Reprint of “Post-buckling analysis for the precisely controlled buckling of thin film encapsulated by elastomeric substrates” [In. J. Solids Struct. 45 (2008) 2014–2023]

The precisely controlled buckling of stiff thin films (e.g., Si or GaAs nano ribbons) on the patterned surface of elastomeric substrate (e.g., poly(dimethylsiloxane) (PDMS)) with periodic inactivated and activated regions was designed by Sun et al. [Sun, Y., Choi, W.M., Jiang, H., Huang, Y.Y., Rogers, J.A., 2006. Controlled buckling of semiconductor nanoribbons for stretchable electronics. Nature Nanotechnology 1, 201–207] for important applications of stretchable electronics. We have developed a post-buckling model based on the energy method for the precisely controlled buckling to study the system stretchability. The results agree with Sun et al.’s (2006) experiments without any parameter fitting, and the system can reach 120% stretchability.     

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.     

MECHANICS ANALYSIS OF TWO-DIMENSIONALLY PRESTRAINED ELASTOMERIC THIN FILM FOR STRETCHABLE ELECTRONICS

Various methods have been developed to fabricate highly stretchable electronics. Recent studies show that over 100% two dimensional stretchability can be achieved by mesh structure of brittle functioning devices interconnected with serpentine bridges. Kim et al show that pressing down an inflated elastomeric thin film during transfer printing introduces two di- mensional prestrain, and therefore further improves the system stretchability. This paper gives a theoretical study of this process, through both analytical and numerical approaches. Simple analytical solutions are obtained for meridional and circumferential strains in the thin film, as well as the maximum strain in device islands, which all agree reasonably well with finite element analysis.     

可延展结构的设计及力学研究新进展

可延展柔性电子器件克服了传统无机电子器件脆、硬的缺点,在保持优异电学性能的同时,以其优秀的可延展性极大拓展了微电子器件的应用范围,备受国内外学术界和电子产业界瞩目. 无机电子器件的可延展柔性化主要通过力学结构设计的方法实现,本文针对近两年具有代表性的三种可延展柔性结构设计,包括分形互联岛桥结构、折纸结构和剪纸结构,简要综述了这些结构的力学研究进展,彰显了力学在可延展柔性电子器件发展中的重要作用,并展望了未来的发展方向.     

Mechanics of stretchable electronics with high fill factors

Mechanics models are developed for an imbricate scale design for stretchable and flexible electronics to achieve both mechanical stretchability and high fill factors (e.g., full, 100% areal coverage). The critical conditions for self collapse of scales and scale contact give analytically the maximum and minimum widths of scales, which are important to the scale design. The maximum strain in scales is obtained analytically, and has a simple upper bound of 3t_scale/(4ρ) in terms of the scale thickness t_scale and bending radius ρ.     

Effect of interfacial stiffness on the stretchability of metal/elastomer bilayers under in-plane biaxial tension

Flexible electronic devices are often subjected to large and repeated deformation, so that their functional components such as metal interconnects need to sustain strains up to tens of percent, which is far beyond the intrinsic deformability of metal materials(～1%). To meet the stringent requirements of flexible electronics, metal/elastomer bilayers, a stretchable structure that consists of a metal film adhered to a stretchable elastomer substrate, have been developed to improve the stretch capability of metal interconnects. Previous studies have predicted that the metal/elastomer bilayers are much more stretchable than freestanding metal films. However, these investigations usually assume perfect bonding between the metal and elastomer layers. In this work, the effect of the metal/elastomer interface with a finite interfacial stiffness on the stretchability of bilayer structures is analyzed. The results show that the assumption of perfect interface(with infinite interfacial stiffness) may lead to an overestimation of the stretchability of bilayer structures. It is also demonstrated that increased adhesion between the metal and elastomer layers can enhance the stretchability of the metal layer.     

Switchable dry adhesive based on shape memory polymer with hemispherical indenters for transfer printing

Transfer printing based on switchable adhesive is essential for developing unconventional systems, including flexible electronics, stretchable electronics, and micro light-emitting diode (LED) displays. Here we report a design of switchable dry adhesive based on shape memory polymer (SMP) with hemispherical indenters, which offers a continuously tunable and reversible adhesion through the combination of the preloading effect and the thermal actuation of SMP. Experimental and numerical studies reveal the fundamental aspects of design, fabrication, and operation of the switchable dry adhesive. Demonstrations of this adhesive concept in transfer printing of flat objects (e.g., silicon wafers), three-dimensional (3D) objects (e.g., stainless steel balls), and rough objects (e.g., frosted glasses) in two-dimensional (2D) or 3D layouts illustrate its unusual manipulation capabilities in heterogeneous material integration applications.     

Buckling of a stiff thin film on a pre-strained bi-layer substrate

Controlled buckling can impart stretchable mechanics to brittle materials when integrated as thin films on soft, elastomeric substrates. Typical elastomers are permeable to fluids, however, and therefor unable to provide robust barriers to entry of water, for instance, into devices built with the supported thin films. In addition, the mechanical strength of a system dominated by a soft substrate is often unsatisfactory for realistic applications. We show that introduction of a bi-layer substrate yields a robust, high strength system that maintains stretchable characteristics, with a soft layer on top of a relatively stiff layer in the substrate. As a mechanical protection, a soft encapsulation layer can be used on top of the device and the stretchability of the encapsulated system is smaller than that of the system without encapsulation. A simple, analytic model, validated by numerical analysis and FEA, is established for stiff thin films on a bi-layer substrate, and is useful to the design of stretchable systems.     

Laser-induced porous graphene on Polyimide/PDMS composites and its kirigami-inspired strain sensor

The laser-induced porous graphene(LIG) prepared in a straightforward fabrication method is presented,and its applications in stretchable strain sensors to detect the applied strain are also explored. The LIG formed on the polyimide/polydimethylsiloxane(PI/PDMS) composite exhibits a naturally high stretchability(over 30%), bypassing the transfer printing process compared to the one prepared by laser scribing on PI films. The PI/PDMS composite with LIG shows tunable mechanical and electronic performances with different PI particle concentrations in PDMS. The good cyclic stability and almost linear response of the prepared LIG's resistance with respect to tensile strain provide its access to wearable electronics. To improve the PDMS/PI composite stretchability, we designed and optimized a kirigami-inspired strain sensor with LIG on the top surface, dramatically increasing the maximum strain value that in linear response to applied strain from 3% to 79%.     

Postbuckling analysis and its application to stretchable electronics

A versatile strategy for fabricating stretchable electronics involves controlled buckling of bridge structures in circuits that are configured into open, mesh layouts (i.e. islands connected by bridges) and bonded to elastomeric substrates. Quantitative analytical mechanics treatments of the responses of these bridges can be challenging, due to the range and diversity of possible motions. Koiter (1945) pointed out that the postbuckling analysis needs to account for all terms up to the 4th power of displacements in the potential energy. Existing postbuckling analyses, however, are accurate only to the 2nd power of displacements in the potential energy since they assume a linear displacement–curvature relation. Here, a systematic method is established for accurate postbuckling analysis of beams. This framework enables straightforward study of the complex buckling modes under arbitrary loading, such as lateral buckling of the island-bridge, mesh structure subject to shear (or twist) or diagonal stretching observed in experiments. Simple, analytical expressions are obtained for the critical load at the onset of buckling, and for the maximum bending, torsion (shear) and principal strains in the structure during postbuckling.     

Stretchability and compliance of freestanding serpentine-shaped ribbons

High-performance stretchable electronics have to utilize high-quality inorganic electronic materials such as silicon, oxide or nitride dielectrics, and metals. These inorganic materials usually crack or yield at very small intrinsic strains, for example, 1%, whereas bio-integrated electronics are expected to at least match the stretchability of bio-tissues (20%) and deployable structure health monitoring networks are expected to expand from wafer scale (several centimeters) to cover macroscopic structures (several meters). To minimize strains in inorganic materials under large deformation, metallic and ceramic films can be patterned into serpentine-shaped ribbons. When the ribbon is stretched, some sections of the ribbon can rotate and/or buckle to accommodate the applied displacement, leaving much smaller intrinsic strain in the materials compared to the applied strain. The choice of the shape of the serpentine depends on systematic studies of the geometric variables. This paper investigates the effect of serpentine shapes on their stretchability and compliance through theoretical, numerical, and experimental means. Our closed-form curved beam solutions, FEM results, and experimental measurements have found good agreement with one another. Our results conclude that in general, the narrower ribbon, the larger arc radius and arc angle, and the longer arm length will yield lower intrinsic strain and effective stiffness. When the arm length approaches infinite, the stretchability can be enhanced by several orders. A few unexpected behaviors are found at arc angles that are close to straight bars. With additional practical constraints such as minimum ribbon width and finite overall breadth, the optimal serpentine shape can be accurately determined using our closed-form analytical solution.     

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.     

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.     

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.     

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。     

A theoretical model of reversible adhesion in shape memory surface relief structures and its application in transfer printing

Transfer printing is an important and versatile tool for deterministic assembly and integration of micro/nanomaterials on unusual substrates, with promising applications in fabrication of stretchable and flexible electronics. The shape memory polymers (SMP) with triangular surface relief structures are introduced to achieve large, reversible adhesion, thereby with potential applications in temperature-controlled transfer printing. An analytic model is established, and it identifies two mechanisms to increase the adhesion: (1) transition of contact mode from the triangular to trapezoidal configurations, and (2) explicit enhancement in the contact area. The surface relief structures are optimized to achieve reversible adhesion and transfer printing. The theoretical model and results presented can be exploited as design guidelines for future applications of SMP in reversible adhesion and stretchable electronics.     

Buckling life estimation of circular tubes under cyclic bending

In this paper, an experimental investigation of the degradation and buckling of circular tubes subjected to cyclic bending is discussed. The machinery specimens (with different diameter-to-thickness ratios but the same inside diameter) and method of testing (cyclic bending) in this study were the same as the ones used by Lee et al. (Lee, K.L., Pan, W.F., Kuo, J.N., 2001. The influence of the diameter-to-thickness ratio on the stability of circular tubes under cyclic bending. International Journal of Solids and Structures 38, 2401–2413.) for 316L stainless steel circular tubes. The experimental investigation was extended to different outside and inside diameters of the same circular tubes subjected to cyclic bending. Based on the experimental findings, the empirical formulation proposed by Lee et al. (2001) was modified so that it can now be used to simulate the relationship between the prescribed curvature and the number of cycles necessary to produce buckling. In addition, it was found that the experimental curve of the ovalization and the number of cycles necessary to produce buckling could be divided into three stages – an initial, secondary and tertiary stage. An empirical relationship, similar to the Bailey–Norton creep formulation, was proposed for simulating the aforementioned curve for the initial and secondary stages in this study. The derived empirical relationship was in good agreement with the experimental data.     

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.     

Minimum theorems in incremental linear elastic fracture mechanics

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. (Nemat-Nasser et al., 1980, Nguyen, 2000, Maugin, 1992, Bourdin et al., 2008, Salvadori, 2008). This approach is here further advanced, by noting that Stress Intensity Factors (SIFs) asymptotic expansion (Amestoy et al., 1986, Amestoy and Leblond, 1992) enjoys a Colonnetti’s decomposition (Colonnetti, 1918, Colonnetti, 1950) interpretation. As a consequence, minimum theorems are derived in terms of crack tip “quasi static velocity”. They are reminiscent of Ceradini’s theorem (Ceradini, 1965, Ceradini, 1966) in plasticity.     

Viscosity approximation with weak contractions for fixed point problem,equilibrium problem,and variational inequality problem

This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction.The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality.Under suitable conditions,some strong convergence theorems are established in the framework of Hilbert spaces.The results presented in the paper improve and extend the corresponding results of Colao et al.（Colao,V.,Acedo,G.L.,and Marino,G.An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings.Nonlinear Anal.71,2708–2715（2009））,Plubtieng and Punpaeng（Plubtieng,S.and Punpaeng,R.A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.J.Math.Anal.Appl.336,455–469（2007））,Colao et al.（Colao,V.,Marino,G.,and Xu,H.K.An iterative method for finding common solutions of equilibrium problem and fixed point problems.J.Math.Anal.Appl.344,340–352（2008））,Yao et al.（Yao,Y.,Liou,Y.C.,and Yao,J.C.Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings.Fixed Point Theory Application 2007,Article ID 64363（2007）DOI 10.1155/2007/64363）,and others.