Coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments

Delaminations in structures may significantly reduce the stiffness and strength of the structures and may affect their vibration characteristics. As structural components, beams have been used for various purposes, in many of which beams are often subjected to axial loads and static end moments. In the present study, an analytical solution is developed to study the coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments. Euler–Bernoulli beam theory and the "free mode" assumption in delamination vibration are adopted. This is the first study of the influences of static end moments upon the effects of delaminations on natural frequencies, critical buckling loads and critical moments for lateral instability. The results show that the effects of delamination on reducing natural frequencies, critical buckling load and critical moment for lateral instability are aggravated by the presence of static end moment. In turn, the effects of static end moments on vibration and instability characteristics are affected by the presence of delamination. The analytical results of this study can serve as a benchmark for finite element method and other numerical solutions.     

Experimental and theoretical study of the buckling of narrow thin plates on an elastic foundation under compression

The paper describes the processes of elastic deformation of thin films under mechanical loading. The film is modeled longitudinally by a compressed plate arranged on an elastic foundation. A computer model of the buckling of the narrow thin plate with a delamination portion located on an elastic foundation is constructed. This paper also touches upon the supercritical behavior of the plate–substrate system. The experiments on the axial compression of a metal strip adhered to a rubber plate are performed, and 2 to 7 buckling modes are obtained therein. The critical loads and buckling modes obtained in the numerical calculations are compared with the experimental data. It is shown that there is the possibility of progressive delamination of the metal plate from the foundation if the critical load is exceeded. It is found that the use of the proposed approach, which, in contrast to other approaches, accounts for the elastic deformation of the substrate, causes the dependence between the critical bending stress and the stiffness of the foundation.     

Dynamic stability analysis of composite plates including delaminations using a higher order theory and transformation matrix approach

A refined higher order shear deformation theory is used to investigate the dynamic instability associated with composite plates with delamination that are subject to dynamic compressive loads. Both transverse shear and rotary inertia effects are taken into account. The theory is capable of modeling the independent displacement field above and below the delamination. All stress free boundary conditions at free surfaces as well as delamination interfaces are satisfied by this theory. The procedure is implemented using the finite element method. Delamination is modeled through the multi-point constraint approach using the transformation matrix technique. For validation purposes, the natural frequencies and the critical buckling loads are computed and compared with three-dimensional NASTRAN results and available experimental data. The effect of delamination on the critical buckling load and the first two instability regions is investigated for various loading conditions and plate thickness. As expected, the natural frequencies and the critical buckling load of the plates with delaminations decrease with increase in delamination length. Increase in delamination length also causes instability regions to be shifted to lower parametric resonance frequencies. The effect of edge delamination on the static and dynamic stability as well as of delamination growth is investigated.     

Modelling of progressive interface failure under combined normal compression and shear stress

The present work is concerned with an analysis of progressive interface failure under normal compressive stress and varying shear stress using the cohesive crack model. The softening model is assumed and frictional linear stress at contact is accounted for. A monotonic loading in anti-plane shear of an elastic plate bonded to a rigid substrate is considered. An analytical solution is obtained by neglecting the effect of minor shear stress component in the plate. The elastic and plate interface compliances are included into the analysis. Three types of solutions are distinguished in the progressive delamination analysis, namely short, medium and long plate solutions. The analysis of quasi-static progressive delamination process clarifies the character of critical points and post-critical response of the plate. The analytical solution provides a reference benchmark for numerical algorithms of analysis of progressive interface delamination. The case of a rigid–softening interface was treated in a companion paper, where also cyclic loading was considered.     

An analysis of axisymmetric buckling and growth of circular-shaped delamination

The buckling and post-buckling of clamped circular plate subjected to distributed radial compressed load is presented by using the high-order perturbation analysis and shooting method. The sixth-order solution shows good agreement with the FEM results in [11]. The results in this paper are applied to investigate the buckling and growth of pressed thin film delamination in the film/substrate system. Under a certain residual pressure in the thin film, two characteristic blister radiiR _c andR _g, the critical radius and growing radius respectively, are obtained. The numerical result shows that the growth criterion of delamination in [9,10] is not perfect. In variant residual stress or interface toughness, the conditions of no growth, stable growth and unstable growth of the delamination are obtained by comparing the driving force at the interface crack tip with the interface toughness.This project is supported by National Natural Science Foundation of China.     

Asymptotic solutions for buckling delamination induced crack propagation in the thin film-compliant substrate system

In a thin film-substrate system in-plane compressive stress is commonly generated in the film due to thermal mismatch in operation or fabrication process. If the stress exceeds a critical value, part of the film may buckle out of plane along the defective interface. After buckling delamination, the interface crack at the ends may propagate. In the whole process, the compliance of the substrate compared with the film plays an important role. In this work, we study a circular film subject to compressive stress on an infinitely thick substrate. We study the effects of compliance of the substrate by modeling the system as a plate on an elastic foundation. The critical buckling condition is formulated. The asymptotic solutions of post-buckling deformation and the corresponding energy release rate of the interface crack are obtained with perturbation methods. The results show that the more compliant the substrate is, the easier for the film to buckle and easier for the interface crack to propagate after buckling.     

Progressive Frictional Delamination of an Infinite Elastic Film on a Rigid Substrate Due to In-Plane Point Loading

The paper presents a solution to a delamination problem of an infinite elastic film resting on a rigid substrate and loaded by a monotonically increasing in-plane point force. A?rigid-slip contact is assumed between the film and the substrate, leading to the development of two regions at the interface: a damaged zone with a relative slip between the materials, and a region where the interface remains intact. Both film natural and essential boundary conditions are zero on the boundary between these two interfacial zones with the shape of the boundary being a part of the solution. Problem??s self-similarity enables us to obtain an approximate distribution of interfacial traction within the delaminated zone and a shape of the zone itself. For film??s Poisson??s ratio ??=?1 the approximate solution becomes exact. It is argued that this can be treated as a special case of a rigid film sliding on a rigid substrate. The presented approach can be used to obtain approximate closed-form solutions to similar delamination problems.     

Deformation problem for an elastically fixed plate modeling a coating partially delaminated from the substrate (Plane Strain)

An asymptotic solution of the problemindicated in the title is obtained at distances large compared with the plate width and some promising methods for its use, in particular, for calculating the coefficients in the boundary conditions of the plate elastic fixation which models a coating partially delaminated from the substrate, are outlined. The possibility of considering the delamination in the approximation of the plate weak bending (the plate approximation) and the possibility of neglecting the tangential stress action along the contact boundary are implemented. The substrate is considered as a half-infinite elastic solid. This solution was obtained by using the Fourier transform and the solution of the resulting equation by the Wiener-Hopf method. The obtained asymptotic solution can be used to study problems related to coating delamination, especially on soft thick substrata.     

超弹性薄膜与可压缩基底双层结构表面失稳分析

当上层超弹性硬质薄膜和下层可膨胀基底构成的双层结构受压时,薄膜的自由表面可通过形成褶皱降低系统能量.研究表明,上下两层的模量比不同时,上层弹性硬质薄膜将表现出不同的表面失稳模式.本文提出了一种新颖的方法可有效抑制双层软材料的表面失稳,即改变基底材料的泊松比,这种方法同时适用于不具有应变硬化的软材料.首先基于Neo-Hookean模型发展了小变形条件下双层结构表面失稳的理论模型,通过半解析的方法得到了表面失稳的临界应变;然后通过有限元计算与模拟,进一步验证了负泊松比基底可延缓表面失稳.结果表明:(1)当双层结构基底泊松比为正且趋于0.5(不可压缩)时,双层结构在较小的压缩应变下出现表面失稳;(2)当基底的泊松比为负且趋于-1时,可被压缩至46%而不出现表面失稳,即可膨胀基底能有效抑制薄膜的表面失稳.本文发展的方法及主要结果可为延展性电子器件的设计提供指导.     

The effect of plastic anisotropy on compressive instability in sheet metal forming

The wrinkling behavior of a thin sheet with perfect geometry is associated with compressive instability. The compressive instability is influenced by many factors such as stress state, mechanical properties of the sheet material, geometry of the body, contact conditions and plastic anisotropy. The analysis of compressive instability in a plastically deforming body is difficult considering all the factors because the effects of the factors are very complex and the instability behavior may show a wide variation for a small deviation of the factors. In this study, the bifurcation theory is introduced for the finite element analysis of puckering initiation and growth of a thin sheet with perfect geometry. All the above mentioned factors are conveniently considered by the finite-element method. The instability limit is found by the incremental analysis and the post-bifurcation behavior is analyzed by introducing the branching scheme proposed by Riks. The finite-element formulation is based on the incremental deformation theory and elastic–plastic material modeling. The finite-element analysis is carried out using the continuum-based resultant shell elements considering the anisotropy of the sheet metal. In order to investigate the effect of plastic anisotropy on the compressive instability, a square plate that is subjected to compression in one direction and tension in the other direction is analyzed by the above-mentioned finite-element analysis. The critical stress ratios above which buckling does not take place are found for various plastic anisotropic modeling methods and discussed. Finally, the effect of plastic anisotropy on the puckering behavior in the spherical cup deep drawing process is investigated. From the results of the finite-element analysis, it is shown that puckering behavior of sheet metal is largely affected by plastic anisotropy.     

Formation of radially expanding liquid sheet by impinging two round jets

A thin circular liquid sheet can be formed by impinging two identical round jets against each other. The liquid sheet expands to a certain critical radial distance and breaks. The unsteady process of the formation and breakup of the liquid sheet in the ambient gas is simulated numerically. Both liquid and gas are treated as incompressible Newtonian fluids. The flow considered is axisymmetric. The liquid-gas interface is modeled with a level set function. A finite difference scheme is used to solve the governing Navier-Stokes equations with physical boundary conditions. The numerical results show how a thin circular sheet can be formed and break at its circular edge in slow motion. The sheet continues to thin as it expands radially. Hence, the Weber number decreases radially. The Weber number is defined as ρu 2 h/σ, where ρ and σ are, respectively, the liquid density and the surface tension, and u and h are, respectively, the average velocity and the half sheet thickness at a local radial location in the liquid sheet. The numerical results show that the sheet indeed terminates at a radial location, where the Weber number reaches one as observed in experiments. The spatio-temporal linear theory predicts that the breakup is initiated by the sinuous mode at the critical Weber number We c =1, below which the absolute instability occurs. The other independent mode called the varicose mode grows more slowly than the sinuous mode according to the linear theory. However, our numerical results show that the varicose mode actually overtakes the sinuous mode during the nonlinear evolution, and is responsible for the final breakup. The linear theory predicts the nature of disturbance waves correctly only at the onset of the instability, but cannot predict the exact consequence of the instability.     

Non-linear buckling and symmetry breaking of a soft elastic sheet sliding on a cylindrical substrate

We consider the axial compression of a thin sheet wrapped around a rigid cylindrical substrate. In contrast to the wrinkling-to-fold transitions exhibited in similar systems, we find that the sheet always buckles into a single symmetric fold, while periodic solutions are unstable. Upon further compression, the solution breaks symmetry and stabilizes into a recumbent fold. Using linear analysis and numerics, we theoretically predict the buckling force and energy as a function of the compressive displacement. We compare our theory to experiments employing cylindrical neoprene sheets and find remarkably good agreement.     

受弯脱层层板的局部失稳临界载荷的有限元分析

含脱层的复合材料层板承受弯曲载荷作用会产生跳跃失稳，还常常引起脱层扩展，从而导致结构失效。本文用基于一阶剪切层板理论的几何非线性有限单元法分析了受弯曲曲载荷作用下含脱层板的人稳的临界载荷。本文指出分叉失稳产生了跳跃失稳，而该跳跃失稳与浅圆拱或薄圆柱壳受向心压力作用下的跳跌 同，在整体平衡路径上没有一个极限点。本文对临界载计算结果比使用能量准则的结果要小，文中给出了原因。     

Buckling-driven delamination in layered spherical shells

An analysis of buckling-driven delamination of a layer in a spherical, layered shell has been carried out. The effects of the substrate having a double curvature compared to previous studies of delamination on cylindrical substrates turn out to be non-trivial in the sense that additional to the effect of the shape of the substrate, a new non-dimensional geometrical parameter enters the conditions for steady-state delamination. It is shown that this additional geometrical parameter in most cases of practical relevance has insignificant influence on the fracture mechanical parameters involved for the problem. The consequence is that solutions need to be mapped as a function of one rather than two dimensionless parameters. Furthermore, the shape of the substrate has profound influence especially on initiation of delamination growth.     

Puncturing a thin elastic sheet

We use membrane theory to analyze the puncturing of a thin solid circular isotropic elastic sheet by a rigid axisymmetric indenter. A solution is obtained in which a hole is formed at the center of the sheet with an interior annulus in frictionless contact with the cylindrical surface. The contacting part is in a state of pure hoop stress with the corresponding hoop stretch exhibiting a strong singularity at the origin. Conditions are given ensuring that the solution has finite total energy and it is shown to be energetically favored over unpunctured states for transverse indenter displacements exceeding a finite critical value.     

圆形脱层的轴对称屈曲及扩展分析

利用高阶摄动结合打靶法，分析了固支圆板在均匀径向力作用下的轴对称屈曲和过屈曲，所得结果与文（６）的ＦＥＭ结构吻合得很好，应用于薄膜－基底结构，研究了受压薄膜脱层的屈曲、扩展问题，得到了在一定的残余压应力作用下，脱层屈曲的临界尺寸Ｒｃ和扩展尺寸Ｒｇ。     

A micro-mechanical model of knitted fabric and its application to the analysis of buckling under tension in wale direction： buckling analysis

With the aid of the micro-mechanical model of knitted fabric proposed in Part 1 we analyze the buckling of a knitted fabric sheet when it is subjected to a tension along the wale direction. The large deformation of the fabric sheet in the critical configuration is considered and, to avoid possible deviation due to the approximation of the theory of thin plate, the three-dimensional theory of instability is used. The fabric sheet is considered as a three-dimensional body and all boundary conditions are satisfied. It is shown that the buckling of the fabric sheet is possible, two buckling modes and the corresponding buckling conditions are obtained, but only the flexural mode is physically possible as observed in experiments.The project supported by the National Natural Science Foundation of China (10272079)     

Localized buckling in sandwich struts with pre-existing delaminations and geometrical imperfections

A recent model for the nonlinear structural response of compression sandwich struts is developed further to account for pre-existing face-core delaminations and initial imperfections in the strut geometry. Whilst the pre-existing delaminations only take effect after the critical bifurcation for overall (Euler) buckling, it is found that the secondary instability associated with localized buckling occurs earlier than for the initially perfectly bonded strut. More severe instabilities can also be promoted by superimposing geometric imperfections. In combination with delamination, the practical structural response can be highly unstable.     

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.     

Analytical solution for buckling of asymmetrically delaminated Reissner’s elastic columns including transverse shear

The exact analytical solution of buckling in delaminated columns is presented. In order to investigate analytically the influence of axial and shear strains on buckling loads the geometrically exact beam theory is employed with no simplification of the governing equations. The critical forces are then obtained by the linearized stability theory. In the paper, we limit the studies to linear elastic columns with a single delamination, but with arbitrary longitudinal and vertical asymmetry of delamination and arbitrary boundary conditions. The studies of quantitative and qualitative influence of transverse shear are shown in detail and extensive results for buckling loads with respect to delamination length, thickness and longitudinal position are presented.