A theory for three-dimensional response of micropolar plates

doi:10.1007/s10483-024-3128-7

Abstract

Abstract:

Through combined applications of the transfer-matrix method and asymptotic expansion technique, we formulate a theory to predict the three-dimensional response of micropolar plates. No ad hoc assumptions regarding through-thickness assumptions of the field variables are made, and the governing equations are two-dimensional, with the displacements and microrotations of the mid-plane as the unknowns. Once the deformation of the mid-plane is solved, a three-dimensional micropolar elastic field within the plate is generated, which is exact up to the second order except in the boundary region close to the plate edge. As an illustrative example, the bending of a clamped infinitely long plate caused by a uniformly distributed transverse force is analyzed and discussed in detail.

Key words: micropolar plate, transfer-matrix, asymptotic expansion, three-dimensional response

2010 MSC Number:

74H45

Dianwu HUANG, Linghui HE. A theory for three-dimensional response of micropolar plates. Applied Mathematics and Mechanics (English Edition), 2024, 45(8): 1403-1414.

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Figures/Tables 3

References 34

1	COSSERAT, E. and COSSERAT, F. Théorie des Corps Déformables, Herman et Fils, Paris (1909)
2	NOWACKI,W.Theory of Asymmetric Elasticity,Pergamon Press,Oxfor(1986)
3	ERINGEN,A. C.Microcontinuum Field Theory, I: Foundations and Solids,Springer,New York(1999)
4	JASIUK,I., andOSTOIA-STARZEWSKI,M.From Lattices and Composites to Micropolar Continua,Springer,Dordrecht(2004)
5	RUEGER,Z., andLAKES,R. S.Strong Cosserat elasticity in a transversely isotropic polymer lattice.Physical Review Letters,120,065501(2018)
6	YAO,Y.,NI,Y., andHE,L. H.Unexpected bending behavior of architected 2D lattice materials.Science Advances,9,3499(2023)
7	SRINIVASA,A. R., andREDDY,J. N.An overview of theories of continuum mechanics with nonlocal elastic response and a general framework for conservative and dissipative systems.Applied Mechanics Reviews,69(3),030802(2017)
8	ALTENBACH,H., andEREMEYEV,V.On the linear theory of micropolar plates.ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik,89(4),242-256(2009)
9	ERICKSEN,J. L., andTRUESDELL,C.Exact theory of stress and strain in rods and shells.Archive for Rational Mechanics and Analysis,1(4),295-323(1958)
10	GREEN,A. E.,NAGHDI,P. M., andWAINWRIGHT,W. L.A general theory of a Cosserat surface.Archive for Rational Mechanics and Analysis,20(4),287-308(1965)
11	GREEN,A. E., andNAGHDI,P. M.Linear theory of an elastic Cosserat plate.Proceedings of the Cambridge Philosophical Society-Mathematical and Physical Sciences,63(2),537-550(1967)
12	GREEN,A. E.,NAGHDI,P. M., andWENNER,M. L.Linear theory of Cosserat surface and elastic plates of variable thickness.Mathematical Proceedings of the Cambridge Philosophical Society,69(1),227-254(1971)
13	COHEN,H., andDESILVA,C. N.Nonlinear theory of elastic directed surfaces.Journal of Mathematical Physics,7(6),960-966(1966)
14	DESILVA,C. N., andTSAI,P. J.A general theory of directed surfaces.Acta Mechanica,18(1-2),89-101(1973)
15	REISSNER, E. Linear and nonlinear theory of shells. Thin-Shell Structures: Theory, Experiment, and Design, Prentice-Hall, New Jersey, 29–44 (1974)
16	NAGHDI,P. M., andRUBIN,M. B.Restrictions on nonlinear constitutive-equations for elastic shells.Journal of Elasticity,39(2),133-163(1995)
17	CHRÓŚCIELEWSKI,J., andWITKOWSKI,W.On some constitutive equations for micropolar plates.ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik,90(1),53-64(2010)
18	ERINGEN,A. C.Theory of micropolar plates.Zeitschrift für Angewandte Mathematik und Physik,18(1),12-30(1967)
19	GEVORKYAN,G. A.The basic equations of flexible plates for a medium of Cosserat.International Applied Mechanics,3(11),41-45(1967)
20	REISSNER,E.A note on pure bending and flexure in plane stress including the effect of moment stresses.Ingenieur-Archiv,39(6),369-374(1970)
21	STEINBERG,L., andKVASOV,R.Enhanced mathematical model for Cosserat plate bending.Thin-Walled Structures,63,51-62(2013)
22	ZOZULYA,V. V.Higher order theory of micropolar plates and shells.ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik,98(6),886-918(2018)
23	SARGSYAN,A., andSARGSYAN,S.Geometrically nonlinear models of static deformation of micropolar elastic thin plates and shallow shells.ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik,101(5),e202000148(2021)
24	CARRERA,E., andZOZULYA,V. V.Carrera unified formulation for the micropolar plates.Mechanics of Advanced Materials and Structures,29(22),3163-3186(2022)
25	FARES,M. E.,SALEM,M. G.,ATTA,D., andELMARGHANY,M. K.Mixed variational principle for micropolar elasticity and an accuratetwo-dimensional plate model.European Journal of Mechanics/A Solids,99,104870(2023)
26	NAMPALLY,P.,KARTTUNEN,A. T., andREDDY,J. N.Nonlinear finite element analysis of lattice core sandwich plates.International Journal of Non-Linear Mechanics,121,103423(2020)
27	GHARAHI,A.Boundary value problems in a theory of bending of thin micropolar plates with surface elasticity.Journal of Elasticity,(2024) doi: 10.1007/s10659-024-10051-2
28	ERBAY,H. A.An asymptotic theory of thin micropolar plates.International Journal of Engineering Science,38(13),1497-1516(2000)
29	SARGSYAN,S. O.Boundary-value problems of the asymmetric theory of elasticity for thin plates.Journal of Applied Mathematics and Mechanics,72(1),77-86(2008)
30	AGANOVIĆ,I.,TAMBAČA,J., andTUTEK,Z.Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity.Journal of Elasticity,84,131-152(2006)
31	ROGERS,T. G.,WASTON,P., andSPENCER,J. M.An exact three-dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness.Proceedings of the Royal Society of London: Series A,437,199-213(1992)
32	HE,L. H.,LIM,C. W., andSOH,A. K.Three-dimensional analysis of an antiparallel piezoelectric bimorph.Acta Mechanica,145(1-4),189-204(2000)
33	GAUTHIER,R. D., andJAHSMAN,W. E.A quest for micropolar elastic constants.Journal of Applied Mechanics-Transactions of the ASME,42(2),369-374(1975)
34	LAKES,R. S.Experimental micro mechanics methods for conventional and negative Poisson's ratio cellular solids as Cosserat continua.Journal of Engineering Materials and Technology-Transactions of the ASME,113(1),148-155(1991)

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