Elsevier

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
International Journal of Solids and Structures

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

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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 Ois 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 byhf, hin and hs, 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 hf hin hs 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。
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Fig. 1. The schematic illustration of an intermediate layer between a thin stiff film and a compliant substrate. (a) A stiff film bonded to a pre-stretched bi-layer substrate; (b) global buckling and (c) local wrinkling due to release of tensile pre-strain.
图 1.刚性薄膜与顺应性基底之间的中间层示意图。(a) 粘接在预拉伸双层基底上的刚性薄膜;(b) 整体屈曲;(c) 拉伸预应变释放导致的局部起皱。

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 εx 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 εx 定义为(Huang 等人,2005 年)、
(1)εx=ufx+12wfx2-ε,where ε is the in-plane compressive strain due to the relaxation of the pre-stretched bi-layer substrate, uf is the longitudinal displacement at the neutral axis upon release of the pre-strain, and wf=Acoskx is the transverse deflection (A and k are the buckling amplitude and wavenumber, λ=2π/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, uf 是释放预应变时中性轴的纵向位移、和 wf=Acoskx 是横向挠度( A k 是屈曲振幅和波数, λ=2π/k 是屈曲波长,稍后确定)。
The normal stress p and shear stress τx 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 τx 可表示为(Huang 等人,2005 年)、
(2)p=E¯fhf3124wfx4-εxE¯fhf2wfx2,(3)τx=E¯fhf2ufx2+wfx2wfx2,where E¯f=Ef/1-νf2 denotes the effective Young’s modulus of the film. Ef and νf 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 τx=τ1sinkx (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 E¯f=Ef/1-νf2 表示薄膜的有效杨氏模量。 Ef νf 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 τx=τ1sinkx (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=p1coskx+p2cos2kx,(5)τx=τ1sinkx+τ2sin2kx,where unknown constants p1, p2, τ1 and τ2 will be determined later.
其中未知常数 p1 , p2 , τ1 τ2 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)σxx+τxzz=0,τxzx+σzz=0,where σx=2F/z2, σz=2F/x2 and τxz=-2F/xz are the stresses in terms of the Airy stress function F=Fx,z, which satisfies (Wang et al., 2020),
其中, σx=2F/z2 σz=2F/x2 τxz=-2F/xz 是艾里应力函数 F=Fx,z 的应力,满足(Wang 等,2020)、
(7)4Fx4+4Fx2z2+4Fz4=4F=0,where is the gradient operator. As the transverse deflection of the film is defined as wf=Acoskx when buckled, and considering the displacement continuity conditions, the Airy stress function of the intermediate layer could be assumed as,
其中 为梯度算子。由于薄膜屈曲时的横向挠度定义为 wf=Acoskx ,考虑到位移连续性条件,中间层的 Airy 应力函数可假定为
(8)Finx,z=n=12Fninzcosnkx,Fninz=C1nenkz+C2nzenkz+C3ne-nkz+C4nze-nkz,where C1n,C2n,C3nand C4n are coefficients to be determined by the boundary conditions of the intermediate layer in Section 2.3.
其中 C1n,C2n,C3n C4n 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)σxx,z=n=12nknkC1n+2C2nenkz+nkC2nzenkz+nkC3n-2C4ne-nkz+nkC4nze-nkzcosnkx,(10)σzx,z=n=12-nk2C1nenkz+C2nzenkz+C3ne-nkz+C4nze-nkzcosnkx,(11)τxzx,z=n=12nknkC1n+C2nenkz+nkC2nzenkz-nkC3n-C4ne-nkz-nkC4nze-nkzsinnkx.
Because the film’s transverse deflection is described by wf=Acoskx, the displacement fields of the intermediate layer can be expressed as,
由于薄膜的横向挠度由 wf=Acoskx 描述,因此中间层的位移场可表示为
(12)u,w=n=12Unzsinnkx,n=12Wnzcosnkx,where Unz and Wnz satisfy the following conditions,
其中 Unz Wnz 满足以下条件、
(13)E1+νUnz=nkC1n+21-νC2nenkz+nkC2nzenkz+nkC3n-21-νC4ne-nkz+nkC4nze-nkz,(14)-E1+νWnz=nkC1n-1-2νC2nenkz+nkC2nzenkz-nkC3n+1-2νC4ne-nkz-nkC4nze-nkz.
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)Fsx,z=n=12Fnszcosnkx,Fnsz=C5nenkz+C6nzenkz+C7ne-nkz+C8nze-nkz,where C5n,C6n,C7nand C8n are coefficients to be determined by the boundary conditions of the substrate in Section 2.3.
其中 C5n,C6n,C7n C8n 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)τxzin=τ1sinkx+τ2sin2kx,win=Acoskxatz=hinwin=w1coskx,uin=u1sinkx+u2sin2kxatz=0,where w1, u1and u2 are to be determined.
其中 w1 u1 u2 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)-kekhin1-2νin-khinekhinke-khin1-2νin+khine-khink2ekhinkekhin1+khin-k2e-khinke-khin1-khink2-2νink2νin-2-k1-2νink1-2νinC11C21C31C41=Ein1+νinAτ1Ein1+νinu1Ein1+νinw1,(18)-2ke2khin1-2νin-2khine2khin2ke-2khin1-2νin+2khine-2khin4k2e2khin2ke2khin1+2khin-4k2e-2khin2ke-2khin1-2khin2k2-2νin2k2νin-2-2k1-2νin2k1-2νinC12C22C32C42=0τ2Ein1+νinu20.
Solving Eqs. (17), (18), the coefficients C1n,C2n,C3n and C4n can be obtained in terms of A, w1, u1, u2, τ1 and τ2. 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 C1n,C2n,C3n C4n 可以通过 A 得到、 w1 u1 u2 τ1 τ2 。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)σzin|z=0=-k2C11+C31coskx-4k2C12+C32cos2kx=kc111u1+c121w1+c131A+c141τ1coskx+2kc112u2+c142τ2cos2kxτxzin|z=0=kkC11+C21-kC31+C41sinkx+2k2kC12+C22-2kC32+C42sin2kx=kc211u1+c221w1+c231A+c241τ1sinkx+2kc212u2+c242τ2sin2kxwhere cijki=1,2,j=1,,4,k=1,2 are given in the Appendix Eq. (A1).
其中 cijki=1,2,j=1,,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)ws=w1coskx,us=u1sinkx+u2sin2kxatz=0σzs=0,τxzs=0atz=-hs.
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)σzs|z=0=kc311u1+c321w1coskx+2kc312u2cos2kxτxzs|z=0=kc411u1+c421w1sinkx+2kc412u2sin2kxwhere cijki=3,4,j=1,2,k=1,2 are given in the Appendix Eq. (A2).
其中 cijki=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)σzin|z=0=σzs|z=0,τxzin|z=0=τxzs|z=0,which leads to  这导致(23)u1=1kγ111kA+γ121τ1,w1=1kγ211kA+γ221τ1,u2=12kγ122τ2.where γ111,γ121,γ211,γ221, and γ122 are given in the Appendix Eq. (A3).
其中 γ111,γ121,γ211,γ221, γ122 在附录公式 (A3) 中给出。
The displacements and normal stresses at the top of the intermediate layer (z=hin) can be obtained as,
中间层顶部( z=hin )的位移和法向应力可按以下公式求得、
(24)uin|z=hin=c511u1+c521w1+c531A+1kc541τ1sinkx+c512u2+12kc542τ2sin2kx,(25)σzin|z=hin=kc611u1+c621w1+c631A+c641τ1coskx+2kc612u2+c642τ2cos2kx,where cijki=5,6,j=1,,4,k=1,2 are given in the Appendix Eq. (A4).
其中 cijki=5,6,j=1,,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)uf=uin|z=hin-hf2wfx=c511u1+c521w1+c531A+1kc541τ1+hf2Aksinkx+c512u2+12kc542τ2sin2kx.
One needs to mention that the forms of the longitudinal displacement of the film uf in previous works by other researchers were either sinkx (Wang et al., 2020) or sin2kx (Huang et al., 2005). However, the analytical form of uf 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 uf 的形式要么是 sinkx (Wang 等人,2020 年),要么是 sin2kx (Huang 等人,2005 年)。然而,在这项工作中, uf 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)τx=E¯fhf(2ufx2+wfx2wfx2)=τ1sinkx+τ2sin2kx.
Substituting wf=Acoskx, Eqs. (23), (26) into Eq. (27), the relations between τ1, τ2 and A can be expressed as follows,
wf=Acoskx 、公式 (23) 和 (26) 代入公式 (27), τ1 τ2 A 之间的关系可表示如下、
(28)τ1=-c511γ111+c521γ211+c531+hf2kc511γ121+c521γ221+c541+1E¯fkhfkA=-ζ1kA,τ2=141c512γ122+c542+12E¯fkhfk2A2=ζ2k2A2.
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)uf=ζ1E¯fhfkAsinkx+12c512γ122+c542ζ2kA2sin2kx=γ1Asinkx+γ2kA2sin2kx.

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 Ub and membrane energy Um of the film per unit length and average strain energy Us 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 Ub 和膜能 Um 以及中间层和软基板的平均应变能