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‘\

BUCKLING OF A COMPOSITE NONLINEARLY
ELASTIC PLATE UNDER UNIAXIAL LOADING

By

John Zhen-hua Song

A DISSERTATION

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

Dacron OF mosopmr
IN
MECHANICS
Department of Metallurgy, Mechanics and Materials Science

1990

ABSTRACT

BUCKLING OF A COMPOSITE NONLINEARLY

ELASTIC PLATE UNDER UNIAXIAL LOADING

BY

John Zhen-hua Song

The buckling of a nonlinearly elastic three-ply composite plate
is studied in the context of finite deformation incompressible
nonlinear elasticity. The buckling of non-composite plates in this
context has been studied previously by other authors. Here a general
class of incompressible materials is introduced through a material
strain energy density dependent variable A(A). Both flexure and
barrelling instabilities are examined. The differential equations and
boundary conditions for the bifurcation analysis of the composite
plate is derived. To simplify the analysis, the neo-Hookean material
is the focus of investigation in the latter portion of this
investigation. Numerical computer programs are then used to predict
the critical loading condition. It is found that the ordering of the
failure thrusts for certain composite configurations is more
complicated than the ordering that occurs in a non-composite plate.
In particular, the thrusts at which barrelling and flexural buckling
modes occur can be interlaced. Furthermore, transitions in the
ordering of the different failure modes may occur as the aspect ratio
of the plate is varied. These results demonstrate the interplay
between the properties of the constituent materials and the geometric

configuration of the plate. Finally, an optimal design problem for

symmetric three-ply sandwich composite plates is studied. The object
is to maximize the resistance to buckling for fixed values of the
volume ratio of two different neo-Hookean materials. It is found that
there is a single aspect ratio dependent transition in the optimal

design.

ACKNOWLEDGEMENT

I would like express my sincere appreciation to my advisor,
Professor Thomas J. Pence for his tremendous guidance and support
throughout this work} I would also like to thank the other members
of the guidance committee, Professor Nicholas Altiero, Professor
Yvonne Jasiuk and Professor David Yen.

While preparing this dissertation, I held teaching assistantships
awarded by the Department of Metallurgy, Mechanics and Materials
Science and a research assistantship from the Composite Materials and
Structures Center of Michigan State University under an REF Program

Grant. The support of these institutions is gratefully acknowledged.

TABLE OF CONTENTS

LIST op FIGURES AND TABLES

 

Section 1 INTRODUCTION

 

Section 2 PROBLEM DESCRIPTION

CD

Section 3 MECHANICAL BEHAVIOR OF THE CLASS OF
INCOMPRESSIBLE NONLINEAR ELASTIC
MATERIALS UNDER CONSIDERATION

 

.21

Section 4 BIFURCATION FROM THE HOMOGENEOUS
SOLUTION
29

 

Section 5 GENERALIZATION OF MATERIAL CONSTANT A
50

 

Section 6 TWO SPECIAL DEFORMATION TYPES FOR THE
NEO-HOOKEAN COMPOSITE PLATE:
FLEXURE AND BARRELLING

 

57

Section 7 BUCKLING AND WRINKLING FAILURE
MODES FOR THE COMPOSITE CONSTRUCTIONS
54

 

Section 8 OPTIMAL DESIGN FOR COMPOSITE
CON STRUCI'IONS

 

APPENDIX A
1 14

 

APPENDIX B
117

 

APPENDIX C
122

 

LIST OF REFERENCES
150

 

ii

LIST OF FIGURES

Figures and Tables

Figure 2-1.
Figure 3-1.
Figure 3-2.
Figure 7-1.
Figure 7-2.
Figure 7-3.
Figure 7-4.
Figure 7-5.
Figure 7-6.
Figure 7-7.
Figure 7-8.
Figure 7-9.
Table 7-1
figure 8-1.
Figure 8-2.
Figure 8-3.
Figure 8-4.
Figure 8-5.

Table 8-1

GEOMETRY OF THE COMPOSITE PLATE
STRESS-STRETCH RELATION FOR MATERIAL (3.1)
SHEAR STRESS RESPONSE FOR MATERIAL (3.1)
FLEXURE AND BARRELLING MODES --

RESPONSE CURVES FOR THE NON-COMPOSITE CASE
RESPONSE CURVES ~-

RESPONSE CURVES

FLEXURE FAILURE PARAMETER PLANE
MAGNIFICATION OF FIGURE 7-5 .-

BARRELLING FAILURE PARAMETER PLANE -

THE FLEXURE AND BARRELLING PARAMETER PLANE
RESPONSE CURVES

TRANSTTION ASPECT RATTOS -

OPTIMAL DESIGN RESPONSE CURVE «-

OPTIMAL DESIGN PARAMETER PAIR REGION --
OPnMAL DESIGN RESPONSE CURVE .-

OPTIMAL DESIGN RESPONSE CURVE -—

OPTIMAL DESIGN RESPONSE CURVE

CROSSOVER VALUE FUNCTION --

iii

Page

20

27

28

78

79

80

81

82

83

84

85

86

87

106

107

108

109

110

Ill

1. INTRODUCTION

Buckling in plates has been of concern for a long time because of
the role it plays in structural failure. Within the theory of finite
deformation elasticity theory, buckling has been studied by numerous
authors, including for example Green and Spencer [4] for a circular
cylinder, Burgess and Levinson [2] for a plate under biaxial load and
Wilkes [13] for a circular tube under end thrust. Recently, the —
buckling of a thick incompressible isotropic elastic plate under
thrust has been studied extensively by K.N. Sawyers and R.S. Rivlin
[10], [11]. These authors employ the theory of small deformations
superposed on finite deformations [9] to determine the critical
conditions under which solutions bifurcate from a well known "trivial"
solution. Sawyers and Rivlin have shown for the non-composite case
that plane-strain buckling can occur involving either flexure or
barrelling mode shapes with an arbitrary integer number m of half-
wavelengths. In this dissertation a similar problem for composite
constructions is examined in detail. Specifically, the buckling
instability of a thick rectangular three-ply sandwich composite plate
is investigated in the context of finite deformation incompressible
non-linear elasticity. The buckling behavior of composite plates in
the classical theory has been the object of extensive previous study
[8]. Here, however, the examination takes place within the fully
three-dimensional theory of nonlinear elasticity. This plate,
consisting of two different incompressible isotropic nonlinear elastic
materials, composed of three stacked rectangular plies with perfect

interfacial bonding, is subjected to a deformation by a thrust T

applied to opposite sides. Some striking differences in the buckling
behavior from that which would otherwiSe occur in a non-composite
plate are obtained. Drastic changes in stability behavior are not
uncommon when one goes from a non-composite construction to a
composite construction. This has previously been demonstrated in [5],
[6] for the physically different, but mathematically similar, void
formation instability mechanism. This dissertation then concludes by
using the buckling results obtained here to study a problem in optimal
design.

In section 2, the basic boundary value problem is formulated.

The composite plate is constructed from 3 plies that are stacked
symmetrically in the Xz-direction and the whole assembly is subjected
to thrusts applied on the Xl-faces. Thus the boundary and interface
conditions are determined. The plies themselves are each assumed to
be constructed from a class of nonlinear materials which are studied
further in section 3. The boundary value problem is then shown to
have a hOmogeneous deformation solution as given in (2.19). The
relation between the thrust and the stretch in the plate is then given
by (2.34).

Section 3 presents the mechanical behavior of the class of
incompressible non-linear elastic materials that are the focus of this
study. These materials are a useful generalization of the neo-Hookean
material in that they provide for a wide range of mechanical behavior,
while giving rise to buckling equations that have a certain useful
analytical form. Finally it is shown that these materials are a
special subclass of a more general class of materials introduced by

Knowles [7].

In section 4, the bifurcation is fully discussed. Here buckling
is defined as the existence of deformed configurations that depart
from the homogeneous deformation solutions obtained in section 2. A
linearized boundary value problem is formulated governing local
bifurcation from these homogeneous deformation solutions. These
bifurcations involve plane strain buckled solutions. A general
buckling equation (4.53) for determining the thrust associated with
these bifurcation modes in the three-ply composite plate is obtained.

Section 5 reveals more detail about the material constant A in
the strain energy density (3.1). As shown by Sawyers and Rivlin [10],
[11], a nonconstant generalization of A -- A(A1, A2, A3) -- plays an
important role in the buckling behavior of a homogeneous plate
composed of an arbitrary incompressible nonlinearly elasic material.
This generaliztion is discussed. It is then shown that the materials
used in the present study, and examined in section 3, are the most
general class of materials with strain energy density W-W(I1) for
which the value of A(A1, A2, A3) is a constant.

In section 6, and for the remainder of this dissertation,
attention is confind to the case in which all plies are composed of a
neo-Hookean material. Barrelling and flexure deformation modes are
introduced and the bifurcation into each such mode is associated with
a “failure” thrust. The buckling equations for each of these modes
are obtained. These equations are in fact special cases of equation
(4.53) for symmetric and anti-symmetric deformation. Computer
programs are developed to solve these two types of problems
numerically.

In section 7, these numerical routines are used to study the

o
flexure and barrelling instabilities in detail. Of particular

interest is a wrinkling instability, cOrresponding to an infinite
number of wavelengths in either the flexure or barrelling case. For
both flexure and barrelling cases, various different qualitative
behaviors in the buckling equations are shown to occur. These
different qualitative behaviors effect the failure mode ordering. It
is shown that the ordering of the failure thrusts for certain
composite configurations is more complicated than the ordering that
occurs in the non-composite case. In particular, under suitable
conditions, the thrusts at which barrelling and flexure buckling modes
occur can be interlaced. Furthermore, transitions in the ordering of
the different failure modes may occur as the aspect ratio of the plate
is varied. Unlike the non-composite case, the lowest, or critical,
thrust need not always correspond to the lowest flexure mode. In
fact, for certain constructions, the critical thrust is associated
with thewrinkling instability.

The last section focuses on optimal design for three-ply
composite plates. A specific design problem is introduced and
studied. The problem involves a three-ply composite laminate plate
that is to be constructed from fixed amounts of two different
neo-Hookean materials. The object is to choose between two competing
symmetric three-ply designs: configuration-1 in which the stiffer
material is utilized for the construction of the outer plies, and
configuration-2 in which the stiffer material is used for the
construction of the central ply. The configuration that gives the
largest critical thrust is then.the optimal design. Cases in which

both configurations are ensured to fail in the lowest flexure mode are

considered first. It is shown for these cases that configuration-l is
the optimal design for plates that are sufficiently long in the
direction of thrust, while configuration-2 is the optimal design for
plates that are sufficient short in the direction of thrust. The
transition aspect ratio is also determined. Then the investigation
turns to consider the optimal design problem for cases in which either
configuration-l or configuration-2 (or both) may fail in a mode other
than the low wavelength flexure mode. It is shown once again that the
optimal design switches from configuration-l to configuration-2 as the
aspect ratio of the plate is varied. However now the determination of
the transition aspect ratio may be quite subtle. In fact, an example
is given in which the failure mode changes from that of low wavelength
flexure to that of wrinkling before the transition aspect ratio.

Then, at the transition aspect ratio, the failure mode switches back
to the low wavelength flexure mode. A unified framework is developed
for the systematic determination of the optimal design in this, and

similar cases.

2. PROBLEM DESCRIPTION

Consider a thick rectangular composite plate with surfaces
perpendicular to the axes of a rectangular cartesian coordinate

system.x -.x (X1,X2,X3). This plate occupies the region

-1 S X. S l , i - l, 2, 3 (2.1)

before the application of loads. It is constructed from 3 plies that

are stacked symmetrically in the X2-direction as follows:

ply-1: -12 < X2 < -R,
ply-2: -R < X.2 < R, (2.2)
ply-3: R < X2 < 12.

The geometry and leading of a particular plate configuration is
depicted in Figure 2-1. Plies 1 and 3 are taken to consist of the
same incompressible, isotropic, homogeneous elastic material (Material
1). Ply 2 is also composed of an incompressible, isotropic,
homogeneous elastic material (Material II) which is in general
different from Material I.

Within each ply we define a deformation

2 - x(X). (2-3)
The corresponding deformation gradient tensor is then given by

F - ax/ax. (2.4)

The requirement of material incompressibility is that

det F - l. (2.5)

The principal stretches are defined to be the eigenvalues of the

1/2

tensor 0 - (FTP) and shall be denoted by A A and A . Note that

l’ 2 3

A1 > 0, i - 1, 2, 3 by virtue of the definition of U and the
nonsingularity of F. The three scalar invariants of the deformation

are then given by

-AA +AA +AA, (2.6)

The incompressibility constraint (2.5) dictates that I - (detF)2 - 1,

3
so that

A1A2A3 - 1. (2.7)

The mechanical response of an incompressible elastic solid is
governed by the strain energy density function W - W(Il,12) of the
material. In view.of (2.6), (2.7) the strain energy density can be
regarded as a function of any two of the three principal stretches.

The Cauchy stress tensor for a general incompressible elastic

material is given by
f - -pI + 2(awyaI1 + Ilaw/arz) n - 2(aw/arz) 32, (2.8)

where p is hydrostatic pressure, B - FFT is Green's deformation
tensor, 11,12 are the first and second invariants of B, and W is the

strain energy density function of the material. Thus in the problem

at hand

WI(I ,I ), in plies 1 and 3,
W _ l 2

II (2.9)
W (11,12), in ply 2.
The Piola-Kirchoff stress tensor is then given by
s - F-1f, (2.10)
and the equilibrium equations can be written in the form
T
div S - O. (2.11)

The plate is subjected to a thrust on each of the faces initially

2 2

traction-free. The surfaces initially at X3 - i 13 are held at this

at X1 - i 11. The surfaces initially at X - i l are taken to be

location by means of frictionless clamp.
Since this plate is subjected to a thrust on each of the surfaces

X1 - i 11, the following boundary conditions will be required to hold:

812 - 813 - 0, on X - i l (2.12)

on X1 - i l (2.13)

corresponding to a normal thrust, frictionless in the tangential
directions, with an overall stretch ratio of p. Here p > 0, rather
than the thrust, will temporarily be regarded as a prescribed
constant. The physically interesting case of compression corresponds

to 0 < p < l.

Since the surfaces X2 - i 12 are assumed to be traction free, the

following boundary conditions are required to hold:

8 - S - S

21 22 23 - O, on X, - i1 . (2.14)

2 2

On the surfaces X3 - i 13, the following boundary conditions are now

required to hold:

- 0, on X - i l (2.15)

x
I
H-
H
O
.‘3
N
I
H-
H

(2.16)

corresponding to a frictionless clamp. Problems of this type for
homogeneous (non-composite) plates have been studied by Sawyers and
Rivlin [10], [11].

In order to treat the composite three-ply laminate, the

additional interface conditions are required to hold:

(1 - 1, 2, 3), X2 - i R (2.17)

 

- x X - i R. (2.18)
+ X

The conditions (2.17) and (2.18) correspond to a case in which the
plies are perfectly bonded across the interfaces.
To within an arbitary displacement in the X2-direction, there is

exactly one pure homogeneous deformation solution to the foregoing

10

boundary value problem. The underlying deformation must be given by

x2 - p'1 x (2.19)

in order to satisfy (2.13), (2.16) and (2.5). Thus the principal

-I
1, A2, A3 are given by A1 - p, A2 - p , A3 - 1. The

material deformation tensor F, Green's deformation tensor B and the

stretches A

first and second invariants of the deformation are in this case given

by

F - diag(p.p-1.1). B - diag(p2.p'2.1).

I - A + A + A 2 - 1 + p2 + p‘z, (2.20)

2 2 2 2 2 2 2 -2
12 - A1 A2 + A2 A3 + A3 A1 - l + p + p

The equilibrium equations (2.11) require that the hydrostatic pressure
p in (2.8) is constant in each ply, say p(1), p(2) and p(3). These
values are found from the requirements (2.14)2 and (2.17)1_2, from
which one obtains with the aid of (2.8), (2.9) and (2.20) that

p(1) _ p(3) _ p(1), p(2) _ p(II), _ (2.21)

where

11

 

p(k) - 2p'2(w (k) + I w (k)) - 2p-4W (k), k - I, 11, (2.22)
1 1 2 2
BW
and W1 - , i - 1, 2. This in turn gives
811

k) + I w2(k)) - 2p-4W2(k) ] I +

r - -[ 2p-2(W1( 1

(k) (k) (k) 2
+ 2011 + 11w2 ) a - 2(w2 ) a , k - I, II. (2.23)

In this work attention will for the most part be restricted to a

class of materials for which the strain energy densities (2.9) are of

the following form

1+Ak/2 Ak

W(k)(11.12) - WWII) - um [(1 + 11) - 4(2) 1 /2 .
k - 1,11. (2.24)

(I) p(II)

here u , , A and A are material constants obeying

I II

(I)

p > o, ”(11) > o, A > -2, A > -2. (2.25)

I II

The special choice of AI - AII - 0 gives the case of two neo-Hookean
materials. Thus this class of materials generalizes the well known
neo-Hookean material in each ply. A thorough discussion of these

materials will be given in the next section.

For this class of materials, the pressure (2.22) simplifies to

12

(k) -2 Ale/2 _p(k>.

p - p 2(l + Ak/Z) (2+p2 +p k-I,II. (2.26)

Substituting from (2.24) into (2.23) and using (2.20) now gives the

Cauchy stress tensor for the deformation (2.19)

-2

/2 p2 -p
p(k) (1 + Ak/2) (2+),2 +p 2 )Ak o - r(k)

k-I,II. (2.27)

Finally using (2.10), (2.27) and (2.20)1, the Piola-Kirchoff stress

tensor becomes

-1
/2 p-p
p(k) (1 + Ak/2) (2+p2 +p 2)Ak [ 0

Note that the remaining conditions (2.12), (2.14), (2.15) and (2.17)
are automatically satisfied.

Let T be the total thrust applied to each of the faces X1 - i 11.
Then

(11)

T - T(I) + T (2.29)

where T(k) (kPI,II) is the thrust applied to material k. Thus for the

homogeneous solution, we have

13

111(k) - T(k)/Area(k), (2.30)

where Area(k) (k-I,II) is the current area of the surface to which
T<k> is applied,

1 (II)

Area(I) - 4(12-R)13p- , Area - 4R1 p'l. (2.31)

3

Using (2.27) and (2.29)-(2.31), it is found that

 

 

_ A /2
(1) (12-R)p(1)(1+AI/2)(2+p2+p 2) I
T _ T, (2.32)
3(9)
_ A /2
(m M‘n’amn/zxzwzw 2) n
T _ T, (2.33)
3(p)
where
T - 413 (p-p'l) 20o) . (2.34)
and
3(9) -

2 -2 "*I/2 (II) 2

(I) -2 All/2
(lz-R)u (1+AI/2)(2+p +p ) + R# (1+AII/2)(2+p +p ) .

(2.35)

14

Note for compression (0 < p < 1) that T < 0 with T + -m as p e 0 and
for elongation (p > 1) that T > 0 with T ~ o as p n w. The undeformed
state (p - 1) occurs if T - O. The following argument shows that T is
monotonically increasing in p if both AI 2 O and A11 2 O.

The monotonicity of T depends On the sign of the derivative of T

with respect of p. Using (2.34) and (2.35)

aT/ap - 413(1+p’2)a<p> + 813(p-p'1)<p-p'3>0<p> . (2.36)

where Q(p) is given by

- (A /2)-1
Q(p) - (lz-R)u(1)(1+AI/2)(AI/2)(2+p2+p 2) I +

2 (“II/2)'1

+ Ru(11’(1+AII/2><AII/2)(2+p +p'2>

Note that (p-p-1)(p-p-3) z 0 and that 3(p) > 0. Thus a sufficient
condition for aT/ap z 0 is that Q(p) z 0. Furthermore Q(p) z 0 if
AI 2 O and A11 2 0. Thus T is monotonically increasing in p if AI z 0
and All 2 0. Thus if AI 2 0 and A11 2 0 then each critical stretch A1
- p > O has associated with it a specific value of thrust, and that
larger thrusts (in either tension or compression) give rise to greater

length changes. If AI < O or/and A < 0, T may cease to be monotone

(I) _ “(11)

II

in p. For example let A - A - -3/2, p

I II a u, then

-1)-3/2 3

I(1+p'2>-<3/2)(p-p'1><p+p’1)‘2<p-p‘ )1.

(2.37)

aT/ap - 1213P(P+p

15

Now under these circumstances T is non-monotone in p as can be

verified numerically, i.e. although for example

aT/ap 2 - 0.1455 1213p > 0, (2.38)
p-

one also finds that

aT/ap - -0.0139 1

13p < 0. (2.39)
p-10

2

For cases in which the load deformation relation (2.34), (2.35) is
non-monotone, nonuniqueness of the homogeneous solution (2.19) can
occur for a given value of p. As will be seen in the next section,
the case of AI - AII - -3/2 leads to an equally unusual response in
simple shear. Consequently this behavior, although of interest in its
own right, will not be investigated further here.

This section will now conclude with some general remarks
regarding generalization of the boundary value problem (2.5), (2.11)-
(2.18) for materials obeying (2.9) and (2.24). This section was
devoted solely to obtaining homogeneous solutions (solutions with
constant deformation gradient F). In view of (2.5), (2.13) and (2.16)
such solutions must be of the form (2.19). The resulting load-
deformation relation is then given by (2.34) and (2.35). Two obvious
generalizations immediately come to mind. The first (generalization
G-l) involves considering classes of material more general than

(2.24). The second (generalization G-2) involves the consideration of

16

ply lay-ups that generalize (2.2), say an arbitrary number of
alternating material plies with no particular thickness symmetry.
Obtaining homogeneous solutions for either generalization 6-1 or 6-2
(or both in concert) presents no great difficulty. In all cases
(2.19) must still hold. In fact for the case of 6-2 one still obtains
the load-deformation relations (2.34), (2.35). In this case (2.28)
continues to hold provided k-I and II apply to the associated material
ply. Turning now to the case of C-1 (acting either alone or in
concert with 6-2), one finds that the strategy employed here will
again allow one to obtain conditions which ensure that the homogeneous
solution (2.19) continues to hold. In this case, however, the
pressures p(k) in (2.26) and the load deformation relation (2.34)-
(2.35) will be correspondingly altered.

The reason that the generalizations G-1 and 6-2 are not developed
here is thus not at all related to difficulties in finding homogeneous
solutions. Rather, it is related to difficulties which would arise
later in connection with obtaining non-homogeneous (buckled)
solutions. Put simply, (G-l) leads to buckling equations that are not
easily analyzed. Further commentary regarding this will be given in
section 5. As shown in the next section, the class of materials
(2.24) allows for a wide range of material response, and so the
strategy of not pursuing (G-l) does not seem to impose a serious
handicap. The consideration of generation G-2 also leads to
formidable difficulties. In this case, as additional plies are added,
additional boundary conditions corresponding to (2.17) and (2.18) are
obtained at each new interface. This brings additional unknowns into

the problem for Obtaining buckled solutions. The level of difficulty

17

C
quickly rises, and one would suspect that "bookkeeping questions"

could quickly dominate the analysis. For these reasons, this
dissertation will henceforth focus attention on the buckling problem

in the absence of generalizations G-1 and 6-2.

18

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19

3. MECHANICAL BEHAVIOR OF THE CLASS OF INCOMPRESSIBLE NONLINEAR

ELASTIC MATERIALS UNDER CONSIDERATION

The purpose of this section is to investigate the nature of the
elastic response for the class of materials with strain energy density

1+A/2

w - u [(1 + I - 4(2">1/2 . (3.1)

1)

where
p > 0, A > -2 . (3.2)

The subscript k and superscript k used in (2.24) will temporarily be
dropped in this section, as attention is here to be focused on the
response of a homogeneous material. Substituting from (2.6) it is
apparent that this strain energy density is given in terms of
principal stretches by

2

2 2 1+A/2
w - p [(1+A1 +A2 +A3 )

- 4(2A)] /2 . ' (3.3)

The inessential additive constant term in (3.1) is chosen to ensure

that W - O in an undeformed state, I1 - 3 or A1 - A2 - A3 - l.

The Baker-Erickson inequality

2

aW/aI1 + Ak

8W/812 > 0 , if Ai # Aj, k e i, k e j, (3.4)
arises from the requirement that if one is given a pair of principal

stretches A A , then the greater of the two corresponding principal

1’3

20

stresses T T occurs in the direction associated with the greater

1’ j

principal stretch ((T1-)(Ai-A )>O if A iiA ). Applying (3.4) to

Ti .‘1

(3.3), one obtains the requirement that

J

p (1+A/2) (1+A12+A2+A3 )A/2

2 (3.5)

Thus (3.5) will hold if and only if p(1 + A/2) > 0, which is the
justification for (3.2). Thus in the composite sandwich construction
(2.9), the restrictions (2.25) ensure that the material in each ply is
consistent with the Baker-Ericksen inequality.

Since W - W(Il), we obtain from (2.8) that

r - -pI + 2 [aw/311] B , (3.6)
where (3.1) gives that

A/2

aw/arl - (u/2) (1+A/2) (1+Il) . (3.7)

Hence (3.6) can be written as

7 - -pI + ”(1 + A/2) (1+11)’V2 a . (3.8)

Consider first a state of uniaxial tension/compression:

l '11
r - 0 , (3-9)

 

21

Then (2.7) gives A - A - A '1/2 so that

with A - A 2 3 1

2 3'

a - A '1 . (3.10)

Combining (3.8) and (3.10) yields

-1 A/2 2
1 +1) A1 ,
-1+1)A/2 Al-l ’

2
111 - -p + p(1+A/2) (A1 +2A

2 (3.11)
0 - -p + p(1+A/2) (A1 +2A

1

Solving (3.11)2 for the pressure and sUbstituting this value into
(3.11)1 gives the uniaxial tension/compression stress-stretch

response:

2 -l)A/2

2 -1
1 - p(1+A/2) (1+A1 +2A1 -A

(*1 1

71 ) . (3.12)
The stress-stretch relations (3.12) are depicted in Figure 3-1 for
different parameters A and p. In particular it is to be noted that
the uniaxial tension/compression response is monotone increasing in A1
from all - -0 when A1 - 0 to a11 - w when A1 - o with all - 0 when A1
- 1. It is also to be noted that the slope of these curves indicates
that the material is "softening in tension" and "hardening in
compression" for all parameter pairs (u,A) obeying (3.2)

In order to obtain the Young's modulus for the associated

linear theory, one may set A1 - 1 + 611 which upon substitution into

22

(3.12) gives

:11 - 2‘“1 p (2+A) (3e11 + 0(e112)) . (3.13)

This provides the Young's modulus of the material in terms of p and A:
E - 3 (2A’1) p (2+A). (3.14)

Consider now a state of simple shear:

x1 - X1 + £12X2,

x2 - X2, (3.15)
x3 - X3,
where :12 is the amount of simple shear. In this case
2
1+612 £12 0
2
B - :12 1 0 , 11 - 3 + £12 . (3.16)
0 O 1
Then (3.6) gives
2
1+:12 612 0
r - -pI + 28W/811 :12 l O , (3.17)

0 0 l

23

which in view of (3.7) yields

2 A/2 c

- u (1+A/2) (4+612 12 .

’12 (3.18)

The shear stress-simple shear relation (3.18) is depicted in Figure
3-2 for different parameters A and p obeying (3.2). In particular it
is to be noted that the shear stress-simple shear response is linear
for the neo-Hookean case A - O. For A > O the material is hardening
in shear and for -2 < A < O the material is softening in shear.
Moreover, only for A > -1 is the shear stress-simple shear response
monotonically inereasing to a. For A - -1 the shear stress is
monotonically increasing to the finite aymptotic value of r12 - p/2.
For -2 < A < -l the material exhibits a collapse in simple shear
whereby it attains its ultimate stress in shear at the value
712-p(2+A)(2)A(A/(1+A))A/2(-2/(1+A))1/2 when e12-2(-1-A)’1/2, before
monotonically decreasing towards 112 - 0 as 512 tends to w. Such a
collapse in shear phenomena is associated with loss of ellipticity in
the equilibrium equations. Stress response functions exhibiting such
a collapse in shear have been used for the continuum mechanical

modeling of shear band formation and phase transitions [1].

Linearizing (3.18) gives

A-l 2
712 - p (2+A) 2 £12 + 0(e12 ) , (3.19)
which provides the infinitesmal shear modulus of the material
A-l
G - p 2 (2+A). (3.20)

Note that the neo-Hookean case A - 0 implies G - u. Furthermore, by

(3.14) and (3.20), it is seen that E - 3C for all values of the

24

parameters A and u obeying (3.2). Thus it is confirmed that the
corresponding value of Poission's ratio u - E/(ZG) - 1 takes on the
value v - 1/2, as it must for the corresponding incompressible linear
theory.

As the above exposition shows, the class of materials (3.1) can
be regarded as a generalization of the neo-Hookean material.
Alternatively, it can be shown that the class of materials (3.1) can
be regarded as a specialization of a class of power law materials
utilized by Knowles [7] in the investigation of certain crack

problems. These materials are characterized by the strain energy

density

W - k/(2b) [(1 + b(Il-3)/n)n - l], (3.21)
where

k > O, b > 0, n > 0. (3.22)

It is found that if
n - 1+A/2, b - (l+A/2)/4 and k - 2(1/4)'A/2(1+A/2)p, (3.23)

then (3.21) reduces to (3.1). Note that the restrictions (3.22) when

used in (3.23) ensure (3.1).

25

 

 

 

 

6”]

 

/

 

 

Fixed A > '2

Figure 3-1:

The stress-stretch relation of the materials with strain energy

density (3.1) for unisxial tension/compression for various values of

A and p.

26

 

 

 

 

 

 

 

 

 

‘7
J Asa? A=l [3:0
_ .u
“2' f
:--I
l
l 3
A:--
I A--2 2
1
£11: 2 8:2
Fixll
T:
12 in?“ ”i
”‘0’ “:10 ’0‘”
41:10
fl:7 .
/J=?
/ AF’ "
1, ,u-o u-O
E: E be
-2< A< o 2 A : o ’2 A>o
Fix A

Figure 3-2:

The shear stress-simple shear relation of the materials with strain

energy density (3.1) for different values of A and u.

27
4. BIFURCATION FROM THE HOMOGENEOUS SOLUTION

Consider now the stability of the homogeneous deformation
solution given in (2.19). Attention shall be restricted to the case
where buckling takes place in the (X1,X2)-p1ane. To do so, let us

consider the fully finite deformation

A

x1 - pX

1 + eu1(X1,X2),

A

-1 -
x2 - p X2 + eu2(X1,X2), (4.1)

where 81, 62 are unknown functions which are independent of X3 by
assumption. In this section, it will be helpful to use a superposed A
to indicate quantities associated with the finite deformation (4.1).
Here 6 is an order parameter which is introduced for the purposes of
obtaining a linearized problem governing bifurcation from the
homogeneous deformation solution (2.19). Finally a superposed ' will
be used to indicate the 0(6) difference between quantities associated
with ; and quantities associated with this homogeneous deformation

solution. Thus, for example, the pressure field corresponding to

(4.1) will be given by

(k)

;(x,e) - p + ep(X1,X2,X3) + C(62), k - I,II, (4.2)

28

and the Piola-Kirchoff stress tensor is given by
S(X,e) - sa‘) + eS(X1,X2,X3) + 0(52), k - 1,11. (4.3)

The material deformation tensor associated with (4.1) is given by

 

p + £ul,l (sul’2 0
A - -1 -
F - euz’1 p + euz’2 0 (4.4)
_0 0 o ,
so that
A - - 2- - - -
det F - [l + e (puz’2 + p ul,l) + e (“1,1u2,2 - u1,2u2’1)].

(4.5)

It is well known that the solution to the corresponding linearized
boundary value problem locates the failure thrusts at which
bifurcation occurs from a homogeneous solution of the type (2.19)
(see Davies [3]) for a rigorous discussion in a probleminvolving a
non-composite compressible elastic material). An analysis of the
linear problem will not reveal the details of the post buckling and
moreover may underestimate the actual thrusts at which instability
occurs for the case of snap-buckling. These more difficult issues

are not treated in this dissertation. Rather the concern is upon the

29

determination of the failure thrusts at which bifurcation takes place
locally from the homogeneous solution (2.19). Thus it follows from

(2.5) and (4.5) that the linearized problem governing local

bifurcation obeys

2,2 -0. (4.6)

One obtains from (4.4), (4.6) that

 

 

 

 

2 - - -l- -
f p +e(2pu1’1) £(pu2’1+p ul,2) O
A - -1- -2 -l- 2
B - e(pu2’1+p ul,2) p +e(2p u2'2) O + 0(e ),
_ O O l 3
(4.7)
_ p.1 + CO -eu 0 .
2,2 1,2
A-l - - 2
P - -eu2’1 p + eu1,1 0 + 0(e ).
_ 0 0 l J
(4.8)

The corresponding Cauchy stress tensor is by virtue of (2.8) and '

30
(2.24) given by

A(k) Ak/2 A

f - -51 + p(k) (1 + Ak/2) (1 + 11) a, k—I,II. (4.9)

Substituting from (4.9) into (2.10) now gives the corresponding

Piola-Kirchoff stress tensor

A AA /2A A
s‘k) - -pr’1 + ”(k) (1 + Ak/2) (1 + 11)Ak F'la, k—I,II.

(4.10)

To obtain an expansion of this tensor in powers of e, note first of

all that

A

I - tr 3 - 1+p2+p'2+e(2pu

1 +2p’1fi2,2)+0(e2), (4.11)

1,1

which in turn gives

2 /2
Ak/ ~2)Ak +

n(k)(1+Ak/2)(1+Il) - 11“" (Mk/2) (2+p2+p

_ -1+ /2 _ _ _
+ ”(k)(1+Ak/2)(2+p2+P 2) Ak (pu1 1+p 1112 2)6 + 0(62).

k-I,II. (4.12)

In addition

31

 

 

'p'lp‘k’ecp‘lsazgp‘k’). 181.21,“), 0'
;(k22'1 - -¢fiz,lp(k2. pP<k2+e(pp+u1 19(k2). 0 + 0(e2).
_ o, 0, p<k>+.;, _
k-I,II, (4.13)
r - 2- - -
p+e(2u1’1+p u2’2), eu2,l’ O
8'18 - .81’2, p-1+£(202’2+p-201’1), 0 + 0(62). (4.14)
_ 0, 0, 1 _

 

 

Now substitution from (4.12)-(4.14) into (4.10) gives upon comparison

with (4.3) that

2 MAR/2 -

 

 

' -p 1p<k)+p2(k)(1+Ak/2)(2+p +p 0, 0
80¢) _ o, _kpp( )+p(k)(1+Ak/2)(2+p2 +p 2),)-12‘1/2 o
_ 0, 0, -p(k)+y(k)(1+Ak/2)(2+p2+p WAR/2 ,
k-I,II (4 15)

where p(k) is given by (2.26) and S has components as follows:

32

- - - k
11 ' '9 p"‘2,2l’( ) +

Ml

_ -1+ /2 _ _ _
k)Ak(l+Ak/2)(3192399 2) Ak (pu1,1+p 1112,2)9 +

2 MAR/2

+ ”(k)(1+Ak/2)(2+p +p u1 21+p2u2’2) ,

-2 WAR/2

8 - ’ (k )+ “(22(1+Ak/2)(2+p +p 92 1 .

12 u1,2p

-2 WAk/Z

S - a ‘2 )+p(k)(1+Ak/2)(2+p2 +p ul 2 ,

2,1p

' _ - '_' (k)
822 ”p u1,1p +

_ -1+ /2 _ _ _ -
k)Ak(1+Ak/2)(2+P2+P 2) AR (pu1’1+p 1112,2)9 1 +

- /2 - - -
+u‘k’(1+Ak/2)(2+p2+p 2)Ak (zu2 2+p 2n, 1) .

_ _ -1+ /2 _ _ -
s33 - -p+u‘k’Ak(1+Ak/2>(2+p2+p 2) Ak (pu1,1+p 1“2,2’ .

k-I,II. (4.16)

The equilibrium equation (2.11) in conjunction with (4.16) then
yields the following system of linear partial differential equations

for the 0(a) problem:

33

-1+ /2
k 2 - Ak - -1-
u‘ ’Ak<1+Ak/2)<2+p +p ) (pal n+9 u2 21» +
2
(k) 2 -2 Ak/ - - -
+ p (1+Ak/2)(2+p +p ) (2u1,11+p u2,21+u1’22) -
-1-
-p p’l - o ,
-1+ /2
k 2 -2 Ak - -1- -
u< )Ak(1+Ak/2)(2+p +p ) (pu1 12+p “2 22)p 1 +
2
(k) 2 -2 Ak/ - -2- -
+11 (1+A'k/2)(2+P +p ) (2u2,22+p u1,12+u2’11) -
'pp’z '- 0 9
P’3 - 0 !
k-I,II. (4.17)
Differentiation of (4.6) yields
- -2 - - 2 -
u2,21 ' 9 ‘11,11 ' u1,12 ‘ ’P 2,22 (“'18)

Substituting from (4.18) into (4.17) gives rise to the following, more

convenient, form

34

-1+ /2 _
k’AkaMk/z)<2+p2+p'2> A“ (p2 -p 2)sz11 +

2Ak/2 - -1-

W‘k’amk/zmw +p 2) <u1,11+u1,22) -,, p 1 - o .

2 -2 '1+Ak/2 -2 2 -

l“’Ak<1+Ak/2)(2+p +p ) (p -p )u u2 22+
/2( _ _
W(k)(l+Ak/2)(2+p +p 2)Ak @2,22+u2’11) -pp’2 - 0 .
5,3 - o .
k-I,II (4.19)

Note that (4.19)3 is satisfied if and only if

13<x1.x2.x3> - 13(x1.x2> . (4.20)

In addition, boundary conditions (2.15) and (2.16) are satisfied
automatically. Thus the linearized problem for local bifurcation from
the homogeneous solution (2.19) is governed by field equations
(a 19)1,2 and (4.6) for functions &1(x1,x2), fi2(x1,x2), 5(x1,x2)
subjected to boundary conditions that follow from (2.12)-(2.14),
(2.17), (2.18). In fact the boundary conditions (2.14), (2.17) and

(2.18) give rise to:

35

-2- -
P “2’1(x2) + u1’2(x2) - o 9
on X - i l

(1 + cz‘I’) 62,2(x2) - D‘I’pfi‘1’<x2> - o .

u2(X2+) - u2(X2-) ,

u1(X2+) - fil(xz-) ,

mm)”1 [p‘282’1(x2+> + a1 2(xzm -

- (D(R’>’1 [p'2fi2,1<x2-) + 61,2<x2->1 .

> on X2-_R.

-p§‘k)(x2+) + w‘k’)’1<1+c2‘k’>fi2 2(x2+> -

 

-(R) (R) -1 (fi) -
— -pp (X2-) + (D ) (1+C2 )u2’2(x2') ' J
(4.21)

where, X2 - -R implies k - II, k - I and X - R implies k - I, k - II.

2
In addition the following "constants“, parameterized by p, have been

introduced:

(122-pank

C1(k) - + l , (4.22)

 

2+p2+p-2

36

 

(P-2_p2)Ak
c2(k) - + 1 , (4.23)
2+p2-i-p.2
- /2 -
D‘k) - [ u‘k’<1+Ak/2)(2+p2+p 2)Ak 1 1
- [ u‘k’<1+Ak/2><p+p'1>Ak 1'1 . k-I,II. (4.24)

We may obtain the solution for the problem at hand by assumming

_ -sin(QX1) _ cos(QX1) _ cos(¢xl)
cos(iX1) sin(flxl) sin(ixl)

(4.25)

where the choices of w - let/l1 (k - 1,2,3...) and W - (j - l/2)1r/11
(j - 1,2,3...) results in the satisfaction of boundary conditions
obtained from linearizing (2.12), (2.13). Let 0 - e or w accordingly
as one considers either the upper or lower terms in (4.25). Thus 0 -

mx/Zl where m - 2k for the upper terms and m - 23 - l for the lower

1
terms. The integer m shall be refered_to as the mode number, since it
determines the number of repeating half-wavelengths in the
Xl-direction of a basic deformation mode.

For both the upper and lower terms in (4.25) the field equations

(4.19)1 2 and (4.6) then become ordinary differential equations:

37
U ~ - c1(k)ozu1 - D‘k)p'lor - o ,

1

c (k)u " - ozu -D(k)pP’ - o , ' (4.26)
2 2 2
, -2

U2 - p 001 - 0 .

Here the superscript ’ denotes differentiation with respect to X2.

The boundary and interface conditions (4.21) will presently be written
in terms of P(X2), U1(X2), U2(X2) as well.

Define the stretch ratio A

-2
A - A2/A1 - p . (4.27)

Note that A > 1 when the ends are compressed (0 < p < l), and A < 1
when the ends are extended (p > 1). One method to solve (4.26) is to
reduce this set of equations into a single ordinary differential

equation for U2 alone. From (4.26)3 one may solve for U1 in terms of

the derivative of the function U2:

. -1 ’
U1 - (A0) U2 . (4.28)

Furthermore, from (4.26)1, one may solve for P in terms of the

functionU1 and it's derivitives, and hence, by virtue of (4.28), in

terms of the derivatives of U2:

P - (D‘k))‘1A'1/20'1 (U1” - c

(k) 2
1 0 U1)

- (D‘k))'lx'3/zn'2 (02” - c1(k)nzuz') . (4.29)

38

Upon using these results in (4.26)2 one obtains a single 4-th order

ordinary differential equation for 02,

(k) , , 2 - ~2 -2 u» _ (k) 2 . _
02 02 0 U2 A 0 (U2 C1 0 U2 ) 0 , (4.30)
which is the same as
.. _ (k) (k) 2 2 .. 2 4 _
U2 (C1 + C2 A ) 0 02 + A 0 U2 0 . (4.31)

The constant coefficient of the middle term is, by virtue of (4.22)

and (4.23), given by

(k) (k) 2 2 2
c1 + c2 A - 1 + A + (1 - A) Ak , k - 1,11. (4.32)

Note that the solution of (4.26) is thus reduced to the solution of
(4.31) for U2, after which U1 and P are obtained directly from (4.28)
and (4.29).

In a similar fashion the boundary and interface conditions (4.21)

together with (4.25), (4.28) and (4.29) now become:

39

(m)2 u2(x2) + U2"(X2) - o ,

on Xz-ilz,
(m)2 [2+A'2+(1-A'1)ZAI] U2’(X2) - U2"’(X2) - o ,
02(x2+) - u2(x2-) ,
on X2 - i R,
u2'(x2+) - U2’(X2-) ,
(1)00)”1 [(A0)2U2(X2+) + U2”(X2+)] - ‘

(R) -1 2 I!
- (D > [(20) U2<x2-> + U2 (x2-)1 .

mm)"1 ((Ao)2[2+A‘2+(1-A'1)ZAR]U2'(x2+) - U2”’(X2+)} -

 

- (D(R)>-1 1(An)2[2+x'2+(1-A'1)2AR1U2'(x2-) - U2"'<x2-)} . .

(4.33)

where, X - -R implies k - II, R - I and X

- R implies k - I, k - II.

2 2

The general solution of (4.31) is

02 - L1(X2)cosh(01(k)xz) + L2(X2)sinh(01(k)

X2)+
+ M1(X2)cosh(02(k)X2) + M2(X2)sinh(02(k)X2), k - 1,11

(4.34)

where four step functions Ln(X2), Mn(X2) for n-l or n-2 have been

40

introduced
L (1). M (1),
n n
(2) (2)
Lh(X2) Ln , Mn(X2) Mn ,
L (3), M (3),
n n

(n-1,2).

In addition the values for 01(k) and 02(k), k - I or
follows
a (k)
l 2 2 2 2 2
0 (k) }- {0 {[1+A +(1-A) Ak] i [(1+A +(1-A) AR)
2

and it is assumed that

01(1) # 02(1), 01(11) # 02(11), whenever A i l

The relation (4.37)1 will hold if and only if

(A + 1)2

AI # -1 and AI # - ,

 

(A - 1)2

and relation (4.37)2 will hold if and only if

(A + 1)2

 

All i -l and AII # -

(A - 1)2

X26(-12, -R),
X26('Ro R) 9
X26(R, 12),

(4.35)

II, are given as

211/2

(4.36)

. (4.37)

(4.38)

(4.39)

’1 41
In the event that either (4.38) or (4.39) (or perhaps both) are not
true, then additional terms of the form chosh(0(k)X2) and
Xzsinh(0(k)x2) make up for the coalescence of terms that takes place
in (4.34).

Note in general that

(A + 1)2
- < -1 , (4.40)

 

(A - 1)2

since A > 0. In particular one finds from (4.36) that the values for

2

both (01(1))2 and (02(1)) are distinct, real and negative if

(A + 1)2
A < - , (4.41)

 

(A - 1)2

and are distinct, real and positive if

A > -1. (4.42)

However the pair 01(I) and 02(1) are complex conjugates if

(A + 1)2

- < AI < -l . (4.43)

 

(A - 1)2

Corresponding results hold for 01(11) and 02(11) when AI is replaced

42
by All in (4.4l)-(4.43). Thus if

A > -1, All > -1, (4.44)
the hyperbolic exponential functions in (4.34) have their standard
meaning, while if either member of (4.44) is not true, then standard
trigonometric functions enter into (4.34). The form (4.34) has been
chosen for convenience of study in the case in which (4.44) holds. As
shown in section 3, the case in which (4.44) holds corresponds to
materials with more or less "usual" engineering behavior. In addition
it is to be noted that in APPENDIX A a fairly elaborate argument is

given establishing that the following relations always hold:

(I) (II) (II) (I)
02 < 02 < “1 < 01 , if 0 < All < AI,

A > O, A # l. (4.45)

Substituting from (4.34) into boundary conditions (4.33) now
gives rise to a 12x12 linear system for the 12 unknown constants

denoted by the L's and M's. This system will be written as the form

J12x12 I‘12x1 " °12x1 ' (“'46)

where

(1) (1) (1) (1) (2) (2) (2) (2) (3) (3) (3) (3) T
IJ-(L1 .L2 .111 .142 .L1 .L2 .241 .112 .L1 .L2 .M1 .112 ).

(4.47)

and J is a 12x12 matrix which will be described in (4.50) and (4.51)

43

in detail. Bifurcation takes place provided that a nontrivial

solution exists for (4.46). This in turn requires that
det J - 0. (4.48)

Equation (4.48) can now be regarded as an equation for those A's at
which buckling can occur for a given value of the mode variable 0 -
mx/(le).

It is convinient to study (4.48) by introducing additional
dimensionless parameters as follows

II I
B - u( )/u( ). a - R/1

n - 012 - mnlz/l (4-49)

1’ 2'
Here n is the mode number of the buckled configuration scaled with
respect to the aspect ratio 12/11; 5 plays a role similar to that of a
stiffness ratio of the two composite materials comprising the
construction; and a is the volume fraction of the central ply within

the complete construction.

It will be convenient to write J as follows

J11 J12 J13

J21 J22 J23 ' (4'50)

 

 

- J31 J32 J33 -

with Jmn (m,n~l,2,3) are 4x4 submatrices whose entries are functions

of the parameters 0, 5, a, A, AI and AII' These submatrices are found

to be

44

J12 ' J23 ' J31 ' °4x4 '

 

 

 

J -
11
'Bllcosh(ol(1)12), -31181nh(01(1)12),lecosh(nz‘11)1z),-321s1nh(02(11’12)‘
-EIIsinh(01(I)12),Ellcosh(01(1)12), -EZIsinh(02(I)12),Ezlcosh(02(1)12)
o, (o, o, o,
_o, o, o, 0,
J13'
10. 0. o. o,
0, 0, 0. 0.
Bllcosh(01(I)12), Bllsinh(01(1)12), BZIcosh(02(I)12), BZIsinh(02(I)12),
_Ellsinh(01(1)12), Ellcosh(01(1)12), EZIsinh(02(I)12), Ezlcosh(02(1)12), .
321'
”-cosh(nl(1)n), sinh(01(I)R), -cosh(02(I)R),
01(I)s(Iinh(01(I)R), -01(I)cosh(01(I)R), 02(I)sinh(02(I)R),
-311cosh(ol(1)n), Bllsinh(01(I)R), -321cosh(02(1)a),
_Ellsinh(01(I)R), -E11cosh(01(I)R), EZIsinh(02(I)R),
sinh(02(I)R), 1

~02cosh(02(I)R),

I (I)
82 sinh(02 R),

-Ezlcosh(02(I)R), .

 

 

 

J

22
'cosh(01(II)R),
-01(II)sinh(01(II)R),

I (11)
631

_-€EIIIsinh(01(II)R),

Icosh(01 R),

 

J

32
'-cosh(01(II)R),

(II)

(II)
-01 sinh(fl1 R),

II (II)
1

11
_-531 sinh(01

-£B cosh“)1 R).

(II)R),

 

45

(II) (II)

 

-sinh(01 R), cosh(02 R),
01(II)cosh(01(II)R), -02(II)sinh(02(II)R),
-€BIIIsinh(01(II)R), eBZIIcosh(02(II)R),
(Elllcosh(nl(II)R), ~6E2118inh(02(II)R),
-sinh(02(II)R),
02(II)cosh(02(II)R),
-£BZIIsinh(02(II)R),
€EZIIcosh(02(II)R),
-sinh(01(II)R), -cosh(02(II)R),
-01(II)cosh(nl(II)R), ~02(II)sinh(02(II)R),
-€BIIIsinh(01(II)R), -€BZIIcosh(02(II)R),
-€E111cosh(01(II)R), -£EZIIsinh(02(II)R),
-sinh(02(II)R),
~02(II)cosh(02(II)R),
-€BZIIsinh(02(II)R),
-gEZIIcosh(02(II)R), j

 

46

J33'
'cosh(01(I)R), sinh(01(I)R), cosh(02(I)R),
01(I)sinh(01(I)R), 01(I)cosh(01(I)R), 02(I)sinh(02(I)R),

 

 

Bllcosh(nl(I)R), BIIsinh(01(I)R), lecosh(02(I)R),
_E11sinh(01(I)R), Ellcosh(01(I)R), EZIsinh(02(I)R),
sinh(02(I)R),
Ozcosh(02(I)R),
lesinh(02(I)R),
EZIcosh(02(I)R),
where
31k - A0 + (An)'1 03(k)2, j-1,2, k—I,II,
k -2 -1 2 (k) -2 (k)3
EJ - ( [2+A +(l-A ) Ak]0j - (An) 01 ), j-1,2, k-I,II,

-1/2+A1/2)(AII'AI)

g . D(I)/D(II) - p (A (1+AII/2)/(1+AI/2).

(4.51)

It is to be noted that (4.48) constitutes a relation for the ordered

variable set (A,n,fi,a,AI,AII). Thus if one defines

i(A,n,fl,a,A All) I det J, (4.52)

I,

47

than (4.48) gives rise to

'(AtflvpvavAlvAII) - 0 (4°53)

as a condition for bifurcations of the type (4.1) away from the
homogeneous solution (2.19). In addition to the conditions (4.44) it
may be concluded from (4.27) and (4.49) that A, q, 6, a are restricted

to the following intervals
A > 0; n > 0; 6 > O; O s a s l. (4.54)

Although 0 must take on discrete values determined by the aspect ratio
12/11, it can be treated as a continuous variable for the purpose of
analysis. Equation (4.53) shall be called the general buckling
equation.

Note that this problem ought to reduce to the non-composite case

for the following two special cases:

(1) 5-1 and AI-Au, ‘

since then material I is identical to material II,
A (4.55)
(ii) either a-O or a-l,

 

since then only one phase is present.

48
5. GENERALIZATION OF THE MATERIAL CONSTANT A

The purpose of this section is to provide additional detail of
the logic which led to the choice of (2.9) as the class of strain
energy densities to consider. The reader will lose no continuity to
the rest of this dissertation if one proceeds directly to section 6.

As already mentioned, Sawyers and Rivlin [10] have already
examined, in great deal, the problem corresponding to that under study
here, for the case in which the body is composed of a single material
(the non-composite case). In fact Sawyers and Rivlin [10] accomplish
the reduction to the problem of a single ordinary differential
equation (corresponding to (4.31) here) for the case of an arbitrary
strain energy density W(Il,12). As might be expected, the associated
linearization, although not difficult in principle, requires very
careful Taylor expansions of the associated stress and deformation
measures, and hence an extraordinary attention to detail. Sawyers and

Rivlin [10] then show that the results so obtained depend heavily on

the expression.

2(A1 + A2)2
A(A1,A2,A3) - (wl1 + 2A3 w12 + A3 1122).

 

(5.1)

Here A A are principal stretches associated with the

1' J‘2' 3
homogeneous solution (2.19) so that A1 - p, A2 - )9-1 and A3 - 1. In
(5.1) the derivatives of W(I1,IZ) are with respect to the principal

invariants

49

8W 62W

"1(11’12) - , w - -———————. 1,3 - 1,2. (5.2)

11
611 311311

 

Thus, in view of (2.20), all of the derivatives appearing in the right
hand side of (5.1) can be regarded as a function of p. Furthermore,

since A - A2/A1 - p'z, (5.1) gives

A(11.A - A(P.P-1.1) - K(p) - 3(A-1/2) - 4(A)- (5.3)

2.A3)

Sawyers and Rivlin [10] show that a great deal of simplification to
the ensueing analysis results for the case in which A(A) is constant,
and, in fact they, for the most part restrict their treatment to this
particular circumstance. The reason for this simplification will be
discussed at the end of this section. Now, as noted in [10], it is
easily verified that A(A) is constant for the case of a neo-Hookean
material. The answer to the following question, however, is not

immediately clear:

What strain energies W(Il,12) give rise to a function A(A) that

is constant?

(Q1)

In view of the anticipated complications that the study of
composite construction would introduce, it was decided to investigate
(Q1) with a view towards using such strain energies for the individual
materials in the problem under consideration here. In fact, there is

apparently no increase in difficulty in investigating the following

50

obvious generalization of (Q1):

For a given function A(A1,A2,A3), and hence a given A(A), what

strain energies W(Il,12) ensure that (5.1) is obtained?

(Q2)

An examination of (5.1) indicates that for a given A(A), the
equation (5.1) can be viewed as a single linear partial differential
equation for W(Il,12). Since there are no associated boundary
conditions, one would suspect that it might be a relatively simple
matter to "solve” (5.1) for W(Il,12) and, moreover, that numerous
solutions W(Il,12) ought to exist. Indeed, this is apparently the
case.

For simplicity, attention will be restricted to the class of

materials that are independent of the second strain invariant 12:

w - («(11) (5.4)
so that (5.1) and (5.3) gives
I<p> - 2(p+p'1>2w11(1+p2+p'2)/w1(1+p2+p'2). (5.5)
and
Au) - 30'1”). (5.6)

It is to be noted that both the power law material (3.21) and
(it's specialization) the material (3.1) utilized in the present
study, are special cases of (5.4).‘ In fact for the power law material

(3.21), one Obtains from (5.5) and (5.6) that

51

_ 2(n-1)(A+1)2
A(A) - (5.7)

A2 + [(n/b)-2]A + 1

 

while for the material (3.1) one similarly obtains

A(A) - A (5.8)

Thus the class of materials (3.1) meets the objective of ensuring
that A(A) is constant. Moreover, in view of (5.7), the material
(3.21) does not meet this objetive unless the denominater in (5.7) is

equal to (A+l)2, which in turn will be true if and only if
(n/b) - 2 - 2 or n - 4b. (5.9)

Thus the number of free constants reduces from three to two for the
material (3.21) under the requirement that A(A) is constant. In fact,
in view of (3.23), the material (3.1) is the most general case of a
material (3.21) that in addition gives a constant value for A(A).
With this background, the question (Q2) will now be investigated

in general for the class of materials (5.4). Let

 

w1(1+p2+p'2) - F(p). (5.10)
Then
dF(p) d1
w11(1+p2+p‘2) - —— / 1 . (5.11)
d d
p p

where

52
dIl/dp - 2p - 2p'3. (5.12)

Thus (5.5) is equivalent to

d? (p-p'3)2i(p)
____ - F - o , (5.13)

62
-1 2
2(2+2 )

 

which in turn will be true if and only if

2(3) (32-1)
1n F(p) - ds . (5.14)

 

 

s (32+1)

The equation (5.14) provides an explicit formula for the determination
of F(p) and hence W(Il). Returning to the case of A(p) - A, a

constant, one obtains from (5.14) that

K(s) (52-1) A 2 _2
1n F(p) - ds - — 1n(2+p +p ) + 1n(c1),
2

 

 

s (52+1)
(5.15)

where 01 is an arbitrary constant. This in turn gives

53

F0!) - c1(2+pz+p'2 M2. (5.16)

which in view of (5.10) gives

dW(I )
_____l__ - c (1+1 )A/Z, (5.17)
1 1
d1
1
Simple integration now yields
1+A/2
W(Il) - (2c1/A) {(1+Il) + c2}, (5.18)

where c1 and c2 are as yet undetermined constants. Thus (5.18) is the
most general form of a strain energy density (5.4) that in addition
gives a constant value of A(A). As is standard, the inessential
additive constant in the strain energy density (5.18) is chosen so

that W’- 0 in the undeformed state (II-3). This gives

c2 - -4(2)A. (5.19)

Finally, by choosing
c1 - uA/4, (5.20)

one obtains (3.1). Thus it may be concluded that (3.1) is the most
general class of strain energy densities that obey (5.4) and in

addition give rise to a constant value for A(A).

As the above procedure indicates, the particular requirement
that A(A) must be a constant did not provide an essential

simplification to obtaining a strain energy density obeying (5.4).

54
Thus the above procedure could be utilized to determine classes of
materials obeying (5.4), and hence (5.1), that give rise to any
given function A(A).

If A(A) is not constant, the work of Sawyers and Rivlin [10] for
the non-composite case would seem to indicate that an equation
analogous to (4.53) would continue to govern buckling provided that AI
and AII were replaced by the associated functions AI(A) and AII(A).
Since complicated expressions are then compounded inside the
hyperbolic functions that comprise the entries of J in (4.50), a
treatment of (4.53) would seemingly present a great deal of

complication. This is the essential reason why attention in this

dissertation has focussed on the class of materials (2.24).

55

6. TWO SPECIAL DEFORHATION TYPES FOR THE NEO-HOOKEAN COMPOSITE PLATE:

FLEXURE AND BARRELLING

The analysis of the problem at hand is quite complicated even if

the material constant A is non-zero. Thus for the rest of this

dissertation, attention shall be restricted to the case in which both

materials are neo—Hookean:

Ak-O’ kP(I,II).

This gives the following simplifications

w(k)(11) - ”(k) (11 - 3)/2 . k-I,II

c (k) - 1. 0“" - (umfl. 1-1.2;

01 } - { An ,
02 n ,
31* 2A0 ,
k - _1 k~1,11
132 (MA ) 0 .
Elk (2+A'2) A0 ,
£2 20 ,

{-ps

(6.

(6

(6.

(6

(6.

(6.

(6.

1)

.2)

3)

.4)

5)

6)

7)

21

22

11'

 

13"

 

 

 

32"

 

2Ac

-As

2Ac

-2Ac

As

°4

-Asa

ZflAc4

-fiAs4 flAca -Zfis3

-ca

-As4

-23Ac4-26Asa-flAc3

-fiAs4 -flAca -2£s3 —

2As Ac

Ac 2s

Ac -s

3 3
~26AshflAc3

-Ac -s

3 3

56

-As

2c

As

2c

 

 

 

 

 

 

 

ca 34 c3 83
Ash Ac“ 33 c3
J33 " ’
2Ac4 2A3“ Ac3 A53
_ As4 AC4 283 2c3 .
(6.8)
where
cl-cosh(n), c2-cosh(An), c3-cosh(na), ca-cosh(Ana),
sl-sinh(n), sz-sinh(An), s3-sinh(na), sh-sinh(Aqa),
A-(A+l/A).
(6.9)
Furthermore, the load-deformation relation (2.34) becomes
-3 II I
2 - 413 (p-p ) [Ru‘ ’ + (12-R)u‘ ’1. (6.10)

For the non-composite case, it is shown in [10], [11] that two
deformation types are possible. The first of which is a flexural
deformation and the second of which is a barrelling deformation.
Moreover it is also shown that these two types exhaust all possible
solutions. For the composite case, both of these deformation types
remain possible. However, analytical difficulties have so far
prevented this investigation from showing that these two types exhaust
all of the possible solutions.

Nevertheless, in what follows, attention is limited to these two
deformation types.

A Flexural deformation is one in which the corresponding

58

solutionU2 is an even function of X2 so that U1 is an odd function of

X. by virtue of (4.28). Considering (4.34) and (4.35), this requires

2
that

> (1),M2(1> (3)"L (3) M (3)’_M2(3)),

(1) (1
(L1 'L2 '"1 ) ' (L1 2 ' 1

(6.11)

(2) (2)
L2 - M2 - on

so that system (4.46) reduces from the 12 x 12 system to the following

6 x 6 system:

 

 

 

 

. 2Ac2 -2As2 Ac1 -As1 o o ‘ - L1(1) 1
'482 A02 -2s1 2c1 0 O L2(1)
-c4 s4 -c3 83 c4 c3 ”1(1) 0
As4 -Ac4 33 -c3 -154 -83 “2(1) - 6XI '
-2Ac4 2Asa -Ac3 As3 2Apc4 Apes L1(2)
. Asa -Ac4 2s3 -2c3 -Afls4 -2653 . _ “1(2) _

(6.12)

In place of equation (4.53) one then obtains a simpler flexural
buckling equation found by setting the determinant of the matrix in

(6.12) equal to zero. Denote this relation as:

wF(Aoflvfloa) - o 0 (6°13)

A barrelling deformation is one in which the corresponding

59

solutionU2 is an odd function of X2 and U1 is an even function of X2.

This requires that

(3) _M

(1) (1) (1) (1) (3) (3) (3)
(L1 9L2 9H1 3M2 ) - ('Ll 9L 9M2 )5

2 1
(6.14)

(2) _ M (2) _ 0,

L1 1

which in turn gives that system (4.46) also reduces from a 12 x 12

system to a 6 x 6 system,

 

 

 

 

- 2A¢2 -2A82 Ac1 -A31 0 o ‘ ' L1(1) -
-As2 Ac2 -2s1 2c1 0 0 L2(1)
-c4 sh -c3 s3 -34 -33 M1(1) 0
Ash -Aca 83 -c3 Aca c3 ”2(1) - 6X1 '
.2Ac4 2As4 -Ac3 A53 -2Afisa-A653 L2(2)
. A84 -Ac4 2s3 -2c3 Aflca 2663 . _ M2(2) .

(6.15)
In this case the associated barrelling buckling equation, found by

setting the determinant of the matrix in (6.15) equal to zero, will be

written as:

*B(A.n.fi.a) - 0 . (6 16)

60

Both .F and i are smooth functions of A, a, fl and a. For a

B
given composite construction, both 6 and a are fixed. Then flexural
and barrelling bifurcations are governed by the n-A relation that
follow from (6.13) and (6.16) respectively.

Consider the flexural case, for a given triple (n,fl,a), one seeks

roots A to (6.13). It is easily seen that

wF(lvflvpva) - 09

since the final two columns in the coefficent matrix of (6.12) are then
identical. Thus A - 1 is always a solution to (6.13). However since
this corresponds to no end displacement and hence no thrust, it is not
of interest and so will not be considered further.

The complicated nature of (6.13) gives rise to formidable
analytical difficulties. Consequently a numerical investigation of
this equation has been pursued. Such an investigation indicates for

fixed n, p, a that V monotonically increases from i - -Q at A - O

F F

through 'F - O at A - l to some maximum value. Then 2F subsequently

is found to monotonically decreases as A 4 0, again passing through WF
- 0. Thus in addition to the root A - l, a second root A > 1 exists
for equation (6.13). Moreover since A > 1, these solutions only exist

for compressive loads. That is, bifurcation instability only occurs

upon thrust. Henceforth denote this nontrivial root by
A - ¢F(n.fi.a) . (6.17)

The barrelling case is similar, namely for all triples (n.6,a), A - l
is always a root of (6.16). In addition one finds that there always

exists exactly one other root A > 1 to the barrelling equation (6.16).

61
Likewise denote this nontrivial root by

A-fihflm). ‘ «Am

Numerical routines were developed, based on simple bisection, to
determine the functions §F(n,fl,a) and §B(n,fi,a). The corresponding
computer programs FLEXUREl and BARRELLINGI can be found in (I) and
(VI) respectively of APPENDIX C. Specifically (I) calculates
QF(n,fi,a) and (VI) calculates §B(n,fi,a).

Although it would be interesting to develop a completely
analytical, as opposed to numerical, treatment of the bifurcation
equations, the size of the matrices involved makes this a very
difficult prospect. That is the general buckling equation (4.53)
stems from a 12 X 12 matrix. In general the size of the matrix for an
arbitrary number of plies N will be 4N x 4N, where, in the problem at
hand N - 3. Sawyers and Rivlin [10] for the non-composite case N - l,
are succesful in developing their analysis to a point much further
than that attained here, before turning to a numerical procedure. In
APPENDIX B, it is shown that the results attained here for the
composite case can, by purely analytical means; be made to yield the
results obtained in [10], for the special case in which the construction

is in fact not a true composite (i.e. if (4.55) holds).

62
7. BUCKLING AND WRINKLING FAILURE MODES FOR THE COMPOSITE CONSTRUCTION

The purpose of this section is to determine the critical
instability, and more generally the ordering of the stability modes,
for the problem under consideration. As mentioned in the previous
section, Sawyers and Rivlin [10], [11] have shown for the
non-composite case that buckling can occur involving either flexural
or barrelled mode shapes with an arbitrary integer number m of
half-wavelengths. A diagram of these failure modes, as they would
occur in the composite consturction under consideration here, is
presented in Figure 7-1, where TmF and TmB (m - 1,2,3...) denote the
corresponding failure thrusts. A central issue is to determine the
ordering of these failure thrusts. For the non-composite plate,

Sawyers and Rivlin [10] have obtained the following result

<Tm <...<T <T . (7.1)

Here Tan is the value associated with an infinite number of wave
lengths and can be identified with a wrinkling instability. For the
composite case some striking differences are obtained, most notable
is a possible change in ordering of the failure thrusts. These
results will be demonstrated in this section and some of their
practical consequences will be discussed.

The failure stretch ratios are now defined as follows:

63

AmF - ¢F(mxlz/211,fi,a) ‘

r , m - 1,2,3... (7.2)

 

AmB - (Emu/211.24) .

once these failure stretch ratios are found, the corresponding failure
thrusts are found from (6.10). In particular it is to be noted that
the failure thrusts are ordered the same as the failure stretch
ratios.

To find the ordering of the flexure failure stretch ratios for
fixed.values of p and a, one plots the points (mxlz/le, AmF) using
(7.2). Programs FLEXUREZ and BARRELLINC2, found in (II) and (VII) of
APPENDIX C, can be used for this purpose. The lowest value of AmF
determines the critical mode number m for flexure as well as the
critical flexure failure stretch ratio and hence the critical flexure
failure thrust. The critical mode number m for barrelling as well as

the critical barrelling failure stretch ratio and the critical

barrelling failure thrust are found similarly.

For all of the non-composite cases (4.55), the numerical method
embodied in those programs consistently gives the (q,A)-re1ation as
found by Sawyers and Rivlin (10) and displayed in Figure 7-2.

Notice in this case that O as a function of n is monotonically

F
increasing from A - l at n - 0 to A - 3.3833... as H e w. Hence
according to (7.2)1 the flexural failure thrusts are ordered as

follows:

F F F F F
T1 < 22 < ... < Tm < Tm+1 < ... 4 Tan , (7.3)

where TmF is found from (6.10) using the asymptotic value A - 3.383...

64

Similarly for the non-composite case, 0 as a function of n is

B
monotonically decreasing from A - a at-q - 0 to A - 3.383... as n 4 0.

Hence according to (7.2)2:

B B B B B
T1 > T2 > ... > Tm > Tn+1 > ... 4 Ton . (7.4)

Finally since both Q and ’8 have the same asymptote as 0 e o, it

F
follows that

T - T I T . (7.5)

Physically Tco gives rise to an instability which corresponds to a
wrinkling failure. Combining (7.3)-(7.5) gives (7.1).

It is found that the orderings (7.3), (7.4) and (7.1) given by
Sawyers and Rivlin (1974,1982), and confirmed by the programs FLEXURE2
and BARRELLING2 for non-composite constructions, will, for certain
composite constructions, cease to hold. The ordering of the failure
thrusts is determined by two factors: (i) the qualitative behavior of
the functions QF(n,fi,e) and ¢B(n,fi,a) for fixed (fi,a) as the mode
parameter 0 is allowed to vary, and (ii) the spacing of the sequence
of n values.

The qualitative behavior of the n-dependence can be characterized
with respect to the parameter pair (fl,a). The spacing of n-values is
then determined by the aspect ratio 12/11. Thus the three parameters:
ply stiffness ratio 5, central ply volume fraction a, and aspect ratio
12/11, completely determine the ordering of the failure thrusts for
the problem at hand. Within this framework the present section is
organized as follows. Begining with the function ¢F(n,fl,a) the

possibilities for the qualitative behavior of the a dependence is

65

documented. Then for each distinct qualitative behavior so obtained
the consequence of different possible q-spacings is examined.

A similar program is followed for the function QB(n,fi,a). In this
fashion the possible new ordering for the failure thrusts are
uncovered and these new orderings are correlated with the associated
composite constructions by means of the parameter pairs (fi,a) and the
aspect ratio 12/11.

First of all, however, it will be expedient to demonstrate those
qualitative behaviors that hold regardless of the pair (B,a). For all
values of (fi,a) one finds that §F(n,fi,a) is initially monotonically
increasing from the value 1 at n - 0 and tends to an asymptotic value
as n 4 o. Similarly for all values of (fi,a) one finds that ¢B(n,fi,a)
is initially monotonically decreasing from a at n - 0 and tends to the
same asymptotic value as n'+ 0. This common asymptotic value shall
be denoted as:

A¢(flta) - 11“ °F("vflua) - 11m ¢B(’79fiva) ° (7'6)
froco groan

Moreover it is found that

for all finite n > 0. Thus (7.5) holds for all composite

configurations where now Tco - Tm(fl,a). In addition (7.7) yieids

3

TF<TB, m-l,2,3... (7.3)
In In

In particular (7.8) indicates that the critical flexure failure thrust
is always less than the critical barrelling failure thrust. Thus the

critical flexure failure thrust gives the first bifurcation for all

66
pair (p,a) and all aspect ratio 12/11 within the class of bifurcations
under consideration.

At this point the investigation shall consider those qualitative
behaviors for the n-dependence of the functions QF(n,fi,a) and
§B(n,p,a) which result in new orderings of the failure thrusts. The
ordering (7.3) of flexure failure thrusts will continue to hold if
QF(n,p,a) is monotonically increasing for all n z 0. Pairs (fi,a)
which give rise to QF(q,fi,a) having this property will be said to
belong to the set P1P.
6 P1P (Figure 7-3). Note in this case that Am - 3.271...

For example, one finds that (fi,a) - (0.5,0.5)

However one finds for certain pairs (fi,a) that QF(*,B,a) is no
longer monotonically increasing over the whole domain 0 z 0. In
particular it is sometimes found that ¢F(n,fl,a) is monotonically
increasing over a finite domain 0 s n < "max - "max(p’a) but is
subsequently monotonically decreasing to Ao(fl,a) for q>nmax(fi,a).
Pairs (fl,a) which give rise to this behavior will be said to belong to

the set P F. For example one finds that (2,0.5) e P F, in which

ii
case "max - 1.977... and Aco - 3.439... (Figure 7-4).

ii

Composite configurations for which (8,0) 6 riiF give rise to
flexure failure thrusts that no longer obey (7.3). To determine the
ordering in such a case one should note for (fi,a) e PiiF that the

equation

¢F(fl.p.0) - Ao(p,a) 9 (7-9)

will have a unique finite root, which shall be denoted by nT(fi,a), and

that this root will obey

0.1.03.0) < nmwm) . (7.10)

67
The program PROJl in.(XI) of APPENDIX C can be used to find this root.

For example, refering to Figure 7-4, note that nT(2,O.5) - l.392...<
1.977... - "max(2’o°5)' Recalling that the discrete values of the
sequence of novalues is determined by 12/11, it follows that, for a
given aspect ratio 12/11, one can then determine integers p, q such

that
pnlz/le < "T s (p + l) n12/211 ,
qxlz/le s "max < (q + 1) «12/211 . (7.11)

It is clear that 0 s p s q.
If p > 0 it follows that «12/211 < "T so that T1F is the critical
flexure failure thrust. In this event one may define the flexure

failure thrust sequences

F F
31 - { T1 ’00. ’Tp ) g
F F
32 - { TP+1 ,. ,Tq ) ,
3 -{1: F 41F) (rovided ) (712)
3 q+1 "O. a ’ p p<q . O

The ordering of the flexure failure thrusts will now consist of 31
followed by an interspersing of 32 with a reverse ordering of the
sequence 33.

On the other hand if p - 0, then «12/211 2 "T' Assume for the
moment that the inequality is strict. It then follows that TmF is the
critical flexure failure thrust. In this event the sequence 31 is
empty and the ordering of the flexure failure thrusts will consist of
an interspersing of 32 with the reverse ordering of the sequence 33.
It is to be emphasized in this interspersing that TmF will in this

68
case lead the sequence of flexure failure thrusts.

In both cases p - 0 and p > 0 the thrust Tan is not an upper bound
for the set of values TmF. Recall now that both the flexure failure
thrusts and the barrelling failure thrusts cluster around Tm. Hence
it may be concluded that some of the barrelling failure thrusts will

be interlaced with some of the flexure failure thrusts.

F
ii ’

with the barrelling failure thrusts regardless of aspect ratio 12/11.

Thus if (fl,a) e P the flexure failure thrusts are interlaced

The critical flexure failure thrust will, depending on the aspect

ratio 12/1 be either the m - l flexure failure or the m - w

1'
wrinkling failure. The transition between the m - 1 flexure failure

and the m - w wrinkling failure occurs at the transition aspect ratio

F F
12/11 - ZnT/«. At this transition aspect ratio T1 - To . Hence fer

a PiiF-plate which is sufficiently short in the direction of thrust

(specifically l < lzt/(ZnT)), wrinkling is the critical flexure

1

instability. Hewever, fer a F F-plate which is sufficiently long in

ii

the direction of thrust (specifically I > lzn/(ZnT)), the critical

1
flexure instability is the m - 1 mode.

11F, that there exist infinitely

many aspect ratios at which a flexure failure thrust from 32 will

It is to be noted for (fi,a) e P

coincide with a flexure failure thrust from 33. To see this choose A
in the interval A < A < O (n ,fl,a). There will exist two
a F max
intersections of the horizontal line corresponding to this value of
A with the graph of QF(n,fi,a). One intersection will occur in ”T < n
< n and the other will occur in n > n . Denote these
max max

intersection values of n by n1 and 02 respectively. The ratio "2/"1

can be made to take on any value greater than 1 by appropriately

69

choosing A in this procedure. If this ratio is a rational number it
then follows that "2/"1 - r/s for infinitely many integer pairs r and
8. Choose one such integer pair, for example the case where r and s
are coprime. Then the aspect ratio 12/11 - 201/(sx) - 2n2/(rx) will
yield TsF - TrF. Moreover this construction will hold for each

rational number greater than 1. Note that there is no possibility

that a single flexure failure thrust could correspond to three or more

flexure failure modes m whenever (fl,a) e riiF'
The regions PiF and PiiF in the semi-infinite strip
lI-{(fl,a)|,620,05asl}, (7.13)

of possible (fi,a) parameter pairs have been determined by means of an
exhaustive numerical sampling procedure and are displayed in Figure 7-
S. This sampling was carried out as follows. First, for a given
parameter pair (fl,a), the corresponding region type was determined by
examining the monotonicity of the A-n flexure relation. This
examination can be accomplished by means of the program FLEXURE3 in
(III) of APPENDIX C. This program has then in turn been incorporated
into the two separate programs FLEXURE4 and FLEXURES (in (IV) and (V)
of APPENDIX C) which sample points (fi,a) on specified horizontal line
segments and vertical line segments of n respectively. In this
fashion, the boundaries between the various regions of n were
determined.

Points to the right (left) of the vertical line segment 5 - 1
correspond to cases in which the central ply is composed of a material
which is more (less) stiff than the material comprising the outer

plies. Points above (below) the horizontal line a - 1/2 correspond to

70
cases in.which the central ply comprises more (less) than half the
construction. A conspicuous feature of Figure 7-5 is the presence of

a region P F corresponding to pairs (fl,a) which belong to neither

iii
P F nor P F. The meaning of this region will be discussed below. In

i ii
Figure 7-4 the points (6,0) 6 P1P, (3,1) 6 P1F and (l,a) e PiF by

virtue of (4.55). However it is found that pairs (fl,a) very close to

these values may in fact not belong to P F For example one finds

1 .
F .. ..

11 11 . The threading of the
F

segment 5 - 1 through the "pass” created by the regions P11 and

rmF near (6.6) - (1.0.0.9) is displayed in Figure 7-6.

The presence of P

that (0.5.0.99) e r F and (1.1.0.5) e r

11F so near the boundary a - l for 0 < 5 < 1

indicates that if the aspect ratio 12/11 is sufficiently large then
the addition of relatively thin and stiff outer layers can suppress
the low wavelength flexure modes enough to lead to the dominance of

the m - a wrinkling instability. Since each pair (fl,a) e P 1? gives

1

rise to an aspect ratio dependence upon the critical flexural

instability, we display the value of nT(fi,a), which determines the

transition aspect ratio, for representative pairs (fi,a) E 1‘11F in

table 7-1. This table was generated with the aid of PROJl in (XI) of
APPENDIX C.

Now turn to consider the buckling behavior of composite
constructions corresponding to parameter pairs (fi,a) E PiiiF' Each
such pair (6,0) gives rise to a function QF(n,fi,a) which contains
both internal maxima and internal minima as n varies from 0 to a. The
ordering of the flexure failure thrusts for composite configurations

in which (fl,a) 6 F1 F may in fact be quite complicated. First of all

ii
it is to be noted that (7.3) might still hold. Specifically if the

71
number of internal maxima is finite and equal to the number of
internal minima, then the asymptotic value Ao will be approached from

below and consequently (7.3) will continue to hold for sufficiently

F
iii

clear that there will exist certain aspect ratios such that (7.3) will

large aspect ratios 12/11. However for each (fl,a) e P it is also

not hold. In particular for each (fi,a) e riiiF there will exist

infinitely many aspect ratios such that certain flexure failure

thrusts will correspond to two distinct flexure mode m by the same

argument used for (fl,a) e riiF' Finally the possibility of a single
flexure failure thrust corresponding to three or more flexure mode m
remains a possibility whenever (fl,a) e riiiF'

The investigation shall now turn to consider the barrelling
failure thrusts. The ordering (7.4) of barrelling failure thrusts
will continue to hold if 08(n,fl,a) is monotonically decreasing in n
for all n z 0. Pairs (fl,a) which give rise to ¢B(n,fi,a) having this

property will be said to belong to the set F13.

For example (2.0.5)
6 F13 (Figure 7-4).

On the other hand if ¢B(n,p,a) is found to be monotonically
decreasing over an interval 0 s n < "min - "min(p'a) and is

subsequently monotonically increasing for n > nm1n(fi,a), then we shall

113. For example (0.5.0.5) e I‘flB (Figure 7-3).
Finally if (5,0) belong to neither P13 nor 1"“B then it is said that

B implies that QB(q,fi,a) has an

say that (fl,a) E P

(fi,a) e P1113. Clearly (8.0) 6 P111

internal maximum.

113, the relation (7.4) will no longer hold.

Specifically, T5 can no longer be the critical barrelling failure

For (fi,a) e P

thrust. In fact, the critical barrelling failure thrust can be made

 

72

to correspond to any mode number m < o by, for example, taking an

aspect ratio 12/11 - an1n(p,a)/(mx). .Coincidence of two failure

barrelling thrusts is also a possibility whenever (fl,a) 6 P113. In

fact for consecutive integeres m and n+1, if TmB - Tm+1B then this

common value of barrelling failure thrust must of necessity be the

critical barrelling failure thrust. Moreover it can be shown that

there exists a unique aspect ratio such that TmB - Tm+1B for any

integer m. Finally, it is to be noted that interlacing of the

flexural and barrelling buckling loads will occur regardless of aspect

8
ii °

then the ordering of the barrelling failure

ratio 12/11 if (fl,a) e P

If (fl,a) e r1113.
thrusts is complicated by the precise placement of the internal maxima
and minima of QB(n,fi,a). In fact, phenomena paralleling the

possibilities outlined previously for riiiF pertain also to P1113.

The partitioning of n into regions P13, P113 and P111B is
displayed in Figure 7-7. This was accomplished with the aid of
BARRELLING3, BARRELLINC4 and BARRELLINGS given in (VIII), (IX), (X)
of APPENDIX C. These programs are the analogues of FLEXUREB,
FLEXURE4 and FLEXURES respectively. Note from Figure 7-7 that the
boundary of the region 1‘11B is at certain points quite close to the
pairs (fi,a) corresponding to the non-composite case (4.55). The
presence of P118 so near the boundary a - l for 8 > 1 indicates that
a plate which includes relatively thin and flexible outer layers can
be “tuned" to any desired critical barrelling mode by appropriately
selecting the aspect ratio 12/11. For such a result to be of

significant practical interest, however, it would seem that it would

be necessary to devise a method to suppress the preceding flexure

 

73
modes.

Since the interlacing of the flexural and barrelling failure
modes is an intriguing result, it would be useful to further
characterize this behavior with respect to 5, a and 12/11. For a
given pair (fl,a) interlacing can be made to occur for at least one
12/11 if for any vfilue of n either ¢F(q,fl,a) > Aco or ¢B(n,fl,a) < A”.
Furthermore interlacing is ensured for all 12/11 if either QF(n,fl,a)
approaches Aco from above as n 4 w or if ¢B(n,fl,a) approaches Aco from

below as n v a. One or the other such asymptotic behaviors giving

F
ii
113- In addition, an asymptotic behavior that ensures
interlacing could also occur if (fi,a) e riiiF u r1113, however for

such points this asymptotic behavior may be extremely sensitive to

rise to interlacing for all 12/11 will occur if either (fl,a) e P

or (fl,a) e P

small changes in (fl,a) and hence difficult to determine. Even so, it

is interesting to note from Figures 7-5 and 7-7 that points (fl,a) e

4
riiF U F113 comprise the major portion of the strip H for both 8 > 1

and 6 < 1. Thus, from this point of view, interlacing is not at all
unusual both when the stiffer material comprises the outer layers (6 <
l, e.g. Figure 7-3), and when the stiffer material comprises the
central ply (B > 1, e.g. Figure 7-4).

It is also interesting to classify the points (fi,a) with respect
to satisfaction ofequations (7.3) and (7.4). There are three
possibilities. First, it may be that both (7.3) and (7.4) --- and
thus (7.1) --- hold for all aspect ratios 12/1 . In this case it

1

will said that (5.5) e E For this to occur the pair (fl,a) must

1.
belong to both 1‘1F and F13. Second, it may be that there is at least

one aspect ratio 12/11 that will result in both (7.3) and (7.4) being

74

violated. For this to occur the pair (fi,a) must belong to neither Pip
nor F13. In this case it will said that (fi.a) e 82. Finally it may
be that for each aspect ratio 12/11 either (7.3) or (7.4) holds, but
that there is also at least one aspect ratio 12/11 for which one of
these relations isfiviolated. This will occur for the remaining case
in which (B.a) belong to either P1F or P13, but not both. In this
case it will said that (fi,a) E 83. Summarizing then these
definitions.

_ F B

-1 - l‘1 n l‘i ,

_ F F B B

-2 - (I‘i1 U I‘111 ) n (I‘ii U riii ) , (7.14)

_ F B B B F F

“3 ' (Pi “ (rii U riii )) U (Pi ” (rii U riii ))

One can determine these regions on the basis of Figures 7-5 and 7-7,
the result of which is given in figure 7-8. This figure indicates
that 51 is confined to a simply connected region containing pairs
(p.a) corresponding to the non-composite case (4.54). Thus. in this
sense, the composite constructions under consideration must be "close"
to a non-composite construction if (7.1) is to hold. Figure 7-8 also
indicates that the region 33 comprises the majority of the semi-
infinite strip n. 'In particular the (fi.a)-pairs (0.5.0.5) and
(2.0,0.5), associated with Figures 7-3 and 7-4 respectively, are each
2, on the other hand, comprises the
least area within the semi-infinite strip n. For pairs (fl,a) e E

a member of 33. The region E
A;

2

both §F(n,fi.a) and QB(Q,p,a) display non-monotone behavior as for

example shown in Figure 7-9 for the point (fl,a) - (0.5.0.8). Finally

it is to be noted from Figure 7L8 that the (fi.a)-classification is far

 

75

more sensitive near a - 1 than it is near a - 0. This confirms one's
intuition as to the effect that placement of thin 'stiffeners" (or
even '10oseners') would have in a much thicker homogeneous plate;
namely that the addition of thin plies on the external XZ-faces of the
plate would have a more pronounced effect on the buckling behavior
than would the insertion of a single double thickness ply on the
plate's midplane. Thus it is found that burying a very thin ply at
the center of a plate will mask its effect upon altering the order of

the failure thrusts.

76

 

 

 

Figure 7-1:

Examples of the flexural and barrelling buckling modes, and failure
thrusts, for m - 1,2 and 3. Higher order buckling modes involve

additional repetition of the basic m - l half-wavelength mode shape.

 

 

:1

 

77

Figure 7-2:

The functions ®F(q,p,a) and 08(n,fl,a) for all (fl.a)-pairs
corresponding to the non-composite case given in (4.38). Here A” =

3.383.

 

 

 

 

 

 

78
A. [ Figure 7-3:
£0“ The functions OF(q,fi,a) and 03(n,fi,0) for (8,0) - (0.5.0.5).
F B .
W (6.0) e [‘1 n I‘u with A” a 3.271 and "min = 1.792.

6.0-

 

 

 

79

Figure 7-4:
\. -
The functions ®F(q,6,o) and ¢B(q,6,0) {or (6.0) - (2.0.5). Here
70.
' F 8
(6,0) G rii n rt with An - 3.439, "max = 1.977 and "T ~ 1.392.

     

 

 

 

0.8

0.0

80

 

 

 

 

 

 

 

e r“ F
(05,015) (2,0,05) ‘11
F
F1
‘ i l I a B
0.0 1.0 2.0 3.0 7.0 ,0
Figure 7-5:

Flexure failure behavior as represented in the semioinfinite strip ,
0 < 0 - R/l2 < l, O < 6 - “(II)/p(1). The region type for each shade
are as displayed. The two points shown correspond to the parameter
pairs associated with Figures 7-3 and 7-4. Although it is not always
obvious from this diagram, the parameter pairs obeying (4.55) are in

the region rip.

 

81

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no.d

. an unseen one ~ I
.Amm.ec so oauts> an 0 e co 6.0 o

.Ao.o.o.sv - A6.qc .46; n.n sunset to nos> essences:

”0.5 0030.0
00.. no.0 00.0

 

 

 

_ p a

(I)
‘3

“0.0

 

00.0

 

 

Lo.o

 

~w.o

 

 

82

 

(o.5,o.S) (2.0.0.5)

"'1

 

 

 

5.0

b:

O
f.
O

()0 1.0 10
Figure 7-7:

Barrelling failure behavior as represented in the semi-infinite strip
(II) (I) .

0 < 0 - R/l2 < l, O < 6 - p [u . The region type for each shade

are as displayed. The two points shown correspond to the parameter

pairs associated with Figures 7-3 and 7-4. Although it is not always

obvious from this diagram, the parameter pairs obeying (4.55) are in

the region F18.

30

83

 

 

 

 

 

(I)

 

 

00 10 10 30 40 “ 50

Figure 7-8:

The partitioning of the semi-infinite strip 0 < a - R/l2 < l, O < 6 -

”(ID/”(1)

into the regions and E . The ordering (7.1) is

‘1' ‘2 3
1., For (6.0) 6 32 U 33 the ordering which

replaces (7.1) is dependent on the aspect ratio 12/11. Although it is

ensured only for (6,0) 6 5

not always obvious from this diagram, the parameter pairs obeying

(4.55) are in the region 51'

84

 

 

 

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a-o.1 0.2 .5 I .3 0.9 1 I 1 1 I 1 5 2 2.5 3 I 4 | 3-5 I
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3 / / I / / / / I3.l$ig IEII. ::~I; ‘5 2 :33 3.0:3I3 525;
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Table 7-1:

Value of the transition aspect ratio qT(6.a) for representative pairs

(5.0) E r

F
ii '

For 12/11 < nT/x, the critical instability is the m - l

flexure DOdG. while for 12/11 > nT/s the critical instability is m - o

wrinkling mode.

 

86
8. OPTIMAL DESIGN FOR COMPOSITE CONSTRUCTIONS

In this section the results obtained in section 7 shall be used
to study a problem in optimal design. This problem is defined as
follows.

It is desired to construct a plate occupying the region ~11 5 X1

5 11 (i - 1,2,3) before the application of loads. To accomplish this

purpose, fixed amounts of two neo-Hookean materials are available.

material A with volume V(A) (A)

(B)

, and material B with
(A) , van

and shear modulus p

(B). -

volume V and shear modulus 6 It is assumed that V

8111213 which is the total volume of the plate. These two materials
shall be used to form a composite construction of the type (2.2).

There are thus only two competing configurations. namely

Configuration-1: in which material-A is used for the central ply and

material-B is used for the outer plies, and

Configuration-2: in which material-B is used for the central ply and
material-A is used for the outer plies.

(8.1)

It shall be said that configuration-l and configuration-2 as given
above are conjugate to each other. The conjugate of a configuration
is obtained by exchanging the roles of the two materials while
maintaining fixed the values V(A) and V(B). Thus

{ v“) /(81113) for configuration-l,

R _ (8.2)

V(B)/(81113) for configuration-2.

87

The relation between T and p corresponding to (6.10) is found to

be given by

2 - (p-p'3)[u‘A’V‘A’+u‘B)V‘B)1/(211). (8.3)

Thus. from the point of view of the thrust deflection properties of
the plate undergoing a pure homogeneous deformation, there is no
particular advantage to be gained by considering either configuration-
1 or configuration-2.

Without loss of generality, let material A be the material with

the lesser shear modulus,

”(A) < 6(3) . (8.4)

Let TmF and TmF* denote the flexure failure thrusts for

configuration-l and configuration-2 respectively. Similarly TmB and
*

Th? will denote the barrelling failure thrusts for configuration-l

and configuration-2. The lowest or critical thrust for each

configuration is thus given by

F B * F* 3*
Tcrit - min {Tm , Tm } and Tcrit - min {Tm , Tm }.
III-1.00 Ill-1,“,

Now it was shown in section 7 that the critical thrust of a symmetric

3-ply sandwich always corresponds to flexure, thus

F
Tcrit - min {Tm }’
m-1,o
(8.5)
F*
Tcrit min {Tm }.

m-l,~

88
The configuration which yields the larger of the two critical thrusts

will be said to be the optimal design.. Thus configuration-l will be
the optimal design if

'A'

Tcrit > Tcrit ’ (3.6)

while configuration-2 will be the optimal design if

Tcrit > Tcrit' (8'7)
It shall be said that configuration-1 and configuration-2 are equally

optimal if

*

Tcrit ' Tcrit ' (8'8)

Recall at this juncture the definition of 6 and 0 as given in (4.49).
For the purposes of this section these values shall be redefined as

follows

6 _ ”(A)/”(B)’ 0 _ V(A)/(V(A)+V(B)). (8.9)

Thus the results obtained in section 7 apply immediately to

configuration-l. In particular, the critical thrust Teri will, by

it
virtue of (8.3) and p - A-l/z, be associated with the value

F
Acrit - min {AIn ). (8.10)
m-l,o

89

Furthermore,
F F F
Acrit - A1 , Tcrit - T1 , for (6.0) 6 P1 . (8.11)
and
F F
A1 , T1 , if 12/11<2qT/«. F
crit ' A crit ' T if 1 /1 >2 /fi f°r (fl'“) 6 rii '
a, Q, 2 1 ”T 9

(8.12)

Thus the parameter plane description exhibited in Figure 7-5
immediately provides useful information regarding the possibilities

for the critical stretch ratio Acr and the critical thrust Tc for

it rit

the case of configuration-l. Recalling now the restriction (8.4), the

definition of 6 given in (8.9) implies that
6 < l. (8.13)

Displayed in Figure 7-5, however, are results found in section 7 that
apply to values 6 > 1. This information is useful in determining the

critical thrust Tcr * for configuration-2. This is because (8.9)

it
* *
indicates that if (6,0) characterizes configuration-l, then (6 ,0 )

characterizes configuration-2 provided that

p - p , 6* - 1 - 0. (8.14)

It will be convenient for the purposes of this investigation to

90

introduce a conjugate function

9.70.9.4) - 9F(n.fi*.a*) - 9F(n.p'1.1-a) . (8.15)

and conjugate failure stretch ratios

AnF* - §F*(m112/211,6.0), m - 1,2,3... . (8.16)

*
Also define A“n (6,0) corresponding in the obvious fashion to (7.6)
s *
and, for (p .0 ) e r *(

define the functions qT*(6,0) and "max 6.0)

F
ii ’
obeying the counterpart to (7.10). Finally, recalling (8.10), let

'A'
- min {Am } . (8.17)

A
crit m-l,o

Then configuration-l will be the optimal design if

*
Acrit: > Acrit ’ (8'18)
while configuration-2 will be the optimal design if
A * > A 8 19
crit crit ° ( ' )
The two configurations are equally optimal if
A A * 8 20
crit - crit ( ' )

91

Since 0F(n.6,0) is initially monotonically increasing from the value 1

at n - 0 for all pairs (6.0) it follows that
9.40.8.0 - 9F*(0.a.a) - 1. (8.21)

It has been found by means of extensive numerical calculations, for

fixed values of (6,0) obeying 0 < 0 <1, 0 < 6 < 1, that there exists a

unique "cross > 0 such that

*
¢F(ncross.fi.a) - 0F (across.fl.a) . (8.22)
The program PROJ2 in (XII) of APPENDIX C can be used to find the

value a It is found without exception that

cross'
0 >0* f 0< <
F(n.fi.a) F (9.3.0) . or 9 across.
(8.23)
*
0F(n.fl.a) < @F (0.6.0) . for n > across-
Note that n - a (6.0); this function shall be called the

0:088 CIOSS

crossover value function between conjugate configurations (6,0) and
(6*,0*). Table 8-1 gives this function for selected values of (6.0).
From (8.23), (8.16) and (7.2)1 it may be concluded that

F F*

A1 > A , if «12/211 < "cross’

(8.24)

A1 < A , if x12/211 > "cross'

92

Thus the configuration with the stiffer material placed in the outer
plies provides the greatest resistance to the m - l flexure mode for
plates that are sufficiently long on the direction of thrust

( specifically 11 > (s/2n )l2 ). while the configuration with the

cross
stiffer material placed in the central ply provides the greatest
resistance to the m - l flexure mode for plates that are sufficiently
short in the direction of thrust ( speCIfically 11 < (n/chrossfl2 ).
This result provides a prescription for the optimal design in

cases in which the critical thrusts for both configuration-l and

configuration-2 are associated with m - l flexure. According to
F

(8.10) this will occur if both (6.0) e PiF and (6*,0*) 6 1"1 . It
follows that if (6.0) e PiF and (6*,0*) e PiF then the optimal design
is

configuration-l if 12/11 < Zflcross/fl’ (8.25)
whereas the optimal design is

configuration-2 if 12/11 > Zflcross/fl' (8.26)
Note that in this case the special aspect ratio 12/11 - ZflcrOSS/W

corresponds to the equally optimal configuration which serves as the
switching point between the two competing configurations, both of
which fail in m - l flexure. The representative case (6.0) -
(0.5, 0.05) for the state of affairs described above is diagramed in
Figure 8-1.

Whether or not (6.0) e F F

i
Figure 7-5. To determine the corresponding information for the

can be determined immediately from

93

conjugate pair (6*.0*) it is useful to map the region 6 > 1 of the
semi-infinite strip 6 z 0, 0 S 0 S l of Figure 7-5 to the square 0 < 6
< 1, 0 < 0 < l by means of the tranformation (8.14). The result of
this mapping is given in Figure 8-2 where, in addition, the relevent
portion of the original Figure 7-5 has also been included. Since
Figure 7-5 was only determined to a value 6 w 5, the result of the
mapping (8.14) is only given for the region 0.2 < 6 < 1. Nevertheless
the behavior of this mapping has been extrapolated by means of the
dashed curves in Figure 8-2. The regions separated by the curves in

Figure 8-2 are each associated with a different possibility as

follows:

F * * F 1
region-a: (6.0) 6 Pi , (6 ,0 ) 6 Pi ;
region-b: (6.0) e Pip, (6*,0*) e FiiF ;

. F * * F
17881011433 (pic) 6 r1 9 (5 9a ) E r111 ;
region-d: (6.6) e riiF, (6*,0*) e riF ; > (8.27)

F * * F
region-e: (6.0) e rii . (6 ,0 ) 6 F11 ;

F * * F
region-f: (6.0) 6 F111 , (6 ,a ) 6 P1 ;

F * * F
region-g: (6,0) 6 P111 , (6 ,0 ) e rii . J

 

94

It is found that the following two cases do not occur:

F * * P
(1) (4.4) e r11 . (a .a ) 6 P111 ;

(8.28)

F * *
(11) (520) 6 F111 9 (p ’a ) e r111

F .
For stiffness ratio 6 and volume fraction 0 correspoding to region-a,
both configuration-l and configuration-2 fail in mpl flexure for all
aspect ratios 12/11. In this region of parameter pairs the optimal
design is governed by (8.25) and (8.26). This region continues to the
boundaries 6 - l. 0 - 0 and 0 - 1 which according to (4.54)
corresponds to the non-composite case in which the optimal design
problem ceases to have meaning. It is to be noted that region-a
comprises only a small part of the square 0 < 0 < l, 0 < 6 < l, and
that the complete extent of this region is dependent on the
extrapolation of the curve in Figure 8-2.

If (6,0) is not in region-a of Figure 8-2, the possibility arises
that either Tcrit fl T1F or Tcrit* # T1F*. It is this possibility to

which the investigation now turns. For points (6.0) not in region-a,

the aspect ratio 12/11 - Zn /x may, but need not, be associated

cross
with a switch in optimal designs. This can be demonstrated by
considering pairs (6,0) belonging to region-d. Then (8.11) indicates
that

F

** 829
Acrit - A1 . ((3 .a ) 6 r1 ). ( . )

whereas (8.12) gives

 

95
F
A if «1 /l < 2n ,
Acrit - I 1 2 1 T ((8.6) e riip). (8.30)
Aco if «12/11 > 2"11

Thus even though (8.24) indicates that a

relative ordering of A1F and A1F*, it may or may not govern a change

governs a change in the
cross

*
in the relative ordering of Act and Acr . To make this clear. let

it it

it first be supposed that

"T > ”cross ' (8‘31)

An example of such a situation occurs if (6,0) - (0.9.0.9) since

F

(0.9.0.9) e r 1 . (1/0.9,1-0.9) z (1.11.0.1) e r1F (see the point p2

i

in Figure 8—2) and nT(0.9,0.9) z 2.1660 > 0.9009 z across(0.9.0.9).

The curve 0F(n,6,0) and QF*(n,6,0) for (6,0) - (0.9.0.9) are given in

Figure 8-3. According to (8.24), (8.29), (8.30) and (8.31) it follows

that
F F* *
Acrit - Al > A1 - Acrit ’ if «12/211 < "cross ’
F F* *
Acrit - Al < Al - Acrit ’ if "cross < "12/211 < "T ’ (8'32)
F* *
Acrit - Ano < A1 - Acrit , if 1112/211 > "T ,

* *
whenever (6.0) e PiiF' (6 ,0 ) 6 P1P provided that "T > n In

cross'

such a case the aspect ratio 12/11 - 2n /« will once again give a

cross

*
change in the relative ordering of Acrit and Acrit . so that the

optimal design is configuration-1 if 12/11 < Zn /n, and the

cross

96

optimal design is configuration-2 if 12/11 > 2n /«. In all cases,

cross

the optimal design fails in m - l flexure. The change in A given

crit
by (8.30) plays no role in the determination of the optimal design. In
the example (6.0) - (0.9.0.9) the equally optimal design case
associated with a switch in optimal design occurs at the aspect ratio

12/11 - Zn

In contrast, now suppose that (6.0) e F

/s = 0.5736.
cross

F
ii

* * F
. (6 ,0 ) 6 Pi and

"T < "cross' (8°33)

An example of such a situation occurs if (6,0) - (0.5.0.95) since

F

(0.5.0.95) e r 1 . (1/0.5,1-0.95) - (2.0.05) e riF (see point p3 in

1
Figure 8-4) and nT(O.5,O.95) = 1.1505 < 1.2884 z "cross(o'5’o'95)'

*
The associated curves §F(n,6,0) and QF (n.6,0) are given in Figure

8-4. According to (8.24), (8.29), (8.30) and (8.33) it follows that

F F* *
A > A - A , if «12/211 < ”T ,

*erit ' 1 1 crit

(8.34)

3* * _
A - Ado < A - A , if x12/211 > n

crit l crit cross '

Thus the optimal design is configuration-l if 12/11 < 2nT/s whereas
/n.

A switch in the optimal design will thus occur for at least one

the optimal design is configuration-2 if 12/11 > Zflcross

aspect ratio 12/11 in the interval 2nT/s < 12/11 < 2n

* F*
Acrit - Ano and Acrit - A1 for all such aspect ratios, a switch in

/«. Since
cross

the optimal design can only occur at an aspect ratio 12/11 which gives

Aco - A1F*. Hence it is necessary to determine roots 0 to the equation

97

9.30.5.» - A,,(p.a). nT < n < n (8.35)

cross'

* *
Since (6*,0 ) e FiF ensures that 0F (n,6.0) is monotonically

increasing in a. there will be a unique root 5 to (8.35) obeying

CIOSS

"T < "cross < ”cross' (8'36)
For the example (6.0) - (0.5.0.95) it is found that "cross(o'5’o'95) =
1.1855 (Figure 8-4). The significance of across is that
F F* *
Acrit - Al > Al - Acrit ’ if "12/211 < "T ’
F* * -
Acrit ' ‘w > A1 ' Acrit ' if "T < ”12/211 < "cross ' (8'37)
F* * -
Acrit - Am < Al - Acrit ’ if '12/211 > "cross °
Th if r F * * r F d h h
us (6.0) e 11 , (6 ,0 ) e 1 an "cross > "T’ t en t ere

exists a unique value a obeying (8.35) and the optimal design is

cross

configuration-l if 12/11 < 26 /« , whereas the optimal design is

cross

configuration-2 if 12/11 > 26 /s. The failure mode of the optimal

CIOSS

design is m - l flexure only if either 0 < 12/11 < ZnT/x or 12/11 >

Zacross/t' whereas the failure mode of the optimal design is the m - o

/«. Note that q plays

wrinkling mode if 20T/* < 12/11 < 2"cross cross

no role in either the determination of the optimal design or in the
determination of the failure mode in this case.

In order to treat this and similar cases in a systematic fashion,

98

F* *
note that A 0F(s12/211,6,0) and A - QF (x12/211.6,0) provide

F-

l. l
*

upper bounds for Act and A respectively. In order to obtain

it crit

- *

lower bounds one may introduce the modified functions 6F and @F as

follows

5F(q,6,0) I max{ F(n) I F non-decreasing, F(n) S ¢F(fl.fl.0)}.
n

5F*(n,6,0) I max{ F(n) I F non-decreasing, F(n) s QF*(n,6,0)}.

0
(8.38)

These definitions ensure that 5F(-.6.0) is the lower monotone envelope
of OF(-,6,0) and that 5F*(-,6,0) is the lower monotone envelope of

* - - *
0P (-,6.0). The value §F(slg/211,6,0) and 9F (312/211,6.0) are

ensured to be lower bounds for Ac and Acr * by virtue of (7.2)1,

rit it
(8.10),(8.l6), (8.17).

F

The definition of IF simplifies in the event that (6.0) 6 Pi U

F
I‘11 . Namely

- F

¢F(n.fi.a) - ¢F(n.fi.a). if (fi.a) E 1‘i . (8 39)
and

- °F('I.5.0). o < a S "T , F

0 (0.3.0) - if (fi.a) 6 F .

F ii

Ac(fl,a) o 'I Z "T 9

(8.40)

99

Thus one obtains from (8.11) and (8.12) that

F

11 . (8.41)

- F
Acrit - ¢F(«12/211,fi,a) , if (5,0) 6 F1 U P

However if (fl,a) e F F then Acr might correspond to a failure mode

iii it
other than m - l or m - o and thus its precise determination is
complicated by the finite spacing of the AmF. One may only conclude

that

- F
§F(u12/211,fi,a) s Acrit s §F(«12/211,fi,a) , if (fi,a) e I‘iii .
(8.42)
Counterparts to (8.39)-(8.42) will of course also hold for the
appropriate conjugate functions and variables:
- * * * * F
¢F (n.fi.a) - éF (n.fi.a) . if (5 .a ) 6 Pi . (8 43)
* *
_ * QF (n.fl.a). 0 < n S nT . * * F
°F (n.fi.a) - * * if (5 .a ) 6 P11 .
Aw . n 2 0T .

(8.44)

 

100

* - * * * F F
Acrit - QF («12/211,fi,a) , if (5 ,a ) 6 P1 U I‘i1 , (8.45)

and

- * * * * * F
éF (112/211,fl,a) s Acrit. s ¢F (x12/211,fi,a), if (B ,a ) 6 P111 .

(8.46)

It now follows from (8.22), (8.23), and (8.38) that for each

(fl,a) obeying 0 < a < 1, 0 < 8 < 1, there exists a unique across > 0
such that
i ' 5 * ' 8 47
F("cross’fl’a) - F ("cr088,fi’a) 9 ( o )
and
6 > 6 * f o < < '
F(21.19.01) F (n.fi.a) . or 0 across.
(8.48)
- - * -
¢F(n.fl.a) < é? (n.fl.a) . for n > across-
Note that "cross - "cross(p’a)’ and that this value may or may not
coincide with "cross' In particular it is seen from the examples
previously under consideration that "cross - "cross for (5,0) —
(0.5.0.05) and (fi,a) - (0.9.0.9), but that "cross < "cross for (B,a) -
(0.5,0.95).

According to (8.20), (8.41), (8.45) and (8.48), configuration-1

and configuration-2 are equally optimal at the aspect ratio

 

101

12/11 - Zicrosswmvn . (8.49)

for all cases in which (fi,a) 6 P1P U riiF and (8*,a*) e Pip U riiF'

In view of (8.27), such points correspond to the union of regions-a,b,

d and e in Figure 8-2. Furthermore, in all such cases, the optimal

design is:

configuration-l if 12/11 < Zflcross/W (8.50)
while the optimal design is

configuration-2 if 12/11 > Zflcross/"' (8.51)
consequently if (fi,o) 6 P1P U P11F and (8*,a*) E FiF U riiF (regions

a,b,d,e) then the optimal design fbr a plate which is relatively long

in the direction of thrust (11 > lzx/Zi ) is given by a

cross
configuration with the stiff material in the outer plies, however the
optimal design for a plate which is relatively short in the direction

of thrust (11 < lzn/Zfi ) is given by the configuration with the

cross
stiff material in the central ply. The critical failure mode in these
cases will be either m - l or m - m and, as shown for the case (fi,a) -
(0.5,0.95), a transition in the failure mode of the optimal design,
but not the optimal design itself, may occur at an aspect ratio other
than 12/11 - 2across/x' Of course in region-a the failure mode of the
optimal design will always correspond to m - 1 flexure.

The remaining regions-c, f and g are interesting because either

(8.42) or (8.46) have potential ramifications regarding the validity

 

102
of (8.50)-(8.51) as a prescription for the optimal design. Consider

a point (fi,a) in region-g, since according to (8.27) this region would
seem to offer the most potential for pathological behavior. An
example corresponding to this case occurs if (fl,a)-(0.5,0.8), and the
associated curves QF(n,fi,a), 5F(n,p,a), QF*(n,fi,a) and 5F*(q,fi,a) are
given in Figure 8-5. For (fi,a) - (0.5.0.8), 5F*(n,fi,a) ceases to
coincide with oF*(n,p,a) only on n > "13* z 2.7110 and 6F(q,p,a)
ceases to coincide with ¢F(q,fl,a) only on n1 < n < "2 where "1 z
1.6216 and n2 z 2.7919. Here "2 is the abscissa of a local minimum
for §F(n,fl,a) and "1 is the unique value of n obeying n < n2 such
that §F(n,fi,a) - ¢F(02,fi,a). Since 5F*(n,fi,a) > ¢F(n,fi,a) for n1 < n
< "2, the upper bound property of QF(n,fi,a) embodied in (8.42) in
conjunction with (8.46) ensures that configuration-2 is the optimal
design for "1 < le/le < "2. Since this is the only interval in n
for which the optimal design was in doubt, it follows that (8.50)-
(8.51) remains the correct prescription for the optimal design in the
particular example (fi,a) - (0.5, 0.8).

The prescription given in (8.50)-(8.51) can only fail for (fi,a)
in region-g if the function 5F*(n,fi,a) "cuts" the function 5F(n,fl,a)
at a value of n at which 5F(n,fi,a) does not coincide with §F(q,fi,a).

The same conclusion holds for region-f. Analytically this gives

.fi.a) - 6F*<6 .p.a), (8.52)

.fi.a) v‘ 45,01 cross

oF("cross cross

as the only condition which could lead to (8.50)-(8.51) becoming an
incorrect prescription for the optimal design whenever (fi,a) is in

either region-f or region-g. Similarly if

 

103

0,?(6 .p.a> w $1.36 ,5...) ," 6,.(6 .p.a>. (8.53)

cross cross cross

for (fi,a) in region-c, then (8.50)-(8.51) could also be an incorrect
prescription for the optimal design. If either (8.52) or (8.53)

occurs, then the finite spacing of the failure thrusts AmF and AmF*
must be taken into consideration and the failure mode may occur for a

finite value of m other than m - 1. However, this investigation has

yet to find an explicit example in which this state of affairs takes

place.

5.0-<

 

104

 

 

 

 

 

FIGURE 8-3

Figure 8-1:

The functions 0F(q,5,o) and °F*("'fl'°) for (5,0) - (0.5.0.05). Here
F
1 I
the optimal design is thus given by (8.25) and (8.26).‘

e e P . .
(5,0) 6 F (5 ,a ) E r, and q = 0.6771. The prescription for

cross

1.0

0.0

 

105

 

 

 

 

 

 

 

 

\
a \ \
\ \
\.\‘ 3
1 . pl
I 1 I I I T I T T
0.0 1.0
Figure 8-2:

The nature of the pairs (5,a) and (5*,a*) for O < 5 < 1. O < a < 1
correspond to the region type as given in (8 27). The solids lines in
this square are determined numerically, and the dashed lines are a
possible extrapolation. The points Pl, P2. P3.
Figure 8-1, 8-3, 8-4 and 8-5 respectively.

P4 correspond to

106

 

 

 

 

. . \
—.

\\ __.

' 1
f
l
“—1 l /
i
?

 

 

1.0 2.0 3.0 4. 7 5,0
-7
CIYCDSSS

Figure 8-3:

The functions 0F(n.5.a) and OF*(q,5,o) for (5,0) - (0.9.0.9). Here

F * * F
(5,0) 6 Fit . (B .0 ) 6 Pi and ”T ~ 2.1660 > 0.9009 = "cross' The

functions °F(n.fl.0) Uhere it differs from OF(q,5.a) is given by the

dashed line.

 

 

 

 

 

 

0.0

 

107

 

 

 

 

0.0

Figure 8-4:

The functions 9F(q,5,o) and °F*(n.5.o) for (5,0) - (0.5,0.95). Here

F

(fl.a) E 71‘ . (5*.0*) 6 F F and n = 1.2884 > 1.1505 = q and

i cross T

across 3 1-1355- The functions 5F(fl.fi.o) where it differs from

OF(q,5,o) is given by the dashed line.

 

(,1

(\3

0.0

 

 

 

 

 

 

 

108

 

 

 

4.0

’I
Figure 8-5:

The functions OF(q,5,0) and 0F*(q,5,0) for (5,0) - (0.5.0.8). Here

F F

1‘ fl
(5,0) 6 riii . (fl .0 ) e Fii . A local minimum of OF(q,5,0) occurs

at 92 = 2.79l9 an 6F(q,5,o) differs from OF(q,5,a) only on 01 < a < 02
as shown by the dashed line segment. Similarly 5F*(n.fl.0) differs

* -
from OF (0,5,0) only on q > "T. = 2.7110 as shown by the dashed line.

1.6216.

It is found that q = 0.9439 and n1 2

CTOSS "CTOSS

 

 

 

1139

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5'-5.o 4'- 5 4'-4.0§8'-3.5 8’-3 0 8'-2.5 4'l2f0 87:1 - 5T-1 1
o-O.9 115994 1 5417 1.4747 1.4014 1.3148 1.2153 1.1074 .9924 .4974 4‘-0.1
4-0.8 1.1821 1.1459 1.1400 1.1044 1.0423 .9900 .9439 .8774 .8238 4'-0.2
4-0.7 .948: .9435 954: .9422 .9253 .9018 .8404 .3210 .7940 4'-0.3
4-0.4 .8354 .8392 8414 .8409 .8348 .8275 .8132 .7941 .7447 4‘-0.4
4-0.5 .7400 .7500 .7412 .7499 .7750 .7787 .7758 .7455 .7554 oi-0.5
o-o.4 .4447 .6808 .7077 .7129 .7241 .7373 .7471 .7534 .7588 4'-0.4
4-0.3 .4039 .4184 4357 .4585 .4833 .7099 .7277 .7500 .7714 o‘-o.7
c-0.2 .5357 .5540 5857 .4117 .4342 .4719 .7123 7513 8039!n'-0.8
o-O.l .4800 .5059 5233' .5529 .4029 4320 .4974 7571 8400|4‘-0.9
8-.2oo 9-.222 5- ZSCifi-.286 8-.333 5-.400 9-.500 9-.404 8-.909
Table 8-1:
Values of the crossover value function across(5,0). The conjugate

pairs (5*,0*) are also shown.

 

 

110

APPENDICES

 

111

APPENDIX A

 

 

112

APPENDIX A: VERIFICATION 0F (4.45)

Throughout this appendix it is assumed that A > O, A # 1, a > 0.

The goal is to prove that

0 (I) < 0 (22) < 0 (12) < 0 (I) 11 0 < A < A

2 2 1 1 ' II I '
Proof:
(II) (II)
(1) 02 s 01 ,
since
(II) 2 (II) 2
- 02 ([1+A2+(1-A)2AII] - [(1+A2+(1-A)2AII)2 41211/21/2 -

2

0 {[1+A2+(l-A)2AII] + [(1+12+(1-1)2A 41211/21/2 -

II)2 ‘

- 2 {[(1+12+(1-1)2 AI )2 41 211/21

- - 2 {[(l-A)2+2(1+A2)(1-A)2AII+(l-A) A112]1/2} < 0
(II) (I)
(11) 01 < 01 ,
since
(II) 2 (1) 2
(01 > - (01 1 -<02 /2) 1 (1- -211 (Au 8,) +

2 2

+ [(1+1 +AI I(1 1) 2) 41211/2 - [(1+1 +AI (1-1) 2) 2-41211/21 < 0 .

 

 

 

113

(I) (117
(iii) 02 < 02 .

Set

211/2

C(y) - 021[1+12+(1-1)2y1-[(1+12+(1-1)2y)2-41 1/2. (A1)

Note that C(Ak) - (02(k) 2, k - I,II. It is found that C(y) is

strictly decreasing in y by the following

G'(y) -

21-1/2

- 021(1-1)2-[(1+12+(1-1)2y)2-41 (1+12+(1-1)2y)(1-1)21/2

(1+12+(1-A)2y)

 

- 02(1-112 [1 - (A2)

1.
(1+A2+(l-A)2y)2-4A2]1/2

Since the second part inside of the bracket on the right hand side in

(A2) is strictly greater than one, it follows that
G'(y) < 0. (A3)
This gives rise to

C(AII) > C(AI), if AI > AII' (A4)

Therefore it is concluded that

(11) (I)
02 > 02 . D

 

 

 

114

APPENDIX B

 

 

 

115

APPENDIX B: THE RELATION OF Lh AND Mn (n-l,2) FOR NON-COMPOSITE CASE

For the non-composite case 5(1) - u(II) - p and AI - AII - A.
Note that in general the results in this appendix shall be developed
for general value of A, not necessarily equal to zero. Therefore,

from (4.35) and (4.47), it follows

(1) L (1) M (1) M (1)) _
I 9 2

(L1 2 ' 1

(2) (2) (2) (2)
- (L1 ILZ 1M1 9M2 ) -

<3) L (3) M (3) M (3)) _
9 9 2

' (L1 2 ' 1

Consider the flexural case so that (6.11)2 gives L2 - M2 - 0. Now

(4.33)1 and (4.34) once again yields

2 2 2 2 2 2 1
(01 +A 0 )cosh(0112)L1 + (02 +A 0 )cosh(0212)M1 - 0,

2

-2 -1 2 2 2
nl[(2+A +(1-A ) A)A 0 -01 ]sinh(0112)L1 + > (82)

2

 

-2 -1 2 2 2
+ 021(2+1 +(1-1 ) A)A 0 -02 ]sinh(0212)M1 - 0 . ,

Now using (4,36), it follows that

”.101; ‘11-‘-

l

i
15
I S
l:
:3.
F
19‘
Li

 

116

01[(2+1’2+(1-1‘1)2A)12n2-0121 -

011 (2+1'2+(1-1'1)2A)1202 -

 

- (02/2)[1+12+(1-1)2A1 - (02/2)[(1+12+(1-1)2A)2 41211/21

011 (02/2)[412+2+2(1-1)2A-1-12-(1-1)2A1 -

- (02/2)[(1+12+(1 1) 2A) 41211/21

2

2 2
2 + A a ) .

One obtains in a similar fashion using (4,36), that
02[(2+A'2+(l-A21)2A)A202-022] -

2 2 2
- 02 (01 + A 0 ) .

Substituting from (B3), (B4) into (B2)2 now yields

2 2 2 2 2 2

01(02 +A 0 )sinh(0112)L1 + 02(01 +A 0 )sinh(0212)M1 -

011 (02/2)[1+A2+(l-A)2A]-(02/2)[(l+A2+(1-A)2A)2 41211/

011 (02/211312+1+(1-1)2A1 - (02/2)[(1+12+(1-1)2A)2 -41211/2

2+A22 0 )-

(B3)

(B4)

(BS)

Note that (B5) is independent of the constant A. Equations (B2)1 and

(B5) therefore give

nu.“ .
' 4

 

 

 

117

 

 

 

 

 

 

 

01 + A O - - cosh(0212) M1 1
2 2 2
02 + A 0 cosh(0112) L1
* . (136)
2 2 2
01 + A 0 - - Olsinh(0112) L1
0 2 + 1202 o sinh(0 1 ) n J
2 2 2 2 l
which, in turn, gives rise to
tanh(0 1 ) 0 2 + A202 2 0
l 2 l 2
tanh(0212) 0 + A O O

and the relation of unknown constants L and M is

 

 

1 1
11 o 2 + 1202 cosh(0 1 )
2 2 2
____ - - 2 2 2 x . (33)
H1 01 + A 0 cosh(0112)

Likewise, for the barrelling case (L1 - M - 0), one finds

l

 

 

 

 

tanh(0 1 ) 0 2 + 1202 2 n
2 2 l 2
- 2 2 2 -—- . (B9)
tanh(0112) 02 + A 0 01
and the relation of unknown constants L2 and M2 is
L 0 2 + A202 sinh(0 l )
2 2 2 2
____ _ - 2 2 2 x . (310)
H2 01 + A 0 sinh(0112)

 

 

118
These are the same results obtained by Sawyers and Rivlin, that is
equations (B7), (38), (B9) and (BIO) are the same as (4.21), (4.25),

(4.20) and (4.24) of [10].

ru_.‘_. .n .M 1 .... =:_.-.___"
'-

 

 

119

Phil..- 3! .I;

an .\ 11.9132011131’

APPENDIX C

.51

120

APPENDIX C: COMPUTER PROGRAMS

(I)

00000

H000 000000

000

PROGRAM FLEXUREI

FOR A GIVEN COMPOSITE CONSTRUCTION CHARACTERIZED BY (BETA,ALPHA) ,
AND A GIVEN VALUE OF ETA, THIS PROGRAM DETERMINES THE VALUE OF
LAMBDA THAT LIES ON THE FLEXU'RE CURVE

CHARACTER ANS*1

REAL*8 M(6,6)

REAL*8 LEMBDA, LMDl, LMD2, LMD
INTEGER*2 IPVT(6), NOUT

REAL*8 DETl, DET2, FAC(6,6)
REAL*8 EPS, ETA, BETA, R, DET

OPEN(6, FILE-'DATAOUT' , STATUS-'NEW')

INPUT

PRINT*, 'INPUT THE RATIO OF MATERIAL PROPERTIES BETA: '
READ(5,*) BETA

PRINT*, 'INPUT THE RATIO OF DEMENSION ALPHA: '
READ(S,*) R

PRINT*, ' INPUT THE BUCKLING WAVE PARAMETER ETA: '
READ(5,*) ETA

THE ROOT IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESET INITIAL VALUES 0F LMDl AND LMD2.

INITIALIZE STEP

LMDl-LOOI

IMD2-6.9

EPS-O . 000000001

LOOP THROUGH A SEQUENCE OF BISECTIONS
CONTINUE

I.MD-(LMD1+IMD2)*0.S

LEMBDA-LMD

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, PAC, 6, IPVT)
COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

CALL DLFDRG (6, PAC, 6, IPVT, DETl, DET2)
DET-- (DET1*10 . 0**DET2)

CHECK WHETHER WE INDEED HAVE A ROOT

- r‘l-‘
K

.AK. \
1“ ‘

 

C

120

999

 

121

IF ( ABS(DET) .LE. EPS ) THEN
PRINT*, 'THE BUCKLING STRETCHING LEMBDA IS:'
PRINT*, LMD
WRITE(6,*) ETA, LMD
PRINT*, 'DO YOU WANT TO GET ANOTHER LEMBDA BY ENTERING NEW',
'SET OF ETA, BETA, ALPHA, ANSWER Y OR N.’
READ(5,'(A1)') ANS
IF (ANS .NE. 'N' .AND. ANS .NE. 'n') THEN
GOTO 10
ELSE
GOTO 999
END IF
ELSE IF (DET .LT. 0.) THEN
LMDl-LMD
GOTO 100
ELSE
(DET .GT. 0.)
LMD2-LMD
IF (LMD2-LMD1 .LT. EPS) THEN
GOTO 120
END IF
GOTO 100 :
END IF b
CLOSE(6)
STOP
END

7?

 

SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
REAL$8 M(6,6), INVLMD, LEMBDA, ETA, BETA, R
REAL*8 T1, T2, T3, T4, CI, C2, C3, CA, SI, 82, S3, 84
TliETA
T2-LEMBDA*ETA
T3-LEMBDA*ETA*R
Th-ETA*R
Sl-SINH(T1)
C1-COSH(T1)
Sl-SINH(T1)
Cl-COSH(T1)
SZ-SINH(T2)
C2-COSH(T2)
S3-SINH(T3)
C3-COSH(T3)
Sh-SINH(T4)
Ch-COSH(T4)
INVLMD-l . O/LEMBDA
M(1,1)-2.0*LEMBDA*C2
M(1,2)--2.0*LEMBDA*SZ
M(l,3)-(INVLMD+LEMBDA)*C1
M(1,4)--(INVLMD+LEMBDA)*SI
M(1,5)-0.0
M(1,6)-0.0

 

 

(II)

00000

, , 122

M(2,1)--(INVLMD+LEMBDA)*82
M(2,2)-(INVLMD+LEMBDA)*CZ
M(2,3)--2.0*SI

M(2,4)-2.0*C1

M(2,5)-0.0

M(2,6)-0.0

M(3,1)--C3

M(3,2)-S3

M(3,3)--C4

M(3,4)-Sh

M(3,5)-CB

M(3,6)-C4

M(4,1)-LEMBDA*S3
M(4,2)--LEMBDA*C3

M(4,3)-84

M(4,4)--Ch

M(4,5)--LEMBDA*S3

M(4,6)--84
M(5,1)--4.0*LEMBDA*C3
M(5,2)-4.0*LEMBDA*S3
M(S,3)--2.0*(INVLMD+LEMBDA)*C4
M(S,4)-2.0*(INVLMD+LEMBDA)*34
M(5,5)-4.0*BETA*LEMBDA*C3
H(5,6)-2.0*BETA*(INVLMD+LEMBDA)*C4
M(6,1)-2.0*(INVLMD+LEMBDA)*S3
M(6,2)--2.0*(INVLMD+LEMBDA)*C3
H(6,3)-4.0*84

M(6,4)--4.0*C4
M(6,5)--2.0*BETA*(INVLMD+LEMBDA)*83
M(6,6)--4.0*BETA*84

RETURN

END

PROGRAM FLEXUREZ

FOR A GIVEN COMPOSITE CONSTRUCTION CHARACTERIZED BY (BETA,ALPHA),
THIS PROGRAM DETERMINES A SET OF ORDERED PAIR (ETA,LAMBDA) THAT
LIES ON THE FLEXURE CURVE

CHARACTER ANS*1

REAL*8 M(6,6)

REAL$8 LEMBDA, LMDl, LMD2, LMD

INTEGER$2 IPVT(6), NOUT

REAL*8 DETl, DET2, FAC(6,6)

REAL$8 EPS, ETA, BETA, R, DET, ETAI, ETAF, ETASTP

.‘A'v :w - A: nabvr ...H
hi

I .1122:- 4” I... v;_ A.'

 

000000 U10

H000

120

 

123

0PEN(6, FILE-'DATAOUT', STATUS-'NEW')

INPUT

PRINT*, 'INPUT THE RATIO OF MATERIAL PROPERTIES BETA:'
READ(5,*) BETA

PRINT*, 'INPUT THE RATIO OF DEMENSION ALPHA:'
READ(5,*) R

PRINT*, 'WHAT IS THE INITIAL VALUE OF ETA:'
READ(5,*) ETAI

PRINT*, 'WHAT IS THE FINAL VALUE OF ETA:'
READ(5,*) ETAF

PRINT*, 'WHAT IS THE STEP SIZE FOR ETA:'
READ(S,*) ETASTP

ETA-ETAI-ETASTP

CONTINUE
ETA-ETA+ETASTP
IF (ETA.GT.ETAF) GOTO 900

THE ROOT IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESET INITIAL VALUES 0F LMDl AND LMD2.

INITIALIZE STEP

LMD1-1.001
LMD2-6 . 9
EPS-0.000000001

LOOP THROUGH A SEQUENCE OF BISECTIONS

CONTINUE

IMD-(LMDI+IMD2)*0.5

LEMBDAPLND

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

CALL DLFDRG (6, FAC, 6, IPVT, DETI, DET2)
DET--(DET1*10.0**DET2)
IF (ABS(DET) .LT. EPS) THEN
PRINT*,LMD
WRITE(6,*) ETA, LMD
GOTO 50
ELSE IF (DET .LT. 0.) THEN
LMDl-LMD
GOTO 100
ELSE
(DET .GT. 0.)
LMDZ-LMD
IF (LMDZ-LMDI .LT. EPS) THEN
GOTO 120
END IF
GOTO 100
END IF

‘usu-au“l up .....- y .
I n

 

124
900 CONTINUE 2
CLOSE(6)
999 STOP
END
0
c
c
c SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
C (See (1))
(III)
PROGRAM FLEXURE3
c
0 FOR A GIVEN COMPOSITE CONSTRUCTION CHARACTERIZED BY (BETA,ALPHA),
c THIS PROGRAM DETERMINES THE FLEXURE REGION TYPE
c
REAL*8 X(50), Y(SO)
REAL#8 M(6,6)
REALSB LEMBDA, LMD1, LMDZ
INTEGER*2 IPVT(6), NOUT
REAL#8 OET1, DET2, FAC(6,6)
REAL$8 EPS, ETA, BETA, R, DET, ETAI, ETASTP
c
LOGIGAL DECR, INCR
c
OEGR-.PALSE.
INGR~.PALSE.
c INPUT
PRINT*, 'INPUT THE RATIO OF MATERIAL PROPERTIES BETAz'
READ(5,*) BETA '
PRINT*, 'INPUT THE RATIO OF DEMENSION ALPHA:'
READ(5,*) R
PRINT*, 'WHAT IS THE INITIAL VALUE OF ETA:'
READ(S,*) ETAI
PRINT*, 'WHAT IS THE STEP SIZE FOR ETA:'
READ(5,*) ETASTP
c
DO 900 Kp1,50
c
X(K)-ETAI+(K-1)*ETASTP
c
c THE ROOT IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
0 BETWEEN PRESET INITIAL VALUES OF LMD1 AND LMD2.
c
c INITIALIZE STEP
c

LMD1-1.001

'r\

In ‘1 I." I.

 

 

 

125

LMD2-6 . 9
EPS-0.000000001

LOOP THROUGH A SEQUENCE OF BISECTIONS

P‘C5CIC3

00 CONTINUE
Y(K)-(LMDI+IMDZ)*0 . 5
LEMBDA§Y(K)
ETA!X(K)
CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

C563!)

CALL DLFDRG (6, FAC, 6, IPVT, DETI, DET2) E
DET--(DET1*10.0**DET2) 1

CHECK WHETHER WE INDEED HAVE A ROOT f

IF (ABS(DET) .LT. EPS) THEN ;
GOTO 890 ;
ELSE IF (DET Lu. 0.) THEN I
LMDldY(K) L
GOTO 100 '
ELSE
C (DET .GT. 0.)
LMD2-Y(x)
IF (LMDz-LMDI .LT. EPS) THEN
GOTO 890
END IF
GOTO 100
END IF
890 CONTINUE
PRINT*, X(x), Y(K)
900 CONTINUE

DO 920 J-2,50

WE NOW ASSUME THAT THERE IS NO FALSE DROP AT THE BEGINING

000

IF (Y(J-I) .GT. Y(J)) DECR-.TRUE.

DECR IS SET EQUAL TO TRUE IF THE FLEXURE CURVE EXHIBITS A
DECREASE

 

ONCE DECR IS SET EQUAL T0 TRUE, IT IS NEVER CHANGED
IF (DECR .AND. Y(J-I) .LT. Y(J)) INCR-.TRUE.

INCR IS SET EQUAL TO TRUE IF THE FLEXURE CURVE EXHIBITS AN
INCREASE AFTER THE DECR, i.e. A SECOND INCREASE

20 CONTINUE

C5\DC1C3C)C§ CICDCDCICIC)

 

126

IF (INCR) THEN
PRINT*, 'GAMMA3F'
ELSE IF (DECR) THEN
PRINT*, 'GAMMAZF'
ELSE
PRINT*, 'GAMMAIF'
END IF
999 STOP
END

SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
(See (1))

00000

b

1
I
C

(IV)

PROGRAM FLEXUREA

FOR A GIVEN VALUE ALPHA, THIS PROGRAM DETERMINES THE FLEXURE
REGION TYPE FOR A SEQUENCE OF BETA

0000

REALSB X(50), Y(SO)

REAL$8 M(6,6)

REAL38 LEMBDA, LMDI, LMDZ

INTEGER*2 IPVT(6), NOUT

REAL$8 DETI, DET2, FAC(6,6)

REALflB EPS, ETA, BETA, R, DET, BETAI, BETAF,BETASTP

LOGICAL DECR, INCR

OPEN (6, FILE-'DATAOUT', STATUS-'NEW')

C INPUT
PRINT*, 'INPUT THE RATIO OF DEMENSION ALPHA:'
READ(5,*) R
PRINT*, 'WHAT IS THE INITIAL VALUE OF BETA:'
READ(5,*) BETAI
PRINT*, 'WHAT IS THE FINAL VALUE OF BETA:'
READ(5,*) BETAF
PRINT*, 'WHAT IS THE STEP SIZE FOR BETA:'
READ(5,*) BETASTP

 

BETA-BETAI-BETASTP

so CONTINUE L
BETA-BETA+BETASTP
IF (BETA .GT. BETAF) GOTO 999
DECR~.FALSE.

 

0

000000

H000

890
900

000

127
INCR- . FALSE .

DO 900 R-1,50
X(K)—O.1+(K-1)*O.1

THE ROOT IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESET INITIAL VALUES OF LMDI AND LMD2.

INITIALIZE STEP

LMDl-l . 001
1202-6 . 9
EPS-O . 000000001

LOOP THROUGH A SEQUENCE OF BISECTIONS

CONTINUE

Y(K)-(LMD1+IMD2)*0 . 5

LEMBDA-Y(K)

ETA-X00

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALLDLFI‘RG (6, M, 6, FAG, 6, IPVT)

COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

CALL DLFDRG (6, FAC, 6, IPVT, DETl, DET2)
DET-- (DET1*10 . O**DET2)

CHECK WHETHER WE INDEED HAVE A ROOT

IF (ABS(DET) .LT. EPS) THEN
GOTO 890
ELSE IF (DET .LT. 0.) THEN
LMDI-Y(K)
GOTO 100
ELSE
(DET .GT. 0.)
LMDZ-Y(K)
IF (IMD2-LMD1 .LT. EPS) THEN
GOTO 890
END IF
GOTO 100
END IF
CONTINUE
CONTINUE

DO 920 J-2,50
WE NOW ASSUME THAT THERE IS NO FALSE DROP AT THE BEGINING
IF (Y(J-l) .GT. Y(J)) DECR-.TRUE.

DECR IS SET EQUAL T0 TRUE IF THE FLEXURE CURVE EXHIBITS A

TW‘:I““‘ ““‘Ew

 

 

0000

000000

999

00000

(V)

0000

128
DECREASE

ONCE DECR IS SET EQUAL TO TRUE, IT IS NEVER CHANGED
IF (DECR .AND. Y(J-l) .LT. Y(J)) INCRF.TRUE.

INCR IS SET EQUAL TO TRUE IF THE FLEXURE CURVE EXHIBITS AN
INCREASE AFTER THE DECR, i.e. A SECOND INCREASE

CONTINUE

IF (INCR) THEN

PRINT*, BETA, R, 'GAMMA3F'

WRITE*, 'BETAF', BETA, 'ALPHA-', R, 'GAMMA3F'
ELSE IF (DECR) THEN

PRINT*, BETA, R, 'GAMMAZF'

WRITE*, 'BETAF', BETA, 'ALPHA-', R, 'GAMMAZF'
ELSE

PRINT*, BETA, R, 'GAMMAIF'

WRITE*, 'BETAP', BETA, 'ALPHA-', R, 'GAMMAlF'
END IF
GOTO 50
CLOSE(6)
STOP
END

SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
(See (1))

PROGRAM FLEXURES

FOR A GIVEN VALUE BETA, THIS PROGRAM DETERMINES THE FLEXURE
REGION TYPE FOR A SEQUENCE OF ALPHA

REAL$8 X(50), Y(SO)

REAL$8 M(6,6)

REAL$8 LEMBDA, LMDI, LMD2

INTEGER*2 IPVT(6), NOUT

REAL$8 DETI, DET2, FAC(6,6)

REAL$8 EPS, ETA, BETA, R, DET, RI, RF, RSTP

IDGICAL DECR , INCR

 

 

0

000000

H000

129
It

OPEN (6, FILE-'DATAOUT', STATUS-'NEW')

INPUT

PRINT*, 'INPUT THE RATIO OF DEMENSION BETA:'
READ(S,*) BETA

PRINT*, 'WHAT IS THE INITIAL VALUE OF ALPHA:'
READ(5,*) RI

PRINT*, 'WHAT IS THE FINAL VALUE OF ALPHA:'
READ(5,*) RF

PRINT*, 'WHAT IS THE STEP SIZE FOR ALPHA:'
READ(5,*) RSTP

' '5

RPRI'RSTP

CONTINUE

RPR+RSTP

IF (R .GT. RF) GOTO 999
DECR-.FALSE.
INCR-.FALSE.

DO 900 K~1,50
X(K)-0.1+(K-1)*0.1

THE ROOT IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESET INITIAL VALUES OF LMDl AND LMD2.

INITIALIZE STEP

LMD1-1.00I
LMD2-6.9
EPS-0.000000001

LOOP THROUGH A SEQUENCE OF BISECTIONS

CONTINUE

Y(K)-(IMDI+LMD2)*0 . 5

LEMBDAiY(K)

ETA—X(K)

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

CALL DLFDRG (6, FAG, 6, IPVT, DETI, DET2)
DET--(DET1*10.0**DET2)

CHECK WHETHER WE INDEED HAVE A ROOT

IF (ABS(DET) .LT. EPS) THEN
GOTO 890

ELSE IF (DET .LT. 0.) THEN
LMDl-Y(K)
GOTO 100

ELSE
(DET .GT. 0.)

“-4 .-
...‘ ‘ 72"

r—“en A

 

 

130

LMD2dY(K) 7
IF (LMD2-LMD1 .LT. EPS) THEN
GOTO 890
END IF
GOTO 100
END IF
890 CONTINUE
900 CONTINUE

C
DO 920 J-2,SO
G
G WE NOW ASSUME THAT THERE IS NO FALSE DROP AT THE BEGINING
c .
IF (Y(J-l) .GT. Y(J)) DECRF.TRUE.
C
G DECR IS SET EQUAL TO TRUE IF THE FLEXURE CURVE EXHIBITS A
C DECREASE
C
G ONCE DECR IS SET EQUAL TO TRUE, IT IS NEVER CHANGED
C
IF (DECR .AND. Y(J-l) .LT. Y(J)) INCR~.TRUE.
C
C INCR IS SET EQUAL TO TRUE IF THE FLEXURE CURVE EXHIBITS AN
C INCREASE AFTER THE DECR, i.e. A SECOND INCREASE
C
920 CONTINUE
C
IF (INCR) THEN
PRINT*, BETA, R, 'GAMMA3F'
WRITE*, 'BETA-', BETA, 'ALPHA-', R, 'GAMMA3F'
ELSE IF (DECR) THEN
PRINT*, BETA, R, 'GAMMAZF'
WRITE*, 'BETAP', BETA, 'ALPHA-', R, 'GAMMAZF'
ELSE
PRINT*, BETA, R, 'GAMMAIF'
WRITE*, 'BETAP', BETA, 'ALPHAP', R, 'GAMMAIF'
END IF
GOTO 50
999 CLOSE(6)
STOP
END
C
C
G
G SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
C (See (1))
(VI)

PROGRAM BARRELLINGI

L-{T _'* T- “— "w

00000

000000

H000

 

131

FOR A GIVEN COMPOSITE CONSTRUCTION CHARACTERIZED BY (BETA,ALPHA),
AND A GIVEN VALUE OF ETA, THIS PROGRAM DETERMINES THE VALUE OF
LAMBDA THAT LIES ON THE BARRELLING CURVE

CHARACTER ANS*1

REAL*8 M(6,6)

REAL*8 LEMBDA, LMDI, LMD2, LMD
INTEGER*2 IPVT(6), NOUT

REALfiB DETI, DET2, FAG(6,6)
REAL#8 EPS, ETA, BETA, R, DET

0PEN(6, FILE-'DATAOUT', STATUS-'NEW')

INPUT

PRINT*, 'INPUT THE RATIO OF MATERIAL PROPERTIES BETA:'
READ(5,*) BETA

PRINT*, 'INPUT THE RATIO OF DEMENSION ALPHA:'
READ(5,*) R

PRINT*, 'INPUT THE BUCKLING WAVE PARAMETER ETA:'
READ(5,*) ETA

THE ROOT IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESET INITIAL VALUES OF LMDI AND LMD2.

INITIALIZE STEP

LMD1-1.001
LMD2-6.9
EPS-0.000000001

LOOP THROUGH A SEQUENCE OF BISECTIONS

CONTINUE

LMD-(LMD1+LMD2)*O.5

LEMBDAPLMD

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

CALL DLFDRG (6, FAG, 6, IPVT, DETI, DET2)
DET--(DET1*10.0**DET2)

CHECK WHETHER WE INDEED HAVE A ROOT

IF ( ABS(DET) .LE. EPS ) THEN
PRINT*, 'THE BUCKLING STRETCHING LEMBDA IS:'
PRINT*, LMD
WRITE(6,*) ETA, LMD
PRINT*, 'DO YOU WANT TO GET ANOTHER LEMBDA BY ENTERING NEW',
'SET OF ETA, BETA, ALPHA, ANSWER Y OR N.’
READ(5,'(A1)') ANS
IF (ANS .NE. 'N' .AND. ANS .NE. 'n') THEN
GOTO 10

 

——T—_i

 

132

 

ELSE
GOTO 999
END IF
ELSE IF (DET .LT. 0.) THEN
LMD1-LMD
GOTO 1OO
ELSE
C (DET .GT. 0.)
LMD2-LMD
IF (IMD2-LMD1 .LT. EPS) THEN
GOTO 120
END IF
GOTO 1OO
END IF
CLOSE(6)
999 STOP I
END ,~

000

SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
REALEB M(6,6), INVLMD, LEMBDA, ETA, BETA, R
REAL*8 T1,.T2, T3, T4, C1, C2, C3, C4, 81, $2, 53, SA
TI-ETA
T2-LEMBDA*ETA
T3-LEMBDA*ETA*R
Th-ETA*R
Sl-SINH(T1)
Cl—COSH(T1)
Sl-SINH(T1)
Cl—COSH(T1)
S2-SINH(T2)
C2-COSH(T2)
S3-SINE(T3)
C3-COSH(T3)
Sh—SINH(T4)
C4-COSH(T4)
INVLMD-1.O/LEMBDA
M(1,1)-2.0*LEMBDA*CZ
M(1,2)--2.0*LEMBDA*82
M(l,3)-(INVLMD+LEMBDA)*CI
M(1,4)--(INVLMD+LEMBDA)*S1
M(1,5)-O.O
M(1,6)-0.0
M(2,1)‘--(INV1MD+LEMBDA)*S2
M(2,2)-(INVLMD+LEMBDA)*CZ
M(2,3)--2.0*Sl
M(2,4)-2.O*C1
M(2,5)-0.0
M(2,6)-0.0
M(3,1)--C3
M(3,2)-S3
M(3,3)--Ca
M(3,4)-Sa

WCI-“Amana - n w .
- ._'A

 

 

133

M(3,5)--S3

M(3,6)--84

M(4,1)-LEMBDA*83
M(4,2)--LEMBDA*C3

M(4,3)-S4

M(4,4)--C4

M(4,5)-LEMBDA*C3

M(4,6)-04
M(5,1)--4.0*LEMBDA*GB
M(5,2)-4.0*LEMBDA*S3
M(5,3)--2.0*(INVLMD+LEMBDA)*C4
M(5,4)-2.0*(INVLMD+LEMBDA)*84
M(S,5)--4.0*BETA*LEMBDA*S3
M(5,6)--2.0*BETA*(INVLMD+LEMBDA)*84
M(6,1)-2.0*(INVLMD+LEMBDA)*S3
M(6,2)--2.0*(INVLMD+LEMBDA)*C3
M(6,3)-4.0*84

M(6,4)--4.0*Gh
M(6,5)-2.0*BETA*(INVLMD+LEMBDA)*G3
M(6,6)-4.0*BETA*C4

RETURN

END

(VII)

PROGRAM BARRELLING2

FOR A GIVEN COMPOSITE CONSTRUCTION CHARACTERIZED BY (BETA,ALPHA),
THIS PROGRAM DETERMINES A SET OF ORDERED PAIR (ETA,LAMBDA) THAT
LIES ON THE BARRELLING CURVE

00000

CHARACTER ANS*1

REAL*8 M(6,6)

REAL*8 LEMBDA, LMDl, LMD2, LMD

INTEGER*2 IPVT(6), NOUT

REAL*8 DETI, DET2, FAG(6,6)

REAL$8 EPS, ETA, BETA, R, DET, ETAI, ETAF, ETASTP

0PEN(6, FILE-'DATAOUT', STATUS-'NEW')
C INPUT

PRINT*, 'INPUT THE RATIO OF MATERIAL PROPERTIES BETA:'

READ(S,*) BETA

PRINT*, 'INPUT THE RATIO OF DEMENSION ALPHA:'

READ(5,*) R

PRINT*, 'WHAT IS THE INITIAL VALUE OF ETA:'

READ(5,*) ETAI

PRINT*, 'WHAT IS THE FINAL VALUE OF ETA:'

READ(5,*) ETAF

PRINT*, 'WHAT IS THE STEP SIZE FOR ETA:'

i
i
1
4
1
I .
3"- ~;
g;

 

 

134

READ ( 5 , *) ETASTP
ETA-ETAI - ETASTP

so CONTINUE
ETA-ETA+ETASTP
IF (ETA.GT.ETAF) GOTO 900

THE ROOT IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESET INITIAL VALUES OF LMDI AND LMD2.

INITIALI ZE STEP

000000

LMDl-l . 001
LMD2-6 . 9
EPS-0 . 000000001

IDOP THROUGH A SEQUENCE OF BISECTIONS .

H000

00 CONTINUE
LMI)-(IMD1+LMD2)*O.5
LEMBDA-LMD
CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

' ‘.—_.‘~’“ Tfi'F” ‘

000

COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

CALL DLFDRG (6, FAG, 6, IPVT, DETI, DET2)
DET-- (DET1*10 . 0**DET2)

C CHECK WHETHER WE INDEED HAVE A ROOT

IF (ABS(DET) .LT. EPS) THEN
120 PRINT*,LMD
WRITE(6,*) ETA, LMD
GOTO 50
ELSE IF (DET .LT. 0.) THEN
LMDI-LMD
GOTO 1OO
ELSE
C (DET .GT. 0.)
LMD2-LMD
IF (LMD2-mm .LT. EPS) THEN
GOTO 120
END IF 6
GOTO 1OO
END IF
900 CONTINUE
CLOSE(6)
999 STOP
END

 

0000

SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)

 

135
(See (VI))

(VIII)

0000

10

N0000

PROGRAM BARRELLING3

FOR A GIVEN COMPOSITE CONSTRUCTION CHARACTERIZED BY (BETA,ALPHA),
THIS PROGRAM DETERMINES THE BARRELLING REGION TYPE !

REALSB X(SQ), Y(SO) ,_
REAL*8 M(6,6) '

REALSB LEMBDA, LMDl, LMD2, LMD i
INTEGER*2 IPVT(6), NOUT y
REALSB DET1, DET2, FAG(6,6) -
REAL$8 EPS, ETA, BETA, R, DET, ETAI, ETAF, ETASTP

LOGICAL DECR, INCR

INPUT

PRINT*, 'INPUT THE RATIO OF MATERIAL PROPERTIES BETA:'
READ(5,*) BETA

PRINT*, 'INPUT THE RATIO OF DEMENSION ALPHA:'
READ(5,*) R

PRINT*, 'WHAT IS THE FINAL VALUE OF ETA:'

READ(5,*) ETAF

DECR-.FALSE.
INCR-.FALSE.
ETAPETAF+O.1 ”
CONTINUE
EIAFETA'O.1
LMDl-1.00I
LMDZ-9 . 9
EPS-0.000000001

 

THE FOLLOWING LOOP IS USED TO WORK BACK FROM THE FINAL VALUE
OF ETA TO DETERMINE AN APPROPRIATE INITIAL VALUE OF ETA.

CONTINUE

LMD-(LMD1+LMD2)*0.5

LEHBDAPLMD

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

CALL DLFDRG (6, FAG, 6, IPVT, DET1, DET2)

 

030000000

00000

H000

136
DET-- (DETI*10 . O**DET2)
CHECK WHETHER WE INDEED HAVE A ROOT

IF (ABS(DET) .LT. EPS) THEN
GOTO 30
ELSE IF (DET .LT. 0.) THEN
LMDl-LMD
GOTO 20
ELSE
(DET .GT. 0.)
LMD2-LMD
IF (DIDZ-LMDI .LT. EPS) THEN
GOTO 30
END IF
GOTO 20
END IF

IF (LMD .GT. 5.9) THEN
ETAI-ETA-O.1
PRINT*, 'ETAI-', ETAI, LMD
ETASTP-(ETAF-ETA) /49 . O
PRINT*, 'ETASTP-', ETASTP
GOTO 50

ELSE .

PRINT*, LMD
GOTO 10

END IF

END OF FIRST LOOP

THIS SECOND LOOP NOW WORKS FORWARD FROM THE INITIAL VALUE OF
ETA TO THE FINAL VALUE OF ETA, DETERMINING LAMBDA FOR EACH
SUCH ETA -

DO 900 K-1,50
X(K)-ETAI+(K-1)*ETASTP

THE ROOT OF IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESENT INITIAL VALUE OF LMDl AND LMD2.

INITIALI 2E STEP
LMDl-l . 001 '
LMDZ-B . 9

EPS-0 . 000000001

LOOP THROUGH A SEQUENCE OF BISECTIONS

CONTINUE

Y(K)-(LMD1+LMD2)*0.5

LEMBDA-Y(K)

ETA-X(K)

CALL GETARRAY (LEMBDA, ETA, BATA, R, M)

T—‘_‘"“ "V‘ ‘“ “‘1'!!—

 

120

000000

000000

999

000

137

CALL DLFTRG (6, 14,26, FAC, 6, IPVT)
CALL DLFDRG (6,-FAG, 6, IPVT, DET1, DET2)
DET-~(DET1*10.0**DET2)

CHECK WHETHER WE INDEED HAVE A ROOT

IF (ABS(DET) .LE. EPS) THEN
PRINT*, X(K), Y(K)
GOTO 890
ELSE IF (DET .LT. 0.) THEN
LMDliY(K)
GOTO 100
ELSE
(DET .GT. 0.)
LMD2#Y(K) '
IF (LMD2-LMDI .LT. EPS) THEN
GOTO 120
END IF
GOTO 100
END IF

END OF SECOND LOOP

CONTINUE
CONTINUE

DO 920 J-2,50
IF (Y(J-l) .LT. Y(J)) INCR-.TRUE.

INCR IS SET EQUAL TO TRUE IF THE BARRELLING CURVE EXHIBITS AN
INCREASE

ONCE INCR IS SET EQUAL TO TRUE, IT IS NEVER CHANGED
IF (INCR .AND. Y(J-I) .GT. Y(J)) DECRP.TRUE.

DECR IS SET EQUAL TO TRUE IF THE BARRELLING CURVE EXHIBITS A
DECREASE AFTER THE INCR, i.e. A SECOND DECREASE

CONTINUE

IF (DECR) THEN
PRINT*, 'GAMMA3B'
ELSE IF (INCR) THEN
PRINT*, 'GAMMA2B'
ELSE
PRINT*, 'GAMMAlB'
END IF
STOP
END

 

 

 

A: 138

SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
(See (VI))

00

(IX)

PROGRAM BARRELLINGA

FOR A GIVEN VALUE ALPHA, THIS PROGRAM DETERMINES THE BARRELLING
REGION TYPE FOR A SEQUENCE OF BETA

0000

REALflB X(50), Y(SO)

REAL*8 M(6,6)

REAL$8 LEMBDA, LMDI, LMD2, LMD

INTEGER*2 IPVT(6), NOUT

REALfiB DET1, DET2, FAG(6,6)

REAL$8 EPS, ETA, BETA, R, DET, BETAI, BETAF,BETASTP
REAL$8 ETAI, ETAF, ETASTP

LOGICAL DECR, INCR

OPEN (6, FILE-'DATAOUT', STATUS-'NEW')

C INPUT
PRINT*, 'INPUT THE RATIO OF DEMENSION ALPHA:'
READ(5,*) R
PRINT*, 'WHAT IS THE FINAL VALUE OF ETA:'
READ(5,*) ETAF
PRINT*, 'WHAT IS THE INITIAL VALUE OF BETA:'
READ(5,*) BETAI
PRINT*, 'WHAT IS THE FINAL VALUE OF BETA:'
READ(5,*) BETAF ’
PRINT*, 'WHAT IS THE STEP SIZE FOR BETA:'
READ(5,*) BETASTP

BETAPBETAI-BETASTP
5 CONTINUE
BETA-BETA+BETASTP
IF (BETA .GT. BETAF) GOTO 999
DECR-.FALSE.
INCR-.FALSE.
ETAPETAF+O.1
10 CONTINUE
ETA-ETA-O.1
LMDl-l . 001
LMD2-9.9
EPS-0.000000001

C THE FOLLOWING LOOP IS USED TO WORK BACK FROM THE FINAL VALUE

 

 

 

0U|0000000

000000

139
OF ETA TO DETERMINE AN APPROPRIATE INITIAL VALUE OF ETA

CONTINUE

IMD-(LMDl-I-LMDZ)*0.5

LEMBDA-LMD

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFI'RG (6, M, 6, FAG, 6, IPVT)

CALL DLFDRG (6, FAG, 6, IPVT, DET1, DET2)
DET-- (DET1*10 . O**DET2)

IF (ABS(DET) .LT. EPS) THEN
GOTO 30
ELSE IF (DET .LT. 0.) THEN
LMDI-LMD
GOTO 20
ELSE
(DET .GT. 0.)
LMD2-LMD
IF (LMD2-LMDI .LT. EPS) THEN
GOTO 30
END IF
GOTO 20
END IF

IF (LMD .GT. 5.9.) THEN
ETAI-ETA-O.1
PRINT*, 'ETAI-', ETAI, LMD
ETASTP-(ETAF- ETA) /49 . O
PRINT*, 'ETASTP-', ETASTP
GOTO 50

ELSE
PRINT*, LMD
GOTO 10

END IF

END OF FIRST LOOP

THIS SECOND LOOP NOW WORKS FORWARD FROM THE INITIAL VALUE OF
ETA TO THE FINAL VALUE OF ETA, DETERMINING LAMBDA FOR EACH
SUCH ETA

DO 900 K-1,50

X(K)-ETAI+(K-1)*ETASTP

THE ROOT IS FOUND BY A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESET INITIAL VALUES OF LMDl AND LMD2.

INITIALIZE STEP
IMDI-I . 001

IMD2-8 . 9
EPS-0 . 000000001

 

— ———T
140 ‘
C LOOP THROUGH A SEQUENCE OF BISECTIONS

100 CONTINUE

Y(K)-(LMD1+LMD2)*O . 5

LEMBDA-Y(K)

ETA-X(K)

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

 

 

C
C COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG
C
CALL DLFDRG (6, PAC, 6, IPVT, DET1, DET2)
DET--(DET1*10.0**DET2)
C
CHECK WHETHER HE INDEED HAVE A ROOT
C
IF (ABS(DET) .LT. EPS) THEN
120 PRINT*, X(K), Y(K) ;"
GOTO 890 ]
ELSE IF (DET .LT. 0.) THEN L
LMDl-Y(K) i
GOTO 100 L
ELSE 3;
C (DET .GT. 0.)
LMD2dY(K)
IF (LMD2-LMD1 .LT. EPS) THEN
GOTO 120
END IF
GOTO 1OO
END IF
C
C END OF SECOND LOOP
C
890 CONTINUE
9OO CONTINUE
0
DO 920 J-2,50
C
IF (Y(J-l) .LT. Y(J)) INCRP.TRUE.
C
C INCR IS SET EQUAL TO TRUE IF THE BARRELLING CURVE EXHIBITS AN
C INCREASE
C
c ONCE INCR IS SET EQUAL TO TRUE, IT Is NEVER CHANGED
C
IF (INCR .AND. Y(J-l) .GT. Y(J)) DECR-.TRUE.
c ,
C DECR IS SET EQUAL TO TRUE IF THE BARRELLING CURVE EXHIBITS A
C DECREASE AFTER THE INCR, i.e. A SECOND DECREASE
C
920 CONTINUE
c

IF (DECR) THEN
PRINT*, BETA, R, 'GAMMA3B'

 

999

00000

(X)

0000

141

WRITE*, 'BETAF', BETA, 'ALPHAF', R, 'GAMMA3B'
ELSE IF (INCR) THEN

PRINT*, BETA, R, 'GAMMA2B' .

WRITE*, 'BETA-', BETA, 'ALPHAF', R, 'GAMMA2B'
ELSE

PRINT*, BETA, R, 'GAMMAIB'

WRITE*, 'BETA-', BETA, 'ALPHAF', R, 'GAMMAIB'
END IF ”
GOTO 5
CLOSE(6)
STOP
END

SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
(See (VI))

{PROGRAM BARRELLINGS

FOR A GIVEN VALUE BETA, THIS PROGRAM DETERMINES THE BARRELLING
REGION TYPE FOR.A SEQUENCE OF ALPHA

REAL*8 X(50), Y(SO)

REAL*8 M(6,6)

REAL$8 LEMBDA, LMDI, LMD2, LMD

INTEGER*2 IPVT(6), NOUT

REAL#8 DET1, DET2, FAG(6,6)

REAL*8 EPS, ETA, BETA, R, DET, RI, RF,RSTP
REAL$8 ETAI, ETAF, ETASTP

LOGICAL DECR, INCR

OPEN (6, FILE-'DATAOUT', STATUS-'NEW')
INPUT

PRINT*, 'INPUT THE RATIO OF DEMENSION BETA:'
READ(5,*) BETA

PRINT*, 'WHAT IS THE FINAL VALUE OF ETA:'
READ(5,*) ETAF

PRINT*, 'WHAT IS THE INITIAL VALUE OF ALPHA:'
READ(5,*) RI

PRINT*, 'WHAT IS THE FINAL VALUE OF ALPHA:'
READ(5,*) RF

PRINT*, 'WHAT IS THE STEP SIZE FOR ALPHA:'
READ(5,*) RSTP

 

 

10

N0000

00000

142

R-RI-RSTP
CONTINUE
R-R+RSTP

IF (R .GT. RF) GOTO 999
DECR-.FALSE.
INCR-.FALSE.
ETA-ETAF+O.1
CONTINUE
ETA-ETA-O.1
LMD1-1.001
LMD2-9.9

EPS-0 . 000000001

THE FOLLOWING IDOP IS USED TO WORK BACK FROM THE FINAL VALUE
OF ETA TO DETERMINE AN APPROPRIATE INITIAL VALUE OF ETA

CONTINUE .

LMD-(LMDI+LMD2)*0.5

LEMBDA-LMD

CALI. GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

CALL DLFDRG (6, FAG, 6, IPVT, DET1, DET2)
DET-- (DET1*10 . O**DET2)

IF (ABS(DET) .LT. EPS) THEN
GOTO 30
ELSE IF (DET .LT. 0.) THEN
LMDl-LMD
GOTO 20
ELSE
(DET .GT. 0.)
LMD2-LMD
'IF (LMD2-LMD1 .LT. EPS) THEN
GOTO 30
END IF
GOTO 20
END IF

IF (LMD .GT. 5.9) THEN
ETAI-ETA-0.1
PRINT*, 'ETAI-', ETAI, LMD
ETASTP-(ETAF- ETA) /h9 . O
PRINT*, 'ETASTP-', ETASTP
GOTO so

ELSE
PRINT*, IMD
GOTO 10

END IF

END OF FIRST IDOP

THIS SECOND LOOP NOW WORKS FORWARD FROM THE INITIAL VALUE OF
ETA TO THE FINAL VALUE OF ETA, DETERMINING LAMBDA FOR EACH

 

H000 000000 0UI00

000

120

143
SUCH ETA

DO 900 KF1,50
X(K)-ETAI+(K-1)*ETASTP

THE ROOT IS FOUND BY’A SIMPLE BISECTOR ROUTINE AND MUST BE
BETWEEN PRESET INITIAL VALUES OF LMDI AND LMD2.

INITIALIZE STEP

LMD1-1.001
LMD2-8.9
EPS-0.000000001

LOOP THROUGH A SEQUENCE OF BISECTIONS

CONTINUE

Y(K)-(LMD1+LMD2)*0 . 5

LEMBDA-Y(K)

ETA-X(K)

CALL GETARRAY (LEMBDA, ETA, BETA, R, M)
CALL DLFTRG (6, M, 6, FAG, 6, IPVT)

COMPUTE THE DETERMINANT USING THE SYSTEM SOFTWARE ROUTINE DLFDRG

CALL DLFDRG (6, FAG, 6, IPVT, DET1, DET2)
DET--(DET1*10.0**DET2)

CHECK WHETHER WE INDEED HAVE A ROOT

IF (ABS(DET) .LT. EPS) THEN
PRINT*, X(K), Y(K)
GOTO 890
ELSE IF (DET .LT. 0.) THEN
LMD15Y(K)
GOTO 100
ELSE
(DET .GT. 0.)
LMD25Y(K)
IF (LMD2-LMD1 .LT. EPS) THEN
GOTO 120
END IF
GOTO 100
END IF

END OF SECOND LOOP

CONTINUE
CONTINUE

DO 920 J-2,50

IF (Y(J-l) .LT. Y(J)) INCR-.TRUE.

 

000000

000000

999

00000

(XI)

 

144

INCR IS SET EQUAL TO TRUE IF THE BARRELLING CURVE EXHIBITS AN
INCREASE , '

ONCE INCR IS SET EQUAL TO TRUE, IT IS NEVER CHANGED
IF (INCR .AND. Y(J-l) .GT. Y(J)) DECRF.TRUE.

DECR IS SET EQUAL TO TRUE IF THE BARRELLING CURVE EXHIBITS A
DECREASE AFTER THE INCR, i.e. A SECOND DECREASE

CONTINUE

IF (DECR) THEN

PRINT*, BETA, R, 'GAMMA3B'

WRITE*, 'BETA-', BETA, 'ALPHA-', R, 'GAMMA3B'
ELSE IF (INCR) THEN

PRINT*, BETA, R, 'GAMMA2B'

WRITE*, 'BETA-', BETA, 'ALPHA-', R, 'GAMMA2B'
ELSE

PRINT*, BETA, R, 'GAMMAIB'

WRITE*, 'BETA-', BETA, 'ALPHA-', R, 'GAMMAIB'
END IF
GOTO 5
CLOSE(6)
STOP
END

SUBROUTINE GETARRAY (LEMBDA, ETA, BETA, R, M)
(See (VI))

PROGRAM PROJI
THIS PROGRAM CALCULATES THE TRANSITION VALUE FOR ETA FROM DATA
FILE CORRESPODING TO (BETA,ALPHA) IN GAMMA2F

INTEGER IMAX, IMIN

REAL X(50), Y(SO), D, YMAX, DMIN
CHARACTER*20 NAME

DO 70 I-I, 1000

PRINT*, 'ENTER FILE NAME OR TYPE END ->'
READ(*,*) NAME

IF (NAME(:3) .EQ. 'END') THEN
CLOSE (5)
STOP

END IF

OPEN (5, FILE-NAME, STATUS-'OLD')

Whig-"m“h If.‘ @33—

145

XMAX-O.o
DO 20 J-l, so
READ(5,*) X(J). Y(J)

FIND THE INTERNAL MAXIMUM

000

IF (YMAX .LT. Y(J)) THEN
YMAXéY(J)
IMAM-J
END IF
20 CONTINUE
PRINT*, 'THE MAXIMUM VALUE OF LAMBDA OCCURS NEAR THE PAIR',
+ '(ETA,LAMBDA)-', X(IMAX), Y(IMAX)

DMIN-1000.0

FIND THE TRANSITION VALUE OF ETA

000

D0 50 LP1,IMAX
D!ABS(Y(L)-Y(50))

IF (DMIN .GT. D) THEN
DMIN-D
IMIN-L
END IF
so CONTINUE ,
PRINT*, 'THE TRANSITION ETA IS NEAR', X(IMIN)

CLOSE(S)

70 CONTINUE
C

STOP

END
(XII)

PROGRAM PROJ2

C
G THIS PROGRAM FINDS THE VALUE OF ETA SWITCH FOR OPTIMUM DESIGN
C BETWEEN (BETA,ALPHA) AND ITS CONJUGATE CONFIGURATION. THIS
G PROGRAM NEEDS THE DATA FILE OF (ETA,LAMBDA)-PAIRS FOR BOTH
C (BETA,ALPHA) AND ITS CONJUGATE.
C

CHARACTER*20 NAMEI, NAME2

REAL X1(50), X2(50), Y1(50), Y2(50), D(50), CHECK, ETASWITCH

LOGICAL SIGI, SIG2
C

DO 30 I-I, 1000

PRINT*, 'ENTER FILE NAMES OR END ->'

READ(*,*) NAMEI, NAME2

IF (NAMEI(:3) .EQ. 'END') THEN
CLOSE(IO)

VEH——. ufi-‘E-s!‘ ‘P
“b n. .

 

150

200

250
40

146

CLOSE(20)

STOP
END IF .
OPEN (10, FILE-NAMEI, STATUS-'OLD')
OPEN (20, FILE-NAME2, STATUS-'OLD')

SIGl-.FALSE.
SIG2-.FALSE.

DO 40 J-I, 50
READ (10,*) X1(J), Y1(J)
READ (20,*) X2(J), Y2(J)

CHECKéY1(1)-Y2(1)
IF (CHECK .GT. 0.) GOTO 200
D0 159 K-1, 50
D(K)dY1(K)-Y2(K)
IF (D(K) .GT. 0.) THEN
IF (SIGI .AND. D(K) .GT. 0.) THEN
GOTO 150
ELSE
SIG-.TRUE.
ETASWITCH-(X1(K)+X1(K-1))/2
END IF E
END IF ’
CONTINUE

CONTINUE
DO 250 L-2, 50
D(L)dY1(L)-Y2(L)
IF (D(L) .LT. 0.) THEN
IF (8102 .AND. D(L) .LT. 0.) THEN
GOTO 250
ELSE
SIG2-.TRUE.
ETASWITCH-(X1(L)+X1(L-1))/2
END IF A
END IF
CONTINUE
CONTINUE
PRINT*, 'THE CROSS POINT IS', ETASWITCH
CONTINUE

STOP
END

 

 

147

LIST OF REFERENCES

_- —r‘m

 

 

148

REFERENCES:

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

Abeyaratne, R. and Knowles, J.K. (1987). Non-elliptic elastic
materials and the modeling of dissipative mechanical behavior: an

example. J. Elasticity 18, 227-278.

Burgess, I.W. and Levinson, M. (1972). The instability of
slightly compressible rectangular rubberlike solids under biaxial

loadings. Int. J. Solids Struct. 8, 133-148.

Davies, P.J. (1989). Buckling and barrelling instabilites in

finite elasticity. J. Elasticity 21, 147-192.

Green, A.E. and Spencer, A.J.M. (1959). The stability of a
circular cylinder under finite extension and torsion. J. Math.

Phys. 37, 316-338.

Horgan, C.O. and Pence, T.J. (1989). Void nucleation in tensile
deadloading of a composite incompressible nonlinearly elastic

sphere. J. Elasticity 21, 61-82.

Horgan, 0.0. and Pence, T.J. (1989). Cavity formation at the
center of a composite nonlinearly elastic sphere. J. Appl. Mech.

111, 302-308.

Knowles, J.K. (1977). The finite anti-plane shear field near the
tip of a crack for a class of incompressible elastic solids.

Int. J. Fracture. 13, 611-639.

Leissa, ATV. (1987). A review of laminated composite plate

buckling. Appl. Mach. Rev. 40, 575-591.

 

[9]

[10]

[11]

[12]

[13]

149
Ogden, R.W. (1984). Nonlinear elastic deformations. Halsted

Press, New YOrk.

Sawyers, K.N. and Rivlin, R.S. (1974). Bifurcation conditions for
a thick elastic plate under thrust. Int. J. Solid Structure 10,

483-501.

Sawyers, K.N. and Rivlin, R.S. (1982). Stability of a thick

elastic plate under thrust. J. Elasticity 12, 101-124.

Simpson, H.C. and Spector, S.J. (1984). On barrelling for a
special material in finite elasticity. Q. Appl. Math. 42, 99-

111.

Wilkes, E.W. (1955). On the stability of a circular tube under

end thrust. Q. J. Mech. Appl. Math. 8, 88-100.

run-Eras; III-Am

TE UNIV. L IBRRRIES

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