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Research Papers

A State Space Method for Surface Instability of Elastic Layers With Material Properties Varying in Thickness Direction

[+] Author and Article Information
Zhigen Wu

School of Civil Engineering,
Hefei University of Technology,
Hefei, Anhui 230009, China
e-mail: zhigenwu@hfut.edu.cn

Jixiang Meng, Yihua Liu, Hao Li

School of Civil Engineering,
Hefei University of Technology,
Hefei, Anhui 230009, China

Rui Huang

Department of Aerospace Engineering
and Engineering Mechanics,
University of Texas,
Austin, TX 78712
e-mail: ruihuang@mail.utexas.edu

1Corresponding author.

Manuscript received March 24, 2014; final manuscript received April 17, 2014; accepted manuscript posted April 22, 2014; published online May 5, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(8), 081003 (May 05, 2014) (10 pages) Paper No: JAM-14-1132; doi: 10.1115/1.4027464 History: Received March 24, 2014; Revised April 17, 2014; Accepted April 22, 2014

A state space method is proposed for analyzing surface instability of elastic layers with elastic properties varying in the thickness direction. By assuming linear elasticity with nonlinear kinematics, the governing equations for the incremental stress field from a fundamental state are derived for arbitrarily graded elastic layers subject to plane-strain compression, which lead to an eigenvalue problem. By discretizing the elastic properties into piecewise constant functions with homogeneous sublayers, a state space method is developed to solve the eigenvalue problem and predict the critical condition for onset of surface instability. Results are presented for homogeneous layers, bilayers, and continuously graded elastic layers. The state space solutions for elastic bilayers are in close agreement with the analytical solution for thin film wrinkling within the limit of linear elasticity. Numerical solutions for continuously graded elastic layers are compared to finite element results in a previous study (Lee et al., 2008, J. Mech. Phys. Solids, 56, pp. 858–868). As a semi-analytical approach, the state space method is computationally efficient for graded elastic layers, especially for laminated multilayers.

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Figures

Grahic Jump Location
Fig. 1

Schematics of an elastic layer on a rigid support: (a) in the stress-free reference state and (b) in the fundamental state subjected to in-plane compression, divided into n sublayers for the state space method

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Fig. 2

The critical strain versus the perturbation wave number for homogeneous elastic layers with various Poisson's ratios. The horizontal dashed lines represent the analytical solutions for ωh→∞, obtained from Eq. (A10).

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Fig. 3

The critical strain versus dimensionless wave number for elastic bilayers with νf = νs = 0.4: (a) Ef/Es = 1000 and (b) hs/hf = 10

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Fig. 4

(a) The critical strain for onset of surface instability and (b) the corresponding critical wavelength versus the thickness ratio for elastic bilayers with νf = νs = 0.4. The solid lines are predicted by the analytical solution [8].

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Fig. 5

(a) The critical strain for onset of surface instability and (b) the corresponding critical wavelength versus the modulus ratio for elastic bilayers. The solid lines are predicted by the analytical solution [8], and the horizontal dashed line in (a) suggests an upper bound for the small-strain approximation.

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Fig. 6

Convergence of the state space solution for an exponentially graded elastic layer: (a) the critical strain and (b) the corresponding wave number, versus the number of sublayers used

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Fig. 7

(a) The critical strain and (b) the corresponding wave number for exponentially graded elastic layers with E¯0/E¯s ranging from 10 to 105, comparing the state space solutions with the finite element results by Lee et al. [14] and the analytical approximation

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Fig. 8

(a) The critical strain and (b) the corresponding wave number, for the graded elastic layers with an error function for the plane-strain modulus, comparing the state space solutions with the finite element results by Lee et al. [14] and the analytical approximation

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Fig. 9

Convergence of the state space solution for linearly graded elastic layers with different modulus ratios: (a) the critical strain and (b) the corresponding wave number, versus the number of sublayers used

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Fig. 10

(a) The critical strain and (b) the corresponding wavelength versus the thickness ratio for a linearly graded elastic layer on a homogeneous substrate. The solid lines are the analytical solution for the elastic bilayers with E¯f = (E¯0+E¯s)/2 for the upper layer.

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