Ductility of thin metal films on polymer substrates modulated by interfacial adhesion

When a laminate of a thin metal film on a tough polymer substrate is stretched, the metal film may rupture at strains ranging from a few percent to a few tens of percent. This variation in the ductility of the metal film is modulated by the adhesion of the metal/polymer interface. To study this modulation, here we use the finite element method to simulate the co-evolution of two processes: debonding along the interface and necking in the metal film. We model the interface as an array of nonlinear springs, and model the metal and the polymer as elastic–plastic solids. The simulation shows that necking of the film is accommodated mainly by interfacial sliding, rather than interfacial opening. Depending on the resistance of the interface to sliding, the metal film can exhibit three types of tensile behavior: the film slides and ruptures at a small strain by forming a single neck, the film slides and deforms to a large strain by forming multiple necks, and the film deforms uniformly to a very large strain without sliding and necking.     

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.     

A piezoelectric field-effect strain gage

A new and highly sensitive strain transducer has been developed using a thin-film semiconductor deposited on a polished piezoelectric ceramic substrate. Field-effect coupling has been found to exist between the substrate and film in which the number of mobile carriers in the semiconductor is dependent on the electric-displacement vector of the substrate. Therefore, the conductivity of the semiconducting film can be altered by piezoelectric charge due to a strain applied to the substrate material. An effective gage constant has been calculated in terms of the piezoelectric and elastic constants of the substrate and electronic properties of the film. Experimental devices were constructed by depositingp type tellurium on polished lead-zirconate-titanate ceramic resulting in experimentally observed gage factors as high as 5800 compared to 100–200 for conventional semiconductor gages. The semiconductor film exhibits an electronic instability that limits its use, at present, to transient measurements with frequencies above 1 Hz. Data will also be presented to show that the gage constant is continuously variable between a positive and negative maximum value by altering the magnitude and direction of the substrate-polarization vector. It is believed that these gages will be useful in those cases where extremely small strains (～10^?7) are to be measured or when moderate strains (～10^?4) are to be determined in an electrically noisy background.     

Propagation of instability in dielectric elastomers

When an electric voltage is applied across the thickness of a thin layer of an dielectric elastomer, the layer reduces its thickness and expands its area. This electrically induced deformation can be rapid and large, and is potentially useful as soft actuators in diverse technologies. Recent experimental and theoretical studies have shown that, when the voltage exceeds some critical value, the homogenous deformation of the layer becomes unstable, and the layer deforms into a mixture of thin and thick regions. Subsequently, as more electric charge is applied, the thin regions enlarge at the expense of the thick regions. On the basis of a recently formulated nonlinear field theory, this paper develops a meshfree method to simulate numerically this instability.     

超弹性薄膜与可压缩基底双层结构表面失稳分析

当上层超弹性硬质薄膜和下层可膨胀基底构成的双层结构受压时,薄膜的自由表面可通过形成褶皱降低系统能量.研究表明,上下两层的模量比不同时,上层弹性硬质薄膜将表现出不同的表面失稳模式.本文提出了一种新颖的方法可有效抑制双层软材料的表面失稳,即改变基底材料的泊松比,这种方法同时适用于不具有应变硬化的软材料.首先基于Neo-Hookean模型发展了小变形条件下双层结构表面失稳的理论模型,通过半解析的方法得到了表面失稳的临界应变;然后通过有限元计算与模拟,进一步验证了负泊松比基底可延缓表面失稳.结果表明:(1)当双层结构基底泊松比为正且趋于0.5(不可压缩)时,双层结构在较小的压缩应变下出现表面失稳;(2)当基底的泊松比为负且趋于-1时,可被压缩至46%而不出现表面失稳,即可膨胀基底能有效抑制薄膜的表面失稳.本文发展的方法及主要结果可为延展性电子器件的设计提供指导.     

确定微纳米尺度金属薄膜拉伸分叉点的实验研究

为了确定微纳米尺度金属薄膜的拉伸分叉点,本文使用磁控溅射镀膜技术,在PI(聚酰亚胺)基底上沉积500nm厚的铜薄膜,制作薄膜/基底结构拉伸试件。在单轴拉伸作用下,通过测量拉伸加载过程中铜薄膜的电阻变化情况,得到薄膜电阻随应变变化的关系,并与理论推导的结果进行对比分析,从而确定了塑性阶段理论曲线与实验曲线分离的点,即铜薄膜的分叉点。以此为基础,研究了铜薄膜在单轴拉伸作用下的分叉行为。研究结果表明,沉积于PI基底上的微纳米尺度铜薄膜在单轴拉伸下,经过弹性变形阶段后,很快就发生分叉,然后产生破坏,而塑性变形阶段和局部化阶段较短;弹性阶段薄膜的电阻变化速率很小,塑性阶段薄膜的电阻变化速率稍有增大,而当薄膜表面开始出现微裂纹后,电阻变化速率急剧增大。     

Spinodal surface instability of soft elastic thin films

When the thicknesses of thin films reduce to microns or even nanometers, surface energy and surface interaction often play a significant role in their deformation behavior and surface morphology. The spinodal surface instability induced by the van der Waals force in a soft elastic thin film perfectly bonded to a rigid substrate is investigated theoretically using the bifurcation theory of elastic structures. The analytical solution is derived for the critical condition of spinodal surface morphology instability by accounting for the competition of the van der Waals interaction energy, elastic strain energy and surface energy. Detailed examinations on the effect of surface energy, thickness and elastic properties of the film show that the characteristic wavelength of the deformation bifurcation mode depends on the film thickness via an exponential relation, with the power index in the range from 0.749 to 1.0. The theoretical solution has a good agreement with relevant experiment results.     

Interfacial slippage effect on the spontaneous instability of soft thin viscoelastic films

Based on the continuum mechanics and the bifurcation theory, a three-dimensional theoretical model for a soft thin viscoelastic film bonded to a rigid substrate is investigated. Considering the effect of interfacial slippage and the competition among van der Waals interaction potential energy, strain energy, and surface energy, a set of three-dimensional governing equations of the spontaneous instability is derived, and the analytical results of time-dependent critical conditions are obtained. Furthermore, the phase diagram of instability due to van der Waals interaction and the variation of the dimensionless characteristic wavenumber and the critical interaction stiffness with the critical time are discussed by considering the effect of interfacial slippage.     

The electro-mechanical response of elastomer membranes coated with ultra-thin metal electrodes

This paper presents experimental and theoretical analyses of the electro-mechanical response of metal/elastomer multilayers. A novel test has been devised to determine the relationship between the mechanical response of clamped elastomer membranes, coated on both sides with metal electrodes, and an applied electric field. The load-deflection response of the multilayers subjected to different voltages was measured using an instrumented spherical indenter having dimensions comparable to the freestanding span. The measurements are used with closed-form solutions for membrane deflection to determine the effective plane-strain modulus of cracked multilayers and electrically induced in-plane strains. The experiments demonstrate that: (i) electrically induced strains vary with the square of the electric field, as expected from electrostatic models of parallel plate capacitors, (ii) the transverse stiffness of membranes can be controlled using applied electric fields, (iii) analytical models accurately predict the relationship between electrode crack spacing, layer properties and effective moduli. Finally, we estimate the toughness of the sub-micron metal electrodes, using cracking models that relate crack spacing, imposed strain and the energy release rate governing channel crack formation.     

Effects of interfacial properties on the ductility of polymer-supported metal films for flexible electronics

Polymer-supported metal films as interconnects for flexible, large area electronics may rupture when they are stretched, and the rupture strain is strongly dependent upon the film/substrate interfacial properties. This paper investigates the influence of interfacial properties on the ductility of polymer-supported metal films by modeling the microstructure of the metal film as well as the film/substrate interface using the method of finite elements and the cohesive zone model (CZM). The influence of various system parameters including substrate thickness, Young’s modulus of substrate material, film/substrate interfacial stiffness, strength and interfacial fracture energy on the ductility of polymer-supported metal films is systematically studied. Obtained results demonstrate that the ductility of polymer-supported metal films increases as the interfacial strength increases, but the increasing trend is affected distinctly by the interfacial stiffness.     

Buckling of a stiff film bound to a compliant substrate—Part II:: A global scenario for the formation of herringbone pattern

We study the buckling of a thin compressed elastic film bonded to a compliant substrate. We focus on a family of buckling patterns, such that the film profile is generated by two functions of a single variable. This family includes the unbuckled configuration, the classical primary mode made of straight stripes, as well the pattern with undulating stripes obtained by a secondary instability investigated in the first companion paper, and the herringbone pattern studied in last companion paper. A simplified buckling model relevant for the analysis of these patterns is introduced. It is solved analytically for moderate or for large residual compressive stress in the film. Numerical simulations are presented, based on an efficient implementation. Overall, the analysis provides a global picture for the formation of herringbone patterns under increasing residual stress. The film shape is shown to converge at large load to a developable shape with ridges. The wavelength of the pattern, selected in a first place by the primary buckling bifurcation, is frozen during the subsequent increase of loading.     

Electro-mechanical coupling bifurcation and bulging propagation in a cylindrical dielectric elastomer tube

This paper explores the critical and post-bulging bifurcation of a cylindrical dielectric elastomer (DE) tube undergoing finite deformation under electro-mechanical coupling loading. Explicit expressions for the critical conditions of electro-mechanical bifurcation are derived by using a simplified mathematical method. The post-bifurcation path is comprehensively investigated by specifying the material model as ideal dielectric elastomer. In the post-bifurcation analysis, we analytically establish conditions for the phase coexistence of steady propagation and analyze the physical implications. We demonstrate a global instability under force or voltage control and a localized instability under volume or charge control. Cylindrical tube experiments have been carried out under electro-mechanical coupling loading to verify the theoretical predictions. Good agreements on the critical conditions as well as the post-bifurcation path are obtained. This work characterizes the bifurcation mechanism of rubber-like materials under complex coupling loading.     

Microscale Experiments at Elevated Temperatures Evaluated with Digital Image Correlation

The methods of uniform heating and resistive (Joule) heating for microscale freestanding surface-micromachined thin metal film specimens were evaluated by a combination of full-field strain measurements by optical microscopy/Digital Image Correlation (DIC) and microscopic infrared (IR) imaging. The efficacy of each method was qualitatively and quantitatively evaluated with the aid of strain fields and IR-obtained temperature distributions along 850 nm thick freestanding microscale specimens subjected to uniaxial tension while heated by each method. The strain and temperature fields were quite uniform in experiments carried out with uniform specimen heating except for minor end-effects at the specimen grips. However, the resistively heated specimens showed highly uneven temperature distribution varying by 50°C along the 1,000 μm specimen gauge length. This high temperature gradient resulted in strain localization and 40% reduction in yield and ultimate tensile strengths of resistively heated specimens compared to the uniformly heated ones. Therefore, it is concluded that resistive heating is not a reliable method for conducting microscale temperature experiments with metallic films.     

Spontaneous instability of soft thin films on curved substrates due to van der Waals interaction

The linear bifurcation theory is used to investigate the stability of soft thin films bonded to curved substrates. It is found that such a film can spontaneously lose its stability due to van der Waals or electrostatic interaction when its thickness reduces to the order of microns or nanometers. We first present the generic method for analyzing the surface stability of a thin film interacting with the substrate and then discuss several important geometric configurations with either a positive or negative mean curvature. The critical conditions for the onset of spontaneous instability in these representative examples are established analytically. Besides the surface energy and Poisson's ratio of the thin film, the curvature of the substrate is demonstrated to have a significant influence on the wrinkling behavior of the film. The results suggest that one may fabricate nanopatterns or enhance the surface stability of soft thin films on curved solid surfaces by modulating the mechanical properties of the films and/or such geometrical properties as film thickness and substrate curvature. This study can also help to understand various phenomena associated with surface instability.     

THEORY OF DIELECTRIC ELASTOMERS

In response to a stimulus, a soft material deforms, and the deformation provides a function. We call such a material a soft active material （SAM）. This review focuses on one class of soft active materials： dielectric elastomers. When a membrane of a dielectric elastomer is subject to a voltage through its thickness, the membrane reduces thickness and expands area, possibly straining over 100%. The dielectric elastomers are being developed as transducers for broad applications, including soft robots, adaptive optics, Braille displays, and electric generators. This paper reviews the theory of dielectric elastomers, developed within continuum mechanics and thermodynamics, and motivated by molecular pictures and empirical observations. The theory couples large deformation and electric potential, and describes nonlinear and nonequilibrium behavior, such as electromechanical instability and viscoelasticity. The theory enables the finite element method to simulate transducers of realistic configurations, predicts the efficiency of electromechanical energy conversion, and suggests alternative routes to achieve giant voltage-induced deformation. It is hoped that the theory will aid in the creation of materials and devices.     

薄膜拉伸褶皱失稳力学进展

薄膜拉伸起皱现象在自然界和现代工业中普遍存在，在过去二十年里，引发了学者们极大的研究兴趣．这种起皱失稳行为源自薄膜能与弯曲能之间的非线性竞争．我们回顾从本世纪初至今薄膜拉伸褶皱失稳力学研究进展，将其分为两个阶段：前十年的研究主要局限于薄膜小应变(~1 %)起皱现象，而后十年的兴趣主要集中在有限应变起皱-再稳定（孤立中心分岔）行为，在过度拉伸(~30 %)时褶皱最终消失．定量理解、预测和追踪这种强非线性力学行为的需求推进了有限应变板壳理论和数值计算方法的发展，不仅促进了对薄膜起皱-消皱机理的深入理解，也为无褶皱膜结构设计和薄膜表面形貌调控提供了新的思路．     

Elastic waves guided by a material interface

The propagation of interfacial small-amplitude waves along a rectilinear thin film separating two pre-stressed, incompressible, elastic media is addressed. The film is modelled as a material surface possessing its own mass density and normal and flexural stiffnesses. It is shown that these features induce dispersion as the obtained secular equations are polynomials of the second degree in the wavenumber when bending stiffness is absent (membrane-like interface), and of the fourth degree otherwise (plate-like interface). In both case, beyond the modified Stoneley mode, a bending mode for the interface, an additional propagating wave can exist, with amplitude polarized along the interface (extensional mode). The associated bifurcation problem is analyzed with focus on the effects of compressive residual forces at the interface. The buckling strain of a compressed metal layer embedded in an elastomeric medium is computed also with an exact approach, to provide the range of validity of the proposed simplified model of material interface.     

A microbeam bending method for studying stress-strain relations for metal thin films on silicon substrates

We have developed a microbeam bending technique for determining elastic-plastic, stress-strain relations for thin metal films on silicon substrates. The method is similar to previous microbeam bending techniques, except that triangular silicon microbeams are used in place of rectangular beams. The triangular beam has the advantage that the entire film on the top surface of the beam is subjected to a uniform state of plane strain as the beam is deflected, unlike the standard rectangular geometry where the bending is concentrated at the support. To extract the average stress-strain relations for the film, we present a method of analysis that requires computation of the neutral plane for bending, which changes as the film deforms plastically. This method can be used to determine the elastic-plastic properties of thin metal films on silicon substrates up to strains of about 1%.Utilizing this technique, both yielding and strain hardening of Cu thin films on silicon substrates have been investigated. Copper films with dual crystallographic textures and different grain sizes, as well as others with strong 〈1 1 1〉 textures have been studied. Three strongly textured 〈1 1 1〉 films were studied to examine the effect of film thickness on the deformation properties of the film. These films show very high rates of work hardening, and an increase in the yield stress and work hardening rate with decreasing film thickness, consistent with current dislocation models.     

非稳态非线性油膜力作用下柔性轴裂纹转子的动力特性分析

分析了在动载轴承非稳态非线性油膜力作用下，具有横向裂纹柔性轴Jeffcott转子在非线性涡动影响下的动力特性。通过数值计算表明，在油膜失稳转速前，随着裂纹轴刚度变化比的增大，系统在低转速区域内具有丰富的非线性动力行为，出现倍周期分叉及混沌现象，涡动振幅随转速升高而减小，直到非稳态非线性油膜失稳，在无裂纹转子油膜临界失稳点处发现了类Hopf分叉现象，系统运动由平衡变为拟周期运动；裂纹转子在油膜临界失稳时的系统运动亦为拟周期运动，裂纹转子轴刚度变化对油膜失稳点及油膜失稳之后转子的运动影响不大，转子系统作拟周期运动。