Vibration and bending of antisymmetrically angle-ply laminated plates with perfectly and weakly bonded layers

Composite Structures 53 (2001) 245±255

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Vibration and bending of antisymmetrically angle-ply laminated plates with perfectly and weakly bonded layers Xiaoping Shu * Department of Mechanical Engineering, Huaihai Institute of Technology, Lianyungang, Jiangsu 222005, People's Republic of China

Abstract A seven degrees-of-freedom model of antisymmetric angle-ply laminated plates with perfectly and weakly bonded layers is developed. The suggested displacement ®eld contains both symmetric and antisymmetric components to the plate middle plane. Based on the three-dimensional elasticity consideration, a set of four shape functions, which are symmetric or antisymmetric to the middle plane, is determined. The interfacial perfectly or weakly bonding conditions and the free traction conditions on the lateral planes are ensured. Consequently, all stresses and displacements, except transverse de¯ection, consist of both symmetric and antisymmetric components. The equations of motion are expressed in a conventional form. The closed-form solutions of simply supported rectangular plates are obtained under the consideration of linear shear slip law. The present model is tested by comparing its numerical results with the existing exact three-dimensional solutions and its high precision can thus be expected. Finally the e€ects of shear slip on free vibration and bending are discussed in numerical examples. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Vibration; Bending; Laminate; Weakly bonding

1. Introduction A large number of publications on the analytical solutions of response of composite laminates may be found in literature. Many approximate two-dimensional theories or methods have been developed and mainly used to analyse the response of cross-ply laminates. The analytical solutions regarding angle-ply laminates cannot be found except those of angle-ply laminates in cylindrical bending and antisymmetric angle-ply laminates. As far as antisymmetric angle-ply laminates are concerned, the relatively simpler theories, such as the classical plate theory, the uniform shear deformation theory and the parabolic shear deformation theory, were mainly adopted in their analyses. For example, Bert and Chen [1], Khdeir [2], Reddy and Phan [3] and Soldatos [4] chose those theories in their analyses of antisymmetric angle-ply laminates. However, their precision had not been tested until Noor [5] published the exact threedimensional elasticity solutions. Soldatos and Watson [6±8] and Soldatos and Shu [9,10] developed a new method for the accurate predication of laminated composite structural elements and

*

Tel.: +86-518-5817613. E-mail address: [email protected] (X. Shu).

its high precision was tested in various cases. Based on the three-dimensional elasticity consideration, a set of `shape functions', which re¯ected the e€ects of transverse shear deformation, was determined. The continuity conditions on interfaces and the loading conditions on lateral planes were ful®lled by choosing those shape functions. However, the shape functions are a€ected by some factors such as the geometrical characteristics of the structural element, its material properties and the stacking pattern and need to be determined case by case. Since this new method is shown its high accuracy in the predictions of through-thickness displacements and stresses, its potential applications in local response analysis, such as interfacial damage, can be expected. Recently Shu and Soldatos [11,12] extended further the method towards modelling of weakly bonded laminates subjected to cylindrical bending. Due to the special structural form of laminates, interfacial bonding between layers is not always perfect. Interfacial bonding strength may be weakened when a laminate is in its manufacturing process or in service. Moreover, since there is always a thin non-rigid layer of matrix, with non-zero thickness and sometimes low shear modulus between two neighbouring layers, natural causes and circumstances (temperature, humidity, etc.) may also weaken the interfacial bonding. Such factors, as well as structural imperfections, degrade the load

0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 0 0 8 - 3

246

X. Shu / Composite Structures 53 (2001) 245±255

carrying capability of a laminate through bigger than anticipated resulting stresses and displacements. Hence, changes in the response of an imperfectly bonded laminate should be predicted as accurately as possible if its life of use should properly be re-evaluated. Shear slip is one of the patterns of interfacial weakly bonding in laminates. Some of the earlier relevant studies [13,14] were unable to model accurately the effects of shear slip because of their relatively poor description of in-plane displacements and transverse shear stresses, which are the key factors that a€ect directly the incorporation of the interfacial shear slip laws. In this respect, Lu et al. [15] improved substantially the precision of modelling shear slip between layers of composite laminates by using a layer-wise displacement ®eld. Cheng et al. [16,17], Schmidt et al. [18,19] and Di Sciuva et al. [20,21] developed later simpler shear slip models for composite laminates by using the so-called zigzag, global displacement ®elds and reformulating them in such a manner that accounts for the interfacial bonding conditions. However, applications that deal with crossply laminated plates have only been considered and discussed in these papers. More general cases, such as angle-ply laminates, have not as yet been considered. The aim of the present paper is to extend the applicability of the method towards the accurate determination of stresses in antisymmetric angle-ply plates with both perfectly and weakly bonded layers. A general seven degrees-of-freedom plate model (G7DOFPT) is developed. Based on the three-dimensional elasticity consideration, a set of four shape functions, which are symmetric or antisymmetric to the plate middle plane, is determined. The suggested displacement ®eld contains both symmetric and antisymmetric components, and ensures the interfacial perfectly or weakly bonding conditions and the free traction conditions on the lateral planes. Consequently, all through-thickness stresses and displacements, except transverse de¯ection, contain both symmetric and antisymmetric components. The equations of motion are expressed in a conventional form. The closed-form solutions of simply supported rectangular plates are obtained under the consideration of a linear shear slip law. The present model is tested by comparing its numerical results with the existing exact threedimensional solutions and its high precision can thus be expected. Finally the e€ects of shear slip on free vibration and bending are discussed in numerical examples. 2. G7DOFPT model for antisymmetric angle-ply laminated plates Consider an elastic plate of thickness h and assume that its middle plane, with area X, lies on the Oxy plane of a Cartesian co-ordinate system Oxyz (the positive Ozaxis is directed upwards). The plate is made of an ar-

bitrary number, N, of elastic layers which are situated antisymmetrically to the plate middle plane. The locations of the upper and lower surfaces of the rth layer are determined by their transverse co-ordinate parameters zr 1 and zr , respectively, …r ˆ 1; 2; . . . ; N †. Consider further that the plate is subjected to a certain loading, q…x; y†, which acts normally on its lateral planes. Finally, denote with U, V and W the plate displacement components along the x, y and z directions, respectively, and employ the usual notation for the corresponding strain and stress components. The concept of two perfectly bonded layers is characterised by the continuity of both the displacements and the interlaminar stresses that act across their interface. In the case of weakly bonded layers, interlaminar stresses are still continuous but some or all of the displacement components are discontinuous across their material interface. The discontinuity of such displacement component(s) can be described with the introduction of corresponding displacement jump(s). As far as shear slip is concerned, there exist some interfacial constitutive relations [22±24] between the transverse shear stresses and the corresponding in-plane displacement jumps at material interfaces. These relations, which in general are complicated and non-linear, have the following general form: F …D…r† ; T…r† † ˆ 0

…1†

which interrelates the vectors,  T T D…r† ˆ ‰DU …r† DV …r† Š ; T…r† ˆ sxz syz zˆzr

…2†

that denote the components of the in-plane displacement jumps and the shear components of the interfacial traction, respectively, at the rth material interface. In more detail, the in-plane components DU …r† and DV …r† of the displacement jump vector, D…r† , characterise the form and the magnitude of the shear slip at the rth material interface. Apparently, there exist two extreme cases of interfacial bonding conditions, namely: (i) perfect bonding, which is characterised by zero values of D…r† , and (ii) complete de-bonding, which is characterised by zero values of T…r† . Hence, any non-zero choice of both D…r† and T…r† is associated with the concept of weakly bonded layers. It should be noted however that the theoretical developments presented prior to Eq. (16), below, are not dependent on the particular form of Eq. (1) (linear or non-linear interfacial constitutive relations), in the same sense that they are not dependent on the form of the elastic law (linear or non-linear) within each particular layer of the laminate. Under these considerations, the present needs of the G7DOFPT suggest that the stress±strain relations in the rth layer of the plate (r ˆ 1; 2; . . . ; N ) should be given as follows …r†

fr…r† g ˆ ‰Q1 Šfeg;

…r†

fs…r† g ˆ ‰Q2 Šfcg;

…3†

X. Shu / Composite Structures 53 (2001) 245±255

"

where T

frg ˆ frx ; ry ; sxy g; T

feg ˆ fex ; ey ; cxy g; Moreover, it 2 Q11 ‰Q1 Š ˆ 4 Q12 Q16

is, Q12 Q22 Q26

0

T

fsg ˆ fsxz ; syz g;

…4†

T

fcg ˆ fcxz ; yyz g: 3



Q16 Q26 5; Q66

‰Q2 Š ˆ

Q55 Q45

 Q45 ; Q44

…5†

zw;x …x; y† ‡ u1 …z†u1 …x; y†

‡ u1 …z†u2 …x; y†; V …x; y; z† ˆ v…x; y†

zw;y …x; y† ‡ u2 …z†v1 …x; y†

‡ u2 …z†v2 …x; y†;

…6†

This displacement ®eld contains seven unknown degrees of freedom, u; v; w; u1 ; v1 ; u2 and v2 , and therefore also involves four shape functions, u1 …z†, u1 …z†, u2 …z† and u2 …z†. Considering the symmetry or anti-symmetry of through-thickness displacements and stresses in antisymmetric angle-ply plates, which were shown and discussed in details in [5], the in-plane displacements U and V are divided to the two parts, symmetric one and antisymmetric one. Hence, u1 …z† and u2 …z† take symmetric form with respect to the middle plane, while u1 …z† and u2 …z† are antisymmetric to that plane. Their roles played in through-thickness in-plane displacements will be discussed in the following numerical examples. The de¯ection W is considered as a constant through the thickness. The choices of the symmetry or antisymmetry of the shape functions will simplify the formulations and ensure the accuracy of the present model. Upon applying the kinematic relations of three-dimensional elasticity to the displacement approximation (6), one obtains the following approximate strain ®eld: feg ˆ fe0 g ‡ zfjg ‡ ‰U…z†Šfe1 g ‡ ‰U…z†Šfe2 g; where

‰u0 …z†Š ˆ

fe0 g ˆ ‰ u;x v;y u;y ‡ v;x Š ; T w;yy 2w;xy Š ; fjg ˆ ‰ w;xx T fe1 g ˆ ‰ u1;x u1;y v1;x v1;y Š ; T fe2 g ˆ ‰ u2;x u2;y v2;x v2;y Š ; T T fc1 g ˆ ‰ u1 v1 Š ; fc2 g ˆ ‰ u2 v2 Š ; 2 3 u1 …z† 0 0 0 ‰U…z†Š ˆ 4 0 0 0 u2 …z† 5; 0 0 u1 …z† u2 …z†

0

0

u02 …z†

u1 …z†

0

# ; 0

0

0 0

0 0 u1 …z† u2 …z† " # 0 u1 …z† 0 0

3

7 u2 …z† 5; 0 …8†

u02 …z†

and a prime denotes ordinary di€erentiation with respect to z. No particular form is currently assigned to each of the shape functions involved, which, for notational convenience, are assumed to have the dimensions of length. However, by imposing the following constraints: u1 …0† ˆ u2 …0† ˆ u1 …0† ˆ u2 …0† ˆ 0

…9†

…7†

du1 du2 du1 du2 ˆ ˆ ˆ ˆ 1; dz zˆ0 dz zˆ0 dz zˆ0 dz zˆ0

…10†

u, v and w become the displacement components of the plate middle plane, whereas …u1 ‡ u2 † and (v1 ‡ v2 ) acquire the meaning of the transverse shear strains acting on it. Hence, on the rth material interface, each one of the components of the displacement jump vector is essentially represented by the interfacial discontinuities of the shape functions …r ˆ 1; 2; . . . ; N 1†, namely: h i h i …r‡1† …r† …r‡1† …r† DU …r† ˆ u1 …zr † u1 …zr † u1 ‡ u1 …zr † u1 …zr † u2 …r†

…r†

ˆ Du1 u1 ‡ Du1 u2 ; h i h …r‡1† …r† …r‡1† DV …r† ˆ u2 …zr † u2 …zr † v1 ‡ u2 …zr † …r†

i …r† u2 …zr † v2

…r†

ˆ Du2 v1 ‡ Du2 v2 :

…11†

Here, the value of the superscript that appears on a shape function indicates the side of the rth interface on which the corresponding layer is situated. Under these considerations, the principle of virtual work of the present model is given as follows: Z t Z Z 0ˆ …frgT dfeg ‡ fsgT dfcg† dX dz X

0

T

u01 …z†

and

W …x; y; z† ˆ w…x; y†:

fcg ˆ ‰u0 …z†Šfc1 g ‡ ‰u0 …z†Šfc2 g;

2 6 ‰U…z†Š ˆ 4

where Qij …i; j ˆ 1; 2; . . . ; 6† denote the appropriate reduced elastic sti€nesses [25]. The present displacement approximation of the G7DOFPT begins in following form: U …x; y; z† ˆ u…x; y†

‰u …z†Š ˆ

247

h

Z X N 1 T ‡ …T…r† † dD…r† dX X rˆ1

Z tZ Z 0

X

h

!

Z X

qdW dX dt

q…U_ dU_ ‡ V_ dV_ ‡ W_ dW_ † dX dz dt;

…12†

where the underlined term accounts for the virtual work due to the interfacial shear slip. The equations of motion of the present model is thus obtained in the following form:

248

X. Shu / Composite Structures 53 (2001) 245±255

Nx;x ‡ Nxy;y ˆ I1  u ‡ I3  u1 ; Nxy;x ‡ Ny;y ˆ I1v ‡ I7v1 ; Mx;xx ‡ 2Mxy;xy ‡ My;yy ‡ q…x; y†  ;yy † ‡ I5   I2 … w;xx ‡ w u2;x ‡ I9v2;y ; ˆ I1 w N 1 X …r† a a Mx;x ‡ Mxy;y Qx Du1 sxz zˆzr ˆ I3  u ‡ I4  u1 ;

 ˆ ff g; ‰LŠfdg ‡ ‰MŠfdg T

T

ff g ˆ ‰0 0 q 0 0 0 0Š ; …13†

rˆ1

a a Myx;x ‡ My;y a M x;x

‡

Qy

a M xy;y

a

Qx

a

M yx;x ‡ M y;y

N 1 X

…r† Du2 syz zˆzr ˆ I7v ‡ I8v1 ;

rˆ1 N 1 X rˆ1 N 1 X

Qy

rˆ1

…r† Du1 sxz zˆzr ˆ

 ;x ‡ I6 u2 ; I5 w

…r† Du2 syz zˆzr ˆ

 ;y ‡ I10v2 ; I9 w

where the appearing force and moment resultants and the non-zero inertia are de®ned as follows: a
a fN g; Z fMg; fM g; fM g
ˆ 1; z; ‰UŠT ; ‰UŠT frg dz; h Z

…14† ‰u0 Š; ‰u0 Š fsg dz; ˆ fQg; fQg h

…I1 ; I2Z; I3 ; I4 ; I5 ; I6 ; I7 ; I8 ; I9 ; I10 † ˆ

h

1; z

2

; u1 ; u21 ; zu1 ; u21 ; u2 ; u22 ; zu2 ; u22

The boundary conditions resulted expressed as follows: un or Nn prescribed; us w or Mn;n ‡ Mns;s prescribed; w;n u1n or Mna prescribed; u1s a prescribed; u2s u2n or M n




q dz:

from (12) may be or or or or

Nns Mn Mnsa a M ns

prescribed; prescribed; prescribed; prescribed: …15†

Here, the subscripts n and s denote the normal and the tangent directions, respectively, along the middle-plane edge curve. It is assumed that interfaces allow sliding, and displacement jumps between layers are assumed to be functions of the interfacial transverse shear stresses. For simplicity the linear shear slip law [13±21] between interfacial displacement jumps and interfacial transverse shear stresses is also used in this work: DU …r† ˆ R…r† sxz ; DV …r† ˆ R…r† syz : …16† x

zˆzr

y

…17†

fdg ˆ ‰u v w u1 v1 u2 v2 Š ;

zˆzr

Here, Rx and Ry are known interfacial compliance constants and their values are allowed to vary from 0 to 1. Hence, in such a particularly simple, linear approximation of the interfacial constitutive relations, the two extreme interfacial bonding conditions, namely the perfect bonding and the complete de-bonding, are described by the zero and the in®nite value of both Rx and Ry , respectively. These de®nitions, in connection with Eqs. (6), (7) and (11), convert the motion equation (13) into the following form

where the operators Lij and Mij (i; j ˆ 1; 2; . . . ; 7) are de®ned in Appendix A. Eq. (17), for a given appropriate set of the shape functions involved, can be solved for the seven unknown displacement functions. Such an appropriate set of shape functions will be determined in Section 3. There, the manner will be further shown in which this solution can be associated to any set of boundary conditions that can be imposed on the edges of the plate. Among the many di€erent sets of such boundary conditions, it is however of particular interest to consider separately the following simply supported (S3) set a

u ˆ w ˆ u1 ˆ v2 ˆ Nxy ˆ Mx ˆ Mxya ˆ M x ˆ 0; at x ˆ 0; a; a

v ˆ w ˆ v1 ˆ u2 ˆ Nxy ˆ My ˆ Myxa ˆ M y ˆ 0;

…18†

at y ˆ 0; b in a rectangular (a  b) plate. The boundary conditions (18) clearly form the two-dimensional analogue of the corresponding three-dimensional set used by Noor [5]. Moreover, it further satis®es exactly the set of the differential equations (17) in the sense that it converts it into a set of simultaneous algebraic equations. For any given set of shape functions, the integrations denoted in Appendix A can be performed either analytically or numerically. Hence, the main concern in accurately predicting response of laminates is the manner in which an appropriate set of shape functions u1 …z†, u2 …z†, u1 …z† and u2 …z† is determined. 3. Determine of shape functions The shape functions will be determined in a similar way [6±12]. Since the G7DOFPT neglects the e€ects of the transverse normal deformation, u1 …z†, u1 …z†, u2 …z† and u2 …z† are determined in this section by making use of the ®rst and second of the three-dimensional equations of equilibrium only. The shape functions will be determined here under the consideration of static response for simplicity. The errors resulted in prediction of free vibration due to this simplicity will be proved to be negligible in the numerical examples. For static analysis these equations are cited as follows: rx;x ‡ sxy;y ‡ sxz;z ˆ 0;

sxy;x ‡ ry;y ‡ syz;z ˆ 0:

…19†

It is now assumed that the plate considered is subjected to S3 set of simply supported boundary conditions (18) which are satis®ed exactly by a displacement choice of the form

X. Shu / Composite Structures 53 (2001) 245±255

and

…u; u1 ; v2 † ˆ …A; C; F † sin ax cos by; …v; v1 ; u2 † ˆ …B; D; E† cos ax sin by; w ˆ G sin ax sin by; a ˆ mpx=a; b ˆ npy=b;

…20†

where any set of m and n represents the half wave numbers in x and y directions, respectively. This would also be understood as being a simple harmonic in the corresponding Fourier sine-series expansion involved in any relevant loading distribution. Connecting with Eqs. (3), (6), (7) and (20), Eq. (19) yields sets of simultaneous ordinary di€erential equations and the shape functions in the rth layer may be solved as follows: 4   X …r† …r† …r† U1 …z† ˆ K1i eki z ‡ K2i e ki z A; iˆ1

4  X …r† …r† …r† U1 …z† ˆ ai K1i eki z ‡ K2i e iˆ1

…r†

U2 …z† ˆ …r†

U2 …z† ˆ

4 X iˆ1 4 X iˆ1

…r†

Ci



…r†

ki z

…r†

K1i eki z ‡ K2i e

…r†

ai Ci



…r†



…r†



ki z

…i ˆ 1; 2; ai ˆ 1; i ˆ 3; 4; ai ˆ

B; 

…21†

‡ bzG;

…r†

…r†

…r†

…r†

…r†

U1 …z† ˆ Eu1 …z†;

…r†

…r†

U2 …z† ˆ F u2 …z†

U2 …z† ˆ Du2 …z†; …r†

…r†

…22†

…r†

for k3 and k4 :   n  …r† …r† …r†2 …r† …r† …r† …r† Q44 Q55 Q45 k4 Q44 Q11 a2 ‡ Q66 b2 2Q16 ab   …r† …r† …r† …r† ‡ Q55 Q66 a2 ‡ Q22 b2 2Q26 ab h  io …r† …r† …r† …r† …r† ‡ 2Q45 Q12 ‡ Q66 ab Q16 a2 Q26 b2 k2    …r† …r† …r† …r† …r† …r† ‡ Q11 a2 ‡ Q66 b2 2Q16 ab Q66 a2 ‡ Q22 b2 2Q26 ab h  i2 …r† …r† …r† …r† Q12 ‡ Q66 ab Q16 a2 Q26 b2 ˆ 0 …24†

…r†

…r†

…r†

2Q16 ab

…r†

Q45 k2i

…i ˆ 1; 2† …r† Ci

…r†

ˆ

…r†

Q55 k2i

…r†

…r†

Q11 a2

…Q12 ‡ Q66 †ab

…r†

…r†

Q66 b2 ‡ 2Q16 ab

…r†

…r†

…25†

…r†

Q26 b2 ‡ Q45 k2i

Q16 a2

…i ˆ 3; 4†: Suppose that the sth layer and the rth layer are antisymmetric to the middle plane. Due to the antisymmetry of stacking pattern involved, the shape functions …r† of the sth layer may thus be expressed in terms of K1i , …r† K2i and ki of the rth layer as follows: 4   X …s† …r† …r† U1 …z† ˆ K2i eki z ‡ K1i e ki z A;

iˆ1

…s†

4 X

…s†

4 X

U2 …z† ˆ

iˆ1

iˆ1

…r†

Ci

ki z

 …r† …r† K2i eki z ‡ K1i e

…r†

ai Ci

…i ˆ 1; 2; ai ˆ

and K1i and K2i …i ˆ 1; 2; 3; 4† represent eight arbitrary constants of integration in the rth layer. The appearing constants ki …i ˆ 1; 2; 3; 4† are the roots of the following quartic algebraic equations: for k1 and k2 :    n  …r† …r† …r†2 …r† …r† …r† …r† Q44 Q11 a2 ‡ Q66 b2 ‡ 2Q16 ab Q44 Q55 Q45 k4   …r† …r† …r† …r† ‡ Q55 Q66 a2 ‡ Q22 b2 ‡ 2Q26 ab h  io …r† …r† …r† …r† …r† 2Q45 Q12 ‡ Q66 ab ‡ Q16 a2 ‡ Q26 b2 k2    …r† …r† …r† …r† …r† …r† ‡ Q11 a2 ‡ Q66 b2 ‡ 2Q16 ab Q66 a2 ‡ Q22 b2 ‡ 2Q26 ab h  i2 …r† …r† …r† …r† Q12 ‡ Q66 ab ‡ Q16 a2 ‡ Q26 b2 ˆ 0 …23† 

…r†

…r†

Q66 b2

…Q12 ‡ Q66 †ab ‡ Q16 a2 ‡ Q26 b2

U2 …z† ˆ

1†;

…r†

…r†

Q11 a2

4  X …s† …r† …r† U1 …z† ˆ ai K2i eki z ‡ K1i e

where U1 …z† ˆ Cu1 …z†;

…r†

Q55 k2i

…r†

Ci ˆ

iˆ1

‡ azG;

ki z

K1i eki z ‡ K2i e

249



‡ azG;

ki z

 …r† …r† K2i eki z ‡ K1i e



ki z

B; 

…26†

‡ bzG:

1; i ˆ 3; 4; ai ˆ 1†:

Hence, the shape functions in (21) and (26) are symmetric or antisymmetric to the middle plane. Only those …r† …r† constants K1i and K2i of the half layers (r ˆ 1; 2; . . . ; N =2) of a laminate need to be determined. Therefore, for an N-layered plate there are 4N integration constants, …r† …r† K1i and K2i (i ˆ 1; 2; 3; 4; r ˆ 1; 2; . . . ; N =2), to be determined according to the interfacial bonding conditions and the shear traction boundary conditions speci®ed on the plate lateral planes. For perfectly bonded interfaces, upon requiring continuity of the in-plane displacement component, U …x; y; z†, at the rth material interface, z ˆ zr , of the laminate, one obtains …r ˆ 1; 2; . . . ; N =2 1†, h i …r‡1† …r† U1 …zr † U1 …zr † sin ax cos by h …r‡1† i …r† ‡ U1 …zr † U1 …zr † cos ax sin by ˆ 0: …27† Noticing that the ®rst term in Eq. (27) is symmetric to the plate middle plane while the second term is antisymmetric to that plane, one obtains …r‡1†

U1

…zr †

…r†

U1 …zr † ˆ 0;

…r‡1†

U1

…zr †

…r†

U1 …zr † ˆ 0

…28†

at the interface. Similarly upon requiring continuity of the in-plane displacement component, V …x; y; z†, at the rth material interface, one obtains …r ˆ 1; 2; . . . ; N =2 1†, …r‡1†

U2

…zr †

…r†

U2 …zr † ˆ 0;

…r‡1†

U2

…zr †

…r†

U2 …zr † ˆ 0: …29†

250

X. Shu / Composite Structures 53 (2001) 245±255

Further upon requiring continuity of the transverse shear stresses at the rth material interface, one obtains …r ˆ 1; 2; . . . ; N =2 1†, …r‡1†

…r‡1†

…r†

Q55 U1;z …zr †

…r‡1†

…34†

ˆ 0;

…r‡1† …r‡1† Q55 U1;z …zr †

…r† …r† Q55 U1;z …zr †

…r† …r† Q45 U2;z …zr †

‡

…r‡1† …r‡1† Q45 U2;z …zr †

ˆ 0;

…r‡1†

…r†

Q45 U1;z …zr † …r†

…r‡1†

Q55 U1;z …zr † ‡ Q45 U2;z …zr †

…r† …r† Q45 U2;z …zr †

…r‡1†

…r†

…r†

…r‡1†

…r‡1†

…r†

…r‡1†

…r‡1†

Q45 U1;z …zr † ‡ Q44 U2;z …zr †

…30†

…r†

Q44 U2;z …zr † ˆ 0; …r‡1†

…r‡1†

…r†

Q45 U1;z …zr † …r†

Q45 U1;z …zr † ‡ Q44 U2;z …zr †

…r†

Q44 U2;z …zr † ˆ 0: At the middle plane (z ˆ 0), namely the interface between the N =2th and N =2 ‡ 1th layers, some of those interfacial conditions (28)±(30) are ful®lled automatically due to the symmetry or antisymmetry of the shape functions and the elastic sti€nesses Qij . The remaining conditions at z ˆ 0 are …N =2†

U1

…0† ˆ 0;

…0† ˆ 0;

…N =2† …N =2† Q45 U2;z …0†

U1;z …0† ‡

…N =2†

U1;z …0† ‡ Q44

Q45

…N =2†

…N =2†

U2

…N =2†

Q55

…N =2†

…N =2†

ˆ 0;

…31†

…N =2†

U2;z …0† ˆ 0:

…1† …1† Q55 U1;z … …1†

h=2† ‡

…1† …1† Q45 U2;z …

h=2† ‡

…1† …1† Q45 U2;z …

…1†

…1†

h=2† ˆ 0;

…1†

h=2† ‡

…1† …1† Q44 U2;z …

Aˆ

4  X iˆ1 4 X iˆ1

Cˆ

…32†

h=2† ˆ 0:

Dˆ

4 X iˆ1

…N =2†

…N =2†

K1i

…N =2†

Ci

4  X iˆ1

h=2† ˆ 0;

Q45 U1;z … h=2† ‡ Q44 U2;z … h=2† ˆ 0; …1† …1† Q45 U1;z …

As detailed in [6,9], upon inserting the particular choice of shape functions (21) into Eqs. (6), the elasticity equations (19) are satis®ed regardless of the values of all the seven unknown constants involved …A; B; C; D; E; F and G†. Hence, values can initially be assigned to all these unknown constants in an almost arbitrary manner. In doing so, the only essential requirement is that non-zero values should be assigned to C; D; E and F , a nulli®cation of which is equivalent to neglecting the e€ects of both the transverse shear deformation and the displacement jump(s). Despite that there is thus a complete freedom in choosing the values of all seven of A; B; C; D; E; F and G some convenient choice to six of them is made by employing the physical role that, according to the constraint equations (9) and (10), u; v; u1 ; v1 ; u2 and v2 are commonly required to play. These constraints yield the following relationships:

Bˆ

Finally upon requiring condition of the free tractions on the plate lateral plane …z ˆ h=2†, one obtains …1† …1† Q55 U1;z …

h i …N =2† …N =2† …N =2† …N =2† …0† ‡ Rx…N =2† Q55 U1;z …0† ‡ Q45 U2;z …0† ˆ 0; h i …N =2† …N =2† …N =2† …N =2† …N =2† 2U2 …0† ‡ Ry…N =2† Q45 U1;z …0† ‡ Q44 U2;z …0† ˆ 0: …N =2†

2U1



‡ K2i …N =2†

K1i

…N =2†

…N =2†



There are total 4N equations in (28)±(32) and then the …r† …r† 4N integration constants, K1i and K2i …i ˆ 1; 2; 3; 4; r ˆ 1; 2; . . . ; N =2†, in shape functions (21) and (26) may be determined. However, for weakly bonded interfaces the in-plane displacement jumps DU …r† and DV …r† (16) occur and the Eqs. (28) and (29) are replaced by the following relations: h i …r‡1† …r† …r† …r† …r† …r† U1 …zr † U1 …zr † ˆ Rx…r† Q55 U1;z …zr † ‡ Q45 U2;z …zr † ; h i …r‡1† …r† …r† …r† …r† …r† U1 …zr † U1 …zr † ˆ Rx…r† Q55 U1;z …zr † ‡ Q45 U2;z …zr † ; h i …r‡1† …r† …r† …r† …r† …r† U2 …zr † U2 …zr † ˆ Ry…r† Q45 U1;z …zr † ‡ Q44 U2;z …zr † ; h i …r‡1† …r† …r† …r† …r† …r† U2 …zr † U2 …zr † ˆ Ry…r† Q45 U1;z …zr † ‡ Q44 U2;z …zr † …33† and the ®rst two equations in (31) are also be replaced by

…N =2†

F ˆ

iˆ1

…N =2†

ai Ci





K2i …N =2†

K2i

 …N =2† K1i

…ai ˆ 1; i ˆ 1; 2; ai ˆ

;

ki ; …N =2†

K1i

iˆ1

4 X

…N =2†

K2i

4  X …N =2† ai K1i Eˆ

;

‡ K2i

…N =2†

K1i

Ci



 ki ;

…35†

 ki ‡ aG; …N =2†

K2i

 ki ‡ bG;

1; i ˆ 3; 4†

in which G is chosen to be a non-zero proportionality factor. Since its value leaves the ®nal numerical results una€ected, G can be left undetermined or set equal to unity without loss of generality [6,9].

4. Numerical results and discussion For simply supported rectangular laminated plates, the equations of motion (17) can be solved analytically. The exact three-dimensional elasticity solutions [5] are used for the test of the present model. A 10-layer antisymmetric angle-ply [h= h=h= h . . .] square laminated plate with perfectly bonded layers is ®rst

X. Shu / Composite Structures 53 (2001) 245±255

considered. The mechanical properties of the orthotropic material used are as follows [5]: EL =ET ˆ 15;

GLT =ET ˆ 0:6;

GTT =ET ˆ 0:5;

mLT ˆ 0:3;

…36†

mTT ˆ 0:49;

where the subscripts L and T denote properties associated with the longitudinal and the transverse ®bre directions, respectively. The fundamental frequencies of the plate with di€erent lay-up angles h and the thicknesslength ratio are tabulated in Table 1. There is exact agreement between the present results and the corresponding exact three-dimensional solution [5] in the case of thin plate. For moderately thick plate …h=a ˆ 0:1† the errors are less than 0.2%. Even in very thick plate …h=a ˆ 0:3† the errors are 0.43, 0.24 and 0.09% with h ˆ 15, 30 and 45, respectively. It is noted that the error magnitude decreases with increasing h. Hence, the present model should be expected to be highly accurate. The present model is also compared with the classical plate theory (CPT), the uniform shear deformational theory with shear correction factor 5/6 (USDT) and the parabolic shear deformation theory (PSDT) in predicting the fundamental frequencies of square plates with perfectly bonded layers in Table 2. Two lay-up patterns, [45°/)45°] and [45°/)45°]4 , are considered. The mechanical properties of the orthotropic material are EL =ET ˆ 40;

GLT =ET ˆ 0:6;

GTT =ET ˆ 0:5;

…37†

mLT ˆ mTT ˆ 0:25:

Obviously the disagreement between the present model and the other three theories increases when the thickness-length ratio increases. When the greatest mismatch of layer materials …N ˆ 2† occurs, the disagreement Table 1 p Fundamental frequencies x ˆ xh q=ET of 10-layer [h= h=a

reaches its maximum value. It is shown that the other three theories, because of their relatively poor description of the transverse shear deformation, always predict sti€er responses than the present model does. The e€ects of shear slip on fundamental frequencies are investigated. The two lay-up patterns, ‰h= hŠ and ‰h= h=h= hŠ, are considered. h=a ˆ 0:2 and b=a ˆ 2. The layer material is same as in (37). Non-dimensional interfacial compliance parameters are employed:

Rx ; Ry ˆ Rx ; Ry ET =a: …38† For simplicity it is assumed that Rx ˆ Ry ˆ R in all following examples. Table 3 shows the degradation of fundamental frequencies when the interfacial bonding condition becomes weaker, namely R increases. The frequencies of the four-layer plate decrease much quickly than those of the two-layer one when R increases. It is noted in this respect that, when the number of layers of a plate with a constant thickness increases, weak bonding exhibits much higher e€ects on the plate responses, even if R is assigned a small value. Hence, it is of importance to note that even a slight weak bonding in all of its interfaces may result in the complete damage of a laminate with a large number of layers. Further the e€ects of shear slip on the frequencies of plates with di€erent thickness-length ratio are displayed in Fig. 1. xp stands for the frequency value of perfectly bonded [45°/)45°/45°/)45°] plates, while xw for that of weakly bonded ones. It is shown that thick plates are a€ected more heavily by shear slip than thin ones. The e€ects of shear slip on static responses of antisymmetric angle-ply laminated plates are next investigated. A two-layer [45°/)45°] plate, with b=a ˆ 2 and

h= . . .] plates h ˆ 15

0.01

Exact [5] Present Exact Present Exact Present Exact Present

0.1 0.2 0.3

251

h ˆ 30

0:1328  10 0:1328  10 0.1162 0.1160 0.3588 0.3577 0.6307 0.6280

2

h ˆ 45

0:1510  10 0:1510  10 0.1296 0.1294 0.3889 0.3880 0.6692 0.6676

2

2

0:1595  10 0:1595  10 0.1351 0.1350 0.3993 0.3987 0.6810 0.6804

2

2 2

Table 2 p Fundamental frequencies x ˆ x…a2 =h† q=ET with various theoriesa

a

h=a

[45°/)45°]

[45°/)45°]4

CPT

USDT

PSDT

Present

CPT

USDT

PSDT

Present

0.01 0.1 0.2 0.25

14.636 14.439 13.885 12.566

14.618 13.044 10.335 9.168

14.621 13.263 10.840 9.759

14.612 12.619 9.625 8.463

25.264 25.052 15.708 12.566

25.176 19.289 12.892 10.842

25.174 19.266 12.972 10.991

25.169 19.020 12.644 10.636

The values of CPT, USDT and PSDT are cited from [3].

252

X. Shu / Composite Structures 53 (2001) 245±255

Table 3 p E€ects of R on the fundamental frequencies x ˆ x…a2 =h† q=ET …h=a ˆ 0:2† R

[h=

h]

[h=

h ˆ 15

30

45

h ˆ 15

30

45

0.0 0.1 0.2 0.3 0.5

8.1351 7.8962 7.7325 7.6146 7.4571

7.1849 7.1066 7.0415 6.9869 6.9007

6.4021 6.3466 6.2985 6.2564 6.1866

9.2550 7.2732 6.4487 5.9850 5.4717

8.9190 7.0629 6.2535 5.7887 5.2666

8.1532 6.5375 5.8010 5.3670 4.8670

Fig. 1. E€ects of shear slip on fundamental freqencies of [45/)45/45/ )45] plates.

h=a ˆ 0:2, is considered. The load on its lateral planes is assumed as follows: q…x; y† ˆ q0 sin…px=a† sin…py=b†:

…39†

In Fig. 2 the following non-dimensional parameters are used: …U^ ; V^ † ˆ 100…U ; V †ET h2 =q0 a3 ; W^ ˆ 100WET h3 =q0 a4 ; …^ rx ; r^y ; s^xy † ˆ …rx ; ry ; sxy †h2 =q0 a2 ; …^ sxz ; s^yz † ˆ …sxz ; syz †h=q0 a;

h]

the e€ects of shear slip (R ˆ 0 and 0.8) on those throughthickness symmetric and antisymmetric quantities of the stresses and displacements. When shear slip occurs, the maximum absolute values of the in-plane stresses at the interface and at the lateral planes increase. At same time the maximum absolute values of the transverse shear stresses increase, while their values at the interface decrease due to the weakly bonding. As far as the displacements are concerned, the displacement jumps between layers are observed, while the de¯ection W increases when shear slip occurs. From the above observations, it is concluded that shear slip will degrade the load-carrying capability of a laminate through bigger than anticipated stresses and de¯ection. Moreover, the roles in which the symmetric and antisymmetric components of the in-plane displacements (6) play are exhibited in Figs. 2(f) and (g). Obviously both symmetric and antisymmetric components are not negligible. The in-plane displacements U and V (6) may be divided to the symmetric components u ‡ u1 …z†u1 ;

v ‡ u2 …z†v1

…40†

Accordingly, in connection with the displacement ®eld (6), these non-dimensional stresses and displacements can ®nally be expressed in terms of their symmetric and antisymmetric quantities with respect to the plate middle plane: …^ rx ; r^y ; s^xy † ˆ …rx ; ry ; s~xy † cos…px=a† cos…py=b† ‡ …~ rx ; r~y ; sxy † sin…px=a† sin…py=b†; …^ sxz ; s^yz ; U^ ; V^ † ˆ …sxz ; s~yz ; U~ ; V † cos…px=a† sin…py=b† ‡ …~ sxz ; syz ; U ; V~ † sin…px=a† cos…py=b†; W^ ˆ W sin…px=a† sin…py=b†; …41† where the bar … † and tilde … † refer to the symmetric and antisymmetric quantities, respectively. Fig. 2 shows

…42†

and the antisymmetric components zw;x ‡ u1 …z†u2 ;

z ˆ z=h:



h=h=

zw;y ‡ u2 …z†v2 :

…43†

However, u1 …z† and u2 …z† equal 0, z and z (1±4z2 =3h2 ), while u1 …z† and u2 …z† vanish in CPT, USDT and PSDT, respectively. Hence, only symmetric components u and v, the values of in-plane displacements at the plate middle plane, are remained in (42) in those theories. This means that the values of the symmetric components of in-plane displacements are uniform through the thickness. However, the corresponding values obtained by the present model in Figs. 2(f) and (g) and by the exact solutions [5] are not uniform and vary obviously through the thickness. Therefore the omission of u1 …z† and u2 …z† in those theories could result in signi®cant errors. This is an important reason why CPT, USDT and PSDT yield much bigger errors than the present model does when predicting the responses of moderately thick and thick plates …h=a 6 0:1† (see Table 2). Moreover, those errors in the case of weakly bonding are bigger than those in the case of perfectly bonding. Hence, it may be an e€ective way to improve the predictions of antisymmetrically angle-ply laminates by

X. Shu / Composite Structures 53 (2001) 245±255

253

Fig. 2. E€ects of shear slip on the symmetric and antisymmetric quantities of stresses and displacements of a [45/)45] laminated plate …h=a ˆ 0:2; b=a ˆ 2†.

including both symmetric and antisymmetric shape functions in their displacement ®eld. 5. Conclusion A highly accurate model of antisymmetric angle-ply laminated plates with perfectly and weakly bonded layers is developed. This G7DOFPT model, which contains seven displacement variables in its displacement ®eld, is an extension of the ®ve-degrees-of-freedom

model (G5DOFPT) for cross-ply laminated plates. Further it can account for interfacial shear slip. Its success is based on three main factors. Firstly the suggested in-plane displacements contain both symmetric and antisymmetric components to the plate middle plane. Unlike CPT, USDT and PSDT theories, whose symmetric components are assumed to be uniform through the thickness, the present model gives more ideal approximation of those components. Secondly the four shape functions in the present model, which are also assumed to be symmetric or antisymmetric to the

254

X. Shu / Composite Structures 53 (2001) 245±255

middle plane, are formulated under the three-dimensional elasticity consideration. Thirdly the interfacial bonding conditions are incorporated into its shape functions and equations of motion. The interfacial bonding conditions and the free traction conditions on lateral planes are ensured. The high accuracy of the present model can thus be expected due to these factors. The present model is tested by comparing its numerical results against the existing exact three-dimensional solutions and shows its high accuracy. It is also compared with other relatively simple theories. The e€ects of shear slip on free vibration and bending are checked in the numerical examples. It is shown that shear slip degrades the load-carrying capability of laminated plates.

L37 ˆ

…F 14 ‡ 2F 33 †… †;xxy

L44 ˆ G11 … †;xx ‡ G22 … †;yy

The research described in this work was sponsored by the Department of Education, China. The author is grateful to Dr. K.P. Soldatos of the University of Nottingham, UK for his helpful suggestions. Appendix A.

L46 ˆ …H12 ‡ H21 †… †;xy ; L47 ˆ H13 … †;xx ‡ H24 … †;yy

B54 ;

L55 ˆ G33 … †;xx ‡ G44 … †;yy

A55

L56 ˆ H31 … †;xx ‡ H42 … †;yy

B45 ;

B26 … †;yyy ;

L14 ˆ E11 … †;xx ‡ E32 … †;yy ; L15 ˆ …E14 ‡ E33 †… †;xy ; L16 ˆ …E12 ‡ E31 †… †;xy ;

rˆ1

…r† 2

…Du1 † =R…r† x ;

M11 ˆ I1 ;

M14 ˆ I3 ;

M33 ˆ I1

I2 … †;xx

rˆ1

M22 ˆ I1 ;

…Aij ; Bij ; Dij † ˆ

Z

h

h

…r† 2

…Du2 † =R…r† y ;

M25 ˆ I7 ;

I2 … †;yy ;

M36 ˆ I5 … †;x ; M37 ˆ I9 … †;y ; M66 ˆ I6 ; M77 ˆ I10 ; Z

N 1 X

D44

M44 ˆ I3 ;

…1; z; z2 †Qij dz

M55 ˆ I8 ;

…i; j ˆ 1; 2; 6†;


um;z ul;z ; um;z ul;z ; um;z ul;z Qij dz

…i; j ˆ 4; m; l ˆ 2; i; j ˆ 5; m; l ˆ 1†; Z T …‰EŠ; ‰F Š; ‰GŠ† ˆ …1; z; ‰UŠ †‰Q1 Š‰UŠ dz; h Z T T …‰EŠ; ‰F Š; ‰GŠ; ‰H Š† ˆ …1; z; ‰UŠ ; ‰UŠ †‰Q1 Š‰UŠ dz; h

L17 ˆ E13 … †;xx ‡ E34 … †;yy ;

some of which take zero value for antisymmetric angleply plates

L22 ˆ A66 … †;xx ‡ A22 … †;yy ; B16 … †;xxx

N 1 X

D55

L77 ˆ G33 … †;xx ‡ G44 … †;yy

…Aij ; Bij ; Dij † ˆ

L23 ˆ

rˆ1

L67 ˆ …G14 ‡ G23 †… †;xy ;

L11 ˆ A11 … †;xx ‡ A66 … †;yy ; 3B16 … †;xxy

N 1 X …r† 2 …Du2 † =R…r† y ;

L57 ˆ …H34 ‡ H43 †… †;xy ;

where

L13 ˆ

rˆ1

L45 ˆ …G14 ‡ G23 †… †;xy ;

The symmetric operators Lij and Mij …i; j ˆ 1; 2; . . . ; 7† in (17) are de®ned as follows:

L12 ˆ …A12 ‡ A16 †… †;xy ;

N 1 X …r† 2 …Du1 † =R…r† x ;

A44

L66 ˆ G11 … †;xx ‡ G22 … †;yy

Acknowledgements

F 24 … †;yyy ;

3B26 … †;xyy ;

L24 ˆ …E21 ‡ E32 †… †;xy ;

A16 ˆ A26 ˆ A45 ˆ B11 ˆ B22 ˆ B12 ˆ B66 ˆ B44 ˆ B55 ˆ D16 ˆ D26 ˆ D45 ˆ 0; E12 ˆ E13 ˆ E22 ˆ E23 ˆ E31 ˆ E34 ˆ E11 ˆ E14

L25 ˆ E33 … †;xx ‡ E24 … †;yy ;

ˆ E21 ˆ E24 ˆ E32 ˆ E33 ˆ 0;

L26 ˆ E31 … †;xx ‡ E22 … †;yy ;

F11 ˆ F14 ˆ F21 ˆ F24 ˆ F32 ˆ F33 ˆ F 12 ˆ F 13

L27 ˆ …E23 ‡ E34 †… †;xy ; L33 ˆ D11 … †;xxxx ‡ 2…D12 ‡ D66 †… †;xxyy ‡ D22 … †;yyyy ;

ˆ F 22 ˆ F 23 ˆ F 31 ˆ F 34 ˆ 0;

F22 … †;yyy ;

G12 ˆ G13 ˆ G21 ˆ G24 ˆ G31 ˆ G34 ˆ G42 ˆ G43 ˆ 0;

F13 … †;xxx

…F23 ‡ 2F34 †… †;xyy ;

G12 ˆ G13 ˆ G21 ˆ G24 ˆ G31 ˆ G34 ˆ G42 ˆ G43 ˆ 0;

F 11 … †;xxx

…F 21 ‡ F 32 †… †;xyy ;

H11 ˆ H14 ˆ H22 ˆ H23 ˆ H32 ˆ H33 ˆ H41 ˆ H44 ˆ 0:

L34 ˆ

…F12 ‡ 2F31 †… †;xxy

L35 ˆ L36 ˆ

X. Shu / Composite Structures 53 (2001) 245±255

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