BUCKLING ANALYSIS OF STRETCHABLE FERROELECTRIC THIN FILM ON ELASTOMERIC SUBSTRATES

The thin stiff films on pre-stretched compliant substrates can form wrinkles, which can be controlled in micro and nanoscale systems to generate smart structures. Recently, buck- led piezoelectric/ferroelectrie nanoribbons have been reported to show an enhancement in the piezoelectric effect and stretchability, which can be applied in energy harvesting devices, sensors and memory devices instead of polymeric polyvinylidine fluoride （PVDF）. This paper studies the buckling and post-buckling process of ferroelectric thin films bonded to the pre-stretched soft layer, which in turn lies on a rigid support. Nonlinear electromechanical equations for the buckling of thin piezoelectric plates are deduced and employed to model the ferroelectric film poled in the thickness direction. Two buckling modes are analyzed and discussed： partially de-adhered buck- ling and fully adhered buckling. Transition from one buckling mode to the other is predicted and the effect of piezoelectricity on the critical buckling condition of piezoelectric film is examined.     

Helical coil buckling mechanism for a stiff nanowire on an elastomeric substrate

When a stiff nanowire is deposited on a compliant soft substrate, it may buckle into a helical coil form when the system is compressed. Using theoretical and finite element method (FEM) analyses, the detailed three-dimensional coil buckling mechanism for a silicon nanowire (SiNW) on a polydimethylsiloxane (PDMS) substrate is studied. A continuum mechanics approach based on the minimization of the strain energy in the SiNW and elastomeric substrate is developed. Due to the helical buckling, the bending strain in SiNW is significantly reduced and the maximum local strain is almost uniformly distributed along SiNW. Based on the theoretical model, the energy landscape for different buckling modes of SiNW on PDMS substrate is given, which shows that both the in-plane and out-of-plane buckling modes have the local minimum potential energy, whereas the helical buckling model has the global minimum potential energy. Furthermore, the helical buckling spacing and amplitudes are deduced, taking into account the influences of the elastic properties and dimensions of SiNWs. These features are verified by systematic FEM simulations and parallel experiments. As the effective compressive strain in elastomeric substrate increases, the buckling profile evolves from a vertical ellipse to a lateral ellipse, and then approaches to a circle when the effective compressive strain is larger than 30%. The study may shed useful insights on the design and optimization of high-performance stretchable electronics and 3D complex nano-structures.     

A shearable and thickness stretchable finite strain beam model for soft structures

Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In a previous work, an Euler–Bernoulli finite strain beam model with thickness stretching effect was proposed for soft thin structures subject to stiff constraint in the width direction. By extending that model to account for the transverse shear effect, a Timoshenko-type finite strain beam model within the plane-strain context is developed in the present work. With some kinematic hypotheses, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. In the limit of infinitesimal strain, the new model degenerates to the classical extensible and shearable elastica model. The corresponding incremental equilibrium equations and associated boundary conditions are also obtained. Based on the new beam model, analytical solutions are given for simple deformation modes, including uniaxial tension, simple shear, pure bending, and buckling under an axial load. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with immovable ends. The results are also compared with the previously developed finite strain Euler–Bernoulli beam model to demonstrate the significance of transverse shear effect for soft beams with a small length-to-thickness ratio. The developed beam model will contribute to the design and analysis of soft robots and soft devices.     

基底上薄膜结构的非线性屈曲力学进展

基底上薄膜结构中的过大残余压应力常常通过屈曲不稳定性诱发薄膜结构和功能的失效。屈曲不稳定性、演化与斑图形成是近年来非线性力学研究的热点。此类屈曲不稳定性受薄膜-基底的力学性质以及界面相互作用影响,进而呈现出复杂的屈曲模式如褶皱、翘曲和折痕等。论文简要综述褶皱、翘曲和折痕等屈曲模式的形成机制、影响因素和后屈曲形貌相关方面的进展。褶皱部分,重点介绍了褶皱的形成、多级褶皱结构、局域化的褶皱、各向异性褶皱和曲面上的褶皱。翘曲部分,介绍了翘曲结构包括一维翘曲结构、“电话线”屈曲泡,网络状屈曲泡等的形成与生长过程,并讨论了曲面几何、界面滑移、开裂等因素的影响。折痕及其它复杂屈曲模式部分,介绍了折痕、叠痕及隆起失稳的形成机制与临界条件.     

Finite width effect of thin-films buckling on compliant substrate: Experimental and theoretical studies

Buckling of stiff thin films on compliant substrates has many important applications ranging from stretchable electronics to precision metrology and sensors. Mechanics plays an indispensable role in the fundamental understanding of such systems. Some existing mechanics models assume plane-strain deformation, which do not agree with experimental observations for narrow thin films. Systematic experimental and analytical studies are presented in this paper for finite-width stiff thin films buckling on compliant substrates. Both experiments and analytical solution show that the buckling amplitude and wavelength increase with the film width. The analytical solution agrees very well with experiments and therefore provides valuable guide to the precise design and control of the buckling profile in many applications. The effect of film spacing is studied via the analytical solutions for two thin films and for periodic thin films.     

Nonlinear analysis of compressed elastic thin films on elastic substrates: From wrinkling to buckle-delamination

Nonlinear buckling of elastic thin films on compliant substrates is studied by modeling and simulations to reveal the roles of pre-strain, elastic modulus ratio, and interfacial properties in morphological transition from wrinkles to buckle-delamination blisters. The model integrates an interfacial cohesive zone model with the Föppl–von Kármán plate theory and Green function method within the general framework of energy minimization. A kinetics approach is developed for numerical simulations. Subject to a uniaxial pre-strain, the numerical simulations confirm the analytically predicted critical conditions for onset of wrinkling and wrinkle-induced delamination, with which a phase diagram is constructed. It is found that, with increasing pre-strain, the equilibrium configuration evolves from flat to wrinkles, to concomitant wrinkles and buckle-delamination, and to an array of parallel straight blisters. The height and width of the buckle-delamination blisters can be approximately described by a set of scaling laws with respect to the pre-strain and interfacial toughness. Subject to an equi-biaxial pre-strain, the critical conditions are determined numerically to construct a similar phase diagram for the buckling modes. Moreover, by varying the pre-strain, modulus ratio, and interfacial toughness, a rich variety of equilibrium configurations are simulated, including straight blisters, and network blisters with or without wrinkles. These results provide considerable insight into diverse surface patterns in layered material systems as a result of the mechanical interactions between the film and the substrate through their interface, which suggests potential control parameters for designing specific surface patterns.     

Non-linear in-plane postbuckling of arches with rotational end restraints under uniform radial loading

This paper presents a thorough and comprehensive investigation of non-linear buckling and postbuckling analyses of pin-ended shallow circular arches subjected to a uniform radial load and which have equal elastic rotational end-restraints. The differential equations of equilibrium for non-linear buckling and postbuckling are established based on a virtual work approach. Exact solutions for the non-linear bifurcation, limit point and lowest buckling loads are obtained; in particular, exact solutions for the non-linear postbuckling equilibrium paths are derived. The criteria for switching between fundamental buckling and postbuckling modes are developed in terms of critical values of a geometric parameter for an arch, with exact solutions for these critical values of geometric parameter being obtained. Analytical solutions of non-linear buckling and postbuckling problems for arches with rotational end-restraints are of great interest, since they constitute one of the very few closed-form analyses of buckling and postbuckling behaviour of continuous structural systems. These exact solutions are a contribution to the non-linear structural mechanics of arches, as well as providing useful benchmark solutions for verifying non-linear numerical analyses.     

Controlled buckling of thin film on elastomeric substrate in large deformation

Electronic systems with large stretchability have many applications. A precisely controlled buckling strategy to increase the stretchability has been demonstrated by combining lithographically patterned surface bonding chemistry and a buckling process. The buckled geometry was assumed to have a sinusoidal form, which may result in errors to determine the strains in the film. A theoretical model is presented in this letter to study the mechanics of this type of thin film/substrate system by discarding the assumption of sinusoidal buckling geometry. It is shown that the previous model overestimates the deflection and curvature in the thin film. The results from the model agree well with finite element simulations and therefore provide design guidelines in many applications ranging from stretchable electronics to micro/nano scale surface patterning and precision metrology.     

Micro-buckling in the nanocomposite structure of biological materials

Nanocomposite structure, consisting of hard mineral and soft protein, is the elementary building block of biological materials, where the mineral crystals are arranged in a staggered manner in protein matrix. This special alignment of mineral is supposed to be crucial to the structural stability of the biological materials under compressive load, but the underlying mechanism is not yet clear. In this study, we performed analytical analysis on the buckling strength of the nanocomposite structure by explicitly considering the staggered alignment of the mineral crystals, as well as the coordination among the minerals during the buckling deformation. Two local buckling modes of the nanostructure were identified, i.e., the symmetric mode and anti-symmetric mode. We showed that the symmetric mode often happens at large aspect ratio and large volume fraction of mineral, while the anti-symmetric happens at small aspect ratio and small volume fraction. In addition, we showed that because of the coordination of minerals with the help of their staggered alignment, the buckling strength of these two modes approached to that of the ideally continuous fiber reinforced composites at large aspect ratio given by Rosen's model, insensitive to the existing “gap”-like flaws between mineral tips. Furthermore, we identified a mechanism of buckling mode transition from local to global buckling with increase of aspect ratio, which was attributed to the biphasic dependence of the buckling strength on the aspect ratio. That is, for small aspect ratio, the local buckling strength is smaller than that of global buckling so that it dominates the buckling behavior of the nanocomposite; for comparatively larger aspect ratio, the local buckling strength is higher than that of global buckling so that the global buckling dominates the buckling behavior. We also found that the hierarchical structure can effectively enhance the buckling strength, particularly, this structural design enables biological nanocomposites to avoid local buckling so as to achieve global buckling at macroscopic scales through hierarchical design. These features are remarkably important for the mechanical functions of biological materials, such as bone, teeth and nacre, which often sustain large compressive load.     

带脱层的复合材料层板屈曲分析中的接触问题

用基于Mindlin板理论的有限元方法进行了带脱层损伤复合材料层板的屈曲载荷分析．为了处理在屈曲模态中上下脱层之间的接触问题，我们提出了一种有效的算法．在这种算法中，首先用一阶灵敏度分析和二次规划方法相结合的选代算法算出接触区域的虚拟力，然后将这种虚拟力转化成一些假想弹簧的刚度系数，并对原始刚度矩阵进行修正．数值算例表明本算法可以有效地克服屈曲模态中上下脱层之间的相互贯穿．同时，还对脱层的大小，形状和位置对屈曲载荷的影响进行了研究．     

软材料粘接结构界面破坏研究综述

软材料已经在软机器人、生物医学及柔性电子等各个领域得到广泛的应用. 实际应用中, 软材料多需要粘附于不同类型的基底上, 与之共同组成工程构件进而实现特定的功能, 粘接界面性能对构件的结构完整性与功能可靠性起着关键性作用. 本文对目前软材料粘接结构界面破坏行为方面的研究进行了系统总结. 首先通过与传统粘接结构的对比, 指出了“软界面”与“软基体”两种软材料粘接结构界面破坏行为的独特性及其物理本质. 接着分别总结了“软界面”与“软基体”两种粘接结构界面破坏行为的实验表征方面的研究成果, 对界面及基体黏弹性耗散对界面破坏机理的影响分别进行了分析. 然后从理论角度, 介绍了针对两种软材料粘接结构界面破坏行为的理论分析方法, 并对已建立的相关理论模型进行了总结. 之后以内聚力模型方法为基础, 介绍了软材料粘接结构界面破坏行为数值模拟方面的相关研究进展. 最后基于已有的研究成果, 提出了目前研究所面临的挑战, 并对可能的软材料粘接结构界面破坏的未来研究方向进行了讨论和展望.      

基于辛Runge-Kutta方法的棋盘形褶皱二维薄膜-基底结构动力学特性研究

基于力学屈曲原理的褶皱薄膜-基底结构已成功应用于制备可延展无机电子器件。然而,该类电子器件在应用时需要服役于复杂动态环境中,针对棋盘形褶皱薄膜结构的动力学问题鲜有研究,此问题又是该类电子器件走向实际应用需要解决的关键问题之一。本文首先采用能量方法,分别计算了二维薄膜的弯曲能、膜弹性能和柔性基底中的弹性能以及薄膜动能;然后采用拉格朗日方程,推导出了该结构的振动控制方程;而该方程为非线性动力学方程,无法给出其解析解;因此,本文采用辛Runge-Kutta方法对其进行数值求解;数值结果表明,辛数值方法具有长期稳定的特性和系统结构特性,为高精度的可延展电子器件的动力学问题研究提供了优异的数值方法。     

Surface layer stability under force and temperature loading

We study stability of an elastic homogeneous ormultilayer plate with a free surface under the action of force and temperature deformations. We show that possible buckling modes can be of two types. For each of these two types, we study the dependence of the critical strain on the wavelength. It was found that, for a homogeneous layer, a decrease in the wavelength leads to a decrease in the critical strain and, for a multilayer material with alternating hard and soft layers, a strain wave of finite length arises in buckling.     

Buckling modes of a compressed plate on an elastic substrate

The buckling modes of a homogeneously compressed elastic plate on a soft elastic substrate are studied. The critical compression is uniquely determined by the bifurcation equation, but this compression is associated with a wide set of buckling modes. It was proved that any solution of the Helmholtz equation satisfies the bifurcation equation. At the same time, in microelectronics, it is required to know which buckling mode is realized. Experimental and theoretical investigations show that the chessboard-like buckling mode should be expected. In what follows, this problem is discussed theoretically. The expected buckling mode can be found by analyzing the energy of the initial postcritical deformation, and the desired mode is determined from the condition of its minimum. The analytic expression of this energy is obtained. Its minimization results in the chessboard-like buckling mode.     

Formulation of strain gradient plasticity with interface energy in a consistent thermodynamic framework

In this work, the strain gradient formulation is used within the context of the thermodynamic principle, internal state variables, and thermodynamic and dissipation potentials. These in turn provide balance of momentum, boundary conditions, yield condition and flow rule, and free energy and dissipative energies. This new formulation contributes to the following important related issues: (i) the effects of interface energy that are incorporated into the formulation to address various boundary conditions, strengthening and formation of the boundary layers, (ii) nonlocal temperature effects that are crucial, for instance, for simulating the behavior of high speed machining for metallic materials using the strain gradient plasticity models, (iii) a new form of the nonlocal flow rule, (iv) physical bases of the length scale parameter and its identification using nano-indentation experiments and (v) a wide range of applications of the theory. Applications to thin films on thick substrates for various loading conditions and torsion of thin wires are investigated here along with the appropriate length scale effect on the behavior of these structures. Numerical issues of the theory are discussed and results are obtained using Matlab and Mathematica for the nonlinear ordinary differential equations (NODE) which constitute the boundary value problem.This study reveals that the micro-stress term has an important effect on the development of the boundary layers and hardening of the material at both hard and soft interface boundary conditions in thin films. The interface boundary conditions are described by the interfacial length scale and interfacial strength parameters. These parameters are important to control the size effect and hardening of the material. For more complex geometries the generalized form of the boundary value problem using the nonlocal finite element formulation is required to address the problems involved.     

Recent development of transient electronics

Transient electronics are an emerging class of electronics with the unique characteristic to completely dissolve within a programmed period of time. Since no harmful byproducts are released, these electronics can be used in the human body as a diagnostic tool, for instance, or they can be used as environmentally friendly alternatives to existing electronics which disintegrate when exposed to water. Thus, the most crucial aspect of transient electronics is their ability to disintegrate in a practical manner and a review of the literature on this topic is essential for understanding the current capabilities of transient electronics and areas of future research. In the past, only partial dissolution of transient electronics was possible,however, total dissolution has been achieved with a recent discovery that silicon nanomembrane undergoes hydrolysis. The use of single- and multi-layered structures has also been explored as a way to extend the lifetime of the electronics. Analytical models have been developed to study the dissolution of various functional materials as well as the devices constructed from this set of functional materials and these models prove to be useful in the design of the transient electronics.     

A strain-isolation design for stretchable electronics

Stretchable electronics represents a direction of recent development in next-generation semiconductor devices.Such systems have the potential to offer the performance of conventional wafer-based technologies,but they can be stretched like a rubber band,twisted like a rope, bent over a pencil,and folded like a piece of paper.Isolating the active devices from strains associated with such deformations is an important aspect of design.One strategy involves the shielding of the electronics from deformation of the substrate through insertion of a compliant adhesive layer. This paper establishes a simple,analytical model and validates the results by the finite element method.The results show that a relatively thick,compliant adhesive is effective to reduce the strain in the electronics,as is a relatively short film.     

Torsional,flexural, and torsional-flexural buckling modes of a cylindrical shell under combined loading

Stability problems for cylindrical shells under various loading modes were considered in numerous papers. A detailed analysis of such problems can be found, e.g., in the monograph [1]. We refer to the solutions presented in this monograph as classical.For long cylindrical shells in axial compression, one of the buckling modes is the purely beam flexural mode similar to the classical buckling mode of a straight rod. It is well known that it can be studied by using the nonlinear or linearized equations of the membrane theory of shells. In [2], it was shown that, on the basis of such equations constructed starting from the noncontradictory version of geometrically nonlinear elasticity relations in the quadratic approximation [3], under the separate action of the axial compression, external pressure, and torsion, there are also previously unknown nonclassical buckling modes, most of which are shear ones.In the present paper, we show that the use of the above equations for cylindrical shells under compression and external pressure with simultaneous pure torsion or bending permits revealing the earlier unknown torsional, beam flexural, and beam torsional-flexural buckling modes, which are nonclassical, just as those found in [2]. The second of these buckling modes is realized when axially compressing forces are formed in the shell with simultaneous torsion, and the third of them is realized under compression combined with pure bending.It was found that, earlier than the classical buckling modes, the torsional buckling modes can be realized for relatively short shells with small shear rigidity in the tangent plane, while the second and third buckling modes can be realized for relatively long shells.     

Volume and surface stability of uniformly compressed transversely isotropic linearly elastic materials

The stability loss of a transversely isotropic linearly elastic medium is studied. The medium is uniformly compressed in both horizontal directions, and the initial stress in the vertical direction is equal to zero. The standard analysis based on the Hadamard condition is used. The bifurcation equation divides into two parts, and therefore, two kinds of buckling modes are possible. The critical initial compression is found, but the buckling modes remain indefinite (as the wave length so the relation between the wave numbers is arbitrary). The stability loss of a compressed half-space with a free surface is studied. Only one kind of buckling mode localized near the free surface is possible, and as for an entire space, the buckling mode and the wave length are indefinite. In these problems, linear as well as non-linear approaches are used. In the linear approach, the pre-buckling deformations are ignored. It is shown that for some values of parameters, the linear approach leads to qualitatively incorrect results. The stability loss of an uniformly compressed plate lying on a soft elastic half-space is studied. By using the non-linear post-critical analysis, it is shown that the buckling mode is a chessboard-like one.     

Mechanics of nonbuckling interconnects with prestrain for stretchable electronics

The performance of the flexibility and stretchability of flexible electronics depends on the mechanical structure design, for which a great progress has been made in past years. The use of prestrain in the substrate, causing the compression of the transferred interconnects, can provide high elastic stretchability. Recently, the nonbuckling interconnects have been designed, where thick bar replaces thin ribbon layout to yield scissor-like in-plane deformation instead of in-or out-of-plane buckling modes. The nonbuckling interconnect design achieves significantly enhanced stretchability. However,combined use of prestrain and nonbuckling interconnects has not been explored. This paper aims to study the mechanical behavior of nonbuckling interconnects bonded to the prestrained substrate analytically and numerically. It is found that larger prestrain,longer straight segment, and smaller arc radius yield smaller strain in the interconnects.On the other hand, larger prestrain can also cause larger strain in the interconnects after releasing the prestrain. Therefore, the optimization of the prestrain needs to be found to achieve favorable stretchability.