# Vibration Analysis of Post-Buckled Thin Film on Compliant Substrates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Buckling Analysis

_{f}on top of a compliant substrate with thickness of h

_{s}subjected to a compressive load $\widehat{P}$ and non-conservative force $\widehat{q}$. Buckling occurs after loading because the thickness of the film is extremely small, which is very similar to a slender beam buckling. In the analytical model, the thin film is considered as Euler–Bernoulli beam with fixed condition at both ends, since the thickness of film h

_{f}is far less than the length L [42]. Planes of the cross sections remain planes after deformation, and the plane of the cross section is still perpendicular to the axis after deformation. It is unnecessary to employ the von Karman plate theory to model the film because it leads to lengthy solutions not convenient for practical use due to the in-plane displacement and the shear traction [44]. A common approximation is to ignore the in-plane displacement and the shear traction [45,46]. The film deformed out of a plane only can be modeled into a beam, which greatly simplifies theoretical analysis [24]. The compliant substrate is considered to be a Winkler elastic foundation [47,48,49,50], whose reaction at any point is proportional to the deflection, with stiffness $\widehat{k}$ and deflection of the thin film $\widehat{w}$ shown in Figure 1b, where $\widehat{w}$ is the function of Cartesian coordinate $\widehat{x}$, which is along the axial direction of the thin film.

_{i}(i = 1,2,3,4) is the coefficient to be determined by the boundary conditions. Substituting Equation (11) into the four boundary conditions of Equation (10) derives

_{i}by solving the Equation (12),

## 3. Vibration Analysis of Post-Buckled System

_{i}are constants, and s

_{i}is

_{i}are constants as below,

_{i}cannot simultaneously be zero, $\mathrm{det}(B)=0$, which provides the condition to obtain the value of ω. Furthermore, the vibrational mode $\xi (x)$ corresponding to natural frequency ω can be obtained.

## 4. Results and Discussion

^{2}. When the pre-stress is chosen as 0.001 very close to zero, the deflection can nearly equal the results obtained without substrate [29], which can verify the accuracy of the analytical model in the post-buckling analysis. As substrate stiffness increases, the Young’s modulus of compliant substrate increases, and the deflection of the thin films decreases. When substrate stiffness changes from 0 to 500, the deflection of the film decreases from 4.9 to 3.0.

^{7}where the number of wrinkles reaches up to 27, which is the typical morphing of local buckling. For comparisons, the number of wrinkles is just two or three when k belongs to the range of 877~6225 or 6225~21,750, respectively. The buckling modes and the first-order critical buckling force obtained from theoretical calculation agree reasonably well with simulation results.

^{2}. When k is close to zero, the vibration mode can be validated by the results obtained by Ref. [29], and if k is larger, the deflections obviously decrease. When k increases from 0 to 500, the deflection of the first-order vibration mode decreases from 57.6 to 40.9.

_{s}, the width b, the thickness h

_{s}, and the length L of PDMS substrate are 2 MPa, 4 mm, 1 mm, and 10 mm, respectively. The elastic modulus E

_{f}and the thickness h

_{f}of copper thin film are 71,000 MPa and 0.01 mm. Meanwhile, the structure with fixed condition at both ends is subjected to a compressive load $\widehat{P}$ = 2.367 × 10

^{−4}N. The dimensionless quantities k and P are obtained by Equation (6),

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of thin film on a compliant substrate and (

**b**) the deformation of the analytical model in the first-order bucking mode with an elastic foundation.

**Figure 2.**The deflection of first-order buckling mode in the film with different substrate stiffness or without substrate when P = 10 × π

^{2}.

**Figure 3.**The first-order critical buckling force versus substrate stiffness comparison between theory and simulation when P = 10

^{6}. The first-order buckling mode with (

**A**) k = 0~877, (

**B**) k = 877~6225, (

**C**) k = 6225~21,750, and (

**D**) when k = 5.5 × 10

^{7}.

**Figure 4.**(

**a**) The first order and (

**b**) the second order vibration modes of the first-order buckling with different substrate stiffness or without substrate when P = 10 × π

^{2}.

**Figure 5.**The first order (blue line) and the second order (red line) natural frequencies in the first two orders buckling modes with different substrate stiffness when P = 10

^{6}. (

**A**,

**B**) are the first order (blue line) and the second order (red line) vibration modes in the first order buckling mode (dot line). (

**C**,

**D**) are the first order (blue line) and the second order (red line) vibration mode in the second order buckling mode (dot line).

**Figure 6.**(

**a**) The first order buckling mode and (

**b**) the first order (blue line) and the second order (red line) vibration modes in the first buckling mode with substrate stiffness k = 3.38 × 10

^{6}when P = 5000.

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**MDPI and ACS Style**

Fan, X.; Wang, Y.; Li, Y.; Fu, H.
Vibration Analysis of Post-Buckled Thin Film on Compliant Substrates. *Sensors* **2020**, *20*, 5425.
https://doi.org/10.3390/s20185425

**AMA Style**

Fan X, Wang Y, Li Y, Fu H.
Vibration Analysis of Post-Buckled Thin Film on Compliant Substrates. *Sensors*. 2020; 20(18):5425.
https://doi.org/10.3390/s20185425

**Chicago/Turabian Style**

Fan, Xuanqing, Yi Wang, Yuhang Li, and Haoran Fu.
2020. "Vibration Analysis of Post-Buckled Thin Film on Compliant Substrates" *Sensors* 20, no. 18: 5425.
https://doi.org/10.3390/s20185425