0
Research Papers

Nonlinear Bending Analysis of First-Order Shear Deformable Microscale Plates Using a Strain Gradient Quadrilateral Element

[+] Author and Article Information
R. Ansari

Department of Mechanical Engineering,
University of Guilan,
P.O. Box 3756,
Rasht, Iran
e-mail: r_ansari@guilan.ac.ir

M. Faghih Shojaei, A. H. Shakouri

Department of Mechanical Engineering,
University of Guilan,
P.O. Box 3756,
Rasht, Iran

H. Rouhi

Department of Engineering Science,
Faculty of Technology and Engineering,
East of Guilan,
University of Guilan,
P.C. 44891-63157,
Rudsar-Vajargah, Iran

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 5, 2014; final manuscript received January 11, 2016; published online February 25, 2016. Assoc. Editor: Daniel J. Segalman.

J. Comput. Nonlinear Dynam 11(5), 051014 (Feb 25, 2016) (18 pages) Paper No: CND-14-1234; doi: 10.1115/1.4032552 History: Received October 05, 2014; Revised January 11, 2016

Based on Mindlin's strain gradient elasticity and first-order shear deformation plate theory, a size-dependent quadrilateral plate element is developed in this paper to study the nonlinear static bending of microplates. In comparison with the classical first-order shear deformable quadrilateral plate element, the proposed element needs 15 additional nodal degrees-of-freedom (DOF) including derivatives of lateral deflection and rotations with respect to coordinates, which means a total of 20DOFs per node. Also, the developed strain gradient-based finite-element formulation is general so that it can be reduced to that on the basis of modified couple stress theory (MCST) and modified strain gradient theory (MSGT). In the numerical results, the nonlinear bending response of microplates for different boundary conditions, length-scale factors, and geometrical parameters is studied. It is revealed that by the developed nonclassical finite-element approach, the nonlinear behavior of microplates with the consideration of strain gradient effects can be accurately studied.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Fleck, N. A. , Muller, G. M. , Ashby, M. F. , and Hutchinson, J. W. , 1994, “ Strain Gradient Plasticity: Theory and Experiment,” Acta Metall. Mater., 42(2), pp. 475–487. [CrossRef]
Nix, W. D. , and Gao, H. , 1998, “ Indentation Size Effects in Crystalline Materials: A Law for Strain Gradient Plasticity,” J. Mech. Phys. Solids, 46(3), pp. 411–425. [CrossRef]
Fleck, N. A. , and Hutchinson, J. W. , 2001, “ A Reformulation of Strain Gradient Plasticity,” J. Mech. Phys. Solids, 49(10), pp. 2245–2271. [CrossRef]
Gurtin, M. E. , 2004, “ A Gradient Theory of Small-Deformation Isotropic Plasticity That Accounts for the Burgers Vector and for Dissipation Due to Plastic Spin,” J. Mech. Phys. Solids, 52(11), pp. 2545–2568. [CrossRef]
McFarland, A. W. , and Colton, J. S. , 2005, “ Role of Material Microstructure in Plate Stiffness With Relevance to Microcantilever Sensors,” J. Micromech. Microeng., 15(5), pp. 1060–1067. [CrossRef]
Mindlin, R. D. , and Tiersten, H. F. , 1962, “ Effects of Couple-Stresses in Linear Elasticity,” Arch. Ration. Mech. Anal., 11(1), pp. 415–448. [CrossRef]
Koiter, W. T. , 1964, “ Couple Stresses in the Theory of Elasticity,” Proc. K. Ned. Akad. Wet. B, 67, pp. 17–44.
Mindlin, R. D. , 1964, “ Micro-Structure in Linear Elasticity,” Arch. Ration. Mech. Anal., 6, pp. 51–78.
Mindlin, R. D. , 1965, “ Second Gradient of Strain and Surface Tension in Linear Elasticity,” Int. J. Solids Struct., 1(4), pp. 417–438. [CrossRef]
Yang, F. , Chong, A. C. M. , Lam, D. C. C. , and Tong, P. , 2002, “ Couple Stress Based Strain Gradient Theory for Elasticity,” Int. J. Solids Struct., 39(10), pp. 2731–2743. [CrossRef]
Lam, D. C. C. , Yang, F. , Chong, A. C. M. , Wang, J. , and Tong, P. , 2003, “ Experiments and Theory in Strain Gradient Elasticity,” J. Mech. Phys. Solids, 51(8), pp. 1477–1508. [CrossRef]
Ke, L. L. , Wang, Y. S. , Yang, J. , and Kitipornchai, S. , 2012, “ Nonlinear Free Vibration of Size-Dependent Functionally Graded Microbeams,” Int. J. Eng. Sci., 50(1), pp. 256–267. [CrossRef]
Wang, Y. G. , Lin, W. H. , and Liu, N. , 2013, “ Nonlinear Free Vibration of a Microscale Beam Based on Modified Couple Stress Theory,” Physica E, 47, pp. 80–85. [CrossRef]
Ke, L. L. , Yang, J. , Kitipornchai, S. , Bradford, M. A. , and Wang, Y. S. , 2013, “ Axisymmetric Nonlinear Free Vibration of Size-Dependent Functionally Graded Annular Microplates,” Compos. Part B: Eng., 53, pp. 207–217. [CrossRef]
Wang, Y. G. , Lin, W. H. , and Liu, N. , 2013, “ Large Amplitude Free Vibration of Size-Dependent Circular Microplates Based on the Modified Couple Stress Theory,” Int. J. Mech. Sci., 71, pp. 51–57. [CrossRef]
Farokhi, H. , Ghayesh, M. H. , and Amabili, M. , 2013, “ Nonlinear Dynamics of a Geometrically Imperfect Microbeam Based on the Modified Couple Stress Theory,” Int. J. Eng. Sci., 68, pp. 11–23. [CrossRef]
Ghayesh, M. H. , Amabili, M. , and Farokhi, H. , 2013, “ Nonlinear Forced Vibrations of a Microbeam Based on the Strain Gradient Elasticity Theory,” Int. J. Eng. Sci., 63, pp. 52–60. [CrossRef]
Ansari, R. , Faghih Shojaei, M. , Mohammadi, V. , Gholami, R. , and Darabi, M. A. , 2014, “ Nonlinear Vibrations of Functionally Graded Mindlin Microplates Based on the Modified Couple Stress Theory,” Compos. Struct., 114, pp. 124–134. [CrossRef]
Xia, W. , Wang, L. , and Yin, L. , 2010, “ Nonlinear Non-Classical Microscale Beams: Static Bending, Postbuckling and Free Vibration,” Int. J. Eng. Sci., 48(12), pp. 2044–2053. [CrossRef]
Zhao, J. , Zhou, S. , Wang, B. , and Wang, X. , 2012, “ Nonlinear Microbeam Model Based on Strain Gradient Theory,” Appl. Math. Model., 36(6), pp. 2674–2686. [CrossRef]
Şimşek, M. , 2014, “ Nonlinear Static and Free Vibration Analysis of Microbeams Based on the Nonlinear Elastic Foundation Using Modified Couple Stress Theory and He's Variational Method,” Compos. Struct., 112, pp. 264–272. [CrossRef]
Ansari, R. , Faghih Shojaei, M. , Gholami, R. , Mohammadi, V. , and Darabi, M. A. , 2013, “ Thermal Postbuckling Behavior of Size-Dependent Functionally Graded Timoshenko Microbeams,” Int. J. Non-Linear Mech., 50, pp. 127–135. [CrossRef]
Ke, L. L. , Yang, J. , Kitipornchai, S. , and Wang, Y. S. , 2014, “ Axisymmetric Postbuckling Analysis of Size-Dependent Functionally Graded Annular Microplates Using the Physical Neutral Plane,” Int. J. Eng. Sci., 81, pp. 66–81. [CrossRef]
Wang, Y. G. , Lin, W. H. , and Liu, N. , 2015, “ Nonlinear Bending and Post-Buckling of Extensible Microscale Beams Based on Modified Couple Stress Theory,” Appl. Math. Model., 39, pp. 117–127. [CrossRef]
Wang, B. , Zhou, S. , Zhao, J. , and Chen, X. , 2011, “ Size-Dependent Pull-In Instability of Electrostatically Actuated Microbeam-Based MEMS,” J. Micromech. Microeng., 21(2), p. 027001. [CrossRef]
Wang, B. , Zhou, S. , Zhao, J. , and Chen, X. , 2011, “ Pull-In Instability Analysis of Electrostatically Actuated Microplate With Rectangular Shape,” Int. J. Precis. Eng. Manuf., 12(6), pp. 1085–1094. [CrossRef]
Ansari, R. , Gholami, R. , Mohammadi, V. , and Faghih Shojaei, M. , 2012, “ Size-Dependent Pull-In Instability of Hydrostatically and Electrostatically Actuated Circular Microplates,” ASME Comput. Nonlinear Dyn., 8(2), p. 021015. [CrossRef]
Kong, S. , 2013, “ Size Effect on Pull-In Behavior of Electrostatically Actuated Microbeams Based on a Modified Couple Stress Theory,” Appl. Math. Model., 37, pp. 7481–7488. [CrossRef]
Papanicolopulos, S. , Zervos, A. , and Vardoulakis, I. , 2009, “ A Three-Dimensional C1 Finite Element for Gradient Elasticity,” Int. J. Numer. Method Eng., 77(10), pp. 1396–1415. [CrossRef]
Engel, G. , Garikipati, K. , Hughes, T. , Larson, M. , Mazzei, L. , and Taylor, R. , 2002, “ Continuous/Discontinuous Finite Element Approximations of Fourth-Order Elliptic Equations in Structural and Continuum Mechanics With Applications to Thin Beams and Plates, and Strain Gradient Elasticity,” Comput. Methods Appl. Mech. Eng., 191(34), pp. 3669–3750. [CrossRef]
Zybell, L. , Muhlich, U. , Kuna, M. , and Zhang, Z. , 2012, “ A Three-Dimensional Finite Element for Gradient Elasticity Based on a Mixed-Type Formulation,” Comput. Mater. Sci., 52(1), pp. 268–273. [CrossRef]
Rudrarajua, S. , Van der Ven, A. , and Garikipatia, K. , 2014, “ Three-Dimensional Isogeometric Solutions to General Boundary Value Problems of Toupin's Gradient Elasticity Theory at Finite Strains,” Comput. Methods Appl. Mech. Eng., 278, pp. 705–728. [CrossRef]
Ansari, R. , Faghih Shojaei, M. , Mohammadi, V. , Bazdid-Vahdati, M. , and Rouhi, H. , 2015, “ Triangular Mindlin Microplate Element,” Comput. Methods Appl. Mech. Eng., 295, pp. 56–76. [CrossRef]
Mousavi, S. M. , Paavola, J. , and Reddy, J. N. , 2015, “ Variational Approach to Dynamic Analysis of Third-Order Shear Deformable Plates Within Gradient Elasticity,” Meccanica, 50(6), pp. 1537–1550. [CrossRef]
Ansari, R. , Gholami, R. , Faghih Shojaei, M. , Mohammadi, V. , and Darabi, M. A. , 2015, “ Size-Dependent Nonlinear Bending and Postbuckling of Functionally Graded Mindlin Rectangular Microplates Considering the Physical Neutral Plane Position,” Compos. Struct., 127, pp. 87–98. [CrossRef]
Ramezani, S. , 2012, “ A Shear Deformation Micro-Plate Model Based on the Most General Form of Strain Gradient Elasticity,” Int. J. Mech. Sci., 57(1), pp. 34–42. [CrossRef]
Gao, X.-L. , and Park, S. K. , 2007, “ Variational Formulation of a Simplified Strain Gradient Elasticity Theory and Its Application to a Pressurized Thick-Walled Cylinder Problem,” Int. J. Solids Struct., 44, pp. 7486–7499. [CrossRef]
Zhang, D. G. , 2014, “ Nonlinear Bending Analysis of FGM Rectangular Plates With Various Supported Boundaries Resting on Two-Parameter Elastic Foundations,” Arch. Appl. Mech., 84(1), pp. 1–20. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic view of the nine-node quadrilateral microplate element

Grahic Jump Location
Fig. 2

Variation of Wmax of microplates with the number of elements based on MSGT (a/h=10,  h/l=2,  and  Q=100)

Grahic Jump Location
Fig. 3

Variation of Wc of microplates with the number of elements based on MSGT (a/h=10,  h/l=2,  and  Q=100)

Grahic Jump Location
Fig. 4

Comparison between the present results and the analytical solutions of Ref. [38] in predicting the dimensionless maximum deflection of aluminum square plates based on the classical elasticity theory

Grahic Jump Location
Fig. 5

Influence of dimensionless length-scale parameter on the nonlinear bending behavior of microplates with different types of boundary conditions based on MCST (a/h=12)

Grahic Jump Location
Fig. 6

Influence of dimensionless length-scale parameter on the nonlinear bending behavior of microplates with different types of boundary conditions based on MSGT (a/h=12)

Grahic Jump Location
Fig. 7

Influence of side length-to-thickness ratio on the nonlinear bending behavior of microplates with different types of boundary conditions based on MCST (h/l=1.5)

Grahic Jump Location
Fig. 8

Influence of side length-to-thickness ratio on the nonlinear bending behavior of microplates with different types of boundary conditions based on MSGT (h/l=1.5)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In