Skip to main content
SpringerLink
Log in

Mathematical Modeling of Elastic Thin Bodies with one Small Size

  • Chapter
  • First Online:
Higher Gradient Materials and Related Generalized Continua

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 120))

Abstract

Some questions on parametrization with an arbitrary base surface of a thin-body domain with one small size are considered. This parametrization is convenient to use in those cases when the domain of the thin body does not have symmetry with respect to any surface. In addition, it is more convenient to find the moments of mechanical quantities than the classical. Various families of bases (frames) and the corresponding families of parameterizations generated by them are considered. Expressions for the components of the second rank unit tensor are obtained. Representations of some differential operators, the system of motion equations, and the constitutive relation (CR) of the micropolar theory of elasticity are given for the considered parametrization of a thin body domain. The main recurrence formulas of system of orthogonal Legendre polynomials are written out and some additional recurrence relations are obtained, which play an important role in the construction of various variants of thin bodies. The definitions of the moment of the kth order of a certain value with respect to an arbitrary system of orthogonal polynomials and system of Legendre polynomials are given. Expressions are obtained for the moments of the kth order of partial derivatives and some expressions for the system of Legendre polynomials. Various representations of the system of motion equations and CR in the moments for the theory of thin bodies are given. Boundary conditions are derived. The CR of the classical and micropolar theory of the zero approximation and approximation of order r in the moments are obtained. The boundary conditions of physical and thermal contents on the front surfaces are given. The statements of dynamic problems in moments of the approximation (r, M) of a micropolar thermomechanics of a deformable thin body, as well as a non-stationary temperature problem in moments are given. It should be noted that using the considered method of constructing a theory of thin bodies with one small size, we obtain an infinite system of equations, which has the advantage that it contains quantities depending on two variables, the base surface Gaussian coordinates x1, x2. So, to reduce the number of independent variables by one we need to increase the number of equations to infinity, which of course has its obvious practical inconveniences. In this connection, the reduction of the infinite system to the finite is made.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as 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 shipping worldwide - see info
Hardcover Book
USD 109.99
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.

Similar content being viewed by others

References

  • Alekseev AE (1994) Derivation of equations for a layer of variable thickness based on expansions in terms of Legendre’s polynomials. Journal of Applied Mechanics and Technical Physics 35(4):612–622

    Google Scholar 

  • Alekseev AE (1995) Bending of a three-layer orthotropic beam. Journal of Applied Mechanics and Technical Physics 36(3):458–465

    Google Scholar 

  • Alekseev AE (2000) Iterative method for solving problems of deformation of layered structures, taking into account the slippage of layers (in Russ.). Dinamika sploshnoy sredy: Sb nauch tr 116:170–174

    Google Scholar 

  • Alekseev AE, Annin BD (2003) Equations of deformation of an elastic inhomogeneous laminated body of revolution. Journal of Applied Mechanics and Technical Physics 44(3):432–437

    Google Scholar 

  • Alekseev AE, Demeshkin AG (2003) Detachment of a beam glued to a rigid plate. Journal of Applied Mechanics and Technical Physics 44(4):577–583

    Google Scholar 

  • Alekseev AE, Alekhin VV, Annin BD (2001) Plane elastic problem for an inhomogeneous layered body. Journal of Applied Mechanics and Technical Physics 42(6):1038–1042

    Google Scholar 

  • Altenbach H (1991) Modelling of viscoelastic behaviour of plates. In: Zyczkowski M (ed) Creep in Structures, Springer, Berlin Heidelberg, pp 531–537

    Google Scholar 

  • Altenbach J, Altenbach H, Eremeyev V (2010) On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Archive of Applied Mechanics 80(1):73–92

    Google Scholar 

  • Ambartsumyan SA (1958) On the theory of bending of anisotropic plates and shallow shells. Izv AN SSSR OTN (5):69–77

    Google Scholar 

  • Ambartsumyan SA (1970) A new refined theory of anisotropic shells. Polymer Mechanics 6(5):766–776

    Google Scholar 

  • Ambartsumyan SA (1974) General Theory of Anisotropic Shells (in. Russ.). Nauka, Moscow

    Google Scholar 

  • Ambartsumyan SA (1987) Theory of Anisotropic Plates (in Russ.). Nauka, Moscow

    Google Scholar 

  • Chepiga VE (1976) To the improved theory of laminated shells (in Russ.). Appl Mech 12(11):45–49

    Google Scholar 

  • Chepiga VE (1977) Construction of the theory of multilayer anisotropic shells with given conditional accuracy of order hN (in Russ.). Mekh Tverdogo Tela (4):111–120

    Google Scholar 

  • Chepiga VE (1986a) Asymptotic error of some hypotheses in the theory of laminated shells (in Russ.). Theory and calculation of elements of thin-walled structures pp 118–125

    Google Scholar 

  • Chepiga VE (1986b) Numerical analysis of equations of the improved theory of laminated shells (in Russ.). 290-B1986, VINITI

    Google Scholar 

  • Chepiga VE (1986c) The study of stability of multilayer shells by an improved theory (in Russ.). 289-B1986, VINITI

    Google Scholar 

  • Chernykh KF (1986) Nonlinear Theory of Elasticity in Engineering Computations (in Russ.). Mashinostroenie, Leningrad

    Google Scholar 

  • Chernykh KF (1988) Introduction into Anisotropic Elasticity (in Russ.). Nauka, Moscow

    Google Scholar 

  • Della Corte A, Battista A, dell’Isola F, et al (2019) Large deformations of Timoshenko and Euler beams under distributed load. Math Phys 70(52)

    Google Scholar 

  • Dergileva LA (1976) Solution method for a plane contact problem for an elastic layer (in Russ.). Continuum Dynamics 25:24–32

    Google Scholar 

  • Egorova O, Zhavoronok S, Kurbatov A (2015) The variational equations of the extended Nth order shell theory and its application to some problems of dynamics (in Russ.). Perm National Polytechnic University Mechanics Bulletin (2):36–59

    Google Scholar 

  • Eremeyey VA, Zubov LM (2008) Mechanics of Elastic Shells

    Google Scholar 

  • Eremeyey VA, Lebedev LP, Altenbach H (2013) Foundations of Micropolar Mechanics. Springer-Verlag

    Google Scholar 

  • Fellers J, Soler A (1970) Approximate solution of the finite cylinder problem using Legendre polynomials. AIAA 8(11)

    Google Scholar 

  • Filin AP (1987) Elements of the Theory of Shells (in Russ.). Stroyizdat, Leningrad

    Google Scholar 

  • Gol’denveizer AL (1976) Theory of Elastic Shells (in Russ.). Nauka, Moscow

    Google Scholar 

  • Gol’denveizer AL (1962) Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity. Journal of Applied Mathematics and Mechanics 26(4):1000–1025

    Google Scholar 

  • Gol’denveizer AL (1963) Derivation of an approximate theory of shells by means of asymptotic integration of the equations of the theory of elasticity. Journal of Applied Mathematics and Mechanics 27(4):903–924

    Google Scholar 

  • Grigolyuk EI, Selezov IT (1973) Nonclassic oscillation theories of rods, plates, and shells (in Russ.), vol 5. VINITI. Itogi nauki i tekniki, Moscow

    Google Scholar 

  • Hencky H (1947) Ãœber die berücksichtigung der schubverzerrung in ebenen platten. Ingenieur-Archiv 16:72–76

    Google Scholar 

  • Hertelendy P (1968) An approximate theory governing symmetric motions of elastic rods of rectangular or square cross section. Trans ASME Journal of Applied Mechanics 35(2):333–341

    Google Scholar 

  • Ivanov GV (1976) Solution of the plane mixed problem of the theory of elasticity in the form of a series in Legendre polynomials (in Russ.). Z Prikl Mekh Tekhn Fiz (6):126–137

    Google Scholar 

  • Ivanov GV (1977) Solutions of plane mixed problems for the Poisson equation in the form of series over Legendre polynomials (in Russ.). Continuum Dynamics 28:43–54

    Google Scholar 

  • Ivanov GV (1979) Reduction of a three-dimensional problem for an inhomogeneous shell to a two-dimensional problem (in Russ.). Dynamic Problems of Continuum Mechanics 39

    Google Scholar 

  • Ivanov GV (1980) Theory of Plates and Shells (in Russ.). Novosib. State Univ., Novosibirsk

    Google Scholar 

  • Javili A, dell’Isola F, Stemmann P (2011) Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J Phys: Conf Ser

    Google Scholar 

  • Kantor MM, Nikabadze MU, Ulukhanyan AR (2013) Equations of motion and boundary conditions of physical meaning of micropolar theory of thin bodies with two small cuts. Mechanics of Solids 48(3):317–328

    Google Scholar 

  • Khoroshun LP (1978) On the construction of equations of layered plates and shells (in russ.). Prikladnaya Mekhanika (10):3–21

    Google Scholar 

  • Khoroshun LP (1985) The concept of a mixture in the construction of the theory of layered plates and shells in russ.). Prikladnaya Mekhanika 21(4):110–118

    Google Scholar 

  • Kienzler R (1982) Eine Erweiterung der klassischen Schalentheorie; der Einfluß von Dickenverzerrungen und Querschnittsverwölbungen. Ingenieur-Archiv 52(5):311–322

    Google Scholar 

  • Kirchhoff G (1850) Ãœber das gleichgewicht und die bewegung einer elastischen scheibe. Journal für die reine und angewandte Mathematik (Crelles Journal) (40):51–88

    Google Scholar 

  • Kupradze VD (ed) (1979) Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics, vol 25. North Holland

    Google Scholar 

  • Kuznetsova E, Kuznetsova EL, Rabinskiy LN, Zhavoronok SI (2018) On the equations of the analytical dynamics of the quasi-3D plate theory of I. N. Vekua type and some their solutions. Journal of Vibroengineering 20(2):1108–1117

    Google Scholar 

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

    Google Scholar 

  • LewiÅ„ski T (1987) On refined plate models based on kinematical assumptions. Ingenieur-Archiv 57(2):133–146

    Google Scholar 

  • Lo KH, Christensen RM, Wu EM (1977a) A high-order theory of plate deformation. Part 1: Homogeneous plates. Trans ASME Journal of Applied Mechanics 44(4):663–668

    Google Scholar 

  • Lo KH, Christensen RM, Wu EM (1977b) A high-order theory of plate deformation. Part 2: Laminated Plates. Trans ASME Journal of Applied Mechanics 44(4):669–676

    Google Scholar 

  • Lurie AI (1990) Non-linear Theory of Elasticity, North-Holland Series in Applied Mathematics and Mechanics, vol 36. North Holland

    Google Scholar 

  • Medick MA (1966) One-dimensional theories of wave propagation and vibrations in elastic bars of rectangular cross section. Trans ASME Journal of Applied Mechanics 33(3):489–495

    Google Scholar 

  • Meunargiya TV (1987) Development of the method of I. N. Vekua for problems of the three-dimensional moment elasticity (in Russ.). Tbilisi State Univ., Tbilisi

    Google Scholar 

  • Mindlin RD, Medick MA (1959) Extensional vibrations of elastic plates. Trans ASME J Appl Mech 26(4):561–569

    Google Scholar 

  • Naghdi PM (1972) The theory of shells and plates. In: Flügge S (ed) Handbuch der Physik, vol VIa/2, Springer, Berlin, Heidelberg, pp 425–640

    Google Scholar 

  • Nikabadze MU (1988a) On the theory of shells with two base surfaces (in Russ.). 8149-B88, VINITI

    Google Scholar 

  • Nikabadze MU (1988b) Parameterization of shells with two base surfaces (in Russ.). 5588-B88, VINITI

    Google Scholar 

  • Nikabadze MU (1989) Deformation of layered viscoelastic shells. In: Actual problems of strength in mechanical engineering (in Russ.), SVVMIU, Sevastopol, p 1

    Google Scholar 

  • Nikabadze MU (1990a) Modeling of nonlinear deformation of elastic shells (in Russ.). PhD thesis, Lomonosov Moscow State University

    Google Scholar 

  • Nikabadze MU (1990b) Plane curvilinear rods (in Russ.). 4509-B90, VINITI

    Google Scholar 

  • Nikabadze MU (1990c) To the theory of shells with two base surfaces (in Russ.). 1859-B90, VINITI

    Google Scholar 

  • Nikabadze MU (1990d) To the theory of shells with two base surfaces (in Russ.). 2676-B90, VINITI

    Google Scholar 

  • Nikabadze MU (1991) New kinematic hypothesis and new equations of motion and equilibrium theories of shells and plane curvilinear rods (in Russ.). Vestn Mosk Univ, Matem Mekhan (6):54–61

    Google Scholar 

  • Nikabadze MU (1998a) Constitutive relations of the new linear theory of thermoelastic shells (in Russ.). In: Actual problems of shell mechanics, UNIPRESS, Kazan, pp 158–162

    Google Scholar 

  • Nikabadze MU (1998b) Different representations of the cauchy-green deformation tensor and the linear deformation tensor and their components in the new theory of shells (in Russ.). Mathematical modeling of systems and processes (6):59–65

    Google Scholar 

  • Nikabadze MU (1999a) Constitutive relations of the new linear theory of thermoelastic shells of TS class (in Russ.). Mathematical modeling of systems and processes (7):52–56

    Google Scholar 

  • Nikabadze MU (1999b) New rod space parametrization (in Russ.). 1663-B99, VINITI

    Google Scholar 

  • Nikabadze MU (1999c) New rod theory (in Russ.). In: 16th inter-republican conference on numerical methods for solving problems of the theory of elasticity and plasticity, Novosibirsk

    Google Scholar 

  • Nikabadze MU (1999d) Various forms of the equations of motion and boundary conditions of the new theory of shells (in Russ.). Mathematical modeling of systems and processes (7):49–51

    Google Scholar 

  • Nikabadze MU (2000a) Some geometric relations of the theory of shells with two basic surfaces (in Russ.). Izv RAN MTT (4):129–139

    Google Scholar 

  • Nikabadze MU (2000b) To the parametrization of the multilayer shell domain of 3d space (in Russ.). Mathematical modeling of systems and processes (8):63–68

    Google Scholar 

  • Nikabadze MU (2001a) Dynamic equations of the theory of multilayer shell constructions under the new kinematic hypothesis (in Russ.). In: Elasticity and non-elasticity, 1, Izd. MGU, pp 389–395

    Google Scholar 

  • Nikabadze MU (2001b) Equations of motion and boundary conditions of the theory of rods with several basic curves (in Russ.). Vestn Mosk Univ, Matem Mekhan (3):35–39

    Google Scholar 

  • Nikabadze MU (2001c) Location gradients in the theory of shells with two basic surfaces (in Russ.). Mech Solids 36(4):64–69

    Google Scholar 

  • Nikabadze MU (2001d) To the variant of the theory of multilayer structures (in Russ.). Izv RAN MTT (1):143–158

    Google Scholar 

  • Nikabadze MU (2002a) Equations of motion and boundary conditions of a variant of the theory of multilayer plane curvilinear rods (in Russ.). Vestn Mosk Univ, Matem Mekhan (6):41–46

    Google Scholar 

  • Nikabadze MU (2002b) Modern State of Multilayer Shell Structures (in Russ.). 2289–B2002, VINITI

    Google Scholar 

  • Nikabadze MU (2003) Variant of the theory of shallow shells (in Russ.). In: Lomonosovskiye chteniya. Section mechanics., Izd. Moscov. Univ., Moscow

    Google Scholar 

  • Nikabadze MU (2004a) Generalization of the Huygens-Steiner theorem and the Boer formulas and some of their applications (in Russ.). Izv RAN MTT (3):64–73

    Google Scholar 

  • Nikabadze MU (2004b) Variants of the theory of shells with the use of expansions in Legendre polynomials (in Russ.). In: Lomonosovskiye chteniya. Section mechanics., Izd. Moscov. Univ., Moscow

    Google Scholar 

  • Nikabadze MU (2005) To the variant of the theory of multilayer curvilinear rods (in Russ.). Izv RAN MTT (6):145–156

    Google Scholar 

  • Nikabadze MU (2006) Application of Classic Orthogonal Polynomials to the Construction of the Theory of Thin Bodies (in Russ.). Elasticity and non-elasticity pp 218–228

    Google Scholar 

  • Nikabadze MU (2007a) Application of Chebyshev Polynomials to the Theory of Thin Bodies. Moscow University Mechanics Bulletin 62(5):141–148

    Google Scholar 

  • Nikabadze MU (2007b) Some issues concerning a version of the theory of thin solids based on expansions in a system of Chebyshev polynomials of the second kind. Mechanics of Solids 42(3):391–421

    Google Scholar 

  • Nikabadze MU (2007c) To theories of thin bodies (in Russ.). In: Non-classical problems of mechanics, Proceedings of the international conference, Kutaisi, vol 1, pp 225–242

    Google Scholar 

  • Nikabadze MU (2008a) Mathematical modeling of elastic thin bodies with two small dimensions with the use of systems of orthogonal polynomials (in Russ.). 722 – B2008, VINITI

    Google Scholar 

  • Nikabadze MU (2008b) The application of systems of Legendre and Chebyshev polynomials at modeling of elastic thin bodies with a small size (in Russ.). 720-B2008, VINITI

    Google Scholar 

  • Nikabadze MU (2014a) Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies (in Russ.). Moscow Univ. Press, Moscow

    Google Scholar 

  • Nikabadze MU (2014b) Method of orthogonal polynomials in mechanics of micropolar and classical elasticity thin bodies (in Russ.). Doctoral dissertation. Moscow, MAI

    Google Scholar 

  • Nikabadze MU(2016) Eigenvalue problems of a tensor and a tensor-block matrix (tmb) of any even rank with some applications in mechanics. In: Altenbach H, Forest S (eds) Generalized continua as models for classical and advanced materials, Advanced Structured Materials, vol 42, pp 279–317

    Google Scholar 

  • Nikabadze MU (2017a) Eigenvalue problem for tensors of even rank and its applications in mechanics Journal of Mathematical Sciences 221(2):174–204

    Google Scholar 

  • Nikabadze MU (2017b) Topics on tensor calculus with applications to mechanics. Journal of Mathematical Sciences 225(1):1–194

    Google Scholar 

  • Nikabadze MU, Ulukhanyan A (2005a) Formulation of the problem for thin deformable 3d body (in Russ.). Vestn Mosk Univ, Matem Mekhan (5):43–49

    Google Scholar 

  • Nikabadze MU, Ulukhanyan A (2005b) Formulations of problems for a shell domain according to three-dimensional theories (in Russ.). 83–B2005, VINITI

    Google Scholar 

  • Nikabadze MU, Ulukhanyan A (2008) Mathematical modeling of elastic thin bodies with one small dimension with the use of systems of orthogonal polynomials (in Russ.). 723 – B2008, VINITI

    Google Scholar 

  • Nikabadze MU, Ulukhanyan AR (2016) Analytical solutions in the theory of thin bodies. In: Altenbach H, Forest S (eds) Generalized continua as models for classical and advanced materials, Advanced Structured Materials, vol 42, pp 319–361

    Google Scholar 

  • Nowacki W (1975) Theory of Elasticity. Mir, Moscow, (Russian translation)

    Google Scholar 

  • Pelekh BL (1973) Theory of shells with finite shear stiffness (in Russ.). Naukova Dumka, Kiev

    Google Scholar 

  • Pelekh BL (1978) The Generalized Theory of Shells (in Russ.). Vischa shkola, Lvov

    Google Scholar 

  • Pelekh BL, SukhorolskiiMA(1977) Construction of the generalized theory of transversal-isotropic shells in application to contact problems (in Russ.). Composites and New Structures pp 27–39

    Google Scholar 

  • Pelekh BL, Sukhorolskii MA (1980) Contact problems of the theory of elastic anisotropic shells (in Russ.). Naukova Dumka, Kiev

    Google Scholar 

  • Pelekh BL, Maksimuk AV, Korovaichuk IM (1988) Contact problems for laminated elements of constructions and bodies with coating (in Russ.). Naukova Dumka, Kiev

    Google Scholar 

  • Pikul VV (1992) To the problem of constructing a physically correct theory of shells (in Russ.). Izv RAN MTT (3):18–25

    Google Scholar 

  • Pobedrya BE (1986) Lectures on tensor analysis (in Russ.). M: Izd. Moscov. Univ.

    Google Scholar 

  • Pobedrya BE (1995) Numerical methods in the theory of elasticity and plasticity (in Russ.). Izd. Moscov. Univ., Moscow

    Google Scholar 

  • Pobedrya BE (2003) On the theory of constitutive relations in the mechanics of a deformable solid (in Russ.). In: Problemy mekhaniki, Fiszmatlit, Moscow, pp 635–657

    Google Scholar 

  • Pobedrya BE (2006) Theory of thermomechanical processes (in Russ.). In: Elasticity and nonelasticity, Izd. MGU, pp 70–85

    Google Scholar 

  • Preußer G (1984) Eine systematische Herleitung verbesserter Plattengleichungen. Ingenieur-Archiv 54(1):51–61

    Google Scholar 

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

    Google Scholar 

  • Reissner E (1944) On the theory of bending of elastic plates. Journal of Mathematics and Physics 23(1-4):184–191

    Google Scholar 

  • Sansone G (1959) Orthogonal Functions. Interscience Publishers Inc, New York

    Google Scholar 

  • Seppecher P, Alibert J, dell’Isola F (2013) Linear elastic trusses leading to continua with exotic mechanical interactions. J of the Mech and Phys of Solids 61(12):2381–2401

    Google Scholar 

  • Sokol’nikov IS (1971) Tensor analysis (in Russ.). Nauka, Moscow

    Google Scholar 

  • Soler AI (1969) Higher-order theories for structural analysis using Legendre polynomial expansions. Trans ASME Journal of Applied Mechanics 36(4):757–762

    Google Scholar 

  • Suyetin PK (1976) Classical orthogonal polynomials (in Russ.). Nauka, Moscow

    Google Scholar 

  • Tvalchrelidze AK (1984) Theory of elastic shells using several base surfaces (in Russ.). In: Theory and numerical methods for calculating plates and shells, Tbilisi

    Google Scholar 

  • Tvalchrelidze AK (1986) Basic equations of the theory of shells, taking into account large deformations and shears (in Russ.). Soobshch AN GruzSSR 121(1):53–56

    Google Scholar 

  • Tvalchrelidze AK (1994) Shell theory using several base surfaces and some applications (in Russ.). PhD thesis, Kutaisi

    Google Scholar 

  • Tvalchrelidze AK, Tvaltvadze DV, Nikabadze MU (1984) To the calculation of large axisymmetric deformations of the shells of rotation of elastomers (in Russ.). In: XXII scientific and technical. conf., Tbilisi

    Google Scholar 

  • Ulukhanyan AR (2011) Dynamic equations of the theory of thin prismatic bodies with expansion in the system of Legendre polynomials. Mechanics of Solids 46(3):467–479

    Google Scholar 

  • Vajeva DV, Volchkov YM (2005) The equations for determination of stress-deformed state of multilayer shells (in Russ.). In: In Proc. 9th Russian–Korean Symp. Sci. and Technol., Novosib. State Univ., Novosibirsk, pp 547–550

    Google Scholar 

  • Vasiliev VV, Lurie SA (1990a) On the problem of constructing non-classical theories of plates (in Russ.). Izv RAN MTT (2):158–167

    Google Scholar 

  • Vasiliev VV, Lurie SA (1990b) To the problem of clarifying the theory of shallow shells (in Russ.). Izv RAN MTT (6):139–146

    Google Scholar 

  • Vekua IN (1955) On a method of calculating of prismatic shells (in Russ.). In: Tr. Tbilis. matem. ins-ta im. A.M.Razmadze, Izd-vo Metsniereba, Tbilisi, vol 21, pp 191–259

    Google Scholar 

  • Vekua IN (1964) The theory of thin and shallow shells of variable thickness (in Russ.). Novosibirsk

    Google Scholar 

  • Vekua IN (1965) Theory of thin shallow shells of variable thickness (in Russ.). In: Tr. Tbilis. matem. ins-ta im. A.M.Razmadze, Izd-vo Metsniyereba, Tbilisi, vol 30, pp 1–104

    Google Scholar 

  • Vekua IN (1970) Variational principles for constructing the theory of shells (in Russ.). Izd-vo Tbil. Un-ta, Tbilisi

    Google Scholar 

  • Vekua IN (1972) On one direction of constructing the theory of shells (in Russ.). In: Mechanics in the USSR for 50 years, vol 3, Nauka, Moscow, pp 267–290

    Google Scholar 

  • Vekua IN (1978) Fundamentals of tensor analysis and the theory of covariant (in Russ.). Nauka Vekua IN (1982) Some common methods for constructing various variants of the theory of shells (in Russ.). Nauka, Moscow

    Google Scholar 

  • Volchkov YM (2000) Finite elements with adjustment conditions on their edges (in Russ.). Dinamika sploshnoy sredi 116:175–180

    Google Scholar 

  • Volchkov YM, Dergileva LA (1977) Solution of elastic layer problems by approximate equations and comparison with solutions of the theory of elasticity (in Russ.). Dinamika sploshnoy sredy 28:43–54

    Google Scholar 

  • Volchkov YM, Dergileva LA (1999) Edge effects in the stress state of a thin elastic interlayer (in Russ.). Journal of Applied Mechanics and Technical Physics 40(2):354–359

    Google Scholar 

  • Volchkov YM, Dergileva LA (2004) Equations of an elastic anisotropic layer. Journal of Applied Mechanics and Technical Physics 45(2):301–309

    Google Scholar 

  • Volchkov YM, Dergileva LA (2007) Reducing three-dimensional elasticity problems to two-dimensional problems by approximating stresses and displacements by Legendre polynomials. Journal of Applied Mechanics and Technical Physics 48(3):450–459

    Google Scholar 

  • Volchkov YM, Dergileva LA, Ivanov GV (1994) Numerical modeling of stress states in two-dimensional problems of elasticity by the layers method (in Russ.). Journal of Applied Mechanics and Technical Physics 35(6):936–941

    Google Scholar 

  • Wunderlich W (1973) Vergleich verschiedener Approximationen der Theorie dünner Schalen (mit numerischen Ergebnissen). Allgemeine Schalentheorien, Techn Wiss Mitteilungen (73):3.1–3.24

    Google Scholar 

  • Zhavoronok S (2014) A Vekua type linear theory of thick elastic shells. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 94(1/2):164–184

    Google Scholar 

  • Zhavoronok SI (2017) On Hamiltonian formulations and conservation laws for plate theories of Vekua–Amosov type. International Journal for Computational Civil and Structural Engineering 13(4):82–95

    Google Scholar 

  • Zhavoronok SI (2018) On the use of extended plate theories of Vekua–Amosov type for wave dispersion problems. International Journal for Computational Civil and Structural Engineering 14(1):36–48

    Google Scholar 

  • Zhilin PA (1976) Mechanics of deformable directed surfaces. Int J Solids Structures 12:635–648

    Google Scholar 

  • Zozulya VV (2017a) Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures 4(1):119–133

    Google Scholar 

  • Zozulya VV (2017b) Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures 4(1):104–118

    Google Scholar 

  • Zozulya VV (2017c) Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures 4(1):221–236

    Google Scholar 

  • Zozulya VV, Saez A (2014) High-order theory for arched structures and its application for the study of the electrostatically actuated MEMS devices. Archive of Applied Mechanics 84(7):1037–1055

    Google Scholar 

  • Zozulya VV, Saez A (2016) A high-order theory of a thermoelastic beams and its application to the MEMS/NEMS analysis and simulations. Archive of Applied Mechanics 86(7):1255–1272

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Bauman Moscow State Technical University, Moscow, Russia

    Mikhail Nikabadze

  2. Bauman Moscow State Technical University, Moscow, Russia

    Armine Ulukhanyan

Authors
  1. Mikhail Nikabadze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Armine Ulukhanyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail Nikabadze .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Technische Universität Berlin, Institut für Mechanik, Berlin, Germany

    Wolfgang H. Müller

  3. Institut für Mechanik, Technische Universität Berlin, Berlin, Germany

    Bilen Emek Abali

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Nikabadze, M., Ulukhanyan, A. (2019). Mathematical Modeling of Elastic Thin Bodies with one Small Size. In: Altenbach, H., Müller, W., Abali, B. (eds) Higher Gradient Materials and Related Generalized Continua. Advanced Structured Materials, vol 120. Springer, Cham. https://doi.org/10.1007/978-3-030-30406-5_9

Download citation

Publish with us

Policies and ethics

Search

Navigation