Generalized layerwise mechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors

A coupled linear layerwise laminate theory and a beam FE are formulated for analyzing delaminated composite beams with piezoactuators and sensors. The model assumes zig-zag fields for the axial displacements and the electric potential and it treats the discontinuities in the displacement fields due to the delaminations as additional degrees of freedom. The formulation naturally includes the excitation of piezoelectric actuators, their interactions with the composite laminate, and the effect of delamination on the predicted sensory voltage. The quasistatic and modal response of laminated composite Gr/Epoxy beams with active or sensory layers having various delamination sizes is predicted. The numerical results illustrate the strong effect of delamination on the sensor voltage, on through the thickness displacement and on the stress fields. Finally, the effect of delamination on modal frequencies and shapes are predicted and compared with previously obtained experimental results.     

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.     

考虑面内破坏过程的含分层损伤复合材料加筋层合板的动力响应

基于复合材料层合板一阶剪切理论，推导了复合材料层合板单元的刚度阵和质量阵列式；同时采用了Adams应交能法与Rayleigh阻尼模型相结合的方法，构造了相应的阻尼阵列式；为了防止在低阶模态中分层处出现的上、下子板不合理的嵌入现象，建立了含分层损伤复合材料加筋层合板动力分析的分层分析模型和虚拟界面联接模型。在上述模型和理论基础上，采用了Tsai提出的刚度退化准则和动力响应分析的精细积分法，对含分层损伤复合材料加筋层合板结构进行了动力响应和破坏分析。通过算例，分别讨论了外载频率、分层位置，以及破坏过程的刚度退化对含损伤复合材料加筋层合板动力响应特征的影响，得到了一些具有理论和工程价值的结论。     

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.     

FRP strengthened walls with delaminated regions – Geometrically nonlinear response

The paper studies the geometrically nonlinear behavior of walls that are strengthened with fiber reinforced polymer (FRP) composite materials but include pre-existing delaminated regions. The paper uses an analytical–numerical methodology. Three specially tailored finite elements that correspond to perfectly bonded regions, to delaminated regions where the debonded layers are in contact, and to delaminated regions where the debonded layers are not in contact are presented. All finite elements are based on a high order multi layered plate theory. The geometrical nonlinearity is introduced by means of the Von Karman nonlinear strains whereas the contact nonlinearity is handled iteratively. The validity and convergence of the finite element models is demonstrated for each type of element through comparison with closed form analytical solutions available for specific cases. The unified model that combines the three types of finite element is then used for studying the nonlinear behavior of a locally delaminated FRP strengthened wall under in-plane normal and in-plane shear loads. Finally, conclusions regarding the effect of the delamination on the response of the strengthening system, on the conditions that evolve in the bonded region that surrounds the delamination, and on the global response of the multi-layered structure are drawn. Additional conclusions regarding the application of the modeling approach to other delamination sensitive layered structural systems close the paper.     

ANALYSIS OF ELASTO-PLASTIC POSTBUCKLING AND ENERGY RELEASE RATE FOR DELAMINATED FIBER METAL LAMINATED BEAMS IN THERMAL ENVIRONMENT

The elasto-plastic postbuckling of fiber metal laminated beams with delamination and the energy release rate along the delamination front are discussed in this paper. Considering geometrical nonlinearity, thermal environment and geometrical initial imperfection, the incremental nonlinear equilibrium equations of delaminated fiber metal laminated beams are established,which are solved using the differential quadrature method and iterative method. Based on these,according to the J-integral theory, the elasto-plastic energy release rate is studied. The effects of some important parameters on the elasto-plastic postbuckling behavior and energy release rate of the aramid reinforced aluminum laminated beams are discussed in details.     

Vibration analysis of rotating fully-bonded and delaminated sandwich beam with CNTRC face sheets and AL-foam flexible core in thermal and moisture environments

Abstract

In the present study, the high-order free vibration analysis of rotating fully-bonded and delaminated sandwich beams; with and without vertical contact; containing AL-foam flexible core and carbon nanotubes reinforced composite (CNTRC) face sheets subjected to thermal and moisture field are investigated by using generalized differential quadrature method (GDQM). The compressible core and face sheets of sandwich beam, respectively, are composed of Aluminum alloy foam with variable mechanical properties in the thickness direction and CNTRC with temperature dependent material properties. In this study, the high-order sandwich panel theory (HSAPT) for AL-foam flexible core and Euler-Bernoulli beam theory for CNTRC face sheets are considered. By employing Hamilton’s principle, the governing partial differential equations of motion and associated boundary and continuity conditions for various types of regions (fully-bonded, delaminated with contact, delaminated without contact) are derived and then discretized by using GDQM. The final formulations lead to 14 partial differential equations for the entire structure including five equations for fully-bonded two-headed parts of AL-foam cored sandwich beam (AL-FCSB) and four equations for delaminated middle part of AL-FCSB beam which are combined in axial and transverse deformations. A parametric study is performed to investigate the influence of some important parameters such as existence of delaminated region, type of delaminated region (with or without contact), longitudinal position of delaminated region, slenderness ratio, face sheet thickness ratio, CNT volume fraction, temperature rise, moisture concentration, rotating speed, and hub radius. The obtained results reveal that the 1st frequency of delaminated AL-FCSB beam, whether with or without vertical contact, is less remarkably than ones of fully-bonded AL-FCSB beam which its value for the case of delaminated ‘with contact’ is larger than that of ‘without contact’. Moreover, the 1st frequency variation of the delaminated AL-FCSB beam is symmetrical with regard to the longitudinal position of the debonded region such that the 1st natural frequency declines with moving the debonded region toward the center of the beam. The study of vibration behavior of rotating sandwich beams is very important in design of rotating structural systems, specially damaged ones, such as airplanes, helicopter rotor blades, and robot arms. One of the most important types of damage encountered in mentioned cases is the decomposition of two layers or delamination. Working these rotating structures in the media, are always along with variations of temperature and humidity and hence their mechanical properties may be changed due to the environment conditions.

Communicated by S. Velinsky     

Energy release rate analysis of delamination in laminates

Delamination growth caused by local buckling of delaminated laminate under compression may lead the structure to failure. In this paper, a two-dimensional model of delaminated laminates was studied using a combination of post-buckling theory and fracture mechanics. The delaminated laminate under compression is first analysed by the first-order deformable plate thoery. Then, the energy release rateG at every point along the delamination front is derived by means of a variation problem with a moving boundary. Finally, the near-surface elliptical delamination is also discussed. There is a large variation of energy release rateG along the delamination front, the direction in which delamination growth may occur depends upon the delamination aspect ratio, the state of applied load and the orientation of plies.Supported by the Chinese Science Foundation of Aeronautics     

复合材料层合梁在横向线载荷下的脱层扩展

基于可动边界变分原理对层合梁脱层扩展进行了分析;考虑了脱层间的接触效应,建立了层合梁在横向线载荷作用下的非线性控制微分方程及相应的定解条件;应用Griffith准则导出了脱层前缘各点处的能量释放率表达式;通过算例讨论了脱层长度、脱层深度、几何尺寸、材料性质等因素对脱层扩展的影响.研究表明:脱层越长、越深、横向载荷越大,脱层越容易扩展;梁的长高比L/h及材料的E_(11)/E_(22)越大,脱层越不易发生扩展.     

Vibration of composite beams with two overlapping delaminations

Delaminations in composite laminates may develop from small cracks due to fabrication and impact loading, or from places of high stress concentration. The locations of the delaminations are not determinate. In this research, an analytical solution for the free vibration of a composite beam with two overlapping delaminations is presented. The delaminated beam is analyzed as seven interconnected beams using the delaminations as their boundaries. The continuity and equilibrium conditions are satisfied between the adjoining regions of the beams. Classical beam theory is applied to each of the beams. Complex vibration behaviors emerge for different sizes and locations of the delaminations. Comparison with analytical results reported in the literature verifies the validity of the present solution.Christian N Della wishes to thank Nangyang Technological University for the scholarship.     

Solutions of beam-shaped-function for analysis of composite plates with embedded delaminations

Based on beam-shaped-function, the analytical solution for composite plates with arbitrary embedded delaminations is presented. The deflection function of the delaminated plate is composed by those of beams with the corresponding loading and support conditions, which can be easily and accurately derived from the beam analysis, and the deflection amplitude is derived by the minimum potential energy principle. The closed form solutions of displacements, stresses, and energy release rate of a composite plate containing an arbitrarily embedded rectangular delamination are obtained and compared with the three-dimensional finite element results to validate the accuracy of this present method. Furthermore, the influences of delamination depth, length, central position, and modulus mismatch ratio (E ₁/E ₂) of the upper and lower sublaminate on the energy release rate are discussed.     

复合材料脱层梁的屈曲分析

本文研究了具有任意位置透型脱层的复合材料梁的屈曲问题。基于弹性理论建立了复合材料脱层梁的基本方程式。对脱层梁进行了分区处理,利用B样条函数作为位移型函数的基函数,方便地描述了脱层长度、脱层位置。考虑边界条件、区间位移连续性条件和弯矩剪力的平衡条件以及纵向内力的附加条件,对基本方程式进行了求解。得出了脱层位置不同,脱层长度不同的屈曲荷载的变化规律,并与轴对称脱层时的屈曲荷载进行了比较,认为层合梁考虑脱层对屈曲的影响是非常必要的。     

A low-dimensional model for delamination in a laterally loaded composite beam: Theory and experiment

Delamination is an important failure mechanism in certain types of composite structures. As layers of the composite become separated from one another, the composite loses some of its structural integrity and may not be capable of supporting the intended loads. Detecting this type of damage is currently a problem of interest to the structural health monitoring community. However, in order to design an appropriate detection strategy, knowledge of the underlying physics of delaminated structures is essential. Here, a low-dimensional model of a delamination in a laminated beam is developed. The model consists of only five elements yet is capable of capturing much of the behavior observed in an accompanying experiment. Both experimental and theoretical results are presented for the static response of a thin, delaminated beam.     

含分层损伤复合材料层合板分层扩展研究

采用基于Ｍｉｎｄｌｉｎ－阶剪切理论的四节点板单元,分析了含椭圆分层合板分层扩展行为。利用虚裂纹闭合技术计算分层前缘处的总能量释放率,并采用总能量释放率准则作为扩展准则,结合自适应网格移动技术,并考虑了分层前缘闭合接角效应,对 合材料层合板的分层扩展行为进行了模拟分析。结果表明,初始分层形状对其扩展有方式有限大影响。     

具脱层的正交铺设圆柱壳的初始后屈曲分析

基于非线性弹性理论,建立了含脱层正交铺设圆柱壳的后屈曲控制方程,应用Koiter初始后屈曲理论和小参数摄动法,导出了系统的一阶和二阶摄动控制方程,以及相应的边界条件、位移连续条件和力平衡条件,然后逐阶求解．算例中,讨论了不同脱层深度和长度对脱层复合材料圆柱壳屈曲和初始后屈曲特性的影响,并与已有文献进行了比较．     

Study on failure process of delaminated stiffened composite plates under compression

Failure behavior of the delaminated stiffened composite plates under compression is studied by the finite element method, based on a Global-Local variational model. A virtual crack closure technique and a self-adaptive grid moving scheme are proposed to predict the delamination growth process. The contact effect along the delamination front is considered. The numerical results show that the influences of the distribution and location of the stiffeners, the configuration and size of the delamination, the boundary condition and the contact upon the failure behavior of the plates are significant. The project supported by the National Natural Science Foundation of China (59975014)     

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.     

Effects of voids on postbuckling delamination growth in unidirectional composites

This work examines the effects of manufacturing induced voids on the postbuckling behavior of delaminated unidirectional composites. In the finite element model developed, a through-width delamination is introduced close to one surface of a flat panel, and a void is placed in the delamination plane ahead of each delamination front. The panel is subjected to compression in the fiber direction. The postbuckling delamination growth is studied by calculating the strain energy release rate (SERR) using the virtual crack closure technique. Local stress analyses of the region near the delamination front are also performed to further investigate the void effects. It is found that although the presence of void does not significantly alter the postbuckling transverse displacement of the delaminated panel, the induced stress perturbation by the void affects the SERR. The Mode II SERR as well as the total SERR increase depending on the size of the void and its distance from the delamination front. Since the Mode I SERR shows non-monotonic behavior with the applied load, the effects of voids are studied on its maximum value.     

Numerical analysis of delamination growth for stiffened composite laminated plates

A study of postbuckling and delamination propagation behavior in delaminated stiffened composite plates was presented. A methodology was proposed for simulating the multi-failure responses, such as initial and postbuckling, delamination onset and propagation, etc. A finite element analysis was conducted on the basis of the Mindlin first order shear effect theory and the von-K6rm~n nonlinear deformation assumption. The total energy release rate used as the criteria of delamination growth was estimated with virtual crack closure technique (VCCT). A self-adaptive grid moving technology was adopted to model the delamination growth process. Moreover, the contact effect along delamination front was also considered during the numerical simulation process. By some numerical examples, the influence of distribution and location of stiffener, configuration and size of the delamination, boundary condition and contact effect upon the delamination growth behavior of the stiffened composite plates were investigated. The method and numerical conclusion provided should be of great value to engineers dealing with composite structures.     

Calculation of energy release rate in composite delaminated cylindrical shells under circumferential concentrated dynamic load

The delamination growth may occur in delaminated cylindrical shells subjected to external dynamic load and it will further cause structural failure. Based on the variational principle of moving boundary and considering the contact effect between delamination regions, in this paper, the nonlinear governing equations for the delaminated cylindrical shells under the action of circumferential concentrated dynamic load are derived, and the corresponding boundary and matching conditions are given. At the same time, according to the Griffith criterion, the formulas of energy release rate along the delamination front are obtained and the delamination growth is studied. In the numerical calculation, the delamination growth of axisymmetrical laminated cylindrical shells is analyzed, and the effects of the delamination sizes and depths, the geometrical parameters, the material properties and the laminate stacking sequences on delamination growth are discussed.