2010_1 (Mater Des) Interlaminar Stress Distribution of Composite Laminated Plates With Functionally Graded Fiber Volume Fraction

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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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Yiming Fu

Hunan University

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Pu Zhang

The University of Manchester

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Author's personal copy

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==


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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==


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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==


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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



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

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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:

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/245371857_Differential_Quadrature_Method_in_Computational_Mechanics_A_Review?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==https://www.researchgate.net/publication/245371857_Differential_Quadrature_Method_in_Computational_Mechanics_A_Review?el=1_x_8&enrichId=rgreq-5f2a7b60dd8f5b453601514a34aa6655-XXX&enrichSource=Y292ZXJQYWdlOzI0ODQ2NTQ1MztBUzoyMTAwMTgxMDk1MzAxMThAMTQyNzA4MzYyNzY1OA==


7/13


rxij XNx 1k2

P1

ik a2ukja3vkj

XNy1l2

Q1

jl a3uila5vil a7rzij

ryij

XNx1

k2

P1

ik a5ukja6vkj

XNy1

l2

Q1


sxyij XNx1k2

P1

ik a3ukja4vkj

XNy1l2

Q1


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.

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8/13


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.

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9/13


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.



10/13


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.



11/13


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.



12/13


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.



13/13


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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Documents

2010_1 (Mater Des) Interlaminar Stress Distribution of Composite Laminated Plates With Functionally Graded Fiber Volume Fraction