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Continuum Mechanics and Thermodynamics

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17-03-2022 | Original Article

On some variational principles in micropolar theories of single-layer thin bodies

Authors: M. Nikabadze, A. Ulukhanyan

Published in: Continuum Mechanics and Thermodynamics | Issue 3/2023

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Abstract

The generalized Reissner-type operator of three-dimensional micropolar mechanics of solids is presented, on the basis of which the generalized Reissner-type operator of three-dimensional micropolar mechanics of thin solids with one small size is obtained under the new parameterization of the domains of these bodies. From the last Reissner-type operator, in turn, the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size is derived under the new parametrization of the domains of these bodies. It should be noted that the advantage of the new parameterization is that it is experimentally more accessible than other parameterizations (Nikabadze in Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies, MSU Publishing House, 2014; Contemp Math. Fundam Dir 55:3–194, 2015; J Math Sci 225:1, 2017). Further, applying the method of orthogonal polynomials (expansion of unknown quantities in series in terms of a system of orthogonal polynomials), from the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size under the new parameterization of the domains of these bodies, the Reissner variational principle of micropolar mechanics of thin solids with one small size in the moments with respect to the system of Legendre polynomials is derived. In addition, the method is described for obtaining the variational principles of Lagrange and Castigliano of micropolar mechanics of thin solid with one small size under the new parametrization of the domains of these bodies in moments with respect to systems of the first and second kind Chebyshev polynomials. The paper is a continuation of the work “Nikabadze, Ulukhanyan, On some variational principles in the three-dimensional micropolar theories of solid”; therefore, before reading this paper, the authors invite the interested reader to familiarize themselves with the work (Nikabadze and Ulukhanyan in On some variational principles in the three-dimensional micropolar theories of solids, submitted).

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A three-dimensional body, one size of which is smaller than the others, is called a thin body with one small size, and a solid body, two sizes of which are small compared to the third dimension, is called a thin body with two small dimensions.

The dependence of the quantities on \(x'\) means their dependence on the curvilinear coordinates \(x^1\) and \(x^2\) of the base surface. The usual rules of tensor calculus used in [2, 3, 21, 24‐27] are applied. The notations and agreements adopted in the work [4] and also in previously published works are preserved [1‐4, 29‐47].

Nikabadze, M.U.: Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies. MSU Publishing House of the Board of Trustees Mech.-Math. Facul. of MSU (2014). (in Russian). https://istina.msu.ru/publications/book/6738800/

Nikabadze, M.U.: On several issues of tensor calculus with applications to mechanics. Contemp. Math. Fundam. Direct. 55, 3–194 (2015). (in Russian). http://istina.msu.ru/media/publications/book/e25/00c/10117043/M.U.Nikabadze.pdf

Nikabadze, M.U.: Topics on tensor calculus with applications to mechanics. J. Math. Sci. 225, 1 (2017). https://doi.org/10.1007/s10958-017-3467-4/MathSciNetCrossRefMATH

Nikabadze, M., Ulukhanyan, A.: Some variational principles in thethree-dimensional micropolar theories of solids and thin solids. Advanced Structured Materials, pp. 193–249 (2022)

Bersani, A.M., Giorgio, I., Tomassetti, G.: Buckling of an elastic hemispherical shell with an obstacle. Cont. Mech. Thermodyn. 25(2), 443–67 (2013)MathSciNetCrossRefMATH

Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z. Angew. Math. Phys. 67(4), 1–28 (2016)

Turco, E., Misra, A., Pawlikowski, M., dell’Isola, F., Hild, F.: Enhanced Piola–Hencky discrete models for pantographic sheets with pivots without deformation energy: numerics and experiments. Int. J. Solids Struct. 15(147), 94–109 (2018)

Cazzani, A., Rizzi, N.L., Stochino, F., Turco, E.: Modal analysis of laminates by a mixed assumed-strain finite element model. Math. Mech. Solids 23(1), 99–119 (2018)MathSciNetCrossRefMATH

Cazzani, A., Serra, M., Stochino, F., Turco, E.: A refined assumed strain finite element model for statics and dynamics of laminated plates. Cont. Mech. Thermodyn. 32(3), 665–92 (2020)MathSciNetCrossRef

10.

Yildizdag, M.E., Demirtas, M., Ergin, A.: Multipatch discontinuous Galerkin isogeometric analysis of composite laminates. Cont. Mech. Thermodyn. 32(3), 607–20 (2020)MathSciNetCrossRef

11.

Greco, L., Cuomo, M., Contrafatto, L.: A reconstructed local B formulation for isogeometric Kirchhoff–Love shells. Comput. Methods Appl. Mech. Eng. 15(332), 462–87 (2018)MathSciNetCrossRefADSMATH

12.

Greco, L., Cuomo, M., Contrafatto, L.: Two new triangular G1-conforming finite elements with cubic edge rotation for the analysis of Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 1(356), 354–86 (2019)CrossRefADSMATH

13.

Greco, L., Cuomo, M., Contrafatto, L.: A quadrilateral G1-conforming finite element for the Kirchhoff plate model. Comput. Methods Appl. Mech. Eng. 1(346), 913–51 (2019)CrossRefADSMATH

14.

Greco, L., Cuomo, M.: An implicit G1-conforming bi-cubic interpolation for the analysis of smooth and folded Kirchhoff–Love shell assemblies. Comput. Methods Appl. Mech. Eng. 1(373), 113476 (2021)CrossRefADSMATH

15.

Wang, F.F., Dai, H.H., Giorgio, I.: A numerical comparison of the uniformly valid asymptotic plate equations with a 3D model: clamped rectangular incompressible elastic plates. Math. Mech. Solids (2021). https://doi.org/10.1177/10812865211025583

16.

Giorgio, I.: A discrete formulation of Kirchhoff rods in large-motion dynamics. Math. Mech. Solids 25(5), 1081–100 (2020)MathSciNetCrossRefMATH

17.

Turco, E., Barchiesi, E., Giorgio, I., dell’Isola, F.: A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory. Int. J. Non-Linear Mech. 1(123), 103481 (2020)

18.

Rektorys, K.: Variational methods in mathematics science and engineering. Springer, Dordrecht. (1977). https://doi.org/10.1007/978-94-011-6450-4

19.

Washizy, K.: Variational Methods in Elasticity and Plasticity, 3rd edn. Pergamon, Oxford (1982)

20.

Vanko, V.I.: Variational principles and problems of mathematical physics. MSU Publishing House of BMSTU (2010). (in Russian)

21.

Vekua, I.N.: Variational Principles of Constructing the Theory of Shells. Tbilisi University Publishing House, Tbilisi (1970). (in Russian)

22.

Pobedrya, B.E.: Mechanics of Composite Materials. MSU Publishing House, Moscow (1984). (in Russian)

23.

Pobedrya, B.E.: Numerical Methods in the Theory of Elasticity and Plasticity, 2nd edn. MSU Publishing House, Moscow (1995). (in Russian)

24.

Vekua, I.N.: Fundamentals of Tensor Analysis and Covariant Theory. Nauka, Moscow (1978). (in Russian)

25.

Vekua, I.N.: Some general methods for constructing various variants of the theory of shells. MSU Publishing House, Nauka (1982). (in Russian)

26.

Lurie, A.I.: Nonlinear Theory of Elasticity. Nauka, Moscow (1980). (in Russian)

27.

Pobedrya, B.E.: Lectures on Tensor Analysis. MSU Publishing House (1986). (in Russian)

28.

Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. Nauka, Moscow (1976). (in Russian)

29.

Nikabadze, M.U.: A variant of the theory of multilayer structures. Mech. Solids 1, 143–158 (2001)

30.

Nikabadze, M.U.: To a version of the theory of multilayer structures. Mech. Solids 36(1), 119–129 (2001)

31.

Nikabadze, M.U., Ulukhanyan, A.R.: Statements of problems for a thin deformable three-dimensional body. Mosc. Univer. Bull., Math. Mech. 5, 43–49 (2005) (in Russian)

32.

Nikabadze, M.U.: A variant of the system of equations of the theory of thin bodies. Mosc. Univer. Bull., Math. Mech. 1, 30–35 (2006). (in Russian)

33.

Nikabadze, M.U.: Application of a system of Chebyshev polynomials to the theory of thin bodies. Mosc. Univer. Bull., Math. Mech. 5, 56–63 (2007). (in Russian)

34.

Nikabadze, M.U.: Application of Chebyshev polynomials to the theory of thin bodies. Mosc. Univer. Mech. Bull. 62(5), 141–148 (2007). https://doi.org/10.3103/S0027133007050056MathSciNetCrossRefMATH

35.

Nikabadze, M.U.: 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 (2007)CrossRefADS

36.

Nikabadze, M.U.: Mathematical modeling of multilayer thin body deformation. J. Math. Sci. 187(3), 300–336 (2012)MathSciNetCrossRefMATH

37.

Nikabadze, M.U., Ulukhanyan, A.R.: 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 (2016). https://doi.org/10.1007/978-3-319-31721-2_15

38.

Nikabadze, M.U., Ulukhanyan, A.R.: Some applications of eigenvalue problems for tensor and tensor-block matrices for mathematical modeling of micropolar thin bodies. Math. Comput. Appl. 24(1), 1–19 (2019). https://doi.org/10.3390/mca24010033MathSciNetCrossRef

39.

Nikabadze, M.U., Ulukhanyan, A.R.: To the modeling of multilayer thin prismatic bodies. IOP Confer. Ser.: Mater. Sci. Eng. 683, 012019 (2019). https://doi.org/10.1088/1757-899X/683/1/012019

40.

Nikabadze, M.U., Ulukhanyan, A.R.: 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, pp. 155–199 (2019). https://doi.org/10.1007/978-3-030-30406-5_9

41.

Nikabadze, M.U., Ulukhanyan, A.R.: Modeling of multilayer thin bodies. Cont. Mech. Thermodyn. 32, 817–842 (2020). https://doi.org/10.1007/s00161-019-00762-6MathSciNetCrossRefMATH

42.

Nikabadze, M.U., Ulukhanyan, A.R.: On the decomposition of equations of micropolar elasticity and thin body theory. Lobachevskii J. Math. 41(10), 2059–2074 (2020). https://doi.org/10.1134/S1995080220100145MathSciNetCrossRefMATH

43.

Nikabadze, M.U., Ulukhanyan, A.R.: On the theory of multilayer thin bodies. Lobachevskii J. Math. 42(8), 1900–1911 (2021). https://doi.org/10.1134/S1995080221080217MathSciNetCrossRefMATH

44.

Nikabadze, M.U.: On compatibility conditions in linear micropolar theory. Mosc. Univer. Bull., Math. Mech. 5, 48–51 (2010). (in Russian)

45.

Nikabadze, M.U.: On compatibility conditions and equations of motion in the micropolar linear theory of elasticity. Mosc. Univ. Bull., Math. Mech. 1, 63–66 (2012)MathSciNetMATH

46.

Nikabadze, M.U.: Compatibility conditions and equations of motion in the linear micropolar theory of elasticity. Mosc. Univ. Mech. Bull. Allerton Press, Inc. 67(1), 18–22 (2012)

47.

Nikabadze, M.U.: 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 (2016). https://doi.org/10.1007/978-3-319-31721-2_14

48.

Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. ZAMM-Z. Angew. Math. Mech. 89(4), 242–56 (2009)MathSciNetCrossRefMATH

49.

Altenbach, H., Eremeyev, V.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Compl. Syst. 3(3), 273–83 (2015)MathSciNetCrossRefMATH

50.

Chróścielewski, J., dell’Isola, F., Eremeyev, V.A., Sabik, A.: On rotational instability within the nonlinear six-parameter shell theory. Int. J. Solids Struct. 1(196), 179–89 (2020)

51.

Giorgio, I., De Angelo, M., Turco, E., Misra, A.: A Biot–Cosserat two-dimensional elastic nonlinear model for a micromorphic medium. Cont. Mech. Thermodyn. 32(5), 1357–69 (2020)MathSciNetCrossRef

52.

Giorgio, I., Spagnuolo, M., Andreaus, U., Scerrato, D., Bersani, A.M.: In-depth gaze at the astonishing mechanical behavior of bone: a review for designing bio-inspired hierarchical metamaterials. Math. Mech. Solids 26(7), 1074–103 (2021)MathSciNetCrossRefMATH

53.

Cosserat, E., Cosserat, F.: Theorie des Corp Dcformablcs. Librairie Scientifique A, Hermann et Fils, Paris (1909)MATH

54.

Le Roux, J.: Etude géométrique de la torsion et de la flexion, dans les déformations infinitésimales d’un milieu continu. Ann. Scient. Ecole Norm. Sup. Sér. 3(28), 523–579 (1911)

55.

Le Roux, J.: Recherches sur géométrie des déformations finies. Ann. Sci. Ecole Norm. Sup. Sér. 3(30), 193–245 (1913)

56.

Jaramillo, T.J.: A generalization of the energy function of elasticity theory. Dissertation, Department of Mathematics, University of Chicago, vol. 98 (1929)

57.

Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16(1), 51–78 (1964)MathSciNetCrossRefMATH

58.

Toupin, R.A.: Theories of elasticity with couple-stresses. Arch. Rat. Mech. Anal. 17(2), 85–112 (1964)CrossRefMATH

59.

Eringen, A.C.: Microcontinuum Field Theories. 1. Foundation and Solids. Springer, New York (1999)

60.

Harm, A., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48(13), 1962–1990 (2011). https://doi.org/10.1016/j.ijsolstr.2011.03.006CrossRef

61.

Aifantis, E.C.: Continuum nanomechanics for nanocrystalline and ultrafine grain materials. IOP Conf. Ser.: Mater. Sci. Eng. 63, 012129 (2014). http://iopscience.iop.org/1757-899X/63/1/012129

62.

dell’Isola, F., Sciara, G., Vidoli, S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. A: Math., Phys. Eng. Sci. 465(2107), 2177–2196 (2009). https://doi.org/10.1098/rspa.2008.0530

63.

Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003). https://doi.org/10.1177/1081286503008001658

64.

dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Abrio Piola. Math. Mech. Solids 20(8), 887–928 (2014). https://doi.org/10.1177/1081286513509811

65.

dell’Isola, F., Seppecher, P., Della Corte, A.: The postulations a la D’Alembert and a la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. A: Math., Phys. Eng. Sci.(2015). https://doi.org/10.1098/rspa.2015.0415

66.

Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids 24(1), 212–34 (2019)MathSciNetCrossRefMATH

67.

Abdoul-Anziz, H., Seppecher, P.: Strain gradient and generalized continua obtained by homogenizing frame lattices. Math. Mech. Compl. Syst. 6(3), 213–50 (2018)MathSciNetCrossRefMATH

68.

Eugster, S., dell’Isola, F., Steigmann, D.: Continuum theory for mechanical metamaterials with a cubic lattice substructure. Math. Mech. Compl. Syst. 7(1), 75–98 (2019)

69.

Novatskiy, V.: Theory of Elasticity. MIR, Moscow (1975). (in Russian)

70.

Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.H.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Cont. Mech. Thermodyn. 31(1), 33–45 (2019)MathSciNetCrossRef

71.

Barchiesi, E., dell’Isola, F., Hild, F.: On the validation of homogenized modeling for bi-pantographic metamaterials via digital image correlation. Int. J. Solids Struct. 1(208), 49–62 (2021)

72.

Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A: Math., Phys. Eng. Sci. 473(2207), 20170636 (2017)MathSciNetCrossRefADSMATH

73.

Giorgio, I.: Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial. Cont. Mech. Thermodyn. 33(4), 1063–82 (2021)MathSciNetCrossRef

74.

Giorgio, I., Varano, V., dell’Isola, F., Rizzi, N.L.: Two layers pantographs: a 2D continuum model accounting for the beams’ offset and relative rotations as averages in SO (3) Lie groups. Int. J. Solids Struct. 1(216), 43–58 (2021)

75.

Scerrato, D., Giorgio, I.: Equilibrium of two-dimensional cycloidal pantographic metamaterials in three-dimensional deformations. Symmetry 11(12), 1523 (2019)CrossRefADS

76.

Giorgio, I., Rizzi, N.L., Andreaus, U., Steigmann, D.J.: A two-dimensional continuum model of pantographic sheets moving in a 3D space and accounting for the offset and relative rotations of the fibers. Math. Mech. Compl. Syst. 7(4), 311–25 (2019)MathSciNetCrossRefMATH

77.

Turco, E., Golaszewski, M., Giorgio, I., D’Annibale, F.: Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos. B Eng. 1(118), 1–4 (2017)

78.

Auger, P., Lavigne, T., Smaniotto, B., Spagnuolo, M., dell’Isola, F., Hild, F.: Poynting effects in pantographic metamaterial captured via multiscale DVC. J. Strain Anal. Eng. Des. (2021). https://doi.org/10.1177/0309324720976625

79.

Yang, H., Ganzosch, G., Giorgio, I., Abali, B.E.: Material characterization and computations of a polymeric metamaterial with a pantographic substructure. Z. Angew. Math. Phys. 69(4), 1–6 (2018)MathSciNetCrossRefMATH

Title: On some variational principles in micropolar theories of single-layer thin bodies
Authors: M. Nikabadze
A. Ulukhanyan
Publication date: 17-03-2022
Publisher: Springer Berlin Heidelberg
Published in: Continuum Mechanics and Thermodynamics / Issue 3/2023
Print ISSN: 0935-1175
Electronic ISSN: 1432-0959
DOI: https://doi.org/10.1007/s00161-022-01089-5

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