From period-doubling to folding in stiff film/soft substrate system: The role of substrate nonlinearity

Uniaxial compressed stiff films on soft substrates can evolve into the period-doubling and folding instabilities, beyond the onset of sinusoidal wrinkling. The substrate is modeled as a neo-Hookean solid with a pre-stretch prior to film attachment, and its nonlinearity is obtained. Both the pre-stretch and the external nominal strain imposed on the film/substrate system can induce different substrate nonlinearity, and thus have different effects on the post-buckling mode evolution of the system. This study shows that the critical strain of period-doubling instability is linear to the pre-stretch. As the compressive nominal strain increases, the folding mode occurs beyond the onset of period-doubling in both the pre-tension and the pre-compression case, due to the softening/hardening effects for the inward/outward displacements generated by the positive substrate nonlinearity.     

Multiple bifurcations in wrinkling analysis of thin films on compliant substrates

Wrinkling phenomena of stiff thin films on compliant substrates are investigated based on a non-linear finite element model. The resulting non-linear equations are then solved by the Asymptotic Numerical Method (ANM) that gives interactive access to semi-analytical equilibrium branches, which offers considerable advantage of reliability compared with classical iterative algorithms. Bifurcation points are detected through computing bifurcation indicators well adapted to the ANM. The effect of boundary conditions and material properties of the substrate on the bifurcation portrait is carefully studied. The evolution of wrinkling patterns and post-bifurcation modes including period-doubling has been observed beyond the onset of the primary sinusoidal wrinkling mode in the post-buckling range.     

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.     

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.     

Effects of tension–compression asymmetry on the surface wrinkling of film–substrate systems

Many soft materials and biological tissues are featured with the tension–compression asymmetry of constitutive relations. The surface wrinkling of a stiff thin film lying on a compliant substrate is investigated through theoretical analysis and numerical simulations. It is found that the tension–compression asymmetry of the soft substrate not only affects the critical strain of buckling but, more importantly, may also influence the wrinkling pattern that occurs in the film–substrate system under specified loading conditions. Due to this mechanism, the thin film subjected to equi-biaxial compression may first buckle into a hexagonal array of dimples or bulges, instead of the checkerboard pattern, and consequently evolve into labyrinths with further loading. Under non-equi-biaxial compression, the system may buckle either into a parallel bead-chain pattern or a stripe pattern, depending on the substrate nonlinearity and the loading biaxiality. Phase diagrams are established for the wrinkling patterns in a wide range of geometric and mechanical parameters, which facilitate the design of surface patterns with desired properties and functions.     

Nonlinear analyses of wrinkles in a film bonded to a compliant substrate

Subject to a compressive membrane force, a film bonded to a compliant substrate often forms a pattern of wrinkles. This paper studies such wrinkles in a layered structure used in several recent experiments. The structure comprises a stiff film bonded to a compliant substrate, which in turn is bonded to a rigid support. Two types of analyses are performed. First, for sinusoidal wrinkles, by minimizing energy, we obtain the wavelength and the amplitude of the wrinkles for substrates of various moduli and thicknesses. Second, we develop a method to simultaneously evolve the two-dimensional pattern in the film and the three-dimensional elastic field in the substrate. The simulations show that the wrinkles can evolve into stripes, labyrinths, or herringbones, depending on the anisotropy of the membrane forces. Statistical averages of the amplitude and wavelength of wrinkles of various patterns correlate well with the analytical solution of the sinusoidal wrinkles.     

Deformability of thin metal films on elastomer substrates

Recent applications in flexible electronics require that thin metal films grown on elastomer substrates be deformable. However, how such laminates deform is poorly understood. While a freestanding metal film subject to tension will rupture at a small strain by undergoing a necking instability, we anticipate that a substrate will retard this instability to an extent that depends on the relative stiffness and thickness of the film and the substrate. Using a combination of a bifurcation analysis and finite element simulations, we identify three modes of tensile deformation. On a compliant elastomer, a metal film forms a neck and ruptures at a small strain close to that of a freestanding film. On a stiff elastomer, the metal film deforms uniformly to large strains. On an elastomer of intermediate compliance, the metal film forms multiple necks, deforms much beyond the initial bifurcation, and ruptures at a large strain. Our theoretical predictions call for new experiments.     

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.     

粘性基底上溅射薄膜褶皱的边界效应研究

复杂的表面微结构在自然界中普遍存在,它们赋予生物组织优异的物化功能.受此启发,各类仿生结构在工程领域(如微机电系统、能源材料、生物传感等)得到了广泛应用.论文利用磁控溅射技术在粘性的胶体基底上沉积金属铁薄膜,研究了薄膜中自发形成的褶皱结构,重点分析了约束边界和膜厚梯度边界对褶皱形貌的调控效应.实验发现,随着膜厚的增加,...     

Periodic patterns and energy states of buckled films on compliant substrates

Thin stiff films on compliant elastic substrates subject to equi-biaxial compressive stress states are observed to buckle into various periodic mode patterns including checkerboard, hexagonal and herringbone. An experimental setting in which these modes are observed and evolve is described. The modes are characterized and ranked by the extent to which they reduce the elastic energy of the film-substrate system relative to that of the unbuckled state over a wide range of overstress. A new mode is identified and analyzed having nodal lines coincident with an equilateral triangular pattern. Two methods are employed to ascertain the energy in the buckled state: an analytical upper-bound method and a full numerical analysis. The upper-bound is shown to be reasonably accurate to large levels of overstress. For flat films, except at small states of overstress where the checkerboard is preferred, the herringbone mode has the lowest energy, followed by the checkerboard, with the hexagonal, triangular, and one-dimensional modes lowering the energy the least. At low overstress, the hexagonal mode is observed in the experiments not the square mode. It is proposed that a slight initial curvature of the film may play role in selecting the hexagonal pattern accompanied by a detailed analysis. An intriguing finding is that the hexagonal and triangular modes have the same energy in the buckled state and, moreover, a continuous transition between these modes exists involving a linear combination of the two modes with no change in energy. Experimental observations of various periodic modes are discussed with reference to the energy landscape. Discrepancies between observations and theory are identified and open issues are highlighted.     

Elastic interaction between force dipoles on a stretchable substrate

The objective of this paper is to study elastic interaction between force dipoles on the surface of a semi-infinite stretchable substrate. The substrate undergoes a uniform, finite pre-stretch, while the additional deformation induced by a force dipole is assumed infinitesimal. By adopting a neo-Hookean constitutive law, the surface Green function for the pre-stretched substrate is obtained. The result is then used to derive the energy of dipolar interaction on the surface. As an application, mutual interaction between two physisorbed molecules is discussed in detail. The dipole moment of a molecule is found from the elementary intermolecular potential, and its dependence on the pre-stretch is established explicitly. Numerical results indicate that pre-stretches can substantially alter the interaction, and thus provide a controllable way to guide the self-assembly of adsorbed molecules on the substrate surface.     

Buckling and postbuckling of a compressed thin film bonded on a soft elastic layer: a three-dimensional analysis

The wrinkling of a stiff thin film bonded on a soft elastic layer and subjected to an applied or residual compressive stress is investigated in the present paper. A three-dimensional theoretical model is presented to predict the buckling and postbuckling behavior of the film. We obtained the analytical solutions for the critical buckling condition and the postbuckling morphology of the film. The effects of the thicknesses and elastic properties of the film and the soft layer on the characteristic wrinkling wavelength are examined. It is found that the critical wrinkling condition of the thin film is sensitive to the compressibility and thickness of the soft layer, and its wrinkling amplitude depends on the magnitude of the applied or residual in-plane stress. The bonding condition between the soft layer and the rigid substrate has a considerable influence on the buckling of the thin film, and the relative sliding at the interface tends to destabilize the system.     

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.     

薄膜破坏过程数值模拟的MPM方法

将连续与不连续本构模型相结合用于薄膜破坏过程的模拟计算. 采用von Mises关联塑性本构模型描述材料的弹塑性变形过程，在塑性硬化和软化阶段通过检查分岔 是否发生, 从而判断是否有破坏出现，即材料出现应变局部化现象. 一旦出现破坏，则在发生 破坏的区域采用位移不连续(decohesion)本构模型进行模拟, 直到破坏面两侧材料完全分离. 分离阶段采用相应的分离算法同时运用黏性边界条件以提高计算效率. 结果表明结合连续与 不连续本构模型的方法, 能很好地模拟材料从局部化到完全破坏的过程，体现了材料破坏连续 建模的思想，可用于预测材料发生破坏的环境和形式, 从而通过改变材料工作环境提高其使用 寿命；同时数值算例也显示了物质点法(material-point-method, 简称MPM)的健壮性和有 效性.     

Wrinkling of a charged elastic film on a viscous layer

A thin metallic film deposited on a compliant polymeric substrate begins to wrinkle under compression induced in curing process and afterwards cooling of the system. The wrinkle mode depends upon the thin film elasticity, thickness, compressive strain, as well as mechanical properties of the compliant substrate. This paper presents a simple model to study the modulation of the wrinkle mode of thin metallic films bonded on viscous layers in external electric field. During the procedure, linear perturbation analysis was performed for determining the characteristic relation that governs the evolution of the plane-strain wrinkle of the thin films under varying conditions, i.e., the maximally unstable wrinkle mode as a function of the film surface charge, film elasticity and thickness, misfit strain, as well as thickness and viscosity of the viscous layer. It shows that, in proper electric field, thin film may wrinkle subjected to either compression or tension. Therefore, external electric field can be employed to modulate the wrinkle mode of thin films. The present results can be used as the theoretical basis for wrinkling analysis and mode modulation in surface metallic coatings, drying adhesives and paints, and microelectromechanical systems (MEMS), etc.     

Analytical study of wrinkling in thin-film-on-elastomer system with finite substrate thickness

Surface wrinkling, a film bonded on a pre-strained elastomeric substrate can form periodic wrinkling patterns, is a common phenomenon in daily life. In existing theoretical models, the film is much thinner than the substrate so that the substrate can be considered to be elastomeric with infinite thickness. In this paper, the effect of finite substrate thickness is analyzed theoretically for free boundary condition cases. Based on the minimum potential energy principle, a theoretical model is established, and the wave length and amplitude of the wrinkling pattern are obtained. When the thickness of the substrate is more than 200 times larger than the thickness of the film, the results of this study agree well with the results obtained from the previous models for infinite substrate thickness. However, for thin substrates, the effect of finite substrate thickness becomes significant. The model given in this paper accurately describes the effect of finite substrate thickness, providing important design guidelines for future thin-film-on-substrate systems such as stretchable electronic devices.     

Nonlinear analysis of photo-induced wrinkling of glassy twist nematic films on compliant substrates

A kinetics approach is developed for the geometrically nonlinear analysis of photo-induced wrinkling of glassy twist nematic films on soft elastic substrates.In this way,the problem is reduced to finding the steady state of an overdamped evolution system according to a kinetic law,rather than directly solving the coupled nonlinear equations.This enables one to account for the complicated director distribution and obtain the precise wrinkling morphology of the film.Though the approach proposed here is for a twist nematic film,it can be extended to study glassy nematic films with other director distributions.     

Influence of nonlinearities on the accuracy of the analytical solution for the shaft loaded blister test

Finite element analysis (FEA) is employed to study the effects of nonlinearities on the accuracy of the analytical solution for the shaft loaded blister test. The FEA model was validated using constrained blister test measurements showing a good correlation between the experimental and the FEA data. The analytical solution is then compared with the energy release rate obtained from J-integral evaluation in the FEA. For small and large shaft displacements deviations larger than 20% are encountered which is explained with the violation of the membrane limit condition and the onset of plasticity for larger displacements, respectively. Simplifications of the analytical solution are discussed using a random sampling method and it is shown that the thickness ratio between film and substrate can be neglected for thin films on rigid substrates. Further, values for the angular quantity, ω, which is required to calculate the mode mix phase angle are tabulated for the case of thin, elastic films on stiff substrates using a crack surface opening displacement extrapolation method.     

Wrinkle patterns of anisotropic crystal films on viscoelastic substrates

This paper presents a nonlinear mathematical model for evolution of wrinkle patterns of an anisotropic crystal film on a viscoelastic substrate layer. The underlying mechanism of wrinkling has been generally understood as a stress-driven instability. Previously, theoretical studies on wrinkling have assumed isotropic elastic properties for the film. Motivated by recent experimental observations of ordered wrinkle patterns in single-crystal thin films, this paper develops a theoretical model coupling anisotropic elastic deformation of a crystal film with viscoelastic deformation of a thin substrate layer. A linear perturbation analysis is performed to predict the onset of wrinkling instability and the initial evolution kinetics. An energy minimization method is adopted to analyze wrinkle patterns in the equilibrium states. For a cubic crystal film under an equi-biaxial compression, orthogonally ordered wrinkle patterns are predicted in both the initial stage and the equilibrium state. This is confirmed by numerical simulations of evolving wrinkle patterns. By varying the residual stresses in the film, numerical simulations show that a variety of wrinkle patterns (e.g., orthogonal, parallel, zigzag, and checkerboard patterns) emerge as a result of the competition between material anisotropy and stress anisotropy.     

Mixed-mode failure of thin films using laser-generated shear waves

A new test method is developed for studying mixed-mode interfacial failure of thin films using laser generated stress waves. Guided by recent parametric studies of laser-induced tensile spallation, we successfully extend this technique to achieve mixed-mode loading conditions. By allowing an initial longitudinal wave to mode convert at an oblique surface, a high amplitude shear wave is generated in a fused silica substrate and propagated toward the thin-film surface. A shear wave is obtained with amplitude large enough to fail an Al film/fused silica interface and the corresponding shear stress calculated from high-speed interferometric displacement measurements. Examination of the interfaces failed under mixed-mode conditions reveals significant wrinkling and tearing of the film, in great contrast to blister patterns observed in similar Al films failed under tensile loading.