Interlaminar stresses in laminated composite plates, cylindrical/spherical shell panels damaged by low-velocity impact

Prediction of damage caused by low-velocity impact in laminated composite plate cylindrical/spherical shell panels is an important problem faced by designers using composites. Not only the in-plane stresses but also the interlaminar normal and shear stresses play a role in estimating the damage caused. The work reported here is an effort in getting better predictions of damage in composite plate cylindrical/spherical shell panels subjected to low-velocity impact.

The low-velocity impact problem is treated as a quasi-static problem. First, the in-plane stresses are calculated by 2-D nonlinear finite element analysis using a 48 degrees of freedom laminated composite shell element. The damage analysis is then carried out using a Tsai-Wu quadratic failure criterion and a maximum stress criteria. Interlaminar normal and shear stresses are predicted after taking into account the in-plane damage caused by low-velocity impact. The interlaminar stresses are obtained by integrating the 3-D equations of equilibrium through the thickness. The deformed geometry is taken into account in the third equation of equilibrium (in the thickness direction). After evaluating the formulation and the computer program developed for correctness, the interlaminar stresses are predicted for composite plates/shell panels which are damaged by low-velocity impact.     

Interlaminar shear stresses in laminated composite plates under thermal and mechanical loading

The combined effects of thermal and mechanical loadings on the distribution of interlaminar shear stresses in composite laminated thin and moderately thick composite plates are investigated numerically using the commercially available software package MSC NASTRAN/PATRAN. The validity of the present finite element analysis is demonstrated by comparing the interlaminar shear stresses evaluated using the experimental measurement. Various parametric studies are also performed to investigate the effect of stacking sequences, length to thickness ratio, and boundary conditions on the interlaminar shear stresses with identical mechanical and thermal loadings. It is observed that the effect of thermal environment on the interlaminar shear stresses in carbon-epoxy fiber-reinforced composite laminated plates are much higher in asymmetric cross-ply laminate and anti-symmetric laminate compared to symmetric cross-ply laminate and unidirectional laminate under identical loadings and boundary conditions.     

考虑三维应力的复合材料层压板疲劳寿命分析

由于层间应力的存在,受面内载荷作用的复合材料层压板实际处于多轴应力状态。构建了由刚性元、弹簧元和二维板元构成的准三维有限元模型,结合单向板在典型应力状态下的疲劳试验结果和疲劳损伤模型,发展了一种考虑三维应力的、预测任意铺层多向层压板疲劳寿命的分析方法,包括应力分析、静力和疲劳累积损伤失效分析及材料性能退化3个主要部分,能够模拟面内和层间损伤产生、发展直至层压板整体破坏的完整过程,并得到疲劳寿命。对2种T300/QY8911多向铺层板进行了实际计算,寿命预测结果与试验结果吻合较好。      

A solution of laminated cylindrical shells using an unconstrained third-order theory

An unconstrained third-order shear deformation theory is presented for the analysis of laminated anisotropic cylindrical shells. Based on the realistic through-thickness distribution of the in-plane displacements, a zig-zag function is used to approximate the piece-wise nature of the displacements. The zero-shearing condition on the laminate surfaces and continuous conditions for the transverse shear stresses on the inter-laminar surfaces have been considered for the final stresses calculation, the displacement functions remain to be unconstrained. This theory is very useful for the finite element analysis because it requests only C⁰ continuity for the assumed displacement fields. By comparing with three-dimensional elasticity theory for laminated orthotropic cylindrical shell, the performance of the present theory is verified. The problems solved in this paper illustrate that the present theory is very accurate for the thin and moderately thick shells.     

Effect of stacking sequence and thickness on the interlaminar stresses for quasi-isotropic laminated plates with a hole

This paper discusses the evaluation of interlaminar stresses, in the presence of in-plane stress gradients, for composite laminates by using three-dimensional equilibrium equations. The stress gradients are calculated by means of an improved finite difference scheme. A quasi-isotropic laminated plate with a circular hole subjected to a uniform tension stress is considered. The effects of stacking sequence and the thickness of the laminate on the interlaminar stresses have been studied.     

Modelling of thick composites using a layerwise laminate theory

The layerwise laminate theory of Reddy¹ is used to develop a layerwise, two-dimensional, displacement-based, finite element model of laminated composite plates that assumes a piecewise continuous distribution of the tranverse strains through the laminate thickness. The resulting layerwise finite element model is capable of computing interlaminar stresses and other localized effects with the same level of accuracy as a conventional 3-D finite element model. Although the total number of degrees of freedom are comparable in both models, the layerwise model maintains a 2-D-type data structure that provides several advantages over a conventional 3-D finite element model, e.g. simplified input data, ease of mesh alteration, and faster element stiffness matrix formulation. Two sample problems are provided to illustrate the accuracy of the present model in computing interlaminar stresses for laminates in bending and extension.     

Prediction of stiffness and stresses in z-fibre reinforced composite laminates

The mechanical properties of z-pinned composite laminates were examined numerically. Finite element calculations have been performed to understand how the through-thickness reinforcement modifies the engineering elastic constants and local stress distributions. Solutions were found for four basic laminate stacking sequences, all having two percent volume fraction of z-fibres. For the stiffness analysis, a micro-mechanical finite element model was employed that was based on the actual geometric configuration of a z-pinned composite unit cell. The numerical results agreed very well with some published solutions. It showed that by adding 2% volume fraction of z-fibres, the through-thickness Young's modulus was increased by 22–35%. The reductions in the in-plane moduli were contained within 7–10%. The stress analysis showed that interlaminar stress distributions near a laminate free edge were significantly affected when z-fibres were placed within a characteristic distance of one z-fibre diameter from the free edge. Local z-fibres carried significant amount of interlaminar normal and shear stresses.     

复合材料开口缝合补强试验及模拟

缝纫能够明显改善层合板的层间力学性能,但缝纫的引入造成了层合板面内的损伤,导致了面内性能的下降.开口的损伤也主要是由孔边的层间失效引发的,开口缝合补强的引入,可以改善孔边的层间力学性能.本文对缝合补强前后开口层合板的静力学拉伸性能进行了试验和有限元的模拟,发现补强后层合板的面内力学性能并没有明显的降低.     

Investigation of anisotropic reinforcement patch corners

Corners at symmetrically attached reinforcement patches of in-plane loaded laminate plates constitute a source for stress localizations. The mechanical in-plane fields of such stress localizations are obtained by means of the boundary finite element method (BFEM). Corresponding in-plane investigations provide the singular in-plane force exponents and their displacement modes characterizing the corner situation. Due to the availability of the in-plane fields in a semi-analytical way their gradients are readily evaluated then. The interlaminar stresses inside the plate are retrieved by equilibrium considerations on a ply-by-ply basis.     

INTERLAMINAR SHEAR STRESS ANALYSIS OF COMPOSITE LAMINATE WITH LAYER REDUCTION TECHNIQUE

Interlaminar stress plays an important role in the delamination failure of laminated composites. A recently presented theory, the Interlaminar Shear Stress Continuity Theory (ISSCT), can directly and accurately predict interlaminar shear stresses in laminated composites by the constitutive equations. The present study further generalizes the derivation to a complete form from which many displacement-based laminate theories can be derived. Most of all, both the single-layer and multiple-layer approaches are incorporated into the analysis in the thickness direction. The laminate is discretized into several sublaminates and, then, a layerwise theory is applied in the analysis of this reduced laminate. This reduction in the number of layers used in the analysis makes the calculation of interlaminar shear stresses on the interested interface in a thick laminate more efficient. In addition, numerical solutions in closed-form and finite element form for laminates under cylindrical bending and bidirectional bending are examined. It is found that in the thickness direction the cubic order of the interpolation function and the discretization with four to six sublaminates can reduce the computational efforts dramatically and retain the accuracy of the predicted stresses within ±8 per cent.     

Dynamic stability of stiffened laminated composite plates and shells subjected to in-plane pulsating forces

The dynamic stability of laminated composite stiffened or non-stiffened plates and shells due to periodic in-plane forces at boundaries is investigated in this paper. A three-dimensional (3-D) degenerated shell element and a 3-D degenerated curved beam element are used to model plates/shells and stiffeners, respectively. The characteristic equations to find the natural frequencies, buckling loads and their corresponding mode shapes are obtained from the finite element equation of motion. Then, the method of Hill's infinite determinants or the method of multiple scales is applied to analyse the dynamic instability regions. Numerical results are presented to demonstrate the effects of various parameters, such as skew angle, lamination scheme, stiffened scheme, in-plane force type and curvature of cylindrical shell, on the dynamic stability of stiffened and non-stiffened plates and shells subjected to in-plane pulsating forces at boundaries.     

螺栓夹持填充孔复合材料层压板层间应力分析

为了研究含螺栓复合材料层压板连接接头的层间应力分布规律,提出了采用填充孔形式和虚拟界面层方法求解层间应力,并建立了三维有限元模型对受面内压缩载荷的螺栓夹持填充孔层压板进行分析。结果表明:层压板层间应力集中不仅发生在孔边,还会在螺栓头边缘附近出现,且夹持力越大螺栓头边缘附近的层间应力集中越严重;合理的螺栓夹持力能改善孔边应力状态,提高孔边抵抗分层的能力,但无法改善螺栓头边缘附近的层间剪切应力集中状态。因此,在进行含螺栓夹持的层压板机械连接结构孔边层间强度设计时,也要考虑螺栓头边缘的层间剪切应力集中问题,以提高复合材料结构的安全性。     

The effects of three-dimensional states of stress on damping of laminated composites

A three-dimensional finite element method is developed for the characterization of the effects of three-dimensional states of stress on the damping of laminated composites. The calculation of laminate damping is performed by the use of a strain energy method and the damping properties of the individual laminae. Particular attention is paid to the effects of interlaminar stresses on laminate damping. These effects are studied by varying the fiber orientation and the laminate width-to-thickness ratio. The predicted damping and natural frequency data from the finite element analysis are compared with experimental results obtained from an impulse-frequency response technique. This study shows that the three-dimensional finite element method is a powerful technique for the determination of damping of laminated composites, and that it provides considerable potential for full three-dimensional characterization in more complex structures and loading situations.     

C<Superscript>0</Superscript>-type global–local theory with non-zero normal strain for the analysis of thick multilayer composite plates

In this paper, an accurate and efficient C⁰-type third-order global–local model incorporating effects of the transverse normal strain is proposed to study the thermal/mechanical behaviors of thick multilayer cross-ply plates. Transverse displacement is assumed to be a linear distribution through the thickness direction, for which the normal strain could be readily computed. Based on the interlaminar continuity conditions of in-plane displacement and transverse shear stresses, layer-dependent variables could be reduced. Employing shear stress free condition at the upper and the lower surfaces, derivatives of transverse displacement are eliminated from the displacement field, so that C⁰ interpolation functions are only required for the finite element implementation. As a result, the number of variables is independent of the number of layers of the laminate. To assess the proposed model, the classical quadratic eight-node isoparametric element is used for the interpolation of all the displacement parameters defined at each nodal point on the composite plate. Comparing with various existing composite plate models, it is found that simple C⁰ finite elements with non-zero normal strain could produce accurate deformations and stresses of thick multilayer composite plates subjected to thermal and mechanical loads.     

基于三维有限元法的层合圆柱壳应力分析

针对空间结构中常见的蜂窝夹芯壳体提出了一种32节点相对自由度三层壳元,以及一种精确计算层间应力的后处理方案。这种32节点壳元可以更好地反映结构固有的特性,易与三维实体单元相连接,使变厚度、带有补强的蜂窝夹芯复合材料壳体等复杂结构问题得以正确建模。本文作者的后处理方案克服了位移有限元层间应力不连续的缺点,保证了应力精确满足边界条件。综合运用以上方法的典型算例表明:计算精度是令人满意的。     

Multi-layer higher-order finite elements for the analysis of free-edge stresses in composite laminates

The analysis of the free-edge stress distributions in composite laminates under uniaxial tension is approached by a finite element technique based on a multi-layer higher-order laminate theory. Several finite elements corresponding to different through-thickness assumed distributions of the displacement unknowns are developed. Numerous stacking sequences are examined in the applications. The results are compared with the ones obtained by various investigators with other modelling approaches. The use of the proposed technique is demonstrated to be simple and effective both for the analysis of in-plane and out-of-plane distributions of intralaminar and, noticeably, interlaminar stress components. © 1998 John Wiley & Sons, Ltd.     

A postprocess method for laminated shells with a doubly curved nine-noded finite element

A numerical method that predicts through-the-thickness stresses accurately by using in-plane displacement of Efficient Higher Order Shell Theory (EHOST) as a postprocessor is implemented in nine-noded doubly curved shell element. In the present study, an efficient postprocess method is developed in the framework of shell finite element without losing the accuracy of solutions. This method consists of two steps. First is to obtain the relationship between shear angles of First Order Shear Deformation Theory (FSDT) and EHOST. Second is to construct accurate displacement and stress fields from the FSDT solution by using EHOST displacement fields as a postprocessor. To obtain accurate transverse shear stresses, integration of equilibrium equation approach is used. In the course of calculating transverse shear stresses, the computation of third derivatives of transverse deflection is required. Simply supported curved panels and finite cylinder problems demonstrate economical and accurate solution of laminate composite shells provided by the present method. The present postprocess method should work as an efficient tool in the stress analysis of multilayered thick shells.     

Nonlinear dynamic stability of laminated composite shells integrated with piezoelectric layers in thermal environment

This paper reports the nonlinear dynamic stability characteristics of laminated composite cylindrical (CYL) and spherical (SPH) shells integrated with piezoelectric layers using the finite element method. The shells are subjected to a thermal environment in addition to the in-plane periodic load and the electric load. The theoretical formulation considers Sanders?? approximation for doubly curved shells, and von Kármán type nonlinear strains are incorporated into the first-order shear deformation theory (FSDT). The formulation includes the effects of transverse shear, in-plane and rotatory inertia. The in-plane periodic load is taken as the parametric excitation in the governing equation. The nonlinear matrix amplitude equation is obtained by employing Galerkin??s method. The correctness of the formulation is established by comparing the authors?? results with those available in the published literature. Detailed parametric studies are carried out to investigate the effects of different parameters on the dynamic stability characteristics of laminated composite shells.     

A multilayer hybrid-stress finite element analysis of a through-thickness edge crack in A ± 45° laminate

A solution is given for the three-dimensional stress field near a through-thickness edge crack in a thin ± 45° laminate having elastic ply moduli typical of graphite/epoxy. The stress distribution was obtained by a three-dimensional multilayer finite element analysis based on the hybrid stress model, formulated through the minimum complementary energy principle. The results indicate that the in-plane stresses of each individual ply follow the classical $\text{1}\text{√r}$ stress singularity, but that the shape of isostress contours in the crack tip region is strongly distorted from predictions based on two-dimensional anisotropic fracture mechanics theory. The interlaminar shear stresses increase rapidly as the crack tip is approached, but are restricted to a local region around the crack tip and flanks. The interlaminar normal stress is assumed to be negligible in the formulation of the analysis.     

LOW-VELOCITY IMPACT RESPONSE OF COMPOSITE LAMINATES CONSIDERING HIGHER-ORDER SHEAR DEFORMATION AND LARGE DEFLECTION

SUMMARY

Low-velocity impact responses of composite laminates are investigated analytically and experimentally. In analytical research, the finite element analyses based on various plate theories and three-dimensional thoeory are performed. For experimental research, a drop weight type impact test system is used. In geometrical non-linear analysis, a displacement field considering higher-order shear deformation and large deflection of the laminate is assumed and the finite element formulation is derived. The modified Hertzian contact law is incorporated into the finite element program to evaluate contact force. Numerical results including impact force histories, deflections, dynamic strains in the laminate from the impact response analysis are presented and compared with the experimental results from impact test. The results of the investigation indicate that higher-order shear deformation and large deflection effects should be considered to accurately describe the low-velocity impact response including interlaminar shear stress of the laminate.