Vibration analysis of two-dimensional structures using micropolar elements

doi:10.1007/s10483-021-2746-8

Abstract

Abstract: Based on the micropolar theory (MPT), a two-dimensional (2D) element is proposed to describe the free vibration response of structures. In the context of the MPT, a 2D formulation is developed within the ABAQUS finite element software. The user-defined element (UEL) subroutine is used to implement a micropolar element. The micropolar effects on the vibration behavior of 2D structures with arbitrary shapes are studied. The effect of micro-inertia becomes dominant, and by considering the micropolar effects, the frequencies decrease. Also, there is a considerable discrepancy between the predicted micropolar and classical frequencies at small scales, and this difference decreases when the side length-to-length scale ratio becomes large.

Key words: finite element method (FEM), quadrilateral element, micropolar theory (MPT), user-defined element (UEL) subroutine, free vibration

2010 MSC Number:

74S05
70J30

M. KOHANSAL-VAJARGAH, R. ANSARI, M. FARAJI-OSKOUIE, M. BAZDID-VAHDATI. Vibration analysis of two-dimensional structures using micropolar elements. Applied Mathematics and Mechanics (English Edition), 2021, 42(7): 999-1012.

TrendMD

References

[1] MINDLIN, R. D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, 51–78(1964)
[2] MINDLIN, R. D. Second gradient of strain and surface tension in linear elasticity. International Journal of Solids and Structures, 1, 417–438(1965)
[3] TOUPIN, R. A. Elastic materials with couple stresses. Archive for Rational Mechanics and Analysis, 11, 385–414(1962)
[4] MINDLIN, R. D. and TIERSTEN, H. F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 11, 415–448(1962)
[5] KOITER, W. T. Couple stresses in the theory of elasticity: I and II. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 67, 17–44(1964)
[6] ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710(1983)
[7] ERINGEN, A. C. Nonlocal Continuum Field Theories, Springer-Verlag, New York (2002)
[8] LESIČAR, T., TONKOVIĆ Z., and SORIĆ J. Two-scale computational approach using strain gradient theory at microlevel. International Journal of Mechanical Science, 126, 67–78(2017)
[9] ANSARI, R., GHOLAMI, R., and SAHMANI, S. Study of small scale effects on the nonlinear vibration response of functionally graded Timoshenko microbeams based on the strain gradient theory. ASME Journal of Computational and Nonlinear Dynamics, 7, 031009(2012)
[10] AKGÖZ, B. and CIVALEK, O. Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mechanica, 224, 2185–2201(2013)
[11] COSSERAT, E. and COSSERAT, F. Théorie des Corps Déformables (in French), Librairie Scientifique A. Hermann Et Fils, Paris (1909)
[12] ERINGEN, A. C. Microcontinuum Field Theories I: Foundations and Solids, Springer-Verlag, New York (1999)
[13] ERINGEN, A. C. Microcontinuum Field Theories II: Fluent Media, Springer-Verlag, New York (2002)
[14] ERINGEN, A. C. An assessment of director and micropolar theories of liquid crystals. International Journal of Engineering Science, 31, 605–616(1993)
[15] PARK, H. C. and LAKES, R. S. Cosserat micromechanics of human bone: strain redistribution by a hydration sensitive constituent. Journal of Biomechanics, 19, 385–397(1986)
[16] CHENA, H., HUA, G., and HUANGB, Z. Effective moduli for micropolar composite with interface effect. International Journal of Solids and Structures, 44, 8106–8118(2007)
[17] WANG, L., CHU, X., WAN, J., and XIU, C. Implementation of micropolar fluids model and hydrodynamic behavior analysis using user-defined function in FLUENT. Advances in Mechanical Engineering, 12, 1–12(2020)
[18] KUMAR, R., MIGLANI, A., and RANI, R. Analysis of micropolar porous thermoelastic circular plate by eigenvalue approach. Archives of Mechanics, 68, 423–439(2016)
[19] ERINGEN, A. C. A continuum theory of rigid suspensions. International Journal of Engineering Science, 22, 1373–1388(1984)
[20] ASKAR, A. Molecular crystals and the polar theories of the continua experimental values of material coefficients for KNO₃. International Journal of Engineering Science, 10, 293–300(1972)
[21] FISCHER, H. Micropolar phenomena in ordered structures. Mechanics of Micropolar Media, World Scientific, Singapore, 1–33(1982)
[22] POUGET, J., ASKAR, A., and MAUGIN, G. A. Lattice model for elastic ferroelectric crystals: microscopic approach. Physical Review B, Condensed Matter, 33(9), 6304–6319(1986)
[23] GAUTHIER, R. D. Experimental investigations of micropolar media, Mechanics of Micropolar Media, World Scientific, Singapore, 395–463(1982)
[24] PRAKASH, J. and SINHA, P. Lubrication theory for micropolar fluids and its application to a journal bearing. International Journal of Engineering Science, 13, 217–232(1975)
[25] ERINGEN, A. C. and SUHUBI, E. Nonlinear theory of simple micro-elastic solids — I. International Journal of Engineering Science, 2, 189–203(1964)
[26] SUHUBI, E. and ERINGEN, A. C. Nonlinear theory of micro-elastic solids — II. International Journal of Engineering Science, 2, 389–404(1964)
[27] ERINGEN, A. C. Simple micro-fluids. International Journal of Engineering Science, 2, 205–217(1964)
[28] ERINGEN, A. C. Mechanics of micromorphic materials. The Eleventh International Congress of Applied Mechanics, Springer-Verlag, New York, 131–138(1966)
[29] ERINGEN, A. C. Theory of micropolar fluids. Journal of Applied Mathematics and Mechanics, 16, 1–18(1965)
[30] EREMEYEV, A., SKRZAT, A., STACHOWICZ, F., and VINAKURAVA, A. On strength analysis of highly porous materials within the framework of micropolar elasticity. Procedia Structural Integrity, 5, 446–451(2017)
[31] SARGSYAN, S. H. and ZHAMAKOCHYAN, K. A. Applied theory of micropolar elastic thin plates with constrained rotation and the finite element method. Materials Physics and Mechanics, 35, 145–154(2018)
[32] SARGSYAN, A. H. and SARGSYAN, S. H. Dynamic model of micropolar elastic thin plates with independent fields of displacements and rotations. Journal of Sound and Vibration, 333, 4354–4375(2014)
[33] WANG, F. Y. On the vibration modes of three-dimensional micropolar elastic plates. Journal of Sound and Vibration, 146, 1–16(1991)
[34] EREMEYEV, A., SKRZAT, A., and STACHOWICZ, F. On finite element computations of contact problems in micropolar elasticity. Advances in Materials Science and Engineering, 2016, 9675604(2016)
[35] ANSARI, R., SHAKOURI, A. H., BAZDID-VAHDATI, M., NOROUZZADEH, A., and ROUHI, H. A nonclassical finite element approach for the nonlinear analysis of micropolar plates. Journal of Computational and Nonlinear Dynamics, 12, 011019(2017)
[36] ANSARI, R., NOROUZZADEH, A., SHAKOURI, A. H., BAZDID-VAHDATI, M., and ROUHI, H. Finite element analysis of vibrating micro-beam and -plates using three-dimensional micropolar element. Thin-Walled Structures, 124, 489–500(2018)
[37] FARAJI-OSKOUIE, M., NOROUZZADEH, A., ANSARI, R., and ROUHI, H. Bending of small-scale Timoshenko beams based on the integral/differential nonlocal-micropolar elasticity theory: a finite element approach. Applied Mathematics and Mechanics (English Edition), 40, 767–782(2019) https://doi.org/10.1007/s10483-019-2491-9
[38] PIETRASZKIEWICZ, W. and EREMEYEV, V. A. On natural strain measures of the non-linear micropolar continuum. International Journal of Solids and Structures, 46, 774–787(2009)
[39] LI, X., LI, L., and HU, Y. Instability of functionally graded micro-beams via micro-structuredependent beam theory. Applied Mathematics and Mechanics (English Edition), 39, 923–952(2018) https://doi.org/10.1007/s10483-018-2343-8
[40] NAMPALLY, P. and REDDY, J. N. Geometrically nonlinear Euler-Bernoulli and Timoshenko micropolar beam theories. Acta Mechanica, 231, 4217–4242(2020)
[41] AKGÖZ, B. and CIVALEK, O. A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mechanica, 226, 2277–2294(2015)
[42] AKGÖZ, B. and CIVALEK, O. A novel microstructure-dependent shear deformable beam model. International Journal of Mechanical Sciences, 99, 10–20(2015)
[43] AKGÖZ, B. and CIVALEK, O. Longitudinal vibration analysis for microbars based on strain gradient elasticity theory. Journal of Vibration and Control, 20, 606–616(2014)
[44] JALAEI, M. H. and CIVALEK. O. On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam. International Journal of Engineering Science, 143, 14–32(2019)
[45] DEMIR, C. and CIVALEK, O. On the analysis of microbeams. International Journal of Engineering Science, 121, 14–33(2017)
[46] SU, Z., WANG, L., SUN, K., and SUN, J. Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams. Applied Mathematics and Mechanics (English Edition), 41, 1303–1320(2020) https://doi.org/10.1007/s10483-020-2662-6
[47] ERINGEN, A. C. Theory of micropolar elasticity. Microcontinuum Field Theories, SpringerVerlag, New York, 621–729(1968)
[48] ABAQUS/Standard Analysis User’s Manual, Simulia, Providence (2012)
[49] LAKES, R. Experimental microelasticity of two porous solids. International Journal of Solids and Structures, 22, 55–63(1986)