Abstract
Some questions about the parametrization of three-dimensional thin body with one small size under an arbitrary base surface and the changing of transverse coordinate from –1 to 1 are considered. The vector parametric equation of the thin body domain is given. In particular, we have defined the various families of bases and geometric characteristics generated by them. Expressions for the components of the second rank isotropic tensor are obtained. The representations of some differential operators, the equations of motion, and the constitutive relations of micropolar elasticity theory under the considered parametrization of the thin body domain are given. The inverse tensor operators to a tensor operator of the equations of motion in terms of displacements for an isotropic homogeneous material and to a stress operator are found. They allow decomposing equations and boundary conditions. The inverse matrix differential tensor operator to the matrix differential tensor operator of the equations of motion in displacements and rotations of the micropolar theory of elasticity is constructed for isotropic homogeneous materials with a symmetry center as well as for materials without a symmetry center. We obtain the equations with respect to displacement vector and rotation vector individually. As a special case, a reduced continuum is considered. Cases in which it is easy to invert the stress and the couple stress operator are found out. From the decomposed equations of classical (micropolar) theory of elasticity, the corresponding decomposed equations of quasistatic problems of theory of prismatic bodies with constant thickness in displacements (in displacements and rotations) are obtained. From these systems of equations, we derive the equations in moments of unknown vector functions with respect to any system of orthogonal polynomials. We obtain the systems of equations of various approximations (from zero to eighth order) in moments with respect to the systems of Legendre and second kind Chebyshev polynomials. The system splits and for each moment of unknown vector function we, obtain a high order elliptic type equation (the system order depends on the order of approximation), the characteristic roots of which can be easily found. Using the method of Vekua, their analytical solution is obtained. For micropolar theory of thin prismatic bodies with two small sizes and a the rectangular cross-section, the decomposed equations in moments of displacement and rotation vectors via an arbitrary system of polynomials (Legendre, Chebyshev) are obtained. Similar equations are also deduced for the reduced medium containing classical equation. The decomposed systems of equations of eight approximations for micropolar theory of multilayer prismatic bodies of constant thickness in moments of displacement and rotation vectors are obtained. Using Vekua method, we can find the analytical solutions for this system and for equations for the reduced medium.
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Notes
- 1.
In the following brief notes as \(\overset{{\scriptscriptstyle {(\sim )}}}{M}\in \overset{{\scriptscriptstyle {(\sim )}}}{S}\), \({\scriptstyle {\sim }}\in \{-,\emptyset ,\wedge ,+\}\) or \({\mathbf {r}}_{\tilde{p}}=g_{\tilde{p}}^{\breve{q}}{\mathbf {r}}_{\breve{q}}\), \({\scriptstyle {\sim }},{\scriptstyle {\smallsmile }}\{-,\emptyset ,\wedge ,+\}\), where \(\emptyset \) is empty set, are applied. The first record means: if \({\scriptstyle {\sim }}=-\) then \(\overset{{\scriptscriptstyle {(-)}}}{M}\in \overset{{\scriptscriptstyle {(-)}}}{S}\); if \({\scriptstyle {\sim }}=\emptyset \) then \(M\in S\); if \({\scriptstyle {\sim }}=\wedge \) than \(\hat{M}\in \hat{S}\); if \({\scriptstyle {\sim }}=+\) then \(\overset{{\scriptscriptstyle {(+)}}}{M}\in \overset{{\scriptscriptstyle {(+)}}}{S}\). The second record means that if, for example, \({\scriptstyle {\sim }}=\emptyset \), \({\scriptstyle {\smallsmile }}=-\) then \({\mathbf {r}}_p=g_p^{\bar{q}}{\mathbf {r}}_{\bar{q}}\); if \({\scriptstyle {\sim }}=\wedge \), \({\scriptstyle {\smallsmile }}=\emptyset \) then \({\mathbf {r}}_{\hat{p}}=g_{\hat{p}}^q{\mathbf {r}}_q\) and soon. Going through all the values, we get all the relations.
- 2.
It means if one of the dummy indices is lowered, then the corresponding index rises and vice versa.
- 3.
Second-rank tensors we mark from the bottom with the wave (), third-rank tensors we mark from below with the wave and hyphen () and fourth-rank tensors we mark from below with two waves ().
- 4.
References
Aero EL, Kuvshinskii EV (1964) Continual theory of asymmetric elasticity. Equilibrium of an isotropic body. Solid State Phys 6(9):2689–2699
Eringen AC (1999) Microcontinuum field theories, vol 1. Foundation and solids, Springer, New York
Galerkin BG (1930) Contribution à la solution générale du problème de la théorie de l’élasticité dans le cas de trois dimensions. CR Acad Sci 190:1047–1048
Galerkin BG (1931) Contribution à la solution générale du problème de la théorie de l’élasticité dans le cas de trois dimensions. CR Acad Sci 193:568–571
Iacovache M (1949) O extindere a metodei lui galerkin pentru sistemul ecuatiilor elasticitěii. Bull Acad Sci RPR, Ser A 1:593
Kupradze VD, Gegelia TG, Basheleishvili MO, Burchuladze TV (1976) Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. Nauka, Moscow (in Russia)
Lurie AI (1990) Nonlinear theory of elasticity. North-Holland, Dordrecht
Nikabadze MU (2007a) Some issues concerning a version of the theory of thin solids based on expansions in a system of chebyshev polynomials of the second kind. Mech Solids 42(3):391–421
Nikabadze MU (2007b) Some problems of tensor calculus, vol II. Moscow State University, Moscow (in Russia)
Nikabadze MU (2008a) Method of orthogonal polynomials in mechanics of micropolar and classical elasticity thin bodies. i., available from VINITI, 135 – B2014 (20.05.2014)
Nikabadze MU (2008b) Method of orthogonal polynomials in mechanics of micropolar and classical elasticity thin bodies. ii., available from VINITI, 136 – B2014 (20.05.2014)
Nikabadze MU (2009a) On some problems of tensor calculus. i. J Math Sci 161(5):668–697
Nikabadze MU (2009b) On some problems of tensor calculus. ii. J Math Sci 161(5):698–733
Nikabadze MU (2014a) Construction of eigentensor columns in the linear micropolar theory of elasticity. Mosc Univ Mech Bull 69(1):1–9
Nikabadze MU (2014b) Development of the method of orthogonal polynomials in mechanics of micropolar and classical elasticity thin bodies. Lomonosov Moscow State University, Moscow (in Russia)
Nikabadze MU (2014c) Method of orthogonal polynomials in mechanics of micropolar and classical elasticity thin bodies. Dsc thesis, Moscow Aviation Institute (National Research University), Moscow
Nikabadze MU (2015) On some questions of tensor calculus with applications to mechanics. In: Tensor analysis, vol 55. PFUR, Moscow, pp 3–194 (in Russia)
Nikabadze MU, Ulukhanyan AR (2008) Mathematical modeling of elastic thin bodies with one small dimension with the use of systems of orthogonal polynomials, available from VINITI, 723 – B2008 (21.08.2008)
Nikabadze MU, Kantor MM, Ulukhanyan AR (2008) On mathematical modeling of elastic thin bodies and numerical realization of some tasks of strip, available from VINITI, 204 – B2011 (29.04.2011)
Nowacki W (1970) Teoria Spreżystości. Państwowe Wydawnictwo Naukowe, Warsaw
Pobedrya BE (1986) Lectures in tensor analysis. Moscow State University, Moscow (in Russia)
Pobedrya BE (1995) Numerical methods in the theory of elasticity and plasticity. Moscow State University, Moscow (in Russia)
Sandru N (1966) On some problems of the linear theory of the asymmetric elasticity. Int J Eng Sci 4(1):81–94
Ulukhanyan AR (2010) Representation of solutions to equations of hyperbolic type. Mosc Univ Mech Bull 65(2):47–50
Ulukhanyan AR (2011) Dynamic equations of the theory of thin prismatic bodies with expansion in the system of Legendre polynomials. Mech Solids 46(3):467–479
Vekua IN (1948) New methods for solving elliptic equations. OGIZ, Moscow (in Russia)
Vekua IN (1978) Fundamentals of tensor analysis and covariant theory. Nauka, Moscow (in Russia)
Vekua IN (1985) Shell theory: general methods of construction. Pitman, Boston
Acknowledgments
The research was supported by the Russian Foundation for Basic Research, projects nos. 15–01–00848-a and 14–01–00317-a.
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Nikabadze, M.U., Ulukhanyan, A.R. (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. Springer, Cham. https://doi.org/10.1007/978-3-319-31721-2_15
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