Inversion of loading time history using displacement response of composite laminates: Three-dimensional cases

Summary An inverse procedure is proposed to reconstruct the time history of transient loads on the surface of composite laminates from the knowledge of dynamic displacement response at only one receiving point. A hybrid numerical method (HNM) is adopted as the forward solver to compute the dynamic displacement response of composite laminates subjected to arbitrary loads. By introducing a kernel displacement function — the dynamic displacement response of composite laminates excited by a point stepimpact load and the displacement response subjected to a load with an arbitrary force function are expressed in a form of convolution integral. The force history is reconstructed by employing an inversion algorithm, in which the least-squares optimization method being adopted to deconvolute the integral. Both point loads and loads with small spatial distribution are investigated and numerical verifications are given. The robustness of the procedure in the presence of noise is also investigated. Good agreements between the identified and true functions for all cases demonstrate the effectiveness of the present inverse procedure. The present inverse procedure is useful for determining impact loads on material surface using response on a point remote to the impact point.     

Analytical and experimental studies on the low-velocity impact response and damage of composite laminates under in-plane loads with structural damping effects

In this paper, low-velocity impact response and damage of composite laminates under in-plane loads are analytically and experimentally investigated. The authors recently proposed a modified displacement field of plate theory, considering the effect of initially loaded in-plane strain, and used a finite element program to analyze the structural behavior of the composite laminate. In this study, the program is upgraded to account for the structural damping effect of the laminate. A pendulum type impact test system and an in-plane loading fixture are constructed for the experimental study. The analytical and experimental impact behaviors are compared at different impact energy levels for cases with an initial in-plane tensile load and a compressive load, as well as cases without the initial in-plane load. The results show good correspondence between the analytical and experimental impact force histories. The effect of the initial in-plane load reduces for higher impact energies. The numerical estimation of the damaged area is in good agreement with the results from C-scanning experiments.     

A new conjugate gradient method for solving multi-source dynamic load identification problem

In this paper, a new conjugate gradient method is suggested to estimate the multi-source dynamic loads acting on a surface of 3D plate. The algorithm reconstructs the distribution functions of the dynamic loads by the displacement responses at two receiving points. The dynamic Green’s function and response function of dynamic forces excitation are obtained by finite element method. Under linear and time-invariant supposition, the response at a receiving point on the structure can be expressed as a convolution integral of the desired unknown force and the corresponding Green’s function, where the continuous convolution functions are spatially and temporally discretized by trapezium formula. The preset method is suggested to identify the dynamic loads by minimizing the error functions. Finally, the multi-source dynamic loads are successfully identified by the proposed method, and the performance of numerical simulations is superior to Landweber iteration method, which demonstrates the efficiency and robustness of the proposed method.     

面内线性变化载荷作用复合材料层合板的振动与屈曲优化

采用广义微分求积（GDQ）法开展了不同边界条件下承受面内线性变化载荷作用复合材料层合板振动与屈曲的分析与优化。针对GDQ法求解面内线性变化载荷工况复合材料层合板屈曲问题存在计算振荡、不收敛现象,提出载荷扰动策略实现了GDQ法对复合材料层合板屈曲问题的稳定高效求解。基于基础圆频率和临界屈曲载荷系数的归一化指标,分析了铺层角度对复合材料层合板综合性能的影响,并结合直接搜索模拟退火算法开展了复合材料层合板的铺层顺序优化。结果表明:铺层角度变化对屈曲性能的影响明显强于频率特性;面内线性变化载荷中,以弯曲载荷作用下复合材料层合板的优化综合性能受边界条件变化的影响最小,而优化铺层角度受边界条件变化的影响最大。研究结果为复杂载荷作用下复合材料层合板的设计提供了参考。      

Probabilistic evaluation of fuselage-type composite structures

A methodology is developed to simulate computationally the uncertain behavior of composite structures. The uncertain behavior includes buckling loads, natural frequencies, displacements, stress/strain, etc., which are the consequences of the random variation (scatter) of the primitive (independent random) variables in the constituent, ply, laminate and structural levels. This methodology is implemented in a computer code IPACS (integrated probabilistic assessment of composite structures). A fuselage-type composite structure is analyzed to demonstrate the code's capability. The probability distribution functions of the buckling loads, natural frequency, displacement, strain and stress are computed. The sensitivity of each primitive (independent random) variable to a given structural response is also identified from the analyses.     

Optimization of laminated composites subject to uncertain buckling loads

Optimal design of composite laminates under buckling load uncertainty is presented. The laminates are subjected to biaxial compressive loads and the buckling load is maximized under worst case in-plane loading which is computed using an anti-optimization approach. The magnitudes of the in-plane loads are not known a priori resulting in load uncertainty subject to the only constraint that the loads belong to a given uncertainty domain. Results are given for continuous and discrete fibre orientations which constitute the optimization problem coupled to load anti-optimization problem leading to a nested solution method. It is observed that the stacking sequence of a laminate designed for a deterministic load case only differs considerably from that of a robust laminate designed taking load uncertainties into account. Consequently the buckling load carried by a deterministic design is considerably less than the one carried by a robust design when both are subjected to uncertain loads.     

Analysis of bolted–bonded composite single-lap joints under combined in-plane and transverse loading

A semi-analytical solution method is developed for stress analysis of single-lap hybrid (bolted/bonded) joints of composite laminates under in-plane as well as lateral loading. The laminate and bolt displacements are based on the Mindlin and Timoshenko beam theories, respectively. For the adhesive, the displacement field is expressed in terms of those of laminates by using the shear-lag model. The derivation of the governing equations of equilibrium of the joint is based on the virtual work principle, where the kinematics of each laminate are approximated by local and global functions and the bolt kinematics is assumed in terms of cubic Hermitian polynomials. The capability of the present approach is justified by validation and demonstration problems, including the analysis of bolted and bonded joints and hybrid joints with and without considering a disbond between the adhesive and laminates.     

Combined tension and bending testing of tapered composite laminates

A simple beam element used at Bell Helicopter was incorporated in the Computational Mechanics Testbed (COMET) finite element code at the Langley Research Center (LaRC) to analyze the responce of tappered laminates typical of flexbeams in composite rotor hubs. This beam element incorporated the influence of membrane loads on the flexural response of the tapered laminate configurations modeled and tested in a combined axial tension and bending (ATB) hydraulic load frame designed and built at LaRC. The moments generated from the finite element model were used in a tapered laminated plate theory analysis to estimate axial stresses on the surface of the tapered laminates due to combined bending and tension loads. Surfaces strains were calculated and compared to surface strains measured using strain gages mounted along the laminate length. The strain distributions correlated reasonably well with the analysis. The analysis was then used to examine the surface strain distribution in a non-linear tapered laminate where a similarly good correlation was obtained. Results indicate that simple finite element beam models may be used to identify tapered laminate configurations best suited for simulating the response of a composite flexbeam in a full scale rotor hub.The U.S. Government right to retain a non-exclusive, royalty-free license in and to any copyright is acknowledged.     

Mode II fracture characterization of a hybrid cork/carbon-epoxy laminate

Fracture characterization under mode II loading of a hybrid laminate composed by a unidirectional carbon fiber-epoxy composite and cork was performed using the End Notched Flexure test. A data reduction scheme based on equivalent crack length concept, specimen compliance and Timoshenko beam theory was applied to evaluate fracture toughness under mode II loading of a composed beam (cork and carbon-epoxy composite). The adopted procedure depends exclusively on the data issuing from load–displacement (P–δ) curve and does not require crack length monitoring during the test which is a difficult task to be accomplished with the necessary accuracy in the ENF test. A numerical analysis using cohesive zone modeling and an inverse procedure was performed to assess the mode II cohesive law that simulates the material fracture under shear loading. It was concluded that hybridization is advantageous relative to monolithic carbon-epoxy laminate in which concerns the observed failure mode, which altered from typically brittle to very ductile thus contributing to avoid sudden shear failures in real applications.     

Response of multilayered composite laminates to dynamic surface loads

A theoretical investigation of the response of multilayered composite laminates to concentrated and distributed dynamic surface loads is carried out. Each layer of the laminate is assumed to be transversely isotropic and dissipative with arbitrarily oriented symmetry axis. The dissipative property of the material is modeled approximately through the introduction of a frequency-dependent damping function. A multiple transform technique is used to calculate the spectra and time histories of the displacements and stresses produced by a variety of dynamic loads, and the quantitative features of the waves produced in the laminate are determined. The methodology developed in this work is expected to be useful in the prediction of the response of composite laminates to impact loads and also in the characterization of acoustic emission (AE) sources in these materials under static and dynamic loads.     

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.     

Lateral buckling analysis of thin-walled laminated channel-section beams

The lateral buckling of a laminated composite beam with channel section is studied. A general analytical model applicable to the lateral buckling of a channel-section composite beam subjected to various types of loadings is derived. This model is based on the classical lamination theory, and accounts for the material coupling for arbitrary laminate stacking sequence configuration and various boundary conditions. The effects of the location of applied loading on the buckling capacity are also included in the analysis. A displacement-based one-dimensional finite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite beam with arbitrary boundary conditions. Numerical results are obtained for thin-walled composites under central point load, uniformly distributed load, and pure bending with angle-ply and laminates. The effects of fiber orientation, location of applied load, and types of loads on the critical buckling loads are parametrically studied.     

Analytical and Experimental Evaluation of Nonlinear Viscoelastic-Viscoplastic Composite Laminates under Creep, Creep-Recovery, Relaxation and Ramp Loading

A numerical procedure to predict long-term laminate properties of fibre reinforced composite materials was developed. In the procedure, we extended the classical laminate theory to include time related response of composite materials for membrane and flexural loading. The material response, dependent on the stress history, was modelled using the Schapery single integral equation. The integrals were handled by an approximate method that uses the Prony's series and only requires the storing of the current stress and some internal strain components. An efficient semi-direct time-integration scheme, providing a stable integration process, was derived to be included in the numerical procedure. Comparisons of theoretical results were made with experiments conducted on composite materials under creep-creep recovery, relaxation and ramp loading.     

纵向稀加筋复合材料园柱层壳的稳定分析

本文引用位移函数,将文献[1]中园柱层壳的经典理论微分方程组转化成一个微分方程式,在此基础上、将W.Nowacki方法[3]推广到复合材料园柱层壳,得到了稀加筋园柱层壳临界载荷的解析解答 给出了算例、若略去筋条时,其数值计算能退化成与文献[1]中相应的数值结果完全一致.      

复合材料非定常温度场的有限元计算

本文借助复合材料的微观特性建立了宏观物理模型,并把空间有限元离散模型和时问-空间有限元离散模型应用于具有任意结构参数和边界条件的复合材料的非定常温度场的有限元计算。计算结果表明,采用集中质量热容矩阵可以得到比采用一致质量热容矩阵更高的计算精度;采用时间-空简有限元可以得到比采用空间有限元更高的计算精度。计算结果还表明,由于复合材料的导热性能极差,因此它是作为航天飞机和空间飞机的热保护装置的理想材料之一。      

Low-velocity impact analysis of composite laminates under initial in-plane load

In this paper, low-velocity impact response and damage of composite laminates is analytically investigated. A modified displacement field of the plate considering initially loaded in-plane strain is proposed. From the displacement field, a finite element equation on the structural behavior of composite laminate is newly induced and a computational program is coded. Numerical results using the FEM code is compared with the numerical ones from reference. Additional numerical analysis is performed on another impact condition, and the effect of initial in-plane load is investigated. Potential delamination damage area in the first inter-ply surface from the bottom of the laminate is approximately estimated, and the effect of the initial in-plane load and the impact condition are also investigated. Consequently, it may be concluded that the initial in-plane load of the laminate does not affect so much on the impact damage area of the laminate.     

复合材料层合板在非保守力作用下的动力稳定性

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,进而导出此问题的有限元基本方程及求解临界力和固有频率的特征方程。用载荷增量法计算了在多种边界条件下不同边长比的复合材料矩形板在面内受随从力作用的临界载荷,分析了不同角铺设方向及两种材料组合板的临界载荷。计算结果表明,边界条件对层合板的动力稳定性有较大影响,复合材料层合板的角铺设方向对临界载荷有较大影响。     

Identification of embedded interlaminar flaw using inverse analysis

The integrity of a composite laminate can be greatly affected by an existence of embedded interlaminar flaw. In general, identification of such a flaw often requires expensive tools and tedious processes. The aim of the present work is to develop a novel method with the aid of an intelligent post-processing scheme, thereby not relying on those sophisticated experiments. Essentially the proposed procedure utilizes an inverse analysis to estimate unknown delamination parameters from limited measurements. The procedure first constructs approximate functions relating the delamination parameters to measurement parameters. Then, a multi-dimensional minimization technique is adopted to search for the best estimates of unknown parameters corresponding to the lowest value of error objective function. In the present verification and simulation analyses, surface strains at discrete locations on a composite laminate under three-point bending are selected as the input measurements. Although reasonable estimates are obtained with these measurements, to increase their accuracy, the deflection at load point is also included as measurement input. Additional improvements are observed when those measurements under multiple loading conditions are included. A detailed error sensitivity analysis is also carried out to confirm the methods robustness. These results suggest the current method to be one of the alternate identification approaches for detecting a single embedded delamination in composite laminates.     

The Displacement Perspective During Ultimate Failure of Composite Laminates

This paper deals with the studies on the state of displacement of symmetric and anti-symmetric angle-ply and cross-ply laminated composite plates during its ultimate failure, subjected to transverse static load. First-order shear deformation theory (FSDT) is employed in conjunction with the finite element approach using eight-noded quadratic isoparametric element. The free vibration analyses of isotropic and laminated composite plates are carried out to ensure the overall validity of the present finite element formulation. The mid surface of the laminate is considered as the reference plane. The principal material directions in different laminae are oriented to produce a laminated structural element capable of resisting loads in several directions. The stiffness of a composite laminate is obtained from the properties of the constituent laminae. The affected stiffness of the failed lamina is discarded completely after the failure of weakest ply. The rigidity matrix of the laminate with remaining laminae is re-established. The re-evaluation process continues until the laminate fails completely. To investigate the displacement behaviour of laminates during the ultimate failure, parametric studies are carried out for different cases by varying the stacking sequences, fiber orientations, layer thicknesses, aspect ratios and the number of layers in the laminate. The comparison of results in terms of non-dimensional natural frequencies and ply-by-ply failure analyses obtained from the present investigation are made with those available in the reported literature.     

Delamination threshold loads for low velocity impact on composite laminates

An investigation of the low velocity impact load level at which a composite laminate will delaminate is presented. The delamination threshold load is described as the load level, obtained from the load–time history or load–displacement plot, at which a sudden load drop occurs due to specimen stiffness loss as a result of laminate level damage. Approximately 500 low velocity impact load–time histories from the Air Force Research Laboratory (AFRL) Low Velocity Impact Database are used to investigate the delamination threshold load. The database contains laminate impact test data for graphite/epoxy (AS4/3501-6), graphite/PEEK (AS4/APC-2), and graphite/BMI (IM7/5260) material systems. The delamination threshold load observations are compared to C-scan damage measurements of impacted specimens to determine if the sudden load drop corresponds to delamination development.