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Zhang2009.pdf
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
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Y.X. Zhang, C.H. Yang / Composite Structures 88 (2009) 147–157
and Robbins [1]. An overall comparison of laminated
theories based on displacement hypothesis was presented
by Liu and Li [2], 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 [3]. A review of
displacement and stress-based refined shear deformation
theories of isotropic and anisotropic laminated plate
was given by Ghugal and Shimpi [4], 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 [5]. A review of shear deformation plate and
shell theories was presented by Reddy and Arciniega [6],
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 [7], and a selective literature survey
on the free-edge effect since 1967 was given by Mittelstedk and Becker [8].
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) [9]. 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 [10]. 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 [11], 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 [12]. 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 [13].
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 [14]. 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
[15]. 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 [16].
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 [17]. 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 [18] 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 [19].
Latheswary et al. [20] 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 [21] 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 [22].
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 [23]. 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 [24] 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 [25]. 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 [29]. 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
[30], 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 [31] 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 [32], and the effect of large amplitude on the
dynamic instability for a simply-supported laminated
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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 [33]
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 [36]. 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 [37].
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 [38], 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
[39], 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 [40], the C0 four
and nine-node finite elements for the transient response
of orthotropic, layered composite sandwich plates [41]
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 [42]. 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 [43]. Han et al. [44]
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 [45]
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 [48]. 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 [49], 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 [50] using
a total Lagrangian description and the principle of virtual
displacements. The large deformation analysis of circular
composite laminated plates [51] 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 [54]. Later a more detailed account of the
research on the buckling and postbukcling before 1995
was presented by Noor [55].
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 [56]. 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 [57]. 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
[58]. 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 [59]. 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 [60].
A generalized layer-wise stochastic finite element formulation was developed for the buckling analysis of both
homogeneous and laminated plates with random material
properties [61]. 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
[62]. 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 [63]. With the nonlinear composite constitutive properties, a few attempts have been
made to study buckling of thin composite laminate panels
[64] and postbuckling of thick-section composite laminate
plates [65]. 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 [68].
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 [69], 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
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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 [70] 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 [73], the
thermal buckling behaviour of composite laminated plates
subjected to uniform or non-uniform temperature fields
[72] and thermal postbuckling behaviour of thick composite laminated plates subjected to a uniform thermal loading
with temperature-dependent properties [74].
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 [75], thermomechanical buckling and postbuckling responses of
flat unstiffened composite panels subjected to combined
temperature change and applied edge displacement [76],
and the buckling and postbuckling responses of flat,
unstiffened composite panels subjected to various combinations of mechanical and thermal loads [77]. 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 [78], and
the influence of boundary conditions, ply orientation, and
plate geometries on the critical buckling temperature was
examined. Prabhu and Dhanaraj [79] 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. [48]. 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 [81]. 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 [82]. The
buckling and postbuckling analysis of shear deformable
composite skew plates subjected to combined uniaxial compression and uniform temperature rise was performed [83].
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 [84], 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 [85]. 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 [86], 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 [87]. 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. [88] 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 [89], 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 [92], and angle-ply multilayered composite and sandwich plates [93] 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 [94], thermal buckling and sensitively
derivatives of temperatures sensitive multi-layered angleply plates [95], thermal buckling of multi-layered anisotropic plates [96], and the response of angle-ply laminated
composite and sandwich plates [97]. 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 [98] 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 [99]
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 [100] 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 [101], 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 [102], 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 [103]. 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 [104], and failure loads were obtained for different laminate thickness, stacking sequences and aspect
ratios and different failure criteria. Kam and Lin [105]
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 [42]. 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 [108]. 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 [109].
The first-ply failure of laminated panels under transverse
loading was analysed using an eight-node isoparametric
154
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quadratic shell element [110], 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 [111]. 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 [114].
Some progressive failure analysis of composite laminates based on the 3D layerwise plate theories have been
carried out. For example, Reddy and Reddy [115] 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 [116] 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 [117] was employed for the computation of
stress and strain components for the first-ply failure analyses of composite laminated plates [118], 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 [107] 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 [119]. 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.
Failure and damage analysis under cyclic loading.
Micromechanical approach for damage analysis.
Analysis of the damage evolution in composite
laminates.
Multiscale modelling of crack initiation, propagation,
and overall structural failure.
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