Composite Structures 174 (2017) 142–157

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Analytical method to investigate nonlinear dynamic responses ofsandwich plates with FGM faces resting on elastic foundationconsidering blast loads

http://dx.doi.org/10.1016/j.compstruct.2017.03.0870263-8223/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: Behzad.alb@gmail.com (B. Mohammadzadeh), Cpebach@

Sejong.ac.kr (H.C. Noh).

Behzad Mohammadzadeh, Hyuk Chun Noh ⇑Department of Civil and Environmental Engineering, Sejong University, Seoul, South Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 January 2017Revised 14 March 2017Accepted 25 March 2017Available online 20 April 2017

Keywords:Blast loadSandwich plateFunctionally graded materialsFiber-reinforced concreteDynamicsInterlaminar stresses

An analytical approach is presented to investigate nonlinear dynamic responses of sandwich plates. Toobtain governing differential equations of motion, the higher order shear deformation theory is employedtogether with Hamilton’s principle. The Navier’s solution and Runge-Kutta method using available math-ematical package software MAPLE 14 are used to solve the governing equations. This method can con-sider any required number of layers through the sandwich plate thickness. To evaluate the methodvalidity a sandwich plate with FGM face sheets and the FRC core resting on an elastic foundation is sub-jected to the blast load due to the burst of 5 kg charge. The maximum plane-normal displacement isobtained by the analytical method and numerical approach. Comparison between results shows goodagreement. Thereafter, time histories obtained from both analytical and numerical approaches are com-pared. The interlaminar stresses are obtained through the sandwich plate thickness. The results showthat neither material failure nor delamination occurs.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Sandwich structures have been widely used in industrial appli-cations. Several studies paid considerable attention to the behaviorof structures in the case of applying blast or impact loads. Explo-sive charges can target different kinds of structures, so it gainsimportance to be studied. An explosion can be defined as a veryfast chemical reaction involving a solid, dust or gas, during whicha rapid release of hot gases and energy takes place [1]. The blastwave pressure distribution, pressure-time history, can be statedas a function of time as Eq. (1) represents [2]:

PðtÞ ¼ Ps0þ 1� tt0þ

� �exp

�bðt � taÞt0þ

� �; ð1Þ

where Ps0þ is incident pressure (Maximum pressure of blast wave),‘t’ time, t0þ positive phase duration, ta blast wave arrival time and bis dimensionless wave decay coefficient. Sandwich structures areincreasingly being applied to various types of industries and sectorssuch as aerospace, marine and automobile engineering, because oftheir superior characteristics and structural performance with light

weight [3]. Functionally graded materials (FGM) has recentlyattracted attentions because of their considerable advantages overconventional materials [4]. The most common FGMs are a composi-tion of metal and ceramic.

The ductile metallic part has superior fracture toughness whilethe high thermal resistance is considerable property of the ceramicpart [5]. FGMs have many applications like those used in spacecraftheat shields, heat exchanger tubes and fusion reactors [6].

In an explosion event the blast wave front together with highelevated temperature affect surrounding stuffs and structures [7].The sandwich plates can be made such that they resist against blastloads, so the need for sandwich plates having FGM faces to sustainhigh temperatures and transfer axial forces and bending momentsas well as a strong core such as fiber reinforced concrete which hasthe ability of carrying compressive stresses comes up.

Recent developments in the analysis of composite laminatedplates point out that the plate thickness has more pronouncedeffects on the behavior of composite laminates than isotropicplates [8]. Also, due to low transverse shear moduli relative tothe in-plane Young’s moduli, transverse shear deformations playa much important role in the kinematics of composite laminates.Exposition of laminated plates to dynamic loads leads to character-ize them by transverse shear deformation. Neglecting the trans-verse shear effects and rotary inertia yields incorrect results,even for thin composite laminated plates [9].

Nomenclature

PðtÞ pressure-time historyPs0þ incident pressuret timet0þ positive phase durationta blast wave arrival timeb wave decay coefficientE modulus of elasticitym Poisson’s ratioq material densityG shear modulus of elasticityj curvaturew charge weightIji mass moment of inertiasqx distributed load along with x-directionqy distributed load along with y-directionqb distributed forces at the bottom layersqt distributed forces at the top of layersN membrane forceM bending momentP higher order bending momentQ shear forceR higher order shear moment�NT thermal force�MT thermal moment�PT thermal higher order momentO0 plate arear stressrT thermal stresss shear stresse strainc shear strainYt tensile strengthS13¼23 interlaminar shear strengthF1 yield stress in X-directionF2 yield stress in Y-direction

F6 shear strengtha length of plate (long side)c width of plate (short side)h thickness of the platehf thickness of the face sheethfrc thickness of the coreti layer height through the plate thicknessPF effective material propertiesPc temperature-dependent properties of ceramicK foundation stiffnessPm temperature-dependent properties of metalVc volume fraction of ceramicVm volume fraction of metalZ depth through the plate thicknessAM cross sectional area of matrixAr cross sectional area of reinforcementAc cross sectional area of compositeN volume fraction indexa thermal expansion coefficient�Qij constitutive stiffness matrix elementU displacement in x-directionV displacement in x-directionW displacement in z-directionWx rotation about y-directionWy rotation about x-direction_U time-derivative of U_V time-derivative of V_W time-derivative of W_Wx time-derivative of Wx_Wy time-derivative of Wy€U second time-derivative of U€V second time-derivative of V€W second time-derivative of W€Wx second time-derivative of Wx€Wy second time-derivative of Wy

B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157 143

To account for the effect of transverse normal strain, two kindsof higher order shear deformation theories have been proposed.One is the layer-wise theory which may trace the local variationsin each layer more accurately, but it is more complicated due tolarge numbers of independent unknowns. Another one is theequivalent single-layer theory which can accurately predict theglobal structural responses including deflection, buckling andvibration of the laminated plates. Further, Reddy[10] developed asimple higher order shear deformation plate theory. This theoryassumes that the transverse shear strains have parabolic distribu-tion across the plate thickness. The advantage of this theory overthe first-order shear deformation theory is that the number ofindependent unknowns is the same as in the first-order sheardeformation theory, but no shear correction factor is required [10].

FGM were proposed by Japanese scientists in 1984. The volumefraction of FGM constituents vary continuously along with thick-ness direction. Therefore, the mechanical properties such as mod-ulus of elasticity E, Poisson’s ratio t, material density q, shearmodulus of elasticity G, smoothly vary through the desirable direc-tion. The FGM have been developed by combining the form offibers, particulates, whiskers or platelets of advanced engineeringmaterials [5]. Numerous researches interested in investigation ofFGM plates for the static, free vibration or buckling problems basedon the first-order shear deformation theory [11] the third-ordershear deformation [6] and three dimensional elasticity [4] withor without foundation interaction effect. Zhen-Xin Wang et al.[10] performed nonlinear analyses of sandwich plates with FGM

face sheets resting on elastic foundation. Very few studies havebeen performed for investigating dynamic responses of FGMplates. Akbarzadeh et al. [11] investigated dynamic responses ofFGM plate using hybrid Fourier-Laplace transform method. DanSun et al. [12] studied dynamic response of the rectangular FGMplates with clamped supports under impulsive load.

According to the above explanations, this study is motivated toemploy the higher order shear deformation theory and virtualwork principal to present an analytical approach to perform non-linear dynamic investigation of sandwich plates having FGM facesresting on elastic foundation subjected to blast load. Since the blastwave is emitted into the surrounding area of sandwich plate, theassumption has been made to consider distributed loads of qx

and qy along with x-direction and y-direction on plate sides,respectively. Besides, uniformly distributed blast pressure is con-sidered to subject to the plate face. Fig. 1 shows an illustration ofapplied loads on the sandwich plate.

2. Characteristics of sandwich plate

A sandwich plate consisting of two FGM face sheets and a coremade of fiber reinforced concrete resting on elastic foundation isconsidered to be subjected to the blast load. The aim is to findthe dynamic responses of plate by solving the corresponding gov-erning differential equations including temperature change effectsand interaction between foundation and sandwich structure. Fig. 2shows an illustration of the sandwich plate layout.

Fig. 1. Illustration of applied loads on sandwich plate.

144 B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157

The FGM face sheet is made from a mixture of ceramic andmetal for which the mixing ratio varies continuously and smoothlyin plane-normal direction, plate thickness, and can be obtained byapplication of simple rule of mixture of composite materials. Theeffective material properties PF including elastic modulus, EF , Pois-son’s ratio, mF and thermal expansion coefficient, aF , are defined asfollows:

PF ¼ PcVc þ PmVm; ð2Þwhere Pc and Pm denotes the temperature-dependent properties ofthe ceramic and metal, respectively. Vc is the volume fraction ofceramic while Vm is volume fraction of metal. Volume fractions ofmetal Vm for top and bottom of FGM face sheets are defined inEqs. (3a) and (3b), respectively [10]:

Vmt ¼ Z � t0t1 � t0

� �N

; ð3aÞ

Vmb ¼ t3 � Zt3 � t2

� �N

; ð3bÞ

where subscript ‘b’ refers to bottom and subscript ‘t’ refers to top. Nis the volume fraction index which dictates the material variationprofile through the FGM layer thickness. We can define thetemperature-dependent material properties as follows [13]:

EFðZ; TÞ ¼ ðEbðTÞ � EtðTÞÞ 2Z þ h2h

� �N

þ Et; ð4aÞ

aFðZ; TÞ ¼ ðabðTÞ � atðTÞÞ 2Z þ h2h

� �N

þ at ; ð4bÞ

Fig. 2. Layout of sandwich plate r

mFðZ; TÞ ¼ ðmbðTÞ � mtðTÞÞ 2Z þ h2h

� �N

þ mt; ð4cÞ

The mass density qF is assumed to be independent of the tem-perature and is only a function of thickness.

qFðZÞ ¼ ðqb � qtÞ2Z þ h2h

� �N

þ qt: ð4dÞ

The core is made of fiber reinforced concrete (FRC). The elasticmodulus of the composite in the loading direction can be obtainedfrom the law of mixtures as follows:

Ec ¼ ErVr þ EMVM; ð5Þwhere Vr is the volume fraction of longitudinal reinforcement, VM isthe volume fraction of matrix, Er is the elastic modulus of the rein-forcement and EM is the elastic modulus of the matrix. Vr and VM

are defined as given in Eqs. (6a) and (6b), respectively:

Vr ¼ Ar

Ac; ð6aÞ

VM ¼ AM

Ac; ð6bÞ

where Ar is the cross sectional area of reinforcement, AM is the crosssectional area of matrix and Ac is the cross sectional area of compos-ite. Poisson’s ratio of composite can be obtained as follows:

#c ¼ mrVr þ mMVM: ð7ÞDensity of FRC, qFrc and thermal coefficient aFrc are constant

through the thickness of the core.The constitutive stress-strain relation can be mentioned as

follows:

rxx

ryy

rxy

8><>:

9>=>; ¼

�Q11�Q12

�Q16

�Q12�Q22

�Q26

�Q16�Q26

�Q66

264

375

exxeyyexy

8><>:

9>=>; ð8aÞ

ryz

rxz

� �¼

�Q44�Q45

�Q45�Q55

" #cyzcxz

� �ð8bÞ

The thermal coefficients of FGM face sheet in X and Y-directionsare equal to aF . The elements of stiffness matrix of Eq. (8), �Qij; canbe stated as follow:

Q11 ¼ Q22 ¼ EFðZ; TÞ1� m2F

; Q12 ¼ mFEFðZ; TÞ1� m2F

; Q16 ¼ Q26 ¼ Q45 ¼ 0;

Q44 ¼ Q55 ¼ Q66 ¼ EFðZ; TÞ2ð1þ mFÞ : ð9Þ

esting on elastic foundation.

B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157 145

For the core, thermal coefficient is aFrc and �Qij are defined asfollows:

Q11 ¼ Q22 ¼ Ec

1� m2Frc; Q12 ¼ mFrcEc

1� m2Frc; Q16 ¼ Q26 ¼ Q45 ¼ 0;

Q44 ¼ Q55 ¼ Q66 ¼ Ec

2ð1þ mFrcÞ : ð10Þ

When an explosion happens the surrounding environment tem-perature is drastically elevated. The plate temperature uniformlyincreases from the initial temperature Ti to the final value of Tf

at which the plate material failure may occur. The temperaturechange is stated as DT ¼ Tf � Ti.

3. Derivation of equation of motion

To perform this part the concepts given in Refs.[10,14,15,16,17,18,19] are used.

A rectangular sandwich plate having two face sheets made offunctionally graded materials (FGM) and a core made of fiber rein-forced concrete is considered. The length of the plate is a, width is cand total thickness of the plate is h. The coordinate system has itsorigin at the corner of the plate on the mid-plane. Let �U, �V and �Wbe the plate displacements parallel to a right-hand set of axes (X, Y,Z), where X is longitudinal direction, Y is along with plate widthand Z is normal to the plate. �Wx and �Wy are the mid-plane rotationsof the normal about Y and X axes, respectively. The displacementcomponents are assumed to have the form as follow [10,15]:

U ¼ �U0ðX;Y ; tÞ þ Z �WxðX;Y; tÞ þ Z2nxðX; Y; tÞ þ Z3fxðX;Y ; tÞ; ð11aÞ

V ¼ �V0ðX;Y; tÞ þ Z �WyðX; Y; tÞ þ Z2nyðX;Y ; tÞ þ Z3fyðX;Y; tÞ; ð11bÞ

W ¼ �W0ðX;Y; tÞ; ð11cÞwhere ‘t’ represents time, �U0; �V0; �W0, �Wx, �Wy, nx, ny, fx, fy areunknowns.

Consideration of clamped boundary conditions can be noted asfollows [14,15]:

�Wy ¼ �W ¼ �U ¼ 0; For X ¼ 0; a; ð12aÞ

�Wx ¼ �W ¼ �V ¼ 0; For Y ¼ 0; b: ð12bÞThe transverse shear stresses ryz and rxz are to be vanished at

the bounding planes of the plate, i.e., stresses at Z ¼ � h2, so the

transverse shear strains e4 and e5 should also vanish there. There-fore, we have:

e4 ¼ @V@Z

þ @W@Y

¼ �Wy þ 2Zny þ 3Z2fy þ@ �W0

@Y¼ 0; ð13aÞ

e5 ¼ @U@Z

þ @W@X

¼ �Wx þ 2Znx þ 3Z2fx þ@ �W0

@X¼ 0; ð13bÞ

From which the following conditions can be inferred:

nx ¼ ny ¼ 0; ð14aÞ

fx ¼ � 4

3h2

@ �W0

@Xþ �Wx

� �; ð14bÞ

fy ¼ � 4

3h2

@ �W0

@Yþ �Wy

� �; ð14cÞ

Substituting Eqs. (13) and (14) into Eq. (11) results in:

U ¼ �U0 þ Z �Wx � 43

Zh

� �2

ð �Wx þ @ �W0

@XÞ

" #; ð15aÞ

V ¼ �V0 þ Z �Wy � 43

Zh

� �2

ð �Wy þ @ �W0

@YÞ

" #; ð15bÞ

W ¼ �W0 ð15cÞThe strain-displacement relations can be stated as follows:

e0xx ¼@ �U0

@Xþ 12

@ �W0

@X

� �2

; e0yy ¼@ �V0

@Yþ 12

@ �W0

@Y

� �2

; e0zz ¼ 0;

e0yz ¼ �Wy þ @ �W0

@Y; e0xz ¼ �Wx þ @ �W0

@X;

e0xy ¼@ �U0

@Yþ @�V0

@Xþ @ �W0

@X@ �W0

@Y; ð16aÞ

e1xx ¼ e0xx þ Zðj01Þ; e1yy ¼ e0yy þ Zðj0

2Þ; e1zz ¼ 0; e1yz ¼ e0yz;

e1xz ¼ e0xz; e1xy ¼ e0xy þ Zðj0xyÞ; ð16bÞ

e2xx ¼ e1xx þ Z3j2xx; e2yy ¼ e0yy þ Z3j2

yy; e2zz ¼ 0; e2yz ¼ e0yz þ Z2j2yz;

e2xz ¼ e0xz þ Z2j2xz; e2xy ¼ e1xy þ Z3j2

xy: ð16cÞwhere

j0xx ¼

@ �Wx

@X; j2

xx ¼ � 4

3h2

@ �Wx

@Xþ @2 �W

@X2

!; j0

yy ¼@ �Wy

@Y;

j2yy ¼ � 4

3h2

@ �Wy

@Yþ @2 �W

@Y2

!; ð17aÞ

j2yz ¼ � 4

h2�Wy þ @ �W

@Y

� �;j2

xz ¼ � 4

h2�Wx þ @ �W

@X

� �; ð17bÞ

j0xy ¼

@ �Wx

@Yþ @ �Wy

@X;j2

xy ¼ � 4

3h2

@ �Wx

@Yþ @ �Wy

@Xþ 2

@2W@X@Y

!: ð17cÞ

The equations of motion of sandwich plate having the FGMfaces in Cartesian Coordinates XYZ are derived by use of virtualwork principle:Z t

0ðdU þ dV � dKÞdt ¼ 0 ð18Þ

where dU is virtual strain energy, dV virtual work done by externalapplied forces and dK virtual kinetic energy. dU is derived asfollows:

dU ¼ZX0

Z h2

�h2

ððrxx þrTx Þdexx þðryy þrT

yÞdeyy þ ðrqygmaxy þrTxyÞdexy

n

þðrxz þrTxzÞdcxz þðryz þrT

yzÞdcyzÞdzodxdy ð19aÞ

Substituting strains from Eq. (15) into Eq. (19a), dU can be rewrittenas follows:

dU ¼ZX0

ðNxx þNTx Þ

@dU0

@xþ 12d

@W0

@x

� �2 !

þ ðSxx þ STx Þd@Wx

@x

� �(

� 4

3h2 ðPxx þ PTx Þd

@2W@x2

þ ðNyy þNTyÞ

@dV0

@yþ 12d

@W0

@y

� �2 !

þðSyy þ STyÞd@Wy

@y

� �� 4

3h2 ðPyy þ PTyÞd

@2W@y2

þðNxy þNTxyÞd

@U0

@yþ @V0

@xþ @W0

@x@W0

@y

� �

þðSxy þ STxyÞ@Wx

@yþ @Wy

@x

� �� 8

3h2 ðPxy þ PTxyÞ

@2W0

@x@y

þ Qx �4

h2 Rx

� �Wx þ @W0

@x

� �þ ðQy �

4

h2 RyÞ Wy þ @W0

@y

� ��dxdy:

ð19bÞ

146 B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157

The virtual work can be stated as given in Eq. (20):

dV ¼ �ZX0

½qxduþ qydv þ ðqb þ qtÞdwþ ðK1W � K2r2WÞdw�;

ð20Þ

where qb and qt are the distributed forces at the bottom and the topof the layers. qx and are distributed loads on sides in x and y direc-tion, respectively.

Virtual kinetic energy is given as Eq. (21a) represents:

dK ¼ZX0

Z t0

t1

qðZÞ ð _Ud _UÞ þ ð _Vd _VÞ þ ð _Wd _WÞh i

dz�

þZ t1

t2

qc½ð _Ud _UÞ þ ð _Vd _VÞ þ ð _Wd _WÞ�dzþZ t2

t3

qðZÞ½ð _Ud _UÞ

þð _Vd _VÞ þ ð _Wd _WÞ�dzodxdy; ð21aÞ

where O0 is the plate area and the superposed dot on a variableindicates rate of change with respect to time. Substituting the strainrelations and mass moments of inertias into Eq. (21a) we canrewrite dK as given in Eq. (21b):

dK ¼ZX0

I0ið _u0d _u0 þ _v0d _v0 þ _w0d _w0Þ þ I1ið _U0d _Wx þ _Wxd _U0

hþ _V0d _Wy þ _Wyd _V0Þ þ PI2i _Wxd _Wx þ _Wyd _Wy

� �

� 4

3h2 I3ið _U0d _Wx þ _U0@d _W0

@xþ _Wxd _U0 þ @ _W0

@xd _U0

þ _V0d _Wy þ _V0@d _W0

@yþ _Wyd _V0 þ @ _W0

@yd _V0Þ

� 4

3h2 I4i 2 _Wxd _Wx þ _Wx@d _W@x

þ @ _W0

@xd _Wx

þ2 _Wyd _Wy þ _Wy@d _W@y

þ @ _W0

@yd _Wy

!

þ 16

9h4 I6i_Wxd _Wx þ _Wx

@d _W0

@xþ @ _W0

@xd _Wx þ @ _W0

@x@d _W0

@x

þ _Wyd _Wy þ _Wy@d _W0

@yþ @ _W0

@yd _Wy þ @ _W0

@y@d _W0

@y

!#dxdy

ð21bÞ

where Iji is the mass moment of inertias which can be defined asgiven in Eq. (21c):

Iji ¼Z ti

ti�1

Z jqiðZÞdz j ¼ 0;1;2;3;4;5;6; i ¼ 1;2;3 ð21cÞ

The membrane force, N, shear force, Q, bending moment, M,higher order bending moment, P and higher order shear force, Rare defined as Eq. (22) represents [15–17]:

ðN;M; PÞ ¼Z h

2

�h2

rð1; Z; Z3ÞdZ; ð22aÞ

ðQx;RxÞ ¼Z h

2

�h2

rxzð1; Z2ÞdZ; ð22bÞ

ðQy;RyÞ ¼Z h

2

�h2

ryzð1; Z2ÞdZ; ð22cÞ

�NT , �MT , �ST and �PT are the thermal forces, moments and higherorder moments caused by elevated temperature, respectively. Theyare defined as given in Eq. (23):

�NTx

�MTx

�PTx

�NTy

�MTy

�PTy

�NTxy

�MTxy

�PTxy

264

375 ¼

Xnk¼1

Z tk

tk�1

Ax

Ay

Axy

264

375

k

ð1; Z; Z3ÞDTdZ ð23aÞ

�STx�STy�STxy

264

375 ¼

�MTx

�MTy

MTxy

2664

3775� 4

3h2

�PTx

PTy

PTxy

2664

3775 ð23bÞ

where DT ¼ T � T0 is the temperature rise from the reference tem-perature T0 at which there are no thermal strains. Matrix A isdefined as follows [15,18]:

Ax

Ay

Axy

264

375 ¼ �

�Q11�Q12

�Q16

�Q12�Q22

�Q26

�Q16�Q26

�Q66

264

375

100

010

264

375 a11

a22

� ð23cÞ

in which a11and a22 are the thermal expansion coefficient measuredin the longitudinal and transverse directions, respectively. Consid-ering Eqs. (18)–(23), a set of governing differential equations ofmotion of sandwich plate resting on elastic foundation subjectedto blast load is obtained as follows:

@Nxx

@xþ @NT

x

@xþ @Nxy

@yþ @NT

xy

@y¼ I0

@2U0

@t2þ I1 � 4

3h2 I3

� �@2Wx

@t2

� 4

3h2 I3@2

@t2@W0

@x

� �þ qx ð24aÞ

@Nyy

@yþ @NT

y

@yþ @Nxy

@xþ @NT

xy

@x¼ I0

@2V0

@t2þ I1 � 4

3h2 I3

� �@2Wy

@t2

� 4

3h2 I3@2

@t2@W0

@y

� �þ qy ð24bÞ

@

@x12ðNxx þNT

x Þ@W0

@x� 4

3h2 ðPxx þ PTx Þ@W0

@xþQx �

4

h2 Rx

�

þðNxy þNTxyÞ

@W0

@y� 8

3h2 ðPxy þ PTxyÞ

@W0

@y

þ @

@y12ðNyy þNT

yÞ@W0

@y� 4

3h2 ðPyy þ PTyÞ@W0

@yþQy �

4

h2 Ry

�

þðNxy þNTxyÞ

@W0

@x� 8

3h2 ðPxy þ PTxyÞ

@W0

@x

¼ Pðx;y; tÞ � ðKW0Þ þ I0@2W0

@t2� 4

3h2 I3@

@x@2U0

@t2

!þ @

@y@2V0

@t2

!" #

� 4

3h2 I4@

@x@2Wx

@t2

!þ @

@y@2Wy

@t2

!" #� 16

9h4 I6@

@x@2Wx

@t2þ @2W0

@t2

!

þ @

@y@2Wy

@t2þ @2W0

@t2

!!ð24cÞ

Qx �4

h2 Rx þ @

@xðSxx þ STx Þ þ

@

@yðSxy þ STxyÞ ¼ I1

@2U0

@t2þ I2

@2Wx

@t2

� 4

3h2 I4 2@2Wx

@t2þ @

@x@2W0

@t2

!" #þ 16

9h4 I6@2Wx

@t2þ @

@x@2W0

@t2

!" #

ð24dÞ

Qx �4

h2 Rx þ @

@xðSxx þ STx Þ þ

@

@yðSxy þ STxyÞ ¼ I1

@2U0

@t2þ I2

@2Wx

@t2

� 4

3h2 I4 2@2Wx

@t2þ @

@x@2W0

@t2

!" #þ 16

9h4 I6@2Wx

@t2þ @

@x@2W0

@t2

!" #

ð24eÞ

B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157 147

Qy �4

h2 Ry þ @

@yðSyy þ STyÞ þ

@

@xðSxy þ STxyÞ ¼ I1

@2V0

@t2þ I2

@2Wy

@t2

� 4

3h2 I4 2@2Wy

@t2þ @

@y@2W0

@t2

!" #þ 16

9h4 I6@2Wy

@t2þ @

@y@2W0

@t2

!" #

ð24fÞ

4. Solution method

In order to find the solution of the equations of motion of sand-wich plate, the Navier solution is employed. To this aim, by consid-ering clamped boundary conditions, the fundamentaldisplacement fields are chosen as given hereunder [15,19]:

U0 ¼ U11ðtÞ sin 2pxa

1� cos2pyc

� �ð25aÞ

V0 ¼ V11ðtÞ 1� cos2pxa

� �sin

2pyc

ð25bÞ

W0 ¼ W11ðtÞ 1� cos2pxa

� �1� cos

2pyc

� �ð25cÞ

Wx ¼ Wx11 ðtÞ sin2pxa

1� cos2pyc

� �ð25dÞ

Wy ¼ Wy11ðtÞ 1� cos2pxa

� �sin

2pyc

ð25eÞ

Substituting the displacements functions from Eq. (25) into Eq.(24) results in a set of time-dependent nonlinear differentialequations:

c1U þ c2V þ c3W þ c4W2 þ c5Wx þ c6Wy � c7 €U � c8 €Wx � c9 €W

� c10DTxi � c11DTyi � qx ¼ 0 ð26aÞ

d1U þ d2V þ d3W þ d4W2 þ d5Wx þ d6Wy � d7

€V � d8€Wy � d9

€W

� d10DTyi � d11DTxi � qy ¼ 0 ð26bÞ

e1UW þ e2VW þ e3WxW þ e4WyW þ e5W þ e6W2 þ e7W

3 þ e8Wx

þ e9Wy � e10 €W þ e11 €U þ e12 €V þ e13 €Wx þ e14 €Wy þ e15WDTxi

þ e16WDTyi � Pðx; y; tÞ þ ðK1W0 � K2r2W0Þ ¼ 0 ð26cÞ

f 1U þ f 2V þ f 3W þ f 4W2 þ f 5Wx þ f 6Wy � f 7 €U � f 8 €Wx � f 9 €W

þ f 10DTxi þ f 11DTyi ¼ 0 ð26dÞ

g1U þ g2V þ g3W þ g4W2 þ g5Wx þ g6Wy � g7

€V � g8€Wy � g9

€W

þ g10DTyi þ g11DTxi ¼ 0 ð26eÞwhere ci; di, ei, f i, gi are given in Appendix A. In order to solve the Eq.(26), to find dynamic responses of the sandwich plate, forth orderRunge-kutta method is used.

5. Investigation into the interlaminar stresses

The composite materials are used in a wide range of industrialapplications and structures because of their special superiormechanical properties. Sandwich plate is one of the compositestructures that is used in variety of structures such as ship hull,containment building, aircraft fuselage, spacecraft fuselage andso on. They are made of different layers, usually three layers, twoface sheets and a core. Face sheets, by having high strength, areable to transfer axial forces, bending moment and in some cases,they have the ability of resisting against high temperature changes

while the core should sustain compression as well as having theability of transferring shear stresses. It is of the great importancethat prefect bonding between the layers in laminar compositesremain intact during the service life of the structure to preventdelamination which causes failure in composite and sandwichstructures. This is the most common type of damage for laminatedcomposite and sandwich plates, which usually occurs in the pro-duction process, or due to the impact forces [9]. Only if this is sat-isfied, the panel will perform on the appropriate level. However,this is not always satisfied, so delamination between the materiallayers often occurs.

The interlaminar stresses have outstanding effects on delamina-tion even in the case that they are much lower than the failurestrength. Delamination may cause deterioration of load carryingcapacity of laminated composite plate such that they cannot actfor their desired function anymore. Delamination is able to developin regions where the interlaminar shear is high and the out-of-plane compression is relatively low. It has been documented thatthe main reason of delamination is fast variation of interlaminarstresses in the vicinity of the free edge [8]. The illustration of inter-laminar stresses is provided in Fig. 3.

Taking look at the Fig. 3, it can be inferred that the interlaminarstresses are the out-of-plane stresses, rz, sxz, and syz, defined at theinterfaces between layers in a laminated composite material. Fromthe figure, it is clear that a tensile stress rz along an interfacewould tend to separate the layers along the interface, and theinterlaminar shear stresses would tend to shear apart the interfacein the corresponding directions. According to the explanationsgiven above, to conceive the importance of interlaminar stresses,in current research the investigation of interlaminar stressesbetween face sheets and core is taken into account.

5.1. Interlaminar stress equations

In order to analyze the interlaminar stresses of the laminatedcomposite plates, in this study, the Reddy’s higher order sheardeformation theory is employed. It assumes that the transverseshear strains have parabolic distribution across the plate thickness.The stress-strain relation can be stated as follow:

rxx ¼ �Q11exx þ �Q12eyy þ �Q16exy ð27aÞ

rxy ¼ �Q16exx þ �Q26eyy þ �Q66exy ð27bÞ

ryy ¼ �Q12exx þ �Q22eyy þ �Q26exy ð27cÞ

ryz ¼ �Q44eyz þ �Q55exz ð27dÞ

rxz ¼ �Q45eyz þ �Q55exz ð27eÞSubstituting displacement fields from Eq. (25) into Eq. (27) and

evaluating for the center of plate(x = a/2, y = b/2), result in equa-tions of interlaminar stresses as given hereunder:

rxx ¼X4i¼1

hxiFi ð28aÞ

ryy ¼X4i¼1

hyiFi ð28bÞ

rxy ¼X4i¼1

hxyiFi ð28cÞ

ryz ¼ hyz1F5 þ hyz2F6 ð28dÞ

Fig. 3. Illustration of interlaminar stresses on sandwich plate [8].

148 B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157

rxz ¼ hxz1F5 þ hxz2F6 ð28eÞ

6. Numerical results and discussion

6.1. Definition of the example problem

A structure subjected to dynamic loading may exhibit a differ-ent behavior compared to a structure loaded statically, especiallyif the applied load has a high peak value and is of short durationsuch as blast load [20,21]. This study considers, as an example, toinvestigate dynamic responses of a sandwich plate with FGM facesheets and a core of made of fiber reinforced concrete(FRC) restingon elastic foundation subjected to blast loads and thermal effectsdue to a specific charge of 5 kg.

FGM face sheets are made of a combination of ceramic andmetal. The material properties of top of FGM sheet is those of cera-mic while its lowest part, adjacent to the core, uses those of metal.Based on the assumption of considering thermal effects, the mod-ulus of elasticity and thermal expansion coefficient are function oftemperature change as expressed in Eq. (29) [10]:

PF ¼ P0ðP�1T�1 þ 1þ P1T þ P2T

2 þ P3T3Þ; ð29Þ

where T is the ambient temperature, P0, P�1, P1, P2 and P3 are thecoefficients of temperature, T(K), and are unique to the constituentmaterials. Table 1 shows the temperature-dependent coefficientsfor ceramic and metal by which the temperature-dependent mate-rial properties are calculated.

Each FGM face sheet has the thickness of 3 mm while the corehas the thickness of 30 mm. the volume fraction index, N, of FGMface sheet is 1. Temperature-dependent characteristics of FGM facesheets are defined by considering the temperature of T = 1000 K. Asthe characteristics of FGM plate change through the plate thick-ness, to achieve a precise model, we consider three plies for each

Table 1Temperature-dependent coefficients for FGM sheet materials [10].

Materials Parameter P0 P�1

Zirconia EF 244.27e9 0.000aF 12.766e�6 0.000

Ti-6Al-4V EF 122.56e9 0.000aF 7.5788e�6 0.000

FGM plate for which the material properties are defined. The mate-rial properties corresponding to the midline of thickness of eachply are considered as representative characteristics. To provide abetter understanding of considering three plies, an illustration ofsandwich plate layout is provided in Fig. 4.

Stiffness matrix elements are obtained by using the definedmaterial properties of sandwich plate plies. The material orienta-tion is set to zero i.e. h = 0 so that �Qij ¼ Qij. The FRC includingPolypropylene fibers with no surfactant is considered for the core.Table 2 shows the representative material properties correspond-ing to each layer and ply.

Table 3 shows the amounts for Qij which are obtained by usingthe data provided in Table 2 and Eqs. (8) and (9).

Equations of motion of the sandwich plate are evaluated for thecenter of the sandwich plate where x = a/2, y = c/2. The length ofplate is a = 1000 mm, while its width is c = 500 mm. The set of non-linear dynamic equations in the time domain is solved by usingforth order Runge-Kutta method and commercial mathematicssoftware Maple 14.

To calculate the blast wave pressure which is considered to beuniformly applied to the plate we need to calculate parametersof Eq. (1). The wave decay coefficient b is defined as follows [22]:

b ¼ z2 � 3:7zþ 4:2 ð30Þwhere z ¼ R=w1=3, w is the charge (explosive) weight and R is stand-off distance. In this study, the explosive weight of w = 5 kg andstandoff distance of R = 1.0 m are considered. Depending on thevalue of b, different pressure-time histories can be described [23].The correlation between the positive phase duration, t0þ , and R isexpressed as given in Eq. (31a):

log10t0þw1=3

� � �2:75þ 0:27log10

Rw1=3

� ð31aÞ

P1 P2 P3

�1.371e�3 1.214e�6 �3.681e�10�1.491e�3 1.006e�5 �6.778e�11

�4.586e�4 0.000 0.0006.638e�4 �3.147e�6 0.000

Table 2Representative material properties of sandwich plate.

Layer Ply NO. EF(GPa) aF(E � 6) qF(kg/m3) tF

Top sheet 1 107.73 99.20 3238.17 0.292 91.18 55.03 3714.50 0.293 74.63 10.85 4190.83 0.29

Core 4 12.00 10.00 1473.70 0.20

5 74.63 10.85 4190.83 0.29Bottom sheet 6 91.18 55.03 3714.50 0.29

7 107.73 99.20 3238.17 0.29

Fig. 4. Sandwich plate layout and ply midline for defining material properties.

Table 4Numerical values for coefficients of Eq. (26) for plate center.

Sub-index i ci di ei fi gi

1 0.000 0.000 0.000 0.000 0.0002 0.000 0.000 0.000 0.000 0.0003 0.000 0.000 0.000 0.000 0.0004 0.000 0.000 0.000 0.000 0.0005 0.000 0.000 �127.695 0.000 0.0006 0.000 �652.131 0.000 0.000 �1509.9337 0.000 0.000 0.000 0.000 0.0008 0.000 0.000 �6774.423 0.000 0.0009 0.000 0.000 �13548.846 0.000 0.00010 51.428 46.233 2.660E�7 33.078 33.07811 0.000 0.000 0.000 0.000 0.00012 – – 0.000 – –13 – – �1.018E�2 – –14 – – �1.000E�2 – –15 – – 0.000 – –16 – – 0.000 – –

B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157 149

The impulse (I) is computed using Eq. (31b), Ps0þ is calculatedfrom Eq. (31c) and t0þ can be obtained by applying Eq. (31d) [24]:

Is0þ ¼ 200w2=3

R2 ð31bÞ

Ps0þ ¼ 0:085w1=3

Rþ 0:3

w1=3

R

� �2

þ 0:8w1=3

R

� �3

ð31cÞ

t0þ ¼ 1:2ffiffiffiffiw6

p:ffiffiffiR

pð31dÞ

Having data given in Tables 2 and 3 as well as the numericalvalues of Eq. (31) the coefficients of Eq. (26) are calculated withrespect to the conditions of the example problem. It is appropriateto note that for the Winkler elastic foundation K = 10. The numer-ical values of coefficients are given in Table 4 as follows:

The interaction of the shock wave with a plate, imparts energyto the plate. The imposed energy is dissipated in the form of defor-mation [25]. Considering all aforementioned conditions of theexample problem and having corresponding coefficients, the

Table 3Stiffness matrix elements.

Stiffness element(GPa) Top sheetply number

1 2 3

Q11 117622.0 99552.4 81482.7Q12 34110.4 28870.2 23630.0Q22 117622.0 99552.4 81482.7Q16 0.0 0.0 0.0Q26 0.0 0.0 0.0Q66 41755.8 35341.1 28926.4Q44 41755.8 35341.1 28926.4Q45 0.0 0.0 0.0Q55 41755.8 35341.1 28926.4

plane-normal displacement at the center of plate, x = a/2, y = c/2,is to be obtained.

6.2. Solution to the problem by use of analytical and numericalmethods

Theoretical methods are still valuable for design purposes, par-ticularly for preliminary design, hazard assessments, security stud-ies and for investigations after accidents [26]. Therefore, to providedesigners and researchers a good insight into the way of use of thepresented approach an example problem is to be solved byemploying the method presented in this study and numericalmethod (FEM) by employing ABAQUS and guidelines given in

Core Bottom sheetply number

4 5 6 7

13101.9 81482.7 99552.4 117622.03799.5 23630.0 28870.2 34110.413101.9 13101.9 99552.4 117622.00.0 0.0 0.0 0.00.0 0.0 0.0 0.04651.2 4651.2 35341.1 41755.84651.2 4651.2 35341.1 41755.80.0 0.0 0.0 0.04651.2 4651.2 35341.1 41755.8

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.00050.0010.00150.0020.00250.0030.00350.0040.00450.0050.0055

w(m

m)

time(sec)

Analytical Numerical

Fig. 5. Comparative time history of sandwich plate obtained from analytical andnumerical methods.

150 B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157

[27]. To solve the problem by present analytical method, the fourthorder Rugne-Kutta method, which requires four evaluations ateach step, is used as it is superior and more accurate than secondorder Runge-Kutta method [28]. First, the presented method is val-idated by making comparison between the results obtained fromtwo methods. Thereafter, time histories of the sandwich plate sub-jected to blast load due to charge of 5 kg, obtained from both meth-ods, are given in a comparative graph. Finally in order to evaluatethe failure condition of sandwich plate the interlaminar stressesare to be found.

6.2.1. Evaluating the validity of suggested approachIn order to evaluate the validity of presented method of this

study, numerical method (FEM) is employed. A sandwich platemade of two FGM face sheets and a FRC core is considered to besubjected to blast load due to explosive charge of 5 kg. Aim is tofind the amount of maximum plane-normal displacement of sand-wich plate by using the presented method and FEM method usingABAQUS. In order to perform numerical method explicit method isused as it is more efficient than implicit method [29]. The compar-ison between results obtained from both approaches is made asgiven in Table 5. The time at which maximum displacement occursto the sandwich plate is given in second column of Table 5 which isindicated by occurrence time. The time step which is consideredfor the analysis is Dt = 0. 082 ms while the loading duration istd = 2.63 ms.

As can be seen from last column of Table 5 the amount of dis-crepancy between results obtained from both methods is as smallas (�4.73%) so the validity of the present approach is approved. toevaluate the error of Runge-Kutta method the one-step error esti-mation is employed. I this approach the error is obtained by sub-tracting the solutions obtained from 4th-order and 5th-orderRunge Kutta methods. The calculations of the error with respectto time step(h) is given in Table 6.

As can be seen from Table 6 the error is proportional to h5

which is agree with theory of Runge- Kutta method.

6.2.2. Time historyIn this part the time histories of sandwich plate subjected the

blast load due to burst of 5 kg charge are presented. To this aim,two approaches, the present analytical method and FEM throughABAQUS, are employed. Comparative graph including time histo-ries obtained from both analytical and numerical methods is illus-trated in Fig. 5 as follows:

As can be seen from time histories given in Fig. 5, line with solidcircles shows the results of the method which is suggested in this

Table 5Maximum plate displacement obtained by analytical method and FEM.

Method Occurrencetime (second)

Thickness(mm)

Charge(kg)

W(mm)Analytical

Error(%)

Analytical Numerical 0.000240.00032

66 5 0.25850.2463

�4.73

Table 6Calculation of the error of Runge Kutta Method.

Step size(h) Error

0.002 0.085420.001 0.036450.0005 0.007430.00025 0.001450.000125 0.000120.0000625 2.56253e�5

study while line with hollow circles shows the results obtainedfrom FEM approach. Taking look into the results it is observed thatFEM underestimates the dynamic responses of sandwich plate sub-jected to blast load.

6.2.3. Interlaminar stressIn this section, an attempt is to be made to obtain the interlam-

inar stresses of sandwich plate through the example problemwhich have been already defined in previous part. The stressesresulted from applied blast load due to charge of 5 kg on sandwichplate can be calculated by applying the stiffness elements, �Qii,which have been already calculated and given in Table 3, as wellas displacements and rotations, which are calculated by solvingthe Eqs. (26) and (28). Table 7 shows the numerical values of coef-ficients of Eq. (28) evaluated for the center of plate. It is worthy tonote that as the plate is symmetric with respect to Z-direction, thecoefficients of ply 1 & 7, 2 & 6, 3 & 4 are the same.

The numerical values of displacements and rotations whichhave been already obtained for plate center, x = a/2, y = c/2, aresubstituted into Eq. (28), so the stresses through the sandwichplate thickness are obtained. The stresses caused by applied loadare to be given as plot through the plate thickness. The plots illus-trate the changes in stresses through the thickness of plate withrespect to plane-normal direction. The variation of stress compo-nents in x and y directions,rx and ry; are plotted for orientationof 0� through the sandwich plate thickness as given in Figs. 6 and7, respectively.

As can be seen from the graphs provided in Figs. 6 and 7 thestresses through the FGM face sheet are higher than those in core.The stress at top face of the FGM sheet which is completely con-

Table 7Numerical values of coefficients of Eq. (28) for the point of X = 500 mm, Y = 250 mm.

Layer Ply Number Sub-index i FGM Face

hx hxy hyz hxz

FGM Sheet 1&7 1 �235,244 �68,221 0 0 02 0 0 0 0 03 �68,221 �235,244 0 – –4 0 0 0 – –

2&6 1 �199,105 �57,740 0 0 02 0 0 0 0 03 �57,740 �199,105 0 – –4 0 0 0 – –

3&5 1 �162,965 �47,260 0 0 02 0 0 0 0 03 �47,260 �162,965 0 – –4 0 0 0 – –

Core 4 1 �26,204 �7599 0 0 02 0 0 0 0 03 �7599 �26,204 0 – –4 0 0 0 – –

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80 90

σ 1(M

Pa)

Section orientation, θ

Fig. 8. Variation of r1 with respect to h.

-18-15-12

-9-6-30369

121518

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30

Plat

e th

ickn

ess(

mm

)

σy(MPa)

Fig. 7. Distribution of ry through the plate thickness (h ¼ 0� , t = 024 ms).

-18-15-12

-9-6-30369

121518

-120-100 -80 -60 -40 -20 0 20 40 60 80 100 120

Plat

e th

ickn

ess(

mm

)

σx(Mpa)

Fig. 6. Distribution of rx through the plate thickness (h ¼ 0� , t = 0.24 ms).

B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157 151

sists of ceramic is lower than the stress in lowest level of face sheetwhich is completely composed of metal. The highest stress valuecorresponds to the FGM plates at levels 15 mm and -15 mm onwhich the core and face sheets are contact. A transition state canbe observed from the face sheet to the core. This part is the mostcritical part to which material failure and delamination may occurand lead to failure of the sandwich structure. So, the need foremploying failure criterion comes up to investigate the stress con-ditions through the thickness of the plate. To evaluate the occur-rence of material failure, the Tsai-hill criterion is employed whileChang-Springer failure criterion is used to inspect the delamina-tion onset.

6.2.4. Tsai-hill criterionEvery material has certain strength, expressed in terms of stress

or strain, beyond which the load carrying capacity is destructed sothat the structure is failed. Both excessive stress and deformationcan cause structural failure. In case of excessive stress, the plasticdeformation or fracture makes structure too weak for service whilethe structure is not usable, in case of excessive deformation, due tomisfit of structural components such as rotation of shafts [30].

A criterion used to hypothesize the failure is recognized as fail-ure criterion while a theory behind a failure criterion is failure the-ory. Hill extended the Von Mises criterion for ductile anisotropicmaterial. Azzi-Tsai extended this equation to anisotropic fiber rein-forced composites. Tsai-Hill criterion can be stated in the form asEq. (32) represents [31]:

Table 8Calculations of LHS of Tsai-Hill criterion formula (Eq. (32)) for section (h ¼ 0�).

Layer ry(MPa) s (Mpa) Z(mm) h

FGM 1100.00 760.00 15.00 0.00Core 38.30 31.50 0.00FGM 1100.00 760.00 �15.00 0.00Core 38.30 31.50 0.00

r1

F1

� �2

þ r2

F2

� �2

� r1r2

F21

þ s12F6

� �2

¼ 1 ð32Þ

where F1 is yield stress in X-direction, F2 yield stress in Y-directionand F6 is shear strength. Failure occurs when LHS (Left-hand side) ofthe Tsai-Hill criterion, Eq. (32), is equal to or greater than one. Eval-uation of stresses through the sandwich plate thickness, to investi-gate the material failure at any level, gains the importance. To thisaim, we calculate the stress components for the layers which arejust next to the interface of core and face sheets (Z=15 mm &�15 mm), since it can be a critical section not only because ofdelamination but also regarding material failure. It is worthy tonote that the material properties which is considered for FGM sheetlayer are those of metal (Ti-6Al-4V). For the case of metal layer theyield stress is F1 ¼ F2 ¼ ry = 1100 MPa while the shear strength is760 MPa. The yield stress for the core is F1 ¼ F2 ¼ ry = 38.3 MPaand its shear strength is 31.5 MPa. Table 8 shows the parameterswhich are required for calculating the left-hand side of Eq. (32)for the section in the direction of h ¼ 0�.

As it can be seen from the last column of Table 8 the amountobtained for LHS of Eq. (32) is lower than 1, the criteria for the fail-ure given in the right hand side of Eq. (32), so the material failuredoes not occur at the aforementioned sections to which the maxi-mum stresses are formed. As the section orientation varies, thestress is changed. To investigate the failure condition, the needfor finding stress in different section orientation comes up. Thevariation of maximum stresses with respect to section orientationin X and Y directions are given in Figs. 8 and 9, respectively.

Taking look at the Figs. 8 and 9 it can be inferred that r1

decreases as the orientation angle, h, increases from 0�, for whichthe maximum stress of r1 = 104.38 MPa occurs, to 90� correspond-ing to minimum amount of stress r1 = 28.76 MPa. An inverse trendis observed for change in r2 with respect to change in h. Since, atno section, the stress exceeds the maximum stress value which isused in evaluation of Tsai-Hill criterion, no material failure isoccurred to the sandwich plate.

6.2.5. DelaminationAs already mentioned one of the major causes of composite and

sandwich structures failure is delamination which results in sepa-ration of the structure layers and finally leads to structural destruc-tion. In order to investigate the delamination occurrence at

r1(MPa) r2(MPa) s12(MPa) LHS

�104.38 �28.76 0.00 72.07E�4�12.84 �2.37 0.00 95.48E�3104.38 28.76 0.00 72.07E�412.84 2.37 0.00 95.48E�3

0

20

40

60

80

100

120

0 10 20 30 40 50 60 70 80 90

σ 2(M

Pa)

Section orientation, θ

Fig. 9. Variation of r2 with respect to h.

152 B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157

intersection of FGM face sheets and FRC core, the Chang-Springerfailure criterion is employed [32]. It predicts onset of delaminationby Eq. (33) as follows:

r33

Yt

� �2

þ s13S13¼23

� �2

þ s23S13¼23

� �2

¼ 1 ð33Þ

where Yt is tensile strength and S13¼23 is interlaminar shearstrength. In this study, the assumption of plane-stress, r33 ¼ 0;has been taken into account to derive the equations of motion. Atthe plate center the out of plane shear stresses vanish. Consideringthe values of r33; s13 and s23 the left hand side of the Eq. (33) is zeroso delamination does not occur at the center of plate. To investigatedelamination at the points far from the center the other point hav-ing the coordinates of X1 = 50 mm, Y1 = 50 mm is considered andthe corresponding stresses at interface of FGM face sheets and core,Z = 15 mm and Z = �15 mm, are calculated. The corresponding coef-ficients of Eq. (28) are given in Table 9.

It is appropriate to note that as the plate is symmetric in alldirections, the calculated stresses at symmetric points,X1 = 50 mm, Y1 = 450 mm, X1 = 950 mm, Y1 = 50 mm andX1 = 950 mm, Y1 = 450 mm, are the same with those of the pointat X1 = 50 mm, Y1 = 50 mm. Therefore, the same stresses happen

Table 9Numerical values of coefficients of Eq. (28) for the point of X = 50 mm, Y = 50 mm.

Layer Ply Number Sub-index i FGM Face

hx

FGM Sheet 1&7 1 21364.352 5.23E�43 1350.644 0.00

2&6 1 18082.252 4.42E�43 1143.154 0.00

3&5 1 14800.162 3.62E�43 935.664 0.00

Core 4 1 2379.772 5.82E�53 150.454 0.00

Table 10Calculations of LHS of Eq. (33).

Layer Coordinates (mm) Yt(MPa) S13 (Mpa) Z

FGM Sheet X = 50,Y = 50 1100.00 760.00 1Core 38.30 31.50FGM Sheet 1100.00 760.00 �Core 38.30 31.50

to the symmetric points and the results of investigation of delam-ination onset can be generalized for them.

Table 10 provides the calculated amounts of stress componentswhich are required for the Eq. (33) to evaluate delamination atinterface of core and face sheets. The amounts of left hand sideof Eq. (33) are calculated for FGM face sheets and core at levelsof Z = 15 mm and �15 mm and provided in Table 10 as well.

According to the results given in Table 10, since the amount ofLHS of Eq. (33) is smaller than 1, it is inferred that delaminationdoes not occur to the sandwich plate at interfaces of core and facesheets.

Considering results of Tables 8 and 10 it can be observed that nofailure, neither material failure nor delamination, occurs to thesandwich plate due to stresses which are caused by blast loadderived from burst of 5 kg charge.

7. Conclusions

In this manuscript, the higher order shear deformation theorywas employed together with Hamilton’s principal to obtain thegoverning differential equations of motion of sandwich plate rest-ing on elastic foundation subjected to blast load. Since during anexplosion event the ambient temperature elevates very fast, sothe effects of high temperature change were included into theobtained governing differential equations. The nonlinear dynamicequations were obtained by employing Navier solution to the gov-erning differential equations of motion. In order to evaluate thevalidity of the suggestedmethod of this study, an example problemwas defined through which the maximum plane-normal displace-ment of sandwich plate was found by employing the presentedmethod and numerical approach with ABAQUS. In this exampleproblem a sandwich plate consists of two FGM face sheets and acore made of fiber reinforced concrete was subjected to a blast loaddue to charge of 5 kg. It is worthy to note that the Runge-Kuttamethod was applied to the nonlinear dynamic equations to solvethe problem. Then, FEM method using ABAQUS was used to findthe maximum displacement. Thereafter the comparison was made

hy hxy hyz hxz

6196.66 0.00 1201.24 0.005.08E�4 0.00 0.00 2464.304657.37 0.00 – –0.00 7584.34 – –5243.85 0.00 1016.70 0.004.30E-4 0.00 0.00 2085.723941.89 0.00 – –0.00 6419.20 – –4292.05 0.00 832.16 0.003.52E�4 0.00 0.00 1707.153226.40 0.00 – –0.00 5254.06 – –

690.13 0.00 133.81 0.005.66E�5 0.00 0.00 274.50518.78 0.00 – –0.00 844.82 – –

(mm) r33 (MPa) s13 (MPa) s23 (MPa) LHS

5.00 0.00 4.46 0.49 3.48E�50.00 0.72 0.08 5.23E�4

15.00 0.00 4.46 0.49 3.48E�50.00 0.72 0.08 5.23E�4

B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157 153

between results of both methods. The small discrepancy betweenthe results showed that a good agreement was achieved so themethod presented in this study was validated and can be used inpractical application. Time histories of sandwich plate obtainedby method presented in this study and FEM method were plottedin a comparative graph. Thereafter the interlaminar stresses werefound as they might lead to delamination onset which is the mostcommon reason of composite and sandwich structures failure.Based on the higher order shear deformation theory the governingequations of interlaminar stresses were derived. Then the numeri-cal values of stresses of desired sandwich plate were calculated. Toevaluate the occurrence of failure to the sandwich plate, the Tsai-Hill failure criterion was employed for inspecting material failureand Chang-Springer failure criterion was applied to investigatethe delamination status. Results showed that no failure occurredto the considered sandwich plate.

Acknowledgment

This work is supported by 1) National Research Foundation ofKorea (NRF-2014R1A1A2056157) and 2) Korea Agency for Infras-tructure Technology Advancement (KAIA) Grant funded by theMinistry of Land, Infrastructure and Transport (Grant 13IFIP-C113546-01).

Appendix A. Coefficients for equations of motion-displacementsand rotations

@ is the number of layers and plies considered through thesandwich plate thickness.

c1 ¼X@i¼1

ð�Q16Þi4p2hi

accos

2pxa

sin2pyc

�ð�Q11Þi4p2hi

a2sin

2pxa

1� cos2pyc

� ��

þð�Q66Þi4p2hi

c2sin

2pxa

cos2pyc

þð�Q16Þi4p2hi

accos

2pxa

sin2pyc

c2 ¼X@i¼1

ð�Q12Þi4p2hi

acsin

2pxa

cos2pyc

�ð�Q26Þi4p2hi

c21� cos

2pxa

� �sin

2pyc

�

þð�Q66Þi4p2hi

acsin

2pxa

cos2pyc

þð�Q16Þi4p2hi

a2 cos2pxa

sin2pyc

c3 ¼X@i¼1

ð�Q16Þi8p3h2

i

3a2ccos

2pxa

sin2pyc

þ ð�Q26Þi8p3h2

i

3c3ð1� cos

2pxa

Þ"

� sin2pyc

� ð�Q66 þ �Q12Þi8p3h2

i

3ac2sin

2pxa

cos2pyc

� ð�Q11Þi8p3h2

i

3a3

�sin2pxa

1� cos2pyc

� �� ð�Q16Þi

16p3h2i

3ac2cos

2pxa

sin2pyc

#

c4 ¼X@i¼1

ð�Q16Þi8p3hi

ca2 sin2 2pxa

1� cos2pyc

� �sin

2pyc

�

þð�Q26Þi8p3hi

c31� cos

2pxa

� �2

cos2pyc

sin2pyc

þ ð�Q66Þi8p3hi

ac2sin

2pxa

1� cos2pxa

� �1� cos

2pyc

� �cos

2pyc

þ ð�Q11Þi8p3hi

a3cos

2pxa

sin2pxa

1� cos2pyc

� �2

þ ð�Q12Þi8p3hi

ac2sin2 2py

csin

2pxa

1� cos2pxa

� �

� ð�Q16Þi8p3hi

a2ccos

2pxa

1� cos2pxa

� �1� cos

2pyc

� �sin

2pyc

c5 ¼X@i¼1

ð�Q16Þi4p2h2

i

3accos

2pxa

sin2pyc

þ ð�Q66Þi2p2h2

i

3c2sin

2pxa

cos2pyc

"

�ð�Q11Þi2p2h2

i

3a2sin

2pxa

1� cos2pyc

� �#

c6 ¼X@i¼1

ð�Q26Þi2p2h2

i

3c21� cos

2pxa

� �cos

2pyc

þ ð�Q66 þ �Q12Þi2p2h2

i

3ac

"

� sin2pxa

cos2pyc

þ ð�Q16Þi2p2h2

i

3a2cos

2pxa

sin2pyc

#

c7 ¼X@i¼1

I0isin2pxa

ð1� cos2pyc

Þ�

c8 ¼X@i¼1

I1i � 4

3h2 I3i

� �sin

2pxa

1� cos2pyc

� ��

c9 ¼X@i¼1

8p3ah2

i

I3isin2pxa

1� cos2pyc

� �" #

c10 ¼X@i¼1

�Q11a11 þ �Q12a22� �

ihi

c11 ¼X@i¼1

�Q16a11 þ �Q26a22� �

ihi

d1 ¼X@i¼1

ð�Q12Þi4p2hi

accos

2pxa

sin2pyc

� ð�Q16Þi4p2hi

a2

�

� sin2pxa

1� cos2pyc

� �þ ð�Q26Þi

4p2hi

c2sin

2pxa

cos2pyc

þð�Q66Þi4p2hi

accos

2pxa

sin2pyc

d2 ¼X@i¼1

ð�Q26Þi4p2hi

acsin

2pxa

cos2pyc

� ð�Q22Þi4p2hi

c21� cos

2pxa

� ��

� sin2pyc

þ ð�Q26Þi4p2hi

acsin

2pxa

cos2pyc

þð�Q66Þi4p2hi

a2cos

2pxa

sin2pyc

d3 ¼X@i¼1

ð�Q12Þi8p3h2

i

3a2ccos

2pxa

sin2pyc

þ ð�Q22Þi8p3h2

i

3c31� cos

2pxa

� �"

� sin2pyc

� ð�Q26Þi16p3h2

i

3ac2sin

2pxa

cos2pyc

� ð�Q16Þi8p3h2

i

3a3

� sin2pxa

1� cos2pyc

� �� ð�Q66Þi

16p3h2i

3ac2cos

2pxa

sin2pyc

#

d4 ¼X@i¼1

ð�Q12Þi8p3hi

ca2 sin2 2pxa

1� cos2pyc

� �sin

2pyc

�

þ ð�Q22Þi8p3hi

c31� cos

2pxa

� �2

cos2pyc

sin2pyc

þ ð�Q26Þi8p3hi

ac2sin

2pxa

1� cos2pxa

� �1� cos

2pyc

� �cos

2pyc

þ ð�Q16Þi8p3hi

a3cos

2pxa

sin2pxa

1� cos2pyc

� �2

þ ð�Q26Þi8p3hi

ac2sin2 2py

csin

2pxa

1� cos2pxa

� �

�ð�Q66Þi8p3hi

a2ccos

2pxa

1� cos2pxa

� �1� cos

2pyc

� �sin

2pyc

154 B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157

d5 ¼X@i¼1

ð�Q12Þi2p2h2

i

3accos

2pxa

sin2pyc

þð�Q26Þi2p2h2

i

3c2sin

2pxa

cos2pyc

"

�ð�Q16Þi2p2h2

i

3a2sin

2pxa

1� cos2pyc

� ��ð�Q66Þi

4p2h2i

3accos

2pxa

sin2pyc

#

d6 ¼X@i¼1

ð�Q22Þi2p2h2

i

3c21� cos

2pxa

� �cos

2pyc

"

þð�Q26Þi4p2h2

i

3acsin

2pxa

cos2pyc

þ ð�Q66Þi2p2h2

i

3a2 cos2pxa

sin2pyc

#

d7 ¼X@i¼1

I0i 1� cos2pxa

� �sin

2pyc

�

d8 ¼X@i¼1

I1i � 4

3h2 I3i

� �1� cos

2pxa

� �sin

2pyc

�

d9 ¼X@i¼1

8p3bh2

i

I3i 1� cos2pxa

� �sin

2pyc

" #

d10 ¼X@i¼1

�Q12a11 þ �Q22a22� �

ihi

d11 ¼X@i¼1

�Q16a11 þ �Q26a22� �

ihi

e1 ¼X@i¼1

ðQ11Þi4p3hi

3a3 ð2hi � 3Þsin2 2pxa

1� cos2pyc

� �2"

þðQ16Þi4p3hi

3a2cð3� 2hiÞsin2pxa 5cos

2pxa

�2� �

sin2pyc

1� cos2pyc

� �

þ Q66 þ12Q12

� �i

8p3hi

3ac2ð3� 2hiÞcos2pxa 1� cos

2pxa

� �sin2 2py

c

þ Q26ð Þi4p3hi

3c3ð3�2hiÞsin2pxa 1� cos

2pxa

� �sin

2pyc

cos2pyc

þðQ66Þi8p3hi

3ac2ð3� 2hiÞsin2 2px

acos

2pyc

1� cos2pyc

� �

e2 ¼X@i¼1

Q66 þ12Q12

� �i

8p3hi

3a2cð3� 2hiÞsin2 2px

acos

2pyc

1� cos2pyc

� ��

þðQ16Þi4p3hi

3a3ð3� 2hiÞsin2pxa cos

2pxa

sin2pyc

1� cos2pyc

� �

þ ðQ26Þi4p3hi

3ac2ð3� 2hiÞsin2pxa 1� cos

2pxa

� �sin

2pyc

5cos2pyc

� 2� �

þðQ66Þi8p3hi

3a2cð3� 2hiÞcos2pxa 1� cos

2pxa

� �sin2 2py

c

þðQ22Þi4p3hi

3c3ð3� 2hiÞsin2pxa 1� cos

2pxa

� �sin

2pyc

1� cos2pyc

� �

e3 ¼X@i¼1

ðQ11Þip3h2

i

315a3ð32hi � 210Þsin2 2px

a1� cos

2pyc

� �2"

þðQ16Þi3p3h2

i

315a2cð210� 32hiÞsin2pxa cos

2pxa

sin2pyc

1� cos2pyc

� �

þ ðQ12 þ 2Q66Þip3h2

315ac2ð210� 32hiÞcos2pxa ð1� cos

2pxa

Þsin2 2pyc

þ ðQ26Þip3h2

315c3ð210� 32hiÞsin2pxa 1� cos

2pxa

� �sin

2pyc

cos2pyc

þ ðQ66Þi2p3h2

i

3ac2ð210� 32hiÞsin2 2px

acos

2pyc

1� cos2pyc

� �

þ ðQ16Þi2p3h2

315a2cð210� 32hiÞsin2pxa 1� cos

2pxa

� �sin

2pyc

� 1� cos2pyc

� �

e4 ¼X@i¼1

ðQ12 þ2Q66Þip3h2

i

315a2cð210�32hiÞsin2 2px

acos

2pyc

1� cos2pyc

� �"

þðQ16Þip3h2

i

315a3 ð210�32hiÞsin2pxa cos2pxa

sin2pyc

1� cos2pyc

� �

þðQ26Þi3p3h2

i

315ac2ð210�32hiÞsin2pxa 1� cos

2pxa

� �sin

2pyc

cos2pyc

þðQ66Þi2p3h2

i

315a2cð210�32hiÞcos2pxa 1� cos

2pxa

� �sin2 2py

c

þðQ22Þip3h2

i

315c3ð210�32hiÞ 1� cos

2pxa

� �2

sin2pyc

cos2pyc

þðQ26Þi2p3h2

i

315ac2ð210�32hiÞsin2pxa 1� cos

2pxa

� �cos

2pyc

� 1� cos2pyc

� �

e5 ¼X@i¼1

ð2Q45Þi92p2hi

15acsin

2pxa

sin2pyc

� ðQ55Þi92p2hi

15a2

�

� cos2pxa

ð1� cos2pyc

Þ þ ðQ44Þi92p2hi

15c21� cos

2pxa

� �cos

2pyc

e6 ¼X@i¼1

ðQ11Þi 8p4h2i

63a4 ð32hi � 21Þsin2 2pxa 1� cos 2py

c

� �2h

þðQ12Þi 8p4h2i63a2c2 ð32hi � 21Þsin2 2px

a cos 2pyc 1� cos 2py

c

� �þðQ16Þi 16p

4h2i63a3c ð32hi � 21Þsin 2px

a ð1� cos 2pxa Þsin 2py

c 1� cos 2pyc

� �þðQ66Þi 32p

4h2i63a2c2 ð32hi � 1Þcos 2px

a 1� cos 2pxa

� �sin2 2py

c

þðQ22Þi 8p4h2i

63c2 ð21� 32hiÞ 1� cos 2pxa

� �2sin2 2pyc

þðQ66Þi 16p4h2i

63a2c2 ð64hi � 1Þsin2 2pxa cos 2py

c 1� cos 2pyc

� �þðQ12Þi 8p4h2i

63a2c2 ð21� 32hiÞcos 2pxa 1� cos 2px

a

� �sin2 2py

c

þðQ26Þi 16p4h2i

3ac3 ð21� 32hiÞsin 2pxa ð1� cos 2px

a Þsin 2pyc 1� cos 2py

c

� �þðQ26Þi 8p

4h2i3ac3 ð64hi � 3Þsin 2px

a ð1� cos 2pxa Þsin 2py

c cos 2pyc

i

e7 ¼X@i¼1

ðQ11Þi 8p4hi

3a4 ð3� 2hiÞsin2 2pxa cos 2px

a ð1� cos 2pyc Þ3

h

þð2Q12Þi 8p4hi

3a2c2 ð3� 2hiÞsin2 2pxa sin2 2py

c 1� cos 2pxa

� �1� cos 2py

c

� �þðQ16Þi 16p

4hi3ac3 ð3� 2hiÞsin 2px

a cos 2pxa 1� cos 2px

a

� �1� cos 2py

c

� �2sin 2py

c

þðQ26Þi 16p4hi

3ac3 ð3� 2hiÞsin 2pxa 1� cos 2px

a

� �2sin3 2py

c

þðQ66Þi 16p4hi

3a2c2 ð3� 2hiÞcos 2pxa 1� cos 2px

a

� �2sin2 2pyc 1� cos 2py

c

� �þðQ22Þi 8p

4hi3c4 ð3� 2hiÞ 1� cos 2px

a

� �3sin2 2pyc cos 2py

c

þðQ26Þi 8p4hi

3ac3 ð3� 2hiÞsin 2pxa 1� cos 2px

a

� �2sin 2pyc cos 2py

c 1� cos 2pyc

� �þðQ16Þi 16p

4hi3a3c ð3� 2hiÞsin3 2px

a sin 2pyc 1� cos 2py

c

� �2þðQ26Þi 16p

4hi3ac3 ð3� 2hiÞsin 2px

a ð1� cos 2pxa Þ2sin 2py

c cos 2pyc ð1� cos 2py

c ÞþðQ66Þi 16p

4hi3a2c2 ð3� 2hiÞsin2 2px

a 1� cos 2pxa

� �cos 2py

c 1� cos 2pyc

� �2i

e8 ¼X@i¼1

ðQ55Þi46phi

15acos

2pxa

1� cos2pyc

� ��

þðQ45Þi46phi

15bsin

2pxa

sin2pyc

c

e9 ¼X@i¼1

ðQ44Þi46phi

15c1� cos

2pxa

� �cos

2pyc

�

þðQ45Þi46phi

15asin

2pxa

sin2pyc

B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157 155

e10 ¼X@i¼1

I0i 1� cos2pxa

� �1� cos

2pyc

� �� 32pI6i

9h4i

"

� 1asin

2pxa

ð1� cos2pyc

Þ þ 1bð1� cos

2pxa

Þsin2pyc

� �

e11 ¼X@i¼1

I3i8p3ah2

i

cos2pxa

ð1� cos2pyc

Þ" #

e12 ¼X@i¼1

I3i8p3bh2

i

1� cos2pxa

� �cos

2pyc

" #

e13 ¼X@i¼1

8pI4i3ah2

i

þ 32pI6i9ah2

i

!cos

2pxa

1� cos2pyc

� �" #

e14 ¼X@i¼1

8pI4i3ah2

i

þ 32pI6i9ah2

i

!1� cos

2pxa

� �cos

2pyc

" #

e15 ¼X@i¼1

phi

3að2hi � 3Þð�Q11a11 þ �Q12a22Þisin

2pxa

1� cos2pyc

� ��

þ2phi

3að2hi � 3Þð�Q16a11 þ �Q26a22Þi 1� cos

2pxa

� �sin

2pyc

e16 ¼X@i¼1

phi

3bð2hi � 3Þð�Q12a11 þ �Q22a22Þi 1� cos

2pxa

� �sin

2pyc

�

þ2phi

3að2hi � 3Þð�Q16a11 þ �Q26a22Þisin

2pxa

1� cos2pyc

� �

f 1 ¼X@i¼1

�ð�Q11Þi2p2h2

i

3a2sin

2pxa

1� cos2pyc

� �"

þð2�Q16Þi2p2h2

i

3accos

2pxa

sin2pyc

þ ð�Q66Þi2p2h2

i

3c2sin

2pxa

cos2pyc

#

f 2 ¼X@i¼1

ð�Q12 þ �Q66Þi2p2h2

i

3acsin

2pxa

cos2pyc

þ ð�Q16Þi2p2h2

i

3a2

"

� cos2pxa

sin2pyc

� ð�Q26Þi2p2h2

i

3c21� cos

2pxa

� �sin

2pyc

#

f 3 ¼X@i¼1

�ð�Q11Þi32p3h3

i

315a3sin

2pxa

1� cos2pyc

� �"

�ð�Q12 þ 2�Q66Þi32p3h3

i

315ab2 sin2pxa

cos2pyc

� ð�Q16Þi32p3h3

i

315a2c

� cos2pxa

sin2pyc

þ ð�Q26Þi32p3h3

i

315c31� cos

2pxa

� �sin

2pyc

þ ð�Q45Þi46phi

15cð1� cos

2pxa

Þ sin 2pyc

�ð�Q55Þi46phi

15asin

2pxa

1� cos2pyc

� �

f 4 ¼X@i¼1

�ð�Q11Þi4p3h2

i

3a3cos

2pxa

sin2pxa

1� cos2pyc

� �2"

þð�Q12Þi4p3h2

i

3ac2sin2 2py

csin

2pxa

1� cos2pyc

� �

�ð�Q16Þi4p3h2

i

3a2ccos

2pxa

ð1� cos2pxa

Þ sin 2pyc

1� cos2pyc

� �

þ ð�Q16Þi4p3h2

i

3a2csin2 2px

asin

2pyc

1� cos2pyc

� �

þ ð�Q26Þi4p3h2

i

3c31� cos

2pxa

� �2

sin2pyc

cos2pyc

þð�Q66Þi4p3h2

i

3ac2sin

2pxa

ð1� cos2pxa

Þ 1� cos2pyb

� �cos

2pyc

#

f 5 ¼X@i¼1

�ð�Q11Þi68p2h3

i

315a2sin

2pxa

1� cos2pyc

� �"

þ ð�Q16Þi136p2h3

i

315accos

2pxa

sin2pyc

þ ð�Q66Þi68p2h3

i

315c2sin

2pxa

cos2pyc

þð�Q55Þi23hi

15sin

2pxa

1� cos2pyc

� �

f 6 ¼X@i¼1

�Q12 þ �Q66� �

i

68p2h3i

315a2sin

2pxa

cos2pyc

þ ð�Q16Þi68p2h3

i

315ac

"

� cos2pxa

sin2pyc

þ ð�Q26Þi68p2h3

i

315c21� cos

2pxa

� �cos

2pyc

þð�Q45Þi23hi

151� cos

2pxa

� �sin

2pyc

f 7 ¼X@i¼1

I1i sin2pxa

1� cos2pyc

� ��

f 8 ¼X@i¼1

ðI2i � 8

3h2i

I4i þ 16

9h4i

I6iÞ sin 2pxa

ð1� cos2pyc

Þ" #

f 9 ¼X@i¼1

32p9ah4

i

I6i � 8p3ah2

i

I4i

!sin

2pxa

1� cos2pyc

� �" #

f 10 ¼X@i¼1

h2i

6ð�Q11a11 þ �Q12a22Þi

" #

f 11 ¼X@i¼1

h2i

6ð�Q16a11 þ �Q26a22Þi

" #

g1 ¼X@i¼1

ð�Q12 þ �Q66Þi2p2h2

i

3accos

2pxa

sin2pyc

þ ð�Q26Þi2p2h2

i

3c2sin

2pxa

"

� cos2pyc

� ð�Q16Þi2p2h2

i

3a2sin

2pxa

ð1� cos2pyc

Þ#

g2 ¼X@i¼1

ð2�Q26Þi2p2h2

i

3acsin

2pxa

cos2pyc

þ ð�Q66Þi2p2h2

i

3a2cos

2pxa

"

� sin2pyc

� ð�Q22Þi2p2h2

i

3c21� cos

2pxa

� �sin

2pyc

#

g3 ¼X@i¼1

ð�Q12 � 2�Q16Þi32p3h3

315a2ccos

2pxa

sin2pyc

"

þð�Q22Þi32p3h3

315c31� cos

2pxa

� �sin

2pyc

� ð3�Q26Þi32p3h3

315ac2sin

2pxa

� cos2pyc

� ð�Q16Þi32p3h3

315a3sin

2pxa

1� cos2pyc

� �

þ ð�Q44Þi46phi

15c1� cos

2pxa

� �sin

2pyc

þð�Q45Þi46phi

15asin

2pxa

1� cos2pyc

� �

156 B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157

g4 ¼X@i¼1

ð�Q12Þi4p3h2

i

3a2csin2 2px

asin

2pyb

1� cos2pyc

� �"

þ ð�Q22Þi4p3h2

i

3c31� cos

2pxa

� �2

cos2pyc

sin2pyc

þ ð�Q26Þi4p3h2

i

3ac2sin

2pxa

ð1� cos2pxa

Þ 1� cos2pyc

� �cos

2pyc

þ ð�Q16Þi4p3h2

i

3a3 sin2pxa

cos2pxa

1� cos2pyc

� �2

þ ð�Q26Þi4p3h2

i

3ac2sin

2pxa

1� cos2pxa

� �sin2 2py

c

�ð�Q66Þi4p3h2

i

3a2ccos

2pxa

1� cos2pxa

� �sin

2pyc

ð1� cos2pyc

Þ#

g5 ¼X@i¼1

ð�Q45Þi23hi

15sin

2pxa

1� cos2pyc

� �þ ð�Q12 þ �Q66Þi

68p2h3i

315ac

"

� cos2pxa

sin2pyc

þ ð�Q26Þi68p2h3

i

315c2sin

2pxa

cos2pyc

�ð�Q16Þi68p2h3

i

315a2sin

2pxa

1� cos2pyc

� �#

g6 ¼X@i¼1

ð�Q44Þi23hi

151� cos

2pxa

� �sin

2pyc

þ ð2�Q26Þi68p2h3

i

315ac

"

� sin2pxa

cos2pyc

þ ð�Q66Þi68p2h3

i

315a2cos

2pxa

sin2pyc

þð�Q22Þi68p2h3

i

315a21� cos

2pxa

� �cos

2pyc

#

g7 ¼X@i¼1

I1i 1� cos2pxa

� �sin

2pyc

�

g8 ¼X@i¼1

ðI2i � 8

3h2i

I4i þ 16

9h4i

I6iÞ 1� cos2pxa

� �sin

2pyc

" #

g9 ¼X@i¼1

32p9bh4

i

I6i � 8p3bh2

i

I4i

!1� cos

2pxa

� �sin

2pyc

" #

g10 ¼X@i¼1

h2i

6ð�Q12a11 þ �Q22a22Þi

" #

g11 ¼X@i¼1

h2i

6ð�Q16a11 þ �Q26a22Þi

" #

Appendix B. Coefficients for equations of motion: Interlaminarstress

hx1 ¼ �Q11 cos2pxa

1� cos2pyc

� �

F1 ¼ 2pa

U þ 2pZa

Wx � 4Z3

3h2

2paWx þ 4p2

a2W

� �

hx2 ¼�Q11

a2sin2 2px

a1� cos

2pyc

� �2

þ�Q12

b2 1� cos2pxa

� �2

sin2 2pyc

F2 ¼ 2p2W2

hx3 ¼ �Q12ð1� cos2pxa

Þ cos 2pyc

F3 ¼ 2pc

V þ 2pZc

Wy � 4Z3

3h2

2pcWy þ 4p2

c2W

� �

hx4 ¼ �Q16 sin2pxa

sin2pyc

F4 ¼ 2pc

U þ 2pa

V þ 4p2

acW2ð1� cos

2pxa

Þ 1� cos2pyc

� �

þ 2pZ Wx

cþWy

a

� �� 8pZ3

3h2

Wx

cþWy

aþ 4pW

ac

� �

hy1 ¼ �Q12 cos2pxa

1� cos2pyc

� �

hy2 ¼�Q12

a2sin2 2px

a1� cos

2pyc

� �2

þ�Q22

c21� cos

2pxa

� �2

sin2 2pyc

hy3 ¼ �Q22 1� cos2pxa

� �cos

2pyc

hy4 ¼ �Q26 sin2pxa

sin2pyc

hxy1 ¼ �Q16 cos2pxa

1� cos2pyc

� �

hxy2 ¼�Q16

a2sin2 2px

a1� cos

2pyc

� �2

þ�Q26

c21� cos

2pxa

� �2

sin2 2pyc

hxy3 ¼ �Q26 1� cos2pxa

� �cos

2pyc

hxy4 ¼ �Q66 sin2pxa

sin2pyc

hyz1 ¼ �Q44 1� cos2pxa

� �sin

2pyc

hyz2 ¼ �Q45 sin2pxa

1� cos2pyc

� �

hxz1 ¼ �Q45 1� cos2pxa

� �sin

2pyc

hxz2 ¼ �Q55 sin2pxa

1� cos2pyc

� �

F5 ¼ Wy þ 2pc

W � 4Z2

h2 Wy þ 2pc

W� �" #

F6 ¼ Wx þ 2pa

W � 4Z2

h2 Wx þ 2pa

W� �" #

B. Mohammadzadeh, H.C. Noh / Composite Structures 174 (2017) 142–157 157

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