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Research Papers

# Analytical and 3D Finite Element Study of the Deflection of an Elastic Cantilever Bilayer Plate

[+] Author and Article Information
M. Chekchaki

CNRS, UPMC Univ Paris 6, Univ Paris-Sud, UMR 7608, Lab FAST, Bât 502, Campus Univ, F-91405 Orsay, Francemourad.chekchaki@etu.upmc.fr

V. Lazarus1

CNRS, UPMC Univ Paris 6, Univ Paris-Sud, UMR 7608, Lab FAST, Bât 502, Campus Univ, F-91405 Orsay, Franceveronique.lazarus@upmc.fr

J. Frelat

CNRS, UPMC Univ Paris 6, UMR 7190, Institut Jean Le Rond d’Alembert, Boite courrier 161-2, 4 Place Jussieu, F-75005, Paris, Francejoel.frelat@upmc.fr

We choose $y=0$ corresponding to the midplane of the bilayer plate. Any other choice would change the value of $ε0$ but not the value of $κ$ and $δ$, which are of interest here.

Finite element code developed by the French Commissariat à l’Energie Atomique http://www-cast3m.cea.fr.

Materials with negative Poisson modulus.

1

Corresponding author

J. Appl. Mech 78(1), 011008 (Oct 13, 2010) (7 pages) doi:10.1115/1.4002306 History: Received November 25, 2009; Revised July 29, 2010; Posted August 03, 2010; Published October 13, 2010; Online October 13, 2010

## Abstract

The mechanical system considered is a bilayer cantilever plate. The substrate and the film are linear elastic. The film is subjected to isotropic uniform prestresses due for instance to volume variation associated with cooling, heating, or drying. This loading yields deflection of the plate. We recall Stoney’s analytical formula linking the total mechanical stresses to this deflection. We also derive a relationship between the prestresses and the deflection. We relax Stoney’s assumption of very thin films. The analytical formulas are derived by assuming that the stress and curvature states are uniform and biaxial. To quantify the validity of these assumptions, finite element calculations of the three-dimensional elasticity problem are performed for a wide range of plate geometries, Young’s and Poisson’s moduli. One purpose is to help any user of the formulas to estimate their accuracy. In particular, we show that for very thin films, both formulas written either on the total mechanical stresses or on the prestresses, are equivalent and accurate. The error associated with the misfit between our theorical study and numerical results are also presented. For thicker films, the observed deflection is satisfactorily reproduced by the expression involving the prestresses and not the total mechanical stresses.

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## Figures

Figure 1

Bilayer cantilever plate

Figure 2

Relative error between the deflection obtained through formulas 9,12 and the numerical values (Ef/Es=10−2, ts/L=10−2, W/L=0.1, and ν=0.3)

Figure 3

Increasing nonuniformity of the film stresses with the film thickness (Ef/Es=10−2, ts/L=10−2, W/L=0.1, and ν=0.3)

Figure 4

Complete analytical formula 9 (solid lines) compared with the FE computations (dashed lines) obtained for ts/L=10−2, W/L=1, and ν=0.3

Figure 5

Relative error between the deflection obtained through the simplified Eq. 11 and the FE computations one for ts/L=10−2, W/L=1, and ν=0.3

Figure 6

Thickness and Young’s modulus ratios influence on the relative error. To the left of the solid line, the relative error between the simplified Eq. 11 and complete Eq. 9 Stoney formula is less than 10%. The points correspond to values of tf/ts, Ef/Es for which the relative error between the simplified Eq. 11 formula and the numerical result is 10% (each point corresponds to different values of ts/L and W/L).

Figure 7

Relative error between Eq. 9 and numerical results for Ef/Es=0.01: tf/ts=0.1, Ef/Es=0.01; tf/ts=1, Ef/Es=0.01; and tf/ts=10, Ef/Es=0.01

Figure 8

Relative error between Eq. 9 and numerical results for Ef/Es=1: tf/ts=0.1, Ef/Es=1; tf/ts=1, Ef/Es=1; and tf/ts=10, Ef/Es=1

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