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.     

Modeling of wrinkling of thin circular sheets

A version of the imperfection method is used to investigate the wrinkling (tension buckling) of thin, elastic, homogeneous, isotropic circular plates of uniform thickness undergoing small deflections without inplane twisting. A numerical procedure based on the finite difference approach is employed to quantitatively predict wrinkling loads and wrinkling patterns for three sets of support conditions. Representative numerical results are presented in tabular and graphical formats and used to illustrate interesting aspects of the predictions.     

Buckling and post-buckling of a stiff film resting on an elastic graded substrate

Systems consisting of a hard layer resting on an elastic graded soft substrate are frequently encountered both in nature and industry. In this paper, we study the surface wrinkling problem of such a composite system subjected to in-plane compression. The Young’s modulus of the elastic substrate is assumed to vary along its depth direction. In particular, we investigate two typical variations in the modulus, described by a power function and an exponential function, respectively. Analytical solutions which permit to determine the critical compressive strain for the onset of wrinkling and the wrinkling wavelength are derived. A series of finite element simulations are performed to validate the theoretical solutions and demonstrate the prominent features of the postbuckling evolution of the system. The results may not only find applications in thin-film metrology and surface patterning but also provide insight into the wrinkling phenomena of various living tissues.     

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.     

软铁磁薄板磁弹性屈曲的理论模型

铁磁弹性薄板的磁弹性屈曲问题一直作为电磁——弹性力学相互作用的一个基本模型进行研究，而作用在其磁介质上的磁力计算则是定量理论预测准确与否的关键．到目前为止，文献上已有的理论模型对悬臂铁磁梁式悬臂板在横向磁场中磁弹性屈曲的理论预测值始终高于实验值，有的甚至相差１００％左右．本文基于电磁力计算的微观安培电流模型，严格给出了软铁磁薄板等效横向磁力的宏观计算表达式．在此基础上，建立了电磁——力学相互耦合作用的非线性理论模型．该模型能描述铁磁薄板结构在非均匀横向磁场环境中的磁弹性失稳（或屈曲）．其定量分析采用了有限元法和有限差分法相结合．数值结果显示：本模型给出的磁弹性屈曲的临界磁场值与实验值符合良好．与此同时，文中还对文献中认为较成功的Ｍｏｏｎ－Ｐａｏ模型的基本假设进行了分析．定量结果发现：Ｍｏｏｎ－Ｐａｏ理论模型的基本假设仅在梁式板的长厚比Ｌ／ｈ比较大时（约在２００左右），是可以接受的，而当Ｌ／ｈ较小时，该假设将导致理论值与实验值的较大误差．Ｌ／ｈ比值越小，理论值与实验值的误差越大     

BIOMECHANICAL MODELING OF SURFACE WRINKLING OF SOFT TISSUES WITH GROWTH-DEPENDENT MECHANICAL PROPERTIES

This paper explores growth induced morphological instabilities in biological soft materials.In view of that the growth of a living tissue not only changes its geometry but also can alter its mechanical properties,we suggest a refined volumetric growth model incorporating the effects of growth on the mechanical properties of materials.Analogy between this volumetric growth model and the conventional thermal stress model is addressed for both small and finite deformation problems,which brings great ease for the finite element analysis based on the suggested model.Examples of growth induced surface wrinkling behavior in soft composites,including coreshell soft cylinders and three-layered soft tissues,are explored.The results and discussions foresee possible applications of the model in understanding the correlation between the morphogenesis and growth of soft biological tissues(e.g.skins and tumors),as well as in evaluating the deformation and surface instability behavior of soft artificial materials induced by swelling/shrinkage.     

A theoretical model of magnetoelastic bucking for soft ferromagnetic thin plates

As an essential model of magnetoelastic interaction between magnetic field and mechanical deformation, the study on magnetoelastic buckling phenomenon of soft ferromagnetic plates in a magnetic environment has been conducted. One of the key steps for the theoretical prediction of the critical magnetic field is how to formulate magnetic force exerted on the magnetized medium. Till today, the theoretical predictions, from theoretical models in publications, of the magnetoelastic buckling of ferromagnetic cantilevered beam-plate in transverse magnetic field are all higher than their experimental data. Sometimes, the discrepancy between them is as high as 100%. In this paper, the macroscope formulation of the magnetic forces is strictly obtained from the microscope Amperion current model. After that, a new theoretical model is established to describe the magnetoelastic buckling phenomenon of ferromagnetic thin plates with geometrically nonlinear deformation in a nonuniform transverse magnetic field. The numerical method for quantitative analysis is employed by combining the finite elemental method for magnetic fields and the finite difference method for deformation of plates. The numerical results obtained from this new theoretical model show that the theoretical predictions of critical values of the buckling magnetic field for the ferromagnetic cantilevered beam-plate are in excellent agreement with their experimental data. By the way, the region of applicability to the Moon-Pao's model, or the couple model, is checked by quantitative results. This project was supported in part by the National Natural Science Foundation of China and the Foundation of the SEdC of China for Returned Chinese Scholars from Abroad.     

Swell-induced surface instability of hydrogel layers with material properties varying in thickness direction

Upon swelling in a solvent, a thin hydrogel layer on a rigid substrate may become unstable, developing various surface patterns. Recent experimental studies have explored the possibilities to generate controllable surface patterns by chemically modifying the molecular structures of the hydrogel near the surface. In this paper, we present a theoretical stability analysis for swelling of hydrogel layers with material properties varying in the thickness direction. As a specialization of the general procedure, hydrogel bilayers with different combinations of the material properties are examined in details. For a soft-on-hard bilayer, the onset of surface instability is determined by the short-wave limit, similar to a homogeneous layer. In contrast, for a hard-on-soft bilayer, a long-wave mode with a finite wavelength emerges as the critical mode at the onset of surface instability, similar to wrinkling of an elastic thin film on a compliant substrate, and the critical swelling ratio is much lower than that for a homogeneous hydrogel layer. A smooth transition of the critical mode is predicted as the volume fraction of the top layer changes, linking surface instability of a homogeneous layer to thin film wrinkling as two limiting cases. The results from the present study suggest that both the critical condition and the instability mode depend sensitively on the variation of the material properties in the thickness direction of the hydrogel layer.     

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.     

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.     

基于不同板理论的嵌于柔性层中薄膜的皱曲分析

论文对于柔性层-薄膜-柔性层三层结构系统,基于经典板理论、一阶剪切变形理论和高阶剪切变形理论,分别推导给出薄膜皱曲的控制方程.对于两个柔性层,则把它们处理成具有有限厚度的平面应变弹性体.针对上下柔性层固支边界或自由边界条件,利用线性扰动方法得到柔性层对薄膜的横向压力差,最终获得确定薄膜具有周期性正弦型皱曲的临界载荷方程...     

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.     

夹层梁总体屈曲及皱曲的有限元计算

皱曲是夹层结构的一种短波屈曲模式,通常发生于夹心较厚或夹心刚度较低的情况。由于模型规模的限制,在常规有限元建模时通常将夹层板模拟为二维板单元,这种方法忽略了面板和夹心在厚度方向上的相互作用,无法计算出皱曲模式。针对上述问题,本文首先介绍了一个计算夹层结构总体屈曲和皱曲的统一理论,并将此理论的计算结果作为理论解。为了同时...     

圆柱壳纯弯曲时塑性失稳临界曲率半径模型

薄壁管材在等曲率矫直生产中,塑性失稳临界曲率半径作为重要的工艺参数,直接决定了设备结构和产品质量。而目前现场仍沿用经验图表结合人工经验和反复试矫对其进行估定,亟待建立针对性的临界曲率半径数学模型以指导生产。在力学建模和分析时,就是确定具有初始曲率的圆柱壳体在纯弯曲条件下塑性失稳的临界曲率半径,为此从旋转壳体一般几何方程出发,基于J2形变理论和能量理论,运用里茨法建立了圆柱壳体在纯弯曲条件下塑性失稳时的临界弯矩,以此确定了临界曲率半径模型,并给出了数值解法。应用ANSYS/LS-DYNA进行了有限元动态仿真试验,证明了模型是近似正确的,并通过仿真对比分析证明了轴向起皱先于截面畸变是圆柱壳体在纯弯曲条件下塑性失稳的主要模态。     

Wrinkling of tubes in bending from finite strain three-dimensional continuum theory

The problem of a tube under pure bending is first solved as a generalised plane strain problem. This then provides the prebifurcation solution, which is uniform along the length of the tube. The onset of wrinkling is then predicted by introducing buckling modes involving a sinusoidal variation of the displacements along the length of the tube. Both the prebuckling analysis and the bifurcation check require only a two-dimensional finite element discretisation of the cross-section with special elements. The formulation does not rely on any of the approximations of a shell theory, or small strains. The same elements can be used for pure bending and local buckling a prismatic beam of arbitrary cross-section. Here the flow theory of plasticity with isotropic hardening is used for the prebuckling solution, but the bifurcation check is based on the incremental moduli of a finite strain deformation theory of plasticity.For tubes under pure bending, the results for limit point collapse (due to ovalisation) and bifurcation buckling (wrinkling) are compared to existing analysis and test results, to see whether removing the approximations of a shell theory and small strains (used in the existing analyses) leads to a better prediction of the experimental results. The small strain analysis results depend on whether the true or nominal stress–strain curve is used. By comparing small and finite strain analysis results it is found that the small strain approximation is good if one uses (a) the nominal stress–strain curve in compression to predict bifurcation buckling (wrinkling), and (b) the true stress–strain curve to calculate the limit point collapse curvature.In regard to the shell theory approximations, it is found that the three-dimensional continuum theory predicts slightly shorter critical wrinkling wavelengths, especially for lower diameter-to-wall-thickness (D/t) ratios. However this difference is not sufficient to account for the significantly lower wavelengths observed in the tests.     

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。     

Towards Understanding Why the Thin Membrane Transducer Deforms:Surface Stress-Induced Buckling

This study is directed towards a comprehensive exploration on the deformation mechanism of the thin membrane transducer(TMT) caused by surface stress variation.We stress that the biomolecular interaction has changed the magnitude of the surface stress;and when the surface stress exceeds a critical value the TMT will buckle and deform.Based upon Gurtin's theory of surface elasticity and principle of finite deformation,we abstract the TMT as a nanobeam with two clamped ends,and the close-formed governing equation set is derived accordingly.A computer code via the shooting method is developed to solve the presented two-point boundary value problem.In succession,the nanobeam deflection and critical parameters for buckling are quantitatively discussed.This investigation lays the theoretical foundation of TMTs;and it is also beneficial to gain deep insight into characterizing mechanical properties of nanomaterials and engineering nano-devices.     

Circumferential buckling instability of a growing cylindrical tube

A cylindrical elastic tube under uniform radial external pressure will buckle circumferentially to a non-circular cross-section at a critical pressure. The buckling represents an instability of the inner or outer edge of the tube. This is a common phenomenon in biological tissues, where it is referred to as mucosal folding. Here, we investigate this buckling instability in a growing elastic tube. A change in thickness due to growth can have a dramatic impact on circumferential buckling, both in the critical pressure and the buckling pattern. We consider both single- and bi-layer tubes and multiple boundary conditions. We highlight the competition between geometric effects, i.e. the change in tube dimensions, and mechanical effects, i.e. the effect of residual stress, due to differential growth. This competition can lead to non-intuitive results, such as a tube growing to be thinner and yet buckle at a higher pressure.     

食管黏膜组织在轴向延伸下的摩擦特性研究

对食管黏膜组织沿轴向分别以不同的延伸率进行延伸,再通过往复摩擦试验近似模拟内镜前端进入食管的过程,研究了食管黏膜组织在不同轴向延伸率伸长下的摩擦性能变化.结果表明:从轴向无延伸到延伸率达到80%时,内镜前端与食管黏膜组织间的摩擦运行行为从食管黏膜组织只发生弹性变形的黏着状态变化到黏着和滑移的混合状态,摩擦阻力和能耗增大,摩擦系数增加.食管黏膜组织的摩擦学性能与其表面结构和力学性能密切相关.在沿轴向延伸率逐渐增加的延伸状态下,食管黏膜表面的皱褶被拉平,维持黏液润滑的特性变差;随着食管黏膜组织轴向延伸率的增加,黏膜组织的弹性模量增加,抵抗摩擦变形的能力增强,摩擦阻力增大.研究结果为临床胃镜诊疗的安全操作提供了理论依据.     

Mechanical behavior and wrinkling of lined pipes

The paper focuses on wrinkling of lined pipes (sometimes referred to as clad pipes) under bending loading, where a corrosion-resistant thin-walled liner is fitted inside a carbon–steel outer pipe. The problem is solved numerically, using nonlinear finite elements to simulate liner pipe deformation and its interaction with the outer pipe. Stresses and strains are monitored throughout the deformation stage, detecting possible detachment of the liner from the outer pipe and the formation of wrinkles. The wrinkling behavior of elastic and elastic–plastic (steel) lined pipes under bending is examined. The results indicate that the lateral confinement of the liner pipe due to the deformable outer pipe and its interaction with the outer pipe has a decisive influence on the wrinkling behavior of the lined pipe. It is also shown that the behavior is characterized by a first bifurcation in a uniform wrinkling pattern, followed by a secondary bifurcation. The values of corresponding buckling curvature are determined and comparison with available experimental results is conducted in terms of wrinkle height development and the corresponding buckling wavelength. The results of the present research can be used for safer design of lined pipes in pipeline applications.