Available online at www.sciencedirect.com Composite Structures 88 (2009) 147–157 www.elsevier.com/locate/compstruct Recent developments in ﬁnite element analysis for laminated composite plates Y.X. Zhang a,*, C.H. Yang b a School of Aerospace, Civil and Mechanical Engineering, The University of New South Wales, Australian Defence Force Academy, Northcott Drive, Canberra, ACT 2600, Australia b School of Engineering and Information Technology, Deakin University, Waurn Ponds, VIC3217, Australia Available online 20 February 2008 Abstract A review of the recent development of the ﬁnite element analysis for laminated composite plates from 1990 is presented in this paper. The literature review is devoted to the recently developed ﬁnite elements based on the various laminated plate theories for the free vibration and dynamics, buckling and postbuckling analysis, geometric nonlinearity and large deformation analysis, and failure and damage analysis of composite laminated plates. The material nonlinearity eﬀects and thermal eﬀects on the buckling and postbuckling analysis, the ﬁrst-ply failure analysis and the failure and damage analysis were emphasized specially. The future research is summarised ﬁnally. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Laminated composite plates; Free vibration; Dynamics; Buckling; Postbuckling; Failure 1. Introduction Composite laminates have been used increasingly in a variety of industrial areas due to their high stiﬀness and strength-to-weight ratios, long fatigue life, resistance to electrochemical corrosion, and other superior material properties of composites. A true understanding of their structural behaviour is required, such as the deﬂections, buckling loads and modal characteristics, the throughthickness distributions of stresses and strains, the large deﬂection behaviour and, of extreme importance for obtaining strong, reliable multi-layered structures, the failure characteristics. Finite element method is especially versatile and eﬃcient for the analysis of complex structural behaviour of the composite laminated structures. Using ﬁnite element method, a signiﬁcant amount of research has been devoted to the analysis of vibration * Corresponding author. Tel.: +61 2 62688169; fax: +61 2 62688276. E-mail address: [email protected] (Y.X. Zhang). 0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.02.014 and dynamics, buckling and postbuckling, failure and damage analysis and etc. A review of the ﬁnite element models developed after 1990 based on the various laminated plate theories for the ﬁnite element analysis of composite laminated plates is presented in this paper. The ﬁnite element analysis reviewed includes the following categories: free vibrations, damping, and transient dynamic response; buckling and postbuckling; geometric nonlinearity and large deformation analysis; damage and failure. Some of the future research on composite laminated plates is also summarized. 2. Laminated composite plate theories The laminated plate theories are essential to provide accurate analysis of laminated composite plates, and a variety of laminated plate theories have been developed and reported in a large amount of literatures. A review of various equivalent single layer and layerwise laminated plate theories was presented by Reddy 148 Y.X. Zhang, C.H. Yang / Composite Structures 88 (2009) 147–157 and Robbins . An overall comparison of laminated theories based on displacement hypothesis was presented by Liu and Li , including shear deformation theories, layerwise theories, generalized Zigzag theories, and the proposed global–local double-superposition theories. A review of theories for laminated and sandwich plates was presented by Altenbach . A review of displacement and stress-based reﬁned shear deformation theories of isotropic and anisotropic laminated plate was given by Ghugal and Shimpi , in which various equivalent single layer and layerwise theories for laminated plates were discussed together with their merits and demerits. A historical review of the zig-zag theories for multi-layered plates and shells up to 2003 were given by Carrera . A review of shear deformation plate and shell theories was presented by Reddy and Arciniega , a selective review and survey of the theories with emphasis on estimation of transverse/interlaminar stresses in laminated composites was given by Kant and Swaminathan , and a selective literature survey on the free-edge eﬀect since 1967 was given by Mittelstedk and Becker . Generally, the laminated plate theories can be broadly divided into the following two categories: (a) Equivalent single layer (ESL) theories, including Classical lamination theory (CLT) The ﬁrst-order shear deformation theory (FSDT) (referred to as Mindlin Plate theory in some literatures) Higher-order shear deformation theories (HSDT)] Layer-wise lamination theory (LLT) (b) Continuum-based 3D elasticity theory The classical lamination theory (CLT) is based on the Kirchhoﬀ plate theory, it is the simplest theory among others, but the shear deformation eﬀects are neglected. The ﬁrst-order shear deformation theories (FSDT) provides a balance between computational eﬃciency and accuracy for the global structural behaviour of thin and moderately thick laminated composite plates, but no accurate prediction for the local eﬀects can be obtained, for example, the interlaminar stress distribution between layers, delaminations, and etc. Various higher-order shear deformation theories have been developed to overcome the limitations in the classical and ﬁrst-order shear deformation theory, and the free boundary conditions of the transverse shear stresses on the upper and lower surfaces can usually be satisﬁed. Layer-wise lamination theory assumes a displacement representation formula in each layer. It can predict accurately the interlaminar stresses, however layerwise models are computational expensive since the number of unknown functions depends on the number of the layers of the laminates. The 3D continuum-based theory can predict the interlaminar stress of a composite laminate, but the computational cost using 3D models is a major concern. 3. Free vibration and dampling analysis of composite laminated plates 3.1. Computational models based on FSDT The ﬁrst-order shear deformation theory (FSDT) has been employed widely to establish ﬁnite element models for free vibration analysis of the composite laminated plates. The eﬀects of lamination and extension–bending coupling, shear and twist-curvature couplings on the lowest frequencies and corresponding mode shapes for free vibration of laminated anisotropic composite plates was investigated using a ﬁnite element method with quadratic interpolation functions and ﬁve engineering degrees of freedom (DOF) . The free and forced vibration response of laminated composite folded plate structures was predicted by a nine-node Lagrangian plate-bending ﬁnite element with ﬁve engineering DOF per node that incorporated rotary inertia . A nine-node isoparametric plate-bending element was used for the analysis of free undamped vibration of rectangular isotropic and ﬁber reinforced laminated composite plates , and an eﬀective mass lumping scheme with rotary inertia was introduced. The free vibration analysis of stiﬀened laminated composite plates was performed using the layered (zigzag) ﬁnite element method based on the ﬁrst-order shear deformation theory . In their work, the layers of the laminated plate were modelled using nine-node isoparametric degenerated ﬂat shell element, and the stiﬀeners were modelled as three-node isoparametric beam elements based on Timoshenko beam theory. Bilinear in-plane displacement constraints were used to maintain the inter-layer continuity, and a special lumping technique was used in deriving the lumped mass matrices. A mixed ﬁnite element formulation with low-order displacement/strain interpolation for plates and shells was used to study the eﬀect of large spatial rotations on the geometric stiﬀness for stability analysis as well as inertia operators for vibrations for laminated composite plates and shells . Damping analysis of composite laminated plates has been carried out using the computational models developed based on the FSDT. The eﬀects of transverse shear deformation on the modal loss factors as well as the natural frequencies of composite laminated plates was investigated using a ﬁnite element method based on the shear deformable plate theory . The complex modules of an orthotropic lamina were employed to model damping eﬀect. A sandwich composite beam and plate ﬁnite superelements with viscoelastic layers were presented for vibration and damping analysis of laminated composite beams or plates . Each layer was considered as simple Timoshenko’s beam or Mindlin-Reissner plate ﬁnite element. The energy dissipation in the viscoelastic layers was taken into account with complex modulus of elasticity theory, and the method of complex eigenvalues and the energy method were considered for damping analysis. This ﬁnite element technique Y.X. Zhang, C.H. Yang / Composite Structures 88 (2009) 147–157 was later utilized to predict the natural frequencies and the modal damping factor of anisotropic ﬁbre-reinforced composite laminated plates . 3.2. Computational models based on HSDT Considerable amount of free vibration analyses of various composite laminated plates has been studied using the ﬁnite element models developed based on diﬀerent kinds of higher-order shear deformation theories. A high-order quadratic isoparametric element using both the eight-node serendipity and the nine-node Lagrangian shape functions was presented for free vibration analysis of laminated composite plates . Various schemes for the generation of the mass matrix were discussed and a comparative study of these schemes was presented. Based on Reddy’s higher-order theory, a ﬁnite element formulation taking into account the parabolic distribution of the transverse shear deformation through the thickness of the plate was presented for vibration analysis of laminated anisotropic plates  with diﬀerent lay-ups and of sandwich plates. A four-node rectangular element with seven-degrees of freedom at each node, whose displacement model was so chosen that the parabolic distribution of transverse shear stresses and the nonlinearity of in-plane displacements across the thickness can be represented adequately was developed for free vibration analysis of laminated composite plate structures having a constant thickness of any individual layer . Latheswary et al.  investigated the static and free vibration analysis of moderately thick laminated composite plates using a 4-node ﬁnite element formulation based on higher-order shear deformation theory, and the transient analysis of layered anisotropic plates using a shear deformable 9-noded Lagrangian element-based on ﬁrst-order shear deformation theory. A C0 continuous ﬁnite element model having ﬁve- and seven-degrees of freedom per node was developed  for free vibration analysis of laminated composite plates, using a higher-order shear deformation theory to account for the parabolic variation of transverse shear stresses through the thickness and linear variation of the normal stresses. The higher-order shear deformation theories (HST6, HST9, HST11, and HST12) and the conventional ﬁrstorder theory (FST) were employed to develop ﬁnite element analysis methods using eight-node isoparametric elements to study the bending, free vibration and impact behaviour of laminated composite plates . Based on a higher-order shear deformation theory, a 4-node, 20-DOF higher-order rectangular plate-bending element was developed for free vibration analysis of laminated composite plates . The transverse displacement was interpolated by using an optimized interpolation function while the additional rotation degrees of freedom were approximated by linear Lagrange interpolation. The con- 149 sistent element mass matrix was used and a damped element was introduced to the ﬁnite element model. A 48-degrees of freedom rectangular ﬁnite element was formulated  based on a kinematics, which exactly ensured the continuity conditions for the displacements, the transverse stresses at the interfaces between the layers of a laminated structure and zero stress conditions at the top and bottom surfaces of the plate, for static and dynamic analyses of moderately thick, multi-layered composite plates. Higher-order trigonometric functions were used to deﬁne the transverse shear deformations and thus the shear correction factors were not required. The free vibration analysis of multi-layered thick composite plates was studied by a ﬁnite element procedure based on an accurate higher-order theory which accounted for the realistic variation of in-plane and transverse displacements through the thickness . The vibration and stability problems of cross-ply and angle-ply laminated composite plates were investigated using general higherorder theories of laminates which took into account the complete eﬀects of transverse shear and normal deformations [26–28]. A simple C0 higher-order facet shell element-based on a shear deformable model of higher-order theory was presented for the free vibration analysis of isotropic, orthotropic and layered anisotropic composite and sandwich laminates . A realistic nonlinear variation of displacements through the shell thickness was incorporated, and shear correction coeﬃcients were eliminated. 3.3. Computational models based on layerwise theories Compared with the computational models for the free vibration and damping analysis developed based on the FSDT and HSDT, relatively few models were developed based on the Layerwise theories. The computational model developed based on the layerwise theories include the 18node, three-dimensional higher-order mixed model for free vibration analysis of multi-layered thick composite plates , in which the continuity of the transverse stress and the displacement ﬁelds were enforced through the thickness of laminated composite plate, the hybrid strain-based layerwise shell element for free vibration of laminated composite plate and shell structures  developed based on several lower-order hybrid strain-based triangular shell elements for the general nonlinear analysis of isotropic shell structures, and etc. 4. Nonlinear dynamic stability and transient response of composite laminated plates The geometric nonlinearity or large amplitude eﬀects on the dynamic stability of the composite laminated plates have been investigated. The dynamic instability of anisotropic laminated composite plates considering geometric nonlinearity , and the eﬀect of large amplitude on the dynamic instability for a simply-supported laminated 150 Y.X. Zhang, C.H. Yang / Composite Structures 88 (2009) 147–157 composite plate were investigated using a C0 shear-ﬂexible, ﬁeld consistent, QUAD-9 plate element. Considering the viscoelastic properties of the material, Kim and Kim  studied the dynamic behavior of laminated composite plates undergoing moderately large deﬂection based on von Karman’s nonlinear deformation theory and Boltzmann’s superposition principle. The eﬀect of large amplitude on the dissipative nature as well as on the natural frequency of viscoelastic laminated plates was examined using ﬁnite element analysis and the method of multiple scales. Based on the ﬁrst-order shear deformation theory, Ribeiro and Petyt [34,35] studied the geometrically nonlinear vibration of thin laminated composite plates using the hierarchical ﬁnite element and the harmonic balance methods. Free and steady-state forced vibration were analysed, and the excitations considered were harmonic plane waves at both normal and grazing incidence. Taking into account the eﬀects of the rotary inertia, transverse shear, and geometrical nonlinearity, a p-version, hierarchical ﬁnite element was presented for free vibration of moderately thick composite laminated plates . The element was later employed to study the large amplitude, geometrically nonlinear periodic vibrations of shear deformable composite laminated plates subjected to harmonic forces applied transversely to the plates . Some higher-order ﬁnite element models have been developed for the ﬁnite element analysis of nonlinear static and dynamic responses of laminated composite plates, such as the ﬁnite element analysis of geometrically nonlinear static and transiently dynamic behaviour of laminated composite plates , in which a higher-order displacement ﬁeld allowing both transverse shear and transverse normal strains was adopted, the ﬁnite element model for the large amplitude free vibration of the laminated composite plates , in which the parabolic variation of transverse shear strains through the thickness of the laminate was accounted for, the continuous ﬁnite element model developed based on a nonlinear higher-order shear deformation theory for nonlinear thermal dynamic analysis of graphite/ aluminium laminated composite plates , the C0 four and nine-node ﬁnite elements for the transient response of orthotropic, layered composite sandwich plates  developed based on a reﬁned form of Reddy’s higher-order theory, in which parabolic variation of the transverse shear stresses was accounted for, and etc. 5. Geometric nonlinear ﬁnite element analysis of laminated composite plates For accurate prediction for the static structural responses of composite laminated plates, geometric nonlinearity should be included in the ﬁnite element analysis. Some literatures on the geometric nonlinear ﬁnite element analysis of laminated composite plates existed. A procedure for the reliability analysis of laminated composite plate structures with large rotations but moder- ate deformation under random static loads was presented via a corotational total Lagrangian ﬁnite element formulation which was based on the von Karman assumption and ﬁrst-order shear deformation theory . An eight-node C0 membrane-plate quadrilateral ﬁnite element-based on the Reissner–Mindlin plate theory was presented to analyse moderately large deﬂection, static and dynamic problems of moderately thick laminates including buckling analysis and membrane-plate coupling eﬀects . Han et al.  used the hierarchical ﬁnite element method to carry out the geometrically nonlinear analysis of laminated composite rectangular plates. Based on the ﬁrst-order shear deformation theory and Timoshenko’s laminated composite beam functions, the current authors developed a uniﬁed formulation of a simple displacement-based 3-node, 18degree-of-freedom ﬂat triangular plate/shell element  and two simple, accurate, shear-ﬂexible displacementbased 4-node quadrilateral elements [46,47] and for linear and geometrically nonlinear analysis of thin to moderately thick laminated composite plates. The deﬂection and rotation functions of the element boundary were obtained from Timoshenko’s laminated composite beam functions. Based on a higher-order shear deformation theory involving four dependent unknowns and satisfying the vanishing of transverse shear stresses at the top and bottom surfaces of the plate, geometrically nonlinear ﬂexural response characteristics of shear deformable unsymmetrically laminated rectangular plates were investigated using a four-node rectangular C1 continuous ﬁnite element having 14 degrees of freedom per node . A high-order plate model which exactly ensured both the continuity conditions for displacements and transverse shear stresses at the interfaces between layers of a laminated structure, and the boundary conditions at the upper and lower surfaces of the plates was used to study the geometrically nonlinear behaviour of multi-layered plates , and based on this reﬁned plate model, a six-node C1 conforming displacement-based triangular ﬁnite element was developed, with the Argyris interpolation used for transverse displacement, the Ganev interpolation used for membrane displacements and transverse shear rotations, and the transverse shear strain distributions represented by cosine functions. A three-dimensional element with two-dimensional kinematic constraints was developed for the geometric nonlinear analysis of laminated composite plates  using a total Lagrangian description and the principle of virtual displacements. The large deformation analysis of circular composite laminated plates  was studied using a 48DOF four-node quadrilateral laminated composite shell ﬁnite element. 6. Buckling and postbuckling analysis of laminated composite plates The buckling of laminated composite plates is an important consideration in the design process, however the critical value of load given by linear buckling analysis may not Y.X. Zhang, C.H. Yang / Composite Structures 88 (2009) 147–157 accurately represent the load-carrying capability of a plate. Although composite laminated plates generally possess less load-carrying capacity after buckling compared to their metallic counterparts, the total load during the postbuckling of a composite laminated plate is still several times that of the critical buckling load. In order to get the practical limits of the load-carrying capability of the composite laminated plates, the postbuckling behaviour has been studied to establish the sustained additional loads after buckling. Considerable eﬀorts have been made for the numerical analysis of the buckling and postbuckling analysis over the years. Leissa [52,53] gave a summary of the buckling and postbuckling studies of composite laminated plates up to 1986, and then he reviewed the development of buckling analysis of laminated composite plates with linear eﬀective constitutive properties . Later a more detailed account of the research on the buckling and postbukcling before 1995 was presented by Noor . 6.1. General buckling and postbuckling analysis of composite laminated plates An assumed hybrid-stress ﬁnite element model together with a composite multilayer element were developed to study the buckling of generally laminated composite plates with arbitrary thickness and edge conditions under an inplane stress system . The equilibrium conditions within each layer, the interlaminar traction reciprocity conditions, and the stress-free boundary conditions on the top and bottom surfaces of the laminate, were satisﬁed by the assumed stress ﬁeld and thus the composite shear correction factors were not required. A shear deformable ﬁnite element was developed for the buckling analysis of laminated composite plates based on Mindlin’s theory in which shear correction factors were derived from the exact expressions for orthotropic materials . The eﬀects of material properties, plate aspect ratio, length-to-thickness ratio, number of layers and lamination angle on the buckling loads of symmetrically and anti-symmetrically laminated composite plates were investigated. An 8-node isoparametric plate ﬁnite element with 5-DOF per node was developed based on the ﬁrst-order shear deformation theory associated with von Karman’s nonlinear strain–displacement relationships to investigate the buckling and post-buckling of moderately thick laminated plates subjected to uni- or bi-axial compression . The eﬀects of boundary conditions, aspect ratio, side to thickness ratio and lay-up sequence on the buckling and post-buckling behaviour were studied in detail. The linear buckling analysis of multilaminated composite plate-shell structures was analysed using a discrete ﬁnite element model based on an eight-node isoparametric element with 10 degrees of freedom per node and the higher-order theory . The geometric stiﬀness matrix was developed taking into consideration the eﬀects of the higher-order terms on the initial in-plane and transverse 151 shear stresses. The element was then used to study the buckling and free vibrations of multilaminated structures of arbitrary geometry and lay-up . A generalized layer-wise stochastic ﬁnite element formulation was developed for the buckling analysis of both homogeneous and laminated plates with random material properties . The pre-buckled stresses were considered in the derivation of geometric stiﬀness matrix and the eﬀect of variation in these stresses on the mean and coeﬃcient of variation of buckling strength was studied. The postbuckling behaviour of laminated composite plates under the combination of in-plane shear, compression and lateral loading was investigated using an element-based Lagrangian formulation based on the assumed natural strain method for composite structures . Natural coordinate-based strains, stresses and constitutive equations were used in the element and the elementbased Lagrangian formulation was computational eﬃcient and had the ability to avoid both membrane and shear locking. 6.2. Eﬀects of material nonlinearity on buckling and postbuckling behaviour of composite laminated plates In the literature, most stability studies of composite laminated plates have been limited to the geometrically nonlinear analysis and the research on the eﬀect of nonlinear eﬀective constitutive material properties on composite structural buckling and postbuckling responses has been very limited. The nonlinearity of in-plane shear is signiﬁcant for composite materials . With the nonlinear composite constitutive properties, a few attempts have been made to study buckling of thin composite laminate panels  and postbuckling of thick-section composite laminate plates . Hu [66,67] investigated the inﬂuence of in-plane shear nonlinearity on buckling and postbuckling responses of composite plates under uniaxial compression and biaxial compression and of shells under end compression and hygrostatic compression. They also investigated the nonlinear buckling of simply-supported composite plates under uniaxial compression, and of composite laminate skew plates under uniaxial compressive loads . The eﬀect of material nonlinearity on buckling and postbuckling of ﬁbre composite laminate plates and shells subjected to general mechanical loading, together with the interaction between the material and geometric nonlinearity was investigated , and it was concluded that the composite material nonlinearity had signiﬁcant eﬀects on the geometrically nonlinearity, structural buckling load, postbuckling structural stiﬀness, and structural failure mode shape of composite laminate plates and shells. 6.3. Buckling and postbuckling analysis of composite laminated plates under thermal eﬀects Considerable literatures have been devoted to the buckling and postbuckling analyses of laminated composite 152 Y.X. Zhang, C.H. Yang / Composite Structures 88 (2009) 147–157 plates subjected to mechanical loads, while the investigations on the postbuckling response of composite plates subjected to thermal or combined thermal and mechanical loadings are rather limited. The thermomechanical buckling and postbuckling response of laminated composite plates is clearly one of practical importance for structures operating at elevated temperatures and thus the understanding of the thermal buckling and postbuckling response of the composite laminated plates is desirable for the design of the composite laminates subjected to high temperatures. Tauchert  presented a comprehensive review of the studies on thermal buckling of composite laminated plates. Finite element method based on classical lamination theory was applied for examining nonlinear/postbuckling analysis of thin laminated plates subjected to uniform temperature distribution [71,72]. Chen and Chen investigated the thermal buckling behaviour of cylindrical laminated plates subjected to a non-uniform temperature , the thermal buckling behaviour of composite laminated plates subjected to uniform or non-uniform temperature ﬁelds  and thermal postbuckling behaviour of thick composite laminated plates subjected to a uniform thermal loading with temperature-dependent properties . The equivalent single layer ﬁrst-order shear deformation theories have been employed widely for the thermal buckling and postbuckling analysis of composite laminated plates. A mixed formulation with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate was used to analyse the thermomechanical buckling of composite plates subjected to combined thermal and axial loadings , thermomechanical buckling and postbuckling responses of ﬂat unstiﬀened composite panels subjected to combined temperature change and applied edge displacement , and the buckling and postbuckling responses of ﬂat, unstiﬀened composite panels subjected to various combinations of mechanical and thermal loads . A 9-node shearﬂexible isoparametric quadrilateral ﬁnite element was used to study the buckling behaviour of laminated composite plates subjected to a uniform temperature ﬁeld , and the inﬂuence of boundary conditions, ply orientation, and plate geometries on the critical buckling temperature was examined. Prabhu and Dhanaraj  also employed a 9node Lagrangian isoparametric element for the thermal buckling analysis of symmetric cross-ply, symmetric angle-ply and quasi-isotropic laminates subjected to uniform temperature distribution. Thermal buckling and postbuckling behaviour of shear deformable laminated composite plates was investigated by employing a fournode rectangular C1-continuous ﬁnite element by Singh et al. . A nonlinear ﬁnite element formulation of a C0-continuity element [80,43] based on the ﬁrst-order shear deformation theory was used to study the postbuckling behaviour of laminated plates induced by a uniform/nonuniform temperature ﬁeld . The nonlinearity due to moderately large deformation of the plate was included in the formulation and the inﬂuences of various parameters such as number of layers, ply-angle, aspect and thickness ratios and boundary conditions on the thermal postbuckling behaviour of laminates subjected to arbitrary temperature distribution were investigated. The thermal buckling and postbuckling analysis has been carried out to skew composite laminates and sandwich plates. Two shear deformable ﬁnite element models based on ﬁrst-order shear deformation theory and the higher-order shear deformation theory, respectively, were employed to study the elastic buckling of both thin and thick skew ﬁbre-reinforced composite and sandwich plates with various skew angles, lamination parameters and boundary conditions subjected to thermal loads . The buckling and postbuckling analysis of shear deformable composite skew plates subjected to combined uniaxial compression and uniform temperature rise was performed . Thermal buckling response of laminated composite square and skew plates was studied using a three-node plate element developed based on the ﬁrst-order shear deformation theory , thermal buckling temperatures including the critical one and mode shapes were numerically investigated and the element showed excellent performance in the moderately thick to very thin plates. A 3-node triangular facet ﬁnite element which accounts for transverse shear deformation was used to examine the bending, buckling, and postbuckling behaviours of laminated composite plates under thermally-induced loads based on a natural thermoelastic theory with a linear through the thickness temperature variation . The material properties were assumed independent of temperature, and the natural mode method was used. Thermal buckling behaviour of composite laminated plate subjected to a uniform temperature ﬁeld was investigated by considering the temperature-dependent elastic and thermal properties , and it was concluded that the inﬂuence of temperature-dependent properties on the thermal buckling behaviour was signiﬁcant. The higher-order shear deformation theories have also been employed for buckling analysis of laminated composite plates. Based on a 9-node Lagrangian isoparametric element and two reﬁned higher-order theories, two discrete ﬁnite element models, with the eﬀect of transverse normal deformation included in one and neglected in the other, were developed for the thermal buckling analysis of composite laminated and sandwich plates . The geometric stiﬀness matrices were developed with the consideration of the eﬀects of the higher-order terms on the initial inplane and transverse shear stresses. Singha et al.  investigated the thermal postbuckling behaviour of graphite/ epoxy multi-layered rectangular plates with various boundary conditions considering the temperature-dependent thermal and elastic properties of the material. A 4-node lock-free rectangular composite plate ﬁnite element having 6-DOF per node based on a bi-cubic representation of the transverse displacement ﬁeld was employed to investigate the post-buckling behaviour of rectangular laminated Y.X. Zhang, C.H. Yang / Composite Structures 88 (2009) 147–157 plates subjected to thermal loads , and the eﬀects of boundary conditions, aspect ratio, number of layers and lay-up sequence on the post-buckling behaviour were studied in detail. After the study of the interlaminar stresses and displacements in cross-ply laminated composite and sandwich plates subjected to mechanical/thermal loading based on the global higher-order theory [28,90,91], Matsunaga analysed thermal buckling problems of cross-ply laminated composite and sandwich plates , and angle-ply multilayered composite and sandwich plates  based on the global higher-order theory with the power series expansions of continuous displacement components. Several sets of truncated Mth-order approximate theories were applied to solve the eigenvalue problems of simply supported laminated composite and sandwich plates. The three-dimensional layerwise analysis has made a contribution to obtain accurate prediction of the free vibration and buckling of thermally stressed mutilayered angleply composite plates , thermal buckling and sensitively derivatives of temperatures sensitive multi-layered angleply plates , thermal buckling of multi-layered anisotropic plates , and the response of angle-ply laminated composite and sandwich plates . Both in-plane and normal displacements were assumed to be C0 continuous in the continuity conditions at the interface between layers in the three-dimensional layerwise theory. The number of unknowns was dependent on the number of layers in a laminate, thus the three-dimensional layerwise analysis are often computationally intractable, especially for laminated plates with a large number of layers. 7. Failure analysis Under normal operating conditions, local failures such as matrix cracks, ﬁbre breakage, ﬁbre matrix debonding and inter-layer delamination, may be developed in the laminated composite structures, and the failure may cause permanent loss of integrity within the laminate and result in loss of stiﬀness and strength of the material. Prediction of the failure process, the initiation and growth of the damages, and the maximum loads that the structures can withstand before failure occurs is essential for assessing the performance of composite laminated plates and for developing reliable and safe design. In particular, the ﬁrst-ply failure analysis of laminated composite plates has been actively investigated in recent years, and the mechanical behaviour and the ﬁrst-ply failure load of laminated composite plates subjected to in-plane loading conditions, such as tension, compression, shear, and out-of-plane loading such as transverse loads have been studied. Compared with the failure analysis of composite laminates subjected to inplane loading, the failure analysis of composite laminates subjected to out-of-plane loading seems more complicated due to material and geometric nonlinearities that come into play when the loads are increased beyond the ﬁrst-ply failure. The diﬀerent laminated plate theories, such as the 153 CLT, FSDT, HSDT and layer-wise theories have been employed for failure analysis. Chang and Lessard  studied the damage in laminated composites containing an open hole, subjected to compressive loading, and the in-plane response of the laminates from initial loading to ﬁnal collapse was studied considering the geometrically nonlinearity. Sahid and Chang  developed a progressive failure model for predicting the accumulated damage and the eﬀects of such damage on the in-plane response of laminated composites subjected to tensile and shear loads. Based on the classical laminated plate theory, Sleight and Knight  studied the damage of composite plates subjected to shear and compressive loading. The postbuckling behaviour and progressive failure response of thin, symmetric laminates under uniaxial compression and uniaxial compression combined with inplane shear loads was studied based on the ﬁrst-order shear deformation theory and geometric nonlinearity , and the 3D Tsai–Hill criterion was used to predict failure of a lamina and the maximum stress criterion was used to predict onset of delamination at the interface of two adjacent layers. Based on ﬁrst-order shear deformation theory and several phenomenological failure criteria, a ﬁnite element model has been developed to ﬁnd linear and nonlinear ﬁrst-ply failure loads of composite laminates subjected to in-plane and transverse loads , and failure analysis and the ﬁrst-ply failure load in both linear and the geometrically nonlinear stage of thin and thick plates under a uniformly distributed transverse load was studied . Firstply failure of laminated composite plates was analysed using the ﬁnite element method developed based on the Reissner–Mindlin plate theory that accounted for moderate rotation , and failure loads were obtained for different laminate thickness, stacking sequences and aspect ratios and diﬀerent failure criteria. Kam and Lin  developed a stochastic ﬁnite element method for the reliability analysis of linear laminated composite plates subjected to transverse loads, and procedures for the reliability analysis of laminated composite plate structures subjected to large deﬂections under random static loads was also presented . The ﬁrst-ply failure probabilities of linear and nonlinear centrally loaded laminated composite plates including the geometric nonlinear eﬀects were examined [106,107], and an 8-node element of the serendipity family and 9-node Lagrangian elements with diﬀerent numerical integration rules were used to study the nonlinear deﬂection and ﬁrst-ply failure load of thin laminated composite plates subject to transverse loading based on several phenomenological failure criteria . The ﬁrstply failure load, progression of damage and ultimate collapse load in the nonlinear deformation regime of laminated composite plates subjected to uniform transverse pressure was studied with the large strain and large rotation included in the geometric nonlinearity analysis . The ﬁrst-ply failure of laminated panels under transverse loading was analysed using an eight-node isoparametric 154 Y.X. Zhang, C.H. Yang / Composite Structures 88 (2009) 147–157 quadratic shell element , and various failure criteria were studied to predict the load of various plates and shells having varying lamination schemes. The ﬁrst-play failure of thin laminated composite plates under combined transverse load with uni-axial compression and transverse load with in-plane shear was studied . An 8-node isoparametric plate-bending element was used to model the progressive failure of laminated composite plates under transverse static loading in linear and elastic range [112,113]. After the failure of the weakest ply, the stiﬀness was reduced by either ﬁbre failure or matrix failure. The stiﬀness of failed lamina was then totally discarded and other existing lamina was considered to remain unchanged after the weakest ply failure. Fewer ﬁnite element models have been developed for failure analysis of composite laminated plate based on HSDT, and one example is the 7-DOF ﬁnite element model including three displacements, two rotations of normal about the plate mid-plane, and two warps of the normal, which was developed to determine the ﬁrst-ply failure and the last-ply failure of laminated composite plates subjected to both in-plane and sinusoidal transverse loads by a progressive stiﬀness reduction technique under conditions of complex loading . Some progressive failure analysis of composite laminates based on the 3D layerwise plate theories have been carried out. For example, Reddy and Reddy  used generalized layerwise plate theory and a progressive failure model to determine ﬁrst-ply and ultimate failure loads of a three-point bending specimen with geometric nonlinearity. The failure mechanism and ultimate failure loads of the cross-ply and quasi-isotropic laminates for diﬀerent stacking sequences with the same thickness subjected to axial extension was conducted  based on the generalized layerwise plate theory (GLPT) in order to consider the local eﬀect near the free edges. A 3D layer-wise mixed ﬁnite element model  was employed for the computation of stress and strain components for the ﬁrst-ply failure analyses of composite laminated plates , and the maximum stress, the maximum strain, Tsai–Hill, Tsai–Wu and Hoﬀman failure theories were used for the failure analysis. The ﬁrst-ply failure of moderately thick laminated composite plates  was studied using a ﬁnite element formulation based on the layerwise linear displacement theory, in which a laminated composite element was divided into a number of mathematical layer groups and displacements were assumed to vary linearly in each layer group. Viscoelastic behaviour of composite materials inﬂuences the failure behaviour, particularly when nonlinear geometrical eﬀects are important. The failure behaviour of composite laminates in the presence of large displacements and creep was modelled including the material behaviour of thermal, hygroscopic and viscoelastic eﬀects . Incremental damage is determined and used to calculate a modiﬁed stiﬀness matrix, and the procedure can be used to analyse buckling, creep buckling and creep buckling including damage. 8. Summary and future research The recent advances of the ﬁnite element analysis of composite laminated plates based on various lamination theories, with the focus on the free vibration and dynamics, buckling and postbuckling analysis, geometric nonlinearity and large deformation analysis and failure and damage analysis of composite laminated plates, are reviewed in this paper. The development of buckling and postbuckling analysis under material nonlinearity and thermal eﬀects are emphasised and in the failure analysis, the concentration is especially on the advances of the ﬁrst-ply failure analysis. Based on the author’s investigation, it has been found that the research on the following aspects of the composite laminated plates is relatively limited and may attract more interests in the future research. Material nonlinearity eﬀects on structural behaviour of composite laminates. Failure and damage analysis under viscoelastic eﬀects such as thermal and creep eﬀects. 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