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7/25/2019 2010_1 (Mater Des) Interlaminar Stress Distribution of Composite Laminated Plates With Functionally Graded Fiber
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Interlaminar stress distribution of compositelaminated plates with functionally graded fiber
volume fraction
Article in Materials and Design June 2010
Impact Factor: 3.5 DOI: 10.1016/j.matdes.2009.12.027
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3 authors, including:
Yiming Fu
Hunan University
112PUBLICATIONS 1,146CITATIONS
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Pu Zhang
The University of Manchester
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Available from: Pu Zhang
Retrieved on: 28 June 2016
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Interlaminar stress distribution of composite laminated plates
with functionally graded fiber volume fraction
Yiming Fu a, Pu Zhang a,*, Fan Yang b
a College of Mechanics and Aerospace, Hunan University, Changsha 410082, PR Chinab Shenzhen Municipal Design and Research Institute Co., Ltd., Shenzhen 518049, PR China
a r t i c l e i n f o
Article history:
Received 20 July 2009
Accepted 16 December 2009
Available online 21 December 2009
Keywords:
Composite material
Fiber volume fraction
Interlaminar stress
a b s t r a c t
Various functionally graded design methods have been proposed recently for fiber reinforced composite
plates. The laminates with variable fiber spacing along the thickness direction are focused on in this
paper. Fiber volume ratio distribution functions are defined separately in each single layer. Classic state
space method as well as differential quadrature state space method are utilized here for different bound-
ary and plied conditions. For the latter method, a sub-layer based scheme, which has both high accuracy
and less numerical capacity, is suggested for functionally graded plates. Numerical examples indicate that
the non-uniform distribution of fibers rearranges the stress field, of which the in-plane stresses are sen-
sitive to the fibers distribution, while the transverse stresses are not affected so much. In-plane stresses
near interfaces would decrease if the fiber ratio reduces in this region, which provides a method to
resolve the interfacial stress concentration problems.
2009 Elsevier Ltd. All rights reserved.
1. Introduction
Nowadays, composite laminates have been widely used in mod-
ern industry due to their high strength-to-weight ratio, high stiff-
ness-to-weight ratio as well as good fatigue resistant properties.
Moreover, the designability of this kind of material makes it have
more development potential than the commonly used metals. Con-
ventional fiber reinforced polymer (FRP) composite laminates are
commonly manufactured by bonding many homogeneous single
layers which haveunifiedfiber orientation andfiber volumefraction
(FVF) together. Of this kind of structures, much research has been
done on their mechanical properties like bending, buckling and
vibration or the failure behaviors such as damage, fracture and fati-
gue.Alongwith this,variouslaminate theorieshave been developed,
forexample, the three-dimensionaltheories, smearedplate theories,
layer-wise models, zigzag models, and global-local models [1,2].
With the occurrence of functionally graded metal-ceramic materi-
als, researchers extend this gradient idea to the design of FRP com-
posites. And during the last two decades, functionally graded FRP
composites have been widely developedfrom the in-plane to
out-of-plane, fromthe gradient distribution of FVFsto fiber orienta-
tions, and from the fibers spatial arrangement to the change of
material properties.
Martin and Leissa [35] are pioneers to this study and they
focused on the effect of the in-plane FVF distribution on the
mechanical properties of plates. Numerical solutions and some ex-
act solutions under specified boundary conditions were obtained
for the plane elasticity problems. Buckling and vibration of the
plate were also studied by them and it was found that the rear-
rangement of fibers can change the critical buckling loads and res-
onant frequencies of structures. After that, Shiau et al. [6,7] used
the finite element method (FEM) to model this plate and found that
the reduction of the in-plane FVF near free edges or holes can re-
duce the stress concentrations there. This kind of FRP plate was
used for the reinforcement of shear walls by Meftah et al. [8,9]
and both the lateral stiffness and vibration characteristics were
studied by using FEM. Nowadays, the gradient design of FVFs is
not limited to the in-plane direction. Through thickness function-ally graded design method was introduced by Benatta et al. in
Ref.[10], of which a single layer composite beam was studied by
using the higher order beam theory and effects of different distri-
bution functions on the bending responses were also discussed.
Oyekoya et al. [11] established a finite element model for plates
with the FVFs gradient distribution along multi-directions and
investigated the buckling and vibration problems. Kuo and Shiau
[12] discussed the effect of different through thickness distribution
functions of the FVF on the critical buckling loads and resonance
frequencies of the plate by using the FEM. The purpose of them
is to design structures with ideal buckling and vibration character-
istics via the non-uniform distribution of FVFs.
Besides the gradient distribution of FVFs, other methods such as
changing the fibers orientations or material properties have also
been proposed in literatures. Batra and Jin[13]found that the res-
0261-3069/$ - see front matter 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.matdes.2009.12.027
*Corresponding author. Tel.: +86 731 88822421; fax: +86 731 88822330.E-mail address:[email protected](P. Zhang).
Materials and Design 31 (2010) 29042915
Contents lists available at ScienceDirect
Materials and Design
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m a t d e s
https://www.researchgate.net/publication/248541287_Seismic_behavior_of_RC_coupled_shear_walls_repaired_with_CFRP_laminates_having_variable_fibers_spacing?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/228645018_A_Selective_Review_on_Recent_Development_of_Displacement-Based_Laminated_Plate_Theories?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/225841582_Theories_and_finite_elements_for_multilayered_anisotropic_composite_plates_and_shells_Arch_Comput_Method_E_987-140?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/223393339_Vibration_and_buckling_of_rectangular_composite_plates_with_variable_fiber_spacing?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/229714590_Application_of_the_Ritz_method_to_plane_elasticity_problems_for_composite_sheets_with_variable_fibre_spacing?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/232389936_Buckling_and_vibration_analysis_of_functionally_graded_composite_structures_using_the_finite_element_method?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/240421129_Lateral_stiffness_and_vibration_characteristics_of_composite_plated_RC_shear_walls_with_variable_fibres_spacing?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/244999415_Some_exact_plane_elasticity_solutions_for_nonhomogeneous_orthotropic_sheets?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/248205468_Buckling_and_vibration_of_composite_laminated_plates_with_variable_fiber_spacing?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/248204652_Stress_concentration_around_holes_in_composite_laminates_with_variable_fiber_spacing?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/256520541_Free-edge_stress_reduction_through_fiber_volume_fraction_variation?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==7/25/2019 2010_1 (Mater Des) Interlaminar Stress Distribution of Composite Laminated Plates With Functionally Graded Fiber
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onant frequency of the plate can be altered by the gradient distri-
bution of fiber orientations along the thickness direction. Han et al.
[14] pointed out that the optimum design of fibers can improve the
interfacial properties of laminates and established a finite element
model. Cho and Rowlands[15]attempted to change the fiber ori-
entations near holes in a short fiber reinforced plate to reduce
the stress concentration. Bouremana et al. [16] proposed a new
idea of structural design in thermal environment, of which the fi-
ber with negative thermal expansion coefficient was used to elim-
inate the thermal stress. Overall, the fibers volume fraction,
orientation and material properties should all be considered in
the composites optimal design process, as illustrated in Fig. 1. To
achieve a comprehensive optimal design, the buckling and vibra-
tion analysis should be carried out together with the failure analy-sis which needs exact calculations of stress fields beforehand.
Evidently, various aspects must be considered such as the environ-
ment, reliability, industrial costs, etc.
Most of the methods used in the above mentioned literatures
are on the basis of higher-order laminate theories and the FEM.
Nevertheless, these conventional appropriate theories encounter
difficulties when handling these functionally graded plates, which
are anisotropic and highly non-homogeneous; exact solutions are
quite difficult to get. Therefore, the state space method, a powerful
three dimensional solution method, is utilized in this paper. Func-
tionally graded laminates, with different boundary and plied con-
ditions, are discussed in the numerical examples. The results
indicate that the reduction of FVFs near interfaces can reduce the
in-plane stress concentration. Nonetheless, the transverse stresseswhich have lower order than in-plane ones affect little.
2. The model
Consider a laminated rectangular plate with length a , widthb
and thickness h placed in the Cartesian coordinate system oxyz
ofFig. 2. It is assumed that the plate hasNsingle layers with equal
thickness h1= h/N. Each interface is assumed to be bonded per-fectly and no initial defect is considered.
For the conventional laminated plate, each layer has unified
FVF (or fiber ratio), i.e. Vkf VkM
f , where VkM
f denotes the
FVF at the mid-plane of the kth layer. However, the fiber ratio
distribution of functionally graded composite laminate is non-
uniform, which is considered to be variable along the thickness
direction in this paper. A through thickness local coordinate sys-
tem f(k)(h1/26 f(k) 6 h1/2) is established in each single layer
with its origin localized at the corresponding mid-plane. The
FVF at interfaces of the kth layer is denoted to be VkI
f , in
distinction with VkM
f at the mid-plane. A modified power law
distribution function is defined for the kth layer as follows:
Vk
f V
kM
f V
kI
f V
kM
f 2jfkj=h1p 1wherep is the power law index, which would be prescribed in the
design process.
The average fiber ratio in the kth layer,Vk
f , often named mean
FVF, is defined as
Vk
f
Rh1=2h1=2
Vk
f dfk
h1
pVkM
f V
kI
f
p 1 2
Another form for formula(2)is
VkM
f Vk
f Vk
f VkI
f
=p 3
The fiber distribution function Vk
f can be easily determined by Eqs.
(1) and (3)after given the mean FVF Vk
f
as well as the interfacial
fiber ratioVkIf .
The FVF at the mid-plane has a restriction 0 6 Vkmin
6 VkMf 6
Vkmax6 1, where Vk
min andVkmax are the minimum and maximum
fiber ratio, respectively. So according to Eq. (3), p should satisfy a
relationship as follows:
pPmaxV
kI
f V
k
f
Vk
f Vk
min
;V
kI
f V
k
f
Vk
f Vkmax
! 4
For instance, when Vkmin 0; V
kmax 0:9, and V
k
f 0:6;pP 2=3 if
VkI
f 0:4, andpP 1/3 ifVkI
f 0:8.
The FVF distribution functions for Vk
f 0:6 are illustrated in
Fig. 3. Functionally graded plates would degrade to the conven-
tional one ifV
kI
f V
k
f . The distribution functions change abruptlynear interfaces for the power law index p= 5 or 10. The largerp is,
Fig. 1. Optimal design process of functionally graded FRP composite structures.
Fig. 2. Sketch of the composite laminated plate and its cross section.
Y. Fu et al. / Materials and Design 31 (2010) 29042915 2905
https://www.researchgate.net/publication/222899953_Controlling_thermal_deformation_by_using_composite_materials_having_variable_fiber_volume_fraction?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/249354613_Optimizing_Fiber_Direction_in_Perforated_Orthotropic_Media_to_Reduce_Stress_Concentration?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/223717249_Non-linear_analysis_of_laminated_composite_and_sigmoid_functionally_graded_anisotropic_structures_using_a_higher-order_shear_deformable_natural_Lagrangian_shell_element?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==7/25/2019 2010_1 (Mater Des) Interlaminar Stress Distribution of Composite Laminated Plates With Functionally Graded Fiber
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the closer of the gradient distribution function to the uniform one
is; a conventional distribution form would be got when p?1.
3. Basic equations
With high non-homogeneity, material properties of the FRP
composite are related to the local fiber ratio at each point. Assume
that the matrix and fiber are both isotropic and their Youngs mod-
ulus and Poissons ratio are denoted as Em, mm andEf, mf, respec-
tively. Chamis rule of mixture [17]is adopted here to obtain the
composites material properties, as
EL VfEf 1 VfEm
ET Em
1 ffiffiffiffiffiffi
Vfp
1 Em=Ef
GLTGTT Gm
1 ffiffiffiffiffiffi
Vfp
1 Gm=Gf
mLT Vfmf 1 Vfmm
mTT ET2GTT
1
5
whereL andTindicate the directions parallel and perpendicular to
the fiber, respectively;Vfdenotes the fiber ratio at a specified point.
The relationship between the shear modulus and other materialparameters is Gf(m)= Ef(m)/ (2 + 2mf(m)). It should be noted that Cha-mis mixed law, due to its exactitude, is more suitable for the 3D
(three dimensional) elasticity analysis than the common rule of
mixture.
Denote 1 to be the direction along the fiber and 2, 3 the direc-
tions perpendicular to the fiber, respectively. Thus the FRP com-
posite is orthotropic in the materials principle coordinate system
o-123 and the stressstrain relation is
r0 C0e0 6
wherer0
, e0
andC0
are the stress, strain and stiffness matrix in the
principle coordinate of materials, respectively, which can be written
as
r0 fr1 r2 r3 s13 s23 s12g
T
e0 fe1 e2 e3 c13 c23 c12g
T
C0
c011 c012 c
013
c022 c023 0
c033c044 0 0
sym: c0
55
0
c066
0BBBBBBB@
1CCCCCCCA
E1L mLTE
1L mLTE
1L
E1T mTTE
1T 0
E1T
G1LT 0 0
sym: G1TT 0
G1LT
0BBBBBBBBB@
1CCCCCCCCCA
1
where E,G and mare the Youngs modulus, shear modulus and Pois-sons ratio, respectively.
The stressstrain relation in the global coordinate systemo-xyz
should be transformed as
r Ce 7
wherer, e,C are the stress, strain and stiffness matrix in the global
coordinate system, which have the form as
r frx ry rzsxzsyzsxygT
e fex ey ezcxzcyzcxygT
C
c11 c12 c13 0 0 c16
c22 c23 0 0 c26
c33 0 0 c36
c44 c45 0
sym: c55 0
c66
0BBBBBBBB@
1CCCCCCCCA
These quantities can be obtained through the coordinate transfor-
mation from Eq.(6), as
r Qr0
e Qe0
C QC0QT8
where Qis the coordinate transformation matrix with expression as
follows:
Q
c2 s2 0 0 0 2cs
s2
c2
0 0 0 2cs0 0 1 0 0 0
0 0 0 c s 0
0 0 0 s c 0
cs cs 0 0 0 c2 s2
0BBBBBBBB@1CCCCCCCCA
wherec, cosh, s , sinh. Eq. (7)is the stressstrain relation of the
laminates with functionally graded FVF for the 3D elasticity
problem.
The 3D equilibrium equations for elastic plates can be written
as
@xrx@ysxy@zsxz 0
@xsxy@yry@zsyz 0
@xsxz@ysyz@zrz 0
9
As infinitesimal deformation theory is considered here, the strain-
displacement relationship is
0.4
0.6
0.8
Vk
f =0.6
Vk I
f =0.6
Vk If =0.4
h1/2
p=5
p=10
Vk
f
k
-h1/2 0
Vk I
f =0.8
Fig. 3. Fiber volume fraction distributions along the thickness direction in a single
layer.
2906 Y. Fu et al. / Materials and Design 31 (2010) 29042915
https://www.researchgate.net/publication/4664242_Mechanics_of_composite_materials_-_Past_present_and_future?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==7/25/2019 2010_1 (Mater Des) Interlaminar Stress Distribution of Composite Laminated Plates With Functionally Graded Fiber
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ex @xu; cxz @zu@xw
ey @yv; cyz @zv@yw
ez @zw; cxy @yu@xv
10
where u, v and w are the displacements along axis x, y and z,
respectively.Substituting Eq.(10)into the stressstrain relation Eq. (7) and
assembling with the equilibrium Eq.(9), the state-space equation
[18]of the plate can be obtained as follows:
@zg Ag 11
where g is the state-space vector and A is the coefficient matrix,
which have can be written as
g u vrzsxzsyzw T
A 0 A1
A2 0
A1
a1c55 a1c45 @x
a1c44 @y
sym: 0
0B@ 1CA
A2
a2@xx 2a3@xy a4@yy a3@xx a4 a5@xy a6@yy a7@x a8@y
a4@xx 2a6@xy a9@yy a8@x a10@y
sym: c133
0B@1CA
a1 1
c44c55c245; a2
c213c33
c11; a3c13c36
c33c16;
a4 c236c33
c66; a5c13c23
c33c12;
a6c23c36
c33c26; a7
c13c33
; a8c36
c33; a9
c223c33
c22;
a10c23
c33:
For the other three stress components rx,ry andsxy, there exists
rxrysxy
8>:9>=>;
a2@xa3@y a3@xa5@y a7
a5@xa6@y a6@xa9@y a10
a3@xa4@y a4@xa6@y a8
0B@1CA uv
rz
8>:9>=>; 12
rx, ry, sxy can be got automatically from the solutions to Eq. (11).Thus the solving methods are focused on in the following part.
4. Solution methodology
4.1. Classic state space method
Classic state space method can obtain excellent analytical
solution; however, both the boundary conditions and material
properties are restrictiveonly simply supported or sliding
boundary conditions as well as orthotropic material are applica-
ble. For the laminate considered in this section which is cross-
ply laminated and simply supported, its boundary conditions
can be written as
x 0; a: rx w v 0
y 0; b : ry w u 0 13
To satisfy these equations, the unknown state space vector is ex-
panded with trigonometric series, as
ux;y;z
vx;y;z
rzx;y;z
sxzx;y;z
syzx;y;z
wx;y;z
8>>>>>>>>>>>>>>>>>>>:
9>>>>>>>>>>=>>>>>>>>>>;
X1
m1 X1
n1
~umnz cosmpn sinnp1
~vmnz sinmpn cosnp1
~rzmnz sinmpn sinnp1
~sxzmnz cosmpn sinnp1
~syzmnz sinmpn cosnp1~wmnz sinmpn sinnp1
8>>>>>>>>>>>>>>>>>>>:
9>>>>>>>>>>=>>>>>>>>>>;
14
wheren =x/a and1 =y/b are two dimensionless variables.Three other stress components, which are not included in the
state space vector, can also be expanded as
rxrysxy
8>:9>=>;
X1m1
X1n1
~rxmn sinmpn sinnp1~rymn sinmpn sinnp1
~sxymn cosmpn cosnp1
8>:9>=>; 15
The mechanical load qapplied on the top surface of the plate can be
set as
qx;y; h X1
m1 X1
n1
~qmn sinmpn sinnp1 16
After substituting Eq.(14)into the state-space Eq.(11), spatial vari-
ablesx andy can be eliminated automatically. Thus for each order
ofm andn, there exists
@zgmn Amngmn 17
where
gmn ~umn ~vmn ~rzmn ~sxzmn ~syzmn ~wmn T
Amn
c144 0 mp
0 0 c155 np
mp np 0
c66n2p2 a2m2p2 c66 a5mnp2 a7mp
c66 a5mnp2 c66m2p2 a9n2p2 a10np 0
a7mp a10np c133
0BBBBBBBBBBBBBB@
1CCCCCCCCCCCCCCAAnd for Eq.(12), there exists
~rxmn
~rymn
~sxymn
8>:
9>=
>;
a2mp a5np a7
a5mp a9np a10
c66np c66mp 0
0B@1CA
~umn
~vmn
~rzmn
8>:
9>=
>;18
It is obvious that Eq.(17)is a matrix differential equation only withvariablez. An analytical solution to Eq.(17)can be obtained for the
case thatAmn is a constant matrix or exponential withzcoordinate;
however, this is impossible in this paperthe material is non-
homogeneous and the through thickness distribution function is
complicated. Therefore, a numerical method is used here to obtain
3D solutions. The laminate is divided into R numerical layers with
equal thickness in zdirection. Denote the z coordinate of the top
and bottom interface of the jth numerical layer as zjI and z
j1I ,
respectively. So for each thin numerical layer the coefficient matrix
Amn can be treated as constant. By using the CayleyHamilton the-
orem[19], it can be obtained from Eq.(17)that
gmnz exp Amn zzj1I
h ig
j1mn 19
For thejth numerical layer, there exists
gjmn exp Amn z
jI z
j1I
h ig
j1mn T
jmng
j1mn 20
Y. Fu et al. / Materials and Design 31 (2010) 29042915 2907
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here gjmn is the state-space vector at the jth numerical interface,
whileTjmnis known as the transfer matrix. Applying Eq.(20)to each
numerical layer then the state-space vector gjmn can also be written
in the form as
gjmn Yji1
Timng0mn 21
For theRth numerical interface, i.e. the top surface of the laminate,
there exists
gRmn
YRi1
Timng0mn 22
here gRmn andg
0mn are the state-space vector at the top and bottom
surfaces of the plate. And the mechanical loads at these two sur-
faces are set as
~s0xzmn ~sRxzmn ~s
0yzmn ~s
Ryzmn ~r
0zmn 0; ~r
Rzmn ~qmn 23
g
R
mn and g0
mn can be determined by solving Eqs. (22) and (23); fur-therly, state-space vectors at other numerical interfaces can also
be obtained from Eq. (21). At last, ~rxmn; ~rymn and ~sxymn can easilybe gained from Eq. (18) automatically. So the 3D solutions to the
static bending problem of the functionally graded laminated plate
are presented.
4.2. Differential quadrature-state space method
It is quite difficult to give the eigenfunctions like Eq. (14) di-
rectly for general cases such as clamped or free edge boundary con-
ditions or the laminate is angle-plied, then the classic state-space
method cannot be used. Fortunately, a semi-analytical method
named DQSSM (differential quadrature-state space method) [20]
is capable to handle these problems. The main idea of this methodis to use the DQM (differential quadrature method) to discrete x
andy variables in Eq. (11), and then solve state-space vectors at
all the discrete points as a whole. In what follows we will present
the details of this method.
The sampling points of DQM are taken as [21]
xi 1 cosi 1p=Nx 1
2 a; i 1; 2;. . .;Nx
yj 1 cosj 1p=Ny 1
2 b; j 1; 2;. . .; Ny
24
wherexiandyjare the coordinates of the sampling points; Nxand Nyare the number of discrete points in the corresponding directions.
For a specified functionW, its partial differential derivative at point
(xi, yj) can be written as
@rsW
@xr@ys
xi ;yi
XNxk1
Pr
ik
XNyl1
Qs
jl Wkl 25
where Prik is the weighting coefficient for rth order derivative re-
spect to x, similar meaning for Qs
jl in y direction. Details for Pr
ik
andQs
jl can be found in Ref.[21], which would not be presented
here.
In this section we only consider the case that the laminate is
four-edge clamped; other boundary cases are similar to this and
would not be introduced. For a full-clamped plate, the boundary
conditions can be written as
x 0; a: u v w 0
y 0; b: u v w 0 26
State space equations at all of the sampling points can be obtained
after substituting Eq.(25)into Eq.(11). Nonetheless, this series of
equations can not be solved directlythey must be incorporated
with the boundary conditions Eq.(26). Finally, the governing equa-
tions for full-clamped plates have the form as follows:
@zuij a1c55sxzija1c45syzij XNx1
k2
P1
ikwkj
@zvij a1c45sxzija1c44syzijXNy1l2
Q1
jl wil
@zrzij XNx1k2
P1
ik sxzkjc44 P1
i1 P1
1k P1
iNxP
1
Nx k
wkj
h i
XNy1l2
Q1
jl syzilc55 Q
1
j1 Q1
1l Q1
jNyQ
1
Ny l
wil
h i@zsxzij
XNx 1k2
P2ik a2ukj a3vkj a7c33 P
1i1 P
11k P
1iNx
P1Nx k
a7ukj a8vkj
h i
XNy 1l2
Q2jl a4uil a6vil a8c33 Q1j1 Q
11l Q
1jNy
Q1Ny l a8uil a10vilh iXNx 1k2
P1ik
XNy 1l2
Q1jl 2a3ukl a4 a5vkl a7
XNx 1k2
P1ik rzkj a8
XNy 1l2
Q1
jl rzil
@zsyzijXNx 1k2
P2
ik a3ukj a4vkj a8c33 P1
i1 P1
1k P1
iNxP
1
Nx k
a7ukj a8vkj
h iXNy 1l2
Q2
jl a6uil a9vil a10c33 Q1
j1 Q1
1l Q1
jNyQ
1
Ny l
h a8uil a10vil X
Nx 1
k2
P1
ik XNy 1
l1
Q1
jl a4 a5ukl 2a6vkl
a8XNx 1k2
P1
ik rzkj a10XNy 1l2
Q1
jl rzil 27
@zwij XNx 1k2
P1
ik a7ukja8vkj XNy1l2
Q1
jl a8uila10vil c133 rzij
where i= 2,3,. . . , Nx 1;j = 2,3,. . . , Ny 1. These equations can also
be written in the matrix form similar to Eq.(17)as
@zgbAgg fuij vij rzij sxzij sxzij wijg
T 28
where g is the total state space vector and bA is the coefficient ma-
trix. It should be noted that gincludes state space variables at all of
the sampling points except boundaries.State space vectors at boundaries have two parts: one for dis-
placements shown in Eq. (26), the other for unknown stresses writ-
ten as
sxzij c44XNx 1k2
P1
ik wkjc45XNy1l2
Q1
jl wil
syzij c45XNx1k2
P1
ik wkjc55XNy1l2
Q1
jl wil
rzij c33XNx1k2
P1
ik a7ukja8vkj c33XNy1l2
Q1
jl a8uila10vil
29
wherei andj indicate the boundary points. Stresses in Eq.(29)can
be obtained from the solutions of Eq. (28).In-plane stress componentsrx,ryandsxycan also be written in
the discrete form, after applying Eq. (25)to Eq.(12), which are as
follows:
2908 Y. Fu et al. / Materials and Design 31 (2010) 29042915
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rxij XNx 1k2
P1
ik a2ukja3vkj
XNy1l2
Q1
jl a3uila5vil a7rzij
ryij
XNx1
k2
P1
ik a5ukja6vkj
XNy1
l2
Q1
jl a6uila9vil a10rzij
sxyij XNx1k2
P1
ik a3ukja4vkj
XNy1l2
Q1
jl a4uila6vil a8rzij
30
wherei = 1,2,. . . , Nx;j = 1,2,. . . , Ny.
At a glance, it seems that Eq. (28) is similar to Eq. (17) and could
be solved as a routine. Conversely, difficulties would be encoun-
tered when programming. As a shortcoming of state space method,
bad conditioned matrix would occur if the coefficient matrix is
quite large, which may leads to significant errors, especially when
the number of discrete numbers increases or the plate becomes
thicker. Fortunately, a joint coupling technique [20] can be used
to overcome this defect, which will be introduced as below.
As illustrated in Fig. 4, the laminate is divided into Rs sub-layers,
each with Rn numerical layers. Moreover, all of the sub-layers or
numerical layers have equal thickness. Similar to the classic state
space case, the z coordinate of the top and bottom interface of
the kth numerical layer in jth sub-layer are denoted as zj;kI and
zj;k1I , respectively. Thus the state space vector g within one sub-
layer has the relationship as follows:
gj;k
Yki1
bTj;igj;0 31where bTj;i is the transfer matrix within the jth sub-layer, which is
defined as
bTj;i exp
bA z
j;iI z
j;i1I
h i 32
Obviously, for variables at the top surface of the jth sub-layer, i.e.
gj;Rn, there exists
gj;Rn
YRni1
bTj;igj;0 j 1; 2;. . .;Rs 33This is the transfer relationship between variables at the two sur-
faces of thejth sub-layer. If the state space vector gj;0 was known,
all the variables in other numerical layers of thejth sub-layer can be
obtained from Eq.(31).
Unlike the classic state space method, joint coupling technique
reserves unknown state space vectors at interfaces between sub-
layers, instead of eliminating them to get a simple relationship as
Eq.(22). The interfacial continuation equation between two adja-
cent sub-layers is
gj;0 gj1;Rn j 2; 3;. . .;Rs 34
For state space vectors at the top and bottom surfaces of the lami-
nate, i.e. g1;0 and gRs ;Rn, there are six variables known as loading
conditions:
z 0 : sxzij syzij rzij 0z h : sxzij syzij 0;rzij q
35
whereq is the applied load on the laminates top surface.
All of the unknown state space vectors at interfaces of sub-lay-
ers can be obtained by solving Eqs. (33)(35); furtherly, state space
variables at each numerical layer can also be gained through Eq.
(31). Note that though the introduction of joint coupling technique
here brings more variables to solve at one time, it truly has high
degree of accuracy, which can be observed in the next part. It
should be mentioned that the sub-layer numberRsis not necessar-
ily equal to the lamina number N; however, two laminas which
have different plied orientations should not be included in one
sub-layer, or may lead to large numerical errors.
Of course, the numerical layers divided in each sub-layer is notneeded for conventional composite laminates, which is homoge-
neous in each single lamina, as studied in Ref. [20]. Nevertheless,
only divide the laminate into sub-layers for functionally graded
plates is computational infeasible: huge amounts of variables must
be solved at one time to get accuracy solutions. So to divide each
sub-layer into many numerical layers is suggested for functionally
graded plates: it on the one hand reduces the number of variables
to be solved at one time, and on the other hand guarantees the
accuracy of solutions.
5. Numerical examples
5.1. Comparison and convergence study
Comparison calculation is presented firstly to validate the accu-
racy and effectiveness of the utilized classic state space method.
Conventional symmetric [0/90/90/0] laminated plates are con-
sidered and it is assumed that the plate is simply supported and
loaded with q ~q11 sinpn sinp1 on top surfaces. The materialproperties are taken as
EL 174:6 GPa; ET7 GPa; GLT 3:5 GPa;
GTT 1:4 GPa; mLT mTT0:25
Some dimensionless variables inTable 1are defined as follows:
wwa=2; b=2; 0ETh
3
~q11a4 100; rx rx
a
2;
b
2;
h
2
h
2
~q11a2;
ry ry a2
; b2
; h4
h2~q11a2
sxy sxy 0; 0;h
2
h
2
~q11a2; sxz sxz 0;
b
2; 0
h~q11a
;
syz syza
2; 0; 0
h~q11a
herea/b= 1. From the comparison inTable 1, it is obvious that our
results agree well with Paganos elasticity solutions[22], for either
thin or thick laminates.
In what follows the reliability and convergence of DQSSM is
studied. A square isotropic plate (m= 0.3) is considered and our re-
sults are compared with Ls (by using DQSSM) and Liews (byusing DQM in three directions). The plate is full-clamped and
loaded with uniform distributed loading 0.5q0 and0.5q0 at the
top and bottom surfaces, respectively. As shown inTable 2, our re-Fig. 4. Illustration of the computational structure in the laminates thickness
direction. (a) Divide of sub-layers. (b) Divide of numerical layers in one sub-layer.
Y. Fu et al. / Materials and Design 31 (2010) 29042915 2909
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sults coincide with Ls except the 13 13 case, which was not
reported formerly. It should be mentioned that bad conditioned
matrix was encountered in our calculation when h= 0.2a and
Nx Ny= 11 11 or 13 13, though the results are acceptable.
Nevertheless, this defect can be overcome by increasing the sub-
layer number Rssix sub-layers are enough to get reliable solutionsfor these cases.
5.2. Laminates with gradient FVF
In this section laminated plates with variable fiber ratio along
the thickness direction are analyzed. The Graphite/Epoxy compos-
ite T300/5208 is studied here with corresponding material proper-
ties shown inTable 3. Effects of FVFs on the composites material
properties are illustrated inFig. 5. It can be seen that the longitu-
dinal Youngs modulus EL is proportional to Vk
f ; however, the
transverse elastic modulus and the shear modulus increase mark-
edly only whenVk
f is quite large.
For convenience, it is assumed that each single layer has the
same fiber distribution function. The power law index is set asp= 5, and the mean fiber ratio Vkf 0:6. Two different cases of
square laminates are considered in this section as
Case 1: Four edges simply supported and with q ~q11 sinpnsinp1 loaded on the top surface; a = b= 4h.
Case 2: Four edges clamped and with q= q0loaded on the top sur-
face;a = b= 10h.
For both two cases some dimensionless variables are defined as
follows:
Case 1:
rx;ry rxa
2 ;
b
2 ;z ;ry a2 ; b2 ;z h2
~q11a2
rz;sxy;sxz;syz rza
2;
b
2;z
;sxy0;0;z;sxz 0;
b
2;z
;syz
a
2;0;z
h~q11a
Case 2:
rx;ry rxa
2;
b
2;z
;ry
a
2;
b
2;z
h
2
q0a2
rz;sxy;sxz;syz rza
2;
b
2;z
;sxy
a
4;
b
4;z
;sxz
a
4;
b
2;z
;syz
a
2;b
4;z
h
q0a
Firstly, results to Case 1 are shown inFigs. 6 and 7, for symmetricand asymmetric laminates, respectively. It can be seen from the
comparisons that different distribution forms of fiber ratios rear-
range the stress fields. The functionally graded plate degrades to
the conventional one when the interfacial fiber ratio VkI
f 0:6.
As the power law indexp= 5 set here, it is evident fromFig. 3that
the material properties of the composite only change abruptly near
interfaces and surfaces for VkIf 0:3 or VkI
f 0:9. Correspond-
ingly, the in-plane normal stresses rx and ry change significantlyin this region after the gradient design. The reduction of the inter-
facial fiber ratio VkI
f decreases these two stress components, which
can reduce the interfacial stress concentrations of laminates. The
in-plane shear stress sxy declines near interfaces when there arefewer fibers in this region, which is similar to the in-plane normal
stresses. However, the redistribution of fiber ratios doesnt affect
the transverse stresses effectively, especially for the transverse
Table 1
Comparison of the deflection and stresses responses of laminated plates [0 /90/90/
0] with Ref. [22]
a/h Source w rx ry syz sxz sxy
4 Pagano 1.954 0.720 0.663 0.292 0.219 0.0467
Present 1.9367 0.7203 0.6519 0.2915 0.2193 0.0466610 Pagano 0.743 0.559 0.401 0.196 0.301 0.0275
Present 0.7370 0.5586 0.3965 0.1959 0.3014 0.02750
20 Pagano 0.517 0.543 0.308 0.156 0.328 0.0230
Present 0.5130 0.5428 0.3052 0.1556 0.3282 0.02302
100 Pagano 0.4385 0.539 0.276 0.141 0.337 0.0216
Present 0.4346 0.5388 0.2683 0.1389 0.3388 0.02135
0.0 0.2 0.4 0.6 0.8 1.0
0
50
100
150
200
250
GLT
=GTT
ET
Materialproperties(GPa)
V k
f
EL
T300/5208
Fig. 5. Effect of different fiber ratios on the material properties of the FRP
composite.
Table 2
Comparison of the results to full-clamped isotropic plates with literatures.
a=h Nx Ny 2Gq10 h1
wa=2; b=2; h=2 q10 rxa=2; b=2; h
Present (Rs= 3) Present (Rs= 6) L[20] Liew[23] Present (Rs= 3) Present (Rs= 6) L[20] Liew[23]
5 5 5 10.96941 10.96941 10.96941 11.15777 4.250688 4.250688 4.25069 4.36199
7 7 11.13728 11.13728 11.13728 11.13736 3.870329 3.870329 3.87033 3.86034
9 9 11.13522 11.13522 11.13522 11.13671 3.887901 3.887901 3.88790 3.89135
11 11 11.17399a 11.17399 11.17399 11.17407 3.814499a 3.814500 3.81450 3.81293
13 13 11.18524a 11.18524 11.18546 3.880341a 3.880341 3.88135
10 5 5 123.8261 123.8261 123.8261 124.4904 15.95727 15.95727 15.9573 16.0563
7 7 124.6102 124.6102 124.6102 124.6105 14.95357 14.95357 14.9536 14.9511
9 9 123.7850 123.7850 123.7850 123.7856 13.66635 13.66635 13.6664 13.666711 11 125.1241 125.1241 125.1241 125.1241 14.42233 14.42233 14.4223 14.4223
13 13 125.0002 125.0002 125.0003 14.10789 14.10789 14.1079
a Cases that encounter bad conditioned matrix.
2910 Y. Fu et al. / Materials and Design 31 (2010) 29042915
https://www.researchgate.net/publication/223342120_Modeling_via_differential_quadrature_method_Three-dimensional_solutions_for_rectangular_plates?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/223342120_Modeling_via_differential_quadrature_method_Three-dimensional_solutions_for_rectangular_plates?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/223605733_Semi-analytical_three-dimensional_elasticity_solutions_for_generally_laminated_composite_plates?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/223605733_Semi-analytical_three-dimensional_elasticity_solutions_for_generally_laminated_composite_plates?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==7/25/2019 2010_1 (Mater Des) Interlaminar Stress Distribution of Composite Laminated Plates With Functionally Graded Fiber
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normal stress rz, which remains the same after the gradientdesign. Transverse shear stresses sxz and syz seem rearrangingwhen the fiber distribution changes but the alteration is very small,
and the values of them decrease a little at the center point when
VkI
f becomes larger. There are mainly three reasons that account
for this phenomenon. First, the softening of the material stiffness
may increase the strain, which generates a mount of stresses. Sec-
ond, different from the longitudinal elastic modulus, the transverse
elastic modulus and the shear modulus change slowly (see Fig. 5)with respect to the FVF so the fiber ratios effect is not obvious.
Moreover, loading transfer between adjacent laminas is commonly
induced by transverse stresses, which makes this kind of design
method not easy to reduce transverse stresses near interfaces as
Table 3
Material properties of the fiber and matrix.
Material E(GPa) G (GPa) m
Graphite T300 231 91 0.27
Epoxy 5208 3.9 1.4 0.35
0.0 0.2 0.4 0.6 0.8 1.0
-0.8
-0.4
0.0
0.4
0.8
z/h
Vk I
f =0.3
Vk I
f =0.6
Vk I
f =0.9
x
0.0
-0.50
-0.25
0.00
0.25
0.50
z/h
y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
z/h
z
0.0 0.2 0.4 0.6 0.8 1.0
-0.8
-0.4
0.0
0.4
0.8
z/h
xy
0.0
0.2 0.4 0.6 0.8 1.0
0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
z/h
xz
0.0 0.2 0.4 0.6 0.8 1.00.00
0.05
0.10
0.15
0.20
z/h
yz
Fig. 6. Interlaminar stress distribution of symmetric [0/90/90/0] simply supported laminated plates with variable fiber spacing.
Y. Fu et al. / Materials and Design 31 (2010) 29042915 2911
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the in-plane stresses. Nonetheless, transverse stresses are usually
in lower order of magnitude than in-plane ones; the change of
in-plane stresses by redistribution the fibers is still significantly,
which provides a method to resolve the interfacial stress concen-
tration problems.
In what follows we will discuss the plates with clamped edges
of Case 2. Both cross-plied and angle-plied laminates are consid-ered with results illustrated in Figs. 8 and 9, respectively. The
laminate is divided into 10 sub-layers within each there are 10
numerical layers to obtain accurate and smooth solutions. Whats
more, Nx Ny= 7 7 meshes are chosen here. The figures show
that the reduction of fiber ratios near interfaces can reduce the
stress concentrations there, which is similar to the simply sup-
ported cases. Transverse stresses rearrange a little but the slight
change can be neglected at interfaces. The reasons have been
pointed out in the former and would not be repeated here. Differ-
ent from the cross-plied laminates, angle-plied ones are moreinteresting and complex. For this kind of laminate, in-plane shear
stress sxy may be quite large, due to the well known stretch-shearcoupling phenomena. As shown in Fig. 9, sxy is very large at the
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
z/h
yz
0.0
-0.4
0.0
0.4
0.8
Vk I
f =0.3
Vk I
f =0.6
Vk I
f
=0.9
z/h
x
0.00.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
-0.8
-0.4
0.0
0.4
z/h
y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
z/h
z
0.0 0.2 0.4 0.6 0.8 1.0
-0.8
-0.4
0.0
0.4
0.8
z/h
xy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
z/h
xz
Fig. 7. Interlaminar stress distribution of asymmetric [90/0/90/0] simply supported laminated plates with variable fiber spacing.
2912 Y. Fu et al. / Materials and Design 31 (2010) 29042915
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interfaces of the 45or 45layers; nevertheless, this stress concen-
tration can still be reduced by the proposed gradient design method.
Note that classic state space can not handle the angle-plied lami-
nates for any boundary conditions, which must be solved by DQSSM.
From another point of view, the redistribution of fiber ratios
leads to the change of not only stress fields but also the strength
of composites, commonly larger FVF has stronger strength. There-fore, the strength criterion seems variational in each point, which
should be carefully treated in the design process. Accurate determi-
nation of the stress field and strength criteria field is necessary for
the failure analysis. The optimum purpose of fiber distributions is
to ensure the structure has both good overall mechanical properties
and sufficient local strengths in eachpoint. Onlysatisfying the buck-
ling or vibration characteristics is not enough.
6. Conclusion
Designability is an advantage of the FRP composite and recently
the gradient design of it attracts many researchers attention. Var-
ious gradient design methods are proposed, including the non-uni-
0.0-0.4
-0.2
0.0
0.2
0.4
z/h
Vk I
f =0.3
Vk I
f =0.6
Vk I
f =0.9
x
0.0 0.2 0.4 0.6 0.8 1.0
-0.30
-0.15
0.00
0.15
0.30
z/h
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.05
0.10
z/h
z
0.0 0.2 0.4 0.6 0.8 1.0
-0.50
-0.25
0.00
0.25
0.50
z/h
xy
0.0
0.2 0.4 0.6 0.8 1.0
0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
z/h
xz
0.0 0.2 0.4 0.6 0.8 1.00.00
0.05
0.10
0.15
0.20
z/h
yz
Fig. 8. Interlaminar stress distribution of cross-ply [0/90/0/90/0] clamped laminated plates with variable fiber spacing.
Y. Fu et al. / Materials and Design 31 (2010) 29042915 2913
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form distributions of the fibers orientation, volume fraction or
material properties, to gain ideal structural properties. Both the
overall mechanical properties and the local strength should be con-
sidered, in the composites design process, of the latter accurate
stress fields is needed for the failure analysis.
The purpose of this paper is to present 3D methods to exactly
determine the stresses in this kind of highly heterogeneous lami-
nated plates. For specified case the classic state space methodcan be used, while a sub-layer based DQSSM is suggested for the
general cases. Numerical examples indicate that the stress fields
rearrange after the functionally graded design. In-plane stresses
near interfaces would decrease if the fiber ratio reduces in this re-
gion, which provides a method to resolve the interfacial stress con-
centration problems. However, transverse stresses which have
lower order of magnitude than the in-plane ones do not change
very much.
Though the 3D methods can obtain accurate solutions, they
have to some extent lost the computational efficiency, especially
when computing the transfer matrix for each single numericallayer. In the future studies, specific laminate theories, which have
both advantages in accuracy and efficiency, must be developed for
heterogeneous laminates which have general boundary and load
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
-0.2
0.0
0.2
0.4
z/h
V k I
f =0.3
V k I
f =0.6
V k I
f =0.9
x
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
0.2
z/h
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.05
0.10
z/h
z
0.0 0.2 0.4 0.6 0.8 1.0
-1.0
-0.5
0.0
0.5
z/h
xy
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
z/h
xz
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.03
0.06
0.09
0.12
0.15
z/h
yz
Fig. 9. Interlaminar stress distribution of angle-ply [0/45/90/45/0] clamped laminated plates with variable fiber spacing.
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conditions. Whats more, the strength criterion of this kind of func-
tionally graded material is also a problem to be resolved later on.
Acknowledgements
Support from the National Natural Science Foundation of Chinathrough Grant No. 10872066 should be acknowledged. The authors
thank for the suggestion from Professor Chaofeng L in Zhejiang
University of China and also the valuable advice of reviewers.
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