Static and Dynamic Micropolar Linear Elastic Beam Finite Element Formulation, Implementation, and Analysis
Publication: Journal of Engineering Mechanics
Volume 141, Issue 8
Abstract
Starting with static and dynamic micropolar linear plane stress elasticity, and applying Timoshenko beam kinematics with axial stretch, a mixed micropolar small-strain beam finite element (FE) formulation results. The mixed formulation is shown to be convergent upon mesh refinement under static and dynamic loading. The acceleration form of the Newmark family of time integration methods is applied to integrate the coupled hyperbolic linear governing equations. Instantaneous axial and transverse step forces are applied and released to analyze the free longitudinal and transverse vibrations with the mixed formulation FE implementation. The transverse displacement and rotational degrees of freedom (DOF) are coupled, but the axial displacement is decoupled from the rotational DOF because the first area moment of inertia is zero. Applied sinusoidal axial and transverse forces lead to axial and transverse displacement and rotational wave patterns that are a combination of low and high frequency waves. The effect of length scale on elastic couple modulus and spin inertia is demonstrated, which shows a transverse and rotational stiffening through upon increasing , yet a decreasing frequency as also increases with .
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Acknowledgments
Funding for this research was provided by National Science Foundation grant CMMI-0700648, Army Research Office grant W911NF-09-1-0111, the Army Research Laboratory, and Office of Naval Research grant N00014-11-1-0691. This funding is gratefully acknowledged.
References
Aganovic, I., Tambaca, J., and Tutek, Z. (2006). “Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity.” J. Elast., 84(2), 131–152.
Eringen, A. (1962). Nonlinear theory of continuous media, 1st Ed., McGraw-Hill, New York.
Eringen, A. (1966). “Linear theory of micropolar elasticity.” J. Math. Mech., 15(6), 909–923.
Eringen, A. (1968). “Theory of micropolar elasticity.” Fracture: An advanced treatise, H. Liebowitz, ed., Vol. 2, Academic Press, New York, 622–729.
Eringen, A. (1999). Microcontinuum field theories. I: Foundations and solids, Springer, Berlin.
Eringen, A., and Suhubi, E. (1964). “Nonlinear theory of simple micro-elastic solids—I.” Int. J. Eng. Sci., 2(2), 189–203.
Gauthier, R., and Jahsman, W. (1976). “Bending of a curved bar of micropolar elastic material.” J. Appl. Mech., 43(3), 502–503.
Gere, J., and Timoshenko, S. (1996). Mechanics of materials, 4th Ed., Nelson Engineering, Boston.
Huang, F.-Y., Yan, B.-H., Yan, J.-L., and Yang, D.-U. (2000). “Bending analysis of micropolar elastic beam using a 3-d finite element method.” Int. J. Eng. Sci., 38(3), 275–286.
Hughes, T. J. R. (1987). The finite element method, Prentice-Hall, Englewood Cliffs, NJ.
Li, L., and Xie, S. (2004). “Finite element method for linear micropolar elasticity and numerical study of some scale effects phenomena in mems.” Int. J. Mech. Sci., 46(11), 1571–1587.
McFarland, A., and Colton, J. (2005). “Role of material microstructure in plate stiffness with relevance to microcantilever sensors.” J. Micromech. Microeng., 15(5), 1060–1067.
Noor, A., and Nemeth, M. (1980). “Micropolar beam models for lattice grids with rigid joints.” Comput. Methods Appl. Mech. Eng., 21(2), 249–263.
Potapenko, S., Schiavone, P., and Mioduchowski, A. (2006). “On the solution of the torsion problem in linear elasticity with microstructure.” J. Math. Mech., 11(2), 181–195.
Ramezani, S., Naghdabadi, R., and Sohrabpour, S. (2009). “Analysis of micropolar elastic beams.” Eur. J. Mech. A. Solids, 28(2), 202–208.
Rayleigh, J. (1945). The theory of sound, 1st Ed., Vol. 1, Dover, New York.
Regueiro, R. (2009). “Finite strain micromorphic pressure-sensitive plasticity.” J. Eng. Mech., 178–191.
Regueiro, R. (2010). “On finite strain micromorphic elastoplasticity.” Int. J. Solids Struct., 47(6), 786–800.
Regueiro, R., Duan, Z., and Yan, B. (2014). “Overlap coupling between spherical discrete elements and micropolar finite elements in one dimension using a bridging-scale decomposition for statics.” in press.
Regueiro, R., and Yan, B. (2013). “Computational homogenization and partial overlap coupling between micropolar elastic continuum finite elements and elastic spherical discrete elements in one dimension.” Handbook of micromechanics and nanomechanics, S. Li, and X. L. Gao, eds., Vol. 1, Pan Stanford, Singapore, 1–45.
Salehian, A., and Inman, D. (2010). “Micropolar continuous modeling and frequency response validation of a lattice structure.” J. Vib. Acoust., 132(1), 011010–7.
Timoshenko, S. (1921). “On the correction for shear of the differential equation for transverse vibrations of prismatic bars.” Philos. Mag., 41(245), 744–746.
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Published In
Journal of Engineering Mechanics
Volume 141 • Issue 8 • August 2015
Copyright
© 2015 American Society of Civil Engineers.
History
Received: Sep 11, 2014
Accepted: Nov 25, 2014
Published online: Apr 29, 2015
Published in print: Aug 1, 2015
Discussion open until: Sep 29, 2015
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