Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-13T02:39:52.920Z Has data issue: false hasContentIssue false
  • English
  • Français

Static deflection analysis of flexural rectangular micro-plate using higher continuity finite-element method

Published online by Cambridge University Press:  06 November 2012

Ali Reza Ahmadi  and
Hamed Farahmand
Ali Reza Ahmadi*
Affiliation:
Kerman Gradate University of Technology, Kerman, Iran
Hamed Farahmand
Affiliation:
Young researcher club, Kerman branch, Islamic Azad University, Kerman, Iran
*
a Corresponding author: a.ahmadi@kgut.ac.ir
Get access

Abstract

In this paper, strain gradient theory is used in developing a mathematical model based on classical flexural Kirchhoff plate theory that can predict static response of rectangular micro-plates. The result of this new model is a sixth order differential equation. Order of differential terms in Galerkin weak form of the equation is reduced so that C2 hierarchical p-version finite elements with second order global smoothness can be used to solve the problem. With different boundary conditions, the computed deflection distribution of micro-plates is compared with those of the classical theory, in which length scale parameters are not present. A series of studies have revealed that when length scale parameters are considered, deflection of a rectangular plate decreases with increasing the length scale effect; in other words micro plates exhibit more rigidity than what is predicted by the classic model. Here, deflections are normalized with respect to results obtained from classical plate theory. Comparison of maximum deflection values obtained from the extended model for micro plates with those available from the classic plate model indicates that classical theory overestimates displacement values and the largest error is observed for square micro plates. The overestimation levels off for plates with aspect ratios greater than three.

Keywords

Flexural micro-platesstrain gradient elasticityhierarchicalp-version finite-element methodhigher continuity finite element
Type
Research Article
Information
Mechanics & Industry , Volume 13 , Issue 4 , 2012 , pp. 261 - 269
Copyright
© AFM, EDP Sciences 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Schijve, J., Note on couple Stresses, J. Mech. Phys. Solids 14 (1966) 113–120 CrossRefGoogle Scholar
Gauthier, R.D., Jahsman, W.E., A quest for micropolar elastic constants, J. Appl. Mech. 42 (1975) 369–374 CrossRefGoogle Scholar
Krishna Reddy, G.V., Venkatasubramanian, N.K., On the flexural rigidity of a micropolar elastic cylinder, J. Appl. Mech. 45 (1978) 429–43l CrossRefGoogle Scholar
R.S. Lakes, The role of gradient effects in the piezoelectricity of bone, IEEE Trans. Biomed. Eng. BME-27 (1980) 282–283
Lakes, R.S., Experimental microelasticity of two porous solids,Int. J. Solids Struc. 22 (1986) 55–63 CrossRefGoogle Scholar
Yang, J.F.C., Lakes, R.S., Experimental study of micropolar and couple stress elasticity in compact bone in bending, J. Biome. 15 (1982) 91–98 CrossRefGoogle ScholarPubMed
Hofstetter, K., Hellmich, C., Eberhardsteiner, J., Development and experimental validation of a continuum micromechanics model for the elasticity of wood, Eur. J. Mech. A/Solids 24 (2005) 1030–1053 CrossRefGoogle Scholar
Fish, J., Kuznetsov, S., Computational continua,Int. J. Numer. Meth. Engng. 84 (2010) 774–802 CrossRefGoogle Scholar
Mindlin, R.D., Micro-structure in linear elasticity, Arch. Rat. Mech. Anal. 16 (1964) 51–78 CrossRefGoogle Scholar
Mindlin, R.D., Eshel, N.N., On first strain-gradient theories in linear elasticity, Int. J. Solids Struct. 4 (1968) 109–124 CrossRefGoogle Scholar
Eringen, C.A., Linear theory of micropolar elasticity, J. Math. Mech. 15 (1966) 909–923 Google Scholar
E. Cosserat, F. Cosserat, Théorie des corps déformables, Hermann et Fils, Paris, 1909
Toupin, R.A., Elastic materials with couple-stresses, Arch. Rat. Mech. Anal. 11 (1962) 385–414 CrossRefGoogle Scholar
Koiter, W.T., Couple stresses in the theory of elasticity, I & II. Proc. K. Ned. Akad. Wet. B 67 (1964) 17–44 Google Scholar
H.F. Tiersten, J.L. Bleustein, Generalized elastic continua, In : G. Herrmann, R.D. Mindlin (eds.), and Applied Mechanics, Pergamon Press, New York, 1974, pp. 67–103
I. Vardoulakis, J. Sulem, Bifurcation Analysis in Geomechanics, Chapman and Hall, London, 1995
R. Lakes, Experimental methods for study of Cosserat elastic solids and other generalized elastic continua, In : Continuum Models for Materials with Microstructure, H.B. Mhlhaus (ed.), Wiley, Chichester, 1995, pp. 1–25
Papargyri-Beskou, S., Beskos, D.E., Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates, Arch. Appl. Mech. 78 (2008) 625–635 CrossRefGoogle Scholar
Park, S.K., Gao, X.-L., Variational formulation of a modified couple stress theory and its application to a simple shear problem, Z. Angew. Math. Phys. 59 (2008) 904–917 CrossRefGoogle Scholar
Farahmand, H., Arabnejad, S., Developing a Novel finite elastic approach in strain gradient theory for microstructures, Int. J. Multiscale Comput. Eng. 8 (2010) 441–446 CrossRefGoogle Scholar
Aifantis, E.C., Strain gradient interpretation of size effects, Int. J. Fract. 95 (1999) 299–314 CrossRefGoogle Scholar
Exadaktylos, G.E., Vardoulakis, I., Microstructure in linear elasticity and scale effects : a reconsideration of basic rock mechanics and rock fracture mechanics, Tectonophysics 335 (2001) 81–109 CrossRefGoogle Scholar
Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P., Experiments and theory in strain gradient elasticity, J. Mech. Phys. Solids 51 (2003) 1477–1508 CrossRefGoogle Scholar
Papargyri-Beskou, S., Giannakopoulos, A.E., Beskos, D.E., Variational analysis of gradient elastic flexural plates under static loading, Int. J. Solids Struc. 47 (2010) 2755–2766 CrossRefGoogle Scholar
Surana, K.S., Petti, S.R., Ahmadi, A.R., Reddy, J.N., On p-version hierarchical interpolation functions for higher order continuity finite element models, Int. J. Comput. Eng. Sci. 2 (2002) 653–673 CrossRefGoogle Scholar
Surana, K.S., Ahmadi, A.R., Reddy, J.N., The k-version of finite element method for self-adjoint operators in BVP, Int. J. Comput. Eng. Sci. 3 (2002) 155–218 CrossRefGoogle Scholar
Surana, K.S., Ahmadi, A.R., Reddy, J.N., The k-version of finite element method for non-self-adjoint operators in BVP, Int. J. Comput. Eng. Sci. 3 (2003) 737–812 CrossRefGoogle Scholar
Surana, K.S., Ahmadi, A.R., Reddy, J.N., The k-version of Finite Element Method for Nonlinear Operators in BVP, Int. J. Comput. Eng. Sci. 5 (2004) 133–207 CrossRefGoogle Scholar
J.N. Reddy, Theory and analysis of elastic plates and shells, Second Edition, Taylor and Francis, Philadelphia, 2006