Nonlinear Analysis of Plates with Rotation Gradient–Dependent Potential Energy for Constrained Microrotation
Publication: Journal of Engineering Mechanics
Volume 144, Issue 2
Abstract
In this study, the weak-form finite-element model has been developed for bending of plates considering the rotation gradient–dependent potential energy along with the conventional strain energy in the case of moderate rotation for Cosserat solid. The microrotation of the material point is considered to be constrained with the macrorotation of the continua. First, the governing equations are obtained from the principle of virtual displacements considering the displacement field as general power (Taylor) series expansion about the displacement of the midplane of the plate, and then the formulations are specialized for the general third-order, first-order, and the classical plate theory. The nonlinear finite-element models have been developed for all the plate theories considered. Further, the analytical solution for a simply supported linear plate is presented. In the numerical examples, the stiffening and anisotropic effects in response to oriented microstructures in the continuum of a microplate are illustrated. The parametric effect of the material length scale on the various components of stress is also studied.
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Acknowledgments
The authors gratefully acknowledge the financial support provided by the Oscar S. Wyatt Endowed Chair, Texas A&M University, for this research.
References
Arbind, A., Reddy, J., and Srinivasa, A. (2017). “Nonlinear analysis of beams with rotation gradient dependent potential energy for constrained micro-rotation.” Eur. J. Mech. A. Solids, 65, 178–194.
Cosserat, E., and Cosserat, F. (1909). Théorie des corps déformables, Hermann Archives, Paris.
Eringen, A. C., and Suhubi, E. (1964). “Nonlinear theory of simple micro-elastic solids—I.” Int. J. Eng. Sci., 2(2), 189–203.
Hoger, A., and Carlson, D. E. (1984). “On the derivative of the square root of a tensor and Guo’s rate theorems.” J. Elast., 14(3), 329–336.
Mindlin, R., and Tiersten, H. (1962). “Effects of couple-stresses in linear elasticity.” Arch. Ration. Mech. Anal., 11(1), 415–448.
Miyamoto, Y., Kaysser, W., Rabin, B., Kawasaki, A., and Ford, R. G. (2013). Functionally graded materials: Design, processing and applications, Vol. 5, Springer, Norwell, MA.
Reddy, J. (2002). Energy principles and variational methods in applied mechanics, 2nd Ed., Wiley, Hoboken, NJ.
Reddy, J. (2013). An introduction to continuum mechanics, 2nd Ed., Cambridge University Press, New York.
Srinivasa, A., and Reddy, J. (2013). “A model for a constrained, finitely deforming, elastic solid with rotation gradient dependent strain energy, and its specialization to von Karman plates and beams.” J. Mech. Phys. Solids, 61(3), 873–885.
Suhubl, E., and Eringen, A. C. (1964). “Nonlinear theory of micro-elastic solids—II.” Int. J. Eng. Sci., 2(4), 389–404.
Thostenson, E. T., and Chou, T.-W. (2002). “Aligned multi-walled carbon nanotube-reinforced composites: Processing and mechanical characterization.” J. Phys. D: Appl. Phys., 35(16), L77.
Toupin, R. A. (1962). “Elastic materials with couple-stresses.” Arch. Ration. Mech. Anal., 11(1), 385–414.
Truesdell, C., and Toupin, R. (1960). The classical field theories, Springer, Berlin.
Warner, M., Modes, C., and Corbett, D. (2010a). “Curvature in nematic elastica responding to light and heat.” Proc., Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 466, The Royal Society, London, 2975–2989.
Warner, M., Modes, C., and Corbett, D. (2010b). “Suppression of curvature in nematic elastica.” Proc., Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 466, The Royal Society, London, 3561–3578.
Warner, M., and Terentjev, E. M. (2003). Liquid crystal elastomers, Vol. 120, Oxford University Press, Oxford, U.K.
Yang, F., Chong, A., Lam, D., and Tong, P. (2002). “Couple stress based strain gradient theory for elasticity.” Int. J. Solids Struct., 39(10), 2731–2743.
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Published In
Journal of Engineering Mechanics
Volume 144 • Issue 2 • February 2018
Copyright
©2017 American Society of Civil Engineers.
History
Received: Sep 29, 2016
Accepted: Jun 7, 2017
Published online: Dec 12, 2017
Published in print: Feb 1, 2018
Discussion open until: May 12, 2018
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