Acta Mechanica Solida Sinica

, Volume 23, Issue 3, pp 260–270 | Cite as

Three-Dimensional Modeling for Thin Plate-Like Structures Including Surface Effects by Using State Space Method

  • Hongyu Sheng
  • Pin Lu


A three-dimensional (3-D) approach based on the state space method is proposed to study size-dependent mechanical properties of ultra-thin plate-like elastic structures considering surface effects. The structure is modeled as a laminate composed of a bulk bounded with upper and bottom surface layers, which are allowed to have different material properties from the bulk layer. State equations, including the surface properties of the structure, can be established on the basis of 3-D fundamental elasticity to analyze the size-dependent static characteristics of the thin plate-like structure. Compared with two-dimensional plate theories based size-dependent models for thin film structures in literature, the present 3-D approach is exact, which can provide benchmark results to assess the accuracy of 2-D plate theories and various numerical approaches.

To show the feasibility of the proposed approach, a 3-D analytical solution for a simply supported plate-like thin structure including surface layers is derived. An algorithm is proposed for the calculation of the state equations obtained to ensure that the numerical results can reveal the surface effects clearly even for extremely thin surface layers. Numerical examples are carried out to exhibit the surface effects and some discussions are provided based on the results obtained.

Key words

micro-structures surface effects size-dependence state equation three-dimensional modeling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Evoy, S., Carr, D.W., Sekaric, L., Olkhovets, A., Parpia, J.M. and Craighead, H.G., Nanofabrication and electrostatic operation of single-crystal silicon paddle oscillators. Journal of Applied Physics, 1999, 86(8): 6072–6077.CrossRefGoogle Scholar
  2. [2]
    Lavrik, N.V., Sepaniak, M.J. and Datskos, P.G., Cantilever transducers as a platform for chemical and biological sensors. Review of Scientific Instruments, 2004, 75(4): 2229–2253.CrossRefGoogle Scholar
  3. [3]
    Ibach, H., The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures. Surface Science Reports, 1997, 29(5–6): 193–263.Google Scholar
  4. [4]
    Muller, P. and Saul, A., Elastic effects on surface physics. Surface Science Reports, 2004, 54(5–8): 157–258.CrossRefGoogle Scholar
  5. [5]
    Gurtin, M.E. and Murdoch, A.I., A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 1975, 57(1): 291–323.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Gurtin, M.E. and Murdoch, A.I., Addenda to our paper: A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 1975, 59(1): 389–390.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Gurtin, M.E. and Murdoch, A.I., Surface stress in solids. International Journal of Solids and Structures, 1978, 14(3): 431–440.CrossRefGoogle Scholar
  8. [8]
    Miller, R.E. and Shenoy, V.B., Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 2000, 11(3): 139–147.CrossRefGoogle Scholar
  9. [9]
    He, L.H., Lim, C.W. and Wu, B.S., A continuum model for size-dependent deformation of elastic films of nano-scale thickness. International Journal of Solids and Structures, 2004, 41(3–4): 847–857.CrossRefGoogle Scholar
  10. [10]
    Lim, C.W. and He, L.H., Size-dependent nonlinear response of thin elastic films with nano-scale thickness. International Journal of Mechanical Science, 2004, 46(8): 1715–1726.CrossRefGoogle Scholar
  11. [11]
    Duan, H.L., Wang, J., Huang, Z.P. and Karihaloo, B.L., Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. Journal of the Mechanics and Physics of Solids, 2005, 53(4): 1574–1596.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Lu, P., He, L.H., Lee, H.P. and Lu, C., Thin plate theory including surface effects. International Journal of Solids and Structures, 2006, 43(13): 4631–4647.CrossRefGoogle Scholar
  13. [13]
    Zhu, H.X., Wang, J.X. and Karihaloo, B., Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms. Journal of Mechanics of Materials and Structures, 2009, 4(3): 589–604.CrossRefGoogle Scholar
  14. [14]
    Srinivas, S. and Rao, A.K., Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. International Journal of Solids and Structures, 1970, 6(8): 14631481.zbMATHGoogle Scholar
  15. [15]
    Sheng, H.Y. and Fan, J.R., A new approach to the thick Laminated plates with clamped edges. Chinese Journal of Computational Physics, 1999, 16(3): 682–687.Google Scholar
  16. [16]
    Sheng, H.Y. and Ye, J.Q., A three-dimensional state space finite element solution for Laminated composite cylindrical shells. Computer Methods in Applied Mechanics and Engineering, 2003, 192(22–24): 2441–2459.CrossRefGoogle Scholar
  17. [17]
    Ye, J.Q., Laminated Composite Plates and Shells: 3D Modelling. London: Springer-Verlag, 2003.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2010

Authors and Affiliations

  1. 1.Institute of Civil EngineeringHefei University of TechnologyHefeiChina
  2. 2.Institute of Materials Research and EngineeringA*STAR (Agency for Science, Technology and Research), 3 Research LinkSingaporeSingapore

Personalised recommendations