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.     

网链缠结对凝胶薄膜起皱的影响

凝胶薄膜在变形时易发生屈曲、起皱等失稳现象,这在凝胶薄膜的应用中是非常重要的．近年来,针对凝胶薄膜的屈曲、起皱失稳行为,越来越多的科研人员尝试从力学角度进行分析．但是大多数的研究是基于Flory-Rehner弹性凝胶理论,未考虑凝胶网链缠结引起的物理交联对凝胶自由能的影响,模型精度不高．本文采用Edwards-Vilgis所提出的Slip-link模型对平面内起皱的凝胶薄膜进行分析,研究了不伸展参数、滑移参数对聚合物凝胶增量模量的影响以及不伸展参数、滑移参数、基底材料泊松比对凝胶薄膜起皱时的临界波长和临界应力的影响．结果表明：化学势在一定范围内变化时,随着化学势的增加,增量模量、临界波长、临界应力减小;不可伸长参数越大,增量模量、临界波长及临界应力越大;滑移参数越大,增量模量、临界波长及临界应力越小．     

Stretch-induced wrinkling of polyethylene thin sheets: Experiments and modeling

This paper presents a study on stretch-induced wrinkling of thin polyethylene sheets when subjected to uniaxial stretch with two clamped ends. Three-dimensional digital image correlation was used to measure the wrinkling deformation. It was observed that the wrinkle amplitude increased as the nominal strain increased up to around 10%, but then decreased at larger strain levels. This behavior is consistent with results of finite element simulations for a hyperelastic thin sheet reported previously (Nayyar et al., 2011). However, wrinkles in the polyethylene sheet were not fully flattened out at large strains (>30%) as predicted for the hyperelastic sheet, but exhibited a residual wrinkle whose amplitude depended on the loading rate. This is attributed to the viscoelastic response of the material. Two different viscoelastic models were adopted in finite element simulations to study the effects of viscoelasticity on wrinkling and to improve the agreement with the experiments, including residual wrinkles and rate dependence. It is found that a parallel network model of nonlinear viscoelasticity is suitable for simulating the constitutive behavior and stretch-induced wrinkling of the polyethylene sheets.     

Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets

Wrinkles are commonly observed in stretched thin sheets and membranes. This paper presents a numerical study on stretch-induced wrinkling of hyperelastic thin sheets based on nonlinear finite element analyses. The model problem is set up for uniaxial stretching of a rectangular sheet with two clamped ends and two free edges. A two-dimensional stress analysis is performed first under the plane-stress condition to determine stretch-induced stress distribution patterns in the elastic sheets, assuming no wrinkles. As a prerequisite for wrinkling, development of compressive stresses in the transverse direction is found to depend on both the length-to-width aspect ratio of the sheet and the applied tensile strain in the longitudinal direction. A phase diagram is constructed with four different distribution patterns of the stretch-induced compressive stresses, spanning a wide range of aspect ratio and tensile strain. Next, an eigenvalue analysis is performed to find the potential buckling modes of the elastic sheet under the prescribed boundary conditions. Finally, a nonlinear post-buckling analysis is performed to show evolution of stretch-induced wrinkles. In addition to the aspect ratio and tensile strain, it is found that the critical condition for wrinkling and the post-buckling behavior both depend sensitively on the sheet thickness. In general, wrinkles form only when both the magnitude and the distribution area of the compressive stresses are sufficiently large. The wrinkle wavelength decreases with increasing strain, in good agreement with the prediction by a scaling analysis. However, as the tensile strain increases, the wrinkle amplitude first increases and then decreases, eventually flattened beyond a moderately large critical strain, in contrast to the scaling analysis.     

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.     

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.     

Ratcheting,wrinkling and collapse of tubes under axial cycling

Circular tubes compressed into the plastic range first buckle into axisymmetric wrinkling modes. Initially the wrinkle amplitude grows with increasing load. The wrinkles gradually induce a reduction in axial rigidity eventually leading to a limit load instability followed by collapse. The two instabilities can be separated by strain levels of a few percent. This work investigates whether a tube that develops small amplitude wrinkles can be subsequently collapsed by persistent cycling. The problem is first investigated experimentally using SAF 2507 super-duplex steel tubes with D/t of 28.5. The tubes are first compressed to strain levels high enough for mild wrinkles to form; they are then cycled axially under stress control about a compressive mean stress. This type of cycling usually results in material ratcheting or accumulation of compressive strain; here it is accompanied by accumulation of structural damage due to the growth of the amplitude of the initial wrinkles. The tube average strain initially grows nearly linearly with the number of cycles, but as a critical value of wrinkle amplitude is approached, wrinkling localizes, the rate of ratcheting grows exponentially and the tube collapses. The rate of ratcheting and the number of cycles to failure depend on the initial compressive pre-strain and on the amplitude of the stress cycles. However, collapse was found to occur when the accumulated average strain reaches the value at which the tube localizes under monotonic compression. A custom shell model of the tube with initial axisymmetric imperfections, coupled to a cyclic plasticity model, are presented and used to simulate the series of experiments performed successfully. A sensitivity study of the formulation to the imperfections and to key constitutive model parameters is then performed.     

Controlled wrinkling analysis of thin films on gradient substrates

The paper investigates continuously changing wrinkle patterns of thin films bonded to a gradient substrate. Three types of gradient substrates including exponential,power-law, and symmetry models are considered. The Galerkin method is used to discretize the governing equation of film bonded to gradient substrates. The wavelength and the normalized amplitude of the wrinkles for substrates of various material gradients are obtained. The numerical simulation based on the finite element method(FEM) is used to evolve the wrinkle patterns. The result agrees well with that of the analytical model.It is concluded that localization of wrinkle patterns strongly depends on the material gradient. The critical membrane force depends on both the minimum value of wrinkle stiffness and the gradient of wrinkle stiffness when the wrinkle stiffness is at its minimum.This work provides a better understanding for local wrinkle formation caused by gradient substrates.     

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.     

基于张力场理论的薄膜褶皱研究评述

张力场理论是较早提出的研究薄膜褶皱的理论,该理论忽略了薄膜的弯曲刚度, 认为在褶皱 区薄膜处于单轴的应力状态,褶皱的方向为大主应力方向,垂直于褶皱方向的小主应力为零. 基于这种方法提出了很多褶皱的分析模型. 这些分析模型通过引入参数对薄膜的本构关系或 变形梯度进行修正,然后通过数值的方法进行求解,可以得到褶皱形成以后的应力分布,及 褶皱的方向. 其主要的缺点是不能得到褶皱的波长、幅度及数量等信息. 基于能量方法的褶 皱分析,首先要假定褶皱的变形模式,然后通过能量关系得到褶皱幅度和波长的表达式. 本 文对比分析薄膜褶皱的不同分析方法,并指出了薄膜褶皱研究的发展趋势.     

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.     

空间薄膜阵面结构褶皱的实验研究

阵面薄膜上的褶皱会显著影响航天器性能,褶皱控制也是薄膜航天器研发的难点。本文采用白光扫描技术,测试了一个正方形薄膜阵面结构在纯剪切作用下的褶皱特性。测试结果显示,随着剪切力的增大,结构膜面褶皱数增加,褶皱幅度增大,膜面平整度降低。随着膜面预应力的增大,膜面上的褶皱数略有增大,褶皱幅度有所减小,膜面平整度有所提高。测试结果还显示,膜面上粘结缝的存在、以及粘结缝与剪切力间的方位关系对膜面褶皱有复杂的影响。论文所得结论对薄膜阵面结构设计研发具有参考价值。     

BUCKLING ANALYSIS OF ELASTIC FILM BONDED ON SMP SUBSTRATE1)

形状记忆聚合物具有形状变化后在特定条件下可恢复的特点,因此作为一种柔性基底材料在柔性电子中得到广泛应用。对于形状记忆聚合物基底和弹性薄膜组成的双层结构,当 基底收缩时,其表面的弹性薄膜可以形成屈曲波形。针对基底收缩过程中波形的变化, 本文实验测得形状记忆聚合物材料在不同温度下的 属性,结合一维应变恢复函数,利用柔性基底表面薄膜屈曲波形参数(波幅、波长等)表达式,求解得到了在基底收缩的过程中,弹性薄膜屈曲波形的变化规律,和实验结果吻合很好。     

Post-buckling evolution of wavy patterns in trapezoidal film/substrate bilayers

This paper investigates quantitatively the post-buckling response of irregular wrinkles in a trapezoidal film/substrate bilayer. The geometric gradient can change the wrinkling profile to create ribbed and graded structural patterns with variations of wave direction, amplitude and wavelength. The tapered angle and edge dimension are examined numerically using a nonlinear shell/solid coupled finite element model that incorporates a path-following continuation technique, which explores their influences on secondary bifurcations, localization and surface mode transition. For instance, the competition between plate-like and beam-like post-buckling behavior is discussed. An analysis of graded amplitude is also provided based on Fourier envelope equations of beam/foundation models, which gives an insightful understanding of these fading wrinkles that differ from the ones usually observed in rectangular geometric cases. The results of this work can be used to guide the design of geometrically gradient film/substrate systems to achieve desired wavy instability patterns.     

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,...     

Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

Instability of an elastic film on a viscous layer

A flat, compressed elastic film on a viscous layer is unstable. The film can form wrinkles to reduce the elastic energy. In this paper, we are interested in the two-dimensional models for thin films bonded to a viscous layer and in particular we focus on generic instabilities evidenced in this context by Suo and coworkers [Huang, Z., Hong, W., Suo, Z., 2005. Non linear analyses of wrinkles in a film bonded to a compliant substrate. J. Mech. Phys. Solids 53, 2101–2118; Lo, Y.H., 1991. New approach to grow pseudomorphic structures over the critical thickness. Appl. Phys. Lett. 59, 2311–2320]. We present a rigorous linear perturbation analysis for anisotropic materials, that allows the prediction of both the orientation of the corrugations of the thin film, and the wavelength that maximize the growth velocity. Finally, we compare our theoretical estimates to experimental results for a In_0.65Ga_0.35As alloy constraint to InP.     

Localized ridge wrinkling of stiff films on compliant substrates

Wrinkling of thin stiff films on thick compliant elastomeric substrates subject to plane strain compression is considered for cases in which the substrate is pre-stretched prior to film attachment. Advanced wrinkling modes are investigated that evolve as the systems are compressed beyond the onset of the primary sinusoidal wrinkling mode. If the substrate pre-stretch is greater than about 40%, an advanced mode in the form of a series of well-spaced ridges separated by relatively flat film is observed in the simulations. Our experiments reveal a localization mode in the form of alternating packets of large and small amplitude wrinkles, but not ridges, while ridge formation has been observed in other recent experiments. Measurements of undulation amplitudes have been made for wrinkle fields of stiff films formed by oxidation of the surface of pre-stretched PDMS substrates. Simulations have been performed with a finite element model and an analytical film/substrate model. The formation of the ridge mode is a consequence of the altered nonlinearity of the substrate produced by the pre-stretch. The role of the tangential substrate stiffness in suppressing localization at the ridges is also highlighted. If there is no substrate pre-stretch, or if the substrate is pre-compressed, the primary sinusoidal mode gives way to an entirely different sequence of advanced modes usually entailing period doubling followed by folding. The nature of substrate nonlinearity that leads to ridges or folds is discussed.     

Wrinkle-crease interaction behavior simulation of a rectangular membrane under shearing

Wrinkling analysis of a rectangular membrane with a single crease under shearing is performed to understand the wrinkle-crease interaction behaviors.The crease is considered by introducing the residual stresses from creasing and the effective modulus into the baseline configuration with assumed circular cross-sectional crease geometry.The wrinkling analysis of the creased membrane is then performed by using the direct perturb-force(DP) simulation technique which is based on our modified displacement components(MDC) method.Results reveal that the crease may influence the stress transfer path in the membrane and further change the wrinkling direction.The crease appears to improve the bending stiffness of the membrane which has an effective resistance on the wrinkling evolution.The effects of the crease orientation on wrinkle-crease interaction are studied toward the end of this paper.The results show that the wrinkling amplitude,wavelength,and direction increase as the crease orientation increases,and the wrinkling number decreases with the increasing crease orientation.These re-sults will be of great benefit to the analysis and the control of the wrinkles in the membrane structures.     

对角受拉方膜褶皱变形幅值的理论预测及实验验证

柔性薄膜结构广泛应用于航天飞行器的关键部件.形面平整度是影响薄膜结构性能的主要因素之一.褶皱幅值是评价薄膜反射面天线形面平整度的重要指标.褶皱幅值的大小与垂直于褶皱方向的横向应变密切相关.本文基于薄板稳定性理论,针对对角受拉方形薄膜建立了一个能够准确预测其褶皱变形幅值的理论模型.该模型考虑了横向拉伸力对薄膜变形的影响,将垂直于褶皱方向的位移分解为由泊松效应造成的横向收缩位移、由面外变形造成的褶皱位移以及由横向拉伸力造成的拉伸位移三个部分,重新推导了褶皱幅值的理论公式. 基于数字图像相关技术,对受拉方形薄膜进行了散斑实验测试. 利用双目视觉三维测量系统,测量了方形薄膜的三维位移场, 获得了薄膜的三维变形形貌和褶皱波形图,研究了褶皱幅值与拉伸载荷之间的非线性关系. 与已有理论模型相比,该模型进一步提高了褶皱幅值的计算精度, 与实验结果吻合良好.本文呈现的理论研究可为精确数值模型的建立及算法实现提供有意义的指导.