Abstract
Using the direct approach the basic relations of the nonlinear micropolar shell theory are considered. Within the framework of this theory the shell can be considered as a deformable surface with attached three unit orthogonal vectors, so-called directors. In other words the micropolar shell is a two-dimensional (2D) Cosserat continuum or micropolar continuum. Each point of the micropolar shell has three translational and three rotational degrees of freedom as in the rigid body dynamics. In this theory the rotations are kinematically independent on translations. The interaction between of any two parts of the shell is described by the forces and moments only. So at the shell boundary six boundary conditions have to be given. In contrast to Kirchhoff-Love or Reissner’s models of shells the drilling moment acting on the shell surface can be taken into account. In the paper we derive the equilibrium equations of the shell theory using the principle of virtual work. The strain measures are introduced on the base of the principle of frame indifference. The boundary-value static and dynamic problems are formulated in Lagrangian and Eulerian coordinates. In addition, some variational principles are presented. For the general constitutive equations we formulate some constitutive restrictions, for example, the Coleman-Noll inequality, the Hadamard inequality, etc. Finally, we discuss the equilibrium of shells made of materials undergoing phase transformations, such as martensitic transformations, and formulate the compatibility conditions on the phase interface.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abeyaratne R, Knowles J.K (2006) Evolution of Phase Transitions. A Continuum Theory. Cambridge University Press, Cambridge
Agrawal A, Steigmann D.J (2008) Coexistent fluid-phase equilibria in biomembranes with bending elasticity. Journal of Elasticity 93(1):63–80
Altenbach H, Eremeyev V.A, Lebedev L.P, Rendón L.A (2010) Acceleration waves and ellipticity in thermoelastic micropolar media. Arch. Appl. Mech 80(3):217–227
Altenbach H, Zhilin P.A (1988) A general theory of elastic simple shells (in Russian). Uspekhi Mekhaniki (Advances in Mechanics) 11(4):107–148
Altenbach J, Altenbach H, Eremeyev V.A (2010) On generalized Cosserat-type theories of plates and shells. A short review and bibliography. Arch. Appl. Mech. 80(1):73–92
Berezovski A, Engelbrecht J, Maugin G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, New Jersey et al. (2008)
Bhattacharya K (2003) Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press, Oxford
Bhattacharya K, James R.D (1999) A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 36:531–576
Boulbitch A.A (1999) Equations of heterophase equilibrium of a biomembrane. Archive of Applied Mechanics 69(2):83–93
Chróścielewski J, Makowski J, Pietraszkiewicz W (2004) Statics and Dynamics of Multyfolded Shells. Nonlinear Theory and Finite Elelement Method. Wydawnictwo IPPT PAN, Warszawa
Chróścielewski J, Pietraszkiewicz W, Witkowski W (2010) On shear correction factors in the non-linear theory of elastic shells. Int. J. Solids Struct. 47(25–26):3537–3545
Chróścielewski J, Witkowski W (2010) On some constitutive equations for micropolar plates. ZAMM 90(1):53–64
Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Hermann Editeurs, Paris (1909) (Reprint, Gabay, Paris, 2008)
Courant R, Hilbert D (1991) Methods of Mathematical Physics, Vol.~1. Wiley, New York
Eremeyev V.A (2005) Acceleration waves in micropolar elastic media. Doklady Physics 50(4):204–206
Eremeyev V.A (2005) Nonlinear micropolar shells: theory and applications. In: Pietraszkiewicz W, Szymczak C (eds.) Shell Structures: Theory and Applications. Taylor & Francis, London pp. 11–18
Eremeyev V.A, Pietraszkiewicz W (2004) The non-linear theory of elastic shells with phase transitions. J. Elasticity 74(1):67–86
Eremeyev V.A, Pietraszkiewicz W (2006) Local symmetry group in the general theory of elastic shells. J. Elasticity 85(2):125–152
Eremeyev V.A, Pietraszkiewicz W (2009) Phase transitions in thermoelastic and thermoviscoelastic shells. Arch. Mech. 61(1):41–67
Eremeyev V.A, Pietraszkiewicz W (2010) On tension of a two-phase elastic tube. In: W. Pietraszkiewicz, I. Kreja (eds.) Shell Structures. Theory and Applications. Vol. 2.. CRC Press, Boca Raton pp. 63–66
Eremeyev V.A, Zubov L.M (1994) On the stability of elastic bodies with couple stresses (in Russian). Izv. RAN. Mekanika Tvedogo Tela (Mechanics of Solids) (3):181–190
Eremeyev V.A, Zubov L.M (2007) On constitutive inequalities in nonlinear theory of elastic shells. ZAMM 87(2):94–101
Eremeyev V.A, Zubov L.M (2008) Mechanics of Elastic Shells (in Russian). Nauka, Moscow
Ericksen J.L, Truesdell C (1958) Exact theory of stress and strain in rods and shells. Arch. Rat. Mech. Analysis 1(1):295–323
Eringen A.C (1999) Microcontinuum Field Theory. I. Foundations and Solids. Springer, New York
Eringen A.C (2001) Microcontinuum Field Theory. II. Fluent Media. Springer, New York
Fichera G (1972) Existence theorems in elasticity. In: S. Flügge (ed.) Handbuch der Physik, vol. VIa/2. Springer, Berlin pp. 347–389
Gibbs, J.W.: On the equilibrium of heterogeneous substances. In: The Collected Works of J. Willard Gibbs. pp. 53–353. Longmans, Green & Co, New York (1928)
Grinfeld,M (1991) Thermodynamics Methods in the Theory of Heterogeneous Systems. Longman, Harlow
Gurtin M.E (2000) Configurational Forces as Basic Concepts of Continuum Physics. Springer-Verlag, Berlin
He Y.J, Sun Q (2009) Scaling relationship on macroscopic helical domains in NiTi tubes. Int. J. Solids Struct. 46(24):4242–4251
He Y.J, Sun Q (2010) Macroscopic equilibrium domain structure and geometric compatibility in elastic phase transition of thin plates. Int. J. Mech. Sci. 52(2):198–211
He Y.J, Sun Q (2010) Rate-dependent domain spacing in a stretched NiTi strip. Int. J. Solids Struct. 47(20):2775–2783
James R.D, Rizzoni R (2000) Pressurized shape memory thin films. J. Elasticity 59:399–436
Kienzler R, Herrman G (2000) Mechanics in Material Space with Applications to Defect and Fracure Mechanics. Springer-Verlag, Berlin
Lebedev L.P, Cloud M.J, Eremeyev V.A (2010) Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey
Li Z.Q, Sun Q (2002) The initiation and growth of macroscopic martensite band in nano-grained NiTi microtube under tension. Int. J. Plasticity 18(11):1481–1498
Libai A, Simmonds J.G (1983) Nonlinear elastic shell theory. Adv. Appl. Mech. 23:271–371
Libai A, Simmonds J.G (1998) The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge
Lions J.L, Magenes E (1968) Problèmes aux limites non homogènes et applications. Dunod, Paris
Lurie A.I (2001) Analytical Mechanics. Foundations of Engineering Mechanics. Springer, Berlin
Lurie A.I (2005) Theory of Elasticity. Foundations of Engineering Mechanics. Springer, Berlin
Maugin G.A (1993) Material Inhomogeneities in Elasticity. Chapman Hall, London
Maugin G.A (2011) A historical perspective of generalized continuum mechanics. In: H. Altenbach, V.I. Erofeev, G.A. Maugin (eds.) Mechanics of Generalized Continua. From the Micromechanical Basics to Engineering Applications. Springer, Berlin
Maugin, G.A., Metrikine, A.V. : (eds.): Mechanics of Generalized Continua: One hundred years after the Cosserats. Springer, New York (2010)
Miyazaki, S., Fu, Y.Q., Huang, W.M. (eds.): Thin Film Shape Memory Alloys: Fundamentals and Device Applications. Cambridge University Press, Cambridge (2009)
Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon-Press, Oxford et al. (1986)
Pietraszkiewicz W (1979) Consistent second approximation to the elastic strain energy of a shell. ZAMM 59:206–208
Pietraszkiewicz, W.: Finite Rotations and Langrangian Description in the Non-linear Theory of Shells. Polish Sci. Publ, Warszawa-Poznań (1979)
Pietraszkiewicz W (1989) Geometrically nonlinear theories of thin elastic shells. Uspekhi Mechaniki (Advances in Mechanics) 12(1):51–130
Pietraszkiewicz W (2001) Teorie nieliniowe powłok. In: C. Woźniak (ed.) Mechanika spre\(\dot{\hbox{z}}\)ystych płyt i powłok. PWN, Warszawa pp. 424–497
Pietraszkiewicz W, Eremeyev V.A (2009) On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46:774–787
Pietraszkiewicz W, Eremeyev V.A (2009) On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. Int. J. Solids Struct. 46(11–12):2477–2480
Pietraszkiewicz W, Eremeyev V.A, Konopińska V (2007) Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM 87(2):150–159
Reissner E (1974) Linear and nonlinear theory of shells. In: Y.C. Fung, E.E. Sechler (eds.) Thin Shell Structures. Prentice-Hall, Englewood Cliffs, NJ pp. 29–44
Truesdell C (1984) Rational Thermodynamics, 2nd edn. Springer, New York
Truesdell C (1991) A First Course in Rational Continuum Mechanics, 2nd edn. Academic Press, New York
Truesdell C, Noll, W (1965) The nonlinear field theories of mechanics. In: S. Flügge (ed.) Handbuch der Physik, Vol. III/3. Springer, Berlin pp. 1–602
Zhilin P.A (1976) Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12:635 – 648
Zubov L.M (1997) Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Altenbach, H., Eremeyev, V.A., Lebedev, L.P. (2011). Micropolar Shells as Two-dimensional Generalized Continua Models. In: Altenbach, H., Maugin, G., Erofeev, V. (eds) Mechanics of Generalized Continua. Advanced Structured Materials, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19219-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-19219-7_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19218-0
Online ISBN: 978-3-642-19219-7
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)