SpringerLink
Log in

Basics of Mechanics of Micropolar Shells

  • Chapter
  • First Online:
Shell-like Structures

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 572))

Abstract

The chapter is devoted to the introduction to the nonlinear theory of micropolar shells called also six-parametric shell theory. Within the theory a shell is described as a deformable directed material surface each point of which has six degrees of freedom (DOF), i.e. three translational and three rotational DOF. In other words the shell kinematics coincides with the kinematics of a two-dimensional (2D) micropolar or Cosserat body. Here we present the basic equations of the micropolar shell theory including variational statements, compatibility conditions, etc.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  • Abeyaratne, R., & Knowles, J. K. (2006). Evolution of phase transitions: A continuum theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Agranovich, M. (1997). Elliptic boundary problems. In M. Agranovich, Y. Egorov, & M. Shubin (Eds.), Partial differential equations IX: Elliptic boundary problems (pp. 1–144)., Encyclopaedia of Mathematical Sciences, volume 79 Berlin: Springer.

    Chapter  Google Scholar 

  • Agrawal, A., & Steigmann, D. J. (2008). Coexistent fluid-phase equilibria in biomembranes with bending elasticity. The Journal of Elasticity, 93(1), 63–80.

    Article  MathSciNet  MATH  Google Scholar 

  • Akay, A., Xu, Z., Carcaterra, A., & Koç, I. M. (2005). Experiments on vibration absorption using energy sinks. The Journal of the Acoustical Society of America, 118(5), 3043–3049.

    Article  Google Scholar 

  • Alijani, F., & Amabili, M. (2014). Non-linear vibrations of shells: A literature review from 2003 to 2013. International Journal of Non-Linear Mechanics, 58, 233–257.

    Article  Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2010). On the effective stiffness of plates made of hyperelastic materials with initial stresses. International Journal of Non-Linear Mechanics, 45(10), 976–981.

    Article  Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2011a) Mechanics of viscoelastic plates made of FGMs. In J. Murín, V. Kompiš, & V. Kutiš (Eds.), Computational modelling and advanced simulations (pp. 33–48), volume 24 of Computational Methods in Applied Sciences. Springer.

    Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (Eds.). (2013a) Generalized continua from the theory to engineering applications, volume 541 of CISM International Centre for Mechanical Sciences. Springer Vienna.

    Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2014a). Vibration analysis of non-linear 6-parameter prestressed shells. Meccanica, 49, 1751–1761.

    Article  MathSciNet  MATH  Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2008). On the analysis of viscoelastic plates made of functionally graded materials. ZAMM, 88(5), 332–341.

    Article  MathSciNet  MATH  Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2009a). On the bending of viscoelastic plates made of polymer foams. Acta Mechanica, 204(3–4), 137–154.

    Article  MATH  Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2009b). On the time-dependent behavior of FGM plates. Key Engineering Materials, 399, 63–70.

    Article  Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2011b). On the shell theory on the nanoscale with surface stresses. The International Journal of Engineering Science, 49, 1294–1301.

    Article  MathSciNet  Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2009c). On the linear theory of micropolar plates. ZAMM, 89(4), 242–256.

    Article  MathSciNet  MATH  Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2011c). Shell-like Structures: Non-classical Theories and Applications, volume 15 of Advanced Structured Materials. Springer.

    Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2013b). Cosserat-type shells. In H. Altenbach & V. A. Eremeyev (Eds.), Generalized continua from the theory to engineering applications (Vol. 541, pp. 131–178). CISM Courses and Lectures Wien: Springer.

    Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2014b). Actual developments in the nonlinear shell theory—state of the art and new applications of the six-parameter shell theory. In W. Pietraszkiewicz & J. Górski (Eds.), Shell structures: Theory and applications (Vol. 3, pp. 3–12). Taylor & Francis.

    Google Scholar 

  • Altenbach, H., & Eremeyev, V. A. (2015). On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Mathematics and Mechanics of Complex Systems, 3(3), 273–283.

    Article  MathSciNet  MATH  Google Scholar 

  • Altenbach, H., & Mikhasev, G. (Eds.). (2014). Shell and Membrane Theories in Mechanics and Biology: From Macro-to Nanoscale Structures, volume 45 of Advanced Structured Materials. Springer.

    Google Scholar 

  • Altenbach, H., & Morozov, N. F. (Eds.). (2013). Surface Effects in Solid Mechanics—Models, Simulations and Applications. Heidelberg: Springer.

    MATH  Google Scholar 

  • Altenbach, H., Eremeyev, V. A., & Morozov, N. F. (2009). Linear theory of shells taking into account surface stresses. Doklady Physics, 54(12), 531–535.

    Article  Google Scholar 

  • Altenbach, H., Eremeev, V. A., & Morozov, N. F. (2010a). On equations of the linear theory of shells with surface stresses taken into account. Mechanics of Solids, 45(3), 331–342.

    Article  Google Scholar 

  • Altenbach, H., Eremeyev, V. A., Lebedev, L. P., & Rendón, L. A. (2010b). Acceleration waves and ellipticity in thermoelastic micropolar media. Archive of Applied Mechanics, 80(3), 217–227.

    Article  MATH  Google Scholar 

  • Altenbach, H., Bîrsan, M., & Eremeyev, V. A. (2012a). On a thermodynamic theory of rods with two temperature fields. Acta Mechanica, 223(8), 1583–1596.

    Article  MathSciNet  MATH  Google Scholar 

  • Altenbach, H., Eremeyev, V. A., & Morozov, N. F. (2012b). Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale. The International Journal of Engineering Science, 59, 83–89.

    Article  MathSciNet  Google Scholar 

  • Altenbach, H., Bîrsan, M., & Eremeyev, V. A. (2013). Cosserat-type rods. In H. Altenbach & V. A. Eremeyev (Eds.), Generalized Continua from the Theory to Engineering Applications (Vol. 541, pp. 179–248). CISM Courses and Lectures Wien: Springer.

    Google Scholar 

  • Altenbach, J., Altenbach, H., & Eremeyev, V. A. (2010c). On generalized Cosserat-type theories of plates and shells: A short review and bibliography. Archive of Applied Mechanics, 80, 73–92.

    Article  MATH  Google Scholar 

  • Amabili, M. (2008). Nonlinear vibrations and stability of shells and plates. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Ambartsumyan, S. A. (1970). Theory of anisotropic plates: Strength, stability, vibration. Stamford: Technomic.

    Google Scholar 

  • Andreaus, U., dell’Isola, F., & Porfiri, M. (2004). Piezoelectric passive distributed controllers for beam flexural vibrations. Journal of Vibration and Control, 10(5), 625–659.

    Article  MATH  Google Scholar 

  • Bauchau, O. A. (2010). Flexible Multibody Dynamics (Vol. 176). Solid Mechanics and its Applications Dordrecht: Springer.

    Google Scholar 

  • Bauchau, O. A., & Trainelli, L. (2003). The vectorial parameterization of rotation. Nonlinear Dynamics, 32(1), 71–92.

    Article  MathSciNet  MATH  Google Scholar 

  • Berdichevsky, V. L. (2009). Variational principles of continuum mechanics. I. Fundamentals. Heidelberg: Springer.

    MATH  Google Scholar 

  • Berezovski, A., Engelbrecht, J., & Maugin, G. A. (2008). Numerical simulation of waves and fronts in inhomogeneous solids. New Jersey: World Scientific.

    MATH  Google Scholar 

  • Bhattacharya, K. (2003). Microstructure of martensite: Why it forms and how it gives rise to the shape-memory effect. Oxford: Oxford University Press.

    MATH  Google Scholar 

  • Bhattacharya, K., & James, R. D. (1999). A theory of thin films of martensitic materials with applications to microactuators. Journal of the Mechanics and Physics of Solids, 36(3), 531–576.

    Article  MathSciNet  MATH  Google Scholar 

  • Bigoni, D., & Drugan, W. J. (2007). Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. Transactions of ASME. Journal of Applied Mechanics, 74(4), 741–753.

    Article  MathSciNet  Google Scholar 

  • Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V. A., & Pietras, D. (2012). Deformation analysis of functionally graded beams by the direct approach. Composites B: Engineering, 43(3), 1315–1328.

    Article  Google Scholar 

  • Boulbitch, A. A. (1999). Equations of heterophase equilibrium of a biomembrane. Archive of Applied Mechanics, 69(2), 83–93.

    Article  MATH  Google Scholar 

  • Carcaterra, A., Akay, A., & Bernardini, C. (2012). Trap** of vibration energy into a set of resonators: Theory and application to aerospace structures. Mechanical Systems and Signal Processing, 26, 1–14.

    Article  Google Scholar 

  • Carrera, E., Brischetto, S., & Nali, P. (2011). Plates and shells for smart structures: Classical and advanced theories for modeling and analysis. Chichester: Wiley.

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Chróścielewski, J., Makowski, J., & Pietraszkiewicz, W. (2004a). Statics and dynamics of multifolded shells: Nonlinear theory and finite element method. Warsaw: IPPT PAN.

    Google Scholar 

  • Chróścielewski, J., Makowski, J., & Pietraszkiewicz, W. (2004b). Statics and dynamics of multyfolded shells: Nonlinear theory and finite elelement method (in Polish). Warszawa: Wydawnictwo IPPT PAN.

    Google Scholar 

  • Chróścielewski, J., Pietraszkiewicz, W., & Witkowski, W. (2010). On shear correction factors in the non-linear theory of elastic shells. International Journal of Solids and Structures, 47(25–26), 3537–3545.

    Article  MATH  Google Scholar 

  • Ciarlet, Ph. (1997). Mathematical elasticity, Volume II: Theory of plates. Amsterdam: Elsevier.

    Google Scholar 

  • Ciarlet, Ph. (2000). Mathematical elasticity, Volume III: Theory of shells. Amsterdam: Elsevier.

    Google Scholar 

  • Courant, R., & Hilbert, D. (1991). Methods of mathematical physics (Vol. 1). New York: Wiley.

    MATH  Google Scholar 

  • de Gennes, P. G., Brochard-Wyart, F., & Quéré, D. (2004). Capillarity and wetting phenomena: Drops, bubbles, pearls, waves. New York: Springer.

    Book  MATH  Google Scholar 

  • dell’Isola, F., & Vidoli, S. (1998). Dam** of bending waves in truss beams by electrical transmission lines with PZT actuators. Archive of Applied Mechanics, 68(9), 626–636.

    Google Scholar 

  • dell’Isola, F., Porfiri, M., & Vidoli, S. (2003). Piezo-ElectroMechanical (PEM) structures: passive vibration control using distributed piezoelectric transducers. Comptes Rendus Mécanique, 331(1), 69–76.

    Google Scholar 

  • dell’Isola, F., Della Corte, A., Greco, L., & Luongo, A. (2016a). Plane bias extension test for a continuum with two inextensible families of fibers: A variational treatment with lagrange multipliers and a perturbation solution. International Journal of Solids and Structures, 81, 1–12.

    Google Scholar 

  • dell’Isola, F., Giorgio, I., Pawlikowski, M., & Rizzi, N. L. (2016b). Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proceedings of the Royal Society of London. Series A, 472(2185), 20150790.

    Google Scholar 

  • Diebels, S., & Steeb, H. (2003). Stress and couple stress in foams. Computational Materials Science, 28(3), 714–722.

    Article  Google Scholar 

  • Duan, H. L., Wang, J., & Karihaloo, B. L. (2008). Theory of elasticity at the nanoscale. In Advances in applied mechanics (Vol. 42, pp. 1–68). Elsevier.

    Google Scholar 

  • Elliott, C. M., & Stinner, B. (2010). A surface phase field model for two-phase biological membranes. SIAM Journal on Applied Mathematics, 70(8), 2904–2928.

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V. A. (2005a). Nonlinear micropolar shells: Theory and applications. In W. Pietraszkiewicz & C. Szymczak (Eds.), Shell structures: Theory and applications (pp. 11–18). London: Taylor & Francis.

    Google Scholar 

  • Eremeyev, V. A. (2005b). Acceleration waves in micropolar elastic media. Doklady Physics, 50(4), 204–206.

    Article  Google Scholar 

  • Eremeyev, V. A., & Lebedev, L. P. (2011). Existence theorems in the linear theory of micropolar shells. ZAMM, 91(6), 468–476.

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V. A., & Pietraszkiewicz, W. (2006). Local symmetry group in the general theory of elastic shells. Journal of Elasticity, 85(2), 125–152.

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V. A., & Pietraszkiewicz, W. (2011). Thermomechanics of shells undergoing phase transition. Journal of the Mechanics and Physics of Solids, 59(7), 1395–1412.

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V. A., & Pietraszkiewicz, W. (2004). The non-linear theory of elastic shells with phase transitions. The Journal of Elasticity, 74(1), 67–86.

    Article  MathSciNet  MATH  Google Scholar 

  • 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, pp. 63–66). Boca Raton: CRC Press.

    Google Scholar 

  • Eremeyev, V. A., & Pietraszkiewicz, W. (2009). Phase transitions in thermoelastic and thermoviscoelastic shells. Archives of Mechanics, 61(1), 41–67.

    MathSciNet  MATH  Google Scholar 

  • Eremeyev, V. A., & Pietraszkiewicz, W. (2012). Material symmetry group of the non-linear polar-elastic continuum. International Journal of Solids and Structures, 49(14), 1993–2005.

    Article  Google Scholar 

  • Eremeyev, V. A., & Pietraszkiewicz, W. (2014). Editorial: Refined theories of plates and shells. ZAMM, 94(1–2), 5–6.

    Article  MathSciNet  Google Scholar 

  • Eremeyev, V. A., & Pietraszkiewicz, W. (2016). Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Mathematics and Mechanics of Solids, 21(2), 210–221.

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V. A., & Zubov, L. M. (1994). On the stability of elastic bodies with couple stresses. Mechanics of Solids, 29(3), 172–181.

    Google Scholar 

  • Eremeyev, V. A., & Zubov, L. M. (2007). On constitutive inequalities in nonlinear theory of elastic shells. ZAMM, 87(2), 94–101.

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V. A., & Zubov, L. M. (2008). Mechanics of elastic shells (in Russian). Moscow: Nauka.

    MATH  Google Scholar 

  • Eremeyev, V. A., Altenbach, H., & Morozov, N. F. (2009). The influence of surface tension on the effective stiffness of nanosize plates. Doklady Physics, 54(2), 98–100.

    Article  MATH  Google Scholar 

  • Eremeyev, V. A., Lebedev, L. P., & Altenbach, H. (2013). Foundations of microplar mechanics. Heidelberg: Springer.

    Book  MATH  Google Scholar 

  • Eremeyev, V. A., Ivanova, E. A., & Morozov, N. F. (2015a). On free oscillations of an elastic solids with ordered arrays of nano-sized objects. Continuum Mechanics and Thermodynamics, 27(4–5), 583–607.

    Article  MathSciNet  MATH  Google Scholar 

  • Eremeyev, V. A., Lebedev, L. P., & Cloud, M. J. (2015b). The Rayleigh and Courant variational principles in the six-parameter shell theory. Mathematics and Mechanics of Solids, 20(7), 806–822.

    Google Scholar 

  • Eringen, A. C. (1967a). Theory of micropolar plates. ZAMP, 18(1), 12–30.

    Article  MathSciNet  Google Scholar 

  • Eringen, A. C. (1967b). Linear theory of micropolar viscoelasticity. International Journal of Engineering Science, 5(2), 191–204.

    Article  MATH  Google Scholar 

  • Eringen, A. C. (1999). Microcontinuum field theory. I: Foundations and solids. New York: Springer.

    Book  MATH  Google Scholar 

  • Eringen, A. C., & Maugin, G. A. (1990). Electrodynamics of continua. New York: Springer.

    Book  Google Scholar 

  • Fichera, G. (1972). Existence theorems in elasticity. In S. Flügge (Ed.), Handbuch der Physik (pp. 347–389), volume VIa/2 Berlin: Springer.

    Google Scholar 

  • Finn, R. (1986). Equilibrium capillary surfaces. New York: Springer.

    Book  MATH  Google Scholar 

  • Fu, Y. B., & Ogden, R. W. (1999). Nonlinear stability analysis of pre-stressed elastic bodies. Continuum Mechanics and Thermodynamics, 11, 141–172.

    Article  MathSciNet  MATH  Google Scholar 

  • Germain, P. (1973a). La méthode des puissances virtuelles en mécanique des milieux continus - première partie, théorie du second gradient. Journal de Mécanique, 12, 235–274.

    MathSciNet  MATH  Google Scholar 

  • Germain, P. (1973b). The method of virtual power in continuum mechanics. part 2: Microstructure. SIAM Journal on Applied Mathematics, 25(3), 556–575.

    Article  MathSciNet  MATH  Google Scholar 

  • Giorgio, I., Grygoruk, R., dell’Isola, F., & Steigmann, D. J. (2015). Pattern formation in the three-dimensional deformations of fibered sheets. Mechanics Research Communications, 69, 164–171.

    Article  Google Scholar 

  • Goda, I., Assidi, M., Belouettar, S., & Ganghoffer, J. F. (2012). A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. Journal of the Mechanical Behavior of Biomedical Materials, 16, 87–108.

    Article  Google Scholar 

  • Goldenveizer, A. L. (1976). Theory of thin elastic shells (in Russ.). Moscow: Nauka.

    Google Scholar 

  • Green, A. E., & Naghdi, P. M. (1979). On thermal effects in the theory of shells. Proceedings of the Royal Society of London Series A, 365(1721), 161–190.

    Article  MathSciNet  MATH  Google Scholar 

  • Green, A. E., & Naghdi, P. M. (1970). Non-isothermal theory of rods, plates and shells. International Journal of Solids and Structures, 6, 209–244.

    Article  MATH  Google Scholar 

  • Gurtin, M. E., & Murdoch, A. I. (1975). A continuum theory of elastic material surfaces. The Archive for Rational Mechanics and Analysis, 57(4), 291–323.

    Article  MathSciNet  MATH  Google Scholar 

  • He, Y. J., & Sun, Q. P. (2009). Effects of structural and material length scales on stress-induced martensite macro-domain patterns in tube configurations. The International Journal of Solids and Structures, 46(16), 3045–3060.

    Article  MATH  Google Scholar 

  • He, Y. J., & Sun, Q. P. (2010). Macroscopic equilibrium domain structure and geometric compatibility in elastic phase transition of thin plates. The International Journal of Mechanical Sciences, 52(2), 198–211.

    Article  Google Scholar 

  • Hörmander, L. (1976). Linear partial differential equations (4th ed., Vol. 116). A Series of Comprehensive Studies in Mathematics Berlin: Springer.

    Google Scholar 

  • Jaiani, G. (2011). Cusped shell-like structures. Heidelberg: Springer.

    Book  MATH  Google Scholar 

  • Jaiani, G., & Podio-Guidugli, P. (2008). IUTAM Symposium on Relations of Shell, Plate, Beam and 3D Models: Proceedings of the IUTAM Symposium on the Relations of Shell, Plate, Beam, and 3D Models Dedicated to the Centenary of Ilia Vekua’s Birth, held Tbilisi, Georgia, April 23–27, 2007, Vol. 9. Springer.

    Google Scholar 

  • James, R. D., & Rizzoni, R. (2000). Pressurized shape memory thin films. The Journal of Elasticity, 59(1–3), 399–436.

    Article  MathSciNet  MATH  Google Scholar 

  • Javili, A., McBride, A., & Steinmann, P. (2012). Thermomechanics of solids with lower-dimensional energetics: On the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Applied Mechanics Reviews, 65, 010802–1–31.

    Google Scholar 

  • Javili, A., dell’Isola, F., & Steinmann, P. (2013). Geometrically nonlinear higher-gradient elasticity with energetic boundaries. Journal of the Mechanics and Physics of Solids, 61(12), 2381–2401.

    Article  MathSciNet  MATH  Google Scholar 

  • Jemielita, G. (2001). Meandry teorii płyt i powłok. In Cz. Woźniak (Ed.), Mechanics of elastic plates and shells (in Polish), volume VIII of Mechanika Techniczna. PWN, Warszawa.

    Google Scholar 

  • Kabrits, S. A., Mikhailovskiy, E. I., Tovstik, P. E., Chernykh, K. F., & Shamina, V. A. (2002). General nonlinear theory of elastic shells (in Russ.). St. Petersburg State University, St. Petersburg.

    Google Scholar 

  • Koç, I. M., Carcaterra, A., Xu, Z., & Akay, A. (2005). Energy sinks: Vibration absorption by an optimal set of undamped oscillators. The Journal of the Acoustical Society of America, 118(5), 3031–3042.

    Article  Google Scholar 

  • Konopińska, V., & Pietraszkiewicz, W. (2007). Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells. The International Journal of Solids and Structures, 44(1), 352–369.

    Article  MATH  Google Scholar 

  • Korteweg, D. J. (1901). Sur la forme que prennent les équations des mouvements des fluides si l’on tient compte des forces capillaires par des variations de densité. Archives Néderlandaises des sciences exactes et naturelles, Sér., II(6), 1–24.

    MATH  Google Scholar 

  • Kreja, I. (2007). Geometrically non-linear analysis of layered composite plates and shells. Gdańsk: Gdańsk University of Technology.

    Google Scholar 

  • Lakes, R. S. (1986). Experimental microelasticity of two porous solids. The International Journal of Solids and Structures, 22(1), 55–63.

    Article  Google Scholar 

  • Laplace, P. S. (1805). Sur l’action capillaire. supplément à la théorie de l’action capillaire. Traité de mécanique céleste (Vol. 4, pp. 771–777). Supplement 1, Livre X Paris: Gauthier-Villars et fils.

    Google Scholar 

  • Laplace, P. S. (1806). À la théorie de l’action capillaire. supplément à la théorie de l’action capillaire. Traité de mécanique céleste (Vol. 4, pp. 909–945). Supplement 2, Livre X Paris: Gauthier-Villars et fils.

    Google Scholar 

  • Lebedev, L. P., Cloud, M. J., & Eremeyev, V. A. (2010). Tensor analysis with applications in mechanics. New Jersey: World Scientific.

    Book  MATH  Google Scholar 

  • Libai, A., & Simmonds, J. G. (1983). Nonlinear elastic shell theory. Advances in Applied Mechanics, 23, 271–371.

    Article  MATH  Google Scholar 

  • Libai, A., & Simmonds, J. G. (1998). The nonlinear theory of elastic shells (2nd ed.). Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Lions, J.-L., & Magenes, E. (1968). Problèmes aux limites non homogènes et applications. Paris: Dunod.

    MATH  Google Scholar 

  • Longley, W. R. & Van Name, R. G. (Eds.). (1928). The collected works of J. Willard Gibbs, PHD., LL.D. Vol. I Thermodynamics. Longmans, New York.

    Google Scholar 

  • Lurie, A. I. (1990). Nonlinear theory of elasticity. Amsterdam: North-Holland.

    MATH  Google Scholar 

  • Lurie, A. I. (2001). Analytical mechanics. Berlin: Springer.

    Google Scholar 

  • Makowski, J., & Pietraszkiewicz, W. (2002). Thermomechanics of shells with singular curves. Zesz. Nauk. No 528/1487/2002, IMP PAN, Gdańsk.

    Google Scholar 

  • Maugin, G. A. (1988). Continuum mechanics of electromagnetic solids. Oxford: Elsevier.

    MATH  Google Scholar 

  • Maurini, C., dell’Isola, F., & Del Vescovo, D. (2004). Comparison of piezoelectronic networks acting as distributed vibration absorbers. Mechanical Systems and Signal Processing, 18(5), 1243–1271.

    Article  Google Scholar 

  • Mindlin, R. D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Transactions of ASME. Journal of Applied Mechanics, 18, 31–38.

    MATH  Google Scholar 

  • Miyazaki, S., Fu, Y. Q., & Huang, W. M. (Eds.). (2009). Thin film shape memory alloys: Fundamentals and device applications. Cambridge: Cambridge University Press.

    Google Scholar 

  • Murdoch, A. I. (1976a). A thermodynamical theory of elastic material interfaces. The Quarterly Journal of Mechanics and Applied Mathematics, 29(3), 245–274.

    Article  MathSciNet  MATH  Google Scholar 

  • Murdoch, A. I. (1976b). On the entropy inequality for material interfaces. ZAMP, 27(5), 599–605.

    Article  MathSciNet  Google Scholar 

  • Naghdi, P. M. (1972). The theory of plates and shells. In S. Flügge (Ed.), Handbuch der Physik (pp. 425–640), volume VIa/2 Heidelberg: Springer.

    Google Scholar 

  • Nirenberg, L. (2001). Topics in nonlinear functional analysis. New York: American Mathematical Society.

    Book  MATH  Google Scholar 

  • Novozhilov, V. V., Chernykh, K. F., & Mikhailovskiy, E. I. (1991). Linear theory of thin shells (in Russ.). Politekhnika, Leningrad.

    Google Scholar 

  • Ogden, R. W. (1997). Non-linear elastic deformations. Mineola: Dover.

    Google Scholar 

  • Pietraszkiewicz, W. (2011). Refined resultant thermomechanics of shells. International Journal of Engineering Science, 49(10), 1112–1124.

    Article  MathSciNet  Google Scholar 

  • Pietraszkiewicz, W. (1979a). Finite rotations and langrangian description in the non-linear theory of shells. Warszawa-Poznań: Polish Sci. Publ.

    MATH  Google Scholar 

  • Pietraszkiewicz, W. (1979b). Consistent second approximation to the elastic strain energy of a shell. ZAMM, 59, 206–208.

    MathSciNet  MATH  Google Scholar 

  • Pietraszkiewicz, W. (1989). Geometrically nonlinear theories of thin elastic shells. Uspekhi Mechaniki (Advances in Mechanics), 12(1), 51–130.

    MathSciNet  Google Scholar 

  • Pietraszkiewicz, W. (2015). The resultant linear six-field theory of elastic shells: What it brings to the classical linear shell models? ZAMM, 10.1002/zamm.201500184.

    Google Scholar 

  • Pietraszkiewicz, W., & Eremeyev, V. A. (2009a). On natural strain measures of the non-linear micropolar continuum. International Journal of Solids and Structures, 46(3–4), 774–787.

    Article  MathSciNet  MATH  Google Scholar 

  • Pietraszkiewicz, W., & Eremeyev, V. A. (2009b). On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. International Journal of Solids and Structures, 46(11–12), 2477–2480.

    Google Scholar 

  • Pietraszkiewicz, W., & Górski, J. (Eds.). (2014). Shell structures: Theory and applications (Vol. 3). Boca Raton: CRC Press.

    Google Scholar 

  • Pietraszkiewicz, W., & Konopińska, V. (2015). Junctions in shell structures: A review. Thin-Walled Structures, 95, 310–334.

    Article  Google Scholar 

  • Pietraszkiewicz, W., & Konopińska, V. (2011). On unique kinematics for the branching shells. The International Journal of Solids and Structures, 48(14), 2238–2244.

    Article  Google Scholar 

  • Pietraszkiewicz, W., & Kreja, I. (Eds.). (2010). Shell structures: Theory and applications (Vol. 2). Boca Raton: CRC Press.

    MATH  Google Scholar 

  • Pietraszkiewicz, W., & Szymczak, C. (Eds.). (2005). Shell structures: Theory and applications. London: Taylor & Francis.

    Google Scholar 

  • Pietraszkiewicz, W., Eremeyev, V. A., & Konopińska, V. (2007). Extended non-linear relations of elastic shells undergoing phase transitions. ZAMM, 87(2), 150–159.

    Article  MathSciNet  MATH  Google Scholar 

  • Reda, H., Rahali, Y., Ganghoffer, J. F., & Lakiss, H. (2016). Wave propagation in 3d viscoelastic auxetic and textile materials by homogenized continuum micropolar models. Composite Structures, 141, 328–345.

    Article  Google Scholar 

  • Reddy, J. N. (2003). Mechanics of laminated composite plates and shells: Theory and analysis (2nd ed.). Boca Raton: CRC Press.

    Google Scholar 

  • Reissner, E. (1944). On the theory of bending of elastic plates. Journal of Mathematical Physics, 23, 184–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Reissner, E. (1985). Reflection on the theory of elastic plates. Applied Mechanics Reviews, 38(11), 1453–1464.

    Article  Google Scholar 

  • Reissner, E. (1977). A note on generating generalized two-dimensional plate and shell theories. ZAMP, 28(4), 633–642.

    Article  MATH  Google Scholar 

  • Rowlinson, J. S., & Widom, B. (2003). Molecular theory of capillarity. New York: Dover.

    Google Scholar 

  • Sargsyan, S. H. (2011). The general dynamic theory of micropolar elastic thin shells. Doklady Physics, 56(1), 39–42.

    Article  MathSciNet  Google Scholar 

  • Sedov, L. I. (1968). Models of continuous media with internal degrees of freedom. Journal of Applied Mathematics and Mechanics, 32(5), 803–819.

    Article  MATH  Google Scholar 

  • Shkutin, L. I. (2007). Analysis of axisymmetric phase strains in plates and shells. Journal of Applied Mechanics and Technical Physics, 48(2), 285–291.

    Article  MathSciNet  MATH  Google Scholar 

  • Simmonds, J. G. (2005). A simple nonlinear thermodynamic theory of arbitrary elastic beams. The Journal of Elasticity, 81(1), 51–62.

    Article  MathSciNet  MATH  Google Scholar 

  • Simmonds, J. G. (2011). A classical, nonlinear thermodynamic theory of elastic shells based on a single constitutive assumption. The Journal of Elasticity, 105(1–2), 305–312.

    Article  MathSciNet  MATH  Google Scholar 

  • Simmonds, J. G. (1984). The thermodynamical theory of shells: Descent from 3-dimensions without thickness expansions. In E. L. Axelrad & F. A. Emmerling (Eds.), Flexible shells (pp. 1–11). Theory and Applications Berlin: Springer.

    Google Scholar 

  • Steigmann, D. J., & Ogden, R. W. (1999). Elastic surface-substrate interactions. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Science, 455(1982), 437–474.

    Google Scholar 

  • Steinberg, L., & Kvasov, R. (2013). Enhanced mathematical model for Cosserat plate bending. Thin-Walled Structures, 63, 51–62.

    Article  Google Scholar 

  • Steinmann, P., & Häsner, O. (2005). On material interfaces in thermomechanical solids. Archive of Applied Mechanics, 75(1), 31–41.

    Article  MATH  Google Scholar 

  • Tovstik, P. E., & Smirnov, A. L. (2001). Asymptotic methods in the buckling theory of elastic shells. Singapore: World Scientific.

    MATH  Google Scholar 

  • Truesdell, C. (1977). A first course in rational continuum mechanics. New York: Academic Press.

    MATH  Google Scholar 

  • Truesdell, C. (1984). Rational thermodynamics (2nd ed.). New York: Springer.

    Book  MATH  Google Scholar 

  • Truesdell, C., & Noll, W. (1965). The nonlinear field theories of mechanics. In S. Flügge (Ed.), Handbuch der Physik (pp. 1–602). III(3) Berlin: Springer.

    Google Scholar 

  • van der Waals, J. D. (1893). The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (Engl. transl. by J. S. Rowlinson). Journal of Statistical Physics, 20, 200–244.

    Article  Google Scholar 

  • Vidoli, S., & dell’Isola, F. (2001). Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks. European Journal of Mechanics: A/Solids, 20(3), 435–456.

    Google Scholar 

  • Wang, C. M., Reddy, J. N., & Lee, K. H. (2000). Shear deformable beams and shells. Amsterdam: Elsevier.

    MATH  Google Scholar 

  • Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., et al. (2011). Surface stress effect in mechanics of nanostructured materials. Acta Mechanica Solida Sinica, 24, 52–82.

    Article  Google Scholar 

  • Wang, Z. Q., Zhao, Y.-P., & Huang, Z.-P. (2010). The effects of surface tension on the elastic properties of nano structures. International Journal of Engineering Science, 48(2), 140–150.

    Article  MathSciNet  Google Scholar 

  • Wiśniewski, K. (2010). Finite rotation shells: Basic equations and finite elements for Reissner kinematics. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Young, T. (1805). An essay on the cohesion of fluids. Philosophical Transactions of the Royal Society of London, 95, 65–87.

    Article  Google Scholar 

  • Zhilin, P. A. (1976). Mechanics of deformable directed surfaces. International Journals of Solids and Structures, 12(9–10), 635–648.

    Article  MathSciNet  Google Scholar 

  • Zubov, L. M. (1997). Nonlinear theory of dislocations and disclinations in elastic bodies. Berlin: Springer.

    MATH  Google Scholar 

  • Zubov, L. M. (2009). Micropolar shell equilibrium equations. Doklady Physics, 54(6), 290–293.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Avionics, Rzeszów University of Technology, Podkarpacie, Rzeszów, Poland

    Victor Eremeyev

  2. South Scientific Center of Russian Academy of Science, Rostov on Don, Russia

    Victor Eremeyev

  3. Department of Mathematics, Mechanics and Computer Science, South Federal University, Rostov on Don, Russia

    Victor Eremeyev

  4. Chair of Engineering Mechanics, Faculty of Mechanical Engineering, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany

    Holm Altenbach

Authors
  1. Victor Eremeyev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Holm Altenbach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Eremeyev .

Editor information

Editors and Affiliations

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

    Holm Altenbach

  2. Technical Mechanics, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany

    Victor Eremeyev

Rights and permissions

Reprints and permissions

Copyright information

© 2017 CISM International Centre for Mechanical Sciences

About this chapter

Cite this chapter

Eremeyev, V., Altenbach, H. (2017). Basics of Mechanics of Micropolar Shells. In: Altenbach, H., Eremeyev, V. (eds) Shell-like Structures. CISM International Centre for Mechanical Sciences, vol 572. Springer, Cham. https://doi.org/10.1007/978-3-319-42277-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-42277-0_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42275-6

  • Online ISBN: 978-3-319-42277-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation