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This is to certify that the
thesis entitled
Buckling of a Two-Ply

Nonlinear Elastic Plate

presented by

Yue Qiu

has been accepted towards fulfillment
of the requirements for

Master ' s Mechanics

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_.-—_-.-___ _._ _

BUCKLING OF A TWO-PLY
NONLINEAR ELASTIC PLATE

By

Yue Qiu

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Materials Science and Mechanics

1992

ABSTRACT

BUCKLING OF A TWO-FLY

NONLINEAR ELASTIC PLATE

By

Yue Qiu

The buckling of a two-ply nonlinear elastic plate is studied in the context of
finite deformation incompressible nonlinear elasticity. The formulation proceeds by
using the theory of superposing an incremental nonhomogeneous deformation onto a
finite homogeneous deformation. Numerical procedures are then used to investigate the
bifurcation and to determine the buckling stretch ratio and buckling thrust. The buck-
ling mode is studied and it is found that the buckling deformation of the two-ply com-

posite plate without symmetry in direction of thickness is of a mixed mode character

involving both flexure and barrelling.

ACKNOWLEDGMENTS

I would like to express sincere appreciation to my advisor, Professor Thomas J.
Pence for his very helpful guidance and support throughout this work. I would also
like to express my special thanks to my parents and my wife. They are always my

sources of encouragements.
The financial support for this work from the Research Excellence Fund for com-

posite materials, administered by the Michigan State University Composite Materials

and Structures Center, is also gratefully acknowledged.

iii

TABLE OF CONTENTS

 

. Introduction ......................................................................................................... 1
. Problem Formulation .......................................................................................... 4
2.1. Problem Description .............................................................................. 4
2.2. Formulation ............................................................................................ 5
2.3. Bifurcation from the Homogeneous Solution ..................................... 10

. Solution of the Buckling Equation for Correlation of

Load Parameter A with Mode Parameter n ..................................................... 17
3.1. Asymptotic Analysis for Large 1 ....................................................... 19

3.2. Asymptotic Analysis for Large 11 ....................................................... 23

3.3. Numerical Analysis ............................................................................. 26

. Deformation ...................................................................................................... 34
. Discussion ......................................................................................................... 41
List of References ............................................................................................ 52
Appendix .......................................................................................................... 54

iv

1. INTRODUCTION

Buckling of load bearing plates is a major type of structrual failure in which
the plate reconfigures itself in such a fashion that often causes it to lose the capability
of carrying loads. So this kind of problem has been of concern for a long time by
numerous researchers. For example, Sawyers and Rivlin [3](1974), '[4](1982)
employed the theory of small deformation superposed onto finite deformation [1] to
determine the critical conditions of bifurcation. These conditions are derived upon gen-
eral strain energy function and their results can be applied to the determination of. bifur-
cation conditions corresponding to any specified strain energy function. They published
their extensive studies on the instability of rectangular plate of incompressible isotro-
pic elastic material with neo—Hookean strain energy function subject to a thrust. Biot’s
[8](1968) study on the edge buckling of a laminated medium is another issue in this
area, according to which, while the top and bottom faces of a semi-infinite laminated
plate are kept from normal displacements, buckling takes place locally along the edge
subject to a compressive stress. Pence and Song [9](l990), [10](1991) have published
their research on the buckling instability in highly deformable composite laminate
plate. They examined in detail the buckling instability of a thick rectangular three-ply
sandwich composite plate with material and geometrical symmetry in direction of thick~
ness. The plate they considered is composed of three stacked rectangular plies with per-
fectly interfacial bonding. These plies are made up of two different incompressible
isotropic nonlinear elastic materials. The top and bottom plies are identical both in
material and in thickness. The buckling instability of this plate under thrust has been
extensively studied, and carried out a lot of valuable results. Unlike the noncomposite
case (Sawyers and Rivlin [3], [4]), this three-ply composite case gives rise to addi-
tional families of buckled solutions. 80 the question arises as to how these new fami-
lies of solutions are correlated with laminate number n. In order to study this question
and, motivated also by the purpose of investigating the buckling behaviors correspond-
ing to the different way the plies stacked, we carried out a study on the buckling insta-

bility of a two-ply composite plate without symmetry in direction of thickness.

2

Generally, in the class of problems of this type, if the plate is composed by n plies, the
possibility of bifurcation requires the solution of a 4n x 4n system of equations. With
the symmetry in direction of thickness, the system can reduce to a 2n x 2n one. There
are various number of ways to stack plies, with or without symmetry. The investiga-
tion on the two-ply problem may have basic meaning. It can give a view on whether
and how, to some extent, the buckling behaviors will vary with the number of plies
and on the situation with or without symmetry. In this thesis, we present our studies
and the results obtained on the instability of a thick rectangular nonlinear elastic com—
posite plate made up of two plies of incompressible isotropic neo-Hookean materials
under a total thrust.

In Chapter 2, the problem is described and the basic boundary value problem is

formulated. The composite plate of dimensions 211 x 212 x 213 composed by two plies

of incompressible neo-Hookean materials are constrained by the requirements that (1)
the top and bottom faces are traction free and (2) the displacement and traction are con-
tinuous across the interface of the two plies corresponding to a perfect bonding. These
give the boundary and interface conditions (2.11)-(2.13). A thrust T acts on the faces
initially at X1 = :11. The incompressibility condition (2.4) and the governing equa-
tion (2.10) together with (2.11)-(2.13) form a complete boundary value problem.
Finally, these yield the bifurcation condition (2.67).

In Chapter 3, asymptotic analyses are conducted for the bifurcation condition.
These give similar results to those obtained by Kim in his study of the three-ply prob-
lem [13]. The numerical procedure used to solve the bifurcation condition and to
obtain the buckling stretch ratio and buckling thrust is described in section 3.3. The
results obtained are also discussed in this chapter.

We present the investigation on deformation modes in Chapter 4. The buckling
deformation of the composite plate is in mixed mode with flexural and barrelling char-
acters. In this chapter we decompose the deformation into four parts, namely, smooth
flexure, smooth barrelling, residual flexure and residual barrelling, and examine the

continuity of these parts across the interface.

3

In the last chapter, we briefly discuss the portions of these four parts of defor-
_ mations occupying a possible total deformation under thrust. We also give pictures of
these decomposed deformation modes. Finally, in closing this thesis, we investigate the

influence of lack of symmetry in direction of thickness on the deformation mode.

2. PROBLEM FORMULATION

Following Sawyers and Rivlin [3] [4] as well as Pence and Song [9] [10], we
shall formulate the solution system for the problem by superposing an incremental non-
homogeneous deformation onto a finite homogeneous deformation solution in this chap-

[612

2.1. PROBLEM DESCRIPTION

sz

 

 

 

 

 

21. i
LI) —> (:33
R2 ply-2 X1
21
R1
ply-1
/x,
211
‘ »

 

 

 

Figure 2.1. The geometrical description of a two-ply composite plate.

The geometrical description of the composite plate under consideration is
shown in Figure 2.1. By setting the origin of the coordinate system at the center of
the interface of the two plies, the plate occupies

--11 s X1 5 11,
ply-1: -R1 SXZSO, ply-2: 0SX25R2’ . (2.1)
—13 S X3 S, 13

in its undeformed configuration, where R, is the thiclmess of ply-1, R2 is the thickness

of ply-2, and R1+R2=212. Ply-1 and ply-2 are both of incompressible neo-Hookean

5
materials. But the shear moduli of ply-1 and ply-2 are different in general.
A thrust T is applied on faces initially at X1 = ill. Assume that the surfaces
initially at x2 = -R1 and x2 = R, are traction free. The surfaces initially at x3 = :13
are assumed to be kept from normal displacements by means of applying certain fi'ic-

tionless forces onto them.

2.2. Formulation

Upon loading on the composite material construction, first, a finite homoge-
neous deformation is taken into account, and second, linearized incremental deforma-
tions are superposed onto the homogeneous deformation following [3] [10].

The deformation within each ply is
x = x (X) , (2.2)

and the corresponding deformation gradient tensor is expressed as

F = 53%. (2.3)
The incompressibility of the material then gives the constraint
det (F) = 1. (2.4)
The left Cauchy-Green strain tensor is defined as
B = FFT, (2.5)

and its eigenvalues are denoted by AZ, 1%, kg, which are the squares of the principal

stretches 11, Lb 7b,. The most general strain energy density function for an incompress-

ible isotropic elastic material is given by
W = W(Il,12) , (2.6)

and the associated Cauchy stress tensor for a general incompressible elastic material is

then given by

 

 

6

aw aw aw 2
I—-pI+2(§:+Il§I-;)B-2(-a—I-2-)B, (2.7)

where p is hydrostatic pressure and 1,, I2 are the first and second invariants of B. We

further denote

W“) = W“) (11.12).

.. ' .. (2.8)
w“”=wW”apqL
for ply-l and ply-2 respectively. The Piola-Kirchoff stress tensor is given by
s = F“:. (2.9)

The equilibrium equations with the absence of body forces and inertia terms in tensor

form are

div sT = o. (2.10)
In the buckling problem considered, normal thrusts of magnitude T act on the two sur-
faces initially at X1 = ill. By assumption, surfaces initially at X2 = - R1 and X2: R2
are traction free and surfaces initially at X3 = :13 are kept in the original planes.

These consequently require the following boundary conditions to be satisfied:

$12=s13=0, onXlzill;
$21: $22 = s23 = 0, onX2 = —R1 and X2 = R2; (211)
S31: $32 = 0, 01) X3 = :13;
x1 — :1th1 , onXl — i11 , (212)
x3—j:l3 , on X3—:l:13 °

Here p is the overall imposed stretch in the X1 direction, it shall shortly be shown that
p is uniquely determined by the thrust T. Across the interface of the two-ply composite

material, both tractions and displacements are required to be continuous. That is,

 

52, , = SZiI . 9 on X2 = O (i=1,2,3);
x2 X2 (2.13)

on X2 = 0.

7

This corresponds to the assumption of a perfect bond.
The governing equation (2.10) with boundary conditions (2.11), (2.12) and
(2.13) and with the incompressibility constraint (2.4) consist of a complete boundary

value problem, which has exactly one pure homogeneous solution as follows:

x1= le ,
x2 = p‘1x2, (2.14)
X3 = X3 ,

and so gives the principal stretches

2.1 = p, 71.2 = p", 7t = 1. (2.15)

p 0 1 o
F: 0 P’ 0 , (2.16)
o o l

and the left Cauchy-Green strain tensor (2.5) then yields

2
P 0 O
B = 0 0‘2 0 , (2.17)
0 o 1

of which the invariants are

I1 =7tf+7t§+7t§= 1+p2+p"2,
I, = rfr§+r§r§+r§rf= 1+p2+p—2, (2.18)
2 2 2
E=MMM=L

The incompressibility constraint (2.4) is satisfied by the homogeneous solution (2.14).

Since F, B are constant tensors, following (2.6), (2.7) and (2.9), the equilibrium
equation (2.10) requires that the hydrostatic pressure p in (2.7) is individually constant
in each ply, it shall be denoted by p“) for ply-l and pa” for ply-2.

The strain energy density function for a neo-Hookean material is given by

 

 

_ “(11’3)
- 2 ,
01'
We 11‘” (Il -3>
2 9
(i1) “(11) (11- 3)
W — 2

for ply-1 and ply—2 respectively.
The Cauchy stress tensor (2.7) thus yields

1 = —p(j)I+u(j)B, j = i,ii.

By substituting (2.17) into (2.21), it yields

 

 

_p(i) +u(j)p2 0 0
z = 0 _p(1) H1(pp—2 o
_ 0 0 __p(i) ”1(1)

The Piola-Kirchoff stress tensor (2.9) becomes

 

 

_p(J) p—1+u(l)p O
s = 0 _p(l)p+u(1) p—l 0
_ 0 0 __p(l') +u(l)‘

From stress boundary conditions (2.11)2, 322 = 0 gives
p0) = u(1') p-Z, j = i, ii.

Substituting (2.24) into (2.22) and (2.23), 1: and 8 become

. p ‘P 0 O .
pp“) 0 0 O _2 as“). j=i,ii
o 0 1-P

and

(2.19)

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

s=u' 0 0 0 Es, j=i,ii.
O 0 l-p‘2

(2.26)

Note that the equilibrium equation (2.10) is satisfied by (2.26) as contributed by
the homogeneous solution (2.14). All the boundary conditions (2.11), (2.12) and (2.13)

are satisfied by the Piola-Kirchoff stress tensor (2.26) for the pure homogeneous defor-

mation (2.14).

Let T be the total thrust applied onto each of the surfaces X1 = i11 and let T0)

be the portions of T applied to material j only (i = i, ii), so
T = T“) +T‘“’.
Thus, for the homogeneous solution

(j) _ _T(j)

1; _ __
11 '
A0)

j = i, ii,

where A0) (i = i, ii) is the current area of the surface to which T07 is applied,
mm = 2R213p'1.

Using (2.25), (2.28) and (2.29), it is obtained from (2.27) that

'r = —21,(p-p‘3) (u‘”R,+u“"R2).

 

 

01'
(i)
(i)_ *1 R1
T ‘" (i) (ii) ’
. u R1+u R2
("1
(ii) ll u R2
T =- (i) (ii) '
u R1+u R2

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

10

2.3. Bifurcation From the Homogeneous Solution

The stability of the foregoing homogeneous solution for the two-ply composite
plate under thrust is to be investigated from now on, using the theory of incremental
deformations superposed onto finite homogeneous solution. Attention is restricted to
buckling that takes place in the Xl-Xz plane. Let u be the incremental deformation that
is to be superposed onto the homogeneous deformation (2.14). Then it has components

u1 = u1 (X1, X2) .
“2 = “2 (X1, X2) ’ (2.32)
U3 = 0.
The fully finite deformation 2 can be expressed as
2 = p—lxz + 8112(X1, X2) s (2.33)

X)

X3=X3 ,

where e is an order parameter which is used to obtain a linearized problem governing
bifurcation from the homogeneous solution (2.14). Following Pence and Song [10], we

use a superposed A to indicate quantities associated with the fully finite deformation
(2.33) and a superposed ' to indicate linearized incremental quantities associated with
the incremental deformations. Hence, the pressure field corresponding to (2.33) is
given by .

13(x, e) = p”) +e§(x,,x2,x3) +0(82), j = i, ii, (2.34)

and the Piola-Kirchoff stress tensor is given by

s(x,e) = s‘” +es(x,,x2,x3) +0032), j = i, ii. (2.35)

From (2.33), the deformation gradient tensor yields

.. p+£uL1 £111,2 0

F = £1.12. 1 p_1 + 8112’ 2 0 ’ (2.36)

O 0 l

11
its determinant is readily obtained as
der 1‘: = 1+e(pu2'2+p-1u1’1)+0(82) . (2.37)

From now on, we restrict attention to the linearized problem and omit any incremental

quantities of 0(82). The incompressibility constraint det F = 1 thus gives
pu2,2+p-1u1,1 = 0. (2.38)

The inverse of F is given by

O 0 l

——1

The Cauchy-Green strain tensor (see (2.5)) can then be expressed as

p2 + 21»:pu1,1 e (puz‘ 1 + p—lul’z) O

E = €(pu2’1+p-1u1’2) p-2+2€p-1u2,2 0' (2'40)
0 O 1
It follows that
s = — (p‘j’ +e§)F“+p0’F“r‘3 j = i, ii. (2.41)
When linearized, (2.35) becomes
s = s+e§. (2.42)

So, by making use of (2.24), (2.26), (2.38), (2.39) and (2.40), the linearized incremen-

tal part s can be written as

_ p-IF + 14(1) (“1'1“ p-Zuz, 2) 110) (“11+ p_2u1,2)
110) (“1,2 + 12-2112, 1) _ 913+ 211“)“; 2 0 . (2.43)
0 0 -13

ml
II

On account of (2.42), the governing equation becomes divs = divs+ediv§. As dis-

cussed previously in connection with (2.11)—(2.l3), the equilibrium equation (2.10) and

12

the boundary conditions (2.11), (2.12) and (2.13) are satisfied by the stress tensor 5 and
deformation x corresponding to the homogeneous deformation (2.14) by applying
(2.24). We shall now focus on the linearized incremental boundary value problem with
equilibrium equation
div 1‘? = 0 , (2.44)
and boundary conditions as in (2.11)-(2.13) are provided by substituting x with u and s
with §. Equation (2.44) then yields
_p—15'1+“(l)
'91—’2 +“(j).(“2,22+“2,11) = 0, (2°45)

(“1,11+ “1.22) = 0’

-i)- .3 = 0.
Equation (2.45)3 can be satisfied if and only if
13(X1: X2: X3) = 13(X1sxz) - (2-46)

Following Sawyers and Rivlin [3], we may obtain solutions for this problem in

the form

uz = cos (<DX1) U2 (X2). (2.47)
f) = cos (<DX1)P(X2) ,
where (D = lat/11 (k = 1, 2, 3, ...), or in the form

u1= cos(‘I’Xl)U1(X2),
u2 = sin (‘I’X1)U2(X2) , (2.48)
5 = sin (‘PX1)P(X2) ,

where ‘1’ = (i-1/2)1t/11 (j = 1, 2, 3, ...). To be common, denote £2 = mic/21l (m = 2k for
(2.47) and m = 2j-1 for (2.48)). Thus m is the number of half wavelength of the base
deformation mode function over the length" of the composite plate in the direction of
the thrust T. The functions U,(X2), U2(X2) and P(X2) are to be determined according to
the equilibrium equation (2.45) and the boundary conditions (2.11)-(2.13). By substitut-
ing either (2.47) or (2.48) into (2.45), the equilibrium equation, together with the

l3

incompressibility constraint (2.38), gives the following set of ordinary differential equa-

tions:

U," (x2) - 92U1(x2) - 21%;}, (x2) = 0,

U2” (x2) — (2211, (X2) - 11%;? (x2) = 0, (2'49)

- p'ZQU1(X2) + U2’ (x2) = 0,
where ’ denotes differentiation with respect to X2. We define a new stretch ratio:
71. = 2.2/1, = p—z. (2.50)

By substituting (2.50) into (2.30), it becomes

T = 2137:1’203- 1) (u(i)R1+u(ii)R2). (2.51)
The thrust T is monotone increasing in A from T = —oo when i=0 to T = oo when

A. = oo, with T=0 when 1:1. Note that A > 1 when the composite plate is compressed
in X,-direction and A. < 1 when extended. And it = 1 when the composite plate is nei-

ther compressed nor extended. From (2.49), we can obtain U,(X2) from U2(X7):
U1(X2) = xlg—zuzxxz). 1r U,(x,) is an even (odd) function then mm is an odd

(even) function. Solving (2.49) for U2(X,), one obtains a single fourth order ordinary

differential equation
U2” — (1+ 23) QZUZ”+ 239%, = 0, (2.52)
and its characteristic equation
q4 - (1+ 73) 9qu + 1.294 = 0 (2.53)
has four real roots i9, :25) which yields a set (A) of four base solution functions

9X, 42X, XQX2 -2LQX2
e ,8 .8 ,e

A = { } . (2.54)

The general solution of equation (2.52) can be any linear combination of the four base

14

solution function in set A. For the purpose of convenience in discussion, we express

the general solution in equivalent hyperbolic form

U2 (x2) = L1 (x2) cos/1(QX2) + L2 (x2) sinh (9x2)

+M1 (X2) cosh (19x2) + M2 (X2) sinh (19x2) , (2.55)

Where L,(X2), L2(X2), M,(X,) and M2(X,_) are step functions which are individually

constant in ply-1 and ply-2 respectively and are denoted as

1:,” Mi" —Rl sx2 so

L,,(X2) == . M (X2) = . . (2.56)
{142) n {M1321 osxzsrr2

where n =1, 2.
The boundary conditions (2.11), (2.12) and interface conditions (2.13),

expressed in terms of U2(X2), become

(mfu2 (x2) + U2” (x2) = o,
(m) 2 (2 + 1/23) U,’ - U2’” (x2) = 0, (2-57)
onX2 = -R1andX2 = R2;
U2 (x2.) = U2 (x2).
U2’ (x2.) = U2’(X2_), (2.58)
oan = O;
u“) [(7.52)2U2 (x2.) + U2” (X2.)]
=11“ [ (m) 202 (x2) + U," (x2) ].
onX2 = 0. (2.59)
11‘” [(2+ 142) (MDZUZ' (x25 -U2'”(x2.)]
=niii>[(2+1/fi) (7.9)2U2'(x2,) —U2”’(X2,)],
The requirement that (2.55) obeys the conditions (2.57)-(2.59) gives rise to a 8 x 8 lin-

ear system for 8 unknown constants denoted by the L’s and M’s. This system shall be

written as

15
J8x8l8x1= 08x1’ (260)
where
T
l: (1.911.111,My),M51),L,<2>,L,<2>,M,<2>,M121} , (2.61)

and J is a 8 x 8 matrix derived from (2.57)-(2.59), which when written in full is

 

 

' ACl —AS1 22C3 —ZKS3 O O 0 O ‘
1 0 1 O -1 O —1 0
o 1 O X 0 -1 O —)t 2 62
J“ -A 0 —2r 0 BA 0 2137. o ’ (')
0 —2 O —A O 28 0 [3A
0 O AC2 AS2 27LC4 2284
_ O O 0 282 2C2 AS 4 AC4 -
where C, = coshma), S, = sinhma),
Cr = comma-01)). 32 = sinh(11(l-a)),
C, = cos/100101), S, = sinhOt'na),
C4 = coshOtnU—a». S4 = sinhOtna—a»,
A = k + 1n.

Here we have three parameters: 11, [3, or, in accompany with 3., whose respective ranges

are
A>0, n>0, B>O, OSaSl. Q63

(1) The mode number 'r]=2§212=m1t1,,/l1 is a dimensionless parameter scaling the buck-
led configuration with respect to the aspect ratio 13/1,. (2) The stifl‘ness ratio [3 = p.59 /
u“) is the ratio of the shear modulus of ply-2 to that of ply-1. (3) The volume fraction
Ot=R,/212, is the ratio of the thickness of ply-l to the thickness of the composite plate,
and so gives that the ratio of the thickness of ply-2 to the thickness of the composite
plate is R1212=1-a. A pair of material parameters (a, B), together with 1,, l2 and 13 spec-
ifies a certain material construction. Note that, by the nature of the problem we are

dealing with, a pair of (or, [3) represents the same material construction as the pair (1-

16

a,l/B) does.
If the material parameters (or, B) take the following values
B=lora=00ra=1, (2.64)

the problem considered reduces to the noncomposite one as studied extensively by
Sawyers and Rivlin [3] [4]. On the other hand, all true composite cases can be
restricted by Bat] and 0<a< 1.

Bifurcation takes place provided that a nontrivial solution exists for (2.60). This

requires that
detJ = 0. (2.65)
dead is a function of 3., n, B and or, and we shall express it as
detJ = ‘I’ (a, B, n, it) . (2.66)

The necessary and sufficient condition for bifurcation to take place (2.65) then can be

written as

TONI, [3.01) = 0- (2.67)
If 7t=l, it is readily seen from (2.62) that A=2, C1=C3, S1=S3, so that the first and the
third columns of J are identical. Thus

‘1’(1.Ti, B, or) E 0. (2.68)

So Ot-l) must be a factor of ‘l’()t,n,B,Ot). Since 1:1 (A=p‘2) gives zero thrust T, no
deformation or buckling takes place, and so this inherently uninteresting case will not

be considered further.

SOLUTION OF THE BUCKLING EQUATION FOR

CORRELATION OF LOAD PARAMETERS A WITH
MODE PARAMETER 11

When expanded, ‘I’(K,n,B,a) is the sum of 25 products of exponents and poly-

nomials:

2
1
\POBTLBJX) : ‘1—6 2 e

where Ki = K10" a) are given by

(2+1),
K =(A+2a—1L

K = (it-201+1),

K4 =(2L-1),

K5 = (2al—A+1),
x6 = (7i.+1)(2a—1),
x7 = (A—l)(2a-1),
K8 = (Zak—A—l),
x9 = (7.+1)a,

K10 =(A—1)a,

x11: (7t+1)(1—a),

x,, = (1-1)(1-a).

my}. a)

Pi(7t, B) . (3.1)

i=-12

(3.2)
1c_1 = —(A+1),
1(_2 = -(7t+20t—1),
rc_3 = —(A—2a+1),
1c_4 = —(}t—1),
1c_5 = -(2aX—l+1),
1(_6 = -(>.+1) (201-1),
x_7 = —(}t-l) (201-1),
“—8 = —(2al—k—1),
1c_9 = —(A+l)a,
1c_10 = -(7L—1)a,
1(_11= —(A+1)(l—a),

1g], =—(A—l)(l—a).

The P,(A,B) are polynomials in A and B, each of which is of the form

17

7
P101. B) = 2 Ind-(13119.

(the “. 5” does not denote differentiation). For example:

F, (1,13) = (13- 1)2A7— (13132—22|3+13)A6
+(66B2—100B+ 66) 2.5- (166B2-260B+ 166) 714
+223([3- 1)2A3— (183B2—498B+ 183) 2.2
+ (140132- 3345+ 140) 7.1 - (116p2 - 184B +116)
+63 ([3- 1f):1 — (27B2-58B+27)3\.'2
+ Hafiz—45+ 13) 2:3— (6B2-4B+6) 71“

+ (13- 1121'5- (13+ If“

and

P20», 13) =- (ls—IVV-‘swz-1)16+14<B-1)27»5
+38 (132-1)).4-31 (p-1)2A3—41(132— l)7t2
+20 (13— l)%.‘- 12(52— 1) + (B- 1W,"
+11 ([32-1)).‘2-2([3-1)22:3+6(132-1)):4
- (B-1)29~’5+ (132—1)):6

In general, it is found that

for i=1,2, ,12 and j=-6, -5, , 0, 1, 2, , 7. This gives

7 7
P10», [3) = 2111(5)”: 2P-,,,(B)xj = 13-41.11) .

i=-6

l8

j=-6

7 .
= Zpl'j(B)2tJ,

j=-6

7
= 2P,j(p)xi.

j=-6

Fwd” = P—i,j(B)s

i=-6

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

19

Note that (3.7) in conjunction with lq(l,a)‘=-K,(X,a) and K00t,a)=0 indicates that (3.1)
can also be written as

12
We». 11. 13.00 = 313130413) +§2 cosh (11190». come». 13). (3.8)

i=1
A systematic study, detailed in the Appendix, uncovers various other relations among
the P,J(B)’s indicating that there are 31 “distinctive forms” from among the 350
(350 = 25 x 14) P1,;(B)’S (i=-12, -11, ..., 0, ..., 11, 12; j=-6, -5, ..., 0, ..., 6, 7). In addi-
tion, it is shown in the Appendix that ‘I’(7t,n,B,01) can be expressed as

‘P(7t,n,B,01) = TIBATPTE, (3.9)

where
A14x1 = [it-6,14,... ,x°,...,>.6,7.7]T, (3.10)
E25x1 = E(A,n,01) = [eT‘K-n,e"K-",... ,e"K°,... ,enKll,e"K12 T, (3.11)

and P25 x14 has entries P,.,(B), which have the property that

12

z PM(B) = 0 j = —6,—5,... ,7. (3.12)
i=-12

In this chapter we conduct an asymptotic analysis of ‘P(A,n,B,0t) both for large

A and for large n. Then we use a numerical approach to obtain the roots of ‘I‘(A,n,B,or.).

3.1. Asymptotic Analysis for large A

If 2. tends to infinity while other parameters 11, B and a are held fixed, then,
since 0 S a S 1, gives -1 S 201-— 1 S 1, it follows fi'om (3.1), (3.2), (3.3) that

“1,71,13,00 = co(n.B.a)e")‘l7+0(eMK7). (3.13)

provided that 0) (n, B, or) at 0. An examination of (3.2) and (3.3) gives that

4‘

20

0101.13.00 = T15[e“P,,7(B> +e‘w'1’“P2,7(B)]

(3.14)
-2a 1 —
+e‘ “ ’93,,(0) +e “P170”-
For true composite material constructions (0<01<1, -l<201-1<1), this yields
0301.13.00 = % (1— i3)2sinh (non) sinh 1n (1 —a)1. (3.15)

because of P1,-, = —P2.-, = —P3'-, = P45, = (l-B)2. In view of r1>0, B>0 and
0<01<1, it follows that c0(n,B,a) = 0 only if B = 1. However, according to (2.64), this is
also a noncomposite material construction. Thus (0(11, B, Ct) at 0 for true composite

material constructions.

Following (3.9)-(3.12), if 11%, then

E25x1(l,0,01) = [1,1,...,1]T, (3.16)
which gives
12
PTE = z Rum) = 0 j: —6, 7 (3.17)
i=-12
so that (3.9) gives
‘1‘(A,0, 0,61) 20. (3.18)

From the definition of n that n = mrt (12/11) , r] = 0 is the extreme case that the com-

posite plate under thrust has no thickness so that any thrust (A) can cause buckling.
A numerical study of ‘I‘(A,n,B,0t) reveals that its magnitude grows extremely

quickly and its sign changes sharply at locations of its roots, especially as it. or 11 gets

large. Mth above restrictions, we can plot curves of ‘P(7t,n,B,or)/(0)(n,B,0t)enxl-I) vs 3.
with given 11, B and 01 as in figure 3.1 and figure 3.2. This gives the advantage of.

scaling these curves into observable graphs. Since (0(1), B, or) enlk7¢0 whenever

21

A>0, T1>O, B at l and 0<a<1, it follows that

‘P (All, B. a)/ (w (n, 15, 00611717) = 0, (3.19)

if and only if equation (2.67) is satisfied.

 

1

 

 

WM. 13, (1)/(0)01, B. alenifl)

 

 

 

416~
-O.8 H a
'10 10 20 3‘0 40 5‘0 6‘0 7‘0 80
. A
Figure 3.1. The curve of ‘P(A,n,B,a)/((o(1l.i3.a) gill?) vs. A at
11:2, B=3. 01:0.5.

 

fl

 

-(18 H

 

‘1' (7., n, 13. a) / (w (n, B. «119117)

 

 

 

_1 r l 1 r
O 10 20 30 4O 50 60 7O 80

A

Figure 3.2. The curve of ‘P(A,n,B,a)/(co(n,B,or)enA'A—I) vs. A at
n=2, [3:2, 01:0.25.

22

All of the curves that have been so plotted show that the graph of ‘I’(A,n,B,a)
has exactly three transversal intersection points with the line ‘P=0. All three of these

intersection points are greater than one and vary continuously with n, B, 01. They shall
be denoted by

1<A(1)<A(2)<A (3.20)

(3)
In addition A=1 is also a root of (3.19) but it is not a transversal intersection, instead
the graph of ‘P(A,n,B,a) is tangent to the line ‘I’=0 at A=1. Finally, as expected from
(3.13), note that the graph of ‘I’(A,n,B,a)/(0)(n,B,a)enAA7) tends to one as A tends to
infinity.

It follows from (3.1) or (3.8) that the partial derivatives with respect to A, 11, B

or 01: g, 37:, 98%, (3)—211:, exist and are continuous. The curves of ‘P(A,n,B,a)/

(0)(r|,B,a)enxA7) vs. A reveal that, for any of the three real roots A(,,>1 (i=1, 2, 3),
Sax‘l’otmm, 13,61) at 0. (3.21)

Hence ‘i’(A(,,,T|,B,01)=0 defines three single value implicit functions
A~(1) = (b101, B: a) r A”) = (b26713 B: a) 9 A13) = (D3 (T1: B: a) (322)
respectively, such that

- 1 < ¢1 (11.13.00 < 4501.13.01) < <I>3(n. 15,01) . (3.23)

Considering the nature of ‘P(A,n,B,a) as given in (3.1) or (3.8), it is diflicult to obtain
explicit expressions for <1>,(n,B,or) (i=1,2,3). Thus we employ a numerical procedure to
find the functions d>,(n,B,0t) and so construct the curve presentations 0‘0) vs. 11) for

given (B, a). Figure 3.5-figure 3.11 are examples of these A 0) vs. 1] curves with sev-

eral material parameter pairs (B, 01). The numerical procedure utilized will be discussed

in section 3.3.

23

. . . . A
For noncomposrte material constructions (B=l, 01:0 or 01:1), Since 811 A7 at 0,

similarly by plotting ‘I’(A,n,B,a)/enAA7 vs. A, it is shown that there are two real roots

AQ,>A(,,>1. These give two implicit functions of

in addition to the trivial root A=1.
It is found, by using the numerical procedure (Section. 3.3), (returning to the

composite material), that
lim (1)1 (71, B, or) =1,
'1'] —) 0
111133201» 13.01) = °°r (3.25)

Iim (D301, B, 0t) = 00.
1’1—90

3.2. Asymptotic Analysis for large n

Since 050151, it follows that —1 5201—1 51, which, since A > 0, gives

(referring to (3.2))

K1 = max (K_12, 1C_11, ..., K0, ..., K11, K12) (3.26)
Hence
T](A+1)
1P (7., n. B. 4) ~ e P, (A. B) as n -—> ..., (3.27)
Since 8110” 1) ¢ 0, it follows that solutions of

P, (A, B) = o (3.28)

yields asymptotic solutions to (2.67) as n —) 00. If there exist roots of A for (3.28),

these roots will be the asymptotes of the A“) -T| curves. We may factor P,(A,B) as

given in (3.4) into

P1 0». B) = -A‘—,<A—1)‘fi(x)f2(i. B). (3.29)

24

where

f, (A) = (1.3-312-1-1), (3.30)

130. B) = (B-llzl3- [3(B-1)2+4B] 12
— [(B-1)2+8B]A- (B-1)2-4B-

The factors f,(A) and f2(B,A) are exactly the same as those obtained by Kim

(3.31)

[13] in his study of the three-ply problem. The equation f,(A)=0 has one real root A”

given by

 

 

- -31+ 312+,14- (4/3)3 + 312—1—44/3)?’ = 3.38297577... . (3.32)

- With the exception of B=1, the factor f2(A,B)=0 also yields exactly one real root A”, B
for all finite B>0. Denote this second asymptotic root by A” B to acknowledge its

dependence on B. To understand this B dependence, consider the cases B=0, B -> co
and B=l. Starting with B=0, note that f2(A,0)=A3-3A2-A-l=f,(A). It follows that
A”, B = 0 = A” as given in (3.32). For large B,
£20., 13) ~BZ().3—37.2—A— 1) +003) . (3.33)
The dominant item in f2(A,B) gives B’i’f’fow = A” again as in (3.32). For B=1, (3.31)
gives that
f2(A, 1) = —4(A+1)2, (3.34)

which has no roots in the range of A>0. We have solved the equation f2(A,B)=0 numeri-

cally and obtained A”, 13 (B) in Figure 3.3. It is found that
Blfllllmfiw) = °°. (3-35)

and

A > A”, for 0 < B < oo. (3.36)

‘39.13

25

SOr

 

451. ..

a
O
"'1

U)
kit
.1."

 

 

b)
O

N
M
_ l-—I

O T_‘ “’T“'-_‘T———T’ M“! ” "'
r ' ; . i

   

N
O

p—0
ll!

..A
O

 

Lit

 

 

O

 

Figure 3.3. The picture of A”, A“ B (B) .

The values Am and AmB give all possibilities for roots A to (2.67) as T1 —->oo and

hence each of the 3 functions <D,(t1,B,a) must approach one of the two of these values

as 11 —) oo. Numerically we find that
Iim (D1 (11, B, 01) = A” ,
Ti —9 °°
nli’f’m‘pz ('11 13’ 0‘) = Aw . (3.37)
Iim (1)3(r1,B,01) =AmB.
Tl -’ °° '
Note from (3.36), (3.25)2 and (3.37)2 that
(D2 (11, B, 01) —Am"3 = 0 (3.38)
mUSt have at least one root. By evaluating ‘1’ (Am, r1, B, a) with changing 11 and fixed
Pairs Of (B,Ot), we find that

‘1’(A..,11, B, 01) > o. (3.39)

Furthermore by evaluating ‘1’(Am B, 11, B, or) with changing 11 and fixed pairs of (B,a)

26

we find that there is exactly one value of T1 = 11 A such that

‘P(}V”’B(B)rnrflaa)>09 ifn<nl;

, (3.40)
W()\”'B(B)rnsflra)<07 11.71)“) '

So 11,, satisfying

<1)2 (11,, 13. a) — A“, = o. (3.41)

is the unique root of equation (3.38). This, in conjunction with (3.36), (3.25);, (3.37);
and (3.39), gives

<I>1(n,B,a)<A.,<<I>2(n,B.a) , and A~,B<<D3(n,13,a). (3.42)

3.3. Numerical Analysis

A numerical procedure is developed to solve ‘P(A,r1,B,a)=0 for each of the
three roots A=<D,('r1,B,a) (i=1,2,3) at each fixed material parameters (B,01) and each
fixed mode parameter 11. The bisection method [11] is employed to do this. It requires
to separate these roots from each other in advance of employing the bisection method.

For example to determine the first root <D,(11,B,a), bisection can proceed starting with

bounds of 1 and A” according to (3.23), (3.42). The major difliculty stems from sepa-

rating the second and third roots, since for small value of 11, both roots are bounded

below by AM}. In this case it is necessary to obtain a “separation value” A”p that is

simultaneously an upper bound for (D2(t1,B,a) and a lower bound for (D3(r1,B,a) before
beginning the bisection process. One such separation value is given by a stationary

value (guaranteed to exist by Rolle’s theorem) which solves
a _
3A? (A, 11, B, or) — 0. (3.43)

A numerical procedure has been developed employing a quadratic approxima-
tion method [12] for finding Aw. This method, at given material parameters (B,0t) and
given mode parameter 11, use a quadratic function iteratively to approximate

‘P(A,r1,B,a) and finally to get the local stationary value. Once such a separation value

27

is obtained, bisection for the second root proceeds by using bounds A“ and Amp, and

bisection for the third root proceeds by using Amp and a sufficiently large number. A

graphic interpretation of this procedure is shown in Figure 3.4.

   
    

 

 

 

 

 

 

 

A
A” B i 1 $301,133)
A” g : fi‘DZO‘irflra)
/ «>1mb
A=1 §
5 Til: 1 n; > T1
T1 : nk ‘ )V 11 = Tl) ‘ l

 

 

< A; 6>

W’n’fl’a) v()‘onrflra)

Figure 3.4. A graphic interpretation of the numerical procedure
used to obtain A=‘1‘,(r1,B,01) (i=1,2,3).

 

 

 

 

 

For the purpose of obtaining complete A—n curves A=<I>,(T1,B,Ot) (i=1,2,3) at
given pairs of (B,Ot), we set up a series of [1151}: 1 such that
0<111 <r12< <r1n, (3.44)

Where T1,, is the upper bound of the range of 11 to be considered. We then use the

approach discussed above to obtain

“in = (Di(r1k,B,01) i = 1,2, 3 (3.45)

28
obeying

‘1’ (Am. 11,. B. a) = 0. (3.46)

k=l,2,...,n. Thus we can get a series { (Aik, 11k) }:= 1 (i=1,2,3) of which the interpola-

tions are our curve presentations of A=<D,(11,B,01) (i=1,2,3). If n is large enough, then
these interpolations are a good approximation of A;<I>,(11,B,a) (i=1,2,3).

Actually, in our numerical procedure 11 changes its value backwards, Le, 11
changes its value in the order of 11,, 11,.1, ..., 111. In implementing the numerical proce-
dure, we use the quadratic approximation method to obtain A"? for the first two values
of 11 (i.e. 11,, and Tin-O- For the sake of saving computer time, we have found that the
following algorithm is able to give a separation point for all subsequent values of 11
(i.e. 11,-2, 11,3, ..., 11,). Namely we use linear extrapolation to approximate the second
and third roots An=<l>,(11,,,B,a), k=n-2, n—3, l and i=2, 3, as

.. A
ik "

A

T11<+1"‘11c+2

ik+l_ ik+2

 

(111,-le“) k=n—2,n-3,...,1 1:23. (3.47)

Then the separation value A,” is given by

A21: '1' x3k

rsep=——2— k=n—2,n—3,...,l. (3.48)

Figure 3.5-3.11 are examples of the curve presentations of A=<l>,(11,B,01)
(i=1,2,3) carried out for pairs of (B,01) specified beneath each picture. The set of pairs
of (B,OL) calculated is the Cartesian product of or e {0.0, 0.1, 0.2, ..., 1.0} and
B e {1, 2, 3, ..., 10}. Among these cases with B = l or or = 0 or 1 reduce to that of a

noncomposite material construction and curves of these cases are precisely the same as

those obtained by Sawyers and Rivlin [3] [4]. The third root ¢3(T],B,a) moves up to
infinity as B —> 1, or —> 0, or a -> 1 as can be seen in these figures. For each pair of

(B,a) the first root (1)1(11,B,01) goes from 1 when 11 -) 0 to AM when 11 —-> co, the sec-

ond root (1)2(11,B,0t) goes from infinity when 11 —) 0 to A” when 11 —> co, and the third

29

root <D,(11,B,a) goes from infinity when 11 —-) 0 to A“ i3 when 11 —> 00. For some pairs

of (B,Ot), the functions d>,(r1,B,01) and d>2(r1,B,a) are not monotone as shown in Figure

3.10 and 3.11.

16

14

 

 

 

 

 

 

 

 

 

 

 

 

 

12 1
10
8
6
4 4
2
0o 1 2 3 4 5 6 7 8 9 lo
11
Figure 3.5. The A vs. 11 curves L=<I>,(11,B,a) at [3:1, 01:0.3.
16 + . . . . , , . L
14 ~ QQMJLG) '
12 ~ -
10 - -
8 _ A=‘I’2(A,Tl,i3,a) .
6 - -
4 _ _
2~fltm -
0o i i 3 A 3 6 5 s 3 10

11
Figure 3.6. The A v3.11 curves 71,-:<I>,(11,B,a) at [3:2, 01:03.

30

 

  

 

 
   

 

 

 

 

 

     

 

    

 

 

 

 

16 1
l4 *- ‘
12 ~ "
10 ~ 1
8 +- A’.=‘113(}'9T19B7a') d
6 P A’:‘I,2(A'9T19B5(1) ‘
4 - .
2 b kwxxanafira)
00 i i 5 4 g e. i a s n
11
Figure 3.7 The A vs. 11 curves A,:<I>,(11,B,a) at B=3. $0.3.
16
14 - a
12 - q
10 r- :
8 _ kw‘flxrnafi’a) _,
6 ’- hwxlvnrfl’a) -
4 - _,
2 L h‘P1(lrn:B3a) . A
0o i 2 5 4 5 8 7 5 «.3 lo

Tl

Figure 3.8. The A vs. 11 curves A,=<D,(r1,B,a) at B=3. 01:0.5.

16

14

12

10

16

31

 

 

 
    
 

 

 

 

 

_ “3001513509 2
A"——ly2()',r1 sBaa)

,.. kwmxanrflaa) d

0 i 2 3 4 5 6 7 8 9 10

“'1

Figure 3.9. The A vs. 11 curves A:<b,(r1,B,a) at [3:2, 01:0.25.

 

10»

 

1.1

12F

2 Lfl(l,n,fi,a) _

Ham

.1

   

A’=‘.PZ()\'7T1 959(1)

 

 

 

 

0

1 2 3 4 5 6 7 8 9 10
11

Figure 3.10. The A vs. 11 curves A1=<1>,(r1,B,01) at B=2, 01:0.1.

32

 

16

 
  

A-'=\IJ3()'2T19B9(X)

121-

10-

 

A’_-”lPZ()"n ,Baa)

 

 

 

 

 

2 _ A=‘P1(A,n9B9a) ‘
/_
O | 1 l L
0 l 2 3 4 5 6 7 8 9 10

Figure 3.11. The A vs. 11 curves A,=<1>,(11,B,a) at B=4, 01:0.9.

In chapter 2 we have formulated the equations for the problem of the buckling
instability of the two-ply composite plate. A set of computer program has been coded
for solving these equations according to the discussion in chapter 3 and 4. VVrth these
computer program we can predict the failure stretch ratio A and then, following (2.51),

the failure thrust T for given material construction ((B,Ot), 11,12,13). Note from (2.51) that

the failure thrust T is monotone increasing with A. Denote Tin for failure thrust at

mode number m and corresponding to A=<I>i(11m=m1t12/1,,B,or.) (i=1,2,3). It follows from
(3.23) and (3.42) that

1 2 3
Tm < T... < Tm < Tm. (3.49)

This gives that the critical failure thrust always corresponds to A=<1>,(11,B,01).
As Sawyers and Rivlin [3] [4] as well as Pence and Song [9] [10] have pointed

out, the failure thrusts for noncomposite material construction are ordered as
0<Ti <T§<... <T°°< <T§<T§. (3.50)

This is the direct deduction from the fact that for noncomposite material construction,

33

A=<D,(r1,B,Ot) is always monotonically increase with 11 and A=<D2(11,B,Ot) is always mono-
tonically decrease with 11. In the case of composite material construction, as can been
seen in figure 3.10 and figure 3.11, some pairs of (B,Ot) no longer gives <D,(11,B,a)
monotonically increasing in 11 and some no longer gives (D2(11,B,01) monotonically

decreasing in 11. These cause a reordering of failure thrusts, i.e., for some m<n

T},<T}n and Tfn<T§. (3.51)

4. DEF ORMATION

The deformation of a thick rectangular nonlinear elastic composite plate consist-
ing of two plies of different shear moduli under thrust has been treated as a homoge—
neous finite deformation superposed with an incremental nonhomogeneous
deformation. The incremental deformations are allowed to take place if and only if
there exists a nontrivial solution for equation (2.60). For a specified material construc-
tion, the material parameter pair (B,Ot) and plate aspect ratio 1,0, are given. For a given
number of half wavelengths m, then the parameter 11 is determined. There are 3 thrusts
T which support bifurcation of solutions for each such half wavelength possibility m.
They are determined by the 3 load parameters A=<r>,(n,B,ot) (i=1,2,3). We now examine
the deformation corresponding to these 3 different possibilities. By substituting A=<1>,(—
r1,B,01) (i=1,2 or 3) into (2.60) and solving it, we can obtain a nontrivial solution vector
1. Then U2(X2) and U,(X2) can be obtained from (2.55) and (2.49),. Furthermore 11, and
u2 are obtained according to (2.47) or (2.48). We rewrite U2(X2) and I here for the con-
venience of discussion:

U2“) : L1“) cosh (0X2) + L,” sinh (9X2)

0 (. (4.1)
+M,J cosh (Anx2) + M,” sinh (19x2),

where j = 1, 2 for ply-l or ply-2. The solution vector
T
} .

1 : {Lf”,L2‘”,Mf”,M§”,L1‘2’,L2‘2’,M1”),M2”) (4.2)

is normalized so that
11111; = 1T1 = 1. (4.3)

Thus the full deformation 2 can be obtained. Figure 4.1 - Figure 4.3 are three exam-
ples of the full deformation corresponding to <1>,(11,B,a), <Dz(11,B,a) and <D,(11,B,01)
respectively for a case of m=2, l,=1.2, l,=1.0. In these figures, 8 is chosen to make the
deformed configurations distinguishable. (In Figure 4.1-4.3, dot line: original configu-

ration, dash line: homogeneous deformation, solid line: buckled deformation).

34

35

 

 

 

 

 

 

 

 

 

 

2
1.5 ~ -
,_ ................................................................................................. -
0.5 — .1
0- _
0.5+ .
,, _ ........................... .
1.5 — -
"—21.5 1 05 0 05 1 15

Figure 4.1. Deformation of the 2-ply composite plate under thrust where
8:0.1, m=2, so that 11=mrr|2/l,=5.236, B=3. 01:0.5 and
A=<I>,(5.236.3.0.5):3.199.

 

 

 

 

 

 

 

 

2 A
1.5 — _
1.. ............................. -
0.5 _ -
0.. ............................. .1
_, _ ................................................................................................. .
-1.5 ~ .
321.5 :1 1 1.5

 

Figure 4.2 Deformation of the 2-ply composite plate under thrust where
8:0.05, m=2, so that 11=m71l,/I,=5.236, B=3. 01:05 and
A=<1>2(5.236.3.0.5):3.457.

36

 

 

 

 

 

 

 

 

 

-3 1 1
-1 .5 -l -0.5 0 0.5 l 1.5

Figure 4.3 Deformation of the 2-ply composite plate under thrust where
8:0.3, m=2, so that 11:mrrl2/l,=5.236, B=3. 01:05 and
A=<I>3(5.23.6,3,0.5):7.345.

Sawyers and Rivlin [3] [4] have shown that all buckled plane deformations of
the type under consideration may be classified as either flexure or barrelling for non-
composite cases. A flexul‘al deformation is defined to be one for which U2 is an even
function with respect to X2, and a barrelling deformation is defined to be one for
which U2 is an odd function with respect to X2. Pence and Song have also shown that
flexural and barrelling deformations take place in the symmetric three-ply problem_
[10]. In the three-ply problem studied in [10], the composite plate considered is sym-

metric in the X2 direction. One can then split the (12 x 12) linear system of the three-

ply problem (similar to (2.60) here) into two separate (6 x 6) subsystems by making
use of the symmetry. One subsystem then gives the flexure deformations and the other
gives barrelling deformations. '

In the two-ply problem studied here, recall from (2.64) that if B=1, Ot=0 or
01:1, then the problem reduces to a noncomposite one. In these cases we have obtained

the same result as found by Sawyers and Rivlin [3] [4]. As mentioned in Section 3.1,

37

there are two roots <1>,(T1,B,01) and (1)2(11,B,01) (given by (3.24)) for equation (2.67) for
noncomposite cases. The solution vecror 1 corresponding to <1>,(11,B,01) and ¢2(T],B,a)
then gives a pure fiexural deformation and a pure barrelling deformation respectively.

In general, if the material construction of the two-ply plate is truly composite
(B at 1, or $0 and out 1), then the symmetry with respect to X2 no longer exists. The
solution vector 1 obtained as mentioned above makes U2, as given by (4.1), neither an
even function of X2 nor an odd function of X2. Neither of the <1>,(r1,B,01) (i=1,2,3) will
then correspond to pure flexure or pure barrelling. In the two-ply problem, buckling
deformations x as given by(2.33) associated with <D,(11,B,a), (D2(r1,B,a) and <D,(11,B,a)
have a mixed mode flexure and barreling character.

In order to examine the characters combining the mixed-mode, we decompose

the vector 1 by

 

 

 

l=a+b+c+d, (4.4)

a = {a,, 0, a3, 0, a5, 0, a7, 0} T, (4.5)

b = {09 b2, 09 b4, 0’ b6: 0: b8} T9 (4'6)

c = {0, c2, 0, c4, 0, c6, 0, c8} T, (4.7)

d = {d}: 0: d3: 0) d5) 0) d7: 0} T7 (48)

where

L1“) +L1(2) M1“) + M1”)
a1:aS=—2_—’ 33:37: 2 ,

Lg” + 1.12) My) . M12
b2:b6=_2——’ b4:b8= 2 ’ (49)

I12(1)__Lz(2) M20)—M§2) '
C2=—06=—__2_—’ C4=—C8= 2 .

 

(1) (2) (1) (2)
d -—d _L1 -L1 d _ d _M1 "Ml
1— 5____T_, 3"7‘ 2 -

This decomposition, in the sense of 2-norm, satisfies

38

Mini = lla||§+llbl|§+IICI|§+IldII§. (4.10)

By this decomposition, the solution for the ordinary difi'erential equation (2.52), U2, is

expressed as

U2 = U3+U§+U§+U§ (4.11)
where
U; = a1 cosh (9x2) + a3 cosh (mxz) , —Rl s x2 s R2, (4.12)
U3 = bzsinh (9x2) + b4sinh (19x2), -R1 5 x2 5 R2, (4.13)
U; = { czsirth (9X2) + c4sirth (XQXZ) , —R1 5 X2 5 0, (4.14)
— czsmh (0X2) - c4smh (AQXZ) , O 5 X2 5 R2 ,
U; _ (11 cash (9X2) + d3 cosh (XQXZ) , -R1 3 X2 5 0, (4.15)

Note that the 4 functions in the decomposition have the following symmetry properties

U;(x2) = U;(—X2), U§(x2) = U§(—x2), (416)
U§(x2) = -U‘2’(-x2), U§(x2) = —U§(-x2).

We shall say that if b=0 and d=0, then the whole deformation is one of pure flexure,

similarly, if a=0 and c=0, then the whole deformation is one of pure barrelling. We

now turn to examine the continuity of U3, U3, U3, U3 and their derivatives across the
interface X2=O. For this purpose it is convenient to introduce the notation [[ ]] I0 to

indicate the jump in value across this interface. Clearly from (4.12), (4.13) it follows

that

[[(Ug) (”mo = 0, [[(U3) (”mo = 0, (4.17)

for n=0,1,2,3, . Here (n) denotes derivatives of order n and (0) indicates the undiffer-

entiated function. On the other hand,

39

[W9 (mm, = 0, [[(U‘z’) ”Ml’fllo = o. (4.13)

_ . . (2n+ 1) (2 )
for n=0,1,2,3,... . To address the possible discontlnulty m (U;) and (ug) "

at the interface X2=0, we note that the interface conditions (2.58)
[[uzmo = o, [[uz'mo = o (4.19)

along with (4.17), (4.18) give

[[(U§)(1)]]IO = 0, [[(U3) (0)]1‘0 = o. (4.20)

2n+1)

( (2n)
However (U2) , (US) for n=1,2,3, are not continuous across the inter-

face X2=0.
Thus by using the boundary and interface conditions, we conclude that in the
range of —R1 5X2 5 R2, UgandUlzJ and their derivatives of any order are continuous,

c (0) c (1) c (2) c (2n) . d (0) d (1)
(U2) , (U2) , (U2) and (U2) are continuous, and (U2) , (U2) ,

d (2) d (2n+1) .
(U2) and (U2) are continuous. Furthermore

11U2‘2"’1 1 IO = [[(U‘z‘) (mm,

[[U2(2n+1)] ] '0 = U (U?) (2n+l)]]l0

where n= 1, 2, 3,... , and a superscript number n inside parentheses indicates taking the

-2 (92"d1+ (An)z"d3) , (4.21)

—2 (92"+1e2+ (M2)2"+1c4), (4.22)

nth derivative, and a superscript number n without parentheses indicates exponentiation

to the power of n. According to the above discussion, we may refer to the four parts of

deformation relating to U3, U3, U3 and U; in the decomposition of U2 as follows:

smooth flexure part, smooth barrelling part, residual flexure part, residual barrelling

Part respectively. According to (4.15) and (4.20);

d1+d3 = o. (4.23)

40
Similarly, according to (4.14) and (4.20)l
c2 + 104 = 0. (4.24)

With equation (4.23) and (4.24), by entering (4.11) into the interface continuity condi-
tion (2.13), we find that

111 = PD-lal+-II)72a3 (4.25)
and
c2 = -%3b2-%b4, (4.26)
where

= ([3+1)(1—22), D1=((3—1)(1+>3), D2=2(B-1)12. (4.27)

On account of (4.9), (4.16), (4.23)-(4.26), we conclude that if the deformation is one
of pure flexure, then c2=c4=cé=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.

5. DISCUSSION

3 In Chapter 4 we have discussed the decomposition (4.4) of the solution vector from
equation (2.60) and the corresponding decomposition (4.11) of the solution to the dif-
ferential equation (2.52). Since any solution vector of equation (2.60) can be decom-
posed in this way, it follows that any possible buckling deformation is the combination

of flexural and barrelling deformations. Note that this decomposition satisfies (4.10),

. . 2 2 2 2 2 2
we further examlne how these portions .IlaIIZ/lllllz, llbllz/lllllz, IchIZ/Illll2 and

u dug/n In; vary with n. A number of material constructions (Bra) have been computed

and these changes have been plotted in curves. Figure 5.2 and 5.3 are included to

give an example of these curves. (In Figure 5.2 and 5.3, solid line: Halli/“Ill; dash

line: “bug/“1113, dash-dot line: Noni/”lug, dot line: (lung/1.111;). They are the varying
portions for l=<bi(n, 3, 0.5) (i=1, 2) (see Figure 5.1) respectively. The numerical algo-
rithm used in calculating these curves in Figure 5.3 only gives accurate results for
n>1, consequently the curves are not shown for n < 1. In general, in the deformations
corresponding to <D,(n,B,0t), smooth flexure dominates the whole deformation; in the
deformations corresponding to (D2(n,[3,a), smooth barrelling dominates the whole defor-
mation. Residual flexure and residual barrelling always occupy small portions. Figure
5.4a-5.5d are examples of these decomposed deformations. In these figures, equation
(4.3) is again used to establish the over all normalization. (In Figure 5.4a—5.5d, dot

line: original configuration, dash line: homogeneous deformation, solid line: buckled

deformation portion).

41

42

 

    
    

 
 

 

 

 

 

 

 

 

 

 

 

 

16
14 ~ _
12 e ..
10 __ k=lP3(l,n9B9a) _.
8 .. _
6 i- .
k=‘I’2(l,n,B,a)
4 '- -i
2 _/ .
A"tqll(7":nrfl’a')
O _1 L L l A 1 1 A 4
0 l 2 3 4 5 6 7 8 9 10
'0
Figure 5.1. The A vs. 1] curves L=<I>i(n,fl,a) at B=3. 01:05.
0.4 ~ g
0.3 - y _
0.2 '- ............. ' ' ~ """".‘.'.-"-'v----— Am" r.....:.-.-.-:...,.....‘..:.-.:r.:,-..'.~’ , .- _-. , --.'- - ..d
0.1 - .' ".",-"'l':::-:"‘: """"""""""""""""""""""""""""""""""" 3
0 - FFFFF ’é... A L
O 1 2 3 4 5 6 7 8 9 10

Figure 5.2. An example of the portions of these four type defor-
mations varying with n for the case of B=3. a=0.5, for the first root
in A vs. 11 curves: M,(n,3.0,0.5).

43

 

 

 

 

 

 

 

0.7
0.6L ................................................................................ 1
0.5 _ q
0.4 - J
0.3 _ ‘
0.2 - fi
0.1 - _
01 2 3 4 5 £3 7 8 9 10

Figure 5.3. An example of the portions of these four type defor-
mations varying with n for the case of B=3. 01:05. for the second
root in it vs. 11 curves: 7t=<l>2(n,3.0,0.5).

 

 

 

2 4 f
1.5- -
1.. ......................................................................................................... .1
0.5~ s
0_ _
,1. .................................................................................................................................. _
-1.5F -

 

 

 

 

 

 

_2 A I A A 1
-l .5 -l -O.5 O 0.5 l 1.5

Figure 5.4a. The smooth flexural deformation portion at 8:0.0005,
m=2, 11:52:36, B=3.0,a=0.5, L4,(5.326,3.0,0.5)=3.199. The over-
all deformation was shown previously in Figure 4.1.

 

 

 

 

 

 

 

 

 

2 4 m fi
1.5» -
1. ........................................................................ _
asl .
0L -
0.5» .
1. ..................................................................................................... g
1.5- -
121.5 1 05 0 05 l 15

Figure 5.4b. The smooth barrelling deformation portion at 8:0.0001 .
m=2, n=5.236, [3:30, 0l=0.5.)v=<bl(5.326,3.0,0.5)=3.199. The over-
all deformation was shown previously in Figure 4.1.

 

 

 

 

 

 

 

 

2
1.5 - 1
1 ....................................................................................................... .
i . .
as. .
0 P ......................................................... a
1 e -l
-1.5 _ a
121.5 :1 i 1.5

 

Figure 5.4c. The residual flexural deformation portion at 2:0.0001,
m=2, 11:5.236, B=3.0, a=0.5, M,(5.236.3.0.0.5)=3.199. The over-
all deformation was shown previously in Figure 4.1.

45

 

 

 

2 4
1.5— _
1_ ........................................................... .
0_ a
-05— _
,1- ............................ -
1.5» .

 

 

 

 

 

 

-2 .
-1.5 -1

Figure 5.4d. The residual barrelling deformation portion at
e=0.0001 , m=2, 11:5.236, B=3.0, oc=0.5, L—<bl(5.236,3.0,0.5)=3.199.
The overall deformation was shown previously in Figure 4.1.

 

 

 

 

 

 

 

 

 

2 a
1.5 F a
1 _ ............................... . ........................................................................ .
0 _ ............................. .
-0.5 ~
-1 )—
1.5 e 1
-2 1 [Ar\- 1
-1.5 -l -0.5 0 0.5 1 1.5

Figure 5.5a. The smooth flexural deformation portion at 8:0.0002,
m=2, n=5.236, B=3.0, «=05, L—<b2(5.326,3.0,0.5)=3.457. The over-
all deformation was shown previously in Figure 4.2.

46

 

 

 

2 f
1.5 — -‘
1,. ............................. 1 .................................................................... 4 ............................... ..
0' i
-O.5 - 1
ll‘ -
-1.5 i- ..

 

 

 

 

 

 

-2 L
-l.5 -1

Figure 5.5b. The smooth barrelling deformation portion at e=0.0002,
m=2, n=5.236, B=3.0, 01:0.5. X=<D2(5.326,3.0,0.5)=3.457. The over-
all deformation was shown previously in Figure 4.2.

 

 

 

 

 

 

 

 

 

2 4
1.5 - .
1L ............................... .
0_ ............................. _
-05 ~ 4
1- ..................................................................... . .
-l.5 » _
121.5 -l l 1.5

 

Figure 5.50. The residual flexural deformation portion at 2:0.0002,
m=2, 11:5.236. [3:30, 0l=0.5. l=<b2(5.236.3.0,0.5)=3.457. The over-
all deformation was shown previously in Figure 4.2.

47

 

 

0.5 ~ -

 

-05 — -

-1.5- "

 

 

 

 

 

 

-2 A
-1.5 -1

Figure 5.5d. The residual barrelling deformation portion at
2:0.0002, m=2, 11:52:36, B=3.0, a=0.5, L—¢2(5.236,3.0,0.5)=3.457.
The overall deformation was shown previously in Figure 4.2.

One difficulty in directly comparing the deformations as discussed here to the
flexural and barrelling deformations discussed in Sawyers and Rivlin’s [3] [4] is due to
the choice of origin of the coordinate system. In the context of Sawyers and Rivlin[3]
[4] as well as Pence and Song [9] [10], the origin is always chosen so as to lie at the
center of the overall plate (and hence in its mid-plane). In contrast, here the origin is
chosen to lie on the interfacial plane, which coincides with the midplane only for the
case 0t=1/2. It is thus instructive to view the deformations as discussed here in an alter-
native coordinate system centered at the mid-plane. To do this we introduce the coordi-

nate transformation
Y1 = X1, Y2 = X2+a, Y3 = X3, (5.1)
where —12 S a $12 is given by

a = (2a- l)12. (5.2)

48

 

 

 

 

 

 

r l
2 P y a > X1
ply-1 > Y1

R, h

 

 

 

Figure 5.6 Coordinate Transformation

The solution for equation (2.53) becomes V2 (and correspondingly V1) with respect to
coordinate system Yl-Yz-Y3. By the coordinate transformation defined in (5.1), it is

readily seen that
V2(Y2) = U2(Y2-a) V1(Y2) = U1(Y2-a). (5.3)

Expressing V2 in full we have

v2 (Y2) = P1“) cosh (9Y2) + P,“ sinh (9Y2)

. . (5.4)
+Q10) cosh (7.523(2) + 02‘” sinh (MIYZ) ,
where j = 1, 2, and
P1“) = L1G) cosh (Qa) - L2“) sinh (Qa) ,
P2“) = L20) cosh (Qa) — L10) sinh ((2a) ,
. . . (5.5)
of” = M1“) cosh (ma) - M2“) sinh (ma) ,
02“) = M23” cosh (ma) - M1“) sinh (ma) .
This then gives
L1“) = Pf” cosh (Qa) + P20) sinh (9a),
L2“) = by) cash (oa) + P10" sinh (Qa) ,
(5.6)

M1“) = of” cosh (ma) + of) sinh (ma) ,
M2“) = of" cash (ma) + of” sinh (ma) .

 

49

Recall from (2.64) that [i=1 is one way to obtain the noncomposite case. How-
ever if [3:1 and a at 1/ 2 then this noncomposite case is being treated in a nonsymmet-
ric coordinate system for the axis system XI'XZ'Xg. However, the treatment is
symmetric in the coordinate system Y,-Y2-Y3. If given k=<b1(n,l,a), generally in this
case [3], [4], [9], [10],

P1“) = P1(2)¢O’ Qiu = 91(2) ‘0’ (5 7)
l 2 1 2 '
Pé ’ =P2( ’ =Q§ ’ =Q§ ’ =0.

These P’s and Q’s in equation (5.7) make V2(Y2) a even function. Then the buckling
deformations corresponding to k=<l>1(n,B,0t) are purely flexural with respect to coordi-
nate system Yl-Yz-Ys. This is the same result as Sawyers and Rivlin [3] [4] as well as
Pence and Song [9] [10] have obtained. On account of (5.6) and (5.7), if a¢0, (i.e.
at at 1/ 2 according to (5.5)) but [3:1, we have

L1“) = le = lecosh (Qa) #0,
Lg” = Lg” = P1“) sinh ((2a) it 0,
M“) _ MO) _ (1) (5.8)

1 — 1 -— Ql cosh (19a) :0,

M2“) = M252) = ansinh (3.9a) #0.

It can be seen from (5.8) that all eight components of l are nonzero and make U2 nei-

ther an even function of X2 nor an odd one. Further, by applying the decomposition
(4.4) (4.9), we have

a1 = P1“) cosh (9a) , a3 = Q1“) cosh (152a) , 9
' (5- )
b2 = Pf1)sinh(fla), b4 = ofl’st'nh (ma),

and
cz=c4=d1=d3=0, or c=d=0. (5.10)

So, even in the noncomposite case, if there is no symmetry in direction of thickness in
a reference system (XI-Xz-X3 here), there exist both smoodl flexural and smooth barrel-

ling deformations. Calculated from (4.10), we have

50
”In; = Ilau§+nbn§. (5.11)

where

2
u an; = 2 { (P1(1)cosh(Qa))2 + (le cosh (man },

(5.12)
2 _ (l) . 2 (l) . 2
llbll2 — 2{ (P1 smh (9a)) + (Ql smh (15%)) }.
Since Q=n/(21,) it follows that
2
“all 1 as -—> 0
——22- = { n . (5.13)
"m2 1/2 asn—wo
and
2
llbll 0 as —)0
—23 = { n . (5.14)
"”2 1/2 asrl—Mo

for the first root l—n curve k=<l>1(n,l,0t) with or at 1/2.

If given h=<l>z(n,[3,ot) for this noncomposite case (B=1 and a a: 1/2), we have

P20) = pg?) :0, Q" = Q2“) :0,

P1“) = P?) = Q1“) =Q1(2) = 0’ (5.15)
in general. In this instance, these P’s and Q’s in equation (5.15) make V2(Y2) an odd
function. The buckling deformations corresponding to 70=<I>2(n,[3,a) are, therefore,
purely barrelling with respect to coordinate system Yl-Yz-Ya. This again is the same
result as in [3] [4] or [9] [10]. Similar to the discussion made above, the eight compo-
nents of I are nonzero and give rise to both smooth flexural and smooth barrelling
deformations and restrain the residual flexural deformation c as well as residual barrel-
ling deformation d, with reference to Xl-Xz-Xa. Analogous to (5.13) (5.14) we have

ll all; 0 as n —) 0

= { , (5.16)
”mg 1/2 as Tl—)oo

as well as

 

51

Ilbllg _ 1 ash—)0

._ . (5.17)
Ill”: 1/2 asn—)oo

for the second root l—n curve k=<b2(n,l,a) with out 1/ 2. Finally recall for the case
[i=1 that the third root l—n curve L=<D3('q,l,0t) has moved up to infinity and hence

does not exist.

In this thesis we have studied the buckling of the 2—ply nonlinear elastic plate,
formulated and solved the problem, fulfilled the procedure for predicting the buckling
loads, and studied the deformation modes. In comparison to the symmetric 3-ply prob—
lem extensively studied by Pence and Song [9] [10] as well as Kim [13], the 2-ply
buckling problem of the composite plate gives one additional family of buckling
stretch (<D3(n,[3,0t)) to the noncomposite cases whereas in the symmetric 3-ply problem
there are two additional families of buckling stretches. Instead of the pure flexural and
pure barrelling deformation modes in the symmetric 3-ply problem, the buckling defor-

mations of the 2-ply problem are in mixed modes having both flexural and barrelling

characters.

[2]

[3]

[4]

[5]

[6]

[7|

[8]

LIST OF REFERENCES

R. W. Ogden, WW. I-Ialsted Press, New York,
1984.

M. A. Biot. W John Wiley &
Sons, New York, 1965.

K. N. Sawyers and R. S. Rivlin, “Bifurcation conditions for a thick elas-
tic plate under thrust”. Int. J. Solids Structures, Vol. 10, pp. 483-501,
1974.

K. N. Sawyers and R. S. Rivlin, “Stability of a thick elastic plate under
thrust”. Journal of Elasticity, Vol. 12, No. 1, January 1982.

P. J. Davies, “Buckling and barrelling instabilities in finite elasticity”.
Journal of Elasticity, Vol. 21, pp. 147-192, 1989.

I. W. Burgess and M. Levinson, “The instability of slightly compressible
rectangular rubberlike solids under bixial loading”. Int. J. Solids Struc-
tures, Vol. 8, pp. 133-148, 1972.

B. Budiansky, “Theory of buckling and post-buckling behavior of elastic
structures”. in Advances of Applied Mechanics, 14, pp. 1-65, 1974, ed.
C. S. Yih, Academic Press.

M. A. Biot, “Edge Buckling of a laminated medium”. Int. J. Solids
Structures, Vol. 4, pp. 125-137, 1968.

52

 

._:L.

[9]

[10]

[111

[12]

[131

53

T. J. Pence and J. Song, “Buckling instabilities in highly deformable
composite laminate plates”. Proceedings of the American Society for
Composites Fifth Technical Congress, pp. 459-468, 1990.

T. J. Pence and J. Song, “Buckling instabilities in a thick elastic three
ply composite plate under thrust”. Int. J. Solids Structures, Vol. 27, pp.
1809-1828, 1991.

G. M. Phillips, P. J. Taylor, 2111ng Ed Appligag’gns Qf Numerical Ana -
m. Academic Press, 1973.

M. Avriel, Ngnlinegr mgamming: Analysis and Methgds. Prentice-
Hall, 1976.

Sang Woo Kim, Research in progress.

 

APPENDIX

THE EXPRESSION OF ‘P()v,n,B,0t)

‘I‘(k,n,B,or) can be expressed as
‘1’ (71.71, B, a) = 1—16ATPTE,
where

T
A = {x—6’ 1-5, "-3 lo! "'3 X6, 17} ’

E = {en K'”, enK‘“, ..., en K°, ..., emc“, enK‘z} T,
and P is a 25 by 14 matrix with entries P,J(B)
P = (Pi.j(B))25xl4'
The Ki = Ki(?t,0t) (i=-12, ,12) are given by
K0 = 0,
K1: (1+1), 1(_1= —(}t+1),
K2=(7t.+2(1-1), K_2=—(l+20t—1),
K3: (it—211+”, K_3=—(}t—20r+1),
K4=(2\.-1), tt_4=-(7t—1),
K5: (Zak—2+1), x_5=—(200t—)t+1),
K6: (7L+1)(20r-1), 1c_6=-()v+l)(201—1),
K7: (k—l)(20l—1), K_7=—(}t—1)(20t—1),

54

(A.l)

(A2)

(A3)

(A4)

(A.5)

 

 

 

55

K8: (Zak—A-l), K_8=—(20l7t-)t-1),
K9: (k+l)0t, 1c_9=—()v+1)01,

x10: (7,—1)ot, tt_10=—(7.—l)ot,
K11=()t+1)(1-(1), 1c_11= —(7L+1)(1—0t),
K12=(K—1)(1-a), tr_12=—(7.—l)(1—ot).

One finds that the 350 P,J(B)’s are constrained by the following relation:
P... = P i = 1,2,...,12, j = —6,—5, .0 ...,6,7. (A6)

The particular values for the Pu([3)’s are given in Table A.1 (the distinctive P,J(B)’s
are denoted by *). The 31 distinctive P,J(B)’s are:

PO’_3 = —128B, - (A.7)

P0,, = 153602—2816B+1536, (A.8)
PO,5 = 512132- 11520+512, (A.9)
132:6 = 132— 1, (A.10)
P4,_6 = B2+ZB+ 1, (All)
P4,_4 = 682—4B+6, (A.12)
P4,_3 = 1802-4B+18, (A.13)
r>,__2 = 2702—58B+27, (A.14)
P4,0 = 116132-184B+ 116, (A.15)
P4,] = 14082—3440+140, (A.16)
P,” = 183132—4988+ 183, (A.17)
P4,4 = 16602-260[i+ 166, (A.18)

56
P45 = 6602-1008+66,
PM, = 1382—220+13,

PM = 02-2044,

P9’_4 = — 16B+16,
P9’_2 = 3202—1128+80,
P9,0 = 25602—416l3+ 160,

P9“ = —320[32+704[3—384,
P = 44882-864(3+416
9,2 . ’

P9, 4 25602 - 592 (3 + 336,

PS,5 = — 16082+2880— 128,
P9,6 = 3282—480+ 16,
P11’_4 = 16(32— 160
P11,_2 = 8002—1128+32,
P11,O = 16002—416B+256,
P1“ = —38402+7048—320,
P11,2 = 41602-864[3+448,
P“,4 = 33602-592[3+256,
PM = —128[32+288[3— 160,

P11,6 = 1682-488+32.

(A. 19)

(A20)

(A21)

(A22)

(A23)

(A24)

(A25)

(A26)

(A27)

(A28)

(A29)

(A.30)

(A.31)

(A.32)

(A.33)

(A.34)

(A35)

(A36)

(A37)

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i=-12

i -6 -5 -4 -3 -2 -1 0
0 0 0 0 * 0 5 12P4.7 0
1 -P4._6 P43 434,4 P4,.3 ' 4.2 631)4.7 'P 4.0
2 * -P4.7 6132.4 '2P4,7 llPZ-é P 4.7 '12P2--6
3 .p26 - 4., -6P24 —2P.,,7 -11P2,_6 P,7 12PM
4 * P 4.7 * * * 63p4,7 *
5 PM -P4,7 6P“ -2P4.7 1 1PH P4,? -12P2._6
6 -13“ p1, -6P.,_7 18p,7 -27P,,, 63H.) -116P..7
7 P4.7 P43 6P.” 18R,7 27P.,_7 63R,7 116P417
8 'Pm -P,,_7 -6Pz., -2P.,.7 -11PM P43 12P2,6
9 0 0 ... -2P,,,,, =1: .128P.,,7 *
10 0 0 “Pg-4 '2P11,-4 ’P9.-2 '128P4.7 "P9.-0
11 0 0 * -2P9,4 * -128P4_7 *
12 O 0 'Pn,.4 ”2139.4 'P 11.4 '128P4,7 'P 11.0

i 1 2 3 4 5 6 7
0 * 0 1536P4_7 0 * O 0
1 P,,_1 -P,,2 223P,,,7 -P4,4 P4,; 4’48 P43
2 20PM -41P2.,6 -31P.,.7 38PM 14R” ~3P2,6 -P,,_7
3 20P47 41P2.,6 -31P4.-, -38Pzg 14P.,,7 3PM5 -P,,'7
4 :1: a: 223p“.7 alt * 2|: a:
5 20R,7 -41P2,,6 -31P4,7 38P2‘45 14P,’7 -3P2,-6 -P4,7
6 140P,,.7 -183P4.7 223R” -l66P.,,7 66R,7 -13P4.7 P417
7 140P.,.7 183P,,_7 223R,7 166P4_7 66PM 13R” P43
8 20R, 7 41PM5 -31P,,.7 -38P2,,g 14R,7 3P,“5 -P4_7
9 * * -384P.,,7 * * * 0
10 P 9.1 439,2 '384P4,7 'P 9.4 P95 'P 9.6 0
11 * * -384P4.7 * * * 0
12 P11,1 ”P113 384134;; 4311.4 P11.5 ‘P11.6 0

Table A.1
A calculation finds that
12
z Pi,j (B) = 0 j = —6, -5, , 7. (A38)

58

Alternatively, ‘P(}t,11,B,a) can be expressed as

1 1‘1
\P(Awn’B9a) = EA PH9

where L is again given by (A2) and

And P is a matrix of 13 x 14 with entries 1311(0) , i=0,1, 12, j=-6,-5, 0,

where

H = {cosh (111(0), cosh (111(1), cosh(nK12)}T.

'3: (fitjwnmxlw

(A.39)

(A.40)

..., 6,7:

(A.41)

(A.42)

(A.43)

"Iilllll'llllilllllllllli