DSpace at VNU: Vibration and nonlinear dynamic response of imperfect three-phase polymer nanocomposite panel resting on elastic foundations under hydrodynamic loads

Vibration and nonlinear dynamic response of imperfect three-phasepolymer nanocomposite panel resting on elastic foundations under hydrodynamic loads a Vietnam National University, Hanoi,

Trang 1

Vibration and nonlinear dynamic response of imperfect three-phase

polymer nanocomposite panel resting on elastic foundations under

hydrodynamic loads

a

Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam

b

School of Mechanical & Automotive Engineering, Kingston University, Roehampton Vale, Friars Avenue, London, SW15 3DW, UK

c

Nha Trang University, 44 Hon Ro, Nha Trang, Khanh Hoa, Viet Nam

a r t i c l e i n f o

Article history:

Available online 11 May 2015

Keywords:

Nonlinear dynamic

Vibration

Laminated three-phase polymer

nanocomposite panel

Hydrodynamics loads

Imperfection

Elastic foundations

a b s t r a c t

An investigation on the nonlinear dynamic response and vibration of the imperfect laminated three-phase polymer nanocomposite panel resting on elastic foundations and subjected to hydrodynamic loads is presented in this paper The formulations are based on the classical shell theory and stress func-tion taking into account geometrical nonlinearity, initial geometrical imperfecfunc-tion and Pasternak type elastic foundation Numerical results for dynamic response and vibration of the three-phase polymer composite panel are obtained by Runge–Kutta method The inﬂuences of ﬁbers and particles, material and geometrical properties, foundation stiffness, imperfection and hydrodynamic loads on the nonlinear dynamic response and nonlinear vibration of the three-phase composite panel are discussed in detail

1 Introduction

Currently, composite materials have become indispensable in

several applications, such as high-performance structures in many

ﬁelds of civil, marine and aerospace engineering, among others

The mechanical behaviors of composite structures, such as

bend-ing, vibration, stability, bucklbend-ing, etc., has attracted attention of

many researchers Rango et al [1]presented the formulation of

an enriched macro element suitable to analyze the free vibration

response of composite plate assemblies Bodaghi et al.[2]

investi-gated thermo-mechanical analysis of rectangular shape adaptive

composite plates with surface bonded shape memory alloy

rib-bons Sahoo and Singh[3]used a new trigonometric zigzag theory

to research the analysis of laminated composite and sandwich

plates Samadpour et al [4] studied nonlinear free vibration of

thermally buckled sandwich plate with embedded pre-strained

shape memory alloy ﬁbers in temperature dependent laminated

composite face sheets Moleiro et al.[5]provided an assessment

of layerwise mixed models using least-squares formulation for

the coupled electromechanical static analysis of multilayered

plates Heydarpour et al.[6]examined the inﬂuences of centrifugal

and Coriolis forces on the free vibration behavior of rotating carbon

nanotube reinforced composite truncated conical shells Lopatin and Morozov[7]considered free vibrations of a cantilever compos-ite circular cylindrical shell Burgueño et al [8] presented approaches for modifying and controlling the elastic response of axially compressed laminated composite cylindrical shells in the far postbuckling regime The vibration and damping characteristics

of free–free composite sandwich cylindrical shell with pyramidal truss-like cores have been conducted by Yang et al.[9]using the Rayleigh–Ritz model and ﬁnite element method

Three-phase composite is a material consisted of matrix, the reinforced ﬁbers and particles which have been investigated by Vanin and Duc[10] Shen et al.[11]] analyzed a coated inclusion

of arbitrary shape embedded in a three-phase composite plate sub-jected to anti-plane mechanical and in-plane electrical loadings Lin et al.[12]presented a solution of magnetoelastic stresses on

a three-phase composite cylinder subjected to a remote uniform magnetic induction Wu et al.[13] developed an effective model

to bound the effective magnetic permeability of three-phase com-posites with coated spherical inclusions There are several claims

on the deflection and the creep for the three-phase composite lam-inates in the bending state[14] These findings have shown that optimal three-phase composite can be obtained by controlling the volume ratios of fiber and particles Afonso and Ranalli [15]

introduced a new general model to calculate the elastic properties

of three-phase composites by means of closed-form analytical solutions is presented Andrianov et al [16] analyzed the

http://dx.doi.org/10.1016/j.compstruct.2015.05.009

⇑ Corresponding author.

E-mail address: ducnd@vnu.edu.vn (N.D Duc).

Contents lists available atScienceDirect

Composite Structures

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 / c o m p s t r u c t

Trang 2

facturing in Vietnam The formulations are based on the classical

shell theory and stress function taking into account geometrical

nonlinearity, initial geometrical imperfection and Pasternak type

elastic foundation Numerical results for dynamic response and

vibration of the three-phase polymer composite panel are obtained

by Runge–Kutta method

It is noted that the present paper is improvement and

supple-ment of the ideas in proceeding paper which we presented at the

Third International Conference on Engineering Mechanics and

Automation[22](ICEMA 3-2014, Hanoi, October-2014), including

SEM structures images of two-phase 2D composite (glass ﬁbers,

polymer matrix) and the three-phase 2Dm composite (glass ﬁbers,

titanium oxide particles and polymer matrix) in order to improve

the paper more convinced

2 Determination of the elastic modules of three-phase

composite

In this paper, the algorithm which is successfully applied in Ref

has been used According to this algorithm, the elastic modules of

3-phase composites are estimated using two theoretical models

of the 2-phase composite consecutively: nDm = Om + nD[17,18]

This paper considers 3-phase composite reinforced with particles

and unidirectional ﬁbers, so the problem’s model will be:

1Dm = Om + 1D Firstly, the modules of the effective matrix Om

which is called ‘‘effective modules’’ are calculated In this step,

the effective matrix consists of the original matrix and added

parti-cles It is considered to be homogeneous, isotropic and have two

elastic modules The next step is estimating the elastic modules

for a composite material consists of the effective matrix and

unidi-rectional reinforced ﬁbers

Assuming that all the component phases (matrix, ﬁber and

par-ticles) are homogeneous and isotropic, we will use

Em;Ea;Ec;mm;ma;mc;wm;wa;wc to denote Young’s modulus and

Poisson ratio and volume fraction for the matrix, ﬁber and

parti-cles, respectively Following[17,18], the modules for the effective

composite can be obtained as below

G ¼ Gm

1 wcð7 5mmÞH

K ¼ Km

1 þ 4wcGmL 3Kð mÞ1

where

L ¼Kc Km

Kcþ4G m

3

; H ¼ Gm=Gc 1

8 10mmþ 7 5ð mmÞG m

G c

E; mcan be calculate from (G; KÞ as below

E ¼ 9K G

3K þ G; m¼3K 2G

The elastic moduli for three-phase composite reinforced with

unidirectional ﬁber are chosen to be calculated using Vanin’s

formulas[20]

G12¼ G

1 waþ 1 þ wð aÞG

a

;

G23¼ G vþ waþ 1 wð aÞG

a

1 wa

ð Þvþ 1 þð vwaÞG

a

;m23

E22

¼ m2 21

E11

þ 1 8G 2

1 wa

ð Þx þ 1 þ wð axÞG

a

x þ waþ 1 wð aÞG

a

"

2 1 wð aÞ x 1ð Þ þ xð a 1Þ x 1 þ 2wð aÞ

G

a

2 waþ xwaþ 1 wð aÞ xð a 1ÞG

a

#

;

m21¼ m ðvþ 1Þ ðmmaÞwa

2 waþvwaþ 1 wð aÞva 1 G

a

in which

To verify the validity of these equations, three-phase composite polymer made of polyester AKAVINA (made in Vietnam), glass ﬁbers (made in Korea) and titanium oxide (made in Australia) with the properties shown inTable 1was investigated[17,18]

By using the SEM instrumentation at the Laboratory for Micro-Nano Technology, University of Engineering and Technology, Vietnam National University, Hanoi,Figs 1 and 2show the images of fabricated samples of composite structures which are made in the Institute of Ship building, Nha Trang University.Fig 1

illustrates a SEM image of 2Dm composite polymer two-phase material (glass ﬁbers volume fraction of 25% without particles)

three-phase material (glass fibers volume fraction of 25% and Titanium oxide particles volume fraction of 3%) Obviously, when the particles are doped, the air cavities significantly reduced and the material is finer In other words, particles enhance the stiffness and the penetration resistance of the materials

3 Governing equations Consider a three-phase composite panel subjected to hydrody-namic loads: hydrodyhydrody-namic lift q1and drag q2as shown inFig 3 The panel is referred to a Cartesian coordinate system x; y; z, where

xy is the mid-plane of the panel and z is the thickness coordinator,

h=2 6 z 6 h=2 The radii of curvatures, length, width and total thickness of the panel are R; a; b and h, respectively

In this study, we assumed that the panel is thin, so the classical laminated shell theory (CLST) is used to establish governing equa-tions and determine the nonlinear response of composite panels In the case of thick panel, we must use higher-order shear deforma-tion theories By choosing of accurate theories can refer to[23] Taking into account the von Karman nonlinearity, the strain– displacement relations are

Table 1 Properties of the component phases for three-phase composite.

Component phase Young modulus E Poisson ratiom

Matrix polyester AKAVINA (Vietnam) 1,43 GPa 0.345

Titanium oxide TiO 2 (Australia) 5,58 GPa 0.20

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ey

cxy

0

B

1

C

A ¼

e0

x

e0

y

c0

xy

0

B

1

C

A þ z

kx

ky

kxy

0 B

1 C

where

e0

x

e0

y

c0

xy

0

B

1

C

A ¼

u;xþ w2

;x=2

v;y w=R þ w2

;y=2

u;yþv;xþ w;xw;y

0

B

1 C A;

kx

ky

kxy

0 B

1 C

A ¼

w;xx

w;yy

2w;xy

0 B

1 C A; ð8Þ

in which u;vare the displacement components along the x; y

direc-tions, respectively

Hooke law for a laminated composite panel is deﬁned as

rx

ry

rxy

0 B

1 C

k

¼

Q0

11 Q0

12 Q0 16

Q0

12 Q0

22 Q0 26

Q0

16 Q0

26 Q0 66

0 B

1 C

k

ex

ey

cxy

0 B

1 C

k

in which k is the number of layers and

Q011¼ Q11cos4

hþ Q22sin4hþ 2ðQ12þ 2Q66Þsin2hcos2

h;

Q012¼ Q12ðcos4

hþ sin4hÞ þ ðQ11þ Q22 4Q66Þ sin2hcos2

h;

Q0

12¼ Q12ðcos4

hþ sin4hÞ þ ðQ11þ Q22 4Q66Þ sin2hcos2

h;

Q0

16¼ ðQ12 Q22þ 2Q66Þsin3hcosh þ ðQ11 Q12 2Q66Þsin hcos3h;

Q0

22¼ Q11sin4hþ Q22cos4

hþ 2ðQ12þ 2Q66Þsin2hcos2

h;

Q0

26¼ ðQ11 Q12 2Q66Þsin3hcosh þ ðQ12 Q22þ 2Q66Þsin hcos3h;

Q0

66¼ Q66ðsin4hþ cos4hÞ þ Q½ 11þ Q22 2ðQ12þ Q66Þ sin2hcos2h;

ð10Þ

and

Q11¼ E11

1 E 22

E 11m2 12

¼ E11

1 m12m21

; Q22¼ E22

1 E 22

E 11m2 12

¼E22

E11

Q11;

Q12¼ E11

1 E22

E 11m2 12

¼m12

Q22

; Q66¼ G12;

ð11Þ

where h is the angle between the ﬁber direction and the coordinate system The force and moment resultants of the laminated compos-ite panels are determined by

Ni¼Xn k¼1

Rh k

hk1½ ri kdz; i ¼ x; y; xy;

Mi¼Xn k¼1

Rhk

hk1z½ rikdz; i ¼ x; y; xy:

ð12Þ

Substitution of Eq.(7)into Eq.(9)and the result into Eq.(12)

give the constitutive relations as

Nx;Ny;Nxy

¼ ðA11;A12;A16Þe0

xþðA12;A22;A26Þe0

yþðA16;A26;A66Þc0

xy

þðB11;B12;B16ÞkxþðB12;B22;B26ÞkyþðB16;B26;B66Þkxy;

Mx;My;Mxy

¼ ðB11;B12;B16Þe0

xþðB12;B22;B26Þe0

yþðB16;B26;B66Þc0

xy

þðD11;D12;D16ÞkxþðD12;D22;D26ÞkyþðD16;D26;D66Þkxy;

ð13Þ

Fig 1 SEM image of 2Dm composite two-phase material (ﬁbers volume fraction is

25% without particles).

Fig 2 SEM image of 2Dm composite three-phase material (ﬁbers volume fraction

is 25% and particles volume fraction is 3%).

Fig 3 Geometry and coordinate system of three-phase composite panels on elastic foundations.

Table 2 The dependency of hydrodynamic lift and drag on the velocities.

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Aij¼Xn

k¼1

ðQ0ijÞkðhk hk1Þ; i; j ¼ 1; 2; 6;

Bij¼1

2

Xn

k¼1

ðQ0ijÞkðh2k h2k1Þ i; j ¼ 1; 2; 6;

Dij¼1

3

Xn

k¼1

ðQ0ijÞkðh3k h3k1Þ i; j ¼ 1; 2; 6:

ð14Þ

The nonlinear motion equation of the composite panels based

on CLST with the Volmir’s assumption [19], u << w; v<<w;

q1@ 2 u

@t 2! 0; q1@ 2v

@t 2! 0 are given by

Mx;xxþ 2Mxy;xyþ My;yyþ Nxw;xxþ 2Nxyw;xyþ Nyw;yy

þq1þ q2 k1w þ k2r2w þNy

R ¼q1@

2 w

@t 2

ð15cÞ

whereq1¼qh withqis the mass density of composite panels and

q1;q2are hydrodynamic lift and drag forces which are

experimen-tally determined and they depend on the velocities according to

the Matveev’s formulas These forces for the ship with length of

16.4 m, width of 3.4 m and volume of occupied water 12000 kg are tabulated inTable 2 [21]

Calculated from Eq.(13)

e0

x¼ A11Nxþ A12Nyþ A16Nxy B11kx B12ky B16kxy;

e0

y¼ A12Nxþ A22Nyþ A26Nxy B

21kx B

22ky B

26kxy;

e0

xy¼ A16Nxþ A26Nyþ A66Nxy B16kx B26ky B66kxy;

ð16Þ

where

Fig 4 Effects of particles volume fraction w c on the dynamic response of

three-phase polymer composite panel.

Fig 5 Effects of ﬁber volume fraction w a on the dynamic response of three-phase polymer composite panel.

Fig 6 Effect of b=a ratio on nonlinear dynamic response of three-phase polymer composite panel.

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11¼A22A66 A

2

26

D ; A

12¼A16A26 A12A66

D ; A

16¼A12A26 A22A16

D ;

A22¼A11A66 A

2

16

D ; A26¼A12A16 A11A26

D ; A66¼A11A22 A

2 12

D ;

A22¼A11A66 A

2

16

D ; A26¼A12A16 A11A26

D ; A66¼A11A22 A

2 12

D ;

D¼ A11A22A66 A11A226þ 2A12A16A26 A212A66 A216A22;

B11¼ A11B11þ A12 B12þ A16B16;B12¼ A11 B12þ A12B22þ A16B26;

B

16¼ A11B16þ A12 B26þ A16B66;B

21¼ A12 B11þ A22B12þ A26B16;

B

22¼ A12B12þ A22 B22þ A26B26;B

26¼ A12 B16þ A22B26þ A26B66;

B

61¼ A16B11þ A26 B12þ A66B16;B

62¼ A16 B12þ A26B22þ A66B26;

B

66¼ A16B16þ A26 B26þ A66B66:

ð17Þ

Substituting once again Eq.(16) into the expression of Mijin

(13), then Mijinto the Eq.(15c)leads to

Nx;xþ Nxy;y¼ 0;

Nxy;xþ Ny;y¼ 0;

P1f;xxxxþ P2f;yyyyþ P3w;xxyyþ P4w;xxxyþ P5w;xyyyþ P6w;xxxxþ P7w;yyyy

þ P8w;xxyyþ P9w;xxxyþ P10w;xyyyþ Nxw;xxþ 2Nxyw;xyþ Nyw;yyþ q1

þ q2 k1w þ k2r2w þNy

R ¼q1@

2

w

where

P1¼ B21; P2¼ B12; P3¼ B11þ B22 2B66; P4¼ 2B26 B61;

P5¼ 2B16 B62; P6¼ B11B11þ B12B21þ B16B61;P7¼ B12B12

þ B22B22þ B26B62;

P8¼ B11B

12þ B12B

22þ B16B

62þ B12B

11þ B22B

21þ B26B 61

þ 4B16B

16þ 4B26B

26þ 4B66B

66;

P9¼ 2ðB11B

16þ B12B

26þ B16B

66þ B16B

11þ B26B

21þ B66B

61Þ;

P10¼ 2ðB12B

16þ B22B

26þ B26B

66þ B16B

12þ B26B

22þ B66B

62Þ: ð19Þ

f ðx; yÞ is stress function deﬁned by

Nx¼ f;yy; Ny¼ f;xx; Nxy¼ f;xy ð20Þ

For an imperfect laminated composite panel, Eq.(18)are modiﬁed to

P1f;xxxxþ P2f;yyyyþ P3w;xxyyþ P4w;xxxyþ P5w;xyyyþ P6w;xxxx

þ P7w;yyyyþ P8w;xxyyþ P9w;xxxyþ þP10w;xyyyþ f;yy w;xxþ w

;xx

2f;xy w;xyþ w

;xy

þ f;xx w;yyþ w

;yy

þ q1þ q2 k1w

þ k2r2w þNy

R ¼q1@

2w

in which wðx; yÞ is a known function representing initial small imperfection of the panel

The geometrical compatibility equation for an imperfect com-posite panel is written as

e0 x;yyþe0 y;xxc0 xy;xy¼ w2

;xy w;xxw;yyþ 2w;xyw

;xy w;xxw

;yy

w;yyw

;xxw;xx

From the constitutive relations(16)in conjunction with Eq.(20)one can write

e0

x¼ A11f;yyþ A12f;xx A16f;xy B

11kx B

12ky B

16kxy;

e0

y¼ A12f;yyþ A22f;xx A26f;xy B

21kx B

22ky B

26kxy;

e0

xy¼ A16f;yyþ A26f;xx A66f;xy B16kx B26ky B66kxy;

ð23Þ

Setting Eq.(23)into Eq.(22)gives the compatibility equation of

an imperfect composite panel as

A22f;xxxxþ E1f;xxyyþ A11f;yyyy 2A26f;xxxy 2A16f;xyyyþ B21w;xxxx

þ B12w;yyyyþ E2w;xxyyþ E3w;xxxyþ E4w;xyyy

fleftð w2

;xy w;xxw;yyþ 2w;xyw

;xy w;xxw

;yy w;yyw

;xxw;xx

R Þ ¼ 0; ð24Þ

where

Fig 7 Effect of b=h ratio on nonlinear dynamic response of three-phase polymer

composite panel.

Fig 8 Effect of R=h ratio on nonlinear dynamic response of three-phase polymer composite panel.

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E1¼ 2A12þ A66; E2¼ B11þ B22 2B66;

E3¼ 2B26 B61; E4¼ 2B16þ B62: ð25Þ

w and f and they can be used to investigate the stability of thin

composite panels on elastic foundations subjected to

hydrody-namic loads

4 Nonlinear dynamical analysis

A three-phase composite panel considered in this paper is assumed to be simply supported and subjected to lift q1, drag q2 forces and axial compression of intensities Pxand Pyrespectively

at its cross section Thus the boundary conditions are

Fig 9 Effect of the linear Winkler foundation on nonlinear dynamic response of

Fig 10 Effect of the Pasternak foundation on nonlinear dynamic response of

Fig 11 Nonlinear dynamic responses of three-phase polymer composite panel with different velocities.

Fig 12 Effect of pre-loaded axial compression P y on nonlinear response of three-phase polymer composite panel.

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w ¼ Nxy¼ Mx¼ 0; Nx¼ Pxh at x ¼ 0; a;

w ¼ Nxy¼ My¼ 0; Ny¼ Pyh at y ¼ 0; b: ð26Þ

The approximate solutions of w; w and f satisfying boundary

condition(26)are assumed to be[17,18]

w; w

ð Þ ¼ W;ð lhÞ sin kmx sin dny; ð27aÞ

f ¼ A1cos 2kmx þ A2cos 2dny þ A3sin kmx sin dny

þA4coskmxcosdny þ1Nx0y2þ1Ny0x2; ð27bÞ

km¼ mp=a; dn¼ np=b; W is amplitude of the deﬂection andlis

imperfection parameter The coefﬁcients Aiði ¼ 1 4Þ are

deter-mined by substitution of Eqs.(27a) and (27b)into Eq.(24)as

A1¼ 1 32A22

d2

k2mWðW þ 2lhÞ; A2¼ 1

32A11

k2m

d2nWðW þ 2lhÞ;

A3¼ðF2F4 F1F3Þ

F22 F21 W; A4¼

ðF2F3 F1F4Þ

F22 F21 W;

ð28Þ

where Fiði ¼ 1 4Þ are given inAppendix A Subsequently, substitution of Eqs.(27a) and (27b)into Eq.(21)

and applying the Galerkin procedure for the resulting equation yield

ab

4½P1

ðF2F4 F1F3Þ

F2

F2 k

4

mþ P2

ðF2F4 F1F3Þ

F2

F2 d

4

þ P3

ðF2F4 F1F3Þ

F2

F2 k

2

md2 P4

ðF2F3 F1F4Þ

F2

F2 P5

ðF2F3 F1F4Þ

F2

þ P6k4mþ P7d4þ P8k2md2ðF2F4 F1F3Þ

F2

k2m

R k1

k2ðk2mþ d2ÞW 2

3kmdn P1

1

A22þ P2

1

A11

1 6RA22

dn

km

W W þ 2ð lhÞ

ab 64

1

A 22

d4þ 1

A 11

k4m

W W þð lhÞ W þ 2ð lhÞ

þ8 3

ðF2F4 F1F3Þ

F2

F2 kmdnW W þð lhÞ þabh

4 Pxk

2

mþ Pyd2

W þlh

þ4ðq1þ q2Þ

kmdn

4h

kmdn

Py

R¼

abq1

4

@2W

where m; n are odd numbers This basic equation is used to investi-gate the nonlinear dynamic response of three-phase polymer com-posite panels under hydrodynamic loads The initial conditions are assumed as Wð0Þ ¼ 0; Wð0Þ ¼ 0 The nonlinear Eq (29) can be solved by the Runge–Kutta method

From Eq.(29), the fundamental frequencies of a perfect panel can be determined approximately by an explicit expression

xmn¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðb1þ b2Þ

q1

s

where biði ¼ 1; 2Þ are given inAppendix A

Fig 13 Effect of imperfection parameterlon nonlinear dynamic response of

Trang 8

density of the panel isq¼ 1550 kg=m.

vol-ume fraction and elastic foundations on natural frequencies of the

three-phase composite polymer panel It can be seen that the value

of the natural oscillation frequency increases when the values k1

and k2 increase Furthermore, the Pasternak elastic foundation

inﬂuences on the natural oscillation frequency larger than the

Winkler foundation The natural frequencies of the panels observed

to be dependent on the particles volume fraction, ﬁber volume

frac-tion, they decreases when increasing the particles volume fraction

wcand ﬁber volume fraction waand the effect of ﬁber to natural

fre-quency is stronger than particle

volume fraction on dynamic response of three-phase polymer

composite panel We can realize that the increase of the particles

and ﬁbers density will decrease the amplitude of the panel

However, the effects of the ﬁbers are stronger

on nonlinear dynamic response of three-phase laminated polymer

composite panel From these ﬁgures, the amplitude of the panel

increases when increasing the ratio b=a and decreasing the ratios

b=h; R=h

the nonlinear dynamic response of three-phase polymer composite

panel It is clear that the panel ﬂuctuation amplitude decreases

when the stiffnesses k1and k2increase, namely, the amplitude of

the panel decreases when it rests on elastic foundations, and the

beneﬁcial effect of the Pasternak foundation is better than the

Winkler one

response of three-phase polymer composite panel It can be seen

that the three-phase polymer composite panel ﬂuctuation

ampli-tudes increase when velocity increases.Fig 12shows the effect of

pre-loaded axial compression Py on the nonlinear dynamic

response of the panel This ﬁgure also indicates that the nonlinear

dynamic response amplitude of the panel increases when the value

of the pre-loaded axial compressive force Pyincreases

response of three-phase polymer composite panel Obviously, the

amplitude of the panel will increase and lose the stability if the

ini-tial imperfection increases

polymer composite panel in two cases: ﬁve-layers asymmetric panel

with the stacking sequence of [0/45/45/45/45] and ﬁve-layers

symmetric panel with the stacking sequence of [45/45/0/45/45]

This comparison is performed on panels with the same plies

orien-tations and the same thickness Clearly, the amplitude of

asymmet-ric panel is higher than symmetasymmet-ric panel

6 Conclusions

This paper presented an analytical approach to investigate the

nonlinear dynamic response and vibration of the imperfect

lami-nated three-phase polymer nanocomposite panel resting on elastic

foundations and subjected to hydrodynamic loads The

formula-tions are based on the classical laminated shell theory (CLST) and

Appendix A

F1¼ A22k4mþ A11d4þ E1k2md2; F2¼ 2A26k3mdnþ 2A16kmd3;

F3¼k

2 m

R B

21k4m B

12d4n E2k2md2n; F4¼ E3k3mdnþ E4kmd3n;

b1¼ P1

ðF2F4 F1F3Þ

F2

F2 k

4

mþ P2

ðF2F4 F1F3Þ

F2

F2 d

4 n

þ P3ðF2F4 F1F3Þ

F22 F21 k

2

md2 P4ðF2F3 F1F4Þ

F22 F21 P5

ðF2F3 F1F4Þ

F22 F21

þ P6k4mþ P7d4nþ P8k2md2nðF2F4 F1F3Þ

F2

k2m

R k1 k2ðk

2

mþ d2nÞ;

b2¼ h P xk2mþ Pyd2n

:

References

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