Delamination modeling in doubly curved laminated shells for free vibration analysis using zigzag theory-based facet shell element and hybrid continuity method

We present a finite element (FE) formulation for the free vibration analysis of doubly curved laminated composite and sandwich shells having multiple delaminations, employing a facet shell element based on the efficient third-order zigzag theory and the region approach of modeling delaminations. The methodology, hitherto not attempted, is general for delaminations occurring at multiple interfacial and spatial locations. A recently developed hybrid method is used for satisfying the continuity of the nonlinear layerwise displacement field at the delamination fronts. The formulation is shown to yield very accurate results with reference to full-field three-dimensional FE solutions, for the natural frequencies and mode shapes of delaminated shallow and deep, composite and highly inhomogeneous soft-core sandwich shells of different geometries and boundary conditions, with a significant computational advantage. The accuracy is sensitive to the continuity method used at the delamination fronts, the usual point continuity method yielding rather poor accuracy, and the proposed hybrid method giving the best accuracy. Such efficient modeling of laminated shells with delamination damage will be of immense use for model-based techniques for structural health monitoring of laminated shell-type structures.     

Analytical Solution of Buckling of Beams with Two Delaminations

A stability problem of composite beams with multiple delamination was tackled. A closed-form solution was found and buckling loads of composite beams with two delaminations were determined in order to obtain their compressive load-carrying capacity. Crack-opening mode was assumed for each detached delaminated region. Seven different regions having different transverse deformations resulted from assumed positions of delaminations. Developing the derived continuity condition equations reduced the number of algebraic equations required to solve the problem analytically. The results of the work were validated by comparing them to those reported in the literature. The effects of length, location, and distribution of multiple delaminations were considered in the comparison, and the results showed very good agreement. Buckling load decreases as delamination size increases. Buckling load for a beam with two delaminations is lower than that for the same beam with a single delamination.     

含矩形内部分层的复合材料层板的 状态分解-片条合成能量解法

根据叠加原理将含有矩形内部分层的层板在横向载荷作用下的受力状态进行分解, 从而 将分层问题归结为在分层表面上的附加剪切载荷作用下层板附加位移与附加应力的分析, 并据此 建立了一个仅包含分层区的力学模型。进而在层板分层区中切取平行于边界的切片, 将切片视为 含分层的层合梁, 其位移模态以相应层合梁的附加位移模态来表示。这样, 可构造层板分层区内满 足位移边界条件的位移场。最后, 应用最小势能原理确定位移幅值的闭合解。计算结果表明, 挠度 幅值远远大于中面位移幅值, 且与由双三角级数能量解法所得挠度幅值吻合很好。      

含矩形内部分层的复合材料圆柱壳块 三维问题的片条合成能量解法

根据叠加原理将含有矩形分层的复合材料圆柱壳块在横向载荷作用下的受力状态进行分解,将分层问题归结为在分层表面受附加剪切载荷作用时附加状态的分析,并据此建立了一个仅包含分层区的力学模型。通过切取纵横切片,利用含分层层合直梁与曲梁的附加位移,构造分层区附加位移模态。最后,应用最小势能原理确定位移幅值的闭合解。     

Delamination buckling of cylindrical shells under axial compression

A model is developed for the study of delamination buckling of axially loaded cylindrical shells. Delamination is assumed to exist before load application, it spans the entire circumference, and it lies on the contact surface of neighboring laminae. The mathematical model employs Donnell-type kinematic relations and linearly elastic material behavior. Furthermore, each lamina is assumed isotropic, and the emphasis is on studying the effect of delamination size and position on the critical load. Two sets of boundary conditions are used with the model: simply supported and clamped. The study reveals several important conclusions. Among them, one may list the following: (a) the critical load is primarily controlled by the position of the delamination from the reference surface, provided that the delamination is not very close to the boundaries; and (b) for long delaminations (relative to the cylinder length), the critical load is not appreciably affected by the boundary conditions.     

Failure analysis of multiple delaminated composite plates due to bending and impact

The present work aims at the first ply failure analysis of laminated composite plates with arbitrarily located multiple delaminations subjected to transverse static load as well as impact. The theoretical formulation is based on a simple multiple delamination model. Conventional first order shear deformation is assumed using eight-noded isoparametric quadratic elements to develop the finite element analysis procedure. Composite plates are assumed to contain both single and multiple delaminations. For the case of impact, Newmark time integration algorithm is employed for solving the time dependent multiple equations of the plate and the impactor. Tsai-Wu failure criterion is used to check for failure of the laminate for both the cases. To investigate the first ply failure, parametric studies are made for different cases by varying the size and number of delaminations as well as the stacking sequences and boundary conditions.     

Numerical and experimental studies on propagation of A0 mode in a composite plate containing semi-infinite delamination: Observation of turning modes

Propagation of the anti-symmetric fundamental Lamb mode (A₀) in a composite laminate containing a semi-infinite delamination was studied through numerical simulations and experimental studies. Here, the wave was generated on a sub-laminate above the delaminated region, and the reflection and transmission of the Lamb modes at the edge of the discontinuity were quantitatively studied. The wave group (named as ‘turning mode’), which was made to propagate from one side of the delamination to the other side (through delaminations) was captured and its transmission factor was estimated. Reflection and transmission factors of wave groups were estimated for three different laminates – quasi-isotropic, unidirectional and cross-ply, containing delamination at various interfaces across thickness. Propagation of ‘turning modes’ were also captured experimentally, employing air-coupled transducers, in quasi-isotropic laminates containing delaminations at different interfaces. Attenuation compensation was incorporated in the experimental signals to improve the comparison of transmission and reflection factors between numerical simulations and experiments.     

含椭圆形分层层板的状态分解-片条合成能量解法

根据叠加原理将含有椭圆形非穿透分层的层板在横向载荷作用下的受力状态进行分解,从而将分层问题归结为在分层表面上的附加剪切载荷作用下层板附加位移与附加应力的分析,并据此建立了一个仅包含分层区的力学模型。进而在层板分层区中切取平行于坐标平面的切片,将切片视为含分层的层合梁,其位移模态以相应层合梁的附加位移模态来表示。这样,可构造层板分层区内满足位移边界条件的位移场。最后,应用最小势能原理确定位移幅值的闭合解。计算结果表明,挠度幅值远远大于中面位移幅值,且与由双三角级数能量解法所得挠度幅值吻合很好。      

Modelling delaminations in postbuckling stiffened composite shear panels

 A method has been developed to predict the effect of delaminations in a postbuckling stiffened structure manufactured from laminated composite materials. The emphasis of the technique, driven by aircraft certification requirements, was towards establishing whether delamination growth would initiate under given loading conditions. A geometric nonlinear finite element analysis was used to calculate the strain energy release rate around the circumference of a circular delamination using the virtual crack closure technique. In order to deal with the complex structural response in a computationally efficient manner, the structure was modelled using plate elements with two layers of plate elements used in the delaminated region. The effect of delamination size on the strength of postbuckling panels was shown to be a complex phenomenon in which trends were difficult to predict. Large delaminations could significantly affect the global and sub-laminate buckling modes and therefore be less critical than smaller delaminations. It was concluded that the method could accurately predict the load and location at which delamination growth would initiate, given suitable critical strain energy release rate data. Received 16 February 2000     

Growth behavior of internal delaminations in composite beam/plates under compression: effect of the end conditions

The effect of the end conditions, i.e. clamped-clamped vs. simply-supported ends on the initial postbuckling and growth behavior of delaminations in plates in studied. The study does not impose any restrictive assumptions regarding the delamination thickness and plate length. First, a closed form solution for the mode mixity, energy release rate and deformation quantities is derived for the case of a clamped-clamped delaminated plate, which complements the already existing solution for a delaminated simply-supported plate. A perturbation procedure is followed, which is based on an asymptotic expansion of the load and deformation quantities in terms of the distortion parameter of the delaminated layer, the latter being considered a compressive elastica. The additional complication in the clamped-clamped case arises because now the amplitude at the clamped end needs to be expanded in terms of the distortion parameter of the delaminated part, in addition to the amplitude at the common section and the distortion parameter of the base plate. The effects of the end conditions on the growth behavior are found to depend on the relative location of the delamination through the thickness. For the same plate length and thickness and the same delamination length and applied strain, delaminations located closer to the surface exhibit nearly the same energy release rate and mode mixity either in a clamped-clamped or a simply supported configuration. However, in delaminations located further away from the surface, for the same applied strain, the energy release rate is larger and there is also a higher mode II component in the simply-supported case. Moreover, the mid-point transverse displacement of the delaminated layer as well as that of the substrate part, is larger in the simply supported case. The same major trend that has been observed in the simply supported case, i.e. the increased growth resistance of the delaminations located near the surface relative to the ones located further inside the plate, is also observed in the clamped-clamped case.     

Straight-sided, buckling-driven delamination of thin films at high stress levels

The fracture mechanics of a straight-sided, thin film delamination at stress levels, which are high compared to the stress required to initiate the delamination is investigated. Buckling at a bifurcation point of the delaminated region, resulting from incompletely relieved stresses in this region, is analysed by a semi- analytical approach for delaminations of infinite extent. The results are compared to numerical predictions based on finite element calculations for finite sized delaminations. The finite element calculations are carried out in the post-buckling regime showing that parts of the crack front will close as a result of bifurcation buckling, while other parts will experience enhanced energy release rate and mode I stress intensity factor. The mode III stress intensity factor is shown to be negligible at the stress levels analysed.     

复合材料梁结构应力波传播特性研究

基于铁木辛柯梁理论,研究含半无限大分层复合材料梁结构中波的反射与透射。考虑表面无接触压力(张开分层)及表面完全接触(闭合分层)两种极端分层条件,对单向复合材料梁建模导出反射与透射矩阵,计算能量反射与透射系数。数值结果表明,能量反射与透射系数同随波频率及分层位置变化,能量传输遵循守恒定律。研究各模态能量分配,描述前两阶弯曲波及第一阶膨胀波间模态转换关系;通过有限元仿真验证该理论分析的正确性。     

Ultrasonic guided wave scattering in a plate overlap

Guided wave scattering in a plate overlap is investigated by numerical calculations and experimental measurements of transmission and reflection factors from the overlap region. In the numerical study, a hybrid boundary element-finite element method is used to calculate the guided wave scattered field from the overlap region. Transmission and reflection factors are calculated for incident A0 and S0 Lamb and n0 shear horizontal waves, including higher modes generated through mode conversion phenomena. In addition, parametric studies of transmission and reflection factors in this problem are performed numerically over various incident modes, frequencies, and overlap lengths. For verification and comparison with numerical results, experiments were conducted to measure the transmission and reflection factors for incident Lamb and shear horizontal waves in steel plates with two different overlap areas. The experimental results agree well with the numerical calculations. The numerical and experimental results show that it is highly feasible to carry out efficient Lamb wave nondestructive evaluation (NDE) in overlapped plates and in multilayer structures with various lap joints by selecting various modes and tuning frequency.     

Fracture Surface Morphology of Delamination Failure of Polymer Fiber Composites Under Different Failure Modes

The delamination of fiber reinforced polymer composites is one of the most common failures encountered in industrial applications. The most unique macroscopic and microscopic fracture surface features of the delaminations under different failure modes are of interests not only for practical failure analysis investigations but also it helps to reveal the physics behind the delamination phenomenon. In this work, fracture surface morphology of the delaminated carbon fiber polymer composites under mode I, mode II, and mixed-mode I/II loading conditions is investigated mainly with scanning electron microscopy. The unique fractographic features are identified and discussed. The results on ductile and brittle matrix composites have shown their own features, and most important of all the alignment angle of fibrils in the resin-rich ductile matrix could be correlated with the delamination mode.     

XFEM simulation of delamination in composite laminates

In this paper, the extended finite element method (XFEM) is extended to simulate delamination problems in composite laminates. A crack-leading model is proposed and implemented in the ABAQUS® to discriminate different delamination morphologies, i.e., the 0°/0° interface in unidirectional laminates and the 0°/90° interface in multidirectional laminates, which accounts for both interlaminar and intralaminar crack propagation. Three typical delamination problems were simulated and verified. The results of single delamination in unidirectional laminates under pure mode I, mode II, and mixed mode I/II correspond well with the analytical solutions. The results of multiple delaminations in unidirectional laminates are in good agreement with experimental data. Finally, using a recently proposed test that characterizes the interaction of delamination and matrix cracks in cross-ply laminates, the present numerical results of the delamination migration caused by the coupled failure mechanisms are consistent with experimental observations.     

多分层对复合材料层合板自振特性的影响

基于Mindlin一阶剪切理论分项等参插值的有限元法, 建立了含多个分层损伤复合材料层合板自由振动分析的有限元模型和分析方法, 并采用线性接触模型模拟分层区域上、 下子板的相互作用。通过典型数值算例, 讨论了分层位置、 数目及板的边界条件诸参数对其振动特性的影响。结果表明: 分层位置沿板长方向变化时, 中间分层的频率变化范围较大, 表面分层变化较小, 但变化趋势基本相同; 沿层合板厚度方向, 多分层中最长分层的位置越靠近层合板中面, 则其对振动特性的影响越大; 多个分层位置较靠近层合板表面, 且板边界条件约束较弱时, 多分层与单分层对振动特性影响的差别不大, 此时, 可将多分层损伤层合板简化为单分层损伤层合板来进行振动分析。      

Simulation and Experiments of Airborne Zero-Group-Velocity Lamb Waves in Concrete Plate

The impact-echo (IE) test method has been widely used to determine the thickness of a plate and to locate delaminations in reinforced-concrete structures. It was later realized that the IE mode is related to the Zero-Group-Velocity S1 Lamb mode (S₁ZGV). An efficient inspection of a structure using rapid, noncontact, air-coupled sensing technology is of great interest. The objective of this research is to investigate the effectiveness of air-coupled sensing to detect the ZGV Lamb mode signals. Numerical simulation and experimental verification were performed to study the in-air ZGV vibration responses induced by a point impact applied on a concrete plate. Two concrete plates, one without delamination and one with delamination, were tested and simulated. For each plate, the simulation results agreed well with the experimental results, and both methods estimated the plate thickness precisely. The numerical simulations also visualized the pressure field in the air and are helpful for interpretation of the experimental results. The numerical and experimental results suggest that the air-coupled sensing technology can serve as an effective and rapid inspection tool for large concrete structures.     

Vibration of beams with two overlapping delaminations in prebuckled states

An analytical solution for the free vibrations of beams with two overlapping delaminations in prebuckled states is presented. The delaminated beam is analyzed as seven interconnected Euler–Bernoulli beams. The continuity and boundary conditions are satisfied between adjoining beams. Both the ‘free mode’ and the ‘constrained mode’ assumption in delamination buckling and vibration are used. A parametric study is performed to investigate the influence of the axial compressive load on the natural frequency and the mode shape of the delaminated beam. A monotonic relation between the natural frequency and the compressive load is observed. Comparisons with the analytical results reported for delamination buckling and vibration verify the validity of the present solution.     

Free vibration analysis of rotating Timoshenko beams with multiple delaminations

Analytical solutions are developed to study the free vibrations of rotating Timoshenko beams with multiple delaminations. The Timoshenko beam theory and both the ‘free mode’ and ‘constrained mode’ assumptions in delamination vibration are adopted. Parametric studies are performed to study the influences of Timoshenko effect and rotating speed on delamination vibration. Results show that the effect of delamination on both modes 1 and 2 natural frequencies is significantly influenced by Timoshenko effect and the rotating speed. Also, the comparison between ‘free mode’ assumption and ‘constrained assumption’ are conducted. Furthermore, the effect of delamination on mode shapes is also influenced by rotating speed and Timoshenko effect. For both Timoshenko effect and rotating speed, the influences on the second vibration mode shape are more significant.