Viscoelastic analysis of von Kármán laminated plates under in-plane compression with initial deflection

Pergamon

Inl. J Non-Lmeor Mechamcs, Vol. 32, No. 6, pp. 1065-1075, 1997 18 1997 Elsevier Scmm Ltd All rights reserved. Printed in Great Britain 002&7462/97 $17.00 + 0.00

PII: SOO20-7462(96)00133-3

VISCOELASTIC ANALYSIS OF VON KARMiiN LAMINATED PLATES UNDER IN-PLANE COMPRESSION WITH INITIAL DEFLECTION Nan-Nong Huang* Department of Mechanical and Marine Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, R.O.C. (Receivedfor

publication 17 October 1996)

Abstract-The creep responses of simply-supported cross-ply and angle-ply viscoelastic laminates, having various magnitudes of imperfections, under in-plane compression are examined. The nonlinear strain-displacement relation is based on the von Karman assumption. The stress function is obtained by solving the compatibility equation by the use of the Laplace transform, and the deflection is calculated from the moment equation by the Galerkin method and a numerical integration scheme. The numerical results of deflection history and edge shortening for the glass/epoxy laminates are presented for illustrating the effect of imperfections and viscoelastic properties on the creep behavior. The solutions based on the quasi-elastic approach are also presented for comparison. 0 1997 Elsevier Science Ltd. Keywords: viscoelastic, laminate, imperfection, quasi-elastic

INTRODUCTION

The non-linear elastic behavior of composite laminates has been treated intensively. Complex structural phenomena such as buckling, bifurcation and snapping are investigated by many researchers. For example, Huang and Tauchert [l] examined the bifurcation and snapping of laminated cylindrical and doubly-curved panels under thermal loading. More recently, Shin et al. [2] analysed the post-buckling response of laminated plates under uniaxial compression, in which the possible change of deformed shape is considered using the concept of minimum total potential energy. Although it is well-known that the polymeric material of fiber-reinforced plastics (FRP) exhibits viscoelastic behavior, the creep buckling and large creep deflection of FRP laminates attract much less attention. Among a few works on this subject, Wilson and Vinson [3] investigated the buckling load reduction of graphite reinforced epoxy. Kim and Hong [4] examined the viscoelastic buckling load of sandwich plates with cross-ply faces. Huang [S] studied the viscoelastic buckling and post-buckling of circular cylindrical laminated shells. These works were conducted within the framework of the quasi-elastic analysis, i.e. the buckling load and post-buckling deflection are obtained by direct substitution of time-varying properties into the elastic formulations of the problem. The quasielastic approximation is applicable for the cases when the internal forces and moments are nearly timewise constant. However, in the process of buckling, the validity of constant loading assumption is questionable. The deflection is induced by the existence of the bending moment, which in turn depends mainly on the magnitude of the deflection. Therefore, the bending moment is time-varying, although the edge compression remains constant. Another shortcoming in the above-mentioned works [3] and [4] is that the calculation of the so-called buckling load reduction is based on the linear classical analysis without considering the effect of initial imperfections. It is known that the imperfections will affect the time at which the deflection grows rapidly [6]. Theoretically, a geometrically perfect viscoelastic plate will never buckle if the applied compressive loading does not exceed the instantaneous buckling load. Therefore, to further clarify the creep behaviors of the FRP

*e-mail: [email protected] 1065

1066

Nan-Nong Huang

laminates under compression, the creep responses of the simply-supported FRP laminates having various degrees of imperfections are examined based on the geometrically non-linear viscoelastic analysis in this study. The laminates under consideration are (1) regular, symmetric cross-ply laminates and (2) regular, antisymmetric angle-ply laminates having a large number of layers. The governing equations, compatibility and moment equations, are expressed in the form of convolution integrals. For typical plastic matrix laminates whose creep compliances obey the power law, the stress function can be obtained from the compatibility equation using the Laplace transform. The deflection parameter is obtained from the moment equation using the Galerkin technique in conjunction with a numerical integration scheme. The creep responses of glass reinforced epoxy (glass/epoxy) laminates are reported. The critical time, if it exists, is estimated from the deflection history. The viscoelatic results are compared with those results obtained using the quasi-elastic approach.

GOVERNING

FORMULATIONS

Consider an FRP laminate as shown in Fig. 1. The laminate under consideration consists of N viscoelastic layers, with each layer taken to be macroscopically homogeneous and orthotropic. The edge widths of the plate in the x and y directions are denoted as a and b, respectively; the thickness is h. The plate considered, having an initial deflection wo, sustains an in-plane compression force P along the x direction. According to the classical plate theory and von Karman flexible plate assumption, the strains si and curvatures Ici of the middle plane are related to the displacements by c71 c, = u,, + (w,X)2/2+ W,XWO,X s, = a*y+ (YJ2/2 + Yxy

=

v,x

+

u,,

Kc

=

-

w,,,,

+

W,YWO,Y

w,xw,y

Icy =

+

-

w,xwo,y

w.yy,

+

IcXY

=

WO,XW,Y -

(1)

2w,,.

where u and u are displacements of the middle plane in the x and y directions, respectively; w is the additional deflection after loading is applied. For the case of regular, symmetric cross-ply laminates with viscoelastic properties, the stress resultants are related with strains and curvatures as [8] Ni=

C j=1,2,6

~~ =

C j=

fs

Aij(t

-

z)

acji(x,Y, aZ

7)drr

c

Aij*s;.

j=1,2,6

-m

D,* IC; (i = 1,2,6)

t,2,6

where Al6 = A26 = Or6 = D26 = 0. Hereafter, the asterisk (*) indicates the convolution integral, and the prime (‘) denotes the derivative with respect to time. The effective relaxation moduli Aij and Dij listed in the above equations are defined by

(Aij, Dij) =

,ils(1, z2)(Qij)k dz

Fig. 1. Laminated plate.

(i, j = 1,2,6)

Viscoelastic

analysis

of von KBrmh

laminated

1067

plates

in which (Qij)k are the transformed reduced relaxation moduli of the kth layer, which can be calculated from the reduced relaxation moduli (Qij)k using the tensor component transformation. It is noted that equations (2) are also applicable to the regular, antisymmetric angle-ply laminates having a large number of layers, for which the coupling effective relaxation moduli Bij can be ignored. The quasi-static analysis is employed in this study. The in-plane gross equilibrium equations are (4)

Nxy.x + Ny,y = 0

N,,. + Nxy,y = 0, and the out-of-plane response is governed by

M x,x.x + 2Mxy.q + My.yy + Nx(w + wo),xx + 2Nx,(w + wo).xy + Ny(w + wo),yy = 0

(5)

Let the stress function F be defined by N, = F,yy, Ny = F,,,,

Nxy = - F.xy

(6)

The force resultants N,, N,, and NXYthus related to the stress function F satisfy the in-plane force equilibrium equations (4). Upon introducing the stress function and equations (1) and (2) into (5) the out-of-plane equilibrium condition is rewritten as Dll

* (w,xxxJ

+

WA2

+

-

246)

F,,,(w

* (w,xxyy)’

+

wo),xx

+

+

&2

* (w,yvvv~’

=‘,,,(w

+

wo),xy

-

F,x,(w

+

~01,~~

=

0

(7)

The compatibility condition for a physically possible strain state, obtained from equation (l), is as follows, &x,yg + ~y.xx- YXY,XY = (YJ2

- W,XXW,YY + 2w,Xywo,xy- w,,,wo,yy -

WO,XXW,YY

(8)

By applying the Laplace transform to equations (l), (6) and (8), the compatibility equation can be expressed as a22p,XXXX + (2ar2 + a66)p,XXYY + alrp,,,,, = s~W,X,)2

- W,XXW,YY + 2W,XYWO,XY - w,XXwo,yy -

wo,xxw,yyl

(9)

where [a] = [Al-l

(10)

Here, the symbol _Y indicates the Laplace operation and a variable with a hat (^) represents the Laplace transform of this variable.

VISCOELASTIC

ANALYSIS

The creep responses of simply-supported FRP laminates under in-plane compression are analysed. The specific boundary conditions are b

atx=O,a:

w=M,=N,,=O,

u=constant

and

N, dy = - PH(t) J*0 a

aty=O,b:

w=My=Nxy=O,

u=constant

and

N,dx =0

(II)

I 0

where H(t) is the Heaviside function. The initial deflection w. and deflection after loading w are expressed as CWO,

WI= CA,,@)I sin(w)

sin(P,y)

(12)

where a, = mx/a and /In = nx/b. The deflection pattern sin (CL,X)sin (/?,,y)is chosen to be the same as the primary instantaneous buckling shape. It can be shown that the assumed deflection satisfies the deflection and moment requirements along edges. Here, only one mode is employed to approximate the deflection. For a laminate with moderately large deflection, one-mode approximation generally yields acceptable results.

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Nan-NongHuang

For the assumed deflection w and the prescribed imperfection wo, the Laplace transform of the stress function (F) can be determined from equation (9). By taking the inverse Laplace transform, the stress function F is obtained as follows, F(x,y,t)

=;

[($11/~2)cos(2p,y)

+ &J2

cos(2a,x)3*(A2

J’Wr) + 2AAo)’ - 74’2

(13)

where

2 = hhm

A=A/h,

Ao=Ao/h,

(Pll(t)=9’-l

(14) The operator _5?-’ appearing in the above formulations represents the inverse Laplace transform. It is not difficult to verify that the stress function F, given in equation (13), satisfies the in-plane force boundary conditions listed in (11). The explicit expressions of functions cjll(t) and $22(t) for regular, symmetric cross-ply and antisymmetric angle-ply laminates whose viscoelastic properties obey the power law are listed in the Appendix. The in-plane displacements u and u can be determined from the following strain-force relations: & = (a,,&

+ a,,&)/s,

$ = (a&

+ a&)/s

Introducing stress function and employing strain-displacement sions become

relations, the above expres-

U,X= (a&,

+

~l2~,&

-

=%4/2

+

w,xwo,x)

?,

+

a22txm

-

3w3

+

w,ywo,y)

=

hzfi,,,

(15)

(16)

Integrating the above equations and performing the inversion of the Laplace transforms yields - -

1 --&(P.h)2(~2 +2&A)*(~dsinW,,x) -6:tiI1 1 -&n(cd)2(A2 +2W)*(bd sinP/Gy) -dg$I2 (18) f + &

U(X, Y, t)/a = - f (amh)2(A2 + 2AoA)

m

[

sin’ (/?,y) sin (2a,x)

(17)

m

- V(X,Y, t)/b = - $ (j?nh)2(A2 + 2AoA)

%+ h

[

with

n

sin2 (a,x) sin(2P.y)

(19) It is noted that the displacements listed in equations (17) and (18) satisfy in-plane movement requirements along the edges. The edge shortening 6 along the x direction is given by -

-

6/u = - u(x = a, y, t)/u = (q,,h)2(A2 + 2A,A)/8

+ (P/bh)lCIll

(20)

Once deflection amplitude A(t) is determined, the edge shortening can be calculated. Next, the expressions for deflections w and w. and stress function F are substituted into the out-of-plane equilibrium equation (7). The equation will not be satisfied exactly but will have a residual R (say) owing to the approximate nature of w. The Galerkin method is

1069

Viscoelastic analysis of von Khrmhn laminated plates

employed here which requires b

(I

Rsin(a,x)sin(j?,y)dxdy

= 0

(21)

ss0 0

Substituting expression (7) into the above equation and performing the integration yields the following non-linear equation: - - (P/P,,)D(O)(A, + A) + D*J’ + (A; + AoA)(4*A’)/8 + (A0 + A)[4*(A2)‘]/16 = 0 (22) where 4 = 411 +

J*4422T

D(t)

=

[Dll

+

212(Di2 + 2Dsfj) + A4D2Jh3,

P,, = &J2bh3D(0) (23)

Parameter P,, in the above equation represents the instantaneous buckling load, which is identical to the elastic buckling load of a laminate whose viscous effect is ignored. With definitions of P,, and D(t), the edge shortening expression (20) can be rewritten as - 6 = (+)/(~nh)~ = (2’ + 2&4)/8 + (P/P,,)D(O)titr (24) Explicit formulation of $ri(t) for a cross-ply laminate is listed in the Appendix. It is unlikely to solve equation (22) exactly. The trapezoid integration rule is employed to obtain approximate solutions for the unknown deflection parameter A. Here, for the purpose of illustration, the numerical integration scheme for D*A’ is presented. Consider the value of D*A’ from t = 0 to t = tj to be calculated. It is convenient to rewrite D*A’ as follows, *, D(tj - r)dA(r) D(tj)A(O) + (25) s0 where tk (k = 0, 1,2, . . . , j) (to = 0) are the time intervals. Applying the trapezoid rule to the above expression, the following numerical formulation can be obtained: j-l

(D*z)(tj) = D(tj)A(to)+ kTokCD(tj- tk) + D(tj -

tk+ 1 )I

CA(tk+ 1) - A(

(26)

Evaluating all the convolution integrals in equation (22) according to the above numerical formulation generates a cubic equation for ii( Therefore, the deflection magnitude at time t = tj can be easily determined. Finally, the quasi-elastic approximation for creep responses can be accomplished by neglecting the heredity effect; that is, by replacing the convolution integrals in each equation with the conventional products. For example, equation (22) becomes - - - (P/P,,)D(O)(A, + A) + DA + (A; + A,A)A@ + (ii, + A)(A2)4/16 = 0 (27) Also, all the time-dependent mechanical properties are related to each other in the same manner as the elastic relationships. For example, the relaxation moduli Qij are calculated from the creep compliances Sij using the following direct inversion:

CQI= CSI-’ and the quasi-elastic approximation 411

=

(28)

for the function 4 1I appearing in equation (14) is (AllA

-

42K422w

(29)

The quasi-elastic approach, based on the above simplifications, is valid for near constant inputs (stress resultants in the current problem). The quasi-elastic solution is identical to the viscoelastic solution if the stress resultants are timewise constant. The so-called viscoelastic buckling load, based on the quasi-elastic approach and linear buckling analysis, can be easily determined from equation (27) by neglecting the non-linear terms in A and without considering the imperfection A,. The critical value of load, designated as P,, is given by P, = P,,D(t)/D(O)

= (r~,,,)~bh~D(t)

(30)

1070

Nan-Nong Huang

The ratio P,/P,, (= D(~)/~(O)) is referred to as the buckling load reduction. The controversy over the quasi-elastic analysis has been discussed in the Introduction. The validity of quasi-elastic approach and P, will be further assessed using the viscoelastic results in the next section.

NUMERICAL

RESULTS

AND DISCUSSIONS

The creep compliances of an FRP lamina generally obey the following power law: S,,(r) = Sir(O),

S,,(t) = S,*(O)

S,,(r) = SX?(O)(l + U~(~/~~)~),S,,(r) = S,,(O)(l + us(~/~~)b~

(31)

The corresponding reduced relaxation moduli Qij(t), calculated using the Laplace transform, are listed in the Appendix. The functions &trI and &,2 appearing in equations (13) and (23) and function y!~ri appearing in the edge shortening formulation (24) are also listed in the Appendix. The glass/epoxy laminates operating in an environment with temperature T = 73 “C and relative humidity M = 96% are selected to study the viscoelastic responses. The material parameters of a unidirectional glass/epoxy lamina, measured at approx. 73 “C and 21% relative humidity, are [9]: Sir(O) = O.O267/GPa, S&O) = 0.233/GPa,

S,,(O) = - O.O0826/GPa, ar = 0.0846,

as = 0.151,

S&O) = O.O942/GPa b = 0.27,

tr = 1 h

(32)

The glass/epoxy material is assumed to be a simple material. The effect of the increase of moisture concentration on the creep behavior is accommodated in the constitutive equation by the use of “reduced time” [lo]. If the variation of moisture is time-independent, the stress-strain relations for a lamina are (33) where the moisture shift factor a, is related to the specific moisture content c as follows, a, = exp(0.26 - 68.73~) meanwhile, the equilibrium moisture concentration M of the en~ronment by the following law [1 1-J: c = 0.018(~/1~)

(34)

c is related to the relative humidity (35)

The creep responses of cross-ply and angle-ply glass/epoxy laminates, predicted based on the viscoelastic analysis and quasi-elastic approximation, are presented below. 1. Regular, symmetric cross-ply The laminates considered have various amplitudes of imperfections with Iz = 1. The lay-up sequence of these regular, 8-layer laminates is [(O*/90a),]s. Figure 2 shows the deflection histories of a perfect laminate and a laminate with imperfection A, = 0.01 both under compressive force P/PCs = 1.05. For this particular case with loading exceeding the instantaneous buckling load, the results based on the quasi-elastic analysis are very close to those based on the viscoelastic analysis. It is noted from Fig. 2 that the instantaneous deflection is very large, and the deflection grows rapidly from the onset of the loading, but gradually flattens out. Such deflection history implies that most of the changes in the deflection-induced stress resultants take place at the very early stage of loading duration, which closely resembles a constant step loading. Consequently, the quasi-elastic approach, which assumes the loading to be time-independent, yields results almost identical to the solutions based on the viscoelastic approach. The creep deflections and edge shortenings of laminates under P/Per = 0.85, having various amplitudes of initial imperfection A, = 0.01, 0.001 and 0.0001, are shown in Figs 3 and 4, respectively. For a laminate with a slightly large initial deflection, for example

1071

Viscoelastic analysis of von KHrmLn laminated plates

0.8 - Viscoel.anaiynis - Quad-al. analysis 0

-I 0

4

8

12

16

Time after loading

20

24

(hour)

Fig. 2. Creep deflection histories of symmetric cross-ply laminates: (P/Per = 1.05, [(O“/90”)Js, I = 1, A = A/h, A, = A,/h).

Viscoel. analy&

0.5 -

0

200

-

400

Time after loading

600

8( 10

(hour)

Fig. 3. Creep deflection histories of symmetric cross-ply laminates with various amplitudes of initial imperfection: (P/Per = 0.85, [(W/900),],, i, = 1, A = A/h, A, = A,/h).

&, = 0.01, it is noted from Fig. 3 that the instantaneous deflection is significant and the deflection creeps fast at the early stage of the loading. The deflection behavior is similar to the previous case, which implies that the variations of stress resultants should be moderate. Therefore, it is observed that the quasi-elastic solutions for both deflection and edge shortening are close to the results predicted using the viscoelastic analysis. However, the deviations between the quasi-elastic and viscoelastic solutions become larger for the cases of laminates having smaller imperfections. The quasi-elastic approach generally predicts a faster growth in the creep deflection. For these cases, the instantaneous deflection is seen to be relatively small, which implies that the deflection-induced moments are also very small at the beginning of the loading. Because the viscoelastic model takes the history of the loading into consideration, the laminate thus will creep slowly. Nonetheless, the quasi-elastic model takes the current values of stress resultants to be the constant loads for the whole loading history; consequently, the creep responses to the loading predicted by the quasi-elastic analysis are greatly exaggerated.

Nan-Nong Huang

1072

Wscael. analysis

0.21

0

-

1 I I, I I I, I I I , I I I e IO 600 400 200 Time after

loading

(hour)

Fig. 4. Edge shortenings of symmetric cross-ply laminates with various amplitudes of initial imperfection: (P/Per = 0.85, [(o”/90”)2]s, i, = 1, 6 = (6/a)/(a,,,h)‘).

0.0 0

1000

2000

Time after

3000 loading

4000

5000

(hour)

Fig. 5. Creep deflection histories of symmetric cross-ply laminates with various amplitudes of initial imperfection: (P/Per = 0.75, [(O”/90”)2]s, I = 1, A = A/h, A, = A,/h).

A further examination of Fig. 3 reveals that for a laminate with a small imperfection, the quasi-elastic analysis predicts a buckling phenomenon occurring at a time approx. 50 h after the load is applied. In other words, the viscoelastic buckling load (PO)reduces to 85% of the instantaneous buckling load (PC,) when the laminate sustains a loading for approx. 50 h. This value of

Viscoelastic analysis of von KBrmln laminated plates

1073

0.8 -

Viscoel. analysis - Quad-el. analyaia- -

0

300 Time after

800

900

loading

1200

(hour)

Fig. 6. Creep deflection histories of antisymmetric angle-ply laminates with various amplitudes of initial imperfection: (P/Per = 0.85, 0 = + 60”, i. = 0.5, 2 = A/h, A, = A,/h).

accurate results for laminates having small imperfections. By comparing Fig. 5 to Fig. 3, it is noted that the creep rate of deflection for a laminate under a smaller loading is much slower than the creep rate of the same laminate subject to a larger loading.

2. Regular antisymmetric angle-ply The deflection histories of antisymmetric angle-ply laminates having various degrees of initial deflections with P/P,, = 0.85 and 1 = 0.5 are plotted against loading duration in Fig. 6. The laminates considered have a large number of layers. The lay-up is regular with fiber angle 8 = f 60”. Large discrepancies between results based on different approaches are observed for laminates with small imperfections. The quasi-elastic analysis predicts a critical time for buckling at approx. 100 h after loading, which is unlikely to be true as compared to the results based on the more realistic viscoelastic analysis.

CONCLUDING

REMARKS

The geometrically non-linear analysis of creep responses for cross-ply and angle-ply FRP laminates with imperfections under in-plane compression has been conducted using the viscoelastic approach. For typical FRP laminates whose creep compliance is described by the power law, the stress function is obtained analytically using the Laplace transform from the compatibility equation. Subsequently, the deflection parameter is calculated from the moment equilibrium equation by applying the Galerkin procedure and a numerical integration scheme. The creep responses of glass/epoxy are reported. The results based on the quasi-elastic analysis are also included for comparison. Based on the numerical examples, it can be concluded that:

(1) The effect of viscoelastic properties of glass/epoxy on the creep response is seen to be significant.

(2) If the instantaneous

deflection is large, the quasi-elastic approach gives acceptable results. This includes the case when the compressive loading exceeds the instantaneous buckling load and the case when there is a large imperfection. (3) For the case of laminates having small imperfections and under a loading smaller than the instantaneous buckling load, there is a large discrepancy between quasi-elastic and viscoelastic solutions. Generally, the quasi-elastic analysis predicts a much faster growth in deflection. The quasi-elastic analysis also reveals a critical time at which a buckling occurs.

Nan-Nong Huang

1074

Of course, it is unlikely to be true according to the more accurate viscoelastic analysis. The creep deflection depends strongly on the magnitudes of imperfections. The deflection may grow fast during a certain time interval, but not be drastic enough to be characterized as a buckling. Acknowledgement-The support of this research by the National Science Council of Taiwan, R.O.C. through Grant NSC82-0401-E-019-18 is gratefully acknowledged. REFERENCES 1. N. N. Huang and T. R. Tauchert, Large deflections of laminated cylindrical and doubly-curved panels under thermal loading. Comput. Struct. 41, 303-312 (1991). 2. D. K. Shin, 0. H. Griffin and Z. Gurdal, Postbuckling response of laminated plates under uniaxial compression. lnt. J. Non-Linear Mech. 28, 95-115 (1993). 3. D. W. Wilson and J. R. Vinson, Viscoelastic analysis of laminated plate buckling. AlAA J. 22,982-988 (1984). 4. C. G. Kim and C. S. Hong, Viscoelastic sandwich plates with cross-ply faces. J. Struct. Engng 114, 150-164 (1988). 5. N. N. Huang, Viscoelastic buckling and postbuckling of circular cylindrical laminated shells in hygrothermal environment. J. Marine Sci. Tech: 2, 9-16 (1994). 6. B. A. Bolev and J. H. Weiner. Theorv of Thermal Stresses. Krieger, FL (1985). 7. M. Uemuia and 0. I. Byon, Secondary buckling of a flat plate under uniaxial compression-Part 1. Theoretical analysis of simply supported flat plate. Int. J. Non-Linear Mech. 12, 355-370 (1977). 8. R. F. Gibson, Principles of Composite Material Mechanics. McGraw-Hill, New York (1994). 9. Y. C. Lou and R. A. Schapery, Viscoelastic characterization of a non-linear fiber-reinforced plastic. J. Composite Materials 5, 208-234 (1971).

10. Y. Weitsman, Effects of fluctuating moisture and temperature on the mechanical response of resin-plates. J. Appl. Mech. 44, 571-576 (1977). _ 11. S. W. Tsai and H. T. Hahn, Introduction to Composite Materials. Technomic Publishing, Westport, CT (1980).

APPENDIX:

RELAXATION

1. Reduced relaxation For an expressed

MODULI

moduli

viscoelastic lamina, Laplace transforms,

the form

41, >422 AND d’, 1

Qij AND FUNCTIONS

relations between

C

s2&ij~jjk ha

moduli and

compliances,

obey equations

the resulting

(i, = 1,2,6)

j= 1.2.6

& is Kronecker delta. moduli calculated

an FRP the above

creep compliances are

QII@, UJL+ Q&)1 =

- ~/JL) (a~)~fk(f) k=O

RZZI,gO

Qs&) =

f (as)‘X(t) t=ll

where &j = Qij(O), JL = SI l(O) 1 aI = J,,RllaT,

Rij.Sj,(0) = &

(i, k = 1,2,6)

h(t) = [ - l-(b + l)]“(t/tOkb/r(kb + 1)

(A3)

Here, symbol F represents the Gamma function and R,, are the corresponding initial elastic reduced stiffnesses. It can be proved that the radius of convergence (for time) in the series expression for Qij(t) is infinite for the case of O
2. Functions

~$11, 422 and *I I for regular, symmetric

4110) = ~JZZ(~) = WJL) + :

cross-ply

laminates

Cdl&? + Mz)klfk@)

k=O

$,,(t) = 25~+ 1 C&(a,)k+ 4&J’lff(t)

(A4)

k=O

W312)* JL(RI, +&z)JL -11,dz= (RI, +&~L-(RII +&ZVL -

where

4(R12)‘5~

11

d3 =

a2

1

RII + &, + =IZ

=-aT R,I

Rll +

, R22

a3=

- JL,

d, =

1

RII +

R22

-

Rll R,, + R,, + 2RIZ aT’

JL

=12

RI1 a4 = RI, + R,, - 2R,, aT

(A5)

Viscoelastic analysis of von K&~&I laminated plates

1075

3. Functions 41 I and 422 for regular, antisymmetri~ angle-ply laminates

having a large number of layers If coupling stiffness Bij is ignored, #J,1 and $22 are as follows,

m

1

41,(t) = +_

Cddd + 4d~d’lX(t) W)

where a5

=

622

+

a7=hl

,,63-=$%2~22),

a6

+J!GFG%~2~,1.),

=

fr22

~~=h,

-

-Jm(2R1,)

4

=

(4,

-

42/%)/(1

-&,/as)>

4

=

h

-

q2hYU

-at&),

RI,

=

Q11(0),

111

=

R~lWr

+

WI,

-f11h

+fital

R22

=

Q22@)>

r22

=

Rt,UT

+

CR22

--fi~h’s

+f,,‘Q

fi,

=

‘%bC,s,‘%,

=

Xl

41

R22kz

cq =

co@

qz -

e,

4

(h1rSz2)

+x2>

xi =

db =

=

s2fZ/4_, s,

=

=

-yi(Js xz

(4,

=

-

(41

&ii=&kzR22)

q2i”b)/(l

-

q2hMl

-@s/as)

-

OT/G)

h,SzdR,~Ws

+ft,h/Jz. =

(R,

L +

Rz2

+

=,2)f11

sin4 8, c2 = cost i3, s2 = sin’ B

The B appearing in the above formulations is the fiber angle.

647)