Buckling behaviour of a stiff thin film on a finite-thickness bi-layer substrate

doi:10.1016/j.ijsolstr.2021.02.013

International Journal of Solids and Structures
国际固体与结构杂志

Volumes 219–220, 1 June 2021, Pages 177-187
第 219-220 卷，2021 年 6 月 1 日，第 177-187 页

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Abstract 摘要

In engineering stretchable electronics, an intermediate layer between a thin stiff film and a substrate can enhance the adhesion of the film and modulate its buckling shape. Capturing buckling behaviour of the tri-layer film/intermediate layer/substrate structure is essential for designing stretchable electronics. In this paper, considering the shear stress between the film and intermediate layer and using the complete mathematical form of the longitudinal displacement of the film, an improved theoretical model of the buckled tri-layer structure is established and the total energy of this tri-layer structure is derived. Based on the total energy, two buckling regimes, global (Euler) buckling and local wrinkling, influenced by the intermediate layer, are studied. Since local wrinkling is a desirable buckling pattern for stretchable electronics, accurate solutions of buckling wavelength and critical strain of local wrinkling verified by finite element analysis are determined. The influence of the intermediate layer on the wrinkling instability of the tri-layer structure is analysed. These results will aid in the design and manufacture of film/substrate-type stretchable electronics.
在工程可拉伸电子器件中，刚性薄膜和基底之间的中间层可以增强薄膜的附着力并调节其屈曲形状。捕捉三层薄膜/中间层/基底结构的屈曲行为对于设计可拉伸电子器件至关重要。本文考虑了薄膜与中间层之间的剪应力，并利用薄膜纵向位移的完整数学形式，建立了一个改进的三层屈曲结构理论模型，并推导出该三层结构的总能量。根据总能量，研究了受中间层影响的两种屈曲状态，即全局（欧拉）屈曲和局部起皱。由于局部起皱是可拉伸电子器件理想的屈曲模式，因此确定了屈曲波长的精确解，并通过有限元分析验证了局部起皱的临界应变。还分析了中间层对三层结构起皱不稳定性的影响。这些结果将有助于薄膜/基板型可拉伸电子器件的设计和制造。

Keywords 关键词

Stretchable electronics

Tri-layer structure

Local wrinkling

Global buckling

可拉伸电子器件三层结构局部起皱全局屈曲

1. Introduction 1.导言

In recent years, tremendous efforts have been dedicated to developing stretchable electronic devices (Chen et al., 2019, Fu et al., 2018, Khang et al., 2006, Kim et al., 2017, Lee et al., 2020, Li et al., 2020a, Li et al., 2019b, Ma et al., 2013, Ma et al., 2020, Rogers et al., 2020, Yokota et al., 2016), which have the characteristics of stretchability/flexibility, bendability and portability and so on. Adhering a stiff thin film on a pre-stretched soft elastic substrate to produce wrinkling patterns is one of the most important and widely-used strategies in the manufacture of stretchable electronic devices (Andres et al., 2018, Chen et al., 2017, Chen et al., 2018, Fu et al., 2019, He et al., 2017, Huang et al., 2005, Kim et al., 2019, Reese et al., 2020, Song et al., 2008, Xu et al., 2015, Xu et al., 2014, Xu and Potier-Ferry, 2016, Zhang and Yin, 2018, Zhang et al., 2019b). In real applications, surface wrinkling of a tri-layer structure (a thin film on a bi-layer compliant substrate) is a dominant structural feature and has more benefits in engineering, compared with a single-layer substrate (Hattori et al., 2014, Li et al., 2020b, Limbert and Kuhl, 2018, Nolte et al., 2006, Won et al., 2020, Yin et al., 2020, Yoo et al., 2018, Zhang et al., 2019a, Zhao et al., 2020). Here, several recent investigations have demonstrated that an intermediate layer is needed to bond the thin film layer to the compliant elastic substrate layer, which can not only enhance the adhesion of the bi-layer substrate (Tran et al., 2019), but also improve the strength and robustness of the tri-layer structure (Cheng et al., 2014). On the other hand, due to the introduction of the intermediate layer, the morphology evolution of film/substrate structure could be unpredictable, and some unexpected phenomena might appear, such as premature wrinkling (Béfahy et al., 2010, Lejeune et al., 2016b). In particular, when the thickness of the substrate is great, the bi-layer compliant substrate has an important influence on the performance of the film-substrate-based stretchable electronic devices (Wang et al., 2020). Hence, for their better design, it is essential to have an understanding of the elasticity of the tri-layer structure.
近年来，人们致力于开发可拉伸电子器件（Chen等人，2019年；Fu等人，2018年；Khang等人，2006年；Kim等人，2017年；Lee等人，2020年；Li等人，2020年a；Li等人，2019年b；Ma等人，2013年；Ma等人，2020年；Rogers等人，2020年；Yokota等人，2016年），这些器件具有可拉伸/柔性、可弯曲和可移植性等特点。在预先拉伸的软弹性基底上粘附硬质薄膜以产生皱纹图案，是制造可拉伸电子器件最重要也是应用最广泛的策略之一（Andres et al、2017、Chen 等人，2018、Fu 等人，2019、He 等人，2017、Huang 等人，2005、Kim 等人，2019、Reese 等人，2020、Song 等人，2008、Xu 等人，2015、Xu 等人，2014、Xu 和 Potier-Ferry，2016、Zhang 和 Yin，2018、Zhang 等人，2019b）。在实际应用中，与单层基底相比，三层结构（双层顺应基底上的薄膜）的表面起皱是一种主要的结构特征，在工程学中具有更多优势（Hattori 等人，2014 年；Li 等人，2020b；Limbert 和 Kuhl，2018 年；Nolte 等人，2006 年；Won 等人，2020 年；Yin 等人，2020 年；Yoo 等人，2018 年；Zhang 等人，2019a；Zhao 等人，2020 年）。在此，最近的一些研究表明，需要一个中间层将薄膜层与顺应性弹性基底层粘合在一起，这不仅可以增强双层基底的粘合力（Tran 等人，2019 年），还可以提高三层结构的强度和稳健性（Cheng 等人，2014 年）。另一方面，由于中间层的引入，薄膜/基底结构的形态演变可能无法预测，可能会出现一些意想不到的现象，如过早起皱（Béfahy 等人，2010 年；Lejeune 等人，2016b）。特别是当基底厚度很大时，双层顺应基底会对基于薄膜基底的可拉伸电子器件的性能产生重要影响（Wang 等人，2020）。因此，为了更好地设计这些器件，了解三层结构的弹性至关重要。

For a thin film-intermediate layer-soft substrate structure, many researchers have studied its wrinkling morphology. By means of a theoretical analysis and numerical simulations, Jia et al. (2012) investigated the wrinkling of a bilayer film resting on an infinite compliant substrate, and their results showed that by modulating the Young’s modulus of the intermediate layer, the surface wrinkling of the composite structure would be transformed from one buckling regime (wrinkling of a bilayer as a whole on a homogeneous substrate) to another (wrinkling of the top single layer resting on a composite substrate), which is a novel approach to realise surface pattern switching. Considering the effect of the intermediate layer on the performance of the stretchable electronics, and based on the energy method, Cheng et al. (2014) established an analytical model for the stiff thin film on a bi-layer compliant substrate structure, and they studied its buckling and post-buckling behaviours. From their results, one can find that with an intermediate layer on the top of the compliant elastic substrate, the characteristics of the stretchable electronics are more robust and stronger. Accounting for the interfacial layer between the film and the substrate, Lejeune et al., 2016a, Lejeune et al., 2016b created a novel model to capture surface instability of the multi-layer structure, and their analytical solutions and numerical results provided an improved understanding of the influence of the multiple interfacial layers on the global behaviour of that structure.
对于薄膜-中间层-软衬底结构，许多研究人员都对其起皱形态进行了研究。Jia 等人（2012）通过理论分析和数值模拟研究了双层薄膜在无限顺应基底上的起皱问题，结果表明，通过调节中间层的杨氏模量，复合结构的表面起皱将从一种屈曲机制（双层整体在均匀基底上的起皱）转变为另一种屈曲机制（顶部单层在复合基底上的起皱），这是实现表面形态切换的一种新方法。考虑到中间层对可拉伸电子器件性能的影响，基于能量法，Cheng 等人（2014）建立了双层顺应基底结构上刚性薄膜的分析模型，并对其屈曲和屈曲后行为进行了研究。从他们的研究结果中可以发现，在顺应弹性基底的顶部加上中间层，可拉伸电子器件的特性会更加坚固和强大。考虑到薄膜与基底之间的界面层，Lejeune 等人，2016a；Lejeune 等人，2016b 创建了一个新模型来捕捉多层结构的表面不稳定性，他们的分析解和数值结果让人们更好地理解了多界面层对该结构全局行为的影响。

Based on the above literature review, one can find that those investigations treated the substrate as a semi-infinite space. There are few works in the open literature which discuss the effect of the finite thickness of a substrate on the wrinkling shape of the tri-layer structure. Recently, Wang et al. (2020) studied the buckling behaviour of the stiff thin film on a bi-layer substrate with a finite thickness. Based on the energy method, they established its theoretical model and discussed the influences of the finite thickness and the Young’s modulus of the bi-layer substrate on buckling wavelength and critical buckling strain. Their results showed that the bi-layer substrate would degenerate to a single layer substrate when the intermediate layer is very thin or very thick. However, as found in Wang et al. (2008), for a relatively thin substrate, global buckling could be observed in experiments, as opposed to local buckling in a thin film on an elastomeric substrate. Li et al. (2019a) established an analytical model for a stiff film bonded to the freestanding single-layer finite-thickness substrate, and they investigated the buckling behaviour of this structure. Their results showed that the local wrinkling and global buckling modes of that structure could be determined in terms of the bending stiffness ratio of the stiff film to the substrate and the ratio of the substrate thickness to the stiff film length.
根据上述文献综述，我们可以发现这些研究将基底视为半无限空间。在公开文献中，讨论基底有限厚度对三层结构皱褶形状影响的著作很少。最近，Wang 等人（2020）研究了有限厚度双层基底上刚性薄膜的屈曲行为。他们基于能量法建立了理论模型，并讨论了双层基底的有限厚度和杨氏模量对屈曲波长和临界屈曲应变的影响。他们的研究结果表明，当中间层很薄或很厚时，双层基底会退化为单层基底。然而，正如 Wang 等人（2008 年）所发现的，对于相对较薄的基底，在实验中可以观察到全局屈曲，而不是弹性基底上薄膜的局部屈曲。Li 等人（2019a）为粘结在独立单层有限厚度基底上的刚性薄膜建立了一个分析模型，并研究了这种结构的屈曲行为。他们的研究结果表明，该结构的局部起皱和整体屈曲模式可根据刚性薄膜与基材的弯曲刚度比以及基材厚度与刚性薄膜长度之比来确定。

From the aforementioned review, one can find that there is no study on the buckling behaviour (especially for distinguishing the buckling regimes between local wrinkling and global buckling) of the tri-layer structure (a film layer on an intermediate layer of finite thickness on a soft substrate), which considers the shear stress between the film and the intermediate layer and the complete form of longitudinal displacement of the film. Hence, this manuscript aims to focus on this topic, and predict the buckling shape of local wrinkling. The outline of this manuscript is organized as the following: in Section 2, taking the shear stress between the film and the intermediate layer and the complete mathematical form of the longitudinal displacement of the film into account, a tri-layer model is established; in Section 3, to distinguish the buckling regimes between local wrinkling and global buckling of this improved tri-layer structure, its buckling behaviour is analysed; the influence of the intermediate layer on its wrinkling instability are discussed in Section 4, and the main conclusions are summarized in Section 5.
从上述综述中可以发现，目前还没有关于三层结构（软基底上的有限厚度中间层上的薄膜层）屈曲行为（尤其是区分局部起皱和整体屈曲的屈曲状态）的研究，其中没有考虑薄膜和中间层之间的剪应力以及薄膜纵向位移的完整形式。因此，本手稿旨在关注这一主题，并预测局部起皱的屈曲形状。本手稿的大纲安排如下：第 2 部分考虑了薄膜与中间层之间的剪应力和薄膜纵向位移的完整数学形式，建立了一个三层模型；第 3 部分为了区分这种改进的三层结构的局部起皱和整体起皱的屈曲状态，分析了其屈曲行为；第 4 部分讨论了中间层对其起皱不稳定性的影响，第 5 部分总结了主要结论。

2. Theoretical modelling 2.理论建模

A theoretical model to characterize the buckling behaviour of the tri-layer structure, which consists of a stiff thin film resting on a finite compliant substrate with a finite intermediate layer, is established. Fig. 1 shows a tensile pre-strain strategy, which is used to realise the fabrication of the stiff thin film on a finite-thickness bi-layer compliant substrate. The thin film is bonded to a flat, pre-strained bi-layer soft substrate (Fig. 1a). Then, upon the releasing of the pre-strain, the bi-layer substrate shrinks, which leads to compression in the film to generate two different buckling regimes (Fig. 1b and c), as observed in experiments (Wang et al., 2008, Li et al., 2019a). A Cartesian coordinate system is chosen and the origin

O

is located at the interface between the bottom of the intermediate layer and the top of the soft substrate. The thicknesses of the film, the intermediate layer, and the substrate are represented by

h_{f}

h_{in}

and

h_{s}

, respectively;

b

represents the film width, and the final length of the structure is

L

. Here, one needs to assume that all the interfaces between any two layers are perfectly attached. To better predict the buckling shape of the tri-layer structure and get more accurate solutions of buckling wavelength and critical strain that causes local wrinkling, the shear stress at the interface of the film and the intermediate layer, and the complete form of longitudinal displacement of the film are taken into account.
我们建立了一个理论模型来描述三层结构的屈曲行为，该结构由刚性薄膜和有限中间层组成。图 1 显示了一种拉伸预应变策略，用于实现在有限厚度的双层顺应基底上制造刚性薄膜。薄膜被粘合到平整的预拉伸双层软基底上（图 1a）。然后，在释放预应变时，双层基底收缩，导致薄膜压缩，产生两种不同的屈曲状态（图 1b 和 c），正如实验中所观察到的（Wang 等人，2008 年；Li 等人，2019a）。选择笛卡尔坐标系，原点

O

位于中间层底部和软基底顶部之间的界面。薄膜、中间层和基底的厚度分别用

h_{f}

h_{in}

和

h_{s}

表示；

b

表示薄膜宽度，结构的最终长度为

L

。在此，我们需要假设任意两层之间的所有界面都是完全连接的。为了更好地预测三层结构的屈曲形状，得到更精确的屈曲波长和导致局部起皱的临界应变的解，需要考虑薄膜和中间层界面处的剪应力以及薄膜纵向位移的完整形式。

2.1. Equilibrium of the thin stiff film
2.1.刚性薄膜的平衡

To derive the equilibrium equation of the thin stiff film, the film is treated as an Euler-Bernoulli beam. The longitudinal strain at the neutral axis of the film

ε_{x}

is defined as (Huang et al., 2005),
为了推导刚性薄膜的平衡方程，薄膜被视为欧拉-伯努利梁。薄膜中性轴上的纵向应变

ε_{x}

定义为（Huang 等人，2005 年）、(1)

ε_{x} = \frac{\partial u^{f}}{\partial x} + \frac{1}{2} {(\frac{\partial w^{f}}{\partial x})}^{2} - ε,

where

ε

is the in-plane compressive strain due to the relaxation of the pre-stretched bi-layer substrate,

u^{f}

is the longitudinal displacement at the neutral axis upon release of the pre-strain, and

w^{f} = A cos (kx)

is the transverse deflection (

A

and

k

are the buckling amplitude and wavenumber,

λ = 2 π / k

is buckling wavelength, to be determined later).
其中，

ε

是由于预拉伸双层基板松弛而产生的面内压应变，

u^{f}

是释放预应变时中性轴的纵向位移、和

w^{f} = A cos (kx)

是横向挠度（

A

和

k

是屈曲振幅和波数，

λ = 2 π / k

是屈曲波长，稍后确定）。

The normal stress

p

and shear stress

τ_{x}

at the interface of the film and the intermediate layer can be expressed as (Huang et al., 2005),
薄膜和中间层界面上的法向应力

p

和剪切应力

τ_{x}

可表示为（Huang 等人，2005 年）、(2)

p = \frac{{\bar{E}}_{f} h_{f}^{3}}{12} \frac{\partial^{4} w^{f}}{\partial x^{4}} - ε_{x} {\bar{E}}_{f} h_{f} \frac{\partial^{2} w^{f}}{\partial x^{2}},

(3)

τ_{x} = {\bar{E}}_{f} h_{f} (\frac{\partial^{2} u^{f}}{\partial x^{2}} + \frac{\partial w^{f}}{\partial x} \frac{\partial^{2} w^{f}}{\partial x^{2}}),

where

{\bar{E}}_{f} = E_{f} / (1 - ν_{f}^{2})

denotes the effective Young’s modulus of the film.

E_{f}

and

ν_{f}

are Young’s modulus and Poisson’s ratio of the film. In previous studies, the shear stress at the interface between the film and the intermediate layer is either ignored (for example, in Cheng et al. (2014)) or assumed to be

τ_{x} = τ_{1} sin (kx)

(for example, in Wang et al. (2020)) that would be incomplete. In order to capture the buckling behaviour of tri-layer structure more accurately, considering the complete form of longitudinal displacement of the film, the normal stress and shear stress acting on the interface can be assumed as the sum of the following harmonic functions,
其中

{\bar{E}}_{f} = E_{f} / (1 - ν_{f}^{2})

表示薄膜的有效杨氏模量。

E_{f}

和

ν_{f}

表示薄膜的杨氏模量和泊松比。在以前的研究中，薄膜与中间层之间界面的剪应力要么被忽略（例如 Cheng 等人（2014 年）），要么被假定为

τ_{x} = τ_{1} sin (kx)

（例如 Wang 等人（2020 年）），这是不完整的。为了更准确地捕捉三层结构的屈曲行为，考虑到薄膜纵向位移的完整形式，可将作用在界面上的法向应力和剪切应力假定为以下谐函数之和、(4)

p = p_{1} c o s (kx) + p_{2} cos (2 k x),

(5)

τ_{x} = τ_{1} sin (kx) + τ_{2} sin (2 k x),

where unknown constants

p_{1}

p_{2}

τ_{1}

and

τ_{2}

will be determined later.
其中未知常数

p_{1}

p_{2}

τ_{1}

和

τ_{2}

将在后面确定。

2.2. Equilibrium of the intermediate layer and the substrate
2.2.中间层和基底的平衡

To obtain the solutions of the plane strain problem for the intermediate layer and the soft substrate, the Airy stress function is adopted. The equilibrium equations for stresses are known as,
为了获得中间层和软基体平面应变问题的解，采用了 Airy 应力函数。应力平衡方程为(6)

\frac{\partial σ_{x}}{\partial x} + \frac{\partial τ_{xz}}{\partial z} = 0, \frac{\partial τ_{xz}}{\partial x} + \frac{\partial σ_{z}}{\partial z} = 0,

where

σ_{x} = \partial^{2} F / \partial z^{2}

σ_{z} = \partial^{2} F / \partial x^{2}

and

τ_{xz} = - \partial^{2} F / \partial x \partial z

are the stresses in terms of the Airy stress function

F = F (x, z)

, which satisfies (Wang et al., 2020),
其中，

σ_{x} = \partial^{2} F / \partial z^{2}

、

σ_{z} = \partial^{2} F / \partial x^{2}

和

τ_{xz} = - \partial^{2} F / \partial x \partial z

是艾里应力函数

F = F (x, z)

的应力，满足（Wang 等，2020）、(7)

\frac{\partial^{4} F}{\partial x^{4}} + \frac{\partial^{4} F}{\partial x^{2} \partial z^{2}} + \frac{\partial^{4} F}{\partial z^{4}} = \nabla^{4} F = 0,

where

\nabla

is the gradient operator. As the transverse deflection of the film is defined as

w^{f} = A cos (kx)

when buckled, and considering the displacement continuity conditions, the Airy stress function of the intermediate layer could be assumed as,
其中

\nabla

为梯度算子。由于薄膜屈曲时的横向挠度定义为

w^{f} = A cos (kx)

，考虑到位移连续性条件，中间层的 Airy 应力函数可假定为(8)

\begin{matrix} F^{in} (x, z) = \sum_{n = 1}^{2} F_{n}^{in} (z) cos (nkx), \\ F_{n}^{in} (z) = C_{1}^{(n)} e^{nkz} + C_{2}^{(n)} z e^{nkz} + C_{3}^{(n)} e^{- n k z} + C_{4}^{(n)} z e^{- n k z}, \end{matrix}

where

C_{1}^{(n)}, C_{2}^{(n)}, C_{3}^{(n)}

and

C_{4}^{(n)}

are coefficients to be determined by the boundary conditions of the intermediate layer in Section 2.3.
其中

C_{1}^{(n)}, C_{2}^{(n)}, C_{3}^{(n)}

和

C_{4}^{(n)}

为系数，由第 2.3 节中中间层的边界条件决定。

By using Eq. (8), the stresses in Eq. (7) can be re-expressed as,
利用公式 (8)，公式 (7) 中的应力可重新表达为(9)

σ_{x} (x, z) = \sum_{n = 1}^{2} n k [\begin{matrix} (nk C_{1}^{(n)} + 2 C_{2}^{(n)}) e^{nkz} + n k C_{2}^{(n)} z e^{nkz} \\ + (nk C_{3}^{(n)} - 2 C_{4}^{(n)}) e^{- n k z} + n k C_{4}^{(n)} z e^{- n k z} \end{matrix}] cos (nkx),

(10)

σ_{z} (x, z) = \sum_{n = 1}^{2} - n k^{2} [\begin{matrix} C_{1}^{(n)} e^{nkz} + C_{2}^{(n)} z e^{nkz} \\ + C_{3}^{(n)} e^{- n k z} + C_{4}^{(n)} z e^{- n k z} \end{matrix}] cos (nkx),

(11)

τ_{xz} (x, z) = \sum_{n = 1}^{2} n k [\begin{matrix} (nk C_{1}^{(n)} + C_{2}^{(n)}) e^{nkz} + n k C_{2}^{(n)} z e^{nkz} \\ - (nk C_{3}^{(n)} - C_{4}^{(n)}) e^{- n k z} - n k C_{4}^{(n)} z e^{- n k z} \end{matrix}] s i n (nkx) .

Because the film’s transverse deflection is described by

w^{f} = A cos (kx)

, the displacement fields of the intermediate layer can be expressed as,
由于薄膜的横向挠度由

w^{f} = A cos (kx)

描述，因此中间层的位移场可表示为(12)

(u, w) = (\sum_{n = 1}^{2} U_{n} (z) sin (nkx), \sum_{n = 1}^{2} W_{n} (z) cos (nkx)),

where

U_{n} (z)

and

W_{n} (z)

satisfy the following conditions,
其中

U_{n} (z)

和

W_{n} (z)

满足以下条件、(13)

\frac{E}{1 + ν} U_{n} (z) = [\begin{matrix} (nk C_{1}^{(n)} + 2 (1 - ν) C_{2}^{(n)}) e^{nkz} + n k C_{2}^{(n)} z e^{nkz} \\ + (nk C_{3}^{(n)} - 2 (1 - ν) C_{4}^{(n)}) e^{- n k z} + n k C_{4}^{(n)} z e^{- n k z} \end{matrix}],

(14)

- \frac{E}{1 + ν} W_{n} (z) = [\begin{matrix} (nk C_{1}^{(n)} - (1 - 2 ν) C_{2}^{(n)}) e^{nkz} + n k C_{2}^{(n)} z e^{nkz} \\ - (nk C_{3}^{(n)} + (1 - 2 ν) C_{4}^{(n)}) e^{- n k z} - n k C_{4}^{(n)} z e^{- n k z} \end{matrix}] .

The Airy stress function of the substrate could be assumed as,
基底的空气应力函数可假定为(15)

\begin{matrix} F^{s} (x, z) = \sum_{n = 1}^{2} F_{n}^{s} (z) cos (nkx), \\ F_{n}^{s} (z) = C_{5}^{(n)} e^{nkz} + C_{6}^{(n)} z e^{nkz} + C_{7}^{(n)} e^{- n k z} + C_{8}^{(n)} z e^{- n k z}, \end{matrix}

where

C_{5}^{(n)}, C_{6}^{(n)}, C_{7}^{(n)}

and

C_{8}^{(n)}

are coefficients to be determined by the boundary conditions of the substrate in Section 2.3.
其中

C_{5}^{(n)}, C_{6}^{(n)}, C_{7}^{(n)}

和

C_{8}^{(n)}

是由第 2.3 节中基底的边界条件决定的系数。

2.3. The boundary conditions
2.3.边界条件

Considering the shear stress between the film and intermediate layer and the complete form of the longitudinal displacement of the intermediate layer due to the complete form of the longitudinal displacement of the film, the boundary conditions of the intermediate layer are given as,
考虑到薄膜和中间层之间的剪应力以及中间层纵向位移的完整形式是由薄膜纵向位移的完整形式引起的，中间层的边界条件为(16)

\{\begin{matrix} τ_{xz}^{in} = τ_{1} sin (kx) + τ_{2} sin (2 k x), w^{in} = A cos (kx) at z = h_{in} \\ w^{in} = w_{1} cos (kx), u^{in} = u_{1} s i n (kx) + u_{2} s i n (2 k x) at z = 0 \end{matrix},

where

w_{1}

u_{1}

and

u_{2}

are to be determined.
其中

w_{1}

、

u_{1}

和

u_{2}

待定。

Submitting Eqs. (9), (10), (11) and Eqs. (13), (14) into Eq. (16), the following algebraic equations can be obtained,
将公式 (9)、(10)、(11) 和公式 (13)、(14) 代入公式 (16)，可以得到以下代数方程、(17)

[\begin{matrix} - k e^{k h_{in}} & (1 - 2 ν_{in} - k h_{in}) e^{k h_{in}} & k e^{- k h_{in}} & (1 - 2 ν_{in} + k h_{in}) e^{- k h_{in}} \\ k^{2} e^{k h_{in}} & k e^{k h_{in}} (1 + k h_{in}) & - k^{2} e^{- k h_{in}} & k e^{- k h_{in}} (1 - k h_{in}) \\ k & 2 - 2 ν_{in} & k & 2 ν_{in} - 2 \\ - k & 1 - 2 ν_{in} & k & 1 - 2 ν_{in} \end{matrix}] [\begin{matrix} C_{1}^{(1)} \\ C_{2}^{(1)} \\ C_{3}^{(1)} \\ C_{4}^{(1)} \end{matrix}] = [\begin{matrix} \frac{E_{in}}{1 + ν_{in}} A \\ τ_{1} \\ \frac{E_{in}}{1 + ν_{in}} u_{1} \\ \frac{E_{in}}{1 + ν_{in}} w_{1} \end{matrix}],

(18)

[\begin{matrix} - 2 k e^{2 k h_{in}} & (1 - 2 ν_{in} - 2 k h_{in}) e^{2 k h_{in}} & 2 k e^{- 2 k h_{in}} & (1 - 2 ν_{in} + 2 k h_{in}) e^{- 2 k h_{in}} \\ 4 k^{2} e^{2 k h_{in}} & 2 k e^{2 k h_{in}} (1 + 2 k h_{in}) & - 4 k^{2} e^{- 2 k h_{in}} & 2 k e^{- 2 k h_{in}} (1 - 2 k h_{in}) \\ 2 k & 2 - 2 ν_{in} & 2 k & 2 ν_{in} - 2 \\ - 2 k & 1 - 2 ν_{in} & 2 k & 1 - 2 ν_{in} \end{matrix}] [\begin{matrix} C_{1}^{(2)} \\ C_{2}^{(2)} \\ C_{3}^{(2)} \\ C_{4}^{(2)} \end{matrix}] = [\begin{matrix} 0 \\ τ_{2} \\ \frac{E_{in}}{1 + ν_{in}} u_{2} \\ 0 \end{matrix}] .

Solving Eqs. (17), (18), the coefficients

C_{1}^{(n)}, C_{2}^{(n)}, C_{3}^{(n)}

and

C_{4}^{(n)}

can be obtained in terms of

A

w_{1}

u_{1}

u_{2}

τ_{1}

and

τ_{2}

. The normal and shear stresses of the intermediate layer at the interface

z = 0

can be expressed as, respectively,
求解公式(17), (18)，系数

C_{1}^{(n)}, C_{2}^{(n)}, C_{3}^{(n)}

和

C_{4}^{(n)}

可以通过

A

得到、

w_{1}

、

u_{1}

、

u_{2}

、

τ_{1}

和

τ_{2}

。界面

z = 0

处中间层的法向应力和剪切应力可分别表示为(19)

\begin{matrix} σ_{z}^{in} |_{z = 0} = - k^{2} (C_{1}^{(1)} + C_{3}^{(1)}) cos (kx) - 4 k^{2} (C_{1}^{(2)} + C_{3}^{(2)}) cos (2 k x) = [k (c_{11}^{(1)} u_{1} + c_{12}^{(1)} w_{1} + c_{13}^{(1)} A) + c_{14}^{(1)} τ_{1}] cos (kx) + (2 k c_{11}^{(2)} u_{2} + c_{14}^{(2)} τ_{2}) cos (2 k x) \\ τ_{xz}^{in} |_{z = 0} = k (k C_{1}^{(1)} + C_{2}^{(1)} - k C_{3}^{(1)} + C_{4}^{(1)}) s i n (kx) + 2 k (2 k C_{1}^{(2)} + C_{2}^{(2)} - 2 k C_{3}^{(2)} + C_{4}^{(2)}) s i n (2 k x) = [k (c_{21}^{(1)} u_{1} + c_{22}^{(1)} w_{1} + c_{23}^{(1)} A) + c_{24}^{(1)} τ_{1}] s i n (kx) + (2 k c_{21}^{(2)} u_{2} + c_{24}^{(2)} τ_{2}) s i n (2 k x) \end{matrix}

where

c_{ij}^{(k)} (i = 1, 2, j = 1, \dots, 4, k = 1, 2)

are given in the Appendix Eq. (A1).
其中

c_{ij}^{(k)} (i = 1, 2, j = 1, \dots, 4, k = 1, 2)

在附录公式 (A1) 中给出。

The boundary conditions of the compliant substrate are given as,
顺应基底的边界条件为(20)

\{\begin{matrix} w^{s} = w_{1} cos (kx), u^{s} = u_{1} s i n (kx) + u_{2} s i n (2 k x) at z = 0 \\ σ_{z}^{s} = 0, τ_{xz}^{s} = 0 at z = - h_{s} \end{matrix} .

Following the similar procedure for the intermediate layer, the normal and shear stresses at the interface of the intermediate layer and the substrate

z = 0

for the substrate can be expressed as,
按照中间层的类似步骤，中间层和基底

z = 0

界面上的法向应力和剪切应力可表示为：

z = 0

、(21)

\begin{matrix} σ_{z}^{s} |_{z = 0} = k (c_{31}^{(1)} u_{1} + c_{32}^{(1)} w_{1}) c o s (kx) + 2 k c_{31}^{(2)} u_{2} c o s (2 k x) \\ τ_{xz}^{s} |_{z = 0} = k (c_{41}^{(1)} u_{1} + c_{42}^{(1)} w_{1}) s i n (kx) + 2 k c_{41}^{(2)} u_{2} s i n (2 k x) \end{matrix}

where

c_{ij}^{(k)} (i = 3, 4, j = 1, 2, k = 1, 2)

are given in the Appendix Eq. (A2).
其中

c_{ij}^{(k)} (i = 3, 4, j = 1, 2, k = 1, 2)

在附录公式 (A2) 中给出。

The equilibrium conditions at the interface between the intermediate layer and the soft substrate require,
中间层与软基质界面的平衡条件要求：(22)

σ_{z}^{in} |_{z = 0} = σ_{z}^{s} |_{z = 0}, τ_{xz}^{in} {|_{z = 0} = τ_{xz}^{s} |}_{z = 0},

which leads to 这导致(23)

u_{1} = \frac{1}{k} (γ_{11}^{(1)} k A + γ_{12}^{(1)} τ_{1}), w_{1} = \frac{1}{k} (γ_{21}^{(1)} k A + γ_{22}^{(1)} τ_{1}), u_{2} = \frac{1}{2 k} γ_{12}^{(2)} τ_{2} .

where

γ_{11}^{(1)}, γ_{12}^{(1)}, γ_{21}^{(1)}, γ_{22}^{(1)},

and

γ_{12}^{(2)}

are given in the Appendix Eq. (A3).
其中

γ_{11}^{(1)}, γ_{12}^{(1)}, γ_{21}^{(1)}, γ_{22}^{(1)},

和

γ_{12}^{(2)}

在附录公式 (A3) 中给出。

The displacements and normal stresses at the top of the intermediate layer (

z = h_{in}

) can be obtained as,
中间层顶部（

z = h_{in}

）的位移和法向应力可按以下公式求得、(24)

u^{in} |_{z = h_{in}} = [c_{51}^{(1)} u_{1} + c_{52}^{(1)} w_{1} + c_{53}^{(1)} A + \frac{1}{k} c_{54}^{(1)} τ_{1}] s i n (kx) + [c_{51}^{(2)} u_{2} + \frac{1}{2 k} c_{54}^{(2)} τ_{2}] s i n (2 k x),

(25)

σ_{z}^{in} |_{z = h_{in}} = [k (c_{61}^{(1)} u_{1} + c_{62}^{(1)} w_{1} + c_{63}^{(1)} A) + c_{64}^{(1)} τ_{1}] cos (kx) + (2 k c_{61}^{(2)} u_{2} + c_{64}^{(2)} τ_{2}) cos (2 k x),

where

c_{ij}^{(k)} (i = 5, 6, j = 1, \dots, 4, k = 1, 2)

are given in the Appendix Eq. (A4).
其中

c_{ij}^{(k)} (i = 5, 6, j = 1, \dots, 4, k = 1, 2)

在附录公式 (A4) 中给出。

The longitudinal displacement at the neutral axis is,
中轴线上的纵向位移为(26)

\begin{matrix} u^{f} = & u^{in} |_{z = h_{in}} - \frac{h_{f}}{2} \frac{\partial w^{f}}{\partial x} \\ = & [c_{51}^{(1)} u_{1} + c_{52}^{(1)} w_{1} + c_{53}^{(1)} A + \frac{1}{k} c_{54}^{(1)} τ_{1} + \frac{h_{f}}{2} A k] s i n (kx) + [c_{51}^{(2)} u_{2} + \frac{1}{2 k} c_{54}^{(2)} τ_{2}] s i n (2 k x) . \end{matrix}

One needs to mention that the forms of the longitudinal displacement of the film

u^{f}

in previous works by other researchers were either

sin (kx)

(Wang et al., 2020) or

sin (2 k x)

(Huang et al., 2005). However, the analytical form of

u^{f}

in this work is complete.
需要提及的是，在其他研究者之前的工作中，薄膜纵向位移

u^{f}

的形式要么是

sin (kx)

（Wang 等人，2020 年），要么是

sin (2 k x)

（Huang 等人，2005 年）。然而，在这项工作中，

u^{f}

的分析形式是完整的。

According to the equilibrium conditions in Eqs. (3), (16) at the interface between the thin film and the intermediate layer, the shear stress can be calculated as,
根据公式 (3)、(16) 中薄膜与中间层界面的平衡条件，剪应力可计算为(27)

τ_{x} = {\bar{E}}_{f} h_{f} (\frac{\partial^{2} u^{f}}{\partial x^{2}} + \frac{\partial w^{f}}{\partial x} \frac{\partial^{2} w^{f}}{\partial x^{2}}) = τ_{1} sin (kx) + τ_{2} sin (2 k x) .

Substituting

w^{f} = A cos (kx)

, Eqs. (23), (26) into Eq. (27), the relations between

τ_{1}

τ_{2}

and

A

can be expressed as follows,
将

w^{f} = A cos (kx)

、公式 (23) 和 (26) 代入公式 (27)，

τ_{1}

、

τ_{2}

和

A

之间的关系可表示如下、(28)

\begin{matrix} τ_{1} = - \frac{(c_{51}^{(1)} γ_{11}^{(1)} + c_{52}^{(1)} γ_{21}^{(1)} + c_{53}^{(1)} + \frac{h_{f}}{2} k)}{(c_{51}^{(1)} γ_{12}^{(1)} + c_{52}^{(1)} γ_{22}^{(1)} + c_{54}^{(1)} + \frac{1}{{\bar{E}}_{f} k h_{f}})} k A = - ζ_{1} k A, \\ τ_{2} = \frac{1}{4} \frac{1}{(c_{51}^{(2)} γ_{12}^{(2)} + c_{54}^{(2)} + \frac{1}{2 {\bar{E}}_{f} k h_{f}})} k^{2} A^{2} = ζ_{2} k^{2} A^{2} . \end{matrix}

Thus, the longitudinal displacement at the neutral axis of the film can be re-written as,
因此，薄膜中性轴上的纵向位移可以改写为(29)

\begin{matrix} u^{f} = & \frac{ζ_{1}}{{\bar{E}}_{f} h_{f} k} A s i n (kx) + \frac{1}{2} [c_{51}^{(2)} γ_{12}^{(2)} + c_{54}^{(2)}] ζ_{2} k A^{2} s i n (2 k x) \\ = & γ_{1} A s i n (kx) + γ_{2} k A^{2} s i n (2 k x) . \end{matrix}

3. Energy method and buckling regime analysis
3.能量法和屈曲机制分析

To determine the buckling regimes of the tri-layer structure, the energy method is utilized. The total energy (per unit length) of the tri-layer structure consists of the bending energy

U_{b}

and membrane energy

U_{m}

of the film per unit length and average strain energy

U_{s}

of the intermediate layer and the soft substrate. Based on the above-derived expressions of the displacements, normal and shear stresses in 2.1 Equilibrium of the thin stiff film, 2.2 Equilibrium of the intermediate layer and the substrate, 2.3 The boundary conditions, those energies can be calculated as, respectively,
为了确定三层结构的屈曲状态，我们采用了能量法。三层结构的总能量（单位长度）由薄膜单位长度的弯曲能

U_{b}

和膜能

U_{m}

以及中间层和软基板的平均应变能

U_{s}

组成。根据上述 2.1 硬薄膜的平衡、2.2 中间层和基底的平衡、2.3 边界条件中位移、法向应力和剪应力的表达式，可分别计算出这些能量、(30)

U_{b} = \frac{k}{2 π} \int_{0}^{2 π / k} \frac{{\bar{E}}_{f} h_{f}^{3} b}{24} {(\frac{\partial^{2} w^{f}}{\partial x^{2}})}^{2} d x = \frac{{\bar{E}}_{f} h_{f}^{3} b}{48} k^{4} A^{2},

(31)

\begin{matrix} U_{m} = & \frac{1}{2} h_{f} b \frac{k}{2 π} \int_{0}^{2 π / k} σ_{x} ε_{x} d x \\ = & \frac{1}{2} {\bar{E}}_{f} h_{f} b [\frac{1}{16} A^{4} k^{4} + \frac{1}{2} γ_{1}^{2} A^{2} k^{2} + \frac{1}{2} {(2 γ_{2} - \frac{1}{4})}^{2} k^{4} A^{4} - \frac{1}{2} A^{2} k^{2} ε + ε^{2}], \end{matrix}

(32)

\begin{matrix} U_{s} = & \frac{1}{2} \frac{k}{2 π} b \int_{0}^{2 π / k} (σ_{z}^{in} w^{in}) |_{z = h_{in}} d x + \frac{1}{2} \frac{k}{2 π} b \int_{0}^{2 π / k} (τ_{xz}^{in} u^{in}) |_{z = h_{in}} d x \\ = & \frac{k A^{2}}{4} [\begin{matrix} (c_{61}^{(1)} γ_{11}^{(1)} + c_{62}^{(1)} γ_{21}^{(1)} + c_{63}^{(1)}) - \\ (c_{51}^{(1)} γ_{11}^{(1)} + c_{52}^{(1)} γ_{21}^{(1)} + c_{53}^{(1)} + c_{61}^{(1)} γ_{12}^{(1)} + c_{62}^{(1)} γ_{22}^{(1)} + c_{64}^{(1)}) ζ_{1} \\ + (c_{51}^{(1)} γ_{12}^{(1)} + c_{52}^{(1)} γ_{22}^{(1)} + c_{54}^{(1)}) ζ_{1}^{2} \end{matrix}] + \frac{(c_{51}^{(2)} γ_{12}^{(2)} + c_{54}^{(2)}) ζ_{2}^{2} k^{4} A^{4}}{8 k}, \end{matrix}

Thus, the total energy of the buckled tri-layer structure is given as,
因此，屈曲三层结构的总能量为(33)

\begin{matrix} U_{total} = & U_{m} + U_{b} + U_{s} \\ = & \frac{{\bar{E}}_{f} h_{f} b ε^{2}}{2} + [\frac{1}{32} + \frac{(c_{51}^{(2)} γ_{12}^{(2)} + c_{54}^{(2)})}{8 {\bar{E}}_{f} k h_{f}} ζ_{2}^{2} + \frac{{(2 γ_{2} - \frac{1}{4})}^{2}}{4}] {\bar{E}}_{f} h_{f} b k^{4} A^{4} + \frac{{\bar{E}}_{f} h_{f} b}{4} (f - ε) k^{2} A^{2}, \end{matrix}

where 其中

f = \frac{h_{f}^{2} k^{2}}{12} + γ_{1}^{2} + \frac{1}{{\bar{E}}_{f} h_{f} k} [\begin{matrix} (c_{61}^{(1)} γ_{11}^{(1)} + c_{62}^{(1)} γ_{21}^{(1)} + c_{63}^{(1)}) \\ - (c_{51}^{(1)} γ_{11}^{(1)} + c_{52}^{(1)} γ_{21}^{(1)} + c_{53}^{(1)} + c_{61}^{(1)} γ_{12}^{(1)} + c_{62}^{(1)} γ_{22}^{(1)} + c_{64}^{(1)}) ζ_{1} \\ + (c_{51}^{(1)} γ_{12}^{(1)} + c_{52}^{(1)} γ_{22}^{(1)} + c_{54}^{(1)}) ζ_{1}^{2} \end{matrix}] .

From Eq. (33), it is easy to see that when

f ⩾ ε

, the total energy becomes minimal at

A = 0

, implying that the stiff thin film does not buckle and remains flat at this condition. When

f < ε

, on the other hand, the tri-layer structure would buckle at,
从公式 (33) 中不难看出，当

f ⩾ ε

时，总能量在

A = 0

处变得最小，这意味着在此条件下，坚硬的薄膜不会发生弯曲并保持平整。反之，当

f < ε

时，三层结构会在以下位置发生屈曲、(34)

A = \frac{2}{k} \sqrt{\frac{f}{1 + g} (\frac{ε}{f} - 1)},

where 其中

g= \frac{4}{{\bar{E}}_{f} k h_{f}} (c_{51}^{(2)} γ_{12}^{(2)} + c_{54}^{(2)}) ζ_{2}^{2} + 8 {(2 γ_{2} - \frac{1}{4})}^{2},

Submitting Eq. (34) into Eq. (33), the total energy could be re-expressed as a function of wavenumber

k

,
将公式 (34) 代入公式 (33)，总能量可以重新表达为波长

k

的函数、(35)

U_{total} = \frac{1}{2} {\bar{E}}_{f} h_{f} b ε^{2} - \frac{1}{2 (1 + g)} {\bar{E}}_{f} h_{f} b {(ε - f)}^{2} .

As stated in Huang et al. (2005), the wavenumber

k

of the tri-layer structure also minimises the total energy, and one can find that the functions

f

and

g

of the total energy are dependent on the wavenumber

k

from Eq. (35). To distinguish the buckling regimes of the structure,

f

and

g

as functions of

k h_{f}

are plotted in Fig. 2, Fig. 3, respectively. The parameters are chosen as:

ν_{f} = 0.27

ν_{in} = 0.48

ν_{s} = 0.48

E_{in} = 0 . 5MPa

E_{s} / E_{in} = 5

E_{f} / E_{in} = 400, 000

ε = 2\%

(Wang et al. 2020).
正如 Huang 等人 (2005) 所述，三层结构的波长

k

也能使总能量最小，并且从式 (35) 中可以发现总能量的函数

f

和

g

与波长

k

有关。为了区分结构的屈曲状态，图 2 和图 3 分别绘制了

f

和

g

与

k h_{f}

的函数关系。参数选取为

ν_{f} = 0.27

ν_{in} = 0.48

ν_{s} = 0.48

E_{in} = 0 . 5MPa

E_{s} / E_{in} = 5

E_{f} / E_{in} = 400, 000

ε = 2\%

(Wang et al. 2020)。

From the results in Fig. 2, it can be noted that when

k h_{f}

increase from

0

0.00 5

, the function

g

is monotonically decreasing and positive. When

k h_{f} > 0.00 5

, the value of the function

g

is 100 times smaller than 1 and can be neglected in Eq. (35). Thus, the expression of the total energy Eq. (35) of the tri-layer structure could be re-expressed as,
从图 2 的结果可以看出，当

k h_{f}

从

0

增加到

0.00 5

时，函数

g

是单调递减且为正值的。当

k h_{f} > 0.00 5

时，函数

g

的值比 1 小 100 倍，在式 (35) 中可以忽略。因此，三层结构的总能量表达式公式 (35) 可以重新表达为(36)

U_{total} = \frac{1}{2} {\bar{E}}_{f} h_{f} b ε^{2} - \frac{1}{2} {\bar{E}}_{f} h_{f} b {(ε - f)}^{2} .

From Eq. (36), it is clear that the total energy is dependent on

f (k h_{f})

, and it is a quadratic function of

f

. Apparently, when

f < ε

, the total energy in Eq. (36) is minimised with respect to the wavenumber

k

, which determines the buckling regimes of the tri-layer structure (Li et al., 2019a).
从公式 (36) 可以看出，总能量取决于

f (k h_{f})

，并且是

f

的二次函数。显然，当

f < ε

时，式 (36) 中的总能量与波长数

k

有关，是最小的，这决定了三层结构的屈曲状态（Li 等人，2019a）。

To distinguish the buckling regimes of the tri-layer structure and evaluate the influences of the thicknesses of the film, the intermediate layer, and the substrate layer on buckling regimes, Fig. 3 is plotted.
为了区分三层结构的屈曲状态，并评估薄膜、中间层和基底层的厚度对屈曲状态的影响，绘制了图 3。

Fig. 3a shows the effect of the thickness ratio

h_{in} / h_{f}

of the intermediate layer to the film on buckling regimes, where the thickness ratio

h_{s} / h_{f}

of the substrate layer to the film is set as 1. From Fig. 3a, it is easy to see that when the value of the ratio

h_{in} / h_{f}

is smaller than 182 (numerically obtained), the function

f

is a monotonically increasing function (blue line with solid circles) of

k h_{f}

(line with solid circles), which implies that when

k = 1

and

λ = L

f

is minimized. In other words, the buckling regime of the tri-layer structure is only global buckling (as Fig. 1b).
图 3a 显示了中间层与薄膜的厚度比

h_{in} / h_{f}

对屈曲机制的影响，其中基底层与薄膜的厚度比

h_{s} / h_{f}

设为 1。3a 中不难看出，当比率

h_{in} / h_{f}

的值小于 182 时（数值求得）、函数

f

是

k h_{f}

的单调递增函数（蓝色实心圆圈线），这意味着当

k = 1

和

λ = L

时，

f

最小。换句话说，三层结构的屈曲机制只是全局屈曲（如图 1b）。

However, when the thickness ratio

h_{in} / h_{f}

is greater than 182 (numerically obtained), the buckling regime of the tri-layer structure is either the global buckling or the local wrinkling, which could be observed from the result (line with solid squares) in Fig. 3a and be determined by comparing the values of a function

f

(equivalent to comparing the total energy) corresponding to two buckling regimes. The function

f

with respect to

k h_{f}

has a minimum point

P

(red pentagon) corresponding to local wrinkling (line with solid squares) (Li et al., 2019a). Meanwhile, the value of the function

f

of the global buckling is determined by

k h_{f} = 2 π h_{f} / L

. If the function value

f

k h_{f}

corresponding to the local wrinkling is smaller than f at

k h_{f}

corresponding to the global buckling (the yellow region), local wrinkling occurs; otherwise, the buckling regime of the tri-layer structure is global buckling (the orange region). In order to distinguish the two buckling patterns more clearly, the critical length

L_{c}

is given, as seen in Fig. 3. When

L > L_{c}

(green dashed line in the orange region), (i.e.,

2 π h_{f} / L < {2 π h_{f} / L}_{c}

), the value of the function

f

is smaller than its minimum value

f_{min}

for local wrinkling, then global buckling occurs; otherwise, the tri-layer structure exhibits local wrinkling (green solid line with solid square in the yellow region).
然而，当厚度比

h_{in} / h_{f}

大于 182 时（数值求得），三层结构的屈曲机制要么是整体屈曲，要么是局部起皱，这可以从图 3a 中的结果（实心方格线）中观察到，并通过比较两个屈曲机制对应的函数

f

的值（相当于比较总能量）来确定。函数

f

相对于

k h_{f}

有一个最小点

P

（红色五边形），对应于局部起皱（实心方格线）（Li 等人，2019a）。同时，全局屈曲的函数值

f

由

k h_{f} = 2 π h_{f} / L

决定。如果局部起皱对应的

k h_{f}

处的函数值

f

小于全局屈曲对应的

k h_{f}

处的函数值 f（黄色区域），则发生局部起皱；否则，三层结构的屈曲机制为全局屈曲（橙色区域）。为了更清楚地区分这两种屈曲模式，给出了临界长度

L_{c}

，如图 3 所示。当

L > L_{c}

（橙色区域中的绿色虚线），（即

2 π h_{f} / L < {2 π h_{f} / L}_{c}

），函数

f

的值小于局部起皱的最小值

f_{min}

，则发生全局屈曲；否则，三层结构会出现局部起皱（黄色区域中的绿色实线与实心方形）。

In Fig. 3b, the thickness ratio

h_{in} / h_{f}

is taken as 1 and

h_{s} / h_{f}

is the variable. Fig. 3b depicts the effect of variable thickness ratio of the substrate to the film on the transition between global buckling and local wrinkling, and shows the result which is similar to the influence of thickness ratio of the intermediate layer to the film.
在图 3b 中，厚度比

h_{in} / h_{f}

取为 1，

h_{s} / h_{f}

为变量。图 3b 描述了基底与薄膜的厚度比变化对整体屈曲和局部起皱之间过渡的影响，其结果与中间层与薄膜的厚度比的影响相似。

To a further clarification about the local wrinkling and global buckling, and a deeper understanding of the relationship between the critical length

L_{c}

and the ratios

h_{s} / h_{f}

and

h_{in} / h_{f}

, the transition between global buckling and local wrinkling is illustrated in Fig. 4.
为了进一步阐明局部起皱和整体屈曲，并深入理解临界长度

L_{c}

与比率

h_{s} / h_{f}

和

h_{in} / h_{f}

之间的关系，图 4 展示了整体屈曲和局部起皱之间的过渡。

In Fig. 4a, the ratio

h_{s} / h_{f}

is taken as one. From this figure, one can clearly observe that when the ratio

h_{in} / h_{f}

is smaller than 182 (numerically obtained), the buckling regime of the structure is global buckling (the blue region). Once that ratio is greater than 182, (that is to say, the thickness of the intermediate is increased), the buckling regime of the structure is determined by the structure’s length

L

. As presented in Fig. 4a, the thicker the intermediate layer, the smaller value of

2 π h_{f} / L_{c}

, implying that to form the global buckling regime, the longer the structure should be.
图 4a 中，比率

h_{s} / h_{f}

取为 1。从图中可以清楚地看到，当比率

h_{in} / h_{f}

小于 182 时（数值计算结果），结构的屈曲状态为整体屈曲（蓝色区域）。当该比率大于 182 时（即中间层厚度增加），结构的屈曲状态由结构长度

L

决定。如图 4a 所示，中间层越厚，

2 π h_{f} / L_{c}

的值越小，这意味着要形成全局屈曲机制，结构的长度应越长。

In Fig. 4b, the ratio

h_{in} / h_{f}

is taken as one and the ratio

h_{s} / h_{f}

is the variable. When comparing the results in Fig. 4b with those in Fig. 4a, one can find that the phase diagram is much more influenced by the thickness ratio

h_{s} / h_{f}

. When the ratio

h_{s} / h_{f}

is greater than 108 (numerically obtained), to produce global buckling, the thickness of the soft substrate layer needs to be increased.
在图 4b 中，比率

h_{in} / h_{f}

取一，比率

h_{s} / h_{f}

为变量。比较图 4b 和图 4a 的结果，可以发现相图受厚度比

h_{s} / h_{f}

的影响更大。当厚度比

h_{s} / h_{f}

大于 108 时（数值结果），要产生全局屈曲，需要增加软基底层的厚度。

In Fig. 4c, the thickness ratio is set as 500, which is 500 times greater than that in Fig. 4a. From Fig. 4c, it is very interesting to note that the critical length

L_{c}

has been increased by more than ten times. Thus, for a fixed structure’s length, the thicker the intermediate layer, the more easily local wrinkling occurs.
在图 4c 中，厚度比设定为 500，是图 4a 中厚度比的 500 倍。从图 4c 中可以发现，临界长度

L_{c}

增加了 10 倍以上。因此，在结构长度固定的情况下，中间层越厚，越容易发生局部起皱。

A similar phenomenon could be observed in Fig. 4d. As the thickness of the intermediate layer is 500 times greater than that of the stiff film, for the fixed intermediate layer’s thickness, the global buckling mode would easily take place when the length of the tri-layer structure

L

is longer.
图 4d 中也观察到类似现象。由于中间层的厚度是刚性薄膜厚度的 500 倍，在中间层厚度固定的情况下，当三层结构

L

的长度较长时，很容易出现全局屈曲模式。

From these results in Fig. 4, it is clear that the intermediate layer could modulate the buckling behaviour of the tri-layer structure, in the form of global buckling and local wrinkling. Since a wrinkling film tolerates large deformation without failure during static and dynamic loading, local wrinkling can help to maintain excellent electronic properties of the film of brittle inorganic materials. Hence, for stretchable electronic devices, the local wrinkling pattern is the more desirable buckling regime, and buckling wavelength and critical strain are two of the most important properties for a local wrinkling pattern. Hence, the effects of the thickness and Young’s modulus of the intermediate layer on those two parameters are investigated in the next section.
从图 4 中的结果可以清楚地看出，中间层可以以整体屈曲和局部起皱的形式调节三层结构的屈曲行为。由于起皱薄膜在静态和动态加载过程中可承受较大的变形而不会失效，因此局部起皱有助于保持脆性无机材料薄膜的优良电子特性。因此，对于可拉伸电子器件来说，局部起皱模式是更理想的屈曲机制，而屈曲波长和临界应变是局部起皱模式最重要的两个特性。因此，下一节将研究中间层的厚度和杨氏模量对这两个参数的影响。

4. Buckling analysis of local wrinkling
4.局部起皱的屈曲分析

In this section, buckling wavelength and critical strain of local wrinkling are further investigated. From Eq. (34), it is easy to see that critical strain

ε_{c}

for the onset of local wrinkling is

ε_{c} = f_{min}

. It should be noted that the point

P

(red pentagon) in Fig. 3 corresponds to local wrinkling. Hence the dimensionless buckling wavelength

λ / h_{f}

can be obtained from the abscissa of point

P

, and the ordinate of the point

P

is the critical strain

ε_{c} = f_{min}

. In addition, the influences of the thickness, Young’s modulus, and Poisson’s ratio of the intermediate layer on the local wrinkling’s wavelength and critical strain are explored.
本节将进一步研究局部起皱的屈曲波长和临界应变。从公式 (34) 中不难看出，局部起皱的临界应变

ε_{c}

为

ε_{c} = f_{min}

。值得注意的是，图 3 中的

P

（红色五边形）点对应于局部起皱。因此，无量纲屈曲波长

λ / h_{f}

可以从点

P

的abscissa 得到，而点

P

的ordinate 就是临界应变

ε_{c} = f_{min}

。此外，还探讨了中间层的厚度、杨氏模量和泊松比对局部起皱波长和临界应变的影响。

4.1. Verification studies
4.1.验证研究

In order to validate the results in this manuscript, the same parameters are chosen from (Jia et al., 2012) and given in Table 1.
为了验证本手稿中的结果，我们从（贾等人，2012 年）中选取了相同的参数，并在表 1 中给出。

Table 1. The parameters of the tri-layer structure (Jia et al., 2012).
表 1.三层结构的参数（Jia 等人，2012 年）。

$E_{f} (GPa)$	$E_{in} (MPa)$	$E_{s} (GPa)$	$ν_{f}$	$ν_{in}$	$ν_{s}$	$h_{f} (μ m)$	$h_{in} (μ m)$	$h_{s} (μ m)$
$77$	$2$	$130$	$0.42$	$0.49$	$0.28$	$0.5$	$40$	$1000$

In Table 2, the wavelengths of the local wrinkling pattern obtained by the proposed model and a finite element analysis (FEA) are compared with those in Jia et al., 2012, Wang et al., 2020. From Table 2, it is easy to find that the result obtained by the proposed model has good agreement with those published papers. In addition, since the shear stress at the interface between the film and the intermediate layer, and the complete form of longitudinal displacement of the film are considered, the numerical wavelength obtained by the proposed model has better agreement with that obtained by the FEA using ABAQUS.
在表 2 中，将所提出的模型和有限元分析（FEA）得到的局部皱纹波长与 Jia 等人，2012 年，Wang 等人，2020 年的结果进行了比较。从表 2 中不难发现，所提模型得到的结果与这些已发表论文的结果有很好的一致性。此外，由于考虑了薄膜与中间层界面处的剪应力以及薄膜纵向位移的完整形式，因此所提模型得到的数值波长与使用 ABAQUS 进行有限元分析得到的数值波长具有更好的一致性。

Table 2. The wrinkling wavelength for the tri-layer structure.
表 2.三层结构的起皱波长。

Method 方法	Wrinkling wavelength $λ (mm)$ 皱纹波长 $λ (mm)$
Jia et al. (2012) 贾等人（2012）	66.38
Wang et al. (2020) 王等人（2020 年）	66.14
Current theoretical results 目前的理论成果	66.84
Finite element analysis 有限元分析	66.82

As illustrated in Fig. 5, a two-dimensional finite element model is established using the commercial software package ABAQUS to validate the accuracy of the solutions obtained in Section 3. Around 20,000 8-node plane-strain quadrilateral reduced integration elements (CPE8R) for the soft substrate, 50,000 CPE8R for the intermediate layer, and 8000 CPE8R for the stiff thin film are used in modelling the tri-layer structure, respectively. The very small element sizes around the interfaces will ensure the convergence of finite element analysis.
如图 5 所示，使用商业软件 ABAQUS 建立了一个二维有限元模型，以验证第 3 节中获得的解决方案的准确性。在模拟三层结构时，分别使用了约 20,000 个 8 节点平面应变四边形缩小积分元素（CPE8R）用于软基底，50,000 个 CPE8R 用于中间层，8,000 个 CPE8R 用于硬薄膜。界面周围非常小的元素尺寸将确保有限元分析的收敛性。

As reported by Béfahy et al., 2010, Lejeune et al., 2016b, the presence of the intermediate layer would introduce some unforeseen effects, such as premature instability. Hence, in the following subsection, the influences of the intermediate layer’s thickness, Young’s modulus, and Poisson’s ratio on local wrinkling behaviour are discussed.
正如 Béfahy 等人，2010 年，Lejeune 等人，2016b 所报告的那样，中间层的存在会带来一些不可预见的影响，如过早失稳。因此，下文将讨论中间层的厚度、杨氏模量和泊松比对局部起皱行为的影响。

4.2. Influences of thickness of intermediate layer
4.2.中间层厚度的影响

To reveal the influence of the intermediate layer’s thickness on the wrinkling instability of the tri-layer structure, Fig. 6, Fig. 7 are shown, where the results are obtained using these parameters:

E_{f} = 200 GPa,

ν_{f} = 0.27,

h_{f} = 0.5 μ m,

E_{in} = 0.5 MPa,

ν_{in} = 0.48,

E_{s} = 2.5 MPa,

ν_{s} = 0.48,

and

h_{s} = 1000 μ m .

To ensure local wrinkling, a very thick substrate is used. Fig. 6a illustrates the variation in buckling wavelength with

h_{in} / h_{f}

for different models. It can be seen that as the value of the ratio

h_{in} / h_{f}

increases, buckling wavelength increases. In addition, from Fig. 6, one can also note that when

h_{in} / h_{f}

is smaller than 1, buckling wavelength and critical strain are independent of the intermediate layer thickness. On the other hand, when

h_{in} / h_{f}

is greater than 300 (numerically obtained), buckling wavelength and critical strain both reach a plateau, and the effect of the substrate is negligible. This can be easily understood since the intermediate layer can be considered infinitely thick for a thin film when the intermediate layer thickness is over three hundred times the film thickness. In both cases, the results of the present model degenerate to a bi-layer structure, and buckling wavelength and critical strain of the proposed model display great agreement with other published bi-layer models (which treated the soft substrate as an infinite-thickness substrate, for example, in Huang et al. (2005).
为了揭示中间层厚度对三层结构起皱不稳定性的影响，图 6、图 7 显示了使用这些参数得到的结果：

E_{f} = 200 GPa,

ν_{f} = 0.27,

h_{f} = 0.5 μ m,

E_{in} = 0.5 MPa,

ν_{in} = 0.48,

E_{s} = 2.5 MPa,

ν_{s} = 0.48,

和

h_{s} = 1000 μ m .

为了确保局部起皱，使用了很厚的基底。图 6a 显示了不同模型的屈曲波长随

h_{in} / h_{f}

的变化情况。可以看出，随着比值

h_{in} / h_{f}

的增大，屈曲波长也随之增大。此外，从图 6 中还可以看出，当

h_{in} / h_{f}

小于 1 时，屈曲波长和临界应变与中间层厚度无关。另一方面，当

h_{in} / h_{f}

大于 300 时（数值求得），屈曲波长和临界应变都达到一个高点，基底的影响可以忽略不计。这一点很容易理解，因为当中间层厚度是薄膜厚度的三百倍以上时，中间层可以被认为是无限厚的。在这两种情况下，本模型的结果都退化为双层结构，而且所提模型的屈曲波长和临界应变与其他已发表的双层模型（将软基底视为无限厚基底，如 Huang 等人 (2005)）非常一致。

However, as shown in the subfigure of Figs. 6a and 7a, if the thickness of the intermediate layer is in the range of

[1, 300]

, wavelengths obtained by the proposed model has better agreement with those obtained by the FEA, which indicates that the intermediate layer’s thickness has a significant influence on buckling wavelength.
但是，如图 6a 和图 7a 的子图所示，如果中间层的厚度在

[1, 300]

的范围内，则拟议模型得到的波长与有限元分析得到的波长具有更好的一致性，这表明中间层的厚度对屈曲波长具有重要影响。

Fig. 7b demonstrates the effect of the thickness ratio

h_{in} / h_{f}

on the critical strain of the local wrinkling pattern. It can be observed that the critical strain

ε_{c}

decreases as the thickness ratio

h_{in} / h_{f}

increases. Clearly, the intermediate layer plays an important role in the critical strain when the thickness ratio is in the range of

[1, 300]

. From Fig. 7b, one can easily see that the critical strains obtained by the proposed model and the FEA have better agreement, compared with the model of the intermediate layer as an infinite substrate.
图 7b 展示了厚度比

h_{in} / h_{f}

对局部皱纹临界应变的影响。可以看出，临界应变

ε_{c}

随着厚度比

h_{in} / h_{f}

的增大而减小。显然，当厚度比在

[1, 300]

范围内时，中间层对临界应变起着重要作用。从图 7b 中不难看出，与中间层为无限基底的模型相比，所提出的模型和有限元分析得出的临界应变具有更好的一致性。

4.3. Influence of Young’s modulus of intermediate layer
4.3.中间层杨氏模量的影响

To explore the influence of the intermediate layer’s Young’s modulus on wavelength and critical strain, the parameters are set as:

E_{f} = 200 GPa,

ν_{f} = 0.27,

h_{f} = 0.5 μ m,

h_{in} = 20 μ m,

ν_{in} = 0.48,

E_{s} = 2.5 MPa,

ν_{s} = 0.48,

h_{s} = 1000 μ m .

为了探究中间层杨氏模量对波长和临界应变的影响，参数设置如下：

E_{f} = 200 GPa,

ν_{f} = 0.27,

h_{f} = 0.5 μ m,

h_{in} = 20 μ m,

ν_{in} = 0.48,

E_{s} = 2.5 MPa,

ν_{s} = 0.48,

h_{s} = 1000 μ m .

Fig. 8 shows the variations of wavelength and critical strain with the intermediate layer’s Young’s modulus for the tri-layer structure. From the results of Fig. 8, it can be observed that the intermediate layer’s Young’s modulus has an important influence on wavelength and critical strain. With the increase of its Young’s modulus, the wavelength has a trough point, while the critical strain would increase. Furthermore, when the intermediate layer is modelled as a finite-thickness substrate in the proposed model, the present result has better agreement with that obtained by FEA, compared with other published models (Jia et al., 2012, Wang et al., 2020).
图 8 显示了三层结构的波长和临界应变随中间层杨氏模量的变化。从图 8 的结果可以看出，中间层的杨氏模量对波长和临界应变有重要影响。随着中间层杨氏模量的增大，波长会出现波谷，而临界应变则会增大。此外，与其他已发表的模型（Jia 等人，2012 年；Wang 等人，2020 年）相比，当中间层在所提出的模型中被模拟为有限厚度基底时，目前的结果与有限元分析得到的结果更为一致。

4.4. Influence of Poisson’s ratio of the intermediate layer
4.4.中间层泊松比的影响

To get a clear understanding of the influence of the Poisson’s ratio of the intermediate layer on the buckling wavelength and the critical strain, Fig. 9 is plotted and the parameters are set as:

E_{f} = 200 GPa,

ν_{f} = 0.27,

h_{f} = 0.5 μ m,

E_{in} = 0.5 MPa,

h_{in} = 20 μ m,

E_{s} = 2.5 MPa,

ν_{s} = 0.48,

h_{s} = 1000 μ m .

为了清楚地了解中间层泊松比对屈曲波长和临界应变的影响，绘制了图 9，并将参数设置为

E_{f} = 200 GPa,

ν_{f} = 0.27,

h_{f} = 0.5 μ m,

E_{in} = 0.5 MPa,

h_{in} = 20 μ m,

E_{s} = 2.5 MPa,

ν_{s} = 0.48,

h_{s} = 1000 μ m .

From the results in Fig. 9, it is found that with the increase of the Poisson’s ratio of the intermediate layer

v_{in}

, the buckling wavelength decreases but the critical strain increases.
从图 9 的结果可以看出，随着中间层泊松比

v_{in}

的增大，屈曲波长减小，但临界应变增大。

From the results of Fig. 6, Fig. 7, Fig. 8, Fig. 9, one can conclude that the intermediate layer would affect the local wrinkling behaviour of the tri-layer structure. The findings of how these geometrical and material properties affect the buckling regimes of the stiff thin films are significant in designing robust stretchable electronics.
从图 6、图 7、图 8 和图 9 的结果可以得出结论，中间层会影响三层结构的局部起皱行为。这些几何和材料特性如何影响刚性薄膜的屈曲状态，这些发现对于设计坚固的可拉伸电子器件具有重要意义。

5. Conclusions 5.结论

In this paper, an analytical model for a stiff thin film bonded on a soft finite-thickness bilayer substrate (an intermediate layer and a compliant substrate) all as linear elastic materials, is established. The mechanisms for global bucking and local wrinkling can be distinguished and determined. The model contains the interfacial shear stress between the film and the intermediate layer, and considers the complete mathematical form of the longitudinal displacement of the film. Thus, more accurate results of critical strain for buckling of the tri-layer structure and buckling wavelength can be obtained. From detailed numerical analyses, some important conclusions are summarized as follows:
本文建立了一个分析模型，用于分析作为线性弹性材料粘结在软有限厚度双层基底（中间层和顺应基底）上的刚性薄膜。可以区分并确定全局降压和局部起皱的机理。该模型包含薄膜与中间层之间的界面剪应力，并考虑了薄膜纵向位移的完整数学形式。因此，可以获得更精确的三层结构屈曲临界应变和屈曲波长结果。通过详细的数值分析，总结出以下一些重要结论：

1.
To distinguish global buckling and local wrinkling observed on finite-thickness tri-layer elastic structures by other researchers, a critical structural length related to the thickness and the Young’s modulus of the film, the intermediate layer and the compliant substrate, is obtained.
为了区分其他研究人员在有限厚度三层弹性结构上观察到的整体屈曲和局部起皱，我们得到了与薄膜、中间层和顺应基底的厚度和杨氏模量相关的临界结构长度。
2.
For local wrinkling, the thickness and the Young’s modulus of the intermediate layer are found to be capable of modulating buckling wavelength and the critical strain. The thicker the intermediate layer, the larger buckling wavelength, and the smaller the critical strain (which was discovered to lead to premature instability by some researchers in the past). With the increase of the Young’s modulus of the intermediate layer, the buckling wavelength of the tri-layer structure is non-monotonic and has a minimum, and the critical strain monotonically increases, which could prevent premature instability.
对于局部起皱，中间层的厚度和杨氏模量可调节屈曲波长和临界应变。中间层越厚，屈曲波长越大，临界应变越小（过去一些研究人员发现这会导致过早失稳）。随着中间层杨氏模量的增加，三层结构的屈曲波长是非单调的，并且有一个最小值，临界应变单调增加，这可以防止过早失稳。
3.
The theoretical results from the presented model are shown to be in better agreement with the finite element results than other published models of the tri-layer structure.
与其他已公布的三层结构模型相比，该模型的理论结果与有限元结果更加吻合。

The results of this paper provide a deeper understanding of the morphology evolution of the tri-layer structure and will aid the design of robust stretchable electronics.
本文的研究结果加深了人们对三层结构形态演变的理解，将有助于设计坚固耐用的可拉伸电子器件。

Declaration of Competing Interest
竞争利益声明

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
作者声明，他们没有任何可能会影响本文所报告工作的已知经济利益或个人关系。

Acknowledgments 致谢

The authors acknowledge the support from the National Natural Science Foundation of China (No. 11802319) and the Natural Science Foundation of Shaanxi Province (2020JQ-123); and the first author also would like to acknowledge the support from Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No. CX202044).
作者感谢国家自然科学基金（编号：11802319）和陕西省自然科学基金（编号：2020JQ-123）的资助；第一作者还感谢西北工业大学博士论文创新基金（编号：CX202044）的资助。

Appendix A. 附录 A.

The expressions of

c_{ij}^{(k)} (i = 1, 2, j = 1, \dots, 4, k = 1, 2)

are given as the following,

c_{ij}^{(k)} (i = 1, 2, j = 1, \dots, 4, k = 1, 2)

的表达式如下、(A1)

\begin{matrix} c_{11}^{(1)} = - \frac{2 ξ_{1} + (2 ν_{in} - 1) sinh (2 ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{12}^{(1)} = - \frac{(ν_{in} - 1) [2 + 2cosh (2 ξ_{1})]}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{13}^{(1)} = - \frac{2 ξ_{1} sinh (ξ_{1}) + 2 (2 - 2 ν_{in}) c o s h (ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{14}^{(1)} = - \frac{2 ξ_{1} sinh (ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})}, \\ c_{11}^{(2)} = - \frac{2 ξ_{2} + (2 ν_{in} - 1) sinh (2 ξ_{2})}{2 ξ_{2} + (4 ν_{in} - 3) sinh (2 ξ_{2})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{14}^{(2)} = - \frac{2 ξ_{2} sinh (ξ_{2})}{2 ξ_{2} + (4 ν_{in} - 3) sinh (2 ξ_{2})}, \\ c_{21}^{(1)} = - \frac{(ν_{in} - 1) [2cosh (2 ξ_{1}) - 2]}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{E_{in}}{1 + ν_{in}}, c_{22}^{(1)} = c_{11}^{(1)}, \\ c_{23}^{(1)} = \frac{2 ξ_{1} cosh (ξ_{1}) + 2 (2 ν_{in} - 1) s i n h (ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{24}^{(1)} = \frac{2 ξ_{1} cosh (ξ_{1}) + 2 (4 ν_{in} - 3) s i n h (ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})}, \\ c_{21}^{(2)} = - \frac{(ν_{in} - 1) [2cosh (2 ξ_{2}) - 2]}{2 ξ_{2} + (4 ν_{in} - 3) sinh (2 ξ_{2})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{24}^{(2)} = \frac{2 ξ_{2} cosh (ξ_{2}) + 2 (4 ν_{in} - 3) s i n h (ξ_{2})}{2 ξ_{2} + (4 ν_{in} - 3) sinh (2 ξ_{2})}, ξ_{n} = n k h_{in} . \end{matrix}

The expressions of

c_{ij}^{(k)} (i = 3, 4, j = 1, \dots, 4, k = 1, 2)

are given as the following,

c_{ij}^{(k)} (i = 3, 4, j = 1, \dots, 4, k = 1, 2)

的表达式如下、(A2)

\begin{matrix} c_{31}^{(1)} = - \frac{(1 - 2 ν_{s}) [c o s h (2 η_{1}) - 1] + 2 η_{1}^{2}}{(3 - 4 ν_{s}) c o s h (2 η_{1}) + 2 η_{1}^{2} + (8 ν_{s}^{2} - 12 ν_{s} + 5)} \frac{E_{s}}{1 + ν_{s}}, \\ c_{32}^{(1)} = \frac{2 (1 - ν_{s}) [sinh (2 η_{1}) - 2 η_{1}]}{(3 - 4 ν_{s}) c o s h (2 η_{1}) + 2 η_{1}^{2} + (8 ν_{s}^{2} - 12 ν_{s} + 5)} \frac{E_{s}}{1 + ν_{s}}, \\ c_{31}^{(2)} = - \frac{(1 - 2 ν_{s}) [c o s h (2 η_{2}) - 1] + 2 η_{2}^{2}}{(3 - 4 ν_{s}) c o s h (2 η_{2}) + 2 η_{2}^{2} + (8 ν_{s}^{2} - 12 ν_{s} + 5)} \frac{E_{s}}{1 + ν_{s}}, \\ c_{41}^{(1)} = \frac{2 (1 - ν_{s}) [sinh (2 η_{1}) + 2 η_{1}]}{(3 - 4 ν_{s}) c o s h (2 η_{1}) + 2 η_{1}^{2} + (8 ν_{s}^{2} - 12 ν_{s} + 5)} \frac{E_{s}}{1 + ν_{s}}, \\ c_{42}^{(1)} = c_{31}^{(1)}, \\ c_{41}^{(2)} = \frac{2 (1 - ν_{s}) [sinh (2 η_{2}) + 2 η_{2}]}{(3 - 4 ν_{s}) c o s h (2 η_{2}) + 2 η_{2}^{2} + (8 ν_{s}^{2} - 12 ν_{s} + 5)} \frac{E_{s}}{1 + ν_{s}}, η_{n} = n k h_{s} . \end{matrix}

The coefficients of the expressions are given as follows,
表达式的系数如下、(A3)

\begin{matrix} γ_{11}^{(1)} = \frac{c_{13}^{(1)} (c_{42}^{(1)} - c_{22}^{(1)}) - c_{23}^{(1)} (c_{32}^{(1)} - c_{12}^{(1)})}{(c_{31}^{(1)} - c_{11}^{(1)}) (c_{42}^{(1)} - c_{22}^{(1)}) - (c_{32}^{(1)} - c_{12}^{(1)}) (c_{41}^{(1)} - c_{21}^{(1)})}, \\ γ_{12}^{(1)} = \frac{c_{14}^{(1)} (c_{42}^{(1)} - c_{22}^{(1)}) - c_{24}^{(1)} (c_{32}^{(1)} - c_{12}^{(1)})}{(c_{31}^{(1)} - c_{11}^{(1)}) (c_{42}^{(1)} - c_{22}^{(1)}) - (c_{32}^{(1)} - c_{12}^{(1)}) (c_{41}^{(1)} - c_{21}^{(1)})}, \\ γ_{21}^{(1)} = \frac{c_{23}^{(1)} (c_{31}^{(1)} - c_{11}^{(1)}) - c_{13}^{(1)} (c_{41}^{(1)} - c_{21}^{(1)})}{(c_{31}^{(1)} - c_{11}^{(1)}) (c_{42}^{(1)} - c_{22}^{(1)}) - (c_{32}^{(1)} - c_{12}^{(1)}) (c_{41}^{(1)} - c_{21}^{(1)})}, \\ γ_{22}^{(1)} = \frac{c_{24}^{(1)} (c_{31}^{(1)} - c_{11}^{(1)}) - c_{14}^{(1)} (c_{41}^{(1)} - c_{21}^{(1)})}{(c_{31}^{(1)} - c_{11}^{(1)}) (c_{42}^{(1)} - c_{22}^{(1)}) - (c_{32}^{(1)} - c_{12}^{(1)}) (c_{41}^{(1)} - c_{21}^{(1)})}, \\ γ_{12}^{(2)} = \frac{c_{14}^{(2)}}{c_{31}^{(2)} - c_{11}^{(2)}} . \end{matrix}

The expressions of

c_{ij}^{(k)} (i = 5, 6, j = 1, \dots, 4, k = 1, 2)

are given as the following,

c_{ij}^{(k)} (i = 5, 6, j = 1, \dots, 4, k = 1, 2)

的表达式如下、(A4)

\begin{matrix} c_{51}^{(1)} = \frac{2 ξ_{1} c o s h (ξ_{1}) + 2 (4 ν_{in} - 3) s i n h (ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})}, \\ c_{52}^{(1)} = - \frac{2 ξ_{1} sinh (ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})}, \\ c_{53}^{(1)} = \frac{1}{2 (ν_{in} - 1)} \frac{(8 ν_{in}^{2} - 10 ν_{in} + 3) c o s h (2 ξ_{1}) - (2 ξ_{1}^{2} + 8 ν_{in}^{2} - 10 ν_{in} + 3)}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})}, \\ c_{54}^{(1)} = \frac{1}{2 (ν_{in} - 1)} \frac{{(3 - 4 ν_{in})}^{2} c o s h (2 ξ_{1}) - 2 ξ_{1}^{2} - {(3 - 4 ν_{in})}^{2}}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{1 + ν_{in}}{E_{in}}, \\ c_{51}^{(2)} = \frac{2 ξ_{2} c o s h (ξ_{2}) + 2 (4 ν_{in} - 3) s i n h (ξ_{2})}{2 ξ_{2} + (4 ν_{in} - 3) sinh (2 ξ_{2})}, \\ c_{54}^{(2)} = \frac{1}{2 (ν_{in} - 1)} \frac{{(3 - 4 ν_{in})}^{2} c o s h (2 ξ_{2}) - 2 ξ_{2}^{2} - {(3 - 4 ν_{in})}^{2}}{2 ξ_{2} + (4 ν_{in} - 3) sinh (2 ξ_{2})} \frac{1 + ν_{in}}{E_{in}}, \\ c_{61}^{(1)} = - \frac{2 ξ_{1} cosh (ξ_{1}) + 2 (2 ν_{in} - 1) s i n h (ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{62}^{(1)} = \frac{2 ξ_{1} sinh (ξ_{1}) + 2 (2 - 2 ν_{in}) c o s h (ξ_{1})}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{63}^{(1)} = \frac{1}{2 (ν_{in} - 1)} \frac{(3 - 4 ν_{in}) c o s h (2 ξ_{1}) + 2 ξ_{1}^{2} + (8 ν_{in}^{2} - 12 ν_{in} + 5)}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{64}^{(1)} = - \frac{1}{2 (ν_{in} - 1)} \frac{(8 ν_{in}^{2} - 10 ν_{in} + 3) c o s h (2 ξ_{1}) - (2 ξ_{1}^{2} + 8 ν_{in}^{2} - 10 ν_{in} + 3)}{2 ξ_{1} + (4 ν_{in} - 3) sinh (2 ξ_{1})}, \\ c_{61}^{(2)} = - \frac{2 ξ_{2} cosh (ξ_{2}) + 2 (2 ν_{in} - 1) s i n h (ξ_{2})}{2 ξ_{2} + (4 ν_{in} - 3) sinh (2 ξ_{2})} \frac{E_{in}}{1 + ν_{in}}, \\ c_{64}^{(2)} = - \frac{1}{2 (ν_{in} - 1)} \frac{(8 ν_{in}^{2} - 10 ν_{in} + 3) c o s h (2 ξ_{2}) - (2 ξ_{2}^{2} + 8 ν_{in}^{2} - 10 ν_{in} + 3)}{2 ξ_{2} + (4 ν_{in} - 3) sinh (2 ξ_{2})} . \end{matrix}

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Cited by (13)

First-order and second-order wrinkling of thin elastic film laminated on a graded substrate
2023, International Journal of Mechanical Sciences
Citation Excerpt :
The total energy of this structure is the sum of the film's strain energy and the substrate's strain energy. As found in Refs. [25,39,44,45,48,69], when a film is resting on an elastic graded substrate and this structure is subjected to an applied strain εapplied, four buckling scenarios may occur, as shown in Fig. 1. When the substrate is a graded elastic material with exponentially decaying modulus, as illustrated in Fig. 1(b), the buckling pattern of this structure is either sinusoidal wrinkling (b1) or global buckling (b2).
Wrinkling of the film-on-substrate structure is widely used in stretchable electronics, and the literature has shown that the substrate with gradient modulus has the advantage of maintaining the structure's integrity. Hence, it is necessary to have a comprehensive understanding of the evolution of the wrinkles in the film-on-elastic-graded-substrate structure. Firstly, a unified theoretical framework is established. Secondly, by using the energy method and the principle of minimum potential energy, the wrinkling behaviour of this structure is investigated, and the analytical expressions of the critical buckling strain, wavelength and wrinkling amplitude of this structure are obtained. Then, based on the energy increment method and Maxwell stability criterion, the post-wrinkling behaviour of this structure is studied, and the theoretical formulations for the critical buckling strain, wavelength and wrinkling amplitude are obtained. Finally, numerical examples are carried out to verify the correctness of the proposed formulations. Through the numerical results, it is found that by modulating the applied strain and decay rate of the exponential substrate, the wrinkling pattern of the film-on-exponential-substrate structure can be transferred from a first-order wrinkling pattern to a second-order wrinkling pattern. Moreover, by tuning the applied strain and characteristic length of the bilayer substrate, the wrinkling pattern of the film-on-bilayer-substrate structure can be determined. These findings are useful for guiding the design of stretchable electronics based on the film-on-elastic-graded-substrate structure.
Post-buckling evolution of compressed thin films adhered to rigid substrates
2022, International Journal of Mechanical Sciences
Compressing a thin film adhered to a substrate can yield delaminated blisters, if the interface adhesion is relatively weak. In this work, through the combination of theoretical modeling and numerical simulations, we investigate the mechanics governing the buckle-delamination and post-buckling evolution of compressed thin films adhered to rigid substrate, where the interface adhesion is attributed by van der Waals interactions. The deformed configurations of thin films can be categorized into five deformation modes depending on the material, geometric and interfacial parameters and the in-plane compressive strain. Through the energy-variational approach, large-deformation beam models are constructed to describe the deformed configurations and energy variations of the arc-shape and self-contact buckling modes. The phase diagrams related to the equilibrium buckling state are constructed by energy analysis, based on which the evolution of the post-buckling behavior is analyzed. Our results may find potential applications in the controlled compressive-buckling assembly of micro-/nano-scale thin films.
Theoretical predictions and evolutions of wrinkles in the film-intermediate layer-substrate structure under compression
2022, International Journal of Solids and Structures
Citation Excerpt :
Furthermore, one can also find that by tuning the geometric and material parameters of the tri-layer structure, various intriguing wrinkling patterns can be generated. In addition, for the highlighting the similarities and differences between this contribution and the two relevant papers (Bi et al., 2021; Chen et al., 2021), a new table is given as the following, In this study, the wrinkling behaviours of a film/intermediate layer/substrate structure under compression are investigated, in the form of theoretical analysis and finite element method simulations.
The strategy of tuning the applied compressive strain on the stiff film/intermediate layer /soft substrate structure (named as the tri-layer structure), which can generate different kinds of surface wrinkling patterns, has been successfully utilized to fabricate stretchable electronic devices. In this paper, theoretical predictions and evolutions of wrinkles in the tri-layer structure have been done. Firstly, the theoretical model of the tri-layer structure is established, which takes the deformation of the intermediate layer into account due to the interfacial shear stress between the film and the intermediate layer. Secondly, the energy method is adopted and the analytical expressions of the critical strain and wavelength of the wrinkles in equilibrium are formulized, based on the principle of minimum potential energy. Then, through the wrinkling analysis, the correctness and superiority of the proposed formulations are verified; the unified theoretical formulations in this paper can be used to predict the first-order wrinkling patterns. Finally, by using the unified theoretical formulations, the evolution from a first-order wrinkling pattern to a second-order wrinkling pattern is first theoretically predicted in this paper. Through the post-buckling analysis, it is found that the second-order wrinkling patterns can be seen as the nested-first-order wrinkling patterns.
Wrinkling of a compressible trilayer domain under large plane deformations
2022, International Journal of Solids and Structures
Citation Excerpt :
The dynamic analysis of a corrugated thin film on a finite-thickness bilayer substrate has been recently conducted (Wang et al., 2021), wherein the effect of the interphase layer on the dynamic behavior of the system has been explored. Finally, Bi et al. (2021) have looked into the trilayer structure with a finite thickness substrate, identifying global and local wrinkling modes in the structure. Other works include the topography-driven delamination of a buckling trilayer structure (Lin et al., 2020), numerical examples on the growth-induced instabilities of trilayer with growth-dependent material properties (Cao et al., 2012b), and the wrinkling behavior of bilayer graphene sheets on top of a substrate (Kim et al., 2019).
Instabilities that arise in layered systems have been a riveting course of study for the past few decades, having found utility in various fields, while also being frequently observed in biological systems. The trilayer structure, composed of a film, interphase and substrate, is employed in several applications where the structure undergoes large deformations and the materials used are far from incompressible. Due to their complex behavior and their potential applications, the instabilities of compressible tri-layered systems; as in how they are initiated and how they can be tuned, yet remain elusive and poorly understood. Hence, the main goal of this contribution is to shed light on the large deformation wrinkling behavior of a compressible, trilayer domain, wherein a theoretical solution which captures the instability behavior of a compressible trilayer system under plane deformations is developed. An excellent agreement is observed between the analytical solutions and numerical findings, obtained using FEM enhanced with eigenvalue analysis, for a wide range of geometrical and material parameters, including compressibility of the domains, stiffness ratios, and interphase thickness. The effect of compressibility is found to be particularly significant for the case of a more compliant interphase compared to the substrate. We rigorously establish a theoretical framework that yields a one-part solution for critical wavelength, which alone captures the different wrinkling modes that have been reported in trilayer structures but previously have been treated as a two-part problem. Finally, at the incompressibility limit, the solution here reduces to its counterparts established in literature.
Nonlinear dynamic analysis for a corrugated thin film on a pre-strained finite-thickness bi-layer substrate
2021, Applied Mathematical Modelling
To improve the robustness of the film/substrate-type stretchable electronics, an intermediate layer has been adopted in their design in recent years. However, the intermediate layer could significantly influence the static and dynamic behaviours of the tri-layer structure (film/intermediate layer/substrate structure). In this paper, considering the shear stress between the film and intermediate layer and the deformation of the intermediate layer and substrate layer, and accounting for the corrugation shape of the film, an improved theoretical model of the tri-layer structure is established and the buckling behaviour of the tri-layer is studied to obtain static buckling amplitude and wavelength. Then, the governing equation of motion of this structure (using the theory of corrugated beam for the buckled film bonded on a finite-thickness bi-layer substrate) is derived, based on the geometrical homogenization theory and the Lagrange equation. By using the numerical results of buckling analysis and the Jacobi elliptic function, the analytical nonlinear frequency of the tri-layer structure with corrugated film is obtained. Finally, numerical examples are analysed to reveal the influences of the Young's modulus and thickness of the intermediate layer on the nonlinear frequency of the corrugated tri-layer structure. From these results, it is concluded that when the thickness of the intermediate layer is hundreds times greater than that of the stiff film (which is the dimension of the current typical design), with the increase of the intermediate layer's Young's modulus, the nonlinear frequency of the tri-layer structure decreases. However, when the intermediate layer is thinner than the film, the nonlinear frequency is independent of the intermediate layer's Young's modulus, which implies that the tri-layer structure degenerates into a bi-layer (film-substrate) structure. These results are helpful for the design of reliable film/substrate-type stretchable electronic devices.
Nonlinear dynamic instability of wrinkled film-substrate structure under axial load
2021, Nonlinear Dynamics

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Outline 概要

Cited by (13) 引用 (13)

Figures (9) 数字 (9)

Tables (2) 表格 (2)

International Journal of Solids and Structures
国际固体与结构杂志

Abstract 摘要

Keywords 关键词

1. Introduction 1.导言

2. Theoretical modelling 2.理论建模

2.1. Equilibrium of the thin stiff film
2.1.刚性薄膜的平衡

2.2. Equilibrium of the intermediate layer and the substrate
2.2.中间层和基底的平衡

2.3. The boundary conditions
2.3.边界条件

3. Energy method and buckling regime analysis
3.能量法和屈曲机制分析

4. Buckling analysis of local wrinkling
4.局部起皱的屈曲分析

4.1. Verification studies
4.1.验证研究

4.2. Influences of thickness of intermediate layer
4.2.中间层厚度的影响

4.3. Influence of Young’s modulus of intermediate layer
4.3.中间层杨氏模量的影响

4.4. Influence of Poisson’s ratio of the intermediate layer
4.4.中间层泊松比的影响

5. Conclusions 5.结论

Declaration of Competing Interest
竞争利益声明

Acknowledgments 致谢

Appendix A. 附录 A.

References 参考资料

Cited by (13)

First-order and second-order wrinkling of thin elastic film laminated on a graded substrate

Post-buckling evolution of compressed thin films adhered to rigid substrates

Theoretical predictions and evolutions of wrinkles in the film-intermediate layer-substrate structure under compression

Wrinkling of a compressible trilayer domain under large plane deformations

Nonlinear dynamic analysis for a corrugated thin film on a pre-strained finite-thickness bi-layer substrate

Nonlinear dynamic instability of wrinkled film-substrate structure under axial load

Buckling behaviour of a stiff thin film on a finite-thickness bi-layer substrate有限厚度双层基底上刚性薄膜的屈曲行为

Abstract 摘要

Keywords 关键词

1. Introduction 1.导言

2. Theoretical modelling 2.理论建模

2.1. Equilibrium of the thin stiff film2.1.刚性薄膜的平衡

2.2. Equilibrium of the intermediate layer and the substrate2.2.中间层和基底的平衡

2.3. The boundary conditions2.3.边界条件

3. Energy method and buckling regime analysis3.能量法和屈曲机制分析

4. Buckling analysis of local wrinkling4.局部起皱的屈曲分析

4.1. Verification studies4.1.验证研究

4.2. Influences of thickness of intermediate layer4.2.中间层厚度的影响

4.3. Influence of Young’s modulus of intermediate layer4.3.中间层杨氏模量的影响

4.4. Influence of Poisson’s ratio of the intermediate layer4.4.中间层泊松比的影响

5. Conclusions 5.结论

Declaration of Competing Interest竞争利益声明

Acknowledgments 致谢

Appendix A. 附录 A.

References 参考资料

Buckling behaviour of a stiff thin film on a finite-thickness bi-layer substrate
有限厚度双层基底上刚性薄膜的屈曲行为

2.1. Equilibrium of the thin stiff film
2.1.刚性薄膜的平衡

2.2. Equilibrium of the intermediate layer and the substrate
2.2.中间层和基底的平衡

2.3. The boundary conditions
2.3.边界条件

3. Energy method and buckling regime analysis
3.能量法和屈曲机制分析

4. Buckling analysis of local wrinkling
4.局部起皱的屈曲分析

4.1. Verification studies
4.1.验证研究

4.2. Influences of thickness of intermediate layer
4.2.中间层厚度的影响

4.3. Influence of Young’s modulus of intermediate layer
4.3.中间层杨氏模量的影响

4.4. Influence of Poisson’s ratio of the intermediate layer
4.4.中间层泊松比的影响

Declaration of Competing Interest
竞争利益声明