3D FEM Analysis of the Buckling Delamination of a Rectangular Viscoelastic Composite Plate with an Embedded Rectangular Crack Under Two-Axial Compression

In Akbarov, Yahnioglu and Karatas (2010) a buckling delamination problem for a rectangular viscoelastic composite plate with a band and edge cracks was investigated under uniaxial compression of the plate. In the present study this investigation is developed for the case where the mentioned rectangular plate contains an embedded rectangular crack and in addition it is assumed that the plate is subjected to two-axial compression.
It is supposed that all end surfaces of the considered plate are simply supported and that these ends are subjected to uniformly distributed normal compressive forces with intensity p₁ and p₃ which act along the O_x1 and O_x3 axes, respectively. Moreover, we assume that the plate contains a rectangular embedded crack, the edge-surfaces of which have initial infinitesimal imperfections before the loading.
The evolution of these initial imperfections with time under two-axial compression of the plate is studied within the framework of the three-dimensional geometrically nonlinear field equations of the theory of viscoelasticity for anisotropic bodies.
For determination of the values of the critical force or of the critical time, as well as those of the buckling delamination mode of the considered plate, the initial imperfection criterion is used.
For the solution to the corresponding boundary-value problems, boundary form perturbation techniques; the Laplace transform; the Schapery method for obtaining the numerical inverse Laplace transform of the sought values; and the 3D FEM are used.
The numerical results of the critical force and critical time, as well as of the buckling delamination modes are presented and discussed.     

Local buckling of the elastic and viscoelastic coating around the penny-shaped interface crack

In the framework of the piecewise-homogeneous body model with the use of the three-dimensional geometrically non-linear exact equations of the theory of elasticity and viscoelasticity the local buckling (delamination) problem of the elastic and viscoelastic coating around a penny-shaped interface microcrack is studied. The method of solution of the considered problem is developed with the use of boundary form perturbation method, FEM and Laplace transform. All investigations are made on the sandwich circular plate and it is assumed that interface crack edges have an insignificant initial rotationally symmetric imperfection and as a buckling criterion the case where this imperfection starts to increase and grows indefinitely is taken. Numerical results illustrating the influence of the mechanical and geometrical parameters to the critical force for elastic coating and to the critical time for the viscoelastic coating are presented.     

Probabilistic analysis of delamination onset in linear anisotropic elastic and viscoelastic composite columns

Previously developed deterministic and stochastic combined load invariant failure criteria are used to determine the onset of delamination in elastic and viscoelastic columns. The analysis includes the effects of initial imperfections as well as offset column loads and transverse shear contributions. The delamination predictions are found to be sensitive to the magnitude of applied loads and of initial imperfections. Illustrative numerical examples are presented for elastic and viscoelastic columns with random combined failure stresses in bending, shear, compression and with normal interlaminar stresses. Probabilities of delamination onset are established for various axial loads and initial imperfections and in the viscoelastic columns additionally as a function of lifetime. Since the failure theories consider the combined effects of bending, shear, compression and normal interlaminar stresses, delamination onset is predicted at smaller axial loads than the critical buckling loads in the elastic case and at shorter viscoelastic lifetimes compared to equivalent columns with no delamination effects.     

含内埋矩形脱层板屈曲分析的传递函数方法

研究了含任意内埋矩形脱层复合材料层合板的屈曲问题。采用一种基于Mindlin一阶剪切理论的条形传递函数方法,将含内埋矩形脱层的复合材料层合板分成含脱层和不含脱层的两种矩形超级单元,然后由各超级单元之间连接结点处的位移连续和力平衡条件得到脱层板屈曲的特征方程,进而得到脱层板的屈曲载荷和屈曲模态。进行参数分析发现,脱层大小、深度、位置以及脱层板的边界条件和复合材料铺层方向对脱层板屈曲载荷的影响较显著。     

Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates

In this article, an analytical method for buckling analysis of thin functionally graded (FG) rectangular plates is presented. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. Based on the classical plate theory (Kirchhoff theory), the governing equations are obtained for functionally graded rectangular plates using the principle of minimum total potential energy. The resulting equations are decoupled and solved for rectangular plate with different loading conditions. It is assumed that the plate is simply supported along two opposite edges and has arbitrary boundary conditions along the other edges. The critical buckling loads are presented for a rectangular plate with different boundary conditions, various powers of FGM and some aspect ratios.     

Buckling analysis of a plate with built-in rectangular delamination by strip distributed transfer function method

Summary. Delamination may reduce the buckling of the laminated plate. In this paper, a semi-analytical, semi-exact method, namely the strip transfer function method based on Mindlins first-order shear deformation theory is developed to investigate the buckling of a delaminated plate. First, a composite plate with bulit-in rectangular delamination is divided into two kinds of rectangular super-units, one with a delamination and one without. The variational principle is used to obtain equilibrium equations and boundary conditions. Then, by displacement continuity and force balance of nodes that connect two super-units, the characteristic equation for the buckling analysis of the delaminated plate is derived. Consequently, buckling load and mode are computed. By comparing the results of the plate with through-the–width delamination to the analytic solutions of a beam and the results of the finite element method, the validity of this method is tested. Furthermore, the influence of length, depth and position of the delamination, the boundary condition and the plying angle of the material on the buckling load is analyzed.     

Buckling and Delamination Growth Analysis of Composite Laminates Containing Embedded Delaminations

The objective of this work is to study the post buckling behavior of composite laminates, containing embedded delamination, under uniaxial compression loading. For this purpose, delamination initiation and propagation is modeled using the softening behavior of interface elements. The full layer-wise plate theory is also employed for approximating the displacement field of laminates and the interface elements are considered as a numerical layer between any two adjacent layers which delamination is expected to propagate. A finite element program was developed and the geometric non-linearity in the von karman sense is incorporated to the strain/displacement relations, to obtain the buckling behavior. It will be shown that, the buckling load, delamination growth process and buckling mode of the composite plates depends on the size of delamination and stacking sequence of the laminates.     

点弹性支承粘弹性矩形薄板的基频分析计算

从三维粘弹性本构关系出发,导出了具有多个点弹性支承的Kelvin型粘弹性矩形薄板的运动微分方程。针对方程中出现的二维广义d函数,采用积分方程法导出了具有多个点弹性支承的四边简支Kelvin型粘弹性矩形薄板自由振动的复特征方程,分析了材料的无量纲延滞时间、点弹性支承的弹性系数和支承位置对矩形薄板的固有频率的影响。     

Experimental study of dynamic buckling of plates under fluid–solid slamming

This paper describes the experimental investigation of the elastic–plastic dynamic buckling properties of rectangular plates under in-plane fluid–solid slamming. Based on the observation of elastic–plastic dynamic response characteristics of the plates, a dynamic buckling criterion and a dynamic yielding criterion are defined. The corresponding critical impulse are determined from the experimental results for each tested plate. The effect of different boundary conditions on the elastic–plastic dynamic buckling properties of plates is also examined. The results indicate that dynamic buckling always takes place elastically for the types of rectangular plates tested under fluid–solid slamming. The dynamic buckling modes of the plates are governed by the plate fundamental transverse free vibration mode. It is also found that boundary conditions strongly affect the dynamic buckling properties of plates subjected to fluid–solid slamming loads. Strengthening plate boundary constraint is a very effective way to enhance the plates’ ability to resist dynamic buckling.     

Delamination buckling growth in laminated composites using layerwise-interface element

In this paper, a numerical investigation on the buckling of composite laminates containing delamination, under in-plane compressive loads, is presented. For this purpose, delamination propagation is modeled using the softening behavior of interface elements. The full layerwise plate theory is applied for approximating the displacement field of laminates and the interface elements are considered as a numerical layer between any two adjacent layers where the delamination is expected to propagate. A non-linear computer code was developed to handle the numerical procedure of delamination buckling growth in composite laminates using layerwise-interface elements. The load/displacement behavior and the contours of embedded and through-the-width delamination propagation for composite laminates are presented. It is shown that delamination growth can be well predicted using this layerwise-interface elements with decohesive law.     

The dynamic elastic buckling of a circular arch with finite displacements and initial imperfections

The equilibrium equations for elastic circular arches are established using the principle of virtual work. The nonlinear partial differential equations of motion are solved using a finite difference method (Park's method for time difference). The dynamic stability of a hinged and a clamped elastic circular arch with a uniform step load is analysed with finite deformations and initial geometric imperfections. Results show that the buckling mode varies with the value of the arch half angle, θ₀. The boundary condition and initial imperfection amplitude also effect the buckling mode. A nearly perfect arch usually buckling with a “direct” buckling form, while an imperfect arch with an “indirect” buckling form. The effect of θ₀ on the ratio p_d/p_s (p_d is the dynamic critical load and p_s the static critical load) is shown for different initial imperfections and different boundary conditions.     

Parametric effects on embedded delamination buckling in composite structures using the EAS three-dimensional element

This study deals with parametric effects on bucking behaviors of laminated composite structures containing an embedded rectangular delamination using the enhanced assumed strain (EAS) three-dimensional element. The three-dimensional finite element (FE) formulation based on the EAS method for composite structures shows excellence from the standpoints of computational efficiency, especially for distorted element shapes. Using the EAS FE formulation developed for this study, the effects of embedded delamination sizes and ply orientations on the elastic buckling behaviors for various aspect ratios and width-to-thickness ratios are studied. The numerical results obtained are in good agreement with those reported by other investigators. Furthermore, the new results reported in this paper show the progression of local buckling and its influence on global buckling and vice versa. Key observation points are discussed and a brief design guideline is given.     

Postbuckling response and failure of symmetric laminates under in-plane shear

This paper summarizes a study aimed at understanding the postbuckling behaviour and progressive failure of thin, simply supported symmetric rectangular laminates with various possible in-plane boundary conditions and under the action of in-plane shear loads. First-order shear deformation theory and geometric non-linearity, in the von-Karman sense, is used with a finite-element procedure. The 3D Tsai–Hill criterion is used to predict failure of lamina and the maximum stress criterion is used to predict the onset of delamination at the interface of two adjacent layers. The effect of in-plane boundary conditions, plate lay-ups, plate aspect ratio, fiber orientations and lamina material properties on the load deflection response, buckling load, first-ply failure load, ultimate load and the maximum transverse displacement associated with failure loads is presented.     

A novel boundary-domain element method of initial stress, finite deformation and discrete cracks in multilayered anisotropic elastic solids

We introduce a novel boundary-domain element method of initial stress, finite deformation (due to large rotation) and discrete cracks in multilayered anisotropic elastic solids. Because the special Green’s function that satisfies the interfacial continuity and surface boundary conditions is employed, the numerical discretization is reduced to be along one side of the cracks and over the subdomains of finite deformation. Two examples are presented. First, the process of interfacial delamination is simulated around a growing through-thickness crack in a pre-stretched film bonded to a flexible substrate. It is shown that the progression of delamination damage is stable but the initiation of delamination crack can be a snap-back instability. This simultaneous damage and fracture process is approached by the cohesive zone model. Second, the postbuckling of a circular delaminated and pre-compressed film is simulated on a flexible substrate. It is shown that the compliance of substrate can play a significant role on the critical behavior of buckling. If the substrate is more compliant or stiffer than the film, the instability initiates as a subcritical hard or a supercritical soft bifurcation. The critical magnitude of pre-strain for the initiation of buckling increases with substrate stiffness. Also, the transition of buckling from the first to the second mode is captured in the simulation.     

具有局部分层的对称铺设层合板在非保守力作用下的屈曲分析

利用弹性非保守系统自激振动的拟固有频率变分原理,推导出复合材料矩形板受非保守随从力作用的变分方程,进而导出复合材料层合板含分层损伤的有限元基本方程及求解临界荷载力和固有频率的特征方程。用载荷增量法计算了不同边长比的复合材料矩形板在面内受随从力作用且含有分层的临界载荷,分析了不同角铺设方向对临界载荷的影响。分析表明,分层对临界载荷有一定影响,复合材料层合板的角铺设方向对临界载荷有较大影响。     

Delaminating buckling model based on nonlocal Timoshenko beam theory for microwedge indentation of a film/substrate system

A novel model built on the basis of nonlocal Timoshenko beam theory is presented for delaminating buckling in the microwedge indentation test of a thin film on an elastic substrate. Two size effects are accounted for in the proposed model. One is the delamination size effect, and the other is the film thickness effect. The influence of the elastic deformation in the substrate and the indentation-induced impression or notch on the buckling behaviors are taken into consideration by employing coupled line springs as the boundary conditions of the buckled film. The critical stress for buckling, the energy release rate and the phase angle of the interface delamination crack are calculated and compared with those by classical beam theories. Sensitivity of the two size effects is observed.     

Influence of interfacial delamination on channel cracking of elastic thin films

Channeling cracks in brittle thin films have been observed to be a key reliability issue for advanced interconnects and other integrated structures. Most theoretical studies to date have assumed no delamination at the interface, while experiments have observed channel cracks both with and without interfacial delamination. This paper analyzes the effect of interfacial delamination on the fracture condition of brittle thin films on elastic substrates. It is found that, depending on the elastic mismatch and interface toughness, a channel crack may grow with no delamination, with a stable delamination, or with unstable delamination. For a film on a relatively compliant substrate, a critical interface toughness is predicted, which separates stable and unstable delamination. For a film on a relatively stiff substrate, however, a channel crack grows with no delamination when the interface toughness is greater than a critical value, while stable delamination along with the channel crack is possible only in a small range of interface toughness for a specific elastic mismatch. An effective energy release rate for the steady-state growth of a channel crack is defined to account for the influence of interfacial delamination on both the fracture driving force and the resistance, which can be significantly higher than the energy release rate assuming no delamination.     

Surface stress and agglomeration effects on nonlocal biaxial buckling polymeric nanocomposite plate reinforced by CNT using various approaches

In this article, the surface stress effect on the biaxial critical buckling load of nonlocal polymeric nanocomposite rectangular plate reinforced by carbon nanotubes (CNTs) is presented. Various approaches such as Eshelby–Mori–Tanaka, the extended mixture rule, Halpin–Tsai, and micromechanical are used to determine the effective material properties of polymeric nanocomposite plate. The governing equations of equilibrium are obtained by using Hamilton’s principle. The Navier’s method is considered to obtain the biaxial critical buckling load of polymeric nanocomposite rectangular plate for simply supported boundary conditions. A detailed parametric study is conducted to explain the effects of aspect ratio, elastic foundation, surface stress and agglomeration on the biaxial buckling of nanocomposite plate. The results show that surface stress effect plays an important role at nanoscale. Also, the biaxial critical buckling load decreases with increasing the CNTs volume fraction in the inclusion (agglomeration effect). The results of this research can be used for micro-electro-mechanical and nano-electro-mechanical devices.     

刚性基底上弹性约束矩形板的屈曲行为分析

对刚性基底上非加载边弹性约束矩形板在固支边均匀面内压力作用下的屈曲行为进行了分析研究。用里兹能量变分法建立确定板屈曲强度的本征值问题,利用满足边界条件的屈曲位移函数导出板屈曲系数计算公式。将该方法用于矩形钢管混凝土钢板局部屈曲的分析和计算,提出了目标板边界约束系数计算公式。试验结果表明,方形钢管混凝土柱钢板局部屈曲强度计算值与试验值吻合良好,钢板边界弹性约束的处理和约束系数的计算方法是有效的。     

3D复合材料中纤维搭桥对脱层屈曲的影响

本文用能量法求解了绑结层合板和编织复合材料等3D复合材料中有纤维搭桥的圆形脱层屈曲问题。搭桥纤维被视为连续分布于脱层面上、可提供与脱层屈曲挠度成正比的恢复力的线性弹簧，根据脱层夹支边界条件以及脱层只能向一个方向挠曲的约束条件，导出了一个特征长度a0，当脱层半径a超出a0时，脱层屈曲将仅发主在r＜a0的子圆域之内，并不跨越整个脱层区域（r≤a）；当脱层半径a小于a0时，脱层屈曲将跨越整个脱层区域，并随着a的减小，脱层屈曲的临界载荷将逐渐提高。