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.     

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.     

On out-of-plane buckling of anisotropic elastic plate subjected to a simple shear

Out-of-plane buckling of anisotropic elastic plate subjected to a simple shear is investigated. From exact 3-D equilibrium conditions of anisotropic elastic body with a plane of elastic symmetry at critical configuration, the eqution for buckling direction (buckling wave direction) parameter is derived and the shape functions of possible buckling modes are obtained. The traction free boundary conditions which must hold on the upper and lower surfaces of plate lead to a linear eigenvalue problem whose nontrivial solutions are just the possible buckling modes for the plate. The buckling conditions for both flexural and barreling modes are presented. As a particular example of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate, which is subjected to simple shear along a direction making an arbitrary angle of θ with respect to an elastic principal axis of materials, is analyzed. The buckling direction varies with θ and the critical amount of shear. The numerical results show that only the flexural mode can indeed exist. Project supported by the National Natural Science Foundation of China (No. 19772032).     

Local-buckling-induced elastic interaction between inclusions in a free-standing film

The local-buckling-induced elastic interaction between two circular inclusions in a free-standing film is reported using numerical simulation. The simulation relies on a continuum model based on the modified Föppl-von Kármán plate theory for a film with arbitrarily distributed eigenstrain and eigencurvature. It is shown that due to the overlapping of the nonlinear local buckling the elastic interaction between the two inclusions with the same eigencurvature is repulsive, while the interaction between them with the opposite eigencurvature is attractive. The interaction strength in both cases decays with their mutual distance. In addition, the inclusion with positive/negative eigenstrain above critical values can trigger an axisymmetric/non-axisymmetric buckling, respectively, and the buckling induced elastic interaction between the two inclusions with eigenstrain shows a nonmonotonic behavior.     

Analytic postbuckling solution of a pre-stressed infinite beam bonded to a linear elastic foundation

The postbuckling deflection of an infinite beam that is bonded to a linear elastic foundation and is subjected to an internal compressive stress is analyzed. The nonlinear equilibrium equation that governs the problem considers extensional deformation of the beam. An analytic solution of the nonlinear equilibrium equation is presented and is found to be in good agreement with numerical simulations of the problem. The numerical simulations confirm that for a linear elastic foundation the postbuckling deflection is periodic. The analytic solution shows that the postbuckling wavelength is unaffected by the level of internal stress, and is equal to the wavelength at the critical state.     

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.     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

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.     

Postbuckling instability of nonlinear nanobeam with geometric imperfection embedded in elastic foundation

In this paper, the static instability of a nanobeam with geometrical imperfections that is embedded in elastic foundation is investigated. Size-dependent effect is included in the nonlinear model. It is argued that nonlocal parameter may render the nanobeam initially unstable. Static response is studied and the condition for instability is stated. The exact postbuckling solution for both the straight and curved nanobeam is presented. It is shown that the bifurcation diagram of a curved nanobeam with initial sinusoidal configuration is similar to that of a straight nanobeam in its nearest buckling mode. The results are verified with pervious relevant works on straight nanobeams and classical theory of curved beams and excellent agreement is shown.     

Buckling of a beam subjected to electromagnetic force and its stabilization by controlling the perturbation of the bifurcation

For a beam subjected to electromagnetic force, magnetoelastic buckling due to the increase of such force is theoretically investigated by taking account of the nonlinearity of the electromagnetic force and the elastic force of the beam. Using Liapunov-Schmidt method and center manifold theory, the equilibrium space, the bifurcation set and the bifurcation diagram are theoretically derived. Also, the effect of the higher modes other than the buckling mode on the mode shape of the postbuckling state is discussed. Furthermore, a control method to stabilize the magnetoelastic buckling is proposed, and the unstable equilibrium state of the beam in the postbuckling state, i.e., the straight position of the beam, is stabilized by controlling the perturbation of the bifurcation.     

冲击载荷作用下矩形薄板的弹性动力屈曲

为了研究冲击载荷作用下考虑应力波效应弹性矩形薄板的动力屈曲,根据动力屈曲发生瞬间的能量转换和守恒准则,导出板的屈曲控制方程和波阵面上的补充约束条件,真实的屈曲位移应同时满足控制方程和波阵面上的附加约束条件。满足上述条件,建立了该问题的完整数值解法,对屈曲过程中冲击载荷、屈曲模态和临界屈曲长度之间的关系进行研究,定量计算了横向惯性效应对提高薄板动力屈曲临界应力的贡献。研究表明:板的厚宽比一定时,临界屈曲长度随冲击载荷的增大而减小;由于屈曲时的横向惯性效应,应力波作用下薄板一阶临界力参数是相应边界板的静力失稳临界力参数的1.5倍;随着边界约束逐渐减弱,板临界力参数逐渐减小,动力特征参数逐渐增大。     

基于三维弹性杆模型的DNA拉伸响应的研究

通过Young-Laplace方程将界面张力引入Kirchhoff方程,并结合Gibbs与Langmuir吸附方程建立了受溶液浓度影响的三维DNA弹性杆模型。基于此模型,引入DNA端部的边界条件,运用打靶法来模拟计算溶液中的DNA链段受端部拉力作用下的几何构型。进一步分析了在界面能与弹性应变能的耦合作用下,DNA链段平衡构型的形状与尺寸的变化规律。     

Postbuckling analysis of columns resting on an elastic foundation

The postbuckling response of perfect and geometrically imperfect elastic columns resting on an elastic Winkler type foundation is thoroughly discussed. This is established by employing an approximate analytic technique leading to very reliable results in the vicinity of the critical state. It was found that the critical state of perfect columns is a stable symmetric bifurcation point and consequently there is no sensitivity to initial geometrical imperfections. Moreover, a simple but readily analyzed mechanical model is proposed to simulate the salient features of buckling mechanism of the column on elastic foundation with those of the model. The simplicity, reliability and efficiency of the proposed analysis as well as the successful modeling of the buckling mechanism of the column by that of a single mode mechanical model are illustrated with the aid of numerical examples.     

Influence of the connecting condition on the dynamic buckling of longitudinal impact for an elastic rod

The stress wave propagation law and dynamic buckling critical velocity are formulated and solved by considering a general axial connecting boundary for a slender elastic straight rod impacted by a rigid body. The influence of connecting stiffness on the critical velocity is investigated with varied impactor mass and buckling time. The influences of rod length and rod mass on the critical velocity are also discussed. It is found that greater connecting stiffness leads to larger stress amplitude, and further results in lower critical velocity. It is particularly noteworthy that when the connecting stiffness is less than a certain value,dynamic buckling only occurs before stress wave reflects off the connecting end. It is also shown that longer rod with larger slenderness ratio is easier to buckle, and the critical velocity for a larger-mass rod is higher than that for a lighter rod with the same geometry.     

Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.     

From wrinkling to global buckling of a ring on a curved substrate

We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler–Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or global buckling of the structure. Our model provides an expression for the critical pressure, as well as a phase diagram that rationalizes the transition between instability modes. Towards assessing the role of curvature, we compare our results for the critical stress and the wrinkling wavelength to their planar counterparts. We show that the critical stress is insensitive to the curvature of the substrate, while the wavelength is only affected due to the permissible discrete values of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we contrast our analytical predictions against finite element simulations.     

Piercing a liquid surface with an elastic rod: Buckling under capillary forces

When a thin elastic structure comes in contact with a liquid interface, capillary forces can be large enough to induce elastic deformations. This effect becomes particularly relevant at small scales where capillary forces are predominant, for example in microsystems (micro-electro-mechanical systems or microfluidic devices) under humid environments. In order to explore the interaction between capillarity and elasticity, we have developed a macroscopic model system in which an initially immersed vertical elastic rod is raised through a horizontal liquid surface. We follow a combined approach of experiments, theory and numerical simulations to study this system. In spite of its apparent simplicity, our experiment reveals a complex phase diagram, involving large hysteretic behaviour. We employ Kirchhoff equations for thin elastic rods and use path-following methods from which we obtain a variety of equilibrium states and associated transitions that are in excellent qualitative and quantitative agreement with those observed experimentally.     

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.     

Post-buckling analysis of slender elastic vertical rods subjected to terminal forces and self-weight

This article presents the behavior of slender elastic rods subjected to axial terminal forces and self-weight. The mathematical formulation is presented, a solution is sought for a double-hinged boundary condition and the analysis is carried out for different values of non-dimensional weight. The formulation derives from geometrical compatibility, equilibrium of forces and moments and constitutive relations yielding a set of six first order non-linear ordinary differential equations with boundary conditions specified at both ends, which characterizes a complex two-point boundary value problem. Furthermore, a perturbation method is used to find the critical buckling loads and initial post-buckling solutions. A numerical integration scheme based on a three parameter shooting method is employed in the post-buckling solutions.     

弹性杆在刚性块轴向撞击下的动力屈曲

基于能量关系,应用功率原理对弹性杆在刚性块轴向撞击下的动力屈曲问题进行了讨论。用幂级数解法,理论上给出了该问题的级数解,同时考虑了应力波传播及反射对屈曲的影响。通过理论分析和数值计算,得到了临界速度与冲击质量以及临界时间的关系,给出了发生屈曲时的临界条件。