Skip to main content
SpringerLink
Log in

Cosserat-Type Shells

  • Chapter
Generalized Continua from the Theory to Engineering Applications

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 541))

Abstract

In this chapter we discuss the Cosserat-type theories of plates and shells. We call Cosserat-type shell theories various theories of shells based on the consideration of a shell base surface as a deformable directed surface, that is the surface with attached deformable or non-deformable (rigid) vectors (directors), or based on the derivation of two-dimensional (2D) shell equations from the three-dimensional (3D) micropolar (Cosserat) continuum equations.

Originally the first approach of such a kind belongs to Cosserat brothers who considered a shell as a deformable surface with attached three unit orthogonal directors. In the literature are known theories of shells which kinematics is described by introduction of the translation vector and additionally p deformable directors or one deformable director or three unit orthogonal to each other directors. Additional vector fields of directors describe the rotational (in some special cases additional) degrees of freedom of the shell. The most popular theories use the one deformable director or three unit directors. In both cases the so-called direct approach is applied.

Another approach is based on the 3D-to-2D reduction procedure applied to the 3D motion or equilibrium equations of the micropolar shell-like body. In the literature the various reduction methods are known using for example asymptotic methods, the-through-the thickness integration procedure, expansion in series, etc.

The aim of the chapter is to present the various Cosserat-type theories of plates of shells and discuss the peculiarities and differences between these approaches.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • I. Aganovi´c, J. Tambača, and Z. Tutek. Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity. J Elasticity, 84:131–152, 2006.

    MathSciNet  MATH  Google Scholar 

  • M. Agranovich. Elliptic boundary problems. In M. Agranovich, Y. Egorov, and M. Shubin, editors, Partial Differential Equations IX, Encyclopaedia of Mathematical Sciences 79, pages 1–144. Springer, Berlin, 1997.

    Google Scholar 

  • H. Altenbach. Eine direkt formulierte lineare Theorie für viskoelastische Platten und Schalen. Ing Arch, 58:215–228, 1988.

    MATH  Google Scholar 

  • H. Altenbach and V. A. Eremeyev. On the shell theory on the nanoscale with surface stresses. Int J Eng Sci, 49(12):1294–1301, 2011.

    MathSciNet  Google Scholar 

  • H. Altenbach and V. A. Eremeyev. Direct approach based analysis of plates composed of functionally graded materials. Arch Appl Mech, 78(10): 775–794, 2008.

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • H. Altenbach and V. A. Eremeyev. On the bending of viscoelastic plates made of polymer foams. Acta Mech, 204(3–4):137–154, 2009b.

    MATH  Google Scholar 

  • H. Altenbach and P. A. Zhilin. A general theory of elastic simple shells (in Russian). Uspekhi Mekhaniki (Advances in Mechanics), 11(4):107–148, 1988.

    MathSciNet  Google Scholar 

  • H. Altenbach and P. A. Zhilin. The theory of simple elastic shells. In

    Google Scholar 

  • R. Kienzler, H. Altenbach, and I. Ott, editors, Critical Review of the Theories of Plates and Shells, volume 16 of Lect. Notes Appl. Comp. Mech., pages 1–12. Springer, Berlin, 2004.

    MATH  Google Scholar 

  • H. Altenbach, V. A. Eremeyev, and L. P. Lebedev. Micropolar shells as twodimensional generalized continua models. In H. Altenbach, G. A. Maugin, and V. Erofeev, editors, Mechanics of Generalized Continua, volume 7 of Advanced Structured Materials, pages 23–55. Springer, Berlin, 2011.

    Google Scholar 

  • S. A. Ambartsumian. The theory of transverse bending of plates with asymmetric elasticity. Mech Compos Mater, 32(1):30–38, 1996.

    Google Scholar 

  • S. A. Ambartsumian. The micropolar theory of plates and shells (in Russian). NAN Armenii, Yerevan, 1999.

    Google Scholar 

  • S. S. Antman. Nonlinear Problems of Elasticity. Springer Science Media, New York, 2nd edition, 2005.

    MATH  Google Scholar 

  • S. S. Antman. Global properties of buckled states of plates that can suffer thickness changes. Arch Ration Mech An, 110(2):103–117, 1990.

    MathSciNet  MATH  Google Scholar 

  • S. S. Antman and D. Bourne. Rotational symmetry vs. axisymmetry in shell theory. Int J Eng Sci, 48(11):991–1005, 2010.

    MathSciNet  Google Scholar 

  • S. S. Antman and W. Lacarbonara. Forced radial motions of nonlinearly viscoelastic shells. J Elasticity, 96(2):155–190, 2009.

    MathSciNet  MATH  Google Scholar 

  • T. Ariman. On circular micropolar plates. Ing Arch, 37(3):156–160, 1968.

    MATH  Google Scholar 

  • Ya. Başar. A consistent theory of geometrically non-linear shells with an independent rotation vector. Int J Solids Struct, 23(10):1401–1415, 1987.

    Google Scholar 

  • J. Badur and W. Pietraszkiewicz. On geometrically non-linear theory of elastic shells derived from pseudo-Cosserat continuum with constrained micro-rotations. In W. Pietraszkiewicz, editor, Finite Rotations in Structural Mechanics, pages 19–32. Springer, Wien, 1986.

    Google Scholar 

  • K. Bhattacharya and R. D. James. A theory of thin films of martensitic materials with applications to microactuators. J Mech Phys Solids, 47 (3):531–576, 1999.

    MathSciNet  MATH  Google Scholar 

  • M. Bîrsan. On the theory of elastic shells made from a material with voids. Int J Solids Struct, 43(10):3106–3123, 2006a.

    Google Scholar 

  • M. Bîrsan. On a thermodynamic theory of porous Cosserat elastic shells. J Therm Stresses, 29(9):879–899, 2006b.

    Google Scholar 

  • M. Bîrsan. On Saint-Venant’s principle in the theory of Cosserat elastic shells. Int J Eng Sci, 45(2–8):187–198, 2007.

    Google Scholar 

  • M. Bîrsan. Inequalities of Korn’s type and existence results in the theory of Cosserat elastic shells. J Elasticity, 90(3):227–239, 2008.

    MathSciNet  MATH  Google Scholar 

  • M. Bîrsan. Thermal stresses in cylindrical Cosserat elastic shells. Eur J Mech A-Solid, 28(1):94–101, 2009a.

    MATH  Google Scholar 

  • M. Bîrsan. On Saint-Venant’s problem for anisotropic, inhomogeneous, cylindrical Cosserat elastic shells. Int J Eng Sci, 47(1):21–38, 2009b.

    Google Scholar 

  • M. Bîrsan. On the problems of Almansi and Michell for anisotropic Cosserat elastic shells. Archive of Mechanics, 61(3–4):195–227, 2009c.

    MATH  Google Scholar 

  • M. Bîrsan. Thermal stresses in anisotropic cylindrical elastic shells. Math Methods Appl Sci, 33(6):799–810, 2010.

    MathSciNet  MATH  Google Scholar 

  • M. Bîrsan. On a problem of Truesdell for anisotropic elastic shells. Analele S¸tiint¸ifice Ale Universit˘at¸ii “Al.I. Cuza” Din Iaşi (S.N.) Matematic˘a,, LVII(1):91–110, 2011.

    Google Scholar 

  • M. Bîrsan. Linear Cosserat Elastic Shells: Mathemathical Theory and Applications. Matrix Rom, Bucureşti, 2009d.

    Google Scholar 

  • M. Bîrsan and H. Altenbach. A mathematical study of the linear theory for orthotropic elastic simple shells. Math Methods Appl Sci, 33(12): 1399–1413, 2010.

    MathSciNet  MATH  Google Scholar 

  • M. Bîrsan and H. Altenbach. On the dynamical theory of thermoelastic simple shells. ZAMM, 91(6):443–457, 2011.

    MathSciNet  MATH  Google Scholar 

  • E. Boschi. Lamb and Love wave-propagation in an infinite micropolar elastic plate. Annali di Geofisica, 26(2-3):341–355, 1973.

    Google Scholar 

  • B. Brank, D. Peric, and F. B. Damjanic. On large deformations of thin elasto-plastic shells: Implementation of a finite rotation model for quadrilateral shell element. Int J Numer Meth Eng, 40(4):689–726, 1997.

    MATH  Google Scholar 

  • C. Chinosi, L. Della Croce, and T. Scapolla. Hierarchic finite elements for thin Naghdi shell model. Int J Solids Struct, 35(16):1863–1880, 1998.

    MATH  Google Scholar 

  • K. L. Chowdhur and P. G. Glockner. Bending of an annular elastic Cosserat plate. B Acad Pol Sci Tech, 21(3):211–218, 1973.

    Google Scholar 

  • J. Chróścielewski. Rodzina elementów sko´nczonych klasy C 0 w nieliniowej sześcioparametrowej teorii powlok. Zesz. Nauk. Politechniki Gda´nskiej, LIII(540):1–291, 1996.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • J. Chróścielewski and W. Witkowski. FEM analysis of Cosserat plates and shells based on some constitutive relations. ZAMM, 91(5):400–412, 2011.

    MathSciNet  MATH  Google Scholar 

  • J. Chróścielewski, J. Makowski, and H. Stumpf. Finite element analysis of smooth, folded and multi-shell structures. Comput Methods Appl Mech Eng, 141:1–46, 1997.

    MATH  Google Scholar 

  • J. Chróścielewski, J. Makowski, and W. Pietraszkiewicz. Non-linear dynamics of flexible shell structures. Comp Assisted Mech Eng Sci, 9:341–357, 2002.

    MATH  Google Scholar 

  • J. Chróścielewski, J. Makowski, and W. Pietraszkiewicz. Statics and Dynamics of Multyfold Shells. Nonlinear Theory and Finite Element Method. Wydawnictwo IPPT PAN, Warszawa, 2004.

    Google Scholar 

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

    MATH  Google Scholar 

  • J. Chróścielewski, I. Kreja, A. Sabik, and W. Witkowski. Modeling of composite shells in 6-parameter nonlinear theory with drilling degree of freedom. Mech Adv Mater Struc, 18(6):403–419, 2011.

    Google Scholar 

  • P. G. Ciarlet. Mathematical Elasticity, volume II. Theory of Plates. Elsevier, Amsterdam, 1997.

    MATH  Google Scholar 

  • P. G. Ciarlet. Mathematical Elasticity, volume III. Theory of Shells. Elsevier, Amsterdam, 2000.

    MATH  Google Scholar 

  • H. Cohen and R. S. D. Thomas. Transient waves in inhomogeneous isotropic elastic plates. Acta Mech, 53(3-4):141–161, 1984.

    MATH  Google Scholar 

  • H. Cohen and R. S. D. Thomas. Transient waves in inhomogeneous anisotropic elastic plates. Acta Mech, 58(1-2):41–57, 1986.

    MathSciNet  MATH  Google Scholar 

  • H. Cohen and C.-C. Wang. A mathematical analysis of the simplest direct models for rods and shells. Arch Ration Mech An, 108(1):35–81, 1989.

    MathSciNet  MATH  Google Scholar 

  • C. Constanda. Complex variable treatment of bending of micropolar plates. Int J Eng Sci, 15(11):661–669, 1977.

    MathSciNet  MATH  Google Scholar 

  • E. Cosserat and F. Cosserat. Théorie des corps déformables. Herman et Fils, Paris, 1909.

    Google Scholar 

  • R. Courant and D. Hilbert. Methods of Mathematical Physics, volume 1. Wiley, New York, 1991.

    Google Scholar 

  • C. N. deSilva and P. J. Tsai. A general theory of directed surfaces. Acta Mech, 18(1-2):89–101, 1973.

    MathSciNet  MATH  Google Scholar 

  • H. A. Erbay. An asymptotic theory of thin micropolar plates. Int J Eng Sci, 38(13):1497–1516, 2000.

    MathSciNet  MATH  Google Scholar 

  • V. A. Eremeyev. Nonlinear micropolar shells: theory and applications. In

    Google Scholar 

  • W. Pietraszkiewicz and C. Szymczak, editors, Shell Structures: Theory and Applications, pages 11–18. Taylor & Francis, London, 2005.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • V. A. Eremeyev and W. Pietraszkiewicz. On tension of a two-phase elastic tube. In W. Pietraszkiewicz and I. Kreja, editors, Shell Structures. Theory and Applications. Vol. 2., pages 63–66. CRC Press, Boca Raton, 2010.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • V. A. Eremeyev and L. M. Zubov. Mechanics of Elastic Shells (in Russian). Nauka, Moscow, 2008.

    Google Scholar 

  • J. L. Ericksen. Wave propagation in thin elastic shells. J Elasticity, 43(3): 167–178, 1971.

    MathSciNet  MATH  Google Scholar 

  • J. L. Ericksen. The simplest problems for elastic Cosserat surfaces. J Elasticity, 2(2):101–107, 1972a.

    MathSciNet  Google Scholar 

  • J. L. Ericksen. Plane infinitesimal waves in homogeneous elastic plates. J Elasticity, 3(3):161–167, 1973a.

    MathSciNet  Google Scholar 

  • J. L. Ericksen. Simpler problems for elastic Cosserat surfaces. J Elasticity, 7(1):1–11, 1977.

    MathSciNet  MATH  Google Scholar 

  • J. L. Ericksen. Introduction to the Thermodynamics of Solids. Springer, New York, 2nd edition, 1998.

    MATH  Google Scholar 

  • J. L. Ericksen. Symmetry transformations for thin elastic shells. Arch Ration Mech An, 47:1–14, 1972b.

    MathSciNet  MATH  Google Scholar 

  • J. L. Ericksen. Apparent symmetry of certain thin elastic shells. J. Me´canique, 12:12–18, 1973b.

    Google Scholar 

  • J. L. Ericksen and C. Truesdell. Exact theory of stress and strain in rods and shells. Arch Ration Mech An, 1(1):295–323, 1958.

    MathSciNet  MATH  Google Scholar 

  • A. C. Eringen. Microcontinuum Field Theory. II. Fluent Media. Springer, New York, 2001.

    Google Scholar 

  • A. C. Eringen. Theory of micropolar fluids. Journal of Mathematics and Mechanics, 16(1):1–18, 1966.

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • A. C. Eringen. Microcontinuum Field Theory. I. Foundations and Solids. Springer, New York, 1999.

    Google Scholar 

  • M. Farshad and B. Tabarrok. Dualities in analysis of Cosserat plate. Mech Res Comm, 3(5):399–406, 1976.

    MATH  Google Scholar 

  • G. Fichera. Existence theorems in elasticity. In S. Flügge, editor, Handbuch der Physik, volume VIa/2, pages 347–389. Springer, Berlin, 1972.

    Google Scholar 

  • D. D. Fox and J. C. Simo. A drill rotation formulation for geometrically exact shells. Comput Methods Appl Mech Eng, 98(3):329—343, 1992.

    MathSciNet  MATH  Google Scholar 

  • D. D. Fox, A. Raoult, and J. C. Simo. A justification of nonlinear properly invariant plate theories. Arch Ration Mech An, 124(2):157–199, 1993.

    MathSciNet  MATH  Google Scholar 

  • G. Friesecke, R. D. James, M. G. Mora, and S. Müller. The Föppl-von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity. C R Acad Sci Paris Ser I, 335:201–206, 2002a.

    MATH  Google Scholar 

  • G. Friesecke, R. D. James, M. G. Mora, and S. Müller. A theorem on geometric rigidity and the derivation of nonlinear plate theory from threedimensional elasticity. Commun Pure Appl Math, LV:1461–1506, 2002b.

    Google Scholar 

  • G. Friesecke, R. D. James, M. G. Mora, and S. Müller. Rigorous derivation of nonlinear plate theory and geometric rigidity. C R Acad Sci Paris Ser I, 334:173—178, 2002c.

    MATH  Google Scholar 

  • G. Friesecke, R. D. James, M. G. Mora, and S. Müller. Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Γ-convergence. C R Acad Sci Paris Ser I, 336 :697–702, 2003.

    MATH  Google Scholar 

  • G. A. Gevorkyan. The basic equations of flexible plates for a medium of Cosserat. Int Appl Mech, 3(11):41–45, 1967.

    Google Scholar 

  • P. G. Glockner and D. J. Malcolm. Cosserat surface – model for idealized sandwich shells. ZAMM, 54(4):T78, 1974.

    MATH  Google Scholar 

  • A. E. Green and P. M. Naghdi. Micropolar and director theories of plates. Q J Mech Appl Math, 20:183–199, 1967a.

    MATH  Google Scholar 

  • A. E. Green and P. M. Naghdi. The linear elastic Cosserat surface and shell theory. Int J Solids Struct, 4(6):585–592, 1968.

    MATH  Google Scholar 

  • A. E. Green and P. M. Naghdi. Non-isothermal theory of rods, plates and shells. Int J Solids Struct, 6:209–244, 1970.

    MATH  Google Scholar 

  • A. E. Green and P. M. Naghdi. On superposed small deformations on a large deformation of an elastic Cosserat surface. J Elasticity, 1(1):1–17, 1971.

    MathSciNet  Google Scholar 

  • A. E. Green and P. M. Naghdi. Derivation of shell theories by direct approach. Trans ASME. J Appl Mech, 41(1):173–176, 1974.

    MathSciNet  MATH  Google Scholar 

  • A. E. Green and P. M. Naghdi. On thermal effects in the theory of shells. Proc R Soc London A, 365A:161–190, 1979.

    MathSciNet  Google Scholar 

  • A. E. Green and P. M. Naghdi. Linear theory of an elastic Cosserat plate. Proc Camb Philos S-M, 63(2):537–550, 1967b.

    MATH  Google Scholar 

  • A. E. Green, P. M. Naghdi, and W. L. Wainwright. A general theory of a Cosserat surface. Arch Ration Mech An, 20(4):287–308, 1965.

    MathSciNet  Google Scholar 

  • E. Grekova and P. Zhilin. Basic equations of Kelvin’s medium and analogy with ferromagnets. J Elasticity, 64:29–70, 2001.

    MathSciNet  MATH  Google Scholar 

  • E. I. Grigolyuk and I. T. Selezov. Nonclassical theories of vibration of beams, plates and shells (in Russian). In Itogi nauki i tekhniki, volume 5 of Mekhanika tverdogo deformiruemogo tela. VINITI, Moskva, 1973.

    Google Scholar 

  • I. T. Gürgöze. Thermostatics of an elastic Cosserat plate containing a circular hole. Proc Camb Philos S-M, 70(JUL):169–174, 1971.

    Google Scholar 

  • M. E. Gurtin and A. I. Murdoch. A continuum theory of elastic material surfaces. Arch Ration Mech An, 57:291–323, 1975.

    MathSciNet  MATH  Google Scholar 

  • D. H. Hodges. Nonlinear Composite Beam Theory, volume 213 of Progress in Astronautics and Aeronautics. American Institute of Aeronautics and Astronautics, Inc., Reston, 2006.

    Google Scholar 

  • D. H. Hodges, A. R. Atilgan, and D. A. Danielson. A geometrically nonlinear theory of elastic plates. Trans ASME. J Appl Mech, 60(1):109–116, 2004.

    Google Scholar 

  • L. Hörmander. Linear Partial Differential Equations. Springer, Berlin, 4th edition, 1976.

    Google Scholar 

  • S. Itou and A. Atsumi. Effect of couple-stresses on elastic Cosserat plate with loads travelling at uniform velocity along bounding surfaces. Arch Mech Stos, 22(4):375–384, 1970.

    MATH  Google Scholar 

  • G. Jemielita. Rotational representation of a plate made of Grioli-Toupin material. Trans ASME. J Appl Mech, 62(2):414–418, 1995.

    MATH  Google Scholar 

  • G. Jemilita. Bibliografia. In Cz. Wo´zniak, editor, Mechanika spr¸e˙zystych plyt i powlok, pages 675–766. PWN, Warszawa, 2001.

    Google Scholar 

  • C. S. Jog. Higher-order shell elements based on a Cosserat model, and their use in the topology design of structures. Comput Methods Appl Mech Eng, 193(23–26):2191–2220, 2004.

    MATH  Google Scholar 

  • Ya. F. Kayuk and A. P. Zhukovskii. Theory of plates and shells based on the concept of Cosserat surfaces. Int Appl Mech, 17(10):922–926, 1981.

    Google Scholar 

  • R. Kienzler. On consistent plate theories. Arch Appl Mech, 72:229 – 247, 2002.

    MATH  Google Scholar 

  • R. Kienzler, H. Altenbach, and I. Ott, editors. Critical review of the theories of plates and shells, new applications, volume 16 of Lect. Notes Appl. Comp. Mech., Berlin, 2004. Springer.

    Google Scholar 

  • G. R. Kirchhoff. Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Crelles Journal für die reine und angewandte Mathematik, 40: 51–88, 1850.

    MATH  Google Scholar 

  • J. K. Knowles and E. Sternberg. On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch Ration Mech An, 63:321–336, 1976.

    MathSciNet  Google Scholar 

  • J. K. Knowles and E. Sternberg. On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. J Elasticity, 8:329–379, 1978.

    MathSciNet  MATH  Google Scholar 

  • V. Konopi´nska and W. Pietraszkiewicz. Exact resultant equilibrium conditions in the non-linear theory of branched and self-intersecting shells. Int J Solids Struct, 44(1):352–369, 2006.

    Google Scholar 

  • T. Korman, F. T. Morghem, and M. H. Baluch. Bending of micropolar plates. Nucl Eng Des, 26(3):432–439, 1974.

    Google Scholar 

  • I. Kreja. Geometrically non-linear analysis of layered composite plates and shells. Gda`nsk University of Technology, Gda`nsk, 2007.

    Google Scholar 

  • S. Krishnaswamy, Z. H. Jin, and R. C. Batra. Stress concentration in an elastic Cosserat plate undergoing extensional deformations. Trans ASME. J Appl Mech, 65(1):66–70, 1998.

    Google Scholar 

  • R. Kumar and S. Deswal. Some problems of wave propagation in a micropolar elastic medium with voids. J Vib Control, 12(8):849–879, 2006.

    MATH  Google Scholar 

  • R. Kumar and G. Partap. Axisymmetric free vibrations of infinite micropolar thermoelastic plate. Appl Math Mech, 28(3):369–383, 2007.

    MathSciNet  MATH  Google Scholar 

  • R. Kumar and G. Partap. Rayleigh Lamb waves in micropolar isotropic elastic plate. Appl Math Mech, 27(8):1049–1059, 2006.

    MathSciNet  MATH  Google Scholar 

  • Z. T. Kurlandz. Derivation of linear shell theory from theory of Cosserat medium. B Acad Pol Sci Tech, 20(10):789–794, 1972.

    Google Scholar 

  • Z. T. Kurlandz. Anisotropic linear Cosserat surface and linear shell theory. Arch Mech, 25(4):613–620, 1973.

    Google Scholar 

  • L. P. Lebedev, M. J. Cloud, and V. A. Eremeyev. Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey, 2010.

    MATH  Google Scholar 

  • M. Levinson. An accurate, simple theory of the statics and dynamics of elastic plates. Mech Res Comm, 7(6):343–350, 1980.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • A. Libai and J. G. Simmonds. The Nonlinear Theory of Elastic Shells. Cambridge University Press, Cambridge, 2nd edition, 1998.

    MATH  Google Scholar 

  • J.-L. Lions and E. Magenes. Problèmes aux limites non homogènes et applications. Dunod, Paris, 1968.

    MATH  Google Scholar 

  • I. Lubowiecka and J. Chróścielewski. On dynamics of flexible branched shell structures undergoing large overall motion using finite elements. Comput Struct, 80:891–898, 2002.

    Google Scholar 

  • A. I. Lurie. Theory of Elasticity. Foundations of Engineering Mechanics. Springer, Berlin, 2005.

    Google Scholar 

  • A. I. Lurie. Analytical Mechanics. Foundations of Engineering Mechanics. Springer, Berlin, 2001.

    Google Scholar 

  • J. Makowski and W. Pietraszkiewicz. Thermomechanics of shells with singular curves. Zesz. Nauk. No 528/1487/2002, Gda´nsk, 2002.

    Google Scholar 

  • J. Makowski and H. Stumpf. Buckling equations for elastic shells with rotational degrees of freedom undergoing finite strain deformation. Int J Solids Struct, 26:353–368, 1990.

    MathSciNet  MATH  Google Scholar 

  • J. Makowski, W. Pietraszkiewicz, and H. Stumpf. Jump conditions in the nonlinear theory of thin irregular shells. J Elasticity, 54:1–26, 1999.

    MathSciNet  MATH  Google Scholar 

  • D. J. Malcolm and P. G. Glockner. Cosserat surface and sandwich shell equilibrium. Trans ASCE. J. Eng Mech Div, 98(EM5):1075–1086, 1972a.

    Google Scholar 

  • D. J. Malcolm and P. G. Glockner. Nonlinear sandwich shell and Cosserat surface theory. Trans ASCE. J Eng Mech Div, 98(EM5):1183–1203, 1972b.

    Google Scholar 

  • T. Merlini and M. Morandini. Computational shell mechanics by helicoidal modeling I: Theory. J Mech Mater Struc, 6(5):659–692, 2011a.

    Google Scholar 

  • T. Merlini and M. Morandini. Computational shell mechanics by helicoidal modeling II: Shell element. J Mech Mater Struct, 6(5):693–728, 2011b.

    Google Scholar 

  • R. D. Mindlin. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Trans ASME. J Appl Mech, 18:31–38, 1951.

    MATH  Google Scholar 

  • A. I. Murdoch and H. Cohen. Symmetry considerations for material surfaces. Arch Ration Mech An, 72(1):61–98, 1979.

    MathSciNet  MATH  Google Scholar 

  • A. I. Murdoch and H. Cohen. Symmetry considerations for material surfaces. Addendum. Arch Ration Mech An, 76(4):393–400, 1981.

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • P. M. Naghdi. On the formulation of contact problems of shells and plates. J Elasticity, 5(3-4):379–398, 1975.

    MathSciNet  MATH  Google Scholar 

  • P. M. Naghdi and M. B. Rubin. Restrictions on nonlinear constitutive equations for elastic shells. J Elasticity, 39:133–163, 1995.

    MathSciNet  MATH  Google Scholar 

  • P. M. Naghdi and A. R. Srinivasa. A dynamical theory of structured solids. 1. Basic developments. Phil Trans Royal Soc London A. Math Phys Eng Sci, 345(1677):425–458, 1993.

    MathSciNet  MATH  Google Scholar 

  • P. M. Naghdi and J. A. Trapp. A uniqueness theorem in the theory of Cosserat surface. J Elasticity, 2(1):9–20, 1972.

    MathSciNet  Google Scholar 

  • R. Nardinocchi and P. Podio-Guidugli. The equations of Reissner-Mindlin plates obtained by the method of internal constraints. Meccanica, 29: 143–157, 1994.

    MATH  Google Scholar 

  • S. A. Nazarov. Asymptotic behavior of the solution of an elliptic boundary value problem in a thin domain. J Math Sci, 64(6):1351–1362, 1993.

    Google Scholar 

  • P. Neff. A geometrically exact Cosserat shell-model including size effects, avoiding degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus. Continuum Mech Therm, 16(6):577–628, 2004.

    MathSciNet  MATH  Google Scholar 

  • P. Neff. The Γ-limit of a finite strain Cosserat model for asymptotically thin domains and a consequence for the Cosserat couple modulus. PAMM, 5 (1):629–630, 2005.

    Google Scholar 

  • P. Neff. A geometrically exact planar Cosserat shell-model with microstructure: Existence of minimizers for zero Cosserat couple modulus. Math Models & Methods Appl Sci, 17(3):363–392, 2007.

    MathSciNet  MATH  Google Scholar 

  • P. Neff and K. Chelmi´nski. A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via Γ-convergence. Interfaces Free Bound, 9:455–492, 2007.

    MathSciNet  MATH  Google Scholar 

  • V. Nicotra, P. Podio-Guidugli, and A. Tiero. Exact equilibrium solutions for linearly elastic plate-like bodies. J Elasticity, 56:231–245, 1999.

    MathSciNet  MATH  Google Scholar 

  • A. K. Noor. Bibliography of monographs and surveys on shells. Appl Mech Rev, 43:223–234, 1990.

    Google Scholar 

  • A. K. Noor. List of books, monographs, and survey papers on shells. In A. K. Noor, T. Belytschko, and J. C. Simo, editors, Analytical and Computational Models of Shells, CED, volume 3, pages vii–xxxiv. ASME, New York, 2004.

    Google Scholar 

  • W. Nowacki. Theory of Asymmetric Elasticity. Pergamon-Press, Oxford et al., 1986.

    MATH  Google Scholar 

  • W. Nowacki and W. K. Nowacki. Propagation of monochromatic waves in an infinite micropolar elastic plate. B Acad Pol Sci Tech, 17(1):45–53, 1969.

    MathSciNet  MATH  Google Scholar 

  • V. A. Pal’mov. Contribution to Cosserat’s theory of plates (in Russian). Trudy LPI, 386:3–8, 1982.

    Google Scholar 

  • W. A. Palmow and H. Altenbach. Über eine Cosseratsche Theorie für elastische Platten. Techn Mech, 3(3):5–9, 1982.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  • W. Pietraszkiewicz. Teorie nieliniowe powlok. In Cz. Wo´zniak, editor, Mechanika spr¸e˙zystych plyt i powlok, pages 424–497. PWN, Warszawa, 2001.

    Google Scholar 

  • W. Pietraszkiewicz. Finite Rotations and Langrangian Description in the Non-linear Theory of Shells. Polish Sci. Publ, Warszawa-Pozna´n, 1979a.

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • W. Pietraszkiewicz. Addendum to: Bibliography of monographs and surveys on shells. Appl Mech Rev, 45:249–250, 1992.

    MathSciNet  Google Scholar 

  • W. Pietraszkiewicz and J. Badur. Finite rotations in the description of continuum deformation. Int J Eng Sci, 21:1097–1115, 1983.

    MathSciNet  MATH  Google Scholar 

  • W. Pietraszkiewicz and V. A. Eremeyev. On natural strain measures of the non-linear micropolar continuum. Int J Solids Struct, 46(3–4):774–787, 2009a.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • W. Pietraszkiewicz, J. Chróścielewski, and J. Makowski. On dynamically and kinematically exact theory of shells. In W. Pietraszkiewicz and C. Szymczak, editors, Shell Structures: Theory and Applications, pages 163–167. Taylor & Francis, London, 2005.

    Google Scholar 

  • W. Pietraszkiewicz, V. A. Eremeyev, and V. Konopi´nska. Extended nonlinear relations of elastic shells undergoing phase transitions. ZAMM, 87 (2):150–159, 2007.

    MathSciNet  MATH  Google Scholar 

  • P. I. Plotnikov and J. F. Toland. Modelling nonlinear hydroelastic waves. Phil Trans Royal Society A, 369(1947):2942–2956, 2011.

    MathSciNet  MATH  Google Scholar 

  • P. Podio-Guidugli. An exact derivation of the thin plate equation. J Elasticity, 22:121–133, 1989.

    MathSciNet  MATH  Google Scholar 

  • A. Pompei and M. A. Rigano. On the bending of micropolar viscoelastic plates. Int J Eng Sci, 44(18-19):1324–1333, 2006.

    MathSciNet  MATH  Google Scholar 

  • J. W. Provan and R. C. Koeller. On the theory of elastic plates. Int J Solids Struct, 6(7):933–950, 1970.

    MATH  Google Scholar 

  • H. Ramsey. Comparison of buckling deformations in compressed elastic Cosserat plates with three-dimensional plates. Int J Solids Struct, 22(3): 257–266, 1986.

    MATH  Google Scholar 

  • H. Ramsey. Axisymmetrical buckling of a cylindrical elastic Cosserat shell under axial-compression. Quart J Mech Appl Math, 40(3):415–429, 1987.

    MATH  Google Scholar 

  • J. N. Reddy. A simple higher-order theory for laminated composite plates. Trans ASME. J Appl Mech, 51:745–752, 1984.

    MATH  Google Scholar 

  • E. Reissner. On the theory of bending of elastic plates. J Math Phys, 23: 184–194, 1944.

    MathSciNet  MATH  Google Scholar 

  • E. Reissner. The effect of transverse shear deformation on the bending of elastic plates. Trans ASME. J Appl Mech, 12(11):A69 – A77, 1945.

    MathSciNet  MATH  Google Scholar 

  • E. Reissner. On bending of elastic plates. Quart Appl Math, 5:55–68, 1947.

    MathSciNet  MATH  Google Scholar 

  • E. Reissner. Reflection on the theory of elastic plates. Appl Mech Rev, 38 (11):1453–1464, 1985.

    Google Scholar 

  • E. Reissner. A note on pure bending and flexure in plane stress including the effect of moment stresses. Arch Mech, 28(6):633–642, 1970.

    Google Scholar 

  • E. Reissner. On sandwich-type plates with cores capable of supporting moment stresses. Acta Mech, 14(1):43–51, 1972.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • H. Rothert. Lineare konstitutive Gleihungen der viskoelastischen Cosseratfläche. ZAMM, 55:647–656, 1975.

    MATH  Google Scholar 

  • H. Rothert. Lösungsmöglichkeiten des Umkehrproblems viskoelastischer Cosseratfächen. ZAMM, 57:429–437, 1977.

    MathSciNet  MATH  Google Scholar 

  • M. B. Rubin. Restrictions on linear constitutive equations for a rigid heat conducting Cosserat shell. Int J Solids Struct, 41(24-25):7009–7033, 2004.

    MATH  Google Scholar 

  • M. B. Rubin. Cosserat Theories: Shells, Rods and Points. Kluwer, Dordrecht, 2000.

    MATH  Google Scholar 

  • M. B. Rubin and Y. Benveniste. A Cosserat shell model for interphases in elastic media. J Mech Phys Solids, 52(5):1023–1052, 2004.

    MathSciNet  MATH  Google Scholar 

  • C. Sansour and H. Bednarczyk. The Cosserat surface as a shell-model, theory and finite-element formulation. Comput Methods Appl Mech Eng, 120(1–2):1–32, 1995.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  • S. H. Sargsyan. Thermoelasticity of thin shells on the basis of asymmetrical theory of elasticity. J Therm Stresses, 32(8):791–818, 2009.

    Google Scholar 

  • S. H. Sargsyan and A. H. Sargsyan. General dynamic theory of micropolar elastic thin plates with free rotation and special features of their natural oscillations. Acoust Phys, 57(4):473–481, 2011.

    Google Scholar 

  • S. O. Sargsyan. On some interior and boundary effects in thin plates based on the asymmetric theory of elasticity. In R. Kienzler, H. Altenbach, and I. Ott, editors, Theories of Plates and Shells: Critical Review and New Applications, pages 201–210. Springer, Berlin, 2005.

    Google Scholar 

  • S. O. Sargsyan. Boundary-value problems of the asymmetric theory of elasticity for thin plates. J Appl Math Mech, 72(1):77–86, 2008.

    MathSciNet  Google Scholar 

  • P. Schiavone. On existence theorems in the theory of extensional motions of thin micropolar plates. Int J Eng Sci, 27(9):1129–1133, 1989.

    MathSciNet  MATH  Google Scholar 

  • P. Schiavone. Uniqueness in dynamic problems of thin micropolar plates. Appl Math Let, 4(2):81–83, 1991.

    MathSciNet  MATH  Google Scholar 

  • P. Schiavone and C. Constanda. Existence theorems in the theory of bending of micropolar plates. Int J Eng Sci, 27(4):463–468, 1989.

    MathSciNet  MATH  Google Scholar 

  • L. I. Shkutin. Nonlinear models of deformable momental continua (in Russian). J Appl Mech Techn Phys, 6:111–117, 1980.

    MathSciNet  Google Scholar 

  • L. I. Shkutin. Mechanics of Deformations of Flexible Bodies (in Russian). Nauka, Novosibirsk, 1985.

    Google Scholar 

  • J. G. Simmonds. The thermodynamical theory of shells: Descent from 3-dimensions without thickness expansions. In E. L. Axelrad and F. A. Emmerling, editors, Flexible shells, theory and applications, pages 1–11. Springer, Berlin, 1984.

    Google Scholar 

  • J. G. Simmonds. Some comments on the status of shell theory at the end of the 20th century: Complaints and correctives. In AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, 38th, and AIAA/ASME/AHS Adaptive Structures Forum, Kissimmee, FL, Apr. 7-10, 1997, Collection of Technical Papers, Pt. 3 (A97-24112 05-39), pages 1912–1921. AIAA, 1997.

    Google Scholar 

  • J. G. Simmonds and D. A Danielson. Nonlinear shell theory with finite rotation and stress-function vectors. Trans ASME. J Appl Mech, 39(4): 1085–1090, 1972.

    MATH  Google Scholar 

  • J. Tambača and I. Velči´c. Evolution model for linearized micropolar plates by the Fourier method. J Elasticity, 96(2):129–154, 2009.

    MathSciNet  MATH  Google Scholar 

  • S. P. Timoshenko and S. Woinowsky-Krieger. Theory of Plates and Shells. McGraw Hill, New York, 1985.

    Google Scholar 

  • S. K. Tomar. Wave propagation in a micropolar elastic plate with voids. J Vib Control, 11(6):849–863, 2005.

    MATH  Google Scholar 

  • C. Truesdell. Die Entwicklung des Drallsatzes. ZAMM, 44(4/5):149–158, 1964.

    MathSciNet  MATH  Google Scholar 

  • C. Truesdell. A First Course in Rational Continuum Mechanics. Academic Press, New York, 2nd edition, 1991.

    MATH  Google Scholar 

  • C. Truesdell. Rational Thermodynamics. Springer, New York, 2nd edition, 1984.

    MATH  Google Scholar 

  • C. Truesdell and W. Noll. The nonlinear field theories of mechanics. In S. Flügge, editor, Handbuch der Physik, Vol. III/3, pages 1–602. Springer, Berlin, 1965.

    Google Scholar 

  • J. R. Turner and D. A. C. Nicol. The Signorini problem for a Cosserat plate. J Inst Math and its Applications, 25(2):133–145, 1980.

    MathSciNet  MATH  Google Scholar 

  • I. Velči´c and J. Tambača. Relaxation theorem and lower-dimensional models in micropolar elasticity. Math Mech Solids, 15(8):812–853, 2010.

    MathSciNet  Google Scholar 

  • C. M. Wang, J. N. Reddy, and K. H. Lee. Shear Deformable Beams and Shells. Elsevier, Amsterdam, 2000.

    Google Scholar 

  • F.-Y. Wang. On the solutions of Eringen’s micropolar plate equations and of other approximate equations. Int J Eng Sci, 28(9):919–925, 1990.

    MATH  Google Scholar 

  • F.-Y. Wang and Y. Zhou. On the vibration modes of three-dimensional micropolar elastic plates. J Sound Vib, 146(1):1–16, 1991. K. Wiśniewski. Finite Rotation Shells: Basic Equations and Finite Elements for Reissner Kinematics. Lecture Notes on Numerical Methods in Engineering and Sciences. Springer, Berlin, 2010.

    Google Scholar 

  • H. T. Y. Yang, S. Saigal, A. Masud, and R. K. Kapania. A survey of recent shell finite elements. Int J Numer Meth Eng, 47(1-3):101–127, 2000.

    MathSciNet  MATH  Google Scholar 

  • L. Zee and E. Sternberg. Ordinary and strong ellipticity in the equilibrium theory of incompressible hyperelastic solids. Arch Ration Mech An, 83 (1):53–90, 1983.

    MathSciNet  MATH  Google Scholar 

  • P. A. Zhilin. Mechanics of deformable directed surfaces. Int J Solids Struct, 12(9–10):635–648, 1976.

    MathSciNet  Google Scholar 

  • P. A. Zhilin. Applied Mechanics. Foundations of the Theory of Shells (in Russian). St. Petersburg State Polytechnical University, Saint Petersburg, 2006.

    Google Scholar 

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

    Google Scholar 

  • L. M. Zubov. On universal deformations of nonlinear isotropic elastic shells. In H. Altenbach and V. A. Eremeyev, editors, Shell-like Structures, volume 15 of Advanced Structured Materials, pages 279–294. Springer, Berlin, 2011.

    Google Scholar 

  • L. M. Zubov. Nonlinear theory of isolated and continuosly distributed dislocations in elastic shells. Arch Civ Eng, XLV(2):385–396, 1999.

    Google Scholar 

  • L. M. Zubov. Semi-inverse solutions in nonlinear theory of elastic shells. Arch Mech, 53(4-5):599–610, 2001.

    MathSciNet  MATH  Google Scholar 

  • L. M. Zubov. Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin, 1997.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl Technische Mechanik, Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach & Victor A. Eremeyev

  2. South Scientific Center of RASci and South Federal University, Rostov on Don, Russian Federation

    Victor A. Eremeyev

Authors
  1. Holm Altenbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor A. Eremeyev
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Otto Von Guericke University, Magdeburg, Germany

    Holm Altenbach

  2. South Federal University and South Scientific Center of Rasci, Rostov on Don, Russian Federation

    Victor A. Eremeyev

Rights and permissions

Reprints and permissions

Copyright information

© 2013 CISM, Udine

About this chapter

Cite this chapter

Altenbach, H., Eremeyev, V.A. (2013). Cosserat-Type Shells. In: Altenbach, H., Eremeyev, V.A. (eds) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences, vol 541. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1371-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-1371-4_3

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-1370-7

  • Online ISBN: 978-3-7091-1371-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation