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.     

压电层合板层间应力新研究

综述了压电层合板数值分析方面的一些进展，在一般层合板理论的基础上提出一个多层压电层合板单元，依据线性压电理论建立压电层合板单元的杂交元模型，利用三维8节点块状单元建立多层压电层合板的有限元列式，解决了压电层合板层间应力的连续性。     

Study of the effect of in-plane and interlaminar stresses on damping in fluid-filled laminated composite cylinders

The modal strain energy method is used in conjunction with a three-dimensional finite element analysis in the characterization of the effects of in-plane and interlaminar stresses in fluid-filled composite laminate cylindrical shells. A semi-analytical, 8-noded isoparametric finite element, which includes both the symmetric and antisymmetric modes in the circumferential direction, is used in the analysis. The effects of fiber angle, contained fluid height, size parameter of the shell, and stacking sequence on the contribution of in-plane and interlaminar stresses to the overall system damping in fluid-filled, composite laminate cylindrical shells are studied.     

Interlaminar stress analysis in general thick composite cylinder subjected to nonuniform distributed radial pressure

Interlaminar stresses in thick a composite cylinder with general layer stacking subjected to uniform and nonuniform distributed radial pressure are studied. The layerwise theory of Reddy is employed for formulation of the problem. An analytical method is presented for solving the governing equations. To increase accuracy, interlaminar stresses are obtained by integrating the equilibrium equation of elasticity. After a convergence study, the accuracy of the layerwise laminate theory is investigated using the predictions of finite element method. Predictions of Hooke's law and integration method for interlaminar stresses are compared. Uniform and nonuniform internal and external loads are considered and a parametric study is done for various cylinders.     

Effect of ellipse orientation on the thermoelastic behaviour of skew laminated composite plate with elliptical cutout

An effort is made to study the thermoelastic behaviour of a cross-ply laminated composite skew plate with elliptical cutout subjected to pressure and non-linearly varying temperature loading in the present analysis. Orientation of the elliptical cut out is varied from 0° to 180° with respect to horizontal at an interval of 30° in the anti clockwise direction is considered for the present analysis. A three-dimensional heat conduction analysis in fibre reinforced composite laminates has been simulated by finite element method to get realistic temperature in the laminate under different thermal boundary conditions. A finite element method, which works on the basis of three-dimensional theory of elasticity, is employed to evaluate the stresses and deformations. The effect of orientation due to pressure loading on the stresses and transverse deflection is observed to be insignificant. The magnitudes of the in-plane normal stresses, σ _x and σ _y, for temperature loading are greatly affected by ellipse orientation and are observed to be minimum at the ellipse orientation of 0° and 90°, respectively. The in-plane and inter-laminar shear stresses are observed to be minimum at the ellipse orientation of 90°.     

A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials under thermal shock

In the present work, a study of thermoelastic analysis of functionally graded cylindrical shells subjected to transient thermal shock loading is carried out. A semi-analytical axisymmetric finite element model using the three-dimensional linear elasticity theory is developed. The three-dimensional equations of motion are reduced to two-dimensional ones by expanding the displacement field in Fourier series in the circumferential direction involving circumferential harmonics. The material properties are graded in the thickness direction according to a power law. The model has been verified with the results of simple analytical isotropic cylindrical shells subjected to a transient thermal loading. Additional FGM results for stresses and displacements are presented.     

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.     

TORSIONAL STRESS ANALYSIS OF CORD-RUBBER COMPOSITE LAMINATES

SUMMARY

The structural response of two-ply cord-rubber composite laminates subjected to torsional loading is investigated using non-linear three-dimensional finite element analysis. The present model treats a two-ply composite as an equivalent three/six-ply system with rubber and cord-rubber elements. The present finite element model is validated by comparing the results obtained to the existing experimental and analytical solutions. The effects of torque and bias angle on twist angle, axial displacement, coupled strain ratio, and interply and interface shear stresses are presented. Also, the effects of interply thickness and rubber elasticity were studied. The presented results illustrate the torsional behaviour of two-ply cord-rubber composites from three-dimensional analysis.     

U-transformation-finite element method in 3-dimensional analysis of orthotropic laminates

For an orthotropic laminate, an equivalent system with doubly cyclic periodicity is introduced. Then a 3-dimensional finite element model for the equivalent system is transformed into the unitary space, where the large finite element matrix equation is decoupled into some small matrix equations. Such a decoupling very efficiently reduces the computational effort. For an orthotropic laminate with four clamped edges, no exact elasticity solution is available, and the deflection values predicted by different methods have a considerable difference each other for a small length-to-thickness ratio. The present predictions are the largest because the present method is a full 3-dimensional finite element analysis without superfluous constraints. Illustrative numerical examples are presented to observe the distributions of stresses through the thickness of the laminates.     

Stress Distributions around Holes in Composite Laminates Subjected to Biaxial Loading

An approximate solution has been derived for the in-plane stresses near a circular hole, in an orthotropic composite laminate under biaxial loading. The derived stresses were found to be in good agreement with the exact analytical solution, for a series of laminates investigated. However, the degree of accuracy of the approximate stress distribution is strongly influenced by the laminate lay-up and the biaxiality ratio. The resultant stresses could be employed in stress based fracture models to investigate the notch sensitivity and fracture mechanisms of composite plates with an open-hole subjected to biaxial loading.     

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.     

A strong solution for a generalized plane strain unsymmetrical piezoelectric bi-layer laminates

Interfacial stresses, electric fields, and electric displacements of a piezoelectric unsymmetrical bi-layer orthotropic laminate are presented. A state space equation for orthotropic piezoelectric material is derived from three-dimensional piezoelectric elasticity directly. With the application of the transfer matrix and recursive solution approach, a strong solution for the unsymmetrical piezoelectric generalized plane strain bi-layer laminated structure is sought after considering all elastic and piezoelectric constants of materials. Electromechanical boundary layer effect is identified quantitatively at free edges. To facilitate the discussion on the results, the corresponding calculations from finite element models are compared with those of the strong solution.     

EFFECT OF MESH DISTORTION ON THE ACCURACY OF TRANSVERSE SHEAR STRESSES AND THEIR SENSITIVITY COEFFICIENTS IN MULTILAYERED COMPOSITES

SUMMARY

A study is made of the effect of mesh distortion on the accuracy of transverse shear stresses and their first-order and second-order sensitivity coefficients in multilayered composite panels subjected to mechanical and thermal loads. The panels are discretized by using a two-field degenerate solid element, with the fundamental unknowns consisting of both displacement and strain components, and the displacement components having a linear variation throughout the thickness of the laminate. A two-step computational procedure is used for evaluating the transverse shear stresses. In the first step, the in-plane stresses in the different layers are calculated at the numerical quadrature points for each element. In the second step, the transverse shear stresses are evaluated by using piecewise integration, in the thickness direction, of the three-dimensional equilibrium equations. The same procedure is used for evaluating the sensitivity coefficients of transverse shear stresses. Numerical results are presented showing no noticeable degradation in the accuracy of the in-plane stresses and their sensitivity coefficients with mesh distortion. However, such degradation is observed for the transverse shear stresses and their sensitivity coefficients. The standard of comparison is taken to be the exact solution of the three-dimensional thermoelasticity equations of the panel.     

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.     

Finite element analysis of free-edge stresses in composite laminates under mechanical an thermal loading

In this paper, a multi-particle finite element [Nguyen VT, Caron JF. A new finite element for free edge effect analysis in laminated composites. Comput Struct, accepted for publication] is applied for general laminated and is shown to be capable of simultaneously predicting global and local responses. The analysis of free-edge stresses of composite laminates subjected to mechanical and thermal loads is performed using this C^o${\mathbb{C}}^o$ eight-node layer-wise finite element after a classical bending validation. Laminates with finite dimensions are considered and three-dimensional out-of-plane stresses in the interior and near the free edges are evaluated. The results obtained with this finite element modelling are compared with those available in the literature. The present calculation provides accurate stresses and can be utilised as and operational tool to predict interlaminar stresses under the loads of mechanical and thermal combined.     

Three-dimensional linear elastic distributions of stress and strain energy density ahead of V-shaped notches in plates of arbitrary thickness

This paper presents an analytical solution, substantiated by extensive finite element calculations, for the stress field at a notch root in a plate of arbitrary thickness. The present approach builds on two recently developed analysis methods for the in-plane stresses at notch root under plane-stress or plane strain conditions, and the out-of-plane stresses at a three-dimensional notch root. The former solution (Filippi et al., 2002) considered the plane problem and gave the in-plane stress distributions in the vicinity of a V-shaped notch with a circular tip. The latter solution by Kotousov and Wang (2002a), which extended the generalized plane-strain theory by Kane and Mindlin to notches, provided an expression for the out-of-plane constraint factor based on some modified Bessel functions. By combining these two solutions, both valid under linear elastic conditions, closed form expressions are obtained for stresses and strain energy density in the neighborhood of the V-notch tip. To demonstrate the accuracy of the newly developed solutions, a significant number of fully three-dimensional finite element analyses have been performed to determine the influences of plate thickness, notch tip radius, and opening angle on the variability of stress distributions, out-of-plane stress constraint factor and strain energy density. The results of the comprehensive finite element calculations confirmed that the in-plane stress concentration factor has only a very weak variability with plate thickness, and that the present analytical solutions provide very satisfactory correlation for the out-of-plane stress concentration factor and the strain constraint factor.     

Fracture simulation of CFRP laminates in mixed mode bending

This paper analyses the progressive mixed mode delamination failure in unidirectional and multidirectional composite laminates using fracture experiments, finite element (FE) simulations and an analytical solution. The numerical model of the laminate is described as an assembly of damageable layers and bilinear interface elements subjected to mixed mode bending. The analytical approach is used to estimate the total mixed mode and decomposed fracture energies for laminates with different stacking sequences, which is also validated through experiments. It is concluded that the interlaminar fracture toughness of multidirectional laminates is considerably higher than that of the unidirectional ones. The effect of initial interfacial stiffness and element size is studied and it is also shown that their value must not exceed a definite limit for the numerical simulations to converge. The model can also be further extended to simulate the mixed mode fracture in hybrid fiber metal laminates.     

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。      

Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis

A partial mixed finite element (FE)–state space method (SSM) semi-analytical approach is presented for the static analysis of piezoelectric smart laminate composite and functionally graded material (FGM) plates. Hence, using the Hamiltonian formalism, the three-dimensional piezoelectricity equations are first worked so that a partial mixed variational formulation, which retains the translational displacements, electric potential, transverse stresses, and transverse electric displacement as primary variables, is obtained; this allows, in particular, straightforward fulfillment of the electromechanical continuity constraints at the laminate interfaces. After an in-plane FE discretization only, the problem is first reduced, for a single layer, to a Hamiltonian eigenvalue problem that is solved using the symplectic approach; then, the multilayer solution is reached via the SSM propagator matrix. The proposed methodology is finally applied to the static analysis of piezoelectric-cross-ply hybrid laminated composite and FGM plates. In a comparison with open literature, available tabulated results show good agreements, thus validating the proposed approach.     

Stress analysis of single-bolt, single-lap, countersunk composite joints with variable bolt-hole clearance

Single-lap, carbon-epoxy joints with countersunk fasteners were modelled using the nonlinear finite element code Abaqus. A highly-detailed analysis of the stress distribution at the countersunk hole boundary is provided. Bolt-hole clearance, which arises due to limitations in manufacturing capabilities, is modelled extensively. Clearance levels both inside and outside typical aerospace fitting tolerances are studied and the finite element model is validated with experimental data. Plots of radial stress in each ply of the countersunk laminate show the load transfer to be severely localised, with only a few plies bearing the majority of the load. The inclusion of clearance in the model was shown to result in far higher radial stresses compared to those in the neat-fit joint model. An associated loss in joint stiffness of more than 10% was recorded for the highest clearance considered (240 μm). Finally compressive through-thickness stresses are shown to be present at the damageable region of the countersunk hole, and increase with bolt-hole clearance. These compressive stresses, which are an indicator of lateral constraint, are seen to suppress “brooming” failure in the countersunk laminate.