Theoretical modelling of laminated composite plates

Formulation of appropriate governing equations, simpler than the three-dimensional equations of elasticity yet capable of predicting, fairly accurately, all important response parameters such as stress and strain, is attempted in modelling a structural component. Several theoretical models are available in the literature for the analyses of plates. The emergence of fibre-reinforced plastics as an attractive form of structural construction, added a new complexity to the modelling considerations of laminates by requiring the estimation of the interlaminar stresses and strains. In this paper, modelling considerations of laminated composite plates are discussed. The classical laminated plate theory and higher-order shear deformation models are reviewed to bring out their interlaminar stress predictive capabilities, and some new modelling possibilities are indicated. This work has been supported by the Aeronautics Research and Development Board, Ministry of Defence, Government of India.     

Finite element analysis of stiffened laminated plates under transverse loading

A finite element model is developed to study the behavior of stiffened laminated plates under transverse loadings. Transverse shear flexibility is incorporated in both beam and plate displacement fields. A laminated plate element with 45 degrees of freedom is used in conjunction with a laminated beam element having 12 degrees of freedom for the bending analysis of eccentrically-stiffened laminated plates. The validity of the formulation is demonstrated by comparing with the available solutions in the literature. The numerical results are presented for eccentrically-stiffened layered plates having various boundary conditions and with stiffeners varying in number.     

Finite volume analysis of laminated composite plates

A numerical procedure for analysis of general laminated plates under transverse load is developed utilizing the Mindlin plate theory, the finite volume discretization, and a segregated solution algorithm. The force and moment balance equations with the laminate constitutive relations are written in the form of a generic transport equation. In order to obtain discrete counterparts of the governing equations, the plate is subdivided into N control volumes by a Cartesian numerical mesh. As a result, five sets of N linear equations with N unknowns are obtained and solved using the conjugate gradient method with preconditioning. For the method validation, a number of test cases are designed to cover thick and thin laminated plates with aspect ratio (width to thickness) from 4 to 100. Simply supported orthotropic, symmetric cross‐ply, and angle‐ply laminated plates under uniform and sinusoidal pressure loads are solved, and results are compared with available analytical solutions. The shear correction factor of 5/6 is utilized throughout the procedure, which is consistent with test cases used in the reviewed literature. Comparisons of the finite volume method results for maximum deflections at the center of the plate and the Navier solutions obtained for aspect ratios 10, 20, and 100 shows a very good agreement. Copyright © 2016 John Wiley & Sons, Ltd.     

Stochastic nonlinear failure analysis of laminated composite plates under compressive transverse loading

This paper present the second ordered statistics of first-ply failure response of laminated composite plate with random material properties under random loading. The basic formulation is based on higher order shear deformation plate theory (HSDT) with the geometrically nonlinearity in the von-Karman. The direct iterative based C⁰ nonlinear finite element method combined with mean centered first order perturbation technique developed by the authors are extended and successfully applied nonlinearity for failure problem with a reasonable accuracy to predict the second order statistics (standard deviation) of first-ply failure response using Tsai-Wu and Hoffman failure criterion with macroscopic analysis. Typical numerical results for various combinations of boundary conditions, plate thickness ratios, aspect ratios, laminates scheme and layers, elastic modulus ratios have been presented to illustrate the application of developed procedure. Some new results are presented and examined which clearly demonstrated the importance of the randomness in the system parameters in the failure response of the structures subjected to transverse loadings.     

An enriched macro finite element for the static analysis of thick general quadrilateral laminated composite plates

This article presents the formulation of an enriched macro finite element based on the trigonometric shear deformation theory for the static analysis of symmetrically laminated composite plates. Shear correction factor is not required because this theory accounts for tangential stress-free boundary conditions on the plate boundary surfaces. The macro element is obtained using the principle of virtual work and Gram-Schmidt orthogonal polynomials as enrichment functions. The implementation of the obtained algorithm is simple and efficient, and allows studying general quadrilateral plates with a single macro element. Several examples are presented to show the capability and applicability of the developed formulation.     

Impact analysis of laminated composite plates and shells by super finite elements

A super finite element method that exhibits coarse-mesh accuracy is used to predict the transient response of laminated composite plates and cylindrical shells subjected to non-penetrating impact by projectiles. The governing equations are based on the classical theories of thin laminated plates and shells taking into account the von Karman kinematics assumptions for moderately large deflections. A non-linear Hertzian-type contact law accounting for curvatures of the colliding bodies is adopted to calculate the impact force . The theoretical basis of the present finite element model is verified by analysing impact-loaded laminated composite plate and shell structures that have previously been studied through analytical or other numerical procedures. The predictive capability of the present numerical approach is successfully demonstrated through comparisons between experimentally-measured and computed force-time histories for impact of carbon fibre-reinforced plastic (CFRP) plates. The current computational model offers a relatively simple and efficient means of predicting the structural impact response of laminated composite plates and shells.     

Interlaminar shear stresses around an internal part-through hole in a laminated composite plate

The equilibrium/compatibility method, which is a semi-analytical post-processing method, is employed for computation of hitherto unavailable through-thickness variation of interlaminar (transverse) shear stresses in the vicinity of the bi-layer interface circumferential re-entrant corner line of an internal part-through circular cylindrical hole weakening an edge-loaded laminated composite plate. A C^o${\text{C}}^{\text{o}}$-type triangular composite plate element, based on the assumptions of transverse inextensibility and layer-wise constant shear-angle theory (LCST), is utilized to first compute the in-plane stresses and layer-wise through-thickness average interlaminar shear stresses, which serve as the starting point for computation of through-thickness distribution of interlaminar shear stresses in the vicinity of the bi-layer interface circumferential re-entrant corner line of the part-through hole. The same stresses computed by the conventional equilibrium method (EM) are, in contrast, in serious error in the presence of the bi-layer interface circumferential re-entrant corner line singularity arising out of the internal part-through hole, and are found to violate the interfacial compatibility condition. The computed interlaminar shear stress can vary from negative to positive through the thickness of a cross-ply plate in the neighborhood of this kind of stress singularity.     

On the axial and interfacial shear stresses due to thermal mismatch in hybrid composite sheets

A simple analytical model has been developed which allows the determination of the axial and the interfacial shear stresses which can occur in hybrid fiber composites as a consequence of the mismatch in coefficients of thermal expansion and Young's modulus. The configuration considered is a finite-width hybrid composite monolayer with alternating high- and low-modulus fibers. To account properly for the interfacial shear between fiber and matrix, a modified shear-lag model is used which permits extensional deformation due to thermal expansion of the matrix in the fiber direction. Typical stresses due solely to temperature changes are calculated, and these show steep boundary-layer edge stresses at free corners.     

Progressive failure analysis of laminated composite with hole under flexural loading

Abstract

Understanding the progressive failure of laminated composite plays an important role in the structural integrity analysis of a structure. Continuum damage mechanics-based approach is one of the powerful tools to analyze the failure of laminated composite structures. The present paper investigates the damage evolution and progression in laminated composites with a hole under flexural loading. The presence of high-stress concentrations along with the free edge stresses at the vicinity of the hole in laminated composite leads to complex failure mechanisms. The influence of the change in thickness and lamina configuration on the strength characteristics is presented.     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.     

Flexural analysis of laminated plates based on trigonometric layerwise shear deformation theory

ABSTRACT

A trigonometric layerwise shear deformation theory is developed for the flexural analysis of laminated plates. The present theory achieves in-plane displacement continuity, transverse shear stress continuity, and traction-free boundary condition. Hence, botheration of shear correction coefficient is neglected. The governing differential equation and boundary conditions are obtained from the principle of virtual work. Although the present analytical method is bounded to a corner supported boundary condition, it neglects the numerical and computational error. Like first-order shear deformation theory, the present theory possesses five numbers of unknowns. Several numerical predictions are carried out and results are compared with those of other existing numerical approaches.     

Interlaminar stresses in thick rectangular laminated plates with arbitrary laminations and boundary conditions under transverse loads

In the present study, interlaminar stresses resulting from bending of thick rectangular laminated plates with arbitrary laminations and boundary conditions are analyzed analytically based on a three-dimensional multi-term extended Kantorovich method (3DMTEKM). Using the principle of minimum total potential energy, three systems of coupled ordinary differential equations with non-homogeneous boundary conditions are obtained. Then an iterative procedure is established to achieve analytical solution. The results obtained from this theory are compared with those of analytical solutions existing in the literature. It is found that the present results have excellent agreements with those obtained by layerwise theory. The results show that the multi-term EKM converges within only three terms of trial functions and the single-term EKM is not able to estimate the local interlaminar stresses near the boundaries of laminates. Finally, the power of the present approach in obtaining the interlaminar stresses in thick rectangular laminated plates with general types of boundary conditions and lay-ups is examined.     

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.     

MITC finite elements for laminated composite plates

Within the framework of the first‐order shear deformation theory, 4‐ and 9‐node elements for the analysis of laminated composite plates are derived from the MITC family developed by Bathe and coworkers. To this end the bases of the MITC formulation are illustrated and suitably extended to incorporate the laminate theory. The proposed elements are locking‐free, they do not have zero‐energy modes and provide accurate in‐plane deformations. Two consecutive regularizations of the extensional and flexural strain fields and the correction of the resulting out‐of‐plane stress profiles necessary to enforce exact fulfillment of the boundary conditions are shown to yield very satisfactory results in terms of transverse and normal stresses. The features of the proposed elements are assessed through several numerical examples, either for regular and highly distorted meshes. Comparisons with analytical solutions are also shown. Copyright © 2001 John Wiley & Sons, Ltd.     

Interlaminar stresses in general thick rectangular laminated plates under in-plane loads

Three-dimensional multi-term extended Kantorovich method (3-D multi-term EKM) is employed to investigate analytically the interlaminar stresses near free edges of finite length general composite laminates subjected to axial and shearing loads. Using the principle of minimum total potential energy, three systems of coupled ordinary differential equations are obtained. Then an iterative procedure is established to achieve analytical solution. Laminates with finite dimensions are considered and full three-dimensional stresses in the interior and the boundary-layer regions are calculated. The results obtained from this theory are compared with those of analytical results existing in the literature or with those obtained by the finite element method when there exist no available results. It is found that interlaminar stresses have excellent agreements with those obtained by layerwise theory in the interior region and near the free edges of both finite and long laminates. Convergence study of the method reveals that the 3-D multi-term EKM converges within only four terms of trial functions and the single-term EKM is not able to estimate the local interlaminar stresses near the boundaries of laminates. Finally, the power of the present approach in obtaining the interlaminar stresses in finite length thick laminated plates with general lay-ups is examined.     

Effect of temperature and humidity on transient hygrothermal stresses during moisture desorption in laminated composite plates

Polymer matrix composites are relevant materials for future supersonic aircraft due to their high specific properties. However, in such aeronautical applications, the material is exposed to severe environmental conditions. The present paper aims at assessing an approximate model to evaluate hygrothermoelastic stress in composite laminated plates during moisture desorption taking into account the change of mechanical characteristics induced by the variation of temperature and moisture. The developed method permits us to calculate such stresses during desorption phase without the computation of the moisture concentration, through laminated plates. It observed through this study that the variation of elasticity modulus due to the temperature causes a stress relaxation. These stresses have to be taken into account for the design of composite structures submitted to a moist environment.

Through the presented study, we hope to contribute to the understanding of hygrothermal behaviour of composite laminated plates.     

剪切载荷作用下复合材料挖补修理层合板试验及有限元分析

为了确定剪切载荷作用下含非穿透损伤复合材料挖补修理层合板的破坏模式和抗剪切能力,进行了复合材料挖补修理层合板的剪切试验,并与未损伤复合材料层合板进行对比。试验结果表明,复合材料挖补修理后的层合板具有较高的强度恢复率,且不影响层合板的后屈曲承载能力。同时,建立了剪切载荷作用下复合材料挖补修理层合板的有限元分析（FEA）模型,复合材料母板和补片采用了三维Hashin准则来判定材料失效,母板层与层之间采用零厚度界面单元以有效模拟剪切载荷作用下复合材料母板上、下子板之间的分层。该模型得到的破坏模式与试验结果基本相符。由于挖补修理的设计与工艺复杂性,理论模拟的破坏载荷与试验结果虽不能完全吻合,但其最大15%左右的差异能够满足修理设计的需要。以上结果说明,该模型对剪切载荷作用下复合材料挖补修理层合板的破坏模式和破坏载荷能够进行工程适用的预测。     

Computation of transient hygroscopic stresses in unidirectional laminated composite plates with cyclic and asymmetrical environmental conditions

External variations in temperature and moisture are of primary importance for the long term behaviour of structures made of polymer matrix composites, because they induce residual stresses within laminated composite plates. It was shown (Hahn et al., 1978; Benkeddad et al., 1995, 1996; Tounsi et al., 2000, 2002, 2004; Adda-Bedia et al., 2001; Tounsi and Adda-Bedia, 2003a, b) that the heterogeneity and the anisotropy of such plates, have an influence on the distribution of transient hygroscopic stresses through the thickness of composite plates. The aim of the present paper is to present a simplified approach for the calculation of transient hygroscopic stresses within unidirectional laminates in the case where these latter are exposed to the cyclic and unsymmetric environmental conditions. Several examples are presented to assess such stresses and to demonstrate the efficiency of the used method. These stresses have to be taken into consideration for the design of composite structures submitted to a moist environment.     

Thermal postbuckling analysis of laminated composite plates

The thermal postbuckling behavior of graphite/epoxy multi-layered rectangular plates of various boundary conditions is studied using the finite element method. Temperature dependent thermal and elastic properties of the material are used in the analysis. The nonlinear finite element equations are solved as a sequence of linear eigenvalue problems to trace the thermal postbuckling paths of 15-layered symmetric angle-ply plates. The presence of secondary instability with an unsymmetric deformation mode has been identified for symmetric laminates under uniform temperature rise. In the case of linearly varying temperature rise through the thickness of the plate, the nonlinear equilibrium equations are solved by the modified Newton–Raphson technique to get the temperature-displacement curves.