Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells

Affiliation(s)

Department of Physical Mathematical Sciences, Gyumri State Pedagogical Institute, Gyumri, Armenia.

Department of Physical Mathematical Sciences, Gyumri State Pedagogical Institute, Gyumri, Armenia.

ABSTRACT

In the present paper asymptotic solution of boundary-value problem of three-dimensional micropolar theory of elasticity with free fields of displacements and rotations is constructed in thin domain of the shell. This boundary-value problem is singularly perturbed with small geometric parameter. Internal iteration process and boundary layers are constructed, problem of their jointing is studied and boundary conditions for each of them are obtained. On the basis of the results of the internal boundary-value problem the asymptotic two-dimensional model of micropolar elastic thin shells is constructed. Further, the qualitative aspects of the asymptotic solution are accepted as hypotheses and on the basis of them general applied theory of micropolar elastic thin shells is constructed. It is shown that both the constructed general applied theory of micropolar elastic thin shells and the classical theory of elastic thin shells with consideration of transverse shear deformations are asymptotically confirmed theories.

In the present paper asymptotic solution of boundary-value problem of three-dimensional micropolar theory of elasticity with free fields of displacements and rotations is constructed in thin domain of the shell. This boundary-value problem is singularly perturbed with small geometric parameter. Internal iteration process and boundary layers are constructed, problem of their jointing is studied and boundary conditions for each of them are obtained. On the basis of the results of the internal boundary-value problem the asymptotic two-dimensional model of micropolar elastic thin shells is constructed. Further, the qualitative aspects of the asymptotic solution are accepted as hypotheses and on the basis of them general applied theory of micropolar elastic thin shells is constructed. It is shown that both the constructed general applied theory of micropolar elastic thin shells and the classical theory of elastic thin shells with consideration of transverse shear deformations are asymptotically confirmed theories.

Cite this paper

Sargsyan, S. (2015) Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells.*Advances in Pure Mathematics*, **5**, 629-642. doi: 10.4236/apm.2015.510057.

Sargsyan, S. (2015) Asymptotically Confirmed Hypotheses Method for the Construction of Micropolar and Classical Theories of Elastic Thin Shells.

References

[1] Friedrichs, K.O. and Dressler, R.F.A. (1961) Boundary Layer Theory for Elastic Plates. Communications on Pure and Applied Mathematics, 1, 1-33.

http://dx.doi.org/10.1002/cpa.3160140102

[2] Green, A.E. (1962) On the Linear Theory of Thin Elastic Shells. Proceedings of the Royal Society Series A, 266, 3-25.

http://dx.doi.org/10.1098/rspa.1962.0053

[3] Vorovich, I.I. (1966) On Some Mathematical Questions of Plate and Shell Theories. Proceedings of the II Union Congress of Theoretical and Applied Mechanics, 3, 116-136. (In Russian)

[4] Goldenvejzer, A.L. (1976) Theory of Elastic Thin Shells. Moscow. (In Russian)

[5] Kaplunov, J.D., Kossovich, L.Yu. and Nolde, E.V. (1998) Dynamics of Thin Walled Elastic Bodies. Academic Press.

[6] Agalovyan, L.A. (1997) The Asymptotic Theory of Anisotropic Plates and Shells. Moscow. (In Russian)

[7] Rogacheva, N.N. (1994) The Theory of Piezoelectric Plates and Shells. Boca Raton, London.

[8] Ustinov, Yu.A. and Shenev, M.A. (1978) On Some Directions of Development of the Asymptotic Method of Plates and Shells. Calculations of Plates and Shells, 3-27. (In Russian)

[9] Sargsyan, S.H. (1992) General Two-Dimensional Theory of Magnetoelasticity of Thin Shells, Yerevan. (In Russian)

[10] Altenbach, H. and Eremeyev, V.A. (2009) On the Linear Theory of Micropolar Plates. Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM), 89, 242-256.

http://dx.doi.org/10.1002/zamm.200800207

[11] Altenbach, J., Altenbach, H. and Eremeyev, V.A. (2009) On Generalized Cosserat-Tape Theories of Plates and Shells: A Short Review and Bibliography. Archive of Applied Mechanics, 80, 73-92.

[12] Sargsyan, S.H. (2012) General Theory of Micropolar Elastic Thin Shells. Journal of Physical Mesomechanics, 15, 69-79.

http://dx.doi.org/10.1134/S1029959912010079

[13] Sargsyan, S.H. (2011) The General Dynamic Theory of Micropolar Elastic Thin Shells. Reports of Physics, 56, 39-42.

[14] Sargsyan, S.H. (2012) Mathematical Model of Micropolar Elastic Thin Plates and Their Strength and Stiffness Characteristics. Journal of Applied Mechanics and Technical Physics, 53, 275-282.

[15] Sargsyan, S.H. (2008) Boundary-Value Problems of Asymmetric Theory of Elasticity for Thin Plates. Journal of Applied Mathematics and Mechanics, 72, 77-86.

[16] Sargsyan, S.H. (2012) The Theory of Micropolar Thin Elastic Shells. Journal of Applied Mathematics and Mechanics, 76, 235-249.

[17] Sargsyan, S.H. (2012) The Construction of Mathematical Model of Micropolar Elastic Thin Beams on the Basis of the Asymptotic Theory. News of Higer Educational Intitutes. The North Caucasus Region. Natural Sciences, 5, 31-37. (In Russian)

[18] Sargsyan, S.H. (2013) The Asymptotic Method of the Construction of Mathematical Models of Micropolar Elastic Thin Plates. Scientific Proceedings of GSPI, 1, 7-37. (In Russian)

[19] Goldenveizer, A.L., Kaplunov, J.D. and Nolde, E.V. (1993) On Timoshenko-Reissner Type Theories of Plates and Shells. International Journal of Solids and Structures, 30, 675-694.

[20] Palmov, V.A. (1964) Basic Equations of the Theory of Asymmetric Elasticity. Applied Mathematics and Mechanics, 28, 1117-1120. (In Russian)

[21] Nowacki, W. (1986) Theory of Asymmetric Elasticity. Pergamon Press, Oxford, New York, Toronto, Sydney, Paris, Frankfurt.

[22] Pelech, P.L. (1977) Stress Concentration around the Holes in Bending Transversely Isotropic Plates. Kiev. (In Russian)

[23] Grigolyuk, E.I. and Kulikov, G.M. (1988) Multilayered Reinforced Shells. Calculation of Pneumatic Tires, Moscow. (In Russian)

[24] Grigorenko, Y.M. and Vasilenko, A.T. (1981) Theory of Shells of Variable Stiffness. Kiev. (In Russian)

[25] Percev, A.K. and Platonov, E.G. (1987) Dynamics of Plates and Shells. Leningrad. (In Russian)

[26] Sargsyan, A.A. (2011) Asymptotic Analysis of Dynamic Initial Boundary-Value Problem of Asymmetric Theory of Elasticity with Free Rotations in Thin Domain of the Shell. News of NAS Armenia, Mechanics, 64, 39-50. (In Russian)

[27] Sargsyan, S.H. (2012) Effective Manifestations of Characteristics of Strength and Rigidy of Micropolar Elastic Thin Bars. Journal of Materials Science and Engineering, 2, 98-108.

[1] Friedrichs, K.O. and Dressler, R.F.A. (1961) Boundary Layer Theory for Elastic Plates. Communications on Pure and Applied Mathematics, 1, 1-33.

http://dx.doi.org/10.1002/cpa.3160140102

[2] Green, A.E. (1962) On the Linear Theory of Thin Elastic Shells. Proceedings of the Royal Society Series A, 266, 3-25.

http://dx.doi.org/10.1098/rspa.1962.0053

[3] Vorovich, I.I. (1966) On Some Mathematical Questions of Plate and Shell Theories. Proceedings of the II Union Congress of Theoretical and Applied Mechanics, 3, 116-136. (In Russian)

[4] Goldenvejzer, A.L. (1976) Theory of Elastic Thin Shells. Moscow. (In Russian)

[5] Kaplunov, J.D., Kossovich, L.Yu. and Nolde, E.V. (1998) Dynamics of Thin Walled Elastic Bodies. Academic Press.

[6] Agalovyan, L.A. (1997) The Asymptotic Theory of Anisotropic Plates and Shells. Moscow. (In Russian)

[7] Rogacheva, N.N. (1994) The Theory of Piezoelectric Plates and Shells. Boca Raton, London.

[8] Ustinov, Yu.A. and Shenev, M.A. (1978) On Some Directions of Development of the Asymptotic Method of Plates and Shells. Calculations of Plates and Shells, 3-27. (In Russian)

[9] Sargsyan, S.H. (1992) General Two-Dimensional Theory of Magnetoelasticity of Thin Shells, Yerevan. (In Russian)

[10] Altenbach, H. and Eremeyev, V.A. (2009) On the Linear Theory of Micropolar Plates. Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM), 89, 242-256.

http://dx.doi.org/10.1002/zamm.200800207

[11] Altenbach, J., Altenbach, H. and Eremeyev, V.A. (2009) On Generalized Cosserat-Tape Theories of Plates and Shells: A Short Review and Bibliography. Archive of Applied Mechanics, 80, 73-92.

[12] Sargsyan, S.H. (2012) General Theory of Micropolar Elastic Thin Shells. Journal of Physical Mesomechanics, 15, 69-79.

http://dx.doi.org/10.1134/S1029959912010079

[13] Sargsyan, S.H. (2011) The General Dynamic Theory of Micropolar Elastic Thin Shells. Reports of Physics, 56, 39-42.

[14] Sargsyan, S.H. (2012) Mathematical Model of Micropolar Elastic Thin Plates and Their Strength and Stiffness Characteristics. Journal of Applied Mechanics and Technical Physics, 53, 275-282.

[15] Sargsyan, S.H. (2008) Boundary-Value Problems of Asymmetric Theory of Elasticity for Thin Plates. Journal of Applied Mathematics and Mechanics, 72, 77-86.

[16] Sargsyan, S.H. (2012) The Theory of Micropolar Thin Elastic Shells. Journal of Applied Mathematics and Mechanics, 76, 235-249.

[17] Sargsyan, S.H. (2012) The Construction of Mathematical Model of Micropolar Elastic Thin Beams on the Basis of the Asymptotic Theory. News of Higer Educational Intitutes. The North Caucasus Region. Natural Sciences, 5, 31-37. (In Russian)

[18] Sargsyan, S.H. (2013) The Asymptotic Method of the Construction of Mathematical Models of Micropolar Elastic Thin Plates. Scientific Proceedings of GSPI, 1, 7-37. (In Russian)

[19] Goldenveizer, A.L., Kaplunov, J.D. and Nolde, E.V. (1993) On Timoshenko-Reissner Type Theories of Plates and Shells. International Journal of Solids and Structures, 30, 675-694.

[20] Palmov, V.A. (1964) Basic Equations of the Theory of Asymmetric Elasticity. Applied Mathematics and Mechanics, 28, 1117-1120. (In Russian)

[21] Nowacki, W. (1986) Theory of Asymmetric Elasticity. Pergamon Press, Oxford, New York, Toronto, Sydney, Paris, Frankfurt.

[22] Pelech, P.L. (1977) Stress Concentration around the Holes in Bending Transversely Isotropic Plates. Kiev. (In Russian)

[23] Grigolyuk, E.I. and Kulikov, G.M. (1988) Multilayered Reinforced Shells. Calculation of Pneumatic Tires, Moscow. (In Russian)

[24] Grigorenko, Y.M. and Vasilenko, A.T. (1981) Theory of Shells of Variable Stiffness. Kiev. (In Russian)

[25] Percev, A.K. and Platonov, E.G. (1987) Dynamics of Plates and Shells. Leningrad. (In Russian)

[26] Sargsyan, A.A. (2011) Asymptotic Analysis of Dynamic Initial Boundary-Value Problem of Asymmetric Theory of Elasticity with Free Rotations in Thin Domain of the Shell. News of NAS Armenia, Mechanics, 64, 39-50. (In Russian)

[27] Sargsyan, S.H. (2012) Effective Manifestations of Characteristics of Strength and Rigidy of Micropolar Elastic Thin Bars. Journal of Materials Science and Engineering, 2, 98-108.