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Modelling and Simulation in Engineering
Volume 2017, Article ID 5246197, 19 pages
https://doi.org/10.1155/2017/5246197
Research Article

Modeling of Size Effects in Bending of Perforated Cosserat Plates

1Department of Mathematics, University of Puerto Rico at Aguadilla, Aguadilla, PR 00604, USA
2Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Mayagüez, PR 00681, USA

Correspondence should be addressed to Lev Steinberg; ude.rpu@grebniets.vel

Received 27 August 2016; Revised 20 November 2016; Accepted 1 December 2016; Published 30 March 2017

Academic Editor: Chung-Souk Han

Copyright © 2017 Roman Kvasov and Lev Steinberg. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Cosserat and F. Cosserat, “Theorie des corps deformables,” 1909.
  2. A. C. Eringen, “Theory of micropolar plates,” Journal of Applied Mathematics and Physics, vol. 18, no. 1, pp. 12–30, 1967. View at Publisher · View at Google Scholar · View at Scopus
  3. A. E. Green, P. M. Naghdi, and M. L. Wenner, “The linear theory of an elastic Cosserat plate,” Proceedings of the Cambridge Philosophical Society, vol. 63, pp. 537–550, 1966. View at Google Scholar · View at Scopus
  4. H. Altenbach and V. A. Eremeyev, “On the theories of plates based on the Cosserat approach,” in Mechanics of Generalized Continua, vol. 21 of Advances in Mechanics and Mathematics, pp. 27–35, Springer, New York, NY, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  5. L. Steinberg, “Deformation of micropolar plates of moderate thickness,” International Journal of Applied Mathematics and Mechanics, vol. 6, no. 17, pp. 1–24, 2010. View at Google Scholar
  6. R. Kvasov and L. Steinberg, “Numerical modeling of bending of Cosserat elastic plates,” in Proceedings of the 5th Computing Alliance of Hispanic-Serving Institutions, pp. 67–70, 2011.
  7. L. Steinberg and R. Kvasov, “Enhanced mathematical model for Cosserat plate bending,” Thin-Walled Structures, vol. 63, pp. 51–62, 2013. View at Publisher · View at Google Scholar · View at Scopus
  8. E. Reissner, “On the theory of bending of elastic plates,” Journal of Mathematics and Physics, vol. 23, pp. 184–191, 1944. View at Publisher · View at Google Scholar · View at MathSciNet
  9. E. Reissner, “The effect of transverse shear deformation on the bending of elastic plates,” Journal of Applied Mechanics, vol. 12, pp. 69–77, 1945. View at Google Scholar · View at MathSciNet
  10. R. Kvasov and L. Steinberg, “Numerical modeling of bending of micropolar plates,” Thin-Walled Structures, vol. 69, pp. 67–78, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. L. Steinberg and R. Kvasov, “Analytical modeling of vibration of micropolar plates,” Applied Mathematics, vol. 6, pp. 817–836, 2015. View at Publisher · View at Google Scholar
  12. P. Solin, Partial Differential Equations and the Finite Element Method, Wiley-Interscience, 2006.
  13. D. Arnold and R. Falk, “Edge effect in the Reissner-Mindlin plate theory,” in Analytic and Computational Models of Shells, pp. 71–90, American Society of Mechanical Engineers, New York, NY, USA, 1989. View at Google Scholar
  14. C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge, UK, 1987. View at MathSciNet
  15. J. B. Conway, A Course in Functional Analysis, vol. 96 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  16. T. Hughes, E. Stein, and R. de Borst, Encyclopedia of Computational Mechanics, vol. 2, Cambridge University Press, Cambridge, UK, 2004.
  17. M. Costabel, M. Dauge, and S. Nicaise, Corner Singularities and Analytic Regularity for Linear Elliptic Systems, Part I: Smooth Domains. v2: Improvement of Layout, 2010.
  18. M. Costabel, M. Dauge, and S. Nicaise, “Analytic regularity for linear elliptic systems in polygons and polyhedra,” Mathematical Models and Methods in Applied Sciences, vol. 22, no. 8, Article ID 1250015, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. P. Lax and A. Milgram, “Parabolic equations,” in Contributions to the Theory of Partial Differential Equations, vol. 33, pp. 167–190, Princeton University Press, 1954. View at Google Scholar
  20. P. Ciarlet and J. Lions, “General preface,” in Finite Element Methods (Part 1), vol. 2 of Handbook of Numerical Analysis, pp. v–vi, Elsevier, 1991. View at Publisher · View at Google Scholar
  21. J. Céa, “Approximation variationnelle des problèmes aux limites,” Annales de l'Institut Fourier, vol. 14, no. 2, pp. 345–444, 1964. View at Publisher · View at Google Scholar · View at MathSciNet
  22. M. Ainsworth and J. T. Oden, “A posteriori error estimation in finite element analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 142, no. 1-2, pp. 1–88, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. R. Duran, Galerkin Approximations and Finite Element Methods, Universidad de Buenos Aires, 2010.
  24. C. Bacuta, V. Nistor, and L. T. Zikatanov, “Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates,” Numerical Functional Analysis and Optimization, vol. 26, no. 6, pp. 613–639, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. R. Lakes, “Experimental methods for study of Cosserat elastic solids and other generalized elastic continua,” in Continuum Models for Materials with Microstructures, H. Mühlhaus, Ed., pp. 1–22, John Wiley & Sons, New York, NY, USA, 1995. View at Google Scholar
  26. R. D. Mindlin, “Influence of couple-stresses on stress concentrations—main features of cosserat theory are reviewed by lecturer and some recent solutions of the equations, for cases of stress concentration around small holes in elastic solids, are described,” Experimental Mechanics, vol. 3, no. 1, pp. 1–7, 1963. View at Publisher · View at Google Scholar · View at Scopus
  27. W. B. Anderson, R. S. Lakes, and M. C. Smith, “Holographic evaluation of warp in the torsion of a bar of cellular solid,” Cellular Polymers, vol. 14, no. 1, pp. 1–13, 1995. View at Google Scholar · View at Scopus