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.     

Buckling of a stiff film bound to a compliant substrate—Part I:: Formulation, linear stability of cylindrical patterns, secondary bifurcations

The buckling of a thin elastic film bound to a compliant substrate is studied: we analyze the different patterns that arise as a function of the biaxial residual compressive stress in the film. We first clarify the boundary conditions to be used at the interface between film and substrate. We carry out the linear stability analysis of the classical pattern made of straight stripes, and point out secondary instabilities leading to the formation of undulating stripes, varicose, checkerboard or hexagonal patterns. Straight stripes are found to be stable in a narrow window of load parameters only. We present a weakly nonlinear post-buckling analysis of these patterns: for equi-biaxial residual compression, straight wrinkles are never stable and square checkerboard patterns are found to be optimal just above threshold; for anisotropic residual compression, straight wrinkles are present above a primary threshold and soon become unstable with respect to undulating stripes. These results account for many of the previously published experimental or numerical results on this geometry.     

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.     

Kinetic wrinkling of an elastic film on a viscoelastic substrate

A compressed elastic film on a compliant substrate can form wrinkles. On an elastic substrate, equilibrium and energetics set the critical condition and select the wrinkle wavelength and amplitude. On a viscous substrate, wrinkle grows over time and the kinetics selects the fastest growing wavelength. More generally, on a viscoelastic substrate, both energetics and kinetics play important roles in determining the critical condition, the growth rate, and the wavelength. This paper studies the wrinkling process of an elastic film on a viscoelastic layer, which in turn lies on a rigid substrate. The film is elastic and modeled by the nonlinear von Karman plate theory. The substrate is linear viscoelastic with a relaxation modulus typical of a cross-linked polymer. Beyond a critical stress, the film wrinkles by the out-of-plane displacement but remains bonded to the substrate. This study considers plane strain wrinkling and neglects the in-plane displacement. A classification of the wrinkling behavior is made based on the critical conditions at the elastic limits, the glassy and rubbery states of the viscoelastic substrate. Linear perturbation analyses are conducted to reveal the kinetics of wrinkling in films subjected to intermediate and large compressive stresses. It is shown that, depending on the stress level, the growth of wrinkles at the initial stage can be exponential, accelerating, linear, or decelerating. In all cases, the wrinkle amplitude saturates at an equilibrium state after a long time. Subsequently, both amplitude and wavelength of the wrinkle evolve, but the process is kinetically constrained and slow compared to the initial growth.     

Imperfection sensitivity of the post-buckling behavior of higher-order shear deformable functionally graded plates

This paper investigates the sensitivity of the post-buckling behavior of shear deformable functionally graded plates to initial geometrical imperfections in general modes. A generic imperfection function that takes the form of the product of trigonometric and hyperbolic functions is used to model various possible initial geometrical imperfections such as sine type, local type, and global type imperfections. The formulations are based on Reddy’s higher-order shear deformation plate theory and von Karman-type geometric nonlinearity. A semi-analytical method that makes use of the one-dimensional differential quadrature method, the Galerkin technique, and an iteration process is used to obtain the post-buckling equilibrium paths of plates with various boundary conditions that are subjected to edge compressive loading together with a uniform temperature change. Special attention is given to the effects of imperfection parameters, which include half-wave number, amplitude, and location, on the post-buckling response of plates. Numerical results presented in graphical form for zirconia/aluminum (ZrO₂/Al) graded plates reveal that the post-buckling behavior is very sensitive to the L2-mode local type imperfection. The influences of the volume fraction index, edge compression, temperature change, boundary condition, side-to-thickness ratio and plate aspect ratio are also discussed.     

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.     

热过屈曲功能梯度壁板的气动弹性颤振

功能梯度材料的宏观物理性能随空间位置连续变化,能充分减少不同组份材料结合部位界面性能的不匹配因素.功能梯度壁板用作高速飞行器的热防护结构,能有效消除气动加热带来的壁板内部热应力集中.本文考虑热过屈曲变形引入的结构几何非线性,分析功能梯度壁板的气动弹性颤振边界.基于幂函数材料分布假设,采用混合定律计算功能梯度材料的等效力学性能.根据一阶剪切变形板理论、冯·卡门应变-位移关系和一阶活塞理论,基于虚功原理建立超声速气流中受热功能梯度壁板的非线性气动弹性有限元方程.采用牛顿-拉弗森迭代法数值求解壁板的热屈曲变形,分析超声速气流对热屈曲变形的影响机理.在壁板热过屈曲的静力平衡位置分析动态稳定性,确定了壁板的颤振边界.研究表明,当陶瓷-金属功能梯度壁板的组份材料沿厚度方向梯度分布时,会破坏结构的对称性导致壁板在面内热应力作用下发生指向金属侧的热屈曲变形.超声速气流中壁板热屈曲变形最大的位置随气流速压增大向下游推移,并伴随屈曲变形量的减小.热过屈曲壁板的几何非线性效应会提高壁板的颤振边界,这种影响在高温、低无量纲速压且壁板发生大挠度热屈曲变形时表现显著.较高无量纲气流速压下由于壁板的热屈曲变形被气动力限定在小挠度范围,几何非线性效应不明显.      

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.     

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.     

功能梯度压杆后屈曲形态的数值计算

基于轴线可伸长杆的过屈曲精确数学模型,采用打靶法对两端简支功能梯度压杆后屈曲行为进行了数值分析.其中假设功能梯度梁的材料性质沿厚度方向按照幂函数连续变化.给出了不同梯度指标下FGM杆的后屈曲特征曲线,并与金属和陶瓷两种单相材料杆的相应特性进行了比较,分析和讨论了载荷、材料的梯度指数、长细比对杆后屈曲平衡路径的影响.     

Elasto-plastic buckling and post-buckling analysis of sandwich plates with functionally graded metal-metal face sheets and interfacial damage简

Based on the elasto-plastic theory, considering the effect of spherical stress tensor on the elasto-plastic deformation and using the slicing treatment to deal with the plasticity of functionally graded coatings, the elasto-plastic increment constitutive equations of the sandwich plates with functionally graded metal-metal face sheets can be derived. Applying the weak bonded theory to the interfacial constitutive relation and taking into account the geometric nonlinearity, the nonlinear increment differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets are obtained by the minimum potential energy principle. The finite difference method and the iterative method are used to obtain the post-buckling path. When the effect of geometrical nonlinearity of the plate is ignored, the elasto-plastic critical buckling load of the sandwich plates with functionally graded metal-metal face sheets can be solved by the Galerkin method and the iterative method. In the numerical examples, the effects of the interface damages, the induced load ratio, the functionally graded index, and the geometry parameters on the elasto-plastic post-buckling path and the elasto-plastic critical buckling load are investigated.     

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.     

Thermo-mechanical post-buckling of FGM cylindrical panels with temperature-dependent properties

This paper presents thermo-mechanical post-buckling analysis of cylindrical panels that are made of functionally graded materials (FGMs) with temperature-dependent thermo-elastic properties that are graded in the direction of thickness according to a simple power law distribution in terms of the volume fractions of the constituents. The panel is initially stressed by an axial load, and is then subjected to a uniform temperature change. The theoretical formulations are based on the classical shell theory with von-Karman–Donnell-type nonlinearity. The effect of initial geometric imperfection is also included. A differential quadrature (DQ) based semi-analytical method combined with an iteration process is employed to predict the critical buckling load (where it is applicable) and to trace the post-buckling equilibrium path of FGM cylindrical panels under thermo-mechanical loading. Numerical results are presented for panels with silicon nitride and nickel as the ceramic and metal constituents. The effects of temperature-dependent properties, volume fraction index, axial load, initial imperfection, panel geometry and boundary conditions on the thermo-mechanical post-buckling behavior are evaluated in detail through parametric studies.     

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.     

Creep buckling and post-buckling analysis of the laminated piezoelectric viscoelastic functionally graded plates

The creep buckling and post-buckling of the laminated piezoelectric viscoelastic functionally graded material (FGM) plates are studied in this research. Considering the transverse shear deformation and geometric nonlinearity, the Von Karman geometric relation of the laminated piezoelectric viscoelastic FGM plates with initial deflection is established. And then nonlinear creep governing equations of the laminated piezoelectric viscoelastic FGM plates subjected to an in-plane compressive load are derived on the basis of the elastic piezoelectric theory and Boltzmann superposition principle. Applying the finite difference method and the Newmark scheme, the whole problem is solved by the iterative method. In numerical examples, the effects of geometric nonlinearity, transverse shear deformation, the applied electric load, the volume fraction and the geometric parameters on the creep buckling and post-buckling of laminated piezoelectric viscoelastic FGM plates with initial deflection are investigated.     

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.     

Post-buckling analysis of FGM annular sector plates based on three dimensional elasticity graded finite elements

In this article, post-buckling and non-linear bending analysis of functionally graded annular sector plates based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor is considered. In-plane normal compressive loads have been applied to either radial, circumferential, or all edges of annular sector plates. Material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while Poisson׳s ratio is assumed to be constant. The governing equations are developed based on the principle of minimum total potential energy and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio, loading condition and two different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to the stretching–bending coupling effects of the FGMs, the post-buckling behavior of movable simply supported FGM plates is not of the bifurcation-type buckling. Moreover, FGM annular sector plates subjected to uniaxial compression at radial edges show a non-linear bending behavior with unique and stable equilibrium paths following a flattening feature.     

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。     

Thermal post-bunkling analyses of functionally graded material rod

The non-linear governing differential equations of immovably simply supported functionally graded material (FGM) rod subjected to thermal loads were derived. The thermal post-buckling behaviors of FGM rod made of ZrO2 and Ti-6Al-4Vwere analyzed by shooting method. Firstly, the thermal post-buckling equilibrium paths of the FGM rod with different gradient index in the uniform temperature field were plotted, and compared with the behaviors of the homogeneous rods made of ZrO2 and Ti-6A1-4V materials, respectively. For given value of end rotation angles, the influence of gradient index on the thermal post-buckling behaviors of FGM rod was discussed. Secondly, the thermal post-buckling characteristics of the FGM rod were analyzed when the temperature difference parameter is changed while the bottom temperature parameter remains constant, and when the bottom temperature parameter is changed while the temperature difference parameter remains constant, and compared with the characteristics of the two homogeneous material rods.     

Thermal buckling and postbuckling of FGM circular plates with in-plane elastic restraints

Based on von Karman's plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material(FGM) circular plates with inplane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.