Dynamic Stability of a Thin Film Bonded to a Compliant Substrate Subjected to a Step Load with Damping

Zhicheng Ou; Xiaohu Yao; Xiaoqing Zhang

doi:10.1515/ijnsns-2015-0091

Article

Dynamic Stability of a Thin Film Bonded to a Compliant Substrate Subjected to a Step Load with Damping

Zhicheng Ou , Xiaohu Yao and Xiaoqing Zhang

Published/Copyright: May 9, 2017

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 18 Issue 3-4

Abstract

The flexible electronic structure is based on the buckling of a thin film on a compliant substrate. This paper studies the dynamic stability of this structure subjected to a uniaxial step load with damping. The equation of motion is derived by Hamilton’s principle and Euler-Lagrange equation. Through the qualitative analysis of the phase portraits of the dynamic equation, as well as the quantitative analysis of the responses according to Budiansky-Roth criterion, the critical dynamic load is determined. It is the same as that in static buckling. Affected by damping, at the stage of pre-buckling, the amplitude of the film vibrates and attenuates, where the maximum response is the initial amplitude. The structural damping can be derived by the logarithm of the ratio between two adjacent peaks of the vibration. At the stage of post-buckling, the amplitude of the film vibrates and tends to a stable response, which is the amplitude in static buckling. The upper and lower bounds of the post-buckling response are asymmetric and solved by modified Krylov-Bogoliubov method.

Keywords: film; substrate; step load; dynamic stability; damping

PACS: 46.32.+x; 46.70.De; 85.85.+j1

Appendix

The approximate upper and lower bounds of eq. (40) when μ>1 can be solved analytically by using modified Krylov-Bogoliubov method [35, 36, 37].

When 0\ltf02<μ−1, the solution of eq. (40) without the damping term is given by

(1)f(τ)=aκ1⋅nd(mτ−ϕ,κ)

where κ12=1−κ2. Under the initial condition eq. (38), the coefficients in eq. (1) are

(2)ω2=12a2,κ2=2+2(1−μ)a2a=2(μ−1)−f02,ϕ=0

The upper and lower bounds of solution (1) are

(3)flow=aκ1=2(μ−1)−a2,fup=a

The time derivative of eq. (1) is

(4)f′(τ)=amκ1κ2⋅cd(mτ−ϕ,κ)⋅sd(mτ−ϕ,k)

The trial solution of eq. (40) with the damping term is assumed in terms of Jacobi elliptic functions as eq. (1), but the parameters are all time-dependent,

(5)f(τ)=a(τ)κ1(τ)⋅nd(M(τ),κ(τ))

with

M(τ)=∫0τm(s)ds−ϕ(τ)

The initial conditions are given by

(6)f(0)=a(0)κ1(0)=a01−κ02=f0κ(0)=κ0,a(0)=a0,ϕ(0)=0

The trial solution eq. (5) should satisfy three additional constraints. The first two constraints are obtained from equation (2) where the parameters are time-dependent,

(7)m(τ)2=12a(τ)2,κ(τ)2=2+2(1−μ)a(τ)2

The third constraint is usual in Krylov-Bogoliubov method: the time derivative of the trial solution must have the same form as that of the generating solution. The third constraint can be rewritten from eq. (4),

(8)f′(τ)=a(τ)m(τ)κ1(τ)κ(τ)2⋅cd(M(τ),κ(τ))⋅sd(M(τ),k(τ))

The time derivative of trial solution eq. (5) is not dependent on the time derivative of the parameters.

For convenience, the Italic symbol pq denotes corresponding Jacobi elliptic function, such as pq=pq(M(τ),μ(τ)). Then eq. (8) can be rewritten as

(9)f′=amκ1κ2cd⋅sd

Differentiating eq. (5) with respect to τ and using the constraint eq. (9) gives

(10)a′κ12−aκκ′cn(cn+Isd)−aκ2κ12ϕ′cn⋅sd=0

with

I=E(M,κ)−κ12M

where E(M,κ) is elliptic integral of the second kind. Differentiating eq. (9) with respect to τ gives

(11)f′′=κ1ma′+am′cd⋅sd+aκ1m(ϕ′−m)(1−2cd2)nd+amκ′κ−1Iκ1−11−2cd2nd+2κ1cd⋅sd

From eqs (5) and (7), we obtain

(12)f3+(1−μ)f=12a3κ1nd[κ12(2nd2−1)−1]

Differentiating the constraints eq. (7) gives

(13)m′m=κκ′2−κ2

Submitting eqs (9), (11) and (12) into eq. (40), and using the constraints eqs (7), (10) and (13), we obtain

(14)κ12cn4κ2−1−2cn2κ12a′a−cn22−2−cn2κ2m′m=2ξκ2κ12cn2sn2

The coefficients of a′/a, m′/m and ξ in eq. (14) are roughly periodic. So eq. (14) can be reduced to a simple form by using the averaging principle method [38],

(15)a′a−R1(κ)m′m=R2(κ)ξ

with

(16)R1(κ)=Q1Q0,R2(κ)=Q2Q0

and

(17a)Q0=κ12cn4κ2−1−2cn2κ12

(17b)Q1=cn22−2−cn2κ2

(17c)Q2=2κ2κ12cn2sn2

The averaging operator is given by

(18)Qi(M,κ)=14K(κ)∫04K(κ)Qi(M,κ)dM

K(κ) here and E(κ) in the following text are the complete elliptic integrals of the first and second kind. The coefficients in eq. (15) are integrated according to equations (16)–(18),

(19)R1(κ)=4−7κ2+3κ4K(κ)+22−κ2E(κ)21−κ22−κ2E(κ)−21−κ2K(κ)R2(κ)=1

R1 can be expanded as a series form

(20)R1(κ)≈1+4κ2+35κ232+O(κ)

Submitting eqs (13), (19) and (20) into eq. (15) gives

(21)a′a−35κ4+32κ2+128κ′32κ2−κ2=ξ

Setting a(τ)=b(τ)eξτ, eq. (21) can be simplified as,

(22)b′b=35κ4+32κ2+128κ′32κ2−κ2

Integrating both sides of eq. (22),

∫b0bdbb=∫κ0κ35κ4+32κ2+12832κ2−κ2dκ

we have

(23)lnbb0=3564κ02−κ2+8332ln2−κ022−κ2+2lnκκ0

Submitting eq. (7) and a=beξτ, eq. (23) can be rewritten as,

(24)ξτ=35(μ−1)321a2−1a02+3532lna02a2+lna02+1−μa2+1−μ

From initial condition (6) and constraint (7), we have

(25)a02=2μ−2−f02

By submitting eq. (25) into eq. (24), the upper bound of eq. (40), fup=a, is satisfied,

(26)ξτ=35(μ−1)3212μ−2−f02−1fup2+358ln2μ−2−f02fup2+lnμ−1−f02fup2+1−μ

According to eq. (3), the lower bound of eq. (40), flow, is satisfied,

(27)ξτ=35(μ−1)3212μ−2−f02−12μ−2−flow2+358ln2μ−2−f022μ−2−flow2+lnμ−1−f02μ−1−flow2

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Received: 2015-7-10

Accepted: 2017-3-1

Published Online: 2017-5-9

Published in Print: 2017-5-24

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Articles in the same Issue

https://doi.org/10.1515/ijnsns-2015-0091

Keywords for this article

film; substrate; step load; dynamic stability; damping