Cloud, M.J.: The Rayleigh and Courant variational principles in the six-parameter shell theory. Math. Mech. Solids 20(7), 806–822 (2015)

    Article  Google Scholar 

  36. Eshelby, J.D.: The force on an elastic singularity. Phil. Trans. R Soc. Lond A 244(877), 87–112 (1951)

    Article  Google Scholar 

  37. Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Royal Soc. Lond. Ser. A Math. Phys. Sci. 241(1226), 376–396 (1957)

    Article  Google Scholar 

  38. Eshelby, J.D.: The elastic energy-momentum tensor. J. Elast. 5(3–4), 321–335 (1975)

    Article  Google Scholar 

  39. 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)

    Article  CAS  Google Scholar 

  40. Giorgio, I., Grygoruk, R., dellIsola F, Steigmann, D.J.: Pattern formation in the three-dimensional deformations of fibered sheets. Mech. Res. Commun. 69, 164–171 (2015)

    Article  Google Scholar 

  41. Giorgio, I., Harrison, P., dell’Isola, F., Alsayednoor, J., Turco, E.: Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches. In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences vol. 474(2216), 20180,063 (2018)

    Article  CAS  Google Scholar 

  42. Giorgio, I., dell’Isola, F., Steigmann, D.J.: Edge effects in hypar nets. Comptes Rendus Mécanique (2019)

    Google Scholar 

  43. Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin (2000)

    Google Scholar 

  44. Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  45. He, Y.J., Sun, Q.P.: Macroscopic equilibrium domain structure and geometric compatibility in elastic phase transition of thin plates. Int. J. Mech. Sci. 52, 198–211 (2010)

    Article  Google Scholar 

  46. Konopińska, V., Pietraszkiewicz, W.: Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells. Int. J. Solids Struct. 44(1), 352–369 (2007)

    Article  Google Scholar 

  47. Konopińska, V., Pietraszkiewicz, W.: On jump conditions at non-material singular curves in the resultant shell thermomechanics. In: Pietraszkiewicz, W., Górski, J. (Eds.) Shell Structures: Theory and Applications, vol. 3, Taylor & Francis Group, London, pp. 117–120 (2014)

    Google Scholar 

  48. Konopińska-Zmysłowska, V., Eremeyev, V.A.: On axially symmetric shell problems with reinforced junctions. In: Owen, R., de Borst, R., Reese, J., Pearce, C. (eds.) ECCM VI-ECFD VII Proceedings, pp. 3681–3688. ECCOMAS, Glasgow (2018)

    Google Scholar 

  49. Kosiński, W.: Field Singularities and Wave Analysis in Continuum Mechanics. Ellis Horwood, Chicester and PWN, Warszawa (1986)

    Google Scholar 

  50. Lagoudas, D.C. (Ed.): Shape Memory Alloys. Modeling and Engineering Applications, Springer, Berlin (2008)

    Google Scholar 

  51. Libai, A., Simmonds, J.G.: Nonlinear elastic shell theory. Adv. Appl. Mech. 23, 271–371 (1983)

    Article  Google Scholar 

  52. Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  53. Makowski, J., Pietraszkiewicz, W.: Thermomechanics of Shells with Singular Curves. Wydawnictwo IMP PAN, Gdańsk (2002)

    Google Scholar 

  54. Makowski, J., Pietraszkiewicz, W., Stumpf, H.: On the general form of jump conditions for thin irregular shells. Arch. Mech. 50(2), 483–495 (1998)

    Google Scholar 

  55. Makowski, J., Pietraszkiewicz, W., Stumpf, H.: Jump conditions in the non-linear theory of thin irregular shells. J. Elast. 54(1), 1–26 (1999)

    Article  Google Scholar 

  56. Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman Hall, London (1993)

    Book  Google Scholar 

  57. Maugin, G.A.: Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford (1999)

    Google Scholar 

  58. Maugin, G.A.: Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. CRC Pressl, Boca Raton (2011)

    Google Scholar 

  59. Miśkiewicz, M.: Structural response of existing spatial truss roof construction based on cosserat rod theory. Contin. Mech. Thermodyn. 31(1), 79–99 (2019)

    Article  Google Scholar 

  60. 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 

  61. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)

    Book  Google Scholar 

  62. Ng, K.L., Sun, Q.P.: Stress-induced phase transformation and detwinning in NiTi polycrystalline shape memory alloy tubes. Mech. Mater. 38(1–2), Sp. Iss. SI):41–56 (2006)

    Article  Google Scholar 

  63. Novozhilov, V.V., Chernykh, K.F., Mikhailovskii, E.I.: Linear Theory of Thin Shells (in Russian). Politekhnika, Leningrad (1991)

    Google Scholar 

  64. Pietraszkiewicz, W.: Refined resultant thermomechanics of shells. Int. J. Eng. Sci. 49(10), 1112–1124 (2011)

    Article  Google Scholar 

  65. Pietraszkiewicz, W.: On a description of deformable junctions in the resultant nonlinear shell theory. In: Advanced Methods of Continuum Mechanics for Materials and Structures, pp. 457–468. Springer (2016)

    Google Scholar 

  66. Pietraszkiewicz, W.: The resultant linear six-field theory of elastic shells: what it brings to the classical linear shell models? J. Appl. Math. Mech. ZAMM 96(8), 899–915 (2016)

    Article  Google Scholar 

  67. Pietraszkiewicz, W., Konopińska, V.: On unique two-dimensional kinematics for the branching shells. Int. J. Solids Struct. 48, 2238–2244 (2011)

    Article  Google Scholar 

  68. Pietraszkiewicz, W., Konopińska, V.: Singular curve in the resultant thermomechanics of shells. Int. J. Eng. Sci. 80, 21–31 (2014)

    Article  Google Scholar 

  69. Pietraszkiewicz, W., Konopińska, V.: Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells. Int. J. Solids Struct. 51, 2133–2143 (2014)

    Article  Google Scholar 

  70. Pietraszkiewicz, W., Konopińska, V.: On refined constitutive equations in the six-field theory of elastic shells. In: Pietraszkiewicz, W., Górski, J. (Eds.) Shell Structures: Theory and Applications, vol. 3, Taylor & Francis Group, London, pp. 137–140 (2014)

    Google Scholar 

  71. Pietraszkiewicz, W., Konopińska, V.: Junctions in shell structures: a review. Thin-Walled Struct. 95, 310–334 (2015)

    Article  Google Scholar 

  72. Pietraszkiewicz, W., Eremeyev, V.A., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. J. Appl. Math. Mech. ZAMM 87(2), 150–159 (2007)

    Article  Google Scholar 

  73. Roytburd, A.L., Slutsker, J.: Theory of multilayer SMA actuators. Mater. Trans. 43(5), 1023–1029 (2002)

    Article  CAS  Google Scholar 

  74. Straughan, B.: Stability and Wave Motion in Porous Media, vol. 165. Springer, New York (2008)

    Google Scholar 

  75. Teng, J.G.: Intersections in steel shell structures. Prog. Struct. Eng. Mater. 2, 459–471 (2000)

    Article  Google Scholar 

  76. Teng, J.G.: Shell junctions. In: Teng, J.G., Rotter, J.M. (Eds.) Buckling of Thin Metal Shells, pp. 369–408. Spon Press, Taylor & Francis (2004)

    Google Scholar 

  77. Truesdell, C.: Rational Termodynamics, 2nd edn. Springer, New York (1984)

    Book  Google Scholar 

  78. Truesdell, C., Toupin, R.A.: Encyclopedia of Physics, vol III/1, Springer, Berlin, chap The classical field theories, pp. 226–858 (1961)

    Google Scholar 

  79. Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4), 85 (2016)

    Article  Google Scholar 

  80. S̆ilhavý, M.: The Mechanics and Thermodynamics of Continuum Media. Springer, Berlin (1997)

    Google Scholar 

  81. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1999)

    Book  Google Scholar 

  82. Witkowski, W.: 4-node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom. Comput. Mech. 43, 307–319 (2009)

    Article  Google Scholar 

  83. Yoneyama, T., Miyazaki, S. (eds.): Shape Memory Alloys and Biomedical Applications. Woodhead P Ltd, Cambridge (2009)

    Google Scholar 

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Acknowledgements

The first author acknowledges financial support from the National Centre of Science of Poland with the grant DEC – 2012/05/D/ST8/02298.

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Authors and Affiliations

  1. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, ul. Gabriela Narutowicza 11/12, 80-233, Gdańsk, Poland

    Violetta Konopińska-Zmysłowska & Victor A. Eremeyev

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  1. Violetta Konopińska-Zmysłowska
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  2. Victor A. Eremeyev
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Corresponding author

Correspondence to Violetta Konopińska-Zmysłowska .

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Editors and Affiliations

  1. Faculty of Mechanical Engineering, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Jacek Chróścielewski

  3. Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland

    Victor A. Eremeyev

  4. Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland

    Krzysztof Wiśniewski

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Konopińska-Zmysłowska, V., Eremeyev, V.A. (2019). Singular Surface Curves in the Resultant Thermodynamics of Shells. In: Altenbach, H., Chróścielewski, J., Eremeyev, V., Wiśniewski, K. (eds) Recent Developments in the Theory of Shells . Advanced Structured Materials, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-030-17747-8_20

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