Buckling analysis of delaminated laminates with consideration of contact in buckling mode

This paper concerns the buckling load analysis of laminates with a pre‐existing delamination, using the finite element method based on the Mindlin plate theory. To deal with the contact problem in the buckling mode, an effective algorithm is presented. In this method, an iterative updating process based on the first‐order sensitivity analysis and the quadratic programming technique is proposed to compute the fictitious forces in contacting areas at first. These fictitious forces are then transferred into the stiffness parameters of some artificial springs. The original stiffness matrix of system can be modified, using these artificial springs. Finally, the penetration between two delaminated layers in the buckling mode can be prevented effectively. Numerical examples show that this method is very efficient to solve the contact problem in eigenvalue analysis from the viewpoint of its accuracy, stability and convergence speed. The effects of contact and delamination size on the buckling load analysis are also investigated. Copyright © 1999 John Wiley & Sons, Ltd.     

有横向纤维搭桥的脱层扩展稳定性分析

本文利用von Karman 非线性薄板理论, 求解了复合材料中有横向纤维搭桥作用的圆形薄膜脱层的后屈曲问题, 获得了桥联脱层均匀扩展的能量释放率和断裂韧性。通过大量的计算和分析得到一个重要结论: 搭桥纤维的存在不仅对脱层的初始扩展起到了增韧作用, 而且避免了脱层扩展的灾难性。同时定量地给出了搭桥作用对脱层初始扩展的增韧程度, 并且确定了桥联脱层扩展稳定与不稳定的分区, 对不稳定脱层还给出了失稳扩展的范围及动态效应的估计。      

Bending behavior of delaminated composite plates with contact effects

Laminated plates with strip-type delamination under pure bending were investigated analytically and experimentally. In the analysis, a two-dimensional nonlinear finite element code based on updated Lagrangian formulation was developed to analyze the bending behavior of the laminated plates and the local buckling phenomenon of the sublaminates in the delaminated region. The formulation includes large displacements and large rotations needed to describe the local buckling phenomenon of the delaminated region. The transformation matrix method, which satisfies the compatibility of displacements between the upper and lower delaminated surfaces, can reduce the number of system unknowns and was used to cope with this contact problem. The modified Newton-Raphson method was used to solve the resulting nonlinear system equation and a load-controlled scheme used in the incremental solution procedure. In each increment, the iteration process was performed until the contact condition was satisfied. In the experiment, a tensile-test machine equipped with a set of four-point bending device was used to record the load-displacement response of the delaminated plates; the bending moment-curvature relation was calculated therefrom. The analytical results are improved by contact analysis. The results shows that the size of delaminated region had no significant influence on the bending strength of laminated plates, no matter whether the local buckling occurred or not. The normal contact force on the surface of the delaminated region kept constant along the delaminated length.     

带脱层的复合材料层板的屈曲分析

用基于Mindlin 板理论的有限元方法进行了带脱层损伤的复合材料层板的屈曲载荷分析。为了获得物理上可能的屈曲模态, 即避免上下脱层的相互贯穿, 在接触区域引入一些假想弹簧, 并给出了这些假想弹簧刚度系数的计算公式和接触计算的迭代格式, 通过这些弹簧对原始刚度矩阵进行修正可以有效地求解屈曲载荷特征值分析中的接触问题。数值计算结果表明了本算法的有效性和引入接触分析对这类屈曲分析的重要性。同时, 还对脱层的大小、形状、位置和脱层的纤维铺层方向对屈曲载荷的影响进行了研究。      

Prediction of thermal post buckling and deduction of large amplitude vibration behavior of spring–hinged beams

A general energy formulation to predict the thermal post buckling behavior of uniform isotropic beams is presented in this paper. The hinged ends of the beam contain elastic rotational restraints to represent the actual practical support situation. The large amplitude vibration behavior of beams is deduced from the post buckling results. The classical hinged and clamped conditions can be obtained as the limiting cases of the rotational spring stiffness. The numerical results, in the form of the ratios of the post buckling to buckling loads for various maximum deflection ratios, are presented in the digital form. An alternate independent formulation, based on the nonlinear finite element formulation, is also used in this paper to validate the numerical results of the present work. Further, the results for the large amplitude vibrations, deduced from the thermal post buckling results are also presented and these results compare very well with the finite element results, available in the literature, for the large amplitude vibration problem. These comparisons show an excellent agreement not only for the present work on the proposed thermal post buckling formulation but also on the deduced results for the large amplitude vibration of beams with the ends elastically restrained against rotation (spring–hinged beams). The numerical results presented confirm the efficacy of the proposed methodology used for predicting the post buckling behavior and deducing the large amplitude vibration behavior of the spring–hinged beams.     

考虑剪切效应的轴对称脱层圆板的后屈曲分析

基于Von Karman板理论,考虑横向剪切变形,建立了具脱层的轴对称层合圆板的后屈曲控制方程。应用正交配点法,将后屈曲控制方程、边界条件、以及连续条件转化为非线性方程组,然后进行迭代求解。讨论了不同脱层深度和脱层半径对层合圆板的屈曲及后屈曲特性影响,且与有关文献的结果进行了比较。     

含界面脱粘及表板基体开裂损伤的复合材料夹层板非线性稳定性的研究

本文作者基于"zig-zag"模型和Mindlin一阶剪切变形板理论,推导了复合材料夹层板屈曲分析的有限元列式,在该列式中考虑了面板的横向剪切变形和芯体的面内刚度对夹层板力学性能的影响。针对具有面板和芯体间界面脱粘和纤维增强树脂基体微裂纹损伤的夹层板损伤特征,分别提出了分层模型和多标量损伤模型,并推导了多标量形式的损伤本构关系。采用修正的 Newton-Raphson迭代格式求解含损复合材料夹层板的非线性稳定性性状。通过算例研究了脱粘面积、基体的损伤演化、表板的铺设方式及载荷形式对复合材料夹层板屈曲性态的影响。本文作者给出的有限元模型和结论,对复合材料夹层板结构设计的损伤容限的制定具有一定的参考价值。     

Strength reduction and delamination growth in thin and thick composite plates under compressive loading

In this paper, the coupled local-global buckling behavior in laminated composite plates with elliptic delaminations and the associated mechanisms of delamination growth under compressive loads are critically examined. The J-integral technique is used for delamination growth prediction in terms of pointwise energy release rate distribution along the delamination edge. A Multi-plate model, in conjunction with a 3-noded quasi-conforming shell element, is used to model the delaminated plates. The incremental equilibrium equations are set up based on total Lagrangian formulation. The solution strategy incorporates Gauss elimination in a cycle of Newton-Raphson iterations and is augmented with automated arc-length controled load incrementation and equilibrium iterations; and with automated post-buckling path tracing based on a linearised asymptotic solution. The effects of structural parameters such as delamination thickness, size and shape, on the post-buckling behavior and on the delamination growth are critically examined.This work was supported by the FAA to the Center of Excellence for Computational Modeling of Aircraft Structures at Georgia Institute of Technology, and in part by a grant from ONR.     

Structural analysis of composite laminates using a mixed hybrid shell element

The structural analysis of thin composite structures requires robust and effective shell elements. In this paper the variational formulation is based on a Hu–Washizu functional with independent displacements, stress resultants and shell strains. For the independent shell strains an additional interpolation part is introduced. This yields an improved convergence behaviour especially for laminated shells with coupled membrane and bending stiffness. The developed mixed hybrid shell element possesses the correct rank and fulfills the in–plane and bending patch test. The formulation is tested by several nonlinear examples including bifurcation and post–buckling response. The essential feature of the new element is the robustness in nonlinear computations with large rigid body motions. It allows very large load steps in comparison to standard displacement models.     

Concomitant wrinkling and buckle-delamination of elastic thin films on compliant substrates

Compressing a thin elastic film attached to a thick compliant substrate can lead to buckling instability. Two commonly observed buckling modes, buckle-delamination and wrinkling, have each been analyzed separately in previous studies. Recent experiments have observed that the two modes can co-exist and co-evolve. In this paper, by analytical and finite element methods, we present a study on concomitant wrinkling and buckle-delamination for an elastic film on a highly compliant substrate. First, without delamination, we present an analytical solution for wrinkling that takes into account the effect of Poisson’s ratio of the substrate. In comparison with a nonlinear finite element analysis, an approximate formula is derived to estimate the normal traction at the interface and to predict initiation of wrinkle-induced delamination. Next, with a pre-existing delamination crack, the critical strain for the onset of buckling instability is predicted by finite element eigenvalue analysis. For an intermediate delamination size, a mixed buckling mode is predicted with the critical compressive strain lower than previous solutions for both wrinkling and buckle-delamination. Post-buckling analyses show a significant shear-lag effect with an effective load transfer length three orders of magnitude greater than the film thickness. Finally, concomitant wrinkling and buckle-delamination is simulated to illustrate the interaction between the two buckling modes, and the results are discussed in view of failure mechanisms and applications in thin film metrology.     

An enriched cohesive zone model for delamination in brittle interfaces

Application of standard cohesive zone models in a finite element framework to simulate delamination in brittle interfaces may trigger non‐smooth load–displacement responses that lead to the failure of iterative solution procedures. This non‐smoothness is an artifact of the discretization; and hence it can be avoided by sufficiently refining the mesh leading to unacceptably high computational costs and a low efficiency and robustness. In this paper, a process‐driven hierarchical extension is proposed to enrich the separation approximation in the process zone of a cohesive crack. Some numerical examples show that instead of mesh refinement, a more efficient enriched formulation can be used to prevent a non‐smooth solution. Copyright © 2009 John Wiley & Sons, Ltd.     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.     

考虑多尺度界面力学特性的损伤复合材料层合板分岔屈曲性能的等代化

提出了一种等效理论来分析含损伤碳纤维增强树脂T300/QY8911复合材料层合板的分层屈曲。针对含贯穿脱层层合板产生面外弯曲后的受力特点,引入损伤界面的接触效应,根据精确模型所给出的多尺度变形失效机制,提出了一个基于刚度等效理论来分析损伤层合板结构失效的力学性能。通过将含脱层的区域等效成一个几何形状和铺设方式完全一致,但刚度相应折减的完善子板,运用三分区模型,根据板壳理论、复合材料力学等基本原理建立各子板的屈曲控制方程,同时结合边界条件和连续性条件求解。算例分析表明,本文所得的屈曲荷载与考虑接触效应精确模型所得的解析解及ABAQUS有限元结果高度吻合。该研究方法充分考虑了脱层带来的刚度降低以及由于分层界面处非线性抗穿透约束的影响,不仅大大简化了繁琐的推导过程和节省了计算量,而且揭示了深层次的力学机制,更为主要的是,该方法可以有效推广到含多分层损伤层合板的非线性力学性能的评估,为航空航天先进复合材料的结构设计和力学分析提供有力的技术支持。      

A Koiter‐Newton approach for nonlinear structural analysis

A new approach termed the Koiter‐Newton is presented for the numerical solution of a class of elastic nonlinear structural response problems. It is a combination of a reduction method inspired by Koiter's post‐buckling analysis and Newton arc‐length method so that it is accurate over the entire equilibrium path and also computationally efficient in the presence of buckling. Finite element implementation based on element independent co‐rotational formulation is used. Various numerical examples of buckling sensitive structures are presented to evaluate the performance of the method. The examples demonstrate that the method is robust and completely automatic and that it outperforms traditional path‐following techniques. This improved efficiency will open the door for the direct use of detailed nonlinear finite element models in the design optimization of next generation flight and launch vehicles. Copyright © 2013 John Wiley & Sons, Ltd.     

Post-buckling analysis of composite plates containing embedded delaminations with arbitrary shape by using higher order shear deformation theory

The compressive post-buckling behavior of composite laminates containing embedded delamination with arbitrary shape is investigated analytically. For modeling the embedded delamination, the laminate is divided into three smaller regions. The higher order shear deformation theory is implemented and the formulation is based on the Rayleigh-Ritz approximation technique by the application of the simple/complete polynomial series for each region. The nonlinear equilibrium equations, which are achieved through the application of the principle of Minimum Potential Energy, are solved by employing the Newton-Raphson iterative procedure. Some interesting results are obtained and compared with those achieved by the finite element method of analysis using ANSYS commercial software. A good agreement is seen to exist between the results. This is while for a given level of accuracy in the results, ANSYS requires a markedly larger number of degrees of freedom compared to that needed by the developed method. Moreover, a considerable reduction in the load carrying capacity of laminate is noticed due to the presence of delamination.     

On the Through-the-Width Multiple Delamination, and Buckling and Postbuckling Behaviors of Symmetric and Unsymmetric Composite Laminates

Multiple delamination causes severe degradation of the stiffness and strength of composites. Interactions between multiple delamination, and buckling and postbuckling under compressive loads add the complexity of mechanical properties of composites. In this paper, the buckling, postbuckling and through-the-width multiple delamination of symmetric and unsymmetric composite laminates are studied using 3D FEA, and the virtual crack closure technique with two delamination failure criteria: B-K law and power law is used to predict the delamination growth and to calculate the mixed-mode energy release rate. The compressive load-strain curves, load-central deflection curves and multiple delamination process for eight composite specimens with different initial delamination sizes and their distributions as well as two angle-ply configurations 0₄//(±θ)₆//0₄ (θ?=?0° and 45°, and “//” denotes the delaminated interface) are comparatively studied. From numerical results, the unsymmetry decreases the local buckling load and initial delamination load, but does not affect the global buckling load compared with the symmetric laminates. Besides, the unsymmetry affects the unstable delamination and buckling behaviors of composite laminates largely when the initial multiple delamination sizes are relatively small.     

Application of the higher‐order Cauchy–Born rule in mesh‐free continuum and multiscale simulation of carbon nanotubes

This paper investigates the application of a recently proposed higher‐order Cauchy–Born rule in the continuum simulation and multiscale analysis of carbon nanotubes (CNTs). A mesh‐free computational framework is developed to implement the numerical computation of the hyper‐elastic constitutive model that is derived from the higher‐order Cauchy–Born rule. The numerical computation reveals that the buckling pattern of a single‐walled carbon nanotube (SWCNT) can be accurately displayed by taking into consideration the second‐order deformation gradient, and fewer mesh‐free nodes can provide a good simulation of homogeneous deformation. The bridging domain method is employed to couple the developed mesh‐free method and the atomistic simulation. The coupling method is used to simulate the bending buckling of an SWCNT and the tensile failure of an SWCNT with a single‐atom vacancy defect, and good computational results are obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

含脱层单向铺设层合梁非线性后屈曲分析

采用四分区模型,将含脱层单向铺设复合材料层合板梁分为4个子梁,根据复合材料层合理论,考虑后屈曲路径上位于脱层界面上、下子梁之间的局部受力与变形机制,建立了子梁之间接触力与变形之间的非线性定量关系。在此基础上,结合可伸长梁的几何非线性理论,推导出了计及接触效应的各子梁的非线性后屈曲控制方程。设定简支板梁的边界条件以及脱层前沿处各子梁之间力和位移的连续性条件,通过对控制方程和定解条件归一化,采用小参数摄动法求解,并根据梁的平衡微分方程的特点,解析其通解与特解的构造,获得了含脱层单向铺设层合梁受轴向压力作用的临界屈曲荷载及后屈曲平衡路径的理论解。通过对含脱层单向铺设的复合材料层合梁进行数值分析,综合讨论了脱层长度和深度等对层合板梁的临界屈曲载荷及接触性能的影响,并将所得的理论解与ABAQUS有限元分析得到的结果进行对比,结果表明二者高度吻合。研究发现梁的屈曲模态包含宏观的整体失效模态和界面的微观屈曲模态。梁的屈曲荷载和接触性能都是其固有属性,前者受梁的几何参数和材料参数的影响较显著,而后者则主要受脱层的位置和大小影响。