# Mathematical Modeling of Elastic Thin Bodies with one Small Size

Chapter
Part of the Advanced Structured Materials book series (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.

## Keywords

Micropolar theory Thin body Constitutive relations Boundary value problems

## References

1. 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):612–622Google Scholar
2. Alekseev AE (1995) Bending of a three-layer orthotropic beam. Journal of Applied Mechanics and Technical Physics 36(3):458–465Google Scholar
3. 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 116:170–174Google Scholar
4. 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):432–437Google Scholar
5. Alekseev AE, Demeshkin AG (2003) Detachment of a beam glued to a rigid plate. Journal of Applied Mechanics and Technical Physics 44(4):577–583Google Scholar
6. Alekseev AE, Alekhin VV, Annin BD (2001) Plane elastic problem for an inhomogeneous layered body. Journal of Applied Mechanics and Technical Physics 42(6):1038–1042Google Scholar
7. Altenbach H (1991) Modelling of viscoelastic behaviour of plates. In: Zyczkowski M (ed) Creep in Structures, Springer, Berlin Heidelberg, pp 531–537Google Scholar
8. 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):73–92Google Scholar
9. Ambartsumyan SA (1958) On the theory of bending of anisotropic plates and shallow shells. Izv AN SSSR OTN (5):69–77Google Scholar
10. Ambartsumyan SA (1970) A new refined theory of anisotropic shells. Polymer Mechanics 6(5):766–776Google Scholar
11. Ambartsumyan SA (1974) General Theory of Anisotropic Shells (in. Russ.). Nauka, MoscowGoogle Scholar
12. Ambartsumyan SA (1987) Theory of Anisotropic Plates (in Russ.). Nauka, MoscowGoogle Scholar
13. Chepiga VE (1976) To the improved theory of laminated shells (in Russ.). Appl Mech 12(11):45–49Google Scholar
14. Chepiga VE (1977) Construction of the theory of multilayer anisotropic shells with given conditional accuracy of order hN (in Russ.). Mekh Tverdogo Tela (4):111–120Google Scholar
15. 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 118–125Google Scholar
16. Chepiga VE (1986b) Numerical analysis of equations of the improved theory of laminated shells (in Russ.). 290-B1986, VINITIGoogle Scholar
17. Chepiga VE (1986c) The study of stability of multilayer shells by an improved theory (in Russ.). 289-B1986, VINITIGoogle Scholar
18. Chernykh KF (1986) Nonlinear Theory of Elasticity in Engineering Computations (in Russ.). Mashinostroenie, LeningradGoogle Scholar
19. Chernykh KF (1988) Introduction into Anisotropic Elasticity (in Russ.). Nauka, MoscowGoogle Scholar
20. 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
21. Dergileva LA (1976) Solution method for a plane contact problem for an elastic layer (in Russ.). Continuum Dynamics 25:24–32Google Scholar
22. 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):36–59Google Scholar
23. Eremeyey VA, Zubov LM (2008) Mechanics of Elastic ShellsGoogle Scholar
24. Eremeyey VA, Lebedev LP, Altenbach H (2013) Foundations of Micropolar Mechanics. Springer-VerlagGoogle Scholar
25. Fellers J, Soler A (1970) Approximate solution of the finite cylinder problem using Legendre polynomials. AIAA 8(11)Google Scholar
26. Filin AP (1987) Elements of the Theory of Shells (in Russ.). Stroyizdat, LeningradGoogle Scholar
27. Gol’denveizer AL (1976) Theory of Elastic Shells (in Russ.). Nauka, MoscowGoogle Scholar
28. 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):1000–1025Google Scholar
29. 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):903–924Google Scholar
30. Grigolyuk EI, Selezov IT (1973) Nonclassic oscillation theories of rods, plates, and shells (in Russ.), vol 5. VINITI. Itogi nauki i tekniki, MoscowGoogle Scholar
31. Hencky H (1947) Über die berücksichtigung der schubverzerrung in ebenen platten. Ingenieur-Archiv 16:72–76Google Scholar
32. 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):333–341Google Scholar
33. 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):126–137Google Scholar
34. Ivanov GV (1977) Solutions of plane mixed problems for the Poisson equation in the form of series over Legendre polynomials (in Russ.). Continuum Dynamics 28:43–54Google Scholar
35. 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 39Google Scholar
36. Ivanov GV (1980) Theory of Plates and Shells (in Russ.). Novosib. State Univ., NovosibirskGoogle Scholar
37. Javili A, dell’Isola F, Stemmann P (2011) Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J Phys: Conf SerGoogle Scholar
38. 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):317–328Google Scholar
39. Khoroshun LP (1978) On the construction of equations of layered plates and shells (in russ.). Prikladnaya Mekhanika (10):3–21Google Scholar
40. 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):110–118Google Scholar
41. Kienzler R (1982) Eine Erweiterung der klassischen Schalentheorie; der Einfluß von Dickenverzerrungen und Querschnittsverwölbungen. Ingenieur-Archiv 52(5):311–322Google Scholar
42. Kirchhoff G (1850) Über das gleichgewicht und die bewegung einer elastischen scheibe. Journal für die reine und angewandte Mathematik (Crelles Journal) (40):51–88Google Scholar
43. 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 HollandGoogle Scholar
44. 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):1108–1117Google Scholar
45. Levinson M (1980) An accurate, simple theory of the statics and dynamics of elastic plates. Mech Res Commun 7(6):343–350Google Scholar
46. Lewiński T (1987) On refined plate models based on kinematical assumptions. Ingenieur-Archiv 57(2):133–146Google Scholar
47. 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):663–668Google Scholar
48. 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):669–676Google Scholar
49. Lurie AI (1990) Non-linear Theory of Elasticity, North-Holland Series in Applied Mathematics and Mechanics, vol 36. North HollandGoogle Scholar
50. 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):489–495Google Scholar
51. Meunargiya TV (1987) Development of the method of I. N. Vekua for problems of the three-dimensional moment elasticity (in Russ.). Tbilisi State Univ., TbilisiGoogle Scholar
52. Mindlin RD, Medick MA (1959) Extensional vibrations of elastic plates. Trans ASME J Appl Mech 26(4):561–569Google Scholar
53. Naghdi PM (1972) The theory of shells and plates. In: Flügge S (ed) Handbuch der Physik, vol VIa/2, Springer, Berlin, Heidelberg, pp 425–640Google Scholar
54. Nikabadze MU (1988a) On the theory of shells with two base surfaces (in Russ.). 8149-B88, VINITIGoogle Scholar
55. Nikabadze MU (1988b) Parameterization of shells with two base surfaces (in Russ.). 5588-B88, VINITIGoogle Scholar
56. Nikabadze MU (1989) Deformation of layered viscoelastic shells. In: Actual problems of strength in mechanical engineering (in Russ.), SVVMIU, Sevastopol, p 1Google Scholar
57. Nikabadze MU (1990a) Modeling of nonlinear deformation of elastic shells (in Russ.). PhD thesis, Lomonosov Moscow State UniversityGoogle Scholar
58. Nikabadze MU (1990b) Plane curvilinear rods (in Russ.). 4509-B90, VINITIGoogle Scholar
59. Nikabadze MU (1990c) To the theory of shells with two base surfaces (in Russ.). 1859-B90, VINITIGoogle Scholar
60. Nikabadze MU (1990d) To the theory of shells with two base surfaces (in Russ.). 2676-B90, VINITIGoogle Scholar
61. 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):54–61Google Scholar
62. Nikabadze MU (1998a) Constitutive relations of the new linear theory of thermoelastic shells (in Russ.). In: Actual problems of shell mechanics, UNIPRESS, Kazan, pp 158–162Google Scholar
63. 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):59–65Google Scholar
64. 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):52–56Google Scholar
65. Nikabadze MU (1999b) New rod space parametrization (in Russ.). 1663-B99, VINITIGoogle Scholar
66. 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, NovosibirskGoogle Scholar
67. 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):49–51Google Scholar
68. Nikabadze MU (2000a) Some geometric relations of the theory of shells with two basic surfaces (in Russ.). Izv RAN MTT (4):129–139Google Scholar
69. Nikabadze MU (2000b) To the parametrization of the multilayer shell domain of 3d space (in Russ.). Mathematical modeling of systems and processes (8):63–68Google Scholar
70. 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 389–395Google Scholar
71. 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):35–39Google Scholar
72. Nikabadze MU (2001c) Location gradients in the theory of shells with two basic surfaces (in Russ.). Mech Solids 36(4):64–69Google Scholar
73. Nikabadze MU (2001d) To the variant of the theory of multilayer structures (in Russ.). Izv RAN MTT (1):143–158Google Scholar
74. 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):41–46Google Scholar
75. Nikabadze MU (2002b) Modern State of Multilayer Shell Structures (in Russ.). 2289–B2002, VINITIGoogle Scholar
76. Nikabadze MU (2003) Variant of the theory of shallow shells (in Russ.). In: Lomonosovskiye chteniya. Section mechanics., Izd. Moscov. Univ., MoscowGoogle Scholar
77. Nikabadze MU (2004a) Generalization of the Huygens-Steiner theorem and the Boer formulas and some of their applications (in Russ.). Izv RAN MTT (3):64–73Google Scholar
78. 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., MoscowGoogle Scholar
79. Nikabadze MU (2005) To the variant of the theory of multilayer curvilinear rods (in Russ.). Izv RAN MTT (6):145–156Google Scholar
80. Nikabadze MU (2006) Application of Classic Orthogonal Polynomials to the Construction of the Theory of Thin Bodies (in Russ.). Elasticity and non-elasticity pp 218–228Google Scholar
81. Nikabadze MU (2007a) Application of Chebyshev Polynomials to the Theory of Thin Bodies. Moscow University Mechanics Bulletin 62(5):141–148Google Scholar
82. 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):391–421Google Scholar
83. 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 225–242Google Scholar
84. 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, VINITIGoogle Scholar
85. 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, VINITIGoogle Scholar
86. 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, MoscowGoogle Scholar
87. Nikabadze MU (2014b) Method of orthogonal polynomials in mechanics of micropolar and classical elasticity thin bodies (in Russ.). Doctoral dissertation. Moscow, MAIGoogle Scholar
88. 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 279–317Google Scholar
89. Nikabadze MU (2017a) Eigenvalue problem for tensors of even rank and its applications in mechanics Journal of Mathematical Sciences 221(2):174–204Google Scholar
90. Nikabadze MU (2017b) Topics on tensor calculus with applications to mechanics. Journal of Mathematical Sciences 225(1):1–194Google Scholar
91. Nikabadze MU, Ulukhanyan A (2005a) Formulation of the problem for thin deformable 3d body (in Russ.). Vestn Mosk Univ, Matem Mekhan (5):43–49Google Scholar
92. Nikabadze MU, Ulukhanyan A (2005b) Formulations of problems for a shell domain according to three-dimensional theories (in Russ.). 83–B2005, VINITIGoogle Scholar
93. 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, VINITIGoogle Scholar
94. 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 319–361Google Scholar
95. Nowacki W (1975) Theory of Elasticity. Mir, Moscow, (Russian translation)Google Scholar
96. Pelekh BL (1973) Theory of shells with finite shear stiffness (in Russ.). Naukova Dumka, KievGoogle Scholar
97. Pelekh BL (1978) The Generalized Theory of Shells (in Russ.). Vischa shkola, LvovGoogle Scholar
98. 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 27–39Google Scholar
99. Pelekh BL, Sukhorolskii MA (1980) Contact problems of the theory of elastic anisotropic shells (in Russ.). Naukova Dumka, KievGoogle Scholar
100. Pelekh BL, Maksimuk AV, Korovaichuk IM (1988) Contact problems for laminated elements of constructions and bodies with coating (in Russ.). Naukova Dumka, KievGoogle Scholar
101. Pikul VV (1992) To the problem of constructing a physically correct theory of shells (in Russ.). Izv RAN MTT (3):18–25Google Scholar
102. Pobedrya BE (1986) Lectures on tensor analysis (in Russ.). M: Izd. Moscov. Univ.Google Scholar
103. Pobedrya BE (1995) Numerical methods in the theory of elasticity and plasticity (in Russ.). Izd. Moscov. Univ., MoscowGoogle Scholar
104. Pobedrya BE (2003) On the theory of constitutive relations in the mechanics of a deformable solid (in Russ.). In: Problemy mekhaniki, Fiszmatlit, Moscow, pp 635–657Google Scholar
105. Pobedrya BE (2006) Theory of thermomechanical processes (in Russ.). In: Elasticity and nonelasticity, Izd. MGU, pp 70–85Google Scholar
106. Preußer G (1984) Eine systematische Herleitung verbesserter Plattengleichungen. Ingenieur-Archiv 54(1):51–61Google Scholar
107. Reissner E (1985) Reflections on the theory of elastic plates. Applied Mechanics Reviews 38(11):1453–1464Google Scholar
108. Reissner E (1944) On the theory of bending of elastic plates. Journal of Mathematics and Physics 23(1-4):184–191Google Scholar
109. Sansone G (1959) Orthogonal Functions. Interscience Publishers Inc, New YorkGoogle Scholar
110. 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):2381–2401Google Scholar
111. Sokol’nikov IS (1971) Tensor analysis (in Russ.). Nauka, MoscowGoogle Scholar
112. Soler AI (1969) Higher-order theories for structural analysis using Legendre polynomial expansions. Trans ASME Journal of Applied Mechanics 36(4):757–762Google Scholar
113. Suyetin PK (1976) Classical orthogonal polynomials (in Russ.). Nauka, MoscowGoogle Scholar
114. Tvalchrelidze AK (1984) Theory of elastic shells using several base surfaces (in Russ.). In: Theory and numerical methods for calculating plates and shells, TbilisiGoogle Scholar
115. Tvalchrelidze AK (1986) Basic equations of the theory of shells, taking into account large deformations and shears (in Russ.). Soobshch AN GruzSSR 121(1):53–56Google Scholar
116. Tvalchrelidze AK (1994) Shell theory using several base surfaces and some applications (in Russ.). PhD thesis, KutaisiGoogle Scholar
117. 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., TbilisiGoogle Scholar
118. 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):467–479Google Scholar
119. 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 547–550Google Scholar
120. Vasiliev VV, Lurie SA (1990a) On the problem of constructing non-classical theories of plates (in Russ.). Izv RAN MTT (2):158–167Google Scholar
121. Vasiliev VV, Lurie SA (1990b) To the problem of clarifying the theory of shallow shells (in Russ.). Izv RAN MTT (6):139–146Google Scholar
122. 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 191–259Google Scholar
123. Vekua IN (1964) The theory of thin and shallow shells of variable thickness (in Russ.). NovosibirskGoogle Scholar
124. 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 1–104Google Scholar
125. Vekua IN (1970) Variational principles for constructing the theory of shells (in Russ.). Izd-vo Tbil. Un-ta, TbilisiGoogle Scholar
126. 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 267–290Google Scholar
127. 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, MoscowGoogle Scholar
128. Volchkov YM (2000) Finite elements with adjustment conditions on their edges (in Russ.). Dinamika sploshnoy sredi 116:175–180Google Scholar
129. 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 28:43–54Google Scholar
130. 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):354–359Google Scholar
131. Volchkov YM, Dergileva LA (2004) Equations of an elastic anisotropic layer. Journal of Applied Mechanics and Technical Physics 45(2):301–309Google Scholar
132. 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):450–459Google Scholar
133. 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):936–941Google Scholar
134. Wunderlich W (1973) Vergleich verschiedener Approximationen der Theorie dünner Schalen (mit numerischen Ergebnissen). Allgemeine Schalentheorien, Techn Wiss Mitteilungen (73):3.1–3.24Google Scholar
135. 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):164–184Google Scholar
136. 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):82–95Google Scholar
137. 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):36–48Google Scholar
138. Zhilin PA (1976) Mechanics of deformable directed surfaces. Int J Solids Structures 12:635–648Google Scholar
139. 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):119–133Google Scholar
140. Zozulya VV (2017b) Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures 4(1):104–118Google Scholar
141. Zozulya VV (2017c) Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models. Curved and Layered Structures 4(1):221–236Google Scholar
142. 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):1037–1055Google Scholar
143. 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):1255–1272Google Scholar