Plane strain buckling and wrinkling of neo-hookean laminates

Inf. J. Solids Strurrure.~ Vol. 31, No. 8. pp. 1149-l 178, I994 Copyright Q 1994 Elsevier Science Ltd Printed in Great Britain. All tights reserved 0020-76X3/94$6.00 + .M)

Pergamon

PLANE

STRAIN BUCKLING AND WRINKLING NEO-HOOKEAN LAMINATES YUE QIU, SANGW~~ KIM

Department

of Materials

Science and Mechanics, 48824-1226. (Received

OF

and THOMAS J. Michigan U.S.A.

PENCE State University. East Lansing,

MI

I 1 March 1993, no revision)

Abstract-The bucklingof layered plates is examined

in the context of finite deformation incompressible nonlinear elasticity. Previous work has shown how buckled configurations are modified, and buckling loads reordered, for those buckled states that are natural extensions of previously known solutions corresponding to an isolated single neo-Hookean layer. Here we examine the possibility of new solutions that have no such isolated layer counterparts. In addition to two families that are extensions of isolated layer solutions, we find two additional families of buckled configurations for the case of a symmetric three-ply laminate and we find one additional family of buckled configurations for the case of a two-ply laminate. All of these new families of solutions have an associated wrinkling load which is a function of the stiffness ratio of the two neo-Hookean materials which form the individual layers. This new wrinkling load is higher than that of the wrinkling load associated with the previously known buckled solutions.

1. INTRODUCTION

The buckling behavior of nonlinearly elastic layered structures can depart significantly from that of a single isolated layer. As regards the latter, Sawyers and Rivlin (1974, 1982) employed the theory of small deformations superposed onto finite deformation (Biot, 1965) to determine buckling loads leading to plane strain inhomogeneous deformations in thick rectangular (single layer) plates. Their analysis, which applies to a fairly general class of isotropic incompressible nonlinearly elastic materials, indicated that the associated buckling can occur in either flexural or barrelled mode shapes with any integer number m of halfwavelength ripples in the direction of thrust. Furthermore the buckling loads associated with the onset of these various deformations are ordered sequentially with the ripple number m, beginning with m = 1 flexure and proceeding to m = 00 flexure, which occurs at a finite value of buckling load and so is identified as a wrinkling instability. At even greater loads the barrelling deformations are initiated, beginning at m = cc barrelling (at the same wrinkling load) and then proceeding in reverse order toward m = 1 barrelling at the largest value of load. The intricate nature of buckling in elastic multi-layers has received attention dating back to studies by Biot (1968) who identified highly localized buckling deformations in layered linearly elastic materials. An extension of the work of Sawyers and Rivlin to layered structures, with the determination of the onset of buckling analysed within the fully nonlinear theory of finite elastic deformations, was given in Pence and Song (1991) for the special case of a symmetric three-ply construction composed of two neo-Hookean constituents. Even within this specialized context, it was shown that the buckling behavior could depart significantly from that of the isolated layer. Here, as for the isolated layer, the symmetry of the construction once again leads to buckling deformations that are either flexural or barrelling in character. However certain constructions did not involve a sequential correlation of buckling load with the ripple number. An examination of the reordering of these extended isolated layer buckled solutions, i.e. the natural continuation of the solutions previously obtained by Sawyers and Rivlin (1974). was the primary subject of Pence and Song (1991). As anticipated, if the various parameters characterizing the composite construction tend to special values associated with isolated layer noncomposite construction, then the results of Pence and Song (1991) reduce to those of Sawyers and Rivlin (1974). However for values of these composite characterization parameters far away from the special noncomposite values, the reordering of the buckling loads could be quite 1149 SAS

31:8-H

1150

YUE

QIU et (11.

drastic, including constructions in which the m = cc wrinkling instability had the lowest buckling load, thereby rendering it the critical instability. Some consequences of these results related to optimal design of symmetric three-ply plates against these buckling instabilities were then addressed in Song and Pence (1992). However left unaddressed in both Pence and Song (1991) and Song and Pence (1992) is the possibility of completely new buckled deformations in the multi-layered construction, by which we mean buckled deformations which have no counterpart in the noncomposite case of the isolated layer. By definition, if such solutions exist for certain layered constructions, they must cease to exist at those special values of the composite characterization parameters associated with noncomposite construction. In this paper, we investigate the possibility of such new solutions, for rectangular plates composed of alternating neo-Hookean layers. Attention is focussed on symmetric three-ply constructions (the subject of Pence and Song, 1991), and on two-ply constructions. As shown in the next section, a general N-ply construction leads to a bifurcation condition for the onset of buckling that is found by requiring that the determinant of a 4N x 4N matrix must vanish. This matrix condition stems from 4N boundary and interface conditions that are not trivially satisfied for the plane strain buckled solution forms under consideration. In Section 3, the symmetric three-ply construction is considered, in which case symmetry reduces the general 12 x 12 matrix to two separate 6 x 6 matrices, one of which governs flexure, and the other of which governs barrelling. For each of these two deformation types, there is a continuation of the isolated layer solutions as previously found in Pence and Song (1991), however. in addition we also uncover the existence of a completely new family of solutions for both flexure and barrelling which have no isolated layer counterpart. The buckling loads associated with these new solutions are found to be higher than those associated with the previously known solution families. In particular, the new solutions also have an associated wrinkling load which is shown to be higher than that of the wrinkling load for the previously known solutions. In fact the analysis that is developed to obtain these new families of solutions clarifies the dependence of both new and old wrinkling loads on the composite characterization parameters that specify the three-ply construction. Section 4 is devoted to the two-ply problem. Here lack of symmetry does not permit any reduction in the general 8 x 8 system governing bifurcation, consequently buckling is neither of pure flexure nor pure barrelling type. In this case we obtain three families of buckled solutions. Two of these families are the extension of the isolated layer families to the two-ply case, and as such these two families have a common wrinkling load. The third family has no isolated layer counterpart, and its wrinkling load is higher than that of the other two families. The results of Section 3 and 4 prompt us to conjecture that, in general. there will be Nf 1 families of buckled solutions for the N-ply construction. The mixed mode (flexurejbarrelling) nature of the buckling deformation in the two-ply case is examined in Section 5. We propose a decomposition of the general mixed mode deformation into four parts : smooth flexure, smooth barrelling, residual flexure and residual barrelling and investigate how their associated percentages vary with the ripple number m. The residual flexure and residual barrelling portions of the deformation are shown to correlate directly with the discontinuity in various high order displacement derivatives across the ply interface.

2. GENERAL

FORMULATION

FOR PLANE

STRAIN

BUCKLING

We consider an N-ply rectangular plate in which the plies are stacked along the X2axis. Each ply is composed of one of two neo-Hookean materials, with shear moduli p”’ and /.L”“, respectively, which strictly alternate by ply. The plate occupies a region g in its undeformed configuration, where C@I is given by

with R,+ R2 = 21?. Perfect bonding is assumed across the ply interfaces. field and the corresponding deformation gradient tensor are written as

The deformation

Buckling

and wrinkling

of neo-Hookean

laminates

1151

LJX

F=dX,

x=x(X),

The incompressibility of the two neo-Hookean

VXEB.

materials then gives the constraint

detF=

1.

(3)

The Piola-Kirchoff stress tensor is given by s

=

’ +p(j)FT

-pF-

, j = I or II for material I or II,

(4)

where p = p(X) is the associated hydrostatic pressure field ; the equilibrium equations are div sT = 0.

(5)

In the buckling problem considered here, frictionless normal thrusts act on the two surfaces at X, = +I, yielding s,2

=

S]3

=

0,

XI

=

onX, = *I,.

kpl,,

(6)

Here p is the overall imposed stretch in the X, direction ; the compressive case of interest corresponds to 0 < p < 1. The surfaces at X2 = -R ,, R2 are taken to be traction free : ~21

=

~22

=

~23

=

on X2 = -RI, R2,

0,

while the surfaces at Xx = +I, are constrained frictionless clamp : s3,

=s32

=o,

x3

=

(7)

to remain in their original planes by a

on X3 = +I,.

fl3,

(8)

Across each interface of the N-ply plate, both tractions and displacements are continuous due to the assumption of perfect bonding :

SlilXz

XIX;

= s2i Xi9

(i = 1,2,3)

=xX;,

on interfaces,

(9)

where X, = Xc,“),k = 1,2, . . . , N- 1 are the ply interface locations. The governing equations (5) with (4) the incompressibility constraint (3), the boundary conditions (6)-(8) and the interface conditions (9) constitute a complete boundary value problem, which, to within a rigid body translation in the X2-direction, has exactly one pure homogeneous solution as follows : XI

=

px,,

x2

=

p-Ix*,

x3

=

x3.

(10)

The associated principal stretches are 1, = p, A2= p- ‘, Lj = 1 and the deformation gradient tensor has the form F = diag (p, p- ‘, 1).

(11)

In conjunction with (4) and (1 I), the traction boundary condition, (7)2 gives p = pp-

Therefore

2

E

$j)

t

j = I, II.

(12)

1152

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YUE Qw et al.

p-p-3 s =

p(”

0

0

0

0

0

0 0

ES(/),

l-p’

(13)

j=I,II.

All other boundary conditions are automatically satisfied by the homogeneous deformation (10). Let T be the total (compressive) thrust applied onto each of the surfaces X, = +I, and let A(‘) be the original area of the surface taken by material j (j = I or II) so that A”‘+A,“, = 41,1,. It then follows from (13) that T

=

~)(p,c~)~“‘+p”“~“‘)),

_(p_p-

(14)

Note that T is monotonically decreasing in p from T = co when p = 0 to T = -m when p = 03, with T = 0 when p = 1. Hence (14) can be inverted to give stretch as a function of thrust : p = p(T), with the compressive case 0 < p < 1 corresponding to T > 0. We now turn to examine the possibility of bifurcation away from any such state by solutions which involve an additional incremental deformation u superposed onto this homogeneous solution. Attention is restricted to buckling that takes place in the (X,, X2)plane in the following plane strain fashion : XI = PX,+EU,(X,,

x2>.

x2 =

p-‘X2+EU2(X,,

X,),

x3

=

x3,

(15)

where the unknown functions u, and uZ are assumed to be independent of X1 and E is an order parameter which is used to obtain a linearized problem governing bifurcation from the homogeneous solution. If the pressure field and the associated Piola-Kirchoff stress tensor are given by

pm,E)= p’j’+-qQ-,, x2,

X,)+0(&‘),

s(X. e) = s”‘+&X,, x2. x,)+0(&Q,

j = I, II,

(16)

respectively, then substituting from (15) into (2)2 and (3) and retaining terms of order E gives (17) while substituting from (15) into (4) and (5) and retaining terms of order E gives div ST = 0,

(18)

where -p~‘~+p”“(U,,,

S=

I

-p-%.2>

/L’qu,,~+p~2u2.,)

0

p”‘(U7., fpm%4,J -p~+2c(“‘u_ 7.2 0

0

0 1 . -P

(19)

Now (18), is satisfied if p(X,, XZ, X,) = p(X,, X,), thus giving that (17) and (IS),., are three linear, homogeneous partial differential equations for U, (X,, X2), u,(X,, X2) and p(X,, X,). Turning now to examine the order E boundary and interface conditions, one verifies that those obtained from (8) are automatically satisfied by virtue of (15) 3 and (19). Following Sawyers and Rivlin (1974), the following forms

Buckling

and wrinkling

of neo-Hookean

1153

laminates

(20) are consistent with both (i) seperation of variables in the system of partial differential equations mentioned above, and (ii) the boundary conditions (6) provided that m71

“=z’

m=

2,4,6,8...

for the upper terms in (20)

1,3,5,7...

for the lower terms in (20).

(21)

Under (20) the system of partial differential equations (17) and (18) ,,* reduce to the single fourth order ordinary differential equation for U,(X,) : u;“-[l

+A*]R*U,;+AWU?

= 0,

(22)

where the prime denotes differentiation with respect to X2 and we have introduced a new stretch ratio

(23) The remaining two functions iJ,(X,) and P(X,) are then given by 1 LJ,G-2)

=

z

u;w,),

P(X,) = P”‘(X,)

= &

(U;ll(X,) -s2W;(x2)),

j = I, II.

(24)

The general solution of (22) is given by U,(X,) = L, (X2) cash (QXJ + L2(X2) sinh (QX,) + M, (X,) cash (10X2) + M2(X2) sinh (izfiX2),

(25)

where L,(X,) and M,(X,), i = 1,2, are constant in each ply. Thus L,(X,) and Mi(X2) are step functions Lj(X2) = Ljk’,

M,(X,)

= &Ilk), i = 1,2

(26)

wherek= l,... , N refer to the in general different values of these constants in each of the N plies. There are thus 4N unknown constants to be determined, and it only remains to satisfy the O(E) boundary and interface conditions generated by (7) and (9). The condition on ~23 in (7) is satisfied automatically, while the remaining conditions in (7) yield four relations binding the 4N unknown constants. Similarly the continuity of ~23 and x3 in (9) is satisfied automatically, while the remaining conditions in (9) yield 4(N- 1) relations binding the 4N unknown constants. In total, this gives 4N relations, which are linear in the unknown constants, and so may be written as

1154

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al.

A& 213

21, Fig. 1. The symmetric three-ply composite plate considered by Pence and Song (1991) and re-examined here for additional buckling solutions. Plane strain buckling takes place in the (X,, X,)-plane.

J 4Nx4N 14Nx

-Omx,,

I -

(27)

where 1 is the vector containing the 4N unknowns in (26). The existence of nontrivial solutions for this system of equations requires that det J 4N

x 4N

- 0. -

(28)

This is the basic bifurcation condition for the buckling problem of the N-ply neo-Hookean plate under thrust.

3. ADDITIONAL

FAMILIES OF BUCKLED SOLUTIONS FOR THE SYMMETRIC THREE-PLY COMPOSITE PLATE

In this section we re-examine the bifurcation condition (28) for the symmetric threeply plate (N = 3) as studied by Pence and Song (199 1). Here R , = R2 = I2 where plies 1, 2 and 3 are, respectively, given by -I? < Xz < -R, -R < X2 < R, R < X2 < lz. If $I’ is the neo-Hookean shear modulus for ply-l and ply-3, and $‘I) is the neo-Hookean shear modulus for ply-2 as illustrated in Fig. 1, then (26) becomes

and the bifurcation condition (28) becomes det J , 2X, z = 0. Now following Sawyers and Rivlin (1974), and Pence and Song (1991), attention is restricted to the two subclasses of flexural buckling deformations and barrelling buckling deformations, which exist as a consequence of the special symmetric nature of the threeply composite construction presently under consideration. A flexural deformation is one in which U&X,) is an even function, whereupon (24) gives that U,(X,) and P(X,) are each odd functions. This will be true provided that (L’,“, L(z’),MC,“, MI”) = (L’:‘, --Li3’, M’?‘, -N3’), a2) = L&*’= 0. Th us a linear system remains for 6 independent constants, which are taken to be I = EL’,‘),L$‘), MC,‘),MI’), L(,Z’,M(,2’]r.The associated 6 x 6 coefficient matrix is given by

Buckling AC,

-2s,

-c3

and wrinkling

laminates

1155

-AS,

2x2

-2AS2

0

0

2c,

-AS,

AC,

0

0

s3

--Cd

s3

-c3

-ACs

AS, -2CJ

2S,

of neo-Hookean

AS, -2x4 AS,

C3

s4 -AC, 2AS4 -AC,

C4

--is,

-sj

1

’

(30)

MC, 2w, -2flS, -A/N,

where

C, = cash(q),

Cz = cash (Q),

S1 = sinh (q),

S2 = sinh (E.?), S3 = sinh (r,roz), S4 = sinh (,?~a),

Cj = cash (ycr),

C4 = cash (iqc~), (31)

A = 1+ l/n, and the parameters u, /I and LXare defined in (35) below. The associated flexural buckling equation is then written as Y,(I,

q, 8, a) = det JF = 0.

(32)

In contrast, a barrelling deformation is one in which U,(X,) is an odd function. In a completely analogous fashion, the barrelling buckling equation is : Y,(i,

q, /?, a) - det

JB = 0,

(33)

where J, is the 6 x 6 matrix :

JB =

AC,

-AS1

2x2

-2x&

0

0

-2s,

2c,

-A&

AC?

0

0

s,

-c,

s4

-ss

-s,

-X4

c,

-c, s3

-cj

AS,

-AC3

AS,

- 21c,

2S,

-2C,

AS,

21s,

-A/B,

AC, -21/R&

- ACq

2BCj

ABC,

>

(34)

and the same notations (31) are used.The corresponding six constants for the barrelling deformation are 1 = [L’,“, L$“, MC,“, M$“, L$“, M’,2’lT. The eqns (32) and (33) relate the four dimensionless parameters :

(35) which are described as follows : (1) the ripple parameter q scales the discrete ripple number m with respect to the aspect ratio 1*/l,, this parameter is often treated in the analysis as a continuous variable; (2) the stz~ness ratio /I is the ratio of the shear modulus of materialII to that of material-I ; (3) the volume fraction a is the ratio of the thickness of the central ply to the thickness of the whole composite plate ; and (4) I is the ratio of principal stretches, which in view of (14) and (23) is the loud parameter with 1 = 1 corresponding to no load and L monotonically increasing from ;1 = 1 as the compressive thrust T increases. Their respective ranges are

1156

YUE QW et ul. q > 0.

p > 0,

0 < x q

1,

3, > 0.

(36)

A pair of material parameters (p, (x), together with I,, f2 and I? specifies a specific symmetric three-ply construction. Each of the special values /I=1

or

LX=O

or

a=1

(37)

correspond to the special case of a noncomposite plate, conversely values of the pair (p, n) obeying (36),., with fi # 1 and x f 0 and a # 1 are associated with genuinely composite constructions. The pair (fl, a) constitute the composite characterization parameters mentioned previously in the Introduction. It is to be noted in all cases that the absence of load (I. = 1) trivially furnishes a solution to both (32) and (33) in view of the coalescence of the final two columns of JF and Jg. Pence and Song (1991) investigated solutions of both (32) and (33) by fixing (t?, /L CL) obeying (36),.,,, and showing numerically that each of these equations then admits a solution 3,obeying iti > 1 and hence corresponding to compression. These solutions, denoted by ,I = Qp(q, /?. a) for (32) and /i = QB(q, /I, CI) for (33) were determined numerically by means of a searching and bisection algorithm and in each case resulted in the minimum value of 1> 1 compatible with (32) or (33) for the given (9, /?, LX).For fixed (p. c(), each of these I-values were found to approach a comnlon asymptotic value )., = i,(p, crf as )I--+ cg ; this value was then computed numerically by taking a large cut-off value of q. Gfi unaddressed in Pence and Song (1991) is the possibility qf additiorlal roots 2. grrutrr thun those given by cDp(q.,5, a) and 4DB(q,/I. a), as well us the possibility thut the asymptotic wlurs A,, associated with wrinkling can be computed by analytical means. This section deals with the further investigation of these two issues. The first issue leads us to analyze (32) and (33) for large I, and the second leads us to analyse (32) and (33) for large g. When the matrix J, in (30) is compared with the matrix JB in (34), the only difrerence is in the last two columns of each matrix. In fact, Y,(L q, fi. a) may be transformed into Y,(j., 9, /I, r) by exchanging r/ and 1.~ in the hyperbolic trigonometric functions of (31), i.e.

(38)

This claim may be verified by first subjecting (30) to the exchanges on the right hand side of the o in (38). and then performing the following row operations : exchange column 1 with column 4, exchange column 2 with column 3, exchange column 5 with column 6, exchange row 1 with row 2, exchange row 3 with row 4, exchange row 5 with row 6, multiply rows 1, 4, and 5 by A, and finally, multiply columns 1, 2. and 5 by L- ‘. Under this set of operations Y I: is transformed into Y B. Relations (38) will enable us to analyse many aspects of the flexural and barrelling cases in a common setting. To this end it will be useful to introduce the subscript * as a placeholder for either the subscript F for flexural deformation or the subscript B for the barrelling deformation. The expressions Y’F and ‘IIB given in (32) and (33) may be expanded into the following sum of products of hyperbolic functions and polynomials :

Y&4, q. & a) =

$H,PA.

(39)

where H, = H,(i,v,a),.

,0 = ~sinh(YK,),sinh(Y]~?), P=

[pi.j(~~l~Ox

. . . . sinh(qk.,O)),

10t

A = A(]“) ,“X , = {a 4,R’ 3,. . (iO, . . . A”, P)‘..

(40) (41) (42)

Buckling

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1157

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In equation (39), the only difference between the flexural and barrelling case are the values forti,=rcCi&a)(i=l ,..., 10) in (40). These values, as well as the elements of the matrix P, are given in Appendix A. It is to be noted that H,(/1,0, a) = 0, so that (39) gives ul*@, 0, /I, ‘X)= 0. We now conduct an analysis of ‘I’*(& q, /I, a), in (39), for large A in order to gain insight as to the total number of roots I to (32) and (33) for fixed (q, #I,a). For large A the leading order terms in (39) will be those containing both 2’ from (42) and e’l’“”‘), e”“- ‘), e~‘“+‘“~” and e?$‘“-‘“+!’from (40) [viz (A.5) and (A.@]. It follows that, for the flexural deformation,

y&4 v, B, Co_.,;

(fi_

1)2 eqe”

_e-q

_,fw-

I) +e-“‘2a-

I)],

(43)

provided that both j3 # 1 and a # 1; and for the barrelling deformation,

provided that p # 1, a # 1 and a # 0. Hence (43) and (44) give

o&q, p, a) = $(/I-l)2cosh

wB(qr /?, a) = $(/I-

(qa)sinh [q(l -a)],

(46)

l)* sinh (qa) sinh [q(l -a)],

(47)

for the flexural defo~ation and the barrelling deformation, respectively. For genuinely composite constructions (fi # 1, a # 1, and a # 0) the expression o,(q, /I, a) eq”R5 does not vanish and as such is a convenient normalization for Y.Jil, q, /I, a). Let

(48)

so that, in a genuinely composite construction, equivalent to

the buckling equations (32) and (33) are

n*G-, v, P, 4 = 0,

(49)

with * = F and * = B, respectively, and I&(2, v, fi, a) + 1 as /z -+ 00. We have numerically studied the functions II,@, 9, /I, EX)for fixed (9, /I, a) in genuinely composite constructions and found that both * = F and * = B will in general each yield two roots L > 1 for (49) before I&(,$ q, p, a) begins to smoothly approach its asymptotic value of one. This is

1158

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QIU et al.

l-I&,1.5,2,0.5)

2-

Fig. 2. The functions rIp(l, q, j?, a) and II,(A, q, 8, a) defined in (48) for the case ‘1 = 1.5, fl = 2. z = 0.5. Each of these functions has two roots I in the range I > 1. and each function approaches the asymptotic value one as I + CXJ.

demonstrated in Fig. 2 for the case (q, B, a) = (1.5,2.0,0.5) which exhibits the ever present root 1 = 1 and the smooth approach to the asymptotic value of 1 for both TI,(L 1.5,2.0.0.5) and II,(n, 1.5, 2.0, 0.5). For 1 < J -C co, this figure indicates the presence of two roots 1 to the equation II,(l, 1.5,2.0,0.5) = 0, the first of which is given by 1” z 3.67 and the second of which is given by 1’ E 11.46. Similarly, the equation lI,(& 1.5, 2.0,0.5) = 0 has two roots /z given by LBz 5.45 and xB z 20.53. A similar situation prevails for all other values (q, j?, CC)associated with genuinely composite construction, namely there exist roots i’ and 1’ for (49) whenever * = F, and there exist roots ,IBand ;i” for (49) whenever * = B, with these roots obeying 1 < /IF < Y,

1 < IbE< LB.

(50)

The functions taking (r], /I, a) to these four roots will be denoted by QF(q, fl, c(), g,,(q, /Y,a), QB(q, /?, a) and gB(v, fl, cc).The roots ,JFand LBand the functions QF(~, j?, LX) and (DB(q,b, (x) were obtained previously in Pence and Song (1991). The study of Pence and Song (1991) however did not uncover the additional roots IF and 1’ owing to numerical instabilities connected with seeking roots ,I to (32) and (33) whenever I* is larger than the respective first roots ,I’and IzB.The scaling (48) suggested by the large Izasymptotic analysis overcomes this difficulty and thus enables us to uncover the new roots 1’ and 1”“.These two new roots are each associated with a new family of buckled configurations for the symmetric threeply construction. The subsequent numerical approach of TI, ---f1 as A ---fCCstrongly suggests that all roots to (32) and (33) are now accounted for. Numerical study of the four functions QF(q, fi, LX),gF(q, p, a), QB(~, b. (w) and GB(q, /I, 01)indicates that each of these functions varies smoothly with (q, /I, cr) and obeys the ordering relation (51) Furthermore, for fixed (p, a) corresponding to a genuinely composite construction, we find as q + 0 that

(52)

Buckling

and wrinkling

of neo-Hookean

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1159

Moreover, as q + co, we find that

(53) where A,(/?, a) < I,(/?, a). Equations (51)-(53) are in accordance with the results given in Pence and Song (1991) for the functions (D&q, 8, a), O&, /I, a) and A,
A 4,

X = %(m&/211,

B, 4,

1: = ~,,(mn12/21,, jl, a),

1: = 6B(m7c12/21,,/?, a),

m = 1,2,3.. .

(54)

owing to the discrete nature of r] as given in (35) ,. The corresponding four families of failure thrusts: TL, Tg, FL, and pi”, m = 1,2,3.. . are then obtained from (14) upon using p = LX’/*. These failure thrusts are ordered the same as the failure stretch ratios. In the event that @&,r, /I, a) is monotonically increasing in q to A,@, a) and a,(~, /I, a) is monotonically decreasing in q to Loo(B, a), then

where T, is a wrinkling thrust associated with the stretch ratio &,(/I, a). In particular, as established by Sawyers and Rivlin (1974), this occurs for the noncomposite case (37) ; the corresponding curves OF(q, p, a) and QB(q, /I, a) are displayed in Fig. 3. As shown in Pence and Song (1991) certain pairs (/I, a) are associated with a loss of monotonicity in (IQ&, /3, a)

0

Fig. 3.

The functions

1

2

3

4

5

6

7

0

@&, /J, a) and a,(~, /?, a) for parameter pairs noncomposite case as given in (37)

910

(B, a) corresponding

to the

1160

YLJE

Qtu et al.

lo-

0

1

2

3

4

5

6

7

6

9

10

9

Fig. 4. The functions @&& z). a,(?, p. E), &(q, /I, a) and &(q, (I, z) for /3 = 0.5, a = 0.5. The two lower curves were displayed previously in Fig. 4 of Pence and Song (1991). The two upper curves correspond to the new families of buckled solutions.

as a function of ripple parameter q. Figure 4 depicts one such example for the pair (j?, M)= (0.5,0.5). Furthermore, certain pairs (p, a) are also associated with a loss of monotonicity in @Qq, & a) as a function of q. Figure 5 depicts one such example for the pair (8, CC)= (2.0,0.5). In such cases the possibility arises that the faiture thrusts T,!&and Tt, will no longer obey (X5), but instead will be interlaced. In all cases (51) ensures that the lowest, or critical, thrust will be one of the Ti for some positive integer m including, possibly, m = co. If OF(rr & a) is monotonically increasing to I, then this critical thrust will be T”;, however if QF(q, & a) is not monotonic, then this critical thrust may occur for m # 1 and indeed may be associated with the wrinkling thrust TF, = T, _ A detailed study of such phenomena as they occur among the two families 7’:, and T& (m = I, 2. __.), with particular emphasis on their correlation to parameter pairs (p. a) and aspect ratio [?/I, is given in Pence and Song (1991). We thus turn our attention to the two new families of failure thrusts p: and pt (m = 1,2,. . .). Consider first the case in which both &&, & a) and &&, j?, a) are monotonically decreasing toward the asymptote xa(fi, a) as for example occurs in Fig. 4. Then ?F is the maximum thrust from among these two families, and f:,. is the minimum thrust,

Fig. 5. The functions Q& 8. a), Os(q. /?, a), ifi&, 8, a) and &(q, j, a) for /? = 2, a = 0.5. The two lower curves were displayed previously in Fig. 5 of Pence and Song (1991), while the two upper curves correspond to the new families of buckled solutions.

Buckling

and wrinkling

of neo-Hookean

laminates

1161

y, is the newly uncovered wrinkling thrust associated with Im. The remaining thrusts will lie between these values, and the two families of thrusts will interlace their values. However for fixed ripple number m, (51) ensures that Fc < Ti. The question arises as to whether or not the thrusts from these two new families can interlace themselves with the thrusts from the two original families. For the case depicted in Fig. 4, no interlacing can occur with the original flexure family, since the interval A, < 1 <& isolates the range of the curve QF(q, p, CC)from the range of the curves &(q, j?, a) and QB(q, j3, a). However no such isolation occurs between the latter two ranges and that of mB(q, p, a) since mB(~, /?, a) -+ cc as r] -+ 0. Thus if the separation between the discrete v]values is sufficiently small, some of the low ripple number thrusts of the original barrelling family will be interlaced with some of the higher ripple number thrusts of the two new families. The separation between discrete q values is determined by the aspect ratio ZJZ,, thus for fixed plate height 12, this interlacing will occur if the plate length in the direction of thrust I, is sufficiently large. Symmetric three-ply plates corresponding to other pairs (fi, a) give rise to fairly similar phenomena, with the departure from the behavior described above determined by the extent to which the functions g,(u, p, a) and 6B(q, 8, a) are not monotonic in q. For example in the case (/?, a) = (2.0,0.5) depicted in Fig. 5, the curve gF(q, /?, a) exhibits a local minimum. Thus in this case ?“‘, is not the minimum failure thrust from among the two new families. Instead this minimum failure thrust will occur at some finite ripple number m within the new flexure thrust family pi. The ordering possibilities for the new thrusts Fc and r,” could certainly be characterized in a systematic fashion by utilizing a methodology similar to that developed in Pence and Song (1991) for the previously known thrusts Ti and Tt. However, since (51) ensures in all cases that the critical failure thrust remains associated with the original flexure family Tc, such a detailed study will not be pursued. In addition, we note that the conclusions of the optimal design study given in Song and Pence (1992) are not affected by the existence of the two new families p: and Fi of buckling thrusts. We now turn to an asymptotic analysis of the curves QF(q, 8, a), gF(q, B, a), QB(q, /I, a) and QB()l, /?, a) as q + 00. This will yield an analytical characterization of the asymptotic values 1, (p. a) and 1, (j?, a) associated with wrinkling, and will also clarify the apparent disappearance of the two new families of buckled solutions for the noncomposite case given by (37). Now (40) and (41) with (A.5) and (A.6), give

where

K~ =

max(Ic,,ic2 ,...,

for both flexural and barrelling deformations.

rcs,KIO) = 1+1,

(56)

Hence

where

pI(l~ B) 3

i

pl,j(B)y,

(58)

j=-4

and the polynomials P,.j(j?) are given in Table A. 1. Since the exponential function is always positive, (57) indicates that solutions of

P,@, B) = 0

(59)

yield asymptotic solutions to the buckling equations Y* = 0 as q + co. In particular, the wrinkling load parameters A,@, a) and n”,(j?, a ) must be roots of (59). We observe that, according to (59), these wrinkling load parameters are independent of the volume fraction a. Evaluating (58) with the aid of Table A.1 gives

1162

(60) where f,(A) = (A3-3A2-A-l),

(61)

f&&B) = (~-l)2~3-[3(~-1)2+4~]A2-[(~-l)2+8~]E.-(~+1)2.

(62)

The equation f,(A) = 0 has one real root A,, given by Cardano’s solution for the cubic polynomial equation, 1, = 1+3$+;‘j-(4/3)3+3J2-j4-(4/3)3

=3.38297577....

(63)

For /? # 1 and /I > 0, the equation f2(A, j?) = 0 also yields exactly one real positive root, Im((B). In fact, we find that the constant A, given in (63) is the asymptotic value A&, CI) of (53) ,, and we also find that the root xm(/I) to f2(A, fl) = 0 is the asymptotic value I”, (/I, a) given in (53)*. Hence (53) can be improved to yield

In particular, (64), is a significant improvement over the numerical cut-off procedure used for computing the wrinkling load parameter A, for the examples presented in Pence and Song (1991). Specifically, it is to be noted that some of the examples presented in Pence and Song (1991) were more successful than others in accurately attaining this constant asymptotic value A, x 3.383. The dependence of IW on /? is depicted in Fig. 6, where it is to be noted that (65) Here (65), and (65), follow from (62) since f2(A, 0) = f,(A) and f2(;l, B) = as /I -+ co, whereas (65), is a consequence of the observation thatf2(A. 1) = no roots for positive A. Since fi = 1 corresponds to the case of noncomposite (65), leads one to anticipate that the two new families of failure thrusts F,’

0

1

2

3

4

5

6

7

6

9

/I’f,(A) +0(/I) -4(1+ l)* has construction, and ft. which

B 10

Fig. 6. The values of the wrinkling load parameters I, and L(j3) which are the roots of (61) and (62) respectively. Here 1, = 3.38297.. . and x,(j) depends only on j?. As fl--) 1, corresponding to a noncomposite plate. 1,(/Q --t 00.

1163

Buckling and wrinkling of neo-Hookean laminates

Fig. 7. The two-ply composite plate. Once again plane strain buckling is assumed to occur in the (X,, X,)-plane.

only exist for the genuinely composite construction, will individually retreat to infinity if the (/?, rz) parameter pair is made to approach any of the values (37) associated with noncomposite construction. 4. FAMILIES OF BUCKLED SOLUTIONS FOR THE TWO-PLY PLATE

It was shown in Pence and Song (1991) that the failure thrusts given in Sawyers and Rivlin (1974) for noncomposite construction, will reorder themselves for certain composite constructions. The results of the previous section indicate that, in addition, composite construction will also give rise to completely new families of failure thrusts. Specifically, the symmetric three-ply construction gives rise to two new families of failure thrusts as given by (54) 3 and (54) 4. The question thus arises as to how such new families will correlate with the total number of plies N in a more general construction. We here consider this question for the case of a two-ply construction as shown in Fig. 7. By setting the origin of the coordinate system at the interface between the two plies, then ply-l occupies -RI < X2 < 0 and ply-2 occupies 0 < X2 < R2.The lack of symmetry in this problem does not allow the type of simplifications as given in Section 3 in which the 12 x 12 system for the symmetric three-ply problem reduced to two separate 6 x 6 systems [(32) and (33)] for flexural and barrelling bifurcations. Instead, for the two-ply problem, one must treat the full 8 x 8 system [(28) with N= 21 and the resulting buckled states will neither have the symmetry associated with flexure nor that associated with barrelling. One finds that if the eight constants in (26) are arranged as follows

(66) then the coefficient matrix J of (27) is given by

AC3

2X,

- 2&

0

0

0

0’

AC, 0

0

0

0

0

-1

2C3 0

-AS4

1 0

1

II

0

-A

0

0 -2;1

0

0

-A

@A 0

0

0

0

0

0

0

0

0

-2s3

J=

-AS,

0

-2

1

0

-1

0

0

-A

0

2pn

0

AC5

28 AS,

0 2X6

BA 2&

2s5

2C5

A&

AC6

where Cj, Cd, S3, S4 and A are again given by (31) and

-1

(67)

1164

YUE Qw et al. C, = cash (~(1 -a)),

C6 = cash (,Jq(l -(w)).

Ss = sinh (fst(1 -a)),

S6 = sinh (Aq(I- CI)).

(68)

Thus, once again, four nondimensional parameters I?, p. c( and ;1 enter into J. Here the stiffness ratio p and the load parameter 1 are again given by (35)? and (35),, however the ripple parameter q and the volume fraction c( are now no longer given by (35), and (32), but are instead given by

(69) Even so, the parameter ranges (36) remain in force. It is to be noted that interchanging the ply ordering results in (a, j3) being transformed into (1 -a, 1/a). Once again, if the material parameters (a, fi) take the values in (37), then the two-ply problem considered here reduces to the noncomposite construction as studied by Sawyers and Rivlin (1974, 1982). In addition, as before, genuinely composite constructions obey j? # 1, a # 0 and IX# 1. The necessary and sufficient condition for bifurcation to take place in the two-ply construction shal1 be written as Y(%, 9.0, a)

E

det J = 0.

(70)

The analysis of (70) bears many similarities to the analysis of either (32) or (33). It is to be noted, however, that the size difference of the coefficient matrix J (64 entries) as compared to either JF or JB (36 entries each) serves to complicate the ensuring analysis. In view of the fairly detailed exposition given in the previous section, we shall here summarize the results of the analysis of (70) and refer the reader to Qiu (1992) for additional details. Once again the no-load case A= 1 always furnishes a solution to (70) as then the first and third column of J are identical. A large i expansion of J can be shown to yield W(%, q, p, a) = w(q, & a)eq”L7+o(eq”A7),

(71)

for genuinely composite constructions where o(q, j, a) = a( I- flJ2 sinh (?a) sinh [q( 1-a)]

f 0,

(721

which we note in fact coincides with the functional form given in (47). Thus we normalize the bifurcation equation (70) so as to give (73) For fixed (vi, 6, a) corresponding to genuinely composite constructions, numerical evaluation of (73) reveals the existence of three roots A obeying A:> 1 and also confirm that II(A, 4, fl, a) -+ 1 as A + 00 (Fig. 8) so as to provide confidence that all possible roots are accounted for. These three roots shall be denoted by 1 < I’ <

I” < L”’

(74)

and the functions taking (q, j, a) to these roots shall be given by cDi(qyp, a), @ii(y* /I, a) and cDiii(rl,j?, a). Numerical procedures for constructing these functions have been developed and curves of load parameter I vs ripple parameter 11for various values of (8, CX) are given in Figs 9-13. Each of the three curves in each figure gives rise to a family of failure load parameters I owing to the discrete nature of the ripple parameter 9 as given in (69) ,. These shall be denoted by

Buckling and wrinkling of neo-Hookean laminates

1165

Fig. 8. The two-ply function II@,, u, /?, a) for the case g = 2, B = 2, a = 0.25. There are three roots, besides L = I, in the two-ply case.

0

1

2

3

4

5

6

7

8

9

tl 10

Fig. 9. The functions CD&,& a), &(q, j?, a) and cb&. & a) for the case fi = 2, a = 0.3.

a

1

2

3

4

5

6

7

8

9

loq

Fig. 10. The functions @;(q, 8, a), @&, /?, a) and @&, /3, a) for the case p = 3, a = 0.3.

1166

0

1

2

3

YUE

QW er

4

5

al.

6

7

0

9

ld

Fig. 11.The functions Q,(a, 8, GI).Q,,(q. /I. a) and Qiii(q, /I. u) for the case /3 = 3, a = 0.5.

0

0

12

3

4

5

6

7

6

9

10

rl

Fig. 12. The functions Q,(v, /I, Go,Qi,(q. /3, c() and Qe(q, p. a) for the case /I = 2, a = 0.1.

16-

14-

12-

2~,~llP,O.~) 0. 0

1

2

3

4

5

6

7

a

9

10

rl

Fig. 13. The functions Q,(q. 8. TV), Q,,(q. /J, c() and Qi,,(q, 8. c() for the case /? = 4, a = 0.9.

Buckling

and wrinkling

of neo-Hookean

21 =

@i(mn12/f19

P9

@),

2;

Qii(t?l7tZz/l,,

fi,

a)*

=

laminates

1167

The corresponding families of failure thrusts (Ti, Tz, If:} with m = 1,2,3, . . . are found from (75) using (14) and p = 1-‘j2. It is found as u]+ 0 that the three curves Q,(q, p, a), %(V, 8, Co and %(q, PTNY) obey

AS q -+ cg it is found that both @i(q, p, a) and @ii(q, fi, a) approach a common asymptote whereas @iii(q,p, a) tends to a different and larger asymptote. The value of these asymptotes can be obtained by a large pl analysis of (70) similar to that conducted in Section 3 [viz. eqns (56)--(G)]. To this end it is found that Y(A, rl, P, a) - eq(“+‘)Z6(Awhere j”,(a) and f#.,

1)4[f, (n)]*f2(A, /I),

(77)

8) are again given by (61) and (62). Thus

where /1, and xm = ad are again as depicted in Fig. 6. Of the functions @i(l, 8, a), ~ii(?, /I?,a), ~iii(V],/?, ~1) it follows from a comparison of (76) with (52), (78) with (53), and Figs 9-13 with Figs 4-5 that : Qi(t/, 8, ct) resembles dDF(q,/I, a), @ii(q, 8, a) resembles mB(q, 8, a), and @iii(q,8, a) resembles both gF(q, p, a) and @B(ff,B, a). Indeed, as either a + 0, a -+ 1 or /I + 1 the functions @i(q, & n) and @ii(qt /3, a) approach the, respective, lower (Bexure) and upper (barrelling) noncomposite curves as given in Fig. 3. Furthermore, since the noncomposite case involves no additional families of buckled solutions, the family of load parameters obtained from ~iii(vl, p, GI)will evidently no longer exist if either a = 0, a = 1 or /3 = 1. If /I + 1 then the asymptote xm(
We note from Figs 4 and 5 that such curve separation by asymptotes is not a feature of the symmetric three-ply problem. To inquire further into the vahdity of (79) we note from (76),, (78)* and I, < I__ that

1168

(80)

must have at least one root II. By numerically evaluating Y(,?,, 9, p, cl) with varying 11and fixed pairs of (p. a), we find no exceptions to the inequality

Furthermore by numerically evaluating Y(n”, , q, @.a) with varying v and fixed pairs of (8, a) we systematically find that there is only one value of of= yli,such that

So q1 = qn(fi, a) satisfying @ii(Vd? p, cx)-;i,(P)= O

(83)

is, in all cases that we have examined, the unique root of eqn (80). This, in conjunction with (76)2, (78)?, (81) and E;, < xx(p), suggests the generality of (79). The results of this section provide a fairly complete description of the families of planestrain buckled solutions for the two-ply neo-Hookean laminate construction. A similar description for the (symmetric) three-ply neo-Hookean laminate construction was provided by the results of the previous section. It is tempting to speculate on the possibilities for the families of plane-strain buckled solutions for general N-ply constructions involving alternating plies of two different neo-Hookean materials. We here note that the following conjecture is consistent with all of our results: for a composite laminate construction consisting of N alternating plies of two different neo-Hookean materials with shear moduli $‘) and $‘I), we conjecture that there will exist N+ 1 families of plane-strain buckled solutions. Two of these families should involve load parameters I that asymptote to 1, as given by (63) and these families should be the extension to the composite case of the solutions obtained by Sawyers and Rivlin (1974). The other N- 1 families should only exist for genuinely composite constructions and all of them should involve load parameters 1” that asymptote to x,(b) as given by the root of (62) with fi = /.&“)/$‘). Here we note from (62) that a necessary consistency relation for this conjecture, xX((B)= x,(p-‘), is satisfied.

5. CHARACTERIZATION

OF THE DEFORMATIDN

MODES FOR THE TWO-PLY

PLATE

In both the noncomposite construction and the symmetric three-ply construction, the construction symmetry with respect to X, gives rise to buckled configurations that correspond to either flexure (U,( -X2) = U2(XZ)) or barrelling ( Uz( - X,) = - U,(X,)). However for the two-ply construction, genuinely composite constructions (fi # 1, 01# 0 and tl # 1) no longer involve construction symmetry with respect to X2. Thus, one would anticipate that the associated buckling configurations should exhibit a mixed-mode flexure and barrelling character. It is the purpose of this section to further examine this issue. In the two-ply construction, U,(X?) is written as : Uy)(X2) = L:” cash (QX,) + L$” sinh (s2X2) + MY’ cash (IoX*) + M$” sinh (;lfzX,), (84) where j = 1, 2 for ply-l or ply-2, and I given by (66). can be normalized so that //I(/” Thus the full (linearized) deformation

E

ITI = 1.

(85)

can then be obtained from (15). (20)2,j and (24) ,.

Buckling and wrinkling of neo-Hookean laminates

1169

2I 1.5 l-

0.5 -

o-

-0.5 -

~---

-1 -

I

! I

-F-----

-1.5 -

-2’ Fig. 14. Deformation of the two-ply composite plate under thrust for 1,/12= 1.2, E = 0.1, m = 2 (so that n = mnlJI, = 5.236) /I = 3, CI= 0.5. Here the m = 2 buckled configuration is from the first family of buckled solutions so that A = ,I’2= @,(X236,3,0.5) = 3.199 (see Fig. 11). and T zx 7.; = 5. t63($“,4”’ +p”“A”“).

As an example, we display in Figs 1416 all three of the deformations for ripple number m = 2 associated with the construction /I = 3, a = 0.5 and 1,/i2 = 1.2. The corresponding families of solutions to the bifurcation equation (73) were shown previously in Fig. 11. In these deformation figures, the value of the order parameter E [refer to (15)J is chosen so as to make the deformed configurations distinguishabIe_ In order to examine the mixed-mode character of the deformation, we decompose the vector 1 as follows : I= a = (a,, 0, q,O, c = (0,

~29%

c-430,

a+b+c+d,

as, 0, Q?, Oj’, c&O,

cs)=,

0361

b = (0, b,

0, bq, 0, bt,, 0, hfT,

d = {LO,

4,0,&

0, 4, O}?

where

Fig. IS. Deformation of the two-ply composite plate under thrust for II/t2 = 1.2, E = 0.05, m = 2 (so that q = ~~Z~/i, = 5.236), @= 3, a = 0.5. Here the m = 2 buckled con~guration is from the second famiiy of buckled solutions so that R = A; = @,,(5.236,3,0.5) = 3.457, and 7’ = T’! = S.g9O(n’“A” +P(“JA”‘J).

(87)

1170 3

2-

Buckled Configuration

l-

Homogeneous Configuration

\

O-

I

-l-

original -2-

Configuration

-3 -1.5

-1

-0.5

0

0.5

1

1.5

Fig. 16. Deformation of the two-ply composite plate under thrust for i,/iz = 1.2, E = 0.3, nt = 2 (so that q = mnl,/l,= 5.236), p = 3. 51= 0.5. Here the m = 2 buckled configuration is from the that d = i.‘; = @,,,(5.236,3.0.5) = 7.345, and of buckled solutions so third family T = 7-e’ = 19.537(P”‘,4”’ +p”“A”‘~).

This decomposition,

in the sense of the Euclidean 1/1112= I/all’+

By this decomposition,

the function

Ilbll’+

norm used previously

in (85), satisfies

llcl12+lldll'.

Uz = U2(X2) is expressed

(89) as (90)

where cash (AQX,),

- R, d A-2 d RI,

(91)

Uh,(X2) = b2 sinh (CD’,) +hl sinh (LQX?),

-_ R, < A-2 d Rz.

(92)

U;(X,)

= a, cash (QX2)+a

Uc,(X,) =

c2 sinh (ax,)

+ c4 sinh (IK2X2),

i - c2 sinh (QX?) - c., sinh (IRX2), d, cash (QX,) + d, cash (AOX?),

GW2)

=

-d,

Note that the four functions

cash (fix,)

- d3 cash (&2X2),

in the decomposition

-R,


0 d X2 d Rz,

-R,

G X2 < 0,

0 Q X, < R?.

have the following

(93)

symmetry

(94)

properties

:

Buckling

and wrinkling

of neo-Hookean

= W-X,),

VW*)

uw,)

laminates

1171

= W-X,),

Ub,(X,) = - Ub,(-AT*), u:(x,) = - U”,(-X2).

(95)

We shall say that if b = 0 and d = 0, then the whole deformation is one of pureflexure ; similarly, if a = 0 and c = 0, then the whole deformation is one of pure barrelling. The four decomposed deformation portions associated with this decomposition for the example presented in Fig. 14 are displayed in Fig. 17. Since the deformation presented in Fig. 14 was associated with the first buckled family of solutions, i.e. the associated load parameter A = A’= ~i(5.236,3.0,0.5) = 3.199, one might anticipate that flexure would dominate in the decomposition. Indeed one finds in this case that llall 2+ /[cl/2 = 0.7049 and 1)b/l 2+ JldJI2 = 0.2951. Similarly, Fig. 18 gives the four decomposed deformations for the example presented in Fig. 15. Since this deformation is associated with the second buckled family of solutions one might anticipate that barrelling would predominate. Indeed for this exampleonefindsthat IIaI(2+II~j12 =0.3859and Ilbl12+lldlj2 = 0.6141.

1.5 -

0.5 -

-0.5 -

Buckled

-1.5 -21 -1.5

I -1

-0.5

0

0.5

1

1.5

Fig. 17a. The smooth flexural deformation portion at E = 0.0001, M = 2, q = 5.236, p = 3.0, c( = 0.5, I, = q(5.236, 3.0,0.5) = 3.199. Here llall’ = 0.5317. The overall deformation was shown previously in Fig. 14.

2

1

1.5 -

l-

0.5 -

O-

-0.5 -

-l-

Homogeneous Configuration

-1.5 -

.-______ __...!L_

‘\original Configuration _______

-2 -1.5

I -1

-0.5

0

0.5

1

1.5

Fig. 17b. The smooth barrelling deformation portion at E = 0.0001, m = 2, q = 5.236, b = 3.0, a = 0.5, 1 = $(X236, 3.0,0.5) = 3.199. Here [lb11* = 0.1166. The overall deformation was shown previously in Fig. 14.

1172

YUE

QlU et al.

2 1.5-

l-

1'

Homogeneous Configuration

___~____~_.

~._

0.5 -

o-

-0.5 -

-1 -

+ original Configuration _

BuCkkd Portion

-1.5 -

_---_A---__ -2 -1.5

-1

-0.5

0

0.5

1

1.5

Fig. 17~.The residual flexural deformation portion at E = 0.0001, m = 2. r~= 5.236, /I = 3.0, a = 0.5, 1, = Qi(5.236, 3.0,0.5) = 3.199. Here llcll’ = 0.1732. The overall deformation was shown previously in Fig. 14.

2 __a \

1.5 -

_______-___-_-

e

Homogeneous Configuration

l-

0.5 -

O-0.5 -

-l-

+ /

1

-1.5 -2 I

,I,

Fig. 17d. The residual barrelling deformation portion at E = 0.0001, m = 2, ‘1= 5.236, b = 3.0, a = 0.5, I = @,(5.236, 3.0,0.5) = 3.199. Here lldll* = 0.1785. The overall deformation was shown previously in Fig. 14.

The four decomposed deformation portions can, as we now show, be used to characterize the interfacial deformation smoothness discontinuity at the ply interface X2 = 0. For this purpose it is convenient to introduce the notation [[ ]]]0 to indicate the jump in value across this interface. Clearly.fiom (91), (92) it follows that

for n = 0, 1,2,3,. . . . Here (n) denotes a derivative of order n with (0) indicating the undifferentiated function. Similarly (91) and (92) also directly give that

[[(U~y2q10 = 0, [[(u”,)““+“lllo = 0,

(97)

for n = 0, 1,2,3,. . . , where the derivatives appearing in (97) are standard one-sided derivatives. To address the possible discontinuity in the (one-sided) derivatives (U1;)(*“+‘) and (U$(zn) at the interface X2 = 0, we note that the interface conditions (9) with (19) give the following conditions, respectively, on U2(X2) and U’&Y2):

1173

Buckling and wrinkling of neo-Hookean laminates

w*lllo = 0, wwl

= 0,

(98)

which along with (96), (97) give

= 0. RmY”Il I0 = 0, [[(U~)(O)lll~

(99)

However (U$)@+ I), (U$““’ for n = I, 2,3,. . . are not continuous across the interface x2 = 0. Thus by using the boundary and interface conditions, we conclude, for - R, < X2 < R?,, that U!$,and U",and their derivatives of any order are continuous ; (U;)(O’, (U;)(‘) and (U#*")are continuous, and (U#", (U<)"'and (U:)(“+‘) are continuous. Furthe~ore one obtains from (90)-(94) that [[U’22”‘]]I0= [[(up’]]~~ W ‘2”’ “]]10 = [[(Up+

q.

= -2(R%,

+(nszpd3),

WV

= -2pn+

‘C2+(AR)2”+ ‘CJ,

(101)

2.

1.5 -

Configuration l0.5 -

O-

-0.5 -

-l-

-1.5 -

-2' -1.5

-1

-0.5

0

0.5

1

1.5

Fig. l8a. The smooth flexural defo~ation portion at E = 0.0002, m = 2, q = 5.236, p = 3.0, a = 0.5.1 = @,,(5.236, 3.0,0.5) = 3.457. Here )/a //’ = 0.0357. The overall deformation was shown previously in Fig. 15.

2

1.5 l-

._____.____ f w

I

Hom~ge~~s Configuration

___-----_

I

0.5-

O-

-0.5 -

-1 -

-1.5 -

Fig. Mb. The smooth barrelling deformation portion at E = 0.0002, m = 2, q = 5.236, p = 3.0, a = 0.5, I = (P,,(5.236, 3.0,0.5) = 3.457. Here llbll* = 0.5835. The overall deformation was shown previously in Fig. 15.

1174

YUE

QILI et al.

Buckled /Portion

-

G&if Configuration 1

-2

I

-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 18~.The residual flexural deformation portion at E = 0.0002. m = 2,~ = 5.236, j = 3.0, c( = 0.5, i, = @,,(5.236, 3.0. 0.5) = 3.457. Here IIc//’ = 0.3502. Theoveralldeformationwasshownpreviously in Fig. 15.

2

%---

---_____--___._--__ ._...

\ Homogeneous

1.5 -

Configuration

-

1

-I _.

l-

1

0.5 -

0

-0.5

I

r : ximtion1

I

t -1 c

-1.5

L_

I

-21 -1.5

-1

~ _____..__.____.__ 0

-0.5

0.5

1

1.5

Fig. lgd. The residual barrelling deformation portion at c = 0.0002. nl = 2. 7 = 5.236, b = 3.0, r = 0.5, i = @,,(5.236. 3.0.0.5) = 3.457. Here /idI/’ = 0.0306. The overall deformation was shown previously in Fig. 15.

where II = 1, 2, 3,. . and a superscript number n inside parentheses indicates the nth derivative, and a superscript number n without parentheses indicates exponentiation to the power of n. In view of the preceding discussion, we shall refer to UU,(X,) as the smooth flexure part of the decomposition, Ub,(X,) as the smooth barrelling part, &(X2) as the residual,flexure part, and U”,(X,) as the residual barrelling part.

With eqn (102), by using the interface continuity condition (98), we find that d, =~a,+~a,

where

and

D,

c2= -ob2-

AD, Db’,

(103)

Buckling and wrinkling of neo-Hookean laminates

Residual flexure

0.4 -

Residual barrelling

Smooth barrelling

0.2 -

0

0

1175

I

12

3

4

5

6

7

6

9

la

51

Fig. 19. The four deformation percentages as they vary with r) for the case of B = 3, a = 0.5, associated with the first root 1’ = Q,(r), 3.0,0.5) of the buckling equation (73). The vertical line at 1 = 5.236 provides correlation with Fig. 17.

D = (p+ l)(l -A’),

D, = (/Il)(l +A*),

D2 = 2(/Y- 1)J2.

(104)

On account of (88), (95), (102)-(104), we conclude that if the deformation is one of pure flexure, then c2 = c4 = c6 = c8 = 0 so that it is in fact smooth flexure. Similarly, if the deformation is one of pure barrelling, then it is in fact smooth barrelling. Under the normalization (89, the values ]]a]12, (1b]] *, ]]c]]’ and lldll* correspond to the percentage of smooth flexure, smooth barrelling, residual flexure and residual barrelling within the overall buckled deformation. We find, for a given two-ply construction characterized by a particular (j?, c(.),that these four deformation type percentages vary with ripple parameter q. Figure 19 displays these changing percentages for the first buckled family of solutions for the case /I = 3, a 4 0.5. Similarly Fig. 20 gives these changing percentages for the second buckled family of solutions, here however we only depict values of v 2 1 in view of numerical difficulties associated with the large values of 1 for this family near q = 0 [see (76),]. These numerical difficulties also prevented us from calculating these percentages for the third buckled family of solutions. As anticipated, smooth flexure is dominant for the first buckling family, depicted in Fig. 19. We note, however, that this

7-----l 0.8 -

Residual Bexure 0.4 -

/

o.i, , ,fs~y7y~::~;~mg f

rl

1

2

3

4

5

6

7

a

9

10

Fig. 20. The four deformation percentages as they vary with r) for the case of /I = 3, a = 0.5, associated with the second root Iz”= UQ(V,3.0,0.5) of the buckling equation (73). The vertical line at r) = 5.236 provides correlation with Fig. 18.

1176

YUE

et al.

QU

dominance lessens as the ripple parameter u increases, resulting in a percentage growth of both the smooth barrelling percentage and the residual deformation types. Similarly, as one would anticipate, the smooth barrelling percentage dominates the second buckling family, depicted in Fig. 20, with a percentage growth in the other deformation percentages for increasing q. Acknowledgemenls-This work was supported by the Composite Materials and Structures Center, Michigan State University, under the REF program, and by the U.S. Army Research Office through Grant DAAL 03-89-G-0089.

REFERENCES Biot, M. A. (1965). Mechanics of Incremental Deformations. John Wiley & Sons, New York. Biot, M. A. (1968). Edge buckling of a laminated medium. Inr. J. Solids Structures 4, 125-137. Pence, T. J. and Song, J. (1991). Buckling instabilities in a thick elastic three-ply composite plate under thrust. Im. J. Solids Struc&res 27, 1809-1828.

Qiu, Y. (1992). Buckling of a two-ply nonlinear elastic plate. Master Thesis, Department of Materials Science and Mechanics, Michigan State University. Sawyers, K. N. and Rivlin, R. S. (1974). Bifurcation conditions for a thick elastic plate under thrust. Inl. J. Solids Structures 10,483-50 1. Sawyers, K. N. and Rivlin, R. S. (1982). Stability of a thick elastic plate under thrust. J. Elasticiry 12, 101-124. Song, J. and Pence, T. J. (1992). On the design of three-ply nonlinearly elastic composite plates with optimal resistance to buckling. Structural Optimization 5,45-54.

APPENDIX

Y,(n. q. B, a) FOR THE SYMMETRIC THREE-PLY

A: THE EXPRESSION

PLATE

Both the flexural and barrelling determinant, ‘+‘,(A,q, 8, a), for the symmetric three-ply plate as given in (32) and (33) can be expressed as

where H,=H,@,q,a),.,,=

{sinh(qK,),sinh(qh-,)

,..., sinh(ph-,,)j,

p = [~,,,(P)l,ox101

(A.3)

.,an,...,a4,a5JT.

, = in-‘, a-3,..

A = A(n),,,

(A.2)

(A.4)

The rci= ~~(1,a) (i = 1.2.. . , 10) in (A.2) have different values for the flexure and barrelling cases, but by using the exchange rule (38), one can convert between these two cases. The K,S for the flexural deformation are given by: ICI

=1+1,

ti>=E.-I,

fc? = i.+2a-I.

lid = I-2a+l,

lie = 2al-1,-

x5 = 2ad-1+1. ti, = 2aE.+2a-I.-

h’8= 2al-2a-i+l h’g= la+a

1,

1 = (A+ 1)(2a- I), = (A-1)(2a-I).

= (A+ 1)a.

fcIO= La-a

= (&])a.

(A.%

and after exchanging 1 c) I, and a tt ai, in the expanded forms of (AS), one obtains for the barrelling deformation that K,

=

a+],

ti* =

li, = 2ai.-I+ K5 =

1+2a-1,

K,

2a+2al-

=

~~=2a-2al-l+1=

K~ = (A+ l)a,

-(,I-

I,

I),

xq = -(2aA-1-l). Kg

1 -I

=

-(rZ-2a+l),

= (A+ 1)(2a- I), -(A-1)(2a-I),

~~~ = -(A-])a.

(‘4.6)

Buckling and wrinkling of neo-Hookean laminates

1177

Table A. 1. The elements of the matrix P of (41) for the symmetric three-ply plate \\j

-4

1 2 3 4 5 6 7 8 9 10

-3

4-3

p’,., PL, P,,?, PL, -PI.-, -Pg.-, 0 0

-PI.-, PI.-, -PI.-, PI,-3 PI,-, -pi,-, 0 0

-2

-I

I

0

t -4P,,-, -4P,.-, -‘PI._ I PI.0 -k 4P,.--4 l 4P,.-‘i 4P,.--4 -4P,,-, P,.o -‘p, I 4P,,-4 4P,.-* P,.o p,:, 4P,.-4 -4P,,-4 P,,o -P3.1 12P,,_, -14P,,_, 22P,,m, -4P,,-, -12P,,_, -14P,,_, -22Pj,_, -4P,.-, -A-

L

* Pm

I

-64P,,_, 64PI,-,

2 -36P,,m, -36P,._, -2P,.,

3

-‘PI., -12P,._,

12p,,-,

-2p3,o

4

iI.,

L

-2PVl -2P,,, -36P,~_,

- 12P,,-, 12P,.-, 28P,._,

P,., P,,, -9P,,_,

-36P,,_, * Pg.2

-28P,,_,

-9P,,_, * Pg.4

-Al.,

5

PI,-, -PI.-, -Pt.-, PI,-, -pt.-, PI,-, PI.-, -PI,-, 0 0

Each element of the 10 x 10 matrix P is a multiple of one of 17 distinctive polynomials which are denoted by * in Table A. 1. Here, and in what follows, the column index j of the matrix entry P,,, is offset by - 5 for consistency with the exponents of I given in (A.4). The 17 distinctive Pi,j(B)s are given by :

PI.-4(B) =

-u+

P,.-,(B) = (8-- lY>

I)*,

P,.,(B) = -WB*- IV+%

P,.-,(P) = ‘Wf-28+3), P,,,(B) = 2(11/32-38~+11),

P,.,(B) = 4(7/9’- 108+7), P,.-4(B) = -(B-l)@+

PW(B) = -(98’-148+9), P,,,(B) = -2(P-

P,.-,(8)

P,.,(B) = -2(8-l)(V+S

1)(38- l),

P,.4(B) = -(B-

l),

Pg.--2(8) = 16/V-l),

1)(8-7),

P,,,(B) = 16(8- 1)(38-2),

= 32(8-u,

P,.,(B) = -32(8-1)(28-l),

Pw(B) = 16(8- 1)(38-4). P&B) = 16(@- 1)(8-2).

APPENDIX

‘I’@, 1. /?, G()FOR THE TWO-PLY

B : THE EXPRESSION

PLATE

Y(i, q. /?,a) can be expressed as Y(A,

rl.

B, Co=

WA

@.I)

where I\={rz-6,L-5

,...,

10,..., 16,117}T.

H = H(1, rl, a) = {cosh(qK,),cosh(qK-,),.

,coshqK,,)},

P.2) (B.3)

and P is a 13 by 14 matrix with entires Pi&?) (B.4) The

Ki = Ki@,

a) (i =

0,. . . , 12) are given by Kg = 0,

K,

=

u2 =

(A+]),

K, = (A-2a+l), KS =

(2al-1+

K,

(A-

=

=

(1+2a-

l),

Kq = (A-- I), l),

1)(2a-

up = (A+ ])a, K,,

(B.5)

@+1)(1-a),

I),

Kg = (A+ 1)(2a- l), Kg =

(2al-A-

tiIO = (I.- l)a, K,2

=

(1-1)(1-a).

l),

1178

YUE Qw et al. Table B. 1. The elements of the matrix P of (B. 1) for the two-ply

i \’ 0 I 2 3 4 5 6 7 8 9 10 II 12

i I

2 3 4 5 6 7 8 9 IO II 12

-5

0

0

-&,

A-6

-p4.7

P 4.7 -P2.-6 0 0 0 0

0

2

1

* P 4.1 2OP,., 2OP,., &., 14OP,, 14OP,,, 2OP,,, * P 9. I ;,

-4

P 4.7 -P4.7 - Pd.7 P 4. 7 -P4.7 P 4.7 P 4.7 -P4.7 0 0 0 0

--Pd.-6

.i 0

-6

--Pd.-4 6Pz,._4

-6Pz.LJ 6&-1 -6Pa.7 6P,., -6P>._4 -p*,.-, -p’,,.-,

3

-3

plate

-2

-1

0

L

256P,,~ 63P,,,

-pt.2 llP2.--h -llP1.~,

-2p4.7 -2p4.7

27P,., -llP2._,

- 128PJ., - 128P4,, -128P,,,

1.-d

-2p,,-,

4

5

768P4.7 0 ; 4.5 223P4., -P4.4 38P,,_, 14P4.1 -3lP4.7 -38P2,_, 14P,, -31P4.7 223P4,, -4;f,,_, -3lP,,, 38*P,-, 14&, -183P4,, 223P,, -166P,,, 66Pa.7 183P,., 223P4., 166P‘l.T 66P,., -38P2._6 4lPz.-0 -3lP,, 14PJ.7 -384P4,, -*P,., -384P4,, --*P%, A.,

0 -Pw -41P2.mh 41PL,

I I.,

-*p, l.2

-384P,,, - 384P4,,

1.4

PO,_, = -648.

Pd._,= PZ+2j+

-A

P 4,m2 = 278’-58fi+27, P,,,= 14O/!F-344jI+140,

Pd.>= 183/C?-4988+183.

P4,4= 166&260~+

Pd.5= 66/I’-

loog+66,

P 4.6 = 13/F-228+

1.

P,,_,= - 168+16.

166, 13,

P,,,= 256/Y*-416/I+

Pv,mL= 32/3’-112/I+80, P,,,= -320fi2+704fl-384,

P,,>= 448/+864fl+416,

P9,4= 2568’-

P9.5= - 16Ofl’+288fl-

5928+ 336,

P9.6= 32p’-488+

16.

P,,,-d = 16fl’-

P 11.1 = -384~2+7@4~-320, P ,,,4 = 336fi2--592/?+256,

P 11.5 = - 128fi2+288b-

P Il.6 = 16+48/S+32.

160,

128,

168,

P 11.0 = 16Oj?-416pf256, , P ,,,? = 4168--8648+448,

P,,.m? = SO/?-1128+32,

1.0

P 4.7 -Pd., -Pa,, -A.; P 4.7 P 4.7 -P‘l., 0 0 0 0

P,,,(@s that are denoted by

P 1,me =/F-l,

Pa,,= 116p12-184fi+116,

Pd.,= fi1-2p+

h

-k

P,,_,= 6/12--4/1+6,

I,

P,,_,= 18/I’-4/3+18.

0

-3k _(, -13p;:, l3P34.7 3PLh

PO,,= 768/S2-- 1408fif768,

Pa5 = 256/?-578/?+256,

0

-3P2:, 3P?.

The particular values for the P,,j(/3)s are given in Table B.1 [in terms of 31 distinctive *I. The 31 distinctive P,,@)sare :

-*p, I.”

7 ____

-pd.,

PI 1.5

-*PM

6

* :P,

0

- p4.0

P 4.7 -12P,,m, P 4.7 12Pz._h 63P,., P 4.7 --I;P,m, 63Pd.7 - 116P,, 63Pd.7 1I@,., P 4.7 l2P,, _h -128P,,,

ilk, -2~4.7- -27P;‘,

18P4.7 18P4.7 - 2p4.7 -2p,,.-4 -2p, -2p,.-4

0

160,