Two-dimensional analyses of delamination buckling of symmetrically cross-ply rectangular laminates

The conventional approach to analysis the buckling of rectangular laminates containing an embedded delamination subjected to the in-plane loading is to simplify the laminate as a beam-plate from which the predicted buckling load decreases as the length of the laminate increases. Two-dimensional analyses are employed in this paper by extending the one-dimensional model to take into consideration of the influence of the delamination width on the buckling performance of the laminates. The laminate is simply supported containing a through width delamination. A new parameter β defined as the ratio of delamination length to delamination width is introduced with an emphasis on the influence of the delamination size. It is found that (i) when the ratio β is greater than one snap-through buckling prevails, the buckling load is determined by the delamination size and depth only; (ii) as the ratio β continues to increase, the buckling load will approach to a constant value. Solutions are verified with the well established results and are found in good agreement with the latter.     

Buckling of delaminated bi-layer beam-columns

An improved analytical model is presented to analyze the delamination buckling of a bi-layer beam-column with a through-the-width delamination. Both the transverse shear deformation and local delamination tip deformations are taken into consideration, and two delaminated sub-layers as well as two substrates in the intact (un-delaminated) regions are modeled as individual Timoshenko beams. A deformable interface is introduced to establish the continuity condition between the two substrates in the intact regions. Consequently, a flexible joint is formed at the delamination tip, and it is different from the conventional rigid joint given in most of studies in the literature, in which the local delamination tip deformations are completely ignored. In contrast to the local delamination buckling in our previous study (Qiao et al., 2010), the present model accounts for the global deformations of the intact region in the delaminated composite beam-column, thus capable of capturing the buckling mode shape transitions from the global, to global–local coexistent, and to local buckling for asymmetric delamination as the interface delamination increases. Good agreement of the present analytical solutions with the full 2-D elastic finite element analysis demonstrates the local deformation effects around the delamination tip and verifies the accuracy of the present model. Parametric studies are conducted to investigate the effects of loading eccentricity, delaminated sub-layer thickness ratio, and interface compliance on the critical buckling load for the delaminated composite beam-column. Transitions of buckling modes from the global to local delamination buckling are also disclosed as the thickness of one sub-layer reduces from the thick sub-layer to a thin film. The developed delamination buckling solution facilitates the design analysis and optimization of laminated composite structures, and it can be used with confidence in buckling analysis of delaminated composite structures.     

含有分层损伤的复合材料加筋层合板的屈曲性态研究

基于Ｍｉｎｄｉｎ假定推导了考虑剪切的复合材料加筋层板的有限元列式,并在此基础上计算出筋间基板含嵌入分层以及筋与基板连接处含穿透分层的加筋层合板在受压缩载荷情况下的屈曲模式和临界力。本文所给出的有限元方法及结论对从事复合材料结构设计的工程人员具有参考价值。     

Parametric studies of failure mechanisms in elastic EB-PVD thermal barrier coatings using FEM

Catastrophic failure of thermal barrier coatings (TBCs), usually occurs due to large scale buckling and spallation, primarily originating at the bond coat and TGO interface. Spallation in TBCs is preceded by a competition between buckling and interface delamination that is stimulated by the waviness of the interface. In the presence of thermal loading, the waviness is responsible for growth of interfacial delamination. In this paper, a finite element model of the two and three layer TBC’s is developed in the commercial code ANSYS to investigate the buckle and interface delamination mechanisms and develop a simplified parametric understanding of these mechanisms. The models for simulation are validated with analytical and experimental results. Parametric relations, in terms of geometric and material parameters representing constituents of the TBC, are developed in this paper for critical stresses and energies causing buckling and debonding initiated instabilities. Through these relations, critical parameters that control failure mechanics are identified for a fail-safe design space.     

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.     

受弯脱层层板后屈曲有限元分析

脱层是复合材料层板结构中主要的缺陷形式之一。当脱层层板受到压力载荷的作用会造成脱层的局部屈曲和扩展,从而使结构的强度和刚度大为降低。含脱层层板的弯曲问题包含了脱层的压缩问题,却比压缩问题更加复杂。本文对含穿透脱层层板在纯弯载荷作用下的后屈曲问题进行了基于一阶剪切层板理论的几何非线性有限元分析,运用虚裂纹闭合技术求解了纯弯载荷作用下的脱层尖端的能量释放率各型分量,并用脱层扩展判据求解了脱层起始扩展载荷。     

薄膜褶皱的非线性屈曲有限元分析

有限元模拟已经成为目前薄膜褶皱预测的重要工具。本文采用ANSYS SHELL63单元对薄膜受剪情况下的褶皱形变进行了非线性屈曲分析。通过本征屈曲分析得到的模态模拟了薄膜的初始缺陷。利用本文的非线性有限元模型分析得到了薄膜褶皱的波长和幅度,并与理论分析结果进行了比较,结果相近。     

Compressive response and failure of braided textile composites: Part 2—computations

In this, the second part of a two part paper, results obtained by using the finite element (FE) method in conjunction with micromechanics to predict the effective elastic stiffness and strength of a carbon 2D triaxially braided composite (2DTBC), are presented. The 3D FE based micromechanics study was carried out on one representative unit cell (RUC) of the carbon 2DTBC (the “micromodel”). The FE models were first used to determine the macroscopic elastic orthotropic stiffnesses of the 2DTBC. The micromodel was deemed acceptable (in terms of the number of elements used in the mesh of the micromodel) if the elastic stiffnesses it displayed were within 5% of the elastic properties found experimentally. Subsequently, buckling eigenmodes were determined for the FE RUC under uniaxial and biaxial loading states, corresponding to the experimental investigation reported in part I of this two part paper. The lowest symmetric modes were identified and these mode shapes were used as imperfections to the FE model for a subsequent nonlinear response analysis using an arc-length method in conjunction with the ABAQUS commercial FE code. The magnitude of the imperfections was left as a parameter and its effect on the predicted response was quantified. The present micromechanics computational model provides a means to assess the compressive and compressive/tensile biaxial strength of the braided composites and its dependence on various microstructural parameters. It also serves as a tool to assess the most significant parameter that affects compressive strength.     

Buckling analysis of pristine and defected graphene

The buckling behavior of monolayer graphene (pristine and vacancy-defected) and bilayer graphene (pristine) loaded in the armchair direction was simulated for different boundary conditions using a truss FE model, representing the exact atomic lattice of graphene, and a FE model of an equivalent 2D plate. The critical buckling stress of pristine monolayer graphene was derived as a function of aspect ratio. The results from the two FE models coincide and are in very good agreement with established analytical solutions. With increasing the aspect ratio, the critical buckling stress of monolayer graphene decreases until the value of 2 from which the effect starts to diminish. Using the truss FE model, the effect of randomly dispersed vacancies on the critical buckling stress and buckling mode of monolayer graphene was studied. It was found that the critical buckling stress decreases dramatically with increasing the defect density: for a defect density of 10%, the critical buckling stress decreases by almost 50%. Moreover, the presence of defects was found to affect the highest buckling modes (above 3) even at low densities. Bilayer graphene has a totally different critical buckling stress than monolayer graphene due to the effect of van der Waals forces which depends on the applied boundary conditions.     

Buckling analysis of two layer delaminated beams with bridging

Stitching has been used as through-thickness reinforcement to reduce the effects of delamination. In stitching, the delamination will be held by stitches in the form of crack/interface bridging. In the present work, the reinforcement of stitching threads is assumed to provide continuous linear restoring tractions opposing the delamination opening. A generalized mathematical model is developed to study the buckling analysis of two layer delaminated beams with bridging by using Rayleigh–Ritz energy method. The delaminated beam is analyzed as four interconnected beams using the delamination as their boundary. Lagrange multipliers are used to enforce the boundary and continuity conditions between the junctions of the interconnected beams. The developed mathematical model is solved as an eigenvalue problem in which the lowest eigenvalue gives the buckling load. Effective-bridging modulus, a new nondimensionalized parameter, is introduced to study the influence of bridging on the delamination buckling. It is shown that bridging strongly influences the buckling load of the delaminated beams and a monotonic relation is observed between the buckling load and the effective-bridging modulus. Parametric studies in terms of delamination sizes and locations along spanwise and thicknesswise positions on the buckling load have been carried out. The bridging is found to be effective for shallow delaminations of moderate length, and for deep and long delaminations. Spanwise positions of delamination strongly influence the buckling loads. In addition, an analytical model for obtaining upper bounds of the buckling load is developed by using Euler–Bernoulli beam theory. Effective-slenderness ratio, a new nondimensionalized parameter is defined and it is found to be controlling the buckling mode configurations, i.e., local, global and mixed modes.     

F E M analysis of delamination buckling in composite plates and shells

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Behaviour of thick composite tubes considering of delamination

The present paper deals with plane finite element analysis of thick composite tubes. Thick composite tubes are commonly used in marine industry and in deep-water offshore applications. Two kinds of interlaminar delamination type defect in a thick walled cylinder subjected to external pressure were confronted; an annular or ring like delamination and a strip delamination. Two finite element models were developed to predict the strain energy release rate at the delamination fronts. In these models the effects of the processing history of the composite material in the form of a uniform thermal load were also included to simulate the state of the residual stress in the composite. The considered defects are studied by means of the effect of buckling, investigating the annular and the strip delamination buckling, and the subsequent loss of load carrying capacity of the delaminated region.     

A p-TYPE SUPERCONVERGENT RECOVERY METHOD FOR FE ELASTIC STABILITY ANALYSIS OF EULER BEAMS1)

本文将有限元p型超收敛算法应用于欧拉梁弹性稳定分析。该法基于有限元解答中失稳载荷和失稳模态结点位移的超收敛特性,建立了单元上失稳模态近似满足的线性常微分方程边值问题,在每个单元上,对该边值问题采用一个高次元进行求解,获得失稳模态的超收敛解,再将失稳模态的超收敛解代入瑞利商的解析表达式,最终获得失稳载荷的超收敛解。该法思路简明,通过少量计算即可显著提高失稳载荷和失稳模态的精度与收敛阶。数值算例表明,该法高效、可靠,值得进一步研究和推广到各类杆系结构。     

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.     

Thermally induced wrinkling in orthotropic layered plates with irregular delaminations

Delaminated regions figure prominently among potential threats to the structural integrity of layered plate configurations. Under a certain thermal loading threshold, geometrically nonlinear local instabilities in the form of buckling or wrinkling across the delaminated region crop up, giving rise to markedly amplified distributions of contour peeling stresses. The present paper aims to shed light on and quantify the manifold aspects and implications of the delamination-thermal-wrinkling trio. The paper faces the challenges of handling the nature of the layered configuration, the inherent geometrical irregularity of delaminated regions, the discontinuous interfacial conditions, the 3D stress state along the delamination contour, and the nonlinear evolution of local instabilities across an orthotropic delamination. For that purpose, a specially tailored 2D multi layered plate model and a corresponding triangular finite element are derived. The original contribution of the proposed model is in its ability to capture the thermally-driven, nonlinear small scale phenomena related to geometrically nonlinear response of the layered structure, using a 2D multi-layered plate theory solved with efficient 2D multi-layered triangular finite elements, as opposed to computationally expensive 3D finite element analysis. This is accomplished via the integration and synergy of methodologies that include: multi-layered high order plate theory to account for the layered layout, geometrically nonlinear strain-displacement relations to account for geometrical nonlinearities, orthotropic and thermo-elastic constitutive laws to account for thermal loads, and interlayer interface modelling which, combined with a the shear-locking free triangular FE, allows accounting for arbitrarily shaped delaminations. The model is validated against a 1D closed form solution and a 3D continuum based finite element analysis and is then used for a numerical study. In the study, the onset and the evolution of local instabilities in an adhesively bonded orthotropic layer across an irregular delamination are looked into. Special attention is given to the significant influence of material orthotropy and the relative directionality of the delamination on the threshold thermal load, the nonlinear wrinkling patterns, and the peeling traction distribution.     

含内埋矩形脱层复合材料圆柱壳屈曲分析的混合变量条形传递函数方法

提出了一种分析含内埋矩形脱层正交各向异性圆柱壳稳定性问题的混合变量条形传递函数方法。首先基于Mindlin一阶剪切壳理论,通过定义圆柱壳的广义力变量和混合变量,建立了壳的改进混合变量能量泛函;然后,为了便于脱层壳的分区求解,通过引入条形单元,创建了基于混合变量条形传递函数解的含脱层和不合脱层两种超级壳单元;在此基础上,将含内埋矩形脱层的复合材料层合壳划分成两种超级壳单元的组合体,通过各超级壳单元相互之间连接结点处的位移连续和力平衡条件得到脱层壳的屈曲方程;最后由屈曲方程计算含内埋矩形脱层壳的屈曲载荷和屈曲模态。算例分析的结果验证了本方法的正确性,并给出了几种因素对屈曲载荷和屈曲模态的影响。     

层板脱层的能量释放率分析

含脱层的层板在承受压缩载荷作用时,很容易发生局部屈曲,导致脱层扩展和结构失效.本文利用可动边界变分问题对脱层扩展进行了分析,导出了脱层前缘各点处的能量释放率表达式.本文还对浅部椭圆脱层进行计算分析,指出:其能量释放率沿脱层边界有很大变化,脱层的扩展方向取决于脱层的形状、受载方式及铺层方向.     

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.     

Postbuckling analysis of laminated composite plates subjected to the combination of in-plane shear,compression and lateral loading

This study deals with postbuckling behavior of laminated composite plates under the combination of in-plane shear, compression and lateral loading using an Element-based Lagrangian formulation. Natural co-ordinate-based strains, stresses and constitutive equations are used in the present shell element. The Element-based Lagrangian formulation described in this paper, in comparison with the traditional approaches, is more attractive not only because it uses only single mapping but also it converges faster. In addition, the finite element (FE) formulation based on the assumed natural strain method for composite structures shows excellence from the standpoints of computational efficiency as well as its ability to avoid both membrane and shear locking behavior. The numerical results obtained are in good agreement with those reported by other investigators. In particular, new results reported in this paper show the influence of various types of loading, materials and number of layers on postbuckling behavior.     

CONTACT ANALYSIS FOR FIBER-REINFORCED, DELAMINATED LAMINATES WITH KINEMATIC NONLINEARITY

A macro-micro analytical approach for the anti-penetrating contact problem at the interfaces of the delamination in symmetrically cross-plied,fiber-reinforced rectangular laminates is presented in this paper.The laminate is simply supported and subjected to a uniform transverse load with a through-width delamination buried at the center position.A contact factor is defined to characterize the contact efect and determined using the micro-mechanics of composite material.By analyzing the kinematics of nonlinear deformation at the interfaces of the delamination,the contact force is derived.Asymptotic solutions from perturbation analysis are presented.It is found that the deformation of the laminate involves a global deflection and a local buckling.The antipenetrating contact efects are characterized by the local buckling and are intrinsic properties of the laminates,relying only on the geometries of the delamination and the material properties.Parametric analyses show that the location and size of the contact areas and the distribution of the contact force are hardly afected by the aspect ratio.