LD5655.V856_1976.R35.pdf - VTechWorks

PROPAGATION OF DELAMINATION IN LAYERED

ANISOTROPIC CYLINDERS

by

Ramaswamy Lakshminarayanan Ramkumar

Dissertation submitted to the Graduate Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in

Engineering Science and Mechanics

APPROVED:

D. Frederick, Chairman

C. W. Smith _____ ]_] ________________ _

l'-• .A. ~Jei s s haa r

-------,---,-------------M. P. Kamat D. P. Telionis

November, 197 6

Blacksburg, Virginia

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Dr. Frederick whose

guidance, encouragement and invaluable suggestions were of immense use

during the course of this investigation.

Thanks are also due to Professor Smith, Dr. Weisshaar, Dr. Kamat

and Dr. Telionis for their interests in, and discussions on, the research

topic. Special mention is made of Dr. Nayfeh who introduced the author to

some powerful mathematical methods.

The author expresses his regards to his parents whose constant advice,

patience, deep understanding and blessings were a perpetual source of in-

spiration and strength. His brother, sister and sister-in-law contributed,

in their own ways, to the progress of the work.

Finally, the author expresses his appreciation for Mrs. Donna Griffith's

cooperation and her excellent typing of the dissertation.

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

CHAPTER I. INTRODUCTION .......................................... .

CHAPTER II. DELAMINATION IN A LAYERED CYLINDER WHEN THE DRIVING

FORCE IS A PRESSURE ON THE CRACK SURFACE ............. ..... .... 8

CHAPTER III. DELAMINATION IN A LAYERED CYLINDER WHEN THE

DRIVING FORCE IS A UNIFORM LOADING ON THE INNER CYLINDRICAL

SURFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

CHAPTER IV. EFFECTS OF THE CRACK TIP GEOMETRY ..................... 46

CHAPTER V. EFFECT OF MIDPLANE AXIAL AND SHEAR STRAINS ............. 79

CHAPTER VI. CONCLUSIONS . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 95

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

APPENDICES: COMPUTER PROGRAMS FOR THE VARIOUS PROBLEMS ............ 106

VITA ............................................................... 167

i ii

a

a .. , b .. , d .. lJ lJ lJ

A. 1

A .. , B .. , D .. lJ lJ lJ

b

c' d

c. 1

hl ' L

M .. lJ

N .. lJ

p

qo

qz r

U,

u ~

u

h2

v, w

x, 8, z

LIST OF SYMBOLS

: a(l:i) are the roots of the homogeneous e8iuation

for shell Il

laminate properties

complex constants in the solutions for the shell

radial displacements

laminate constants

: b(l:i) are the roots of the homoqeneous equation for

shell I2 + + . -(c-id) are the roots of the homogeneous equation for

shell II

real constants in the radial displacement solutions

for the cylinder

thicknesses of the inner and outer layers

initial half-crack length

moment resultants for the laminate

inplane stress resultants for the laminate

pressure on the crack surface

uniform loading on the inner cylindrical surface

loading on the cylinder in the z direction

inner radius of the cylinder

shell displacements in the axial, circumferential

and radial directions

Airy stress function

total internal energy in the cylinder

the axial, circumferential and radial coordinates for

the cylinder

iv

x defined as (x-L)

the slope of region II at the crack tip, when multi-

plied by these constants, gives the slopes of regions

Il and I2 at the crack tip

adhesive fracture surface energy

v

LIST OF FIGURES

l. Cylinder geometry and loading: Pressure on the crack

surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. Effect of h1/h 2 on PcR for an isotropic cylinder ........... 24

3. Effect of h1;h2 on PcR for an anisotropic cylinder:

(l) B-E(0°); (2) B-Al(45°) ............................ 26

4. Effect of h1/h2 on PcR for an anisotropic cylinder:

(l) B-E(0°); (2) G-E(45°) ............................. 27

5. Effect of h1;h2 on PcR for an anisotropic cylinder:

(l) B-Al(0°); (2) G-E(45°) ............................ 28

6. Effect of h1/h 2 on PcR for an anisotropic cylinder:

(1) G-E(0°); (2) B-E(45°) ............................. 29

7. Comparison among PcR values for isotropic, orthotropic

and anisotropic cylinders 30

8. Cylinder geometry and loading: uniform loading on inner

9.

l 0.

11.

12.

cylindrical surface ................................... 32

Effect of h1/h 2 on q for an isotropic cylinder .......... 37 °CR

Effect of h1/h 2 on q for an anisotropic cylinder: 0 cR

(1) B-Al (0°); (2) G-E(45°) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Effect of h1;h2 on q for an anisotropic cylinder: OCR

(l) B-E(0°); (2) G-E(45°) ............................. 41

Eff~ct of h1!h2 on q for an anisotropic cylinder: 0cR

(1) B-E(0°); (2) B-Al(45°) ............................ 42

13. Effect of h1/h 2 on q for an anisotropic cylinder: OCR

(l) G-E(0°); (2) B-E(45°) .......................... · ·. 43

vi

LIST OF FIGURES (con't.)

14. Comparison among q values for isotropic, orthotropic °CR

and anisotropic cylinders ............................... 44

15. Effect of 8l, 82 and continuity in Nxz on PcR:

(1) B-E(0°); (2) B-Al(45°); h1/h 2=4 ..................... 59

16. Effect of 81, 82 and continuity in Nxz on PcR:

(1) G-E(0°); (2) B-E(45°); h1/h2=4 ...................... 60

17. Effect of 81, s2 and continuity in ~lxz on PcR:

(1) B-Al(0°); (2) G-E(45°); h1/h2=4 ..................... 61

18. Effect of 81, 82 and continuity in Nxz on PcR:

(1) B-E(0°); (2) G-E(45°); h1/h2=4 ...................... 62

19. Effect of 81, s2 and continuity in N on q : xz OCR

(1) G-E(0°); (2) B-E(45°); h1/h2=1 ...................... 71

20. Effect of s1, 82 and continuity in N on q : xz OCR

(1) B-E(0°); (2) B-Al(45°); h1/h2=1 ..................... 72

21. Effect of 81, s2 and continuity in N on a : xz ·ocR

(1) G-E(0°); (2) B-E(45°); h1/h2=4 ...................... 73

22. Effect of 81, s2 and continuity in N on q : xz OCR

(1) B-Al(0°); (2) G-E(45°); h1/h2=4 ..................... 74

23. Effect of s1, s2 and continuity in N on q : xz OCR

(1) B-E(0°); (2) G-E(45°); h1/h2=1 ...................... 75

24. Effect of s1, s2 and continuity in N on q : xz OCR

(1) B-E(0°); (2) G-E(45°); h1!h2=4 ...................... 76

25. Effect of 81, 82 and continuity in N on q : xz OCR

(1) B-E(0°); (2) B-Al (45°); h1/h2=4 . . .. . .. . . . . . . . . . . . . . . 77

vii

LIST OF FIGURES (con't.)

26. Effect of midplane axial and shear strains on PcR:

(1) B-Al(0°); (2) G-E(45°); h1Jh2=1 ................... 83


(1) B-E(0°}; (2) G-E(45°); h1Jh2=1 .................... 84

28. Effect of midplane axial and shear strains on q : °CR

(1) B-Al(0°); (2) G-E(45°); h1Jh 2=1 ................... 89

29. Effect of midplane axial and shear strains on a : 'OCR

(1) B-E(0°); (2) G-E(45°); h1Jh2=1 .................... 90


(1) G-E(0°); (2) B-E(45°); h1Jh2=1 .................... 91

viii

LIST OF TABLES

1/2 l. Effect of 81 and 82 on PcR/(ya) :

(l) B-E(0°); (2) G-E(45°); hifh 2=1 ...................... 51

2. Effect of s1 and 82 on PcR/(ya) 1/ 2:

(l) B-E(0°); (2) G-E(45°); h1/h2=1/4 .................... 52

3. Effect of 81 and 82 on PcR/(ya) l/ 2:

(l) B-Al (0°); (2) G-E(45°); h1/h 2=1 . . . . . . . . . . . . . . . . . . . . . 53

4. Effect of 61 and 82 on PcR/(ya) 1/ 2:

(l) B-Al(0°); (2) G-E(45°); h1/h2=1/4 ................... 54

5. Effect of 81 and 82 on PcR/(ya) 112 :

(l) G-E(0°); (2) B-E(45°); h1!h2=1 ...................... 55 l/? 6. Effect of s1 and 82 on PcR/(ya) -:

(l) G-E(0°); (2) B-E(45°); h1/h2=1/4 .................... 56 1/2 7. Effect of 81 and 82 on PcR/(ya) :

(l) B-E(0°); (2) B-Al(45°); h1/h2=1 ..................... 57

8. Effect of 81 and 82 on PcR/(ya) l/ 2:

(l) B-E(0°); (2) B-Al(45°); h1/h2=1/4 ................... 58

9. Effect of 81 and 62 on q /(y ) 112: oCR a

(l) B-E(OQ); (2) G-E(45°); h1/h2=1/4 .................... 66

10. Effect of 81 and 82 on q /(y )112: oCR a

(l) B-Al(0°); (2) G-E(45°); h1/h2=1/4 ................... 67 . 1/2 11. Effect of s1 and 82 on q /(y) :

OCR a (l) B-Al(0°); (2) G-E(45°); h1/h2=1/4 ................... 68

12. Effect of 61 and 82 on q /(y )112: OCR a

(l) G-E(0°); (2) B-E(45°); h1!h2=1/4 .................... 69

ix

LIST OF TABLES (can't.)

13. Effect of s1 and s2 on q /(y )112: OCR a

(1) B-E(0°); (2) B-Al(45°); h1/h2=1/4 ..................... 70


(1) G-E(0°); (2) B-E(45°); h1/h2=1 ........................ 85


(1) B-E(0°); (2) B-Al(45°); h1/h2=1/4 ..................... 86


(1) B-E(0°); (2) B-Al(45°); h1/h 2=1 ....................... 87


(1) B-E(0°); (2) B-Al(45°); h1/h2=4 ....................... 88


(1) B-E(0°); (2) B-Al(45°); h1/h 2=1 ....................... 92


(1) B-E(0°); (2) B-Al(45°); h1/h2=1/4 ..................... 93


(1) B-E(0°); (2) B-Al(45°);_h1/h2=4 ....................... 94

x

CHAPTER I. INTRODUCTION

Cylindrical pressure vessels are commonplace in most industries.

With the many recent advances in technology and the advent of compos-

ites, the behavior of a composite cylinder under various operating

conditions is a subject of current interest. Notwithstanding the

sophisticated levels technology has achieved, internal flaws in struc-

tural components are still existent on a large scale. These internal

flaws may not be easily detectable and could, in many cases, lead to

crack nucleation. If such a crack is assumed to be present along the

bond line in a laminated cylinder, then the safety of the structure

under extreme operating conditions is of utmost concern.

The increase in strength and the improvement in performance of

composites over conventional metallic components have triggered a wide

application of these materials in the aerospace, automotive, shipbuild-

ing and other industries. Stambler [l]* forecasts a bright future for

composite materials in the aerospace industry, and lists many interest-

ing facts that justify his predictions. A few of these are recalled

below.

Lockheed engineers estimated that substitution of an organic compos-

ite in a 50% resin content by volume could provide a weight reduction of

25 to 30 %. Six different primary and secondary parts for Northrop's

F-5A fighter were designed and fabricated from various graphite systems.

Most of these parts including a landing gear door, leading and trailing

edge flaps, and the horizontal stabilizer performed well. A landing

* Numbers in brackets [ ] indicate references.

,., c.

gear door, flown in an actual aircraft, was 36% lighter than its aluminum

counterpart and exceeded all static and flight test structural require-

ments. The F-15 fighter has a composite horizontal stabilizer with

boron/epoxy skins. Many F-111 test parts have been made from a number

of composites including boron and graphite. A graphite box beam assembly

has been designed by Grumman Aircraft. The demand for fiberglass has

been predicted to increase in the next decade. Composite materials may

also be used in aircraft engine parts soon. The use of boron/polymide

turbine blades instead of titanium blades could effect a weight saving

of about 50%. Space applications have led to the development of compos-

ites using a resin system called Ryton which is practically insoluble

and appears to be completely inert. These successes are bound to boost

the application of composite materials in the aerospace industry in the

future.

The literature on composite materials is ever-expanding and there

are journals such as the Journal of Composite Materials, dedicated

solely to research on the behaviour of these materials in a microscopic

as well as a macroscopic sense. References [2]-[18] list a number of

well-documented texts on composite materials. Despite the many research

programs that are being conducted there are several investigative areas

such as the failure analysis of composites that are still not understood

properly and where reliable data are far from complete. References [19]

and [20] present an excellent discussion on such aspects of composite

materials. A major effort launched recently by the U.S. Army, the U.S.

Navy, the U.S. Air Force, NASA and FAA in collaboration with the various

industries and universities, was called "Composites Recast" [21]. The

3

Design Criteria Panel for "Composites Recast" has recommended, among

other things, a study of the delamination problem under static and

fatigue loading and the applicability of classical fracture mechanics to

the prediction of fracture from discontinuities, fillets, holes and

joints.

The various plies in a laminated cylinder are generally held to-

gether by an adhesive bonding. Adhesives can effect a considerable

weight reduction which is crucial in the aerospace industry in improving

performance. But the adverse environmental effects on adhesives and

insufficient understanding of their failure mechanisms could retard

their application. Two such cases, presented in reference [22], involve

composites used in payload shrouds on two launch vehicles. The Atlas-

Agena shroud, a phenolic glass honeycomb sandwich construction, failed

during launch and was replaced by a ring stiffened magnesium shroud

which served successfully in the launch of Mariner IV to Mars. The

shroud on Titan III, a fiberglass honeycomb construction, was replaced,

upon failure, by one made of stiffened aluminum. In these examples, the

structural efficiency that could have been provided by the composite

material had to be sacrificed because the shrouds delaminated due to

increased differential pressure and temperature. In an effort to offset

such disadvantages, the Materials Panel of "Composites Recast" [21]

expressed the need for a sound understanding of what is necessary to

make and maintain a good interfacial bond in fiber-reinforced and lay-

ered composites. Composites suffer from poor wet strength because of

improper surface preparation, and inadequate degassing of finely dis-

persed air bubbles. This creates voids which accelerate cohesive failure

in the adhesive. Sharp corners and voids act as stress concentrators

and cause cracks to propagate in the structural component under certain

loading conditions. A layered cylinder delaminates when a crack in the

adhesive propagates along the bond line between two adjacent laminae.

The determination of the operating conditions under which the crack

propagates in a stable fashion - contrary to the layman's feeling that

all crack extensions are catastrophic - is the object of the present

study.

Griffith [23] published his concept of crack propagation in 1920

stating that an existing crack will propagate if, in so doing, the total

energy of the system is lowered. Irwin [24] and Orowan [25] showed,

many years later, that the Griffith energy balance condition, when

modified to account for plastic work, was a necessary and sufficient

condition for predicting brittle fracture. By 1959, the so-called

Griffith-Irwin concept had become well known and the American Society

for Testing Materials created a special committee to launch a broad inves-

tigative program to achieve a better understanding of fracture. Shortly

thereafter Neuber [26-271, Kuhn [28-29], and Barenblatt [30] received

much attention with their differing concepts of essentially the same

limiting condition causing failure. Their different approaches yielded

the same result because the theoretical foundation for Fracture Mecha-

nics may be soundly based on the linear, small strain theory of elasti-

city, regardless of the phenomenon occurring within the plastic zone at

the crack tip which precipitates fracture.

Even though the subject has passed the initial stage and is still

in its development stage, a few texts already have been written on the

5

various theories related to fracture mechanics [31-36]. Many problems

have been solved using a continuum approach, and a few of these that are

relevant to the present study are listed in references [37-115]. A

careful examination of these publications reveals the many weaknesses

that are present in the modelling of the different problems. Unfortu-

nately, almost always, a study of the simplest model invariably precedes

a more realistic formulation giving rise to a numerical - if not a

closed-form-solution. Although this implies that a certain amount of

crudeness has been introduced into the model, the results that are

obtained give a fair idea of the real behaviour of the actual problem.

Also, the effort required to solve the simpler model would differ from

that required to solve the actual problem, if it can be done, by an

order of magnitude. This justifies the simplifications introduced in

most fracture analyses. The amount of confidence placed on the solutions

obtained for the various problems depends directly on the measure of

correlation of these solutions with existing experimental data. Numeri-

cal techniques, such as the finite element technique, have also been

employed to solve fracture problems [116-121].

Most of the problems in fracture mechanics tend to become extremely

complex if the actual direction of propagation of the crack, the inhomo-

geneity in the individual orthotropic plies, the effect of the adhesives,

the presence of interlaminar shear and the crack tip geometry are made

as realistic as experiments show them to be. The references alluded to

earlier discuss many of these effects in detail. There is still a

feeling of uncertainty when one tries to generalize these effects for

any general laminated cylinder. This is due to the fact that failure

mechanisms in composites are not properly understood and, even if they

6

are, not modelled in a compatible fashion. Nevertheless, the many

assumptions made in this area reduce the problem to a tractable form,

yielding solutions that give a basic understanding of the actual behav-

iour.

The present investigation aims at determining the growth of delam-

ination in a layered, anisotropic cylinder and to delineate regions of

unstable crack propagation. To obtain a closed-form solution, an infi-

nitely long cylinder is treated as a thin shell, assuming the presence

of circumferentially symmetric crack and loading. This reduces the

shell equations to a fourth-order, non-homogeneous, ordinary differen-

tial equation in terms of the radial displacement. The cylinder, after

delamination, is treated as three shell elements joined at the crack

tip - two shell elements above and below the cracked region and the

secure element to the right of the crack tip. Proper boundary conditions

and matching conditions at the crack tip are imposed to solve for the

radial displacements of the cylinder in the three regions in terms of

the crack length. The total strain energy in the cylinder may then be

obtained, using Clapeyron's theorem, and thus results in an expression

for the rate of change of the strain energy in terms of the crack length.

Once this quantity is found, Griffith's energy balance criterion may be

used to evaluate the critical loading condition under which the delamin-

ation would tend to propagate. The Griffith criterion states that a

crack would tend to propagate if the decrease in the total strain energy

due to a differential increase in the crack length is equal to the

amount of adhesive surface fracture energy required to create the differ-

ential increase in the crack surface area.

7

Two types of loading are considered separately: in the first case,

the driving force is the pressure acting on the crack surface, and, in

the second case, the loading is applied on the inner cylindrical surface.

The effect of the crack tip geometry is also studied based on an elastic

analysis, though the region near the crack tip is known to be plastic.

The effects of the adhesive layers, the hoop strain and the axial strain

are also studied.

The present analysis employs classical thin shell theory. Interla-

minar shear stresses, which are of large magnitudes near the crack tip,

are ignored. In other words, only a global picture of the crack pheno-

menon is considered, and the crack tip near field effects are neglected.

The determination of the extent to which these shear stresses affect the

crack growth in the cylinder is an interesting topic for future research.

CHAPTER II. DELAMINATION IN A LAYERED CYLINDER

WHEN THE DRIVING FORCE IS A PRESSURE ON THE CRACK SURFACE

The qrowth of a delamination in an infinitely long cylinder under

the action of a pressure acting on the crack surface is studied in this

section. The effect of the adhesive layer of known material properties

may also be included if that layer is treated as an isotropic layer

sandwiched between two anisotropic plies when laminate theory is used.

Calcote [8] has derived the governing equations for the static analysis

of a laminated cylinder using thin shell assumptions [123-126]. Further

simplifications are introduced by assuming circumferential symmetry in

the crack qeometry and the loading. Kulkarni and Frederick [84] ana-

lyzed this problem for an isotropic cylinder, neglecting the effect of

the adhesive layer.

Fiqure l shows the cylindrical model with the delamination extend-

inq over a lenqth of 2L and the pressure p acting on the crack surface.

Regions Il, I2 and II are considered as three different anisotropic

cylindrical shells which join together at the crack tip. Each region

may be composed of many unidirectional laminae qlued together by thin

adhesive layers. The total thickness of the cylinder is assumed to be

very small in comparison to the radius of the cylinder. This justifies

the use of the assumptions of thin shell theory and Donnell approxima-

tions [8]. It will be assumed that the delamination in the cylinder

will propagate alonq the interface between two adjacent layers, in the

adhesive layer. The validity of this assumption depends on the fiber

orientation in the individual plies. the laminate qeometry, and the

location of the crack. Also, the delamination will be assumed to exist

8

z

---L--

r

x

FIGURE l. CYLINDER GEOMETRY AND LOADING: PRESSURE ON THE CRACK SURFACE

10

all around the circumference of the cylinder, giving rise to symmetry in

the crack geometry in that direction. The uniform pressure p, actinq as

a driving force on the crack surface, will be taken to be axially symme-

tric as well. As a result of the above assumptions, the radial displace-

ment, w, of the cylinder is seen to be independent of the circumferential

location.

Equations (4.86) and (4.87) in Calcote r8J are the compatibility

and equilibrium equations for a laminated cylinder. These are rewritten

here as Equations (1) and (2). The shell compatibility equation is:

all U + b + (2b23 - b31) -4 'oooo 21"''xxxx r r

w + 'XXXO (b22 + bll - 2b33)

r2

(2b13 - b32) bl2 w + ---..,.-.-- w + - w

'XXOO 3 'XOOO 4 '0000 r r 1 r w, xx = 0

The shell equilibrium equation is:

(2b13 - b32) + bl2 u dl3 ----=----Li +d w +4-v.J + r3 'xooo r4 '0000 11 'XXXX r 'XXXO

2(dl2 + 2d33) d23 d22 2 "'' xxoo + 4 -3- w, xooo + -4- w, 0000 r r r

= -q . z

In the above equations a .. , b .. and d .. are laminate constants depending lJ lJ lJ on the laminate geometry and the material properties of the different

layers of the laminate. The radial displacement of the cylindrical shell

( 1 )

( 2)

11

is wand r is its inner radius. The axial, circumferential and radial

coordinates of the cylinder are x, 0, and z. As defined in reference

[8], U is the Airy stress function and q2 is the loading per unit area

in the positive z direction. Due to the circumferential symmetry in the

proposed model, Equations (1) and (2) reduce to the following forms,

respectively:

(3)

(4)

Equations (3) and (4) may be combined to give the following equation: 2

1 b21 (dll

b21 1 ( 5) - a)w'xxxx + - -- w, - - u = -q 22 r a22 xx r 'xx z

The stress resultants for the laminate, defined in terms of the

Airy stress function, are related to the midplane strains and its rates

of change of curvature as presented in Calcote [8]. In these expressions,

Donnell's approximations are introduced to simplify the expressions for

rates of chanqe of curvature [8]. Furthermore, the axial strain and the

shear strain are neglected in comparison to the radial strain in the

cylinder. These simplifications give rise to the following expression

for theinplane stress resultant, U,xx in terms of the radial displace-

ment and its second derivative with respect to the axial coordinate [8]:

In Equation (6), Aij and Bij are laminate constants [8].

this expression into Equation (5) one obtains:

b2 b A 21 + l (_1.l + . 22 (dll - a22 )w,xxxx r a22 812)w,xx - 7 w = -qz

(6)

Substituting

( 7)

12

Equation (7) is the governing equation for a cylindrical shell with

a loading of qz in the positive z direction. Since the delaminated

cylinder is modelled as three different cylindrical shells joined at the

crack tip, the above shell equation has to be applied and a solution has

to be obtained for each of the three regions Il, I2 and II. The circum-

ferential symmetry of the problem and Donnell's approximations reduce

the shell equations from two coupled, fourth order, partial differential

equations in terms of U and w to one fourth order, ordinary differential

equation in terms of w alone, Equation (7). The coefficients in this

shell equation are functions of the laminate constants and the radius of

the cylinder. Hence the three regions Il, I2 and II each have the same

form of governing equation, but with different coefficients, for each

region, in the general case where each is a different anisotropic cylin-

der. The inhomogeneity, qz, is the loading function and is dependent on

the type of loading. In the present study the loading is taken to be a

uniform pressure p acting on the crack surface, and therefore:

qz =

qz =

qz =

p

-p

0

for the shell Il

for the shell I2, and

for the shell II.

(Sa)

(Sb)

(Sc)

The homogeneous part of Equation (7) gives rise to an auxiliary

equation which yields four roots that are complex, in general. If the

discriminant of the auxiliary equation: b A b2

-1 (_n+ s12 )2 + 4 _IT (d _ _n) r2 a22 r2 11 a22

is positive, the roots may be real or/and complex. But the bij and Bij

terms that represent a measure of the coupling between the membrane

13

and bending actions of the laminate are normally very small in compari-

son to the dij terms [8]. Furthermore, the term d11 has a negative

value. Hence the discriminant for most laminates of practical interest

is negative. This implies that the roots of the auxiliary equation are

complex conjugate pairs. Proper precaution must be taken when a compu-

ter program is written to solve the shell equation to warn the program-

mer of any choice of laminate leading to positive discriminant values.

Assuming the discriminant to be negative, the roots of the homo-

geneous equation are:

b - lc_1l + 812) + i r a22

+ + = - (c - id).

2 b21

2(dll - -) a22 (9)

If the cylinder is assumed to have only two unidirectional plies glued

together by an adhesive forming an interface of infinitesimal thickness,

regions Il and I2 will have only one ply each. In this case, c and d

would have the same value as the bij and Bij terms vanish. If a is the

value c and d take for region 11, and bis the value they take for

region I2, the roots of the homogeneous equation for the three shells

may be written as:

~ a ( l + i ) for she 11 I l

~ b(l + i) for shell I2

+ (c ~ id) for shell II

( l Oa)

( l Ob)

( l Oc)

Once the above roots are determined, the homogeneous solutions for the

three regions may be written as:

14

Hil = eax(Al cos ax + i A2 sin ax) + h e-ax(A3 cos ax + i A4 sin ax) ( 11 a)

wI2 = ebx(A5 cos bx + i A6 sin bx) + h -bx( . e A7 cos bx + 1 A8 sin bx) ( 11 b)

WII ......

= ecx(A9 cos dx + i A10 sin dx) + h ,.., e-cx(A11 cos dx+ i A12 sin dx) ( 11 c)

where A1, A2, ... ,A12 are complex constants and x = x-L.

Shells Il and 12 have a pressure loading which gives rise to a

particular solution, while shell II has no loading and, hence, no

nontrivial particular solution. The particular solutions for the three

shells are:

w~l = - en ( l 2a)

wr2 = _p__ p CI2 (12b)

wII = 0 p ( l 2c)

All AI2 22 22 where Cil = and CI2 = -7 -7 ( 13)

The homogeneous and the particular solutions for each shell may be

combined to give the general or complete solution for that shell.

Before this is done, the homogeneous solutions in terms of complex

constants may be rewritten in terms of real constants for convenience.

This results in eight real constants for each shell. Adding the parti-

cular solutions to these homogeneous solutions in terms of real con-

stants, the general or complete solutions for the three shells are:

1,-.1

wil = eax[(c 1 cos ax - c2 sin ax) + i(c3 cos ax + c4 sin ax)] +

e-ax[(c 5 cos ax - c6 sin ax) + i (c 7 cos ax + c8 sin ax)] - _B_ ( l 4a) Cil

12 bx[ W = e (c9 cos bx - c10 sin bx) + i(c11 cos bx+ c12 sin bx)J + -bx ( e L c13 cos bx - c14 sin bx) + i(c 15 cos bx+ c16 sin bx)]

_Q_ + CI2 ( l 4b)

WI I -= ecx[(c17 cos dx - cl8 sin dx) + i(cl9 cos dx + c20 sin dx) J + -e-cx[(c21 cos dx - c22 sin dx) + i(c23 cos dx + c24 sin dx) J.

( l 4c)

The three fourth order, ordinary differential equations that govern

the motion of the three shells have solutions containing twelve inter-

dependent complex constants or twenty four real constants. Twelve bound-

ary conditons have to be prescribed to solve for these constants. Since

the problem under consideration is analyzed as a conservative problem with

no net energy dissipation, the radial displacement in the cylinder has to

be real. This realization simplifies the evaluation of the constants in

Equations (14). The solution for the constants contributing to the ima-

ginary part of the displacements involves a set of homogeneous equations

that yield only a trivial solution. Therefore, even before the boundary

conditions are specified, only the real part of the radial displacement

solution for each shell will be retained. This reduces Equations (14)

to the forms shown below:

w11 = eax(c1 cos ax - c2 sin ax) + e-ax(c5 cos ax - c6 sin ax) -

_B_ CI 1 ( l 5a)

16

vJI 2 bx( = e c9 cos bx - c10 sin bx) + e-bx(c13 cos bx - c14 sin bx) +& ( l 5b)

wll ,..,

ex( cos dx - sin dx) + = e cl7 cl8 -e-cx(c 21 cos dx - c22 sin dx) ( l 5c)

Boundary Conditions

Twelve boundary conditions must be specified in order to evaluate

the constants in Equations (15). Four of these are prescribed at the

line of symmetry, x=O, and two at the boundary at infinity. The six re-

maining conditions are the matching conditions at the crack tip. It is

through these matching conditions that the present model is made to resem-

ble the actual problem. The first six conditions may be imposed initially

to reduce the number of unresolved constants to six. Symmetry in the load-

ing and geometry makes the radial displacement w a maximum at the line of

symmetry. This implies that the slope in the axial direction vanishes at

the line of symmetry. 11 w,x = 0 at x=O ( 16)

wI2 = 0 at x=O ( 17) 'x

The symmetry property also implies that the normal shear force resultant,

Nxz' has to vanish at the line of symmetry.

Nil xz = 0 at x=O

NI2 xz = 0 at x=O

The radial displacement in shell II is bounded at infinity.

half-crack length, L, this finiteness condition is:

wII is bounded at x + 00 or x + 00

0

( 18)

( 19)

For a finite

(20).

17

The assumptions made earlier lead to an expression for Nxz in terms

of the radial displacement and its derivatives [8]:

( 21 )

Imposing conditions (16) to (lg) into the equations (15a) and (15b),

one obtains:

c1 - c2 - c5 - c6 = O

cg - clo - cl3 - cl4 = 0

B Il _J1_ ( c

r 1 = 0

(22)

(23)

(24)

812 ~ (cg - clo - cl3 - c14) - 2b2 oii (- cg - clo + cl3 - cl4)

= 0 (25)

The four equations above yield the results:

c = 1 C5 (26)

c2 = -c6 ( 27)

Cg = cl3 (28)

clo = -cl4 (29)

The finiteness condition,Equation (20),demands that in Equation ( l 5c)

C17 = Cl 8 = 0 (30)

The results listed above reduce Equations ( 15) to the following form: wil = 2 c1 cos ax cosh ax - 2 c2 sin ax sinh ax - p/Cil ( 31 )

wI2 = 2 cg cos bx cosh bx - 2 c10 sin bx sinh bx + p/CI2 (32)

WI I --ex cos dx - c22 sin dx) (33) = e ( c21

18

To solve for the remaining six constants the matching conditions

are specified at the crack tip. Continuity in displacement and bending

moment at the tip account for three conditions. Since the crack tip

region is actually plastic and the crack tip geometry is not clearly

defined, the slopes of the three regions at that point are arbitrarily

set to zero. The effect of having non-zero slopes at the crack tip is

studied in a later section. The six matching conditions may then be

expressed as:

At x=l, Il \'.JI I w =

At x=l, WI2 II = w

At x=l, Mil + MI2 = MI I x x x

At x=l, I l I 2 II = 0 w,x = w,x = w,x

An expression for M , compatible with earlier assumptions, may be x written in terms of w and its derivatives [8]:

(34)

(35)

(36)

(37)

(38)

Imposing the conditions in Equations (37) on Equations (31), (32)

and (33), the following relations are obtained:

c1(H-I)-c2(H+I) = 0

c9(E-F)-c 10 (E+F) = 0

cc 21 + dc 22 = 0

(39)

(40)

( 41 )

In the above expressions and in subsequent equations, the following

definitions were introduced for convenience:

A = cos bl cosh bl

B = sin bl sinh bl

E = cos bl sinh bl

19

F = sin bl cosh bl

G = cos al cash al (42)

H = cos al sinh al

I = sin al cosh al

J = sin al sinh al

The three shell equations now have the form:

wil - 2c1[cos ax cosh ax - (H-I) sin ax sinh ax] - p/Cil - (H+I) ( 43)

WI2 = [ (E-F) · J I 2c9 cos bx cosh bx - (E+F) sin bx s1nh bx + p/C 2 (44)

-II -ex[ - c ] W = c21 e cos dx + Cf sin ax (45)

The Equations (34), (35) and (36) yield the following expressions for the

remaining three constants:

c21 /p = (RKP5 + RKP6)/RKP4

c9/p = (c21 /p - l/CI2)/RJ2

c1/p = (c21 ;p + l/Cil)/RJl

where RJl = 2[G - J(H-I)/(H+I)]

RJ2 = 2[A - B(E-F)/(E+F)]

RJ3 = 4a2 D~~[J + G(H-I)/(H+I)]

RJ4 = 4b2 D~~LB + A(E-F)/(E+F)] Bil

RKPl = __lg_ RJl + RJ3 r

812 RKP2 = __lg_ RJ2 + RJ4 r

(46)

( 47)

(48)

(49)

20

RKP4 = RKPl/RJl + RKP2/RJ2 - RKP3

RKP5 = (B{~/r - RKPl/RJl)/Cil

RKP6 = (RKP2/RJ2 - B{~/r)/CI2

At this point, the radial displacements of the three shell ele~ents can

be superimposed to give the radial displacement of the entire cylinder

under the action of a pressure p acting on the crack surface. This dis-

placement is a function of the applied pressure, the material properties

of the plies in the layered cylinder, the laminate geometry and the half-

crack length. In the next step, the Griffith energy balance criterion

may be used to determine the criticality condition under which the delam-

ination would tend to propagate.

Griffith's Energy Balance Criterion

The Griffith energy balance criterion states that an adhesive or

cohesive fracture would tend to propagate at a critical value of the

applied loading when the incremental loss in the total strain energy of

deformation with an infinitesimal increase in the fracture area just

exceeds the fracture surface energy required to create the additional

fracture area. Using the displacements in each region, the total strain ,..,

energy in the cylinder, U, may be evaluated through Clapeyron's theorem.

Using the symmetry in the problem and considering only one half of the

cylinder,

0 w1 l p 2nrdx + ~- ] H12 (-p) 2nrdx

0

L or, U = 21rr p J (Wil-w12 )dx

0

21

(50)

When the delamination propagates by an infinitesimal amount dl, the

fracture area increases by an infinitesimal amount of 2nr(2dl) or

4nr dl. This is because the crack extends by an amount dl on either

side of the line of symmetry. Suppose Ya is the specific adhesive

fracture surface energy, which is the energy required to create unit

adhesive fracture surface area. The Griffith energy balance criterion

may then be stated as:

( 51 )

Equations (43) and (44) are substituted into Equation (50) to give an

expression for U in terms of the pressure p, the half-crack length L,

and the inner radius of the cylinder. When this expression is differ-

entiated with respect to Land substituted into Equation (51) it gives

the critical value of the pressure, PcR' at which propagation of the de-

lamination is imminent. Omitting the various steps in this tedious pro-

cedure, and grouping repetitive terms under different names, the final

result may be written in a compact manner as:

PcR[PJRl + (DCl) PJR2 + PJR3 + (DC9) PJR4] 1/ 2 = [2ya] 1/ 2 (52)

where PJRl = 2(c1/p)[4IHG - 2J(H2-I 2)]/(H+I) 2 - l/Cll

PJR2 = 2(H2+I 2)/[a(H+I)]

PJR3 = 2(c 9/p)[2B(E2-F2) - 4AEF]/(E+F) 2 - l/CI2

PJR4 = -2(E2+F2)/[b(E+F)]

DC21 = [CP6 + CP7 - CP5(c 21 ;p)]/RKP4

DC9 = [DC21 CP2(c9/p)]/RJ2

DCl = [DC21 CPl(c1/p)]/RJl

CPl = J (RJ3)/[a oi~(H+I)]

CP2 = B (RJ4)/[b D~~(E+F)]

22

CP3 = 2a[4a2oi~(H 2+I 2 )-G(RJ3)]/(H+I)

CP4 = 2b[4b2oii(E2+F2)-A(RJ4)]/(E+F)

CP6 = -[CP3 (RJl) - (RJ3) CPl]/[Cil (RJ1) 2]

CP7 = [CP4 (RJ2) - (RJ4) CP2]/[CI2 (RJ2) 2J CP5 = -Cil (CP6) + CI2 (CP7)

Results

(53)

From Equation (5~ it is seen that the critical pressure is a func-

tion of the specific adhesive fracture energy, the material properties

of the layers, and the half-crack length. The specific adhesive frac-

ture energy is a quantity that is obtained through data obtained from

experiments on crack propagation along the interface between two dissi-

milar media. Assuming this to be a known constant, PcR/[YaJ 112 may be

studied as a function of the half-crack length and the ply properties.

This analysis may easily be altered to incorporate the effect of the

finite adhesive layer thickness, even though the results derived hith-

erto are valid only for negligible adhesive layer thicknesses. In order

to obtain this solution, the homogeneous part of the displacement solu-

tion for regions Il and I2 must be made general. This is done by assuming

the roots of the homogeneous equation for shells Il and I2 of the same

form as those for shell II (Equations 10).

In the present study, the effects of the thickness ratio of the two

layers in the cylinder and the degree of anisotropy on the critical pres-

sure are studied. Figure 2 shows the effect of the thickness ratio on the

critical pressure for an isotropic cylinder. Figures 3 to 6 show the sa~e

23

for an anisotropic cylinder. The effect of various degrees of aniso-

tropy on PcR for a cylinder with layers of the same thickness is shown

in Figure 7. An isotropic cylinder, an orthotropic cylinder and a

generally anisotropic cylinder are compared in this figure. In every

case, the variation in PcR/[yaJ1/ 2 with the non-dimensionalized half-

crack length, L/r, is studied.

The results for the isotropic cylinder, Figure 2, shows that, for a

very small half-crack length, there is a region of instability 1·1here the

crack propagates with no additional pressure. This region extends to a

value of L/r around 0. 1 for a thickness ratio of 1/4 or 4, and up to an

L/r value of about 0.2 for a thickness ratio of 2/3, l or 3/2. Due to

the sym~etry in the loading, the reversal of the thicknesses of the two

layers does not alter the results for the isotropic case. The initial

unstable region is followed by a stable region where additional pressure

is required to propagate the delamination. This stable region extends

from about L/r=0.2 to near L/r=0.35 for a thickness ratio of 2/3, 1 or

3/2. For a thickness ratio of 1/4 or 4, this region extends roughly

between L/r=O.l and L/r~O. 16. In the case of the cylinder with thick-

ness ratios of 2/3, l or 3/2, the stable region is followed by one where

the crack tends to propagate at a constant pressure. However, for

thickness ratios of 1/4 or 4, there is another unstable region around

0. 16 to 0.27, followed by a stable region, around 0.27 to 0.45, before

the crack starts to propagate at a constant pressure. This interesting

behaviour calls for experimental observations regarding the physical

cause for the two unstable regions. It should be noted that the pressure

at which the crack eventually tends to propagate in a quasi-static

24

900

700

500

0 h, = h2 = 0 05

o{ h1= .04; h2 = .06 h I= .06 ; h2 = .04

{ h,=.02; h2=.08 6 h,=.08;h2=.02

E = 30 x 106 psi

l/= 0.3

300 .__ __ ___,_ ____ _.__ __ __..__ ___ L ____ L __ __,

0 0.1 0.2 0.3 0.4 0.5

L/r

FIGURE 2. EFFECT OF h1/h2 on PcR FOR AN ISOTROPIC CYLINDER

0.6

25

manner depends on the thickness ratio of the two layers in the cylinder.

The trend in these results is comprehensible for a physical problem.

Layers of equal thickness are more difficult to peel than those of une-

qual thicknesses; the greater the inequality the easier it is to separate

the layers.

The effect of the thickness ratio on the critical delamination pres-

sure for an anisotropic cylinder is shown in Figures 3 to 6. The case with h = l 0.08 in. and h2=0.02 in., a thickness ratio of 4, is interesting. It also

has two successive regions of stable-unstable behaviour as discussed be-

fore, but the initial stable region, for about the same range of L/r values,

extends over a larger range of pressure values. Figure 7 shows the effect

of anisotropy on the critical pressure for a constant thickness ratio of l.

No inferences can be made without a knowledge of "equivalent" isotropic

1 ayers.

600

500

400

300

200

26

~---.--------,----.--·---i

LAYERS: (])BORON - EPOXY (0°) @BORON - ALUMINUM (45°)

e h 1 = 0.08 ; h2 = 0.02

100 i.__ __ ~..n..i..:=----'---~--

0 0.2 0.4 0.6

L/r 0.8 1.0

FIGURE J. EFFECT OF h1/h2 on PcR FOR AN ANISOTROPIC CYLINDER:

(1) B-E(0°); (2) B-Al(45°)

1 I

27

LAYERS 450 ( I) BORON - EPOXY ( 0 °)

( 2) GRAPHITE - EPOXY ( 45 °)

350

250

150

50,__--~_.._--~___. ____ ~-'--~----'-------..____,

o.o 0.2 0.4 0.6 0.8 1.0

L/r

FIGURE 4. EFFECT OF h1/h2 on PcR FOR AN ANISOTROPIC CYLINDER:

(1) B-E(0°); (2) G-E(45°)

28

500

LAYERS

( I ) BORON -ALUMINUM ( 0°) ( 2) GRAPHITE - EPOXY ( 45 °)

400

300

200

1001..-~~-'-~~--L~~~--1--~~-L-~~~-4--~~....J

0.0 0.1 0.2 0.3 0.4 0.5

L/r

FIGURE 5: EFFECT OF h1/h2 on PcR FOR AN ANISOTROPIC CYLINDER: (l) B-Al(0°); (2) G-E(45°)

cEI~

500

400

300

200

100

'IQ L.

LAYERS

(I) GRAPHITE-EPOXY (0°)

( 2) BORON- EPOXY ( 45 °)

oL--~~-'-~~---L-~~~"--~~_._~~------

o.o 0.2 0.4 0.6 0.8 1.0

L/r

FIGURE 6: EFFECT OF h1/h2 on PcR FOR AN ANISOTROPIC CYLINDER: (l) G-E(0°); (2) B-E(45°)

900

800

700

600

~1~500

I I

400 r 300

200

30

h 1 = h2 = 0.05 11

ISOTROPIC ~ E = 30 x 106 psi; ZI= 03

ANISOTROPIC L(DB-AL(0°);® G-E (45°)

ORTHOTROPIC /WB-E(0°J;(f) B-E(0°)

I

I I

I ~

I 00 L_ __ _J_ __ __,1, ____ __L_ ___ __.._ ____ -L--__ ~

0 0.1 0.2 0.3 L/r

0.4 0.5

FIGURE 7: COMPARISON AMONG PcR VALUES FOR ISOTROPIC, ORTHOTROPIC AND ANISOTROPIC CYLINDERS

0.6

CHAPTER III. DELAMINATION IN A LAYERED CYLINDER

WHEN THE DRIVING FORCE IS A UNIFORM LOADING ON THE

INNER CYLINDRICAL SURFACE

The model chosen for this problem is identical to the one analyzed

in the previous section except that the loading is now applied on the

inner cylindrical surface. This problem finds application in pressure

vessel safety design. Here the propagation of the delamination is not

easy to visualize in a physical sense. The crack growth is not due to a

Mode I behaviour as it was in the previous problem, but a combination of

Modes I and II. If this were to be analyzed according to the classical

Fracture Me~hanics concept of evaluating stress intensity factors for

the various modes of deformation, the present problem would involve a

greater degree of complexity. The use of Clapeyron's theorem to compute

the total strain energy in the cylinder, using the displacements obtained

through the solution of laminated shell equations, circumvents the

problem of dealing with the stress singularities at the crack tip.

Furthermore, it makes the present case very similar to the previous one

in methodology and no more tedious.

The cylinder with the circumferentially symmetric crack geometry

and the uniform loading is shown in Figure 8. Thin shell assumptions

are made to simplify the shell equations through Donnell's approxima-

tions. The cylinder is again divided into three shell elements that are

joined together at the crack tip. The resulting shell equations are the

same as those for the cylinder with pressure acting on the crack surface,

except for the inhomogeneity introduced by the loading. The uniform load-

ing, q0 , yields particular solutions to regions I2 and II, while the homo-

31

32

z

r

FIGURE 8: CYLINDER GEOMETRY AND LOADING: UNIFORM LOADING ON INNER CYLINDRICAL SURFACE

x

33

geneous solutions remain unchanged in the three regions.

\~Il = 0 p for shell Il (54a)

wI2 = q/CI2 p for shell 12 (54b)

WII = /CI I p qo for she 11 II (54c)

(55)

With the above changes in the particular solutions, the complete or

general solutions to the three shell equations with modified inhomo-

geneity may be written as:

wI 1 = eax(c l cos ax - c2 sin ) -ax( ax + e c5 cos ax - c6 sin ax)

(56a) vi I 2 = ebx(c9 cos bx - c10 sin ) -bx( bx + e c13 cos bx - c14 sin bx)

+ q0 /CI2 (56b)

\./II "' "' ex( cos dx - sin dx) + -ex( cos dx - sin dx) = e cl7 c18 e c21 c22

+ q/CII (56c)

The boundary and matching conditions used to solve for the twelve real

constants are the same as before:

At x=O, Il 12 = Nil NI2 0 w,x = w,x = = xz xz

At x->-oo' WI I is finite

At x=L, 11 12 I I 0 ( 57) 'ti' x = w,x = w 'x =

At x=L, wil = \.JI 2 = vJ I I

At x=L, Mll + MI2 . I I = M x x x

34

Imposing the above boundary conditions, the radial displacement in the

three shells are obtained as functions of the ply properties, the applied

loading, the half-crack length and the axial coordinate along the cylin-

der.

w11 = 2c1[cos ax cosh ax ~sin ax sinh ax] (58a)

w12 = 2c9[cos bx cosh bx - ~~~~~ sin bx sinh bx] + q0 /CI2 (58b)

-w11 = c21 e-cx[cos cfX + (c/d) sin ax]+ q0 /CII (58c)

where c21 /q0 = (RKP5/CII + RKP6/CI2)/RKP4 (59)

c1/q0 = (c 21 ;q0 + l/CII)/RJl (60)

c I q = ( c 1 I q + 1 IC I I - 1 IC I 2 ) I RJ 2 9 0 2 0 ( 61 )

In the above equations RJl, RJ2, RJ3, RJ4, RKPl, RKP2, RKP3 and RKP4

are the quantities defined in the previous chapter. RKP5 and RKP6 are

redefined here as:

RKP5 = Bi~/r - RKPl/RJl RKP2/RJ2

RKP6 = RKP2/RJ2 - s;~/r = RJ4/RJ2

(62)

(63)

With the above expressions for the radial displacements in the three

regions of the cylinder, the total strain energy U may be obtained by

using Clapeyron's theorem:

L oo

1 "" 1 f I2 1 f II ~ U = ~ w (-q0 )2nrdx + ~ W (-q 0 )2urdx

u

0 L

= L I2 J ~o dx + J

00

0 0

II vJ --dx qo

or,

(64)

3 r-.J

The above expression for the total strain energy in the cylinder, when

differentiated with respect to the half-crack length, L, gives a mea-

sure of the rate of change of strain energy with a change in the crack

length. The Griffith energy balance criterion relates this change in

the strain energy with a growth in the delamination to the adhesive

fracture surface energy Ya.

(65)

On carrying out the tedious differentiation of U and substituting it

into the Griffith criterion, the critical loading q , at which the OCR

delamination tends to propagate, is obtained.

q [2(c9/q ){2B(E2-F2)-4AEF}/(E+F) 2 -OCR o

{b(E+F)} - 2c(DC21)/(c2±d2)J 112 = (2Y )1/2 a

where DC21 = [CP6/CI2 - (CP5)c 21 /o 0 - CP5/CII]/RKP4

DC9 = [DC21 - (CP2)c9 /q 0 ]/RJ2

(66)

( 67)

(68)

Whereas CPl, CP2, CP3 and CP4 were defined in Chapter 2, CP5 and CP6 are

redefined as:

CP6 = [CP4(RJ2)

CP5 = [CP3(RJl)

(RJ4)CP2]/(RJ2) 2

(RJ3)CPl]/(RJl) 2 + CP6

(69)

(70)

The critical loading is a function of the half-crack length, the material

properties of the layers, and the adhesive fracture surface energy. The

value of Ya for any combination of dissimilar media is obtained experi-

mentally. If this is assumed to be a known quantity, q /IY may be oCR a

studied as a function of the cylinder geometry, the material properties

of the layers and the half-crack length.

36

The various simplifications introduced into the analysis of this

problem, and in Chapter 2, have led to closed-form solutions for the

critical loading at which delamination tends to propagate. If a more

realistic model is desired, such closed-form solutions do not seem

plausible. A numerical technique, with the accompanying round-off

errors, may have to be resorted to in such cases. This justifies the

choice of the present model whose closed-form solution would form a good

basis for a more refined model.

Results

The effects of the half-crack length, the cylinder geometry and

various combinations of layers on the critical value of the internal

loading are studied for cylinders with varying degrees of anisotrooy.

q ;;y- is plotted as a function of L/r in each case. Figure 9 shows OCR a

the effect of the thickness ratio of the two layers on the critical

loading for an isotropic cylinder. Unlike the results that were ob-

tained for the cylinder with a pressure acting on the crack surface

(Figure 2), the results obtained in the present problem for a particular

thickness ratio are not the same when the reciprocal of this thickness

ratio is considered. This is due to the fact that the loading in the

present case acts directly on shells I2 and II along the inner cylindrical

surface. This makes the thickness ratio h1;h2 (Figure 8) crucial in

determining the stability or instability in the propagation of the

delamination. For a thickness ratio of 1/4 and 2/3, the curve shows an

initial unstable crack growth region followed by a stable region before

the crack tends to propagate at a constant value of the applied loading.

37

V=0.3

850 E = 30 x 106 psi

0 hi= 0.06; h2 = 0.04

I 0 hi;: 0.05; h2 = 0.05 I I

750 ~ <> hi= 004; h2= 006 -I I CJ h1= 002; h2= 0.08 !

i !

I

650 ~ I

! i

JI~ I I

I 550 r

i i I

450 f j [

350

.._ ____ L _______ J _ - __ L.

03 I

04

I . •. J ------ -

0

FIGURE 9:

0.1 0.2 0.5 0.6

L/r

EFFECT OF h1/h 2 on q0 CR FOR AN ISOTROPIC

CYLINDER

38

When the thicknesses of the two layers are equal, the initial unstable

region is again followed by a stable region. But, this stable region is

succeeded by one where the crack tends to extend at a constant loading

value and the second unstable region is not present. The behaviours

mentioned above were seen in the previous problem too.

An interesting phenomenon takes place if the thickness ratio is

made larger than one. Two such cases, with h1!h2 = 1.5 and 4.0, are

studied. For a thickness ratio of 1.5, the initial region of crack

growth is seen to be stable. The slope of the curve in this region is a

measure of the stability. The larger the slope, a greater increase in

the applied loading is required to extend the crack by the same amount,

and hence greater stability. The stability is seen to be proportional

to the thickness ratio; the greater the ratio, the greater the stability.

This initial stable region is followed by an unstable region and a

second stable region, terminating in a region where the crack extends at

a constant value of the applied loading. The constant value of the

critical loading in the final region is seen to be dependent on the

thickness ratio h1!h2; the greater the ratio, the larger the value of

this loading.

The behaviour of the cylinder for a thickness ratio of 4 was seen

to be similar to that of a cylinder with a thickness ratio of 1.5. The

stability of this cylinder in the initial range of L/r values is so

great that very large changes in the loading are required to cause a

small extension of the crack. If this behaviour were to be shown in

Figure 9 the other curves would be moved too close to one another to

get a good understanding of the behaviour they predict. Also, within a

39

small range of values of L/r, somewhere between 0. 14 and 0.2, a very

interesting phenomenon occurs. While the critical loads just outside

this range are very high, the values within this range of L/r values

become imaginary. This predicts a state where the crack is arrested

and no real value of the loading would cause it to propagate. Though

intuitively questionable, this behaviour is an outcome of the type of

model chosen, the thickness ratio of the two layers, and the type of

loading. One possible explanation for this 'crack arrest' could be that,

due to the outer layer being thin compared to the inner layer, a state

of compression may be realized by the outer layer tending to close the

crack.

The effect of the thickness ratio on the critical loading for an

anisotropic cylinder is shown in Figures 10 to 13. For low values of the

crack length, the crack growth is highly destabilizing for a thickness

ratio of 1/4, slightly destabilizing for a ratio of 1, and highly stabi-

lizing for a ratio of 4. The other regions follow the pattern described

earlier. Figure 14compares the responses of isotropic, orthotropic and

anisotropic cylinders. Unlike the examples cited hitherto, the aniso-

tropic cylinders chosen for the results shown in Figures 11 to 13 have

an initial stable region for a thickness ratio of one. These cylinders

also have a 11 crack arrest" region as explained before. If the initial

delamination in the cylinder is of a length corresponding to the L/r

values in the "crack arrest" region, the delamination will not propagate

for any increase in the applied loading. A second explanation for this

behaviour, apart from the crack-closing compressive region speculated

earlier, may be ventured here. Depending on the fiber orientation in

700

600

500

400

300

0

FIGURE 10:

40

+ 1696

0.1 0.2

LAYERS Q) BORON - ALUMINUM (0°) Cf) GRAPHITE - EPOXY (45°)

0.3

L/r

04 05

EFFECT OF h1!h2 on q FOR AN ANISOTROPIC 0 cR

CYLINDER: (1) B-Al(0°); (2) G-E(45°)

0.6

41

I I l..c.i ,.I

icRACK I ARREST LAYERS

550 : REGION

I

(I) BORON -EPOXY ( 0°)

(2) GRAPHITE - EPOXY (45°) I I

450

350

250

150

50'--~~--'-~~---''--~~--'-~~--'-~~~_._~~~

0.0 0.2 0.4 0.6 0.8 1.0

L/r

FIGURE 11: EFFECT OF h1/h2 on q0 FOR AN ANISOTROPIC CR

CYLINDER: (1) B-E(0°); (2) G-E(45°)

900

800

100

600

JI~ 500

400

300

200

100

0.0

FIGURE 12:

CRACK ARREST REGION

0.2

42

LAYERS (I) BORON -EPOXY (0°) (2) BORON -ALUMINUM ( 45 °)

0.4 0.6 0.8 t.O

L/r

EFFECT OF h1/h2 on q FOR AN ANISOTROPIC 0 cR

CYLINDER: (1) B-E(0°); (2) B-Al (45°)

500

400

300

200

100

o.o

FIGURE 13:

CRACK. ARREST REGION

0.2

43

LAYERS

(I) GRAPHITE - EPOXY (0°)

(2) BORON- EPOXY (45°)

0.4 0.6 0.8

L/r

1.0

EFFECT OF h1/h2 on q FOR AN ANiSOTROPIC 0cR

CYLINDER: (l) G-E(0°); (2) B-E(45°)

44

900 ~--~----,-----

800

700

600

Jl~o 500

400

300

200

ANISOTROPIC [(J)B-AI (0°l;® G-E {45°)

ORTHOTROPI C f Q)B- E (0°) ;Cf) B-E (0°)

I 00 L__ __ __.L ___ L.._ __ __L_ ___ _,___ __ __.__ __ ~

0 0.1 02 03 L/r

0.4 05

FIGURE 14: COMPARISON AMONG q VALUES FOR ISOTROPIC, OCR

ORTHOTROPIC AND ANISOTROPIC CYLINDERS

06

4,-.)

each uni-directional ply used in the cylinder, the crack may propagate

either along the interface of the two plies or across the plies. If a

proper choice of the two layers is made, "inter-laminar cracking" or

delamination would occur as assumed in the model. Otherwise "thru-

cracking" could manifest itself through the region hitherto called the

"crack arrest" region. This calls for a closer collaboration between

the analyst and the experimentalist so that the exact behaviour may be

properly understood.

CHAPTER IV. EFFECTS OF THE CRACK TIP GEOMETRY

The region immediately around the cr·ack tip is plastic and the

geometry of the crack here is difficult to define. This results in an

approximation in the elastic analysis when the crack tip geometry has to

be specified to solve the problem. In the problems considered in the

last two sections a sharp, flat crack tip was assumed. The three shell

elements in the model were constrained to have a zero-slope condition at

the crack tip. While this assumption made possible a closed-form solu-

tion, its validity remains to be established. The zero-slope conditions

at the crack tip may be replaced by non-zero rotations of the three

shell elements in a compatible manner. The slopes of any two shells are

specified to be constant multiples of the slope of the remaining shell,

and the effect of the variation in these constants on the critical

loading is studied. In this case an extra matching condition - the

continuity of the normal shear force resultant across the crack tip -

has to be imposed. The six conditions at the crack tip may then be

written as:

At x=L, Il 12 II w = w = w

At x=L, I 1 s WI I w = ,x 1 ,x At x=L, 12 s WII ( 71 ) \ti = ,x 2 'x At x=L, MI l + MI2 = MI I

x x x At x=L, N Il

xz + NI2 = NII

xz xz

where B1 and s2 are non-zero constants.

The generalization of the matching conditions complicates the radial

displacement solutions to such an extent that recourse to a numeri-

46

47

cal procedure seems to be the easiest path to be taken. Closed-form

solutions for the radial displacements, in terms of the axial coordinate

x, the half-crack length L and the material properties of the cylinder,

are not easily obtainable and involve the solution of six simultaneous

equations. These six equations yield the six constants in the displace-

ment expressions for the three regions. Once these are obtained, the

total strain energy U in the cylinder is obtained through Claypeyron's

theorem. This integral representation for U has to be differentiated

with respect to the half-crack length L in order that the Griffith

energy balance criterion may be used to determine the critical loading

condition. Even if one were enthusiastic enough to solve the six

simultaneous equations in closed-form, the tedious differentiation

involved in obtaining aU/aL justifies the use of a numerical procedure,

and, indeed, makes it advi sab 1 e.

Cylinder with a Pressure on the Crack Surface

It was seen in Chapter 2 that the application of symmetry and

finiteness conditions, at x=O and x->-<», reduces the displacement ex-

pressions to the form: Il 2c1cos co sh 2c2sin ax sinh ax - p/C!l

~ vJ = ax ax

!2 2c 9cos co sh bx 2c10s in bx sinh bx + p/CI2 ( 72) w = bx ~· II -cX( c22sin dx) w = e c21 cos dx

The six matching conditions at the crack tip may be imposed on the above

displacement expressions to yield the following equations respectively:

2c1G - 2c2J - c21 = p/Cil

2cqA - 2c10B - c21 = -p/CI2

(73)

(74)

48

81 (H-I)c1 - (H+I)c2 + 2a (cc21 + dc 22 ) = 0 (75)

82 (E-F)c9 - (E+F)c10 + 2b (cc 21 + dc22 ) = 0 (76}

(XXA)c1 + (XXB)c2 + (XXC)c9 + (XXD)c 10 - (XXF)c 21 -

(XXG)c22 = -(XXE}p (77}

(YYA)c1 + (YYB)c2 + (YYC)c9 + (YYD)c 10 - (YYE)c 21 -

(YYF)c22 = 0 (78)

11 2 11 where XXA = 2G B12;r + 4a J 011 11 2 11 XXB = -2J B12;r + 4a G 011

xxc = 2A Bi~/r + 4b2B oii

XXD = -2B Bi~/r + 4b2A oii

I 1 11 I2 I2 XXE = rB 12;A22 - rB12/A22

XXF = Bi~/r - (c 2 -d 2 )oi~ I I XXG = -2cd 011

YYA = 2a(H-I)Bi~/r + 4a 3 (H+I)oi~

YYB = -2a(H+I)Bi~/r + 4a3(H-I)o 11 Il

YYC = 2b(E-F)Bi~/r + 4b3(E+F)oii

YYD = -2b(E+F)Bi~/r + 4b 3 (E-F)oi~ II 2 2 II YYE = -cB12;r - c(3d -c )o11 II 2 2 II YYF = -dB12/r + d(3c -d )o,,

(79)

The six simultaneous equations are written in matrix form with the un-

knowns as c1/p, c2/p, c9/p, c10;p, c21 ;p and c22;p. The terms in the

coefficient matrix are functions of the material properties, cylinder

49

geometry and the half-crack length. The inhomogeneous terms are func-

tions of the cylinder geometry and the material properties.

Once the constants in the radial displacement expressions are found,

the total strain energy in the cylinder may be found as explained in

Chapter 2:

u 2 211rp

L

=f 0

Il I2 (~ - ~) dx p p (80)

When this expression for U is differentiated with respect to L, one ob-

ta ins:

Since (c1/p) etc. have been obtained numerically, their derivatives with

respect to the half-crack length Lare also obtained numerically. After

the constants are evaluated for a particular value of L, L is changed to

L + 6L and the new values of the constants obtained. The difference be-

tween the new value and the previous value of each constant, divided by

6L, gives the derivative of that constant with respect to L. 6L is

chosen to be very small to give a fairly good approximation for the

values of the derivatives sought. - 2 The numerical value of (au;aL)/(2nrp ) in the above expression is

substituted into the Griffith energy balance criterion to evaluate the

critical value of the pressure, pCR' at which propagation of the dela-

mination is imminent.

50

The effect of the crack tip geometry is studied for four different

sets of s1, s2 values. These are chosen in such a manner so as to in-

clude crack geometries ranging from a flat, sharp crack to a well-rounded

crack tip. The results are obtained for the layered cylinders consi-

dered in Chapter 2, and a comparison is made between the two sets of

critical pressure values. It is seen that, with a reasonable degree of

accuracy, the results for the different sets of s1, s2 values are the

same for all the cases. This could be true only if the slopes of the

three shell elements at the crack tip are indeed very close to zero,

justifying the assumptions made in Chapter 2. Tables l to 8 show the

results obtained for a few cylinders where this observation may be made.

An added advantage in the present formulation, compared to the one

in Chapter 2, is that the continuity of normal shear across the crack tip

is also enforced. When the slopes of the three shell elements are speci-

fied to be zero identically, the shear continuity is not enforced. This

has a drastic effect on the prediction of the delamination behaviour of

certain cylinders (Figures 15 to 18). The results for cylinders with a

thickness ratio of 1/4 or l show good agreement with those obtained in

Chapter 2 (Tables 1 to 8). But, when a ratio of 4 is considered, a

marked difference in the delamination behaviour of the cylinder from the

predicted behaviour of Chapter 2 is observed. This is predominant in

the region where the half crack length is small (Figures 15 to 18). While

the earlier formulation resulted in two stable regions for such cylinders,

including one in the initial region, the present analysis yields an ini-

tial unstable region, a stable region and a constant pressure crack growth

region. The initial stable region obtained in Chapter 2 is transformed,

51

Table 1

Effect of B1 and B2 on PcR/(ya) 112:(l)B-E(0°);(2)G-E(45°);h1Jh2=1

PcR/(ya)l/2 for Comparison L/r

s1= 0.01 Bl= 0. 10 s1= 0.50 s1= 0.20 (Chapter II) s2=-0. 01 s2=-0.50 B2=-0.50 B2=-0.80

0.067 759.9 771 .0 776. 2 779. 7 776.0

0 .100 360.4 369. 1 374.6 376.9 369.4

0 .133 232.5 240.8 245 .1 247.9 249.3

0 .167 182.3 189. 9 193. 1 196. 7 205.7

0.200 160.3 166. 9 168.8 172 .8 182.2

0.233 149.2 153.9 154. 9 158.3 163.3

0.267 143.2 146. 1 146.5 148.8 149.2

0.300 140 .1 141 .4 141 . 5 142. 7 141 . 0

0.333 138. 9 139.2 139. 1 139.6 137.4

0.367 139 .0 138.7 138.6 138.6 136. 8

0.400 139.7 139. 3 139. 1 138.9 137.7

0.433 140.7 140.2 140. l 139.8 139. l

0.467 141. 6 141 . 2 141 . l 140.8 140.5

0.500 142. 3 141 . 9 141 . 9 141 . 7 141 . 6

0.533 142.7 142.5 142.5 142.3 142.4

0.567 143.0 142.9 142.8 142.7 142.9

0.600 143. l 143.0 143.0 143.0 143. l

0.667 143. l 143. l 143 .0 143.0 143.2

0.833 142.9 142.9 142.9 142.9 143.0

l .000 142.9 142.9 142.9 142.8 142.9

52

Table 2

Effect of s1 and 82 on PcR/(ya) 112:(l)B-E(0°);(2)G-E(450);h,/h2=1/4

1/2 Pc RI (Ya) for Comparison

L/r s1= 0.01 81= 0.10 s1= o.so s1= 0.20 (Chapter I I) 82=-0.0l f32=-0.50 s2=-0.50 s2=-0.80

0.067 461.0 460.6 460.5 460.1 463.0

0.100 215.9 215.6 215.5 215 .5 216. 7

. 0 .133 137.7 137.5 137.4 137 .4 138. l

0.167 108. 9 l 08.8 108.7 l 08. 7 109. 1

0.200 99.5 99.4 99.4 99.4 99.6

0.233 98.4 98.3 98.3 98.3 98.3

0.267 100.0 100.0 l 00.0 100.0 100.0

0.300 102 .0 102. 0 l 02.0 l 02.0 101. 9

0.333 103 .3 l 03 .3 103.3 l 03 .3 l 03 .2

0.367 l 03. 9 103.9 l 03. 9 103. 9 103.8

0.400 104.0 104.0 104.0 104.0 l 04 .o 0.433 l 04 .o l 04.0 104.0 l 04. 0 l 04 .0

0.467 103.9 103. 9 103.9 103. 9 103. 9

0.500 103.8 l 03.8 103.8 l 03.8 l 03.8

0.533 l 03 .8 l 03.8 l 03.8 l 03.8 103.8

0.567 l 03.8 l 03.8 l 03 .8 l 03.8 l 03.8

0.600 103.8 103.8 103 .8 103 .8 103.8

0.667 l 03.8 103.8 103 .8 103.8 103 .8

0.833 103.8 103.8 103 .8 103.8 103 .8

l. 000 103 .8 103.8 l 03.8 103.8 l 03.8

53

Table 3

Effect of s1 and s2 on Pcr/(ya) 112:(l)B-Al(00);(2)G-E(45°);h1/h2=1

1/2 PcR/(ya) for Comparison L/r

s1= 0.01 s1=0.10 s1= o.50 s1= 0.20 (Chapter II) s1=-o.01 s1=-o.5o s1=-0.50 s1=-0.80

0.067 768.8 771 .5 773.3 774.2 772.6

0. 100 372.2 374.4 375.7 376.2 374.1

0 .133 253 .1 254.8 255.8 256.4 255.0

0.167 215.9 217.0 217 .6 218 .1 217 .5

0.200 208.9 209.4 209.6 209.9 209.5

0.233 212. 1 212 .1 212 .1 212. 1 211 .8

0.267 216.5 216.3 216.2 216 .1 216.0

0.300 219.3 219.2 219 .1 219.0 219.0

0.333 220.5 220.4 220.3 220.3 220.3

0.367 220.7 220.7 220.6 220.6 220.7

0.400 220.6 220.6 220.6 220.6 220.6

0.433 220.4 220.4 220.4 220.4 220.4

0.467 220.3 220.3 220.3 220.3 220.3

0.500 220.3 220.3 220.3 220.3 220.3

0.533 220.3 220.3 220.3 220.3 220.3

0.567 220.3 220.3 220.3 220.2 220.3

0.600 220.3 220.3 220.3 220.3 220.3

0.667 220.3 220.3 220.3 220.3 220.3

0.833 220.3 220.3 220.3 220.3 220.3

1.000 220.3 220.3 220.3 220.3 220.3

54

Table 4

Effect of s1 and s2 on PcR/(ya)l/ 2:(l)B-Al(0°);(2)G-E(45°);h1/h2=1/4


s1= 0.01 s1= 0.10 s1= o.5o s1= 0.20 (Chapter II) s2=-0.0l 62=-0.50 s2=-0.50 s2=-0.80

0.067 507.4 502.7 500.3 499.7 514.7

0.100 280.1 276.3 273.8 273.7 295.l

0 .133 225.6 223.0 220.7 221 .2 247.3

0 .167 213.2 211 .8 210.2 210.9 228.2

0.200 210.2 209.8 209 .1 209.5 213.8

0.233 209.7 209.8 209.7 209.8 207.4

0.267 210.3 210.5 210.8 210.7 207.1

0.300 211.4 211 .6 211. 9 211 .8 209. 1

0.333 212.5 212.6 212 .8 212.7 211.2

0.367 213 .1 213.2 213.3 213. 3 212.6

0.400 213.4 213. 5 213.5 213.5 213.4

0.433 213.5 213.5 213.6 213.6 213.6

0.467 213.5 213.5 213.5 213.5 213.6

0.500 213.4 213.4 213.5 213.5 213.5

0.533 213.4 213.4 213.4 213.4 213.4

0.567 213.3 213.3 213. 3 213.3 213.3

0.600 213.3 213. 3 213. 3 213.3 213.3

0.667 213.3 213.3 213.3 213.3 213.3

0.833 213.3 213. 3 213.3 213.3 213.3

l .000 213.3 213.3 213.3 213.3 213.3

55

Table 5

Effect of s1 and B2 on PcR/(ya) 112:(l)G-E(OD);(2)B-E(45D);h1/h2=l


f31= 0.01 f31= 0.10 f31= 0.50 f31= 0.20 (Chapter II) f32=-0. 01 32=-0.50 62=-0.50 62=-0.80

0.067 865.5 885.2 894.5 901 .6 870.2

0. l 00 398.9 414. 1 422.4 426.5 413.6

0.133 242.4 255.7 261 .9 266.2 269.4

0 .167 173.9 185.0 189. l 194 .o 207.3

0.200 137. 9 146.8 149. l 153. 9 166. l

0.233 115. 7 122. 2 123. 4 127. 3 134.3

0.267 101 . 6 106.0 106. 5 109. 3 112 .0

0.300 92.4 95. 1 95.3 97.2 97.5

0.333 86.6 88 .1 88.2 89.3 88.7

0.367 83.3 84.0 84.0 84.6 83.8

0.400 81. 7 82.0 81.9 82.2 81.4

0.433 81.3 81.3 81 .2 81. 2 80.6

0.467 81.6 81'.4 81.3 81.2 80.7

0.500 82.2 81. 9 81.9 81. 7 81.4

0.533 82.9 82.7 82.6 82.5 82.3

0.567 83.6 83.4 83.4 83.2 83. 1

0.600 84.2 84.0 84.0 83.9 83.9

0.667 85.0 84.9 84.9 84.8 84.9

0.833 85.3 85.3 85.3 85.3 85.3

l. 000 85.2 85.2 85 .1 85.2 85.2

56

Table 6

Effect of s1 and s2 on PcR/{ya) 112:(1)G-E(0°);{2)B-E(45°);h1/h2=1/4

1/2 PcR/{ya) for Comparison L/r

s1= 0.01 81= o. 10 81= o.50 81= 0.20 (Cha pt er I I) 82=-0.0l 82=-0.50 82=-0.50 82=-0.80

0.067 369.0 367.8 367.8 367.8 369.4

o. 100 167.4 167. 5 167.7 167.7 168. 1

o. 133 100. 1 100. 1 100.2 100.2 100.4

o. 167 71. 8 71. 8 71.9 71. 9 71. 9

0.200 59. 1 59. 1 59.2 59.2 59.2

0.233 53.8 53.8 53.8 53.9 53.9

0.267 52.4 52.4 52.4 52.4 52.4

0.300 52.6 52.6 52.6 52.6 52.6

0.333 53.5 53.5 53.5 53.4 53.4

0.367 54.3 54.3 54.3 54.3 54.3

0.400 54.9 54.9 54.9 54.9 54.9

0.433 55.3 55.3 55.3 55.3 55.3

0.467 55.4 55. 4· 55.4 55.4 55.4

0.500 55.4 55.4 55.4 55.4 55.4

0.533 55.4 55.4 55.4 55.4 55.4

0.567 55.4 55.4 55.4 55.4 55.4

0.600 55.3 55.3 55.3 55.3 55.4

0.667 55.3 55.3 55.3 55.3 55.3

0.833 55.3 55.3 55.3 55.3 55.3

1. 000 55.3 55.3 55.3 55.3 55.3

57

Table 7

Effect of s1 and s2 on PcR/(ya) 112:(l)B-E(0°);(2)B-Al(45°);h1/h2=1

1/2 PcR/(ya) for Comparison L/r

s1= o. 01 s1= o. io s1= o.50 s1= 0.20 (Chapter II) s2=-0. 01 82=-0.50 82=-0.50 82=-0.80

0.067 1267.4 1288. l 1301. 7 1302. 8 . 1277. 8

o. 100 583.6 600.7 610.0 613.6 604.7

o. 133 355.9 370. l 376.3 380.6 392. 5

o. 167 257.4 268.5 272. 5 277. 0 301. 1

0.200 208.5 216.5 218.6 222.6 241. 0

0.233 181. 7 186.7 187. 5 190.4 199. 1

0.267 167.5 170. 1 170.4 172. 1 174.2

o. 300 160.4 161. 5 161. 5 162.3 161. 5

0.333 157.7 157. 9 157.8 158. l 156.4

0.367 157. 7 157.5 157.3 157.3 155.6

0.400 159. 0 158.6 158.5 158. 3 157.0

0.433 160.7 160.2 160.2 159. 9 159. 1

0.467 162.3 161. 9 161. 8 161. 6 161. 2

0.500 163.5 163.2 163.2 163.0 162.8

0.533 164.3 164. 1 164. 1 164.0 164.0

0.567 164. 8 164.7 164.6 164.6 164. 7

0.600 165.0 164.9 164.9 164.9 165.0

0.667 165. 0 165.0 165. 0 165.0 165. 1

0.833 164.7 164. 7 164.7 164. 7 164.7

1. 000 164.7 164.7 164. 6 164.6 164. 7

58

Table 8

Effect of 81 and 82 on PcR/(ya) 1/ 2:(l)B-E(0°);(2)B-Al(45°);h1/h2=1/4

1/2 PcR/(y a) for Comparison L/r

81= 0.01 81= o. 10 s1= o.5o 81= 0.20 (Chapter II) 82=-0. 01 82=-0.50 82=-0.50 82=-0.80

0.067 475.0 474.9 475.0 474.7 477.2

0.100 222.8 222.7 222.7 222.6 223.6

o. 133 , 42. 4 142. 4 142.3 142.3 142.8

o. 167 , , 3. 0 113.0 112. 9 112. 9 113. 2

0.200 103.6 103. 6 103.6 103. 5 103.7

0.233 102.6 102. 6 102.6 102.6 102.6

0.267 104. 5 104.5 104. 5 104.5 104.4

0.300 106. 5 , 06. 5 106.5 106.5 106.4

0.333 107.8 107.8 107.8 107.8 107.8

0. 367 , 08. 4 108. 4 108.4 108.4 108.4

0.400 108.5 108.5 108.5 108.5 108.5

0.433 108.5 108.5 108.5 108.5 108.5

0.467 108. 4 108.4 , 08. 4 108.4 108.4

0.500 108.3 108.3 108.3 108. 3 108.3

0.533 108.3 108.3 108.3 108.3 108.3

0.567 108.3 108. 3 108.3 108.3 108.3

0.600 , 08. 3 108. 3 108.3 108. 3 108.3

0.667 108.3 108.3 108.3 108.3 108.3

0.833 , 08. 3 108. 3 108.3 108.3 108.3

1. 000 108.3 , 08. 3 108.3 108.3 108. 3

59

500

LAYERS

( I ) BORON - EPOXY ( 0° ) (2) BORON-ALUMINUM (45°)

h1 I h2 = 4

400 0 /31 = 0.1 !32 = - 0.5 !:>. /31 = /32 = 0

300

200

100--~~--~~--'~~~~~~-'-~~~..l.--I

o.o 0.2 0.4 0.6

L/r

0.8 1.0

FIGURE 15: EFFECT OF ~, B2 AND CONTINUITY IN Nxz ON PcR:

(l) B-E(0°); (2) B-Al(45°); h1/h 2=4

60

350

250

150

LAYERS

(I) GRAPHITE-EPOXY ( 0°)

(2) BORON- EPOXY ( 45°)

hi /h2 = 4

0 f31 = 0.2 I f32 =-Q.8 6 !31 = !32 = 0

50L-...~~-1-~~--'-~~~l--~~~~~-'----

o.o 0.2 0.4 0.6

L/r

0.8 1.0

FIGURE 16: EFFECT OF '1' S2 AND CONTI NU ITV IN Nxz ON PcR:

(1) G-E(0°); (2) B-E(45°); h1/h2=4

61

250

200

150

LAYERS

(I) BORON-ALUMINUM (0°)

( 2) GRAPHITE- EPOXY ( 45°)

h 1 I h 2 = 4

0 /31 = 0.1 i /32 = -0.5 A /31 = /32 = 0

100L-~~...L..~~-L~~~L--~~-1-~~-'----'

o.o 0.2 0.4 0.6

L/r

0.8 1.0

FIGURE 17: EFFECT OF s,' B2 AND CONTINUITY IN Nxz ON PcR:

(l) B-Al(0°); (2) G-E(45°); h1/h2=4

62

250

200

150

LAYERS

( I) BORON - EPOXY ( 0°) ( 2) GRAPHITE - EPOXY ( 45°)

h, /h2= 4

0 f3, = 0.01 i f32 = - 0.01

A /31 = /32 = 0

IOOL-~~-L-~~..._J,..~~~i--~~-"-~~-----

0.0 0.2 0.4 o.G 0.8 1.0

L/r

FIGURE 18: EFFECT OF ~' B2 AND CONTINUITY IN Nxz ON PcR:

(1) B-E(0°); (2) G-E(45°); h1/h2=4

63

through the inclusion of shear.continuity and arbitrary slopes at the

crack tip, to an unstable region of delamination. In the case of cy-

1 inders with a thickness ratio of one or less, the inclusion of shear

continuity does not seem to alter the delamination behaviour, and the

critical pressure values for various crack lengths are comparable.

The effect of the inclusion of shear force continuity across the crack

tip is of a larger magnitude in the case of a cylinder loaded on the

inner surface.

Loading on the Inner Cylindrical Surface

The symmetry conditions at x=O and the finiteness condition at

x~ reduce the radial displacement expressions for this problem to

the form shown in Chapter 3:

wil = 2c1cos ax cosh ax - 2c2sin ax sinh ax

wI 2 = 2c9cos bx cosh bx - 2c10sin bx sinh bx + q0 /CI2 (82)

wII = e-cx(c21 cos dx - c22sin dx) + q0/CII

Imposing the six matching conditions with the above expressions for the

shell displacements, one obtains:

2c1G - 2c2J - c21 = q0/CII

2c9A - 2c10B - c21 = q0 /CII - q0 /CI2 S1 (H-I)c1 - (H+I)c2 + 2a (cc21 + dc22 ) = 0

S2 (E-F)c9 - (E+F)c10 + 2b (cc21 + dc22 ) = 0

(XXA)c1 + (XXB)c2 + (XXC)c9 + (XXD)c10 - (XXF)c21 -

(XXG)c22 = (XXE)q0

(YYA)c1 + (YYB)c2 + (YYC)c9 + (YYD)c10 - (YYE)c21 -

(YYF)c22 = 0

I2 I2 II II where XXE = rB12!A22 - rB12!A22

(83)

(84)

(85)

(86)

(87)

(88)

(89)

The quantities XXA, XXB, XXC, XXD, XXF, XXG, YYA, YYB, YYC, YYD, YYE and

YYF were defined earlier.

Comparing these equations with the ones obtained for the previous

problem, it is seen that the only changes are in the inhomogeneities in

the six equations. When the equations are cast in matrix form, the co-

efficient matrix is seen to be identical for both the problems. As ex-

64

plained earlier, the values for c1/q0 , etc. and their derivatives with

respect to the half-crack length, ~L (c1/q0 ) etc., may be obtained nu--merically. The total strain energy in the cylinder, U, is given by

Clapeyron's theorem (Chapter 3):

dx + (90)

-If the derivative aU/aL is found after substituting the corresponding -displacement expressions into the integral representation for U, one

obtains:

( 91 )

The critical value of the loading, q , at which delamination tends to OCR

propagate is obtained by substituting the numerical value for (aU/aL)/

(-2nrq~) into the Griffith energy balance criterion (Chapter 3).

The propagation of the delamination in the various cylinders studied

in Chapter 3 is examined through the present method of analysis. The re-

sults are presented in Tables 9 to 13 and Figures 19 to 25. As in the

previous problem, the different sets of s1 and s2 values yielded approxi-

mately the same results, reinforcing the claim that the crack tip slopes

are very close to zero. The critical values for the loading obtained in

this analysis agree well with those in Chapter 3 for cylinders with a

thickness ratio of 1/4 (Tables 9 to l~. A few cases show reasonable

65

66

Table 9

Effect of 13 1 and s2 on q0CR/(ya}l/2:(l}B-E(0°};(2}G-E(45°};h1/h2=1/4

qo /(ya}l/2 for Comparison L/r CR

s1= 0.01 s1= o. io s1= o.50 s1= 0.20 (Chapter I II} 132=-0.0l 132=-0.50 s2=-0.50 132=-0.80

0.067 280.2 281.6 281.8 282.5 290. 1

0.100 195.7 196. l 196.2 196.3 197. 1

0. 133 139. 9 139. 9 139. 8 139.8 138.7

0. 167 114. 5 114. 4 114.4 114.3 113. 1

0.200 105.7 105.6 105. 6 105.6 104.8

0.233 104.6 104.6 104.6 104. 6 104.2

0.267 106.2 106.2 106.2 106.2 106. 2

0.300 108. 1 108. l 108. l 108. 1 108.2

0.333 109.3 109.3 109.3 109.3 109.5

0.367 109.9 109.9 109. 9 109.9 110.0

0.400 11o.0 110.0 110.0 110. 0 110.0

0.433 11o.0 11o.0 11o.0 110.0 110.0

0.467 109.9 l 09. 9 l 09. 9 l 09. 9 109.9

0.500 109.8 109. 8 109.8 109. 8 109. 8

0.533 109. 8 109.8 109. 8 109.8 109.8

0.567 109.8 109. 8 109.8 109.8 109.8

0.600 109. 8 109.8 109.8 109.8 109.8

0.667 109.8 109.8 109.8 109.8 109.8

0.833 l 09. 8 109.8 109.8 109.8 109.8

1.000 109. 8 109.8 109.8 109.8 109.8

67

Table 10

Effect of e1 and e2 on q0 /{ya) 112:(l)B-Al(0°);{2)G-E(45°);h1/h2=1/4 CR

qo /{ya)l/2 for Comparison L/r CR

e1= 0.01 e1= o. 10 e1= o.50 e1= 0.20 (Chapter III) e2=-0. 01 e2=-0. 50 82=-0.50 e2=-0.80

0.067 383.8 383.0 382.5 382.5 360.9

o. 100 321.6 319.6 318.5 318. 3 280.3

o. 133 292. 9 291. l 289.8 289.9 253.8

0. 167 286.9 285.9 284.8 285.2 259. l

0.200 285.9 285.6 285. l 285.4 273.6

0.233 285.6 285.6 285.6 285.7 285.2

0.267 285.6 285.8 285.9 285.9 290. l

0.300 286. l 286.2 286.4 286.4 290.5

0.333 286.6 286.7 286.8 286.8 289. 5

0.367 287.0 287. l 287. l 287. l 288.3

0.400 287.2 287.2 287.2 287.3 287.6

0.433 287.2 287.3 287.3 287.3 287.2

0.467 287.2 287.3 287.3 287.3 287.0

0.500 287.2 287.3 287.2 287.3 287.0

0.533 287.2 287.2 287.2 287.3 287. l

0.567 287.2 287.2 287.2 287.2 287. l

0.600 287.2 287.2 287.2 287.2 287. l

0.667 287.2 287.2 287.2 287.2 287.2

0.833 287.2 287.2 287.2 287.2 287.2

l. 000 287.2 287.2 287.2 287.2 287.2

68

Table 11

Effect of s1 and s2 on q0 /{ya) 112:(l)B-Al(0°);(2)G-E(45°);h1/h2=1 CR

qo /(ya)l/2 for Comparison L/r CR

s1= 0.01 s1= o. 10 s1= o.50 s1= 0.20 (Chapter III) (32=-0.0l s2=-0.50 s2=-0.50 (32=-0.80

0.067 513.3 513. l 513. l 512.9 514.4

o. 100 494.3 494. 4 494. 5 494.5 507.7

o. 133 468.3 468.9 469. 2 469.4 498. 7

o. 167 451. 9 452.5 452.8 453.0 480.3

0.200 447.8 448.0 448. l 448.3 459.0

0.233 449.4 449.4 449.4 449.4 448.6

0.267 451. 8 451. 7 451. 7 451. 6 447.4

0.300 453.4 453.3 453.3 453.2 449.5

0.333 454. l 454.0 454.0 454.0 451.9

0.367 454.3 454.2 454.2 454.2 453.4

0.400 454.2 454.2 454.2 454.2 454.2

0.433 454. l 454. l 454. l 454. l 454.3

0.467 454.0 454.0 454.0 454.0 454.3

0.500 454.0 454.0 454.0 454.0 454.2

0.533 454.0 454.0 454.0 454.0 454. l

0.567 454.0 454.0 454.0 454.0 454.0

0.600 454.0 454.0 454.0 454.0 454.0

0.667 454.0 454.0 454.0 454.0 454.0

0.833 454.0 454.0 454.0 454.0 454.0

1. 000 454.0 454.0 454.0 454.0 454.0

69

Table 12

Effect of s1 and s2 on q /(y )112:(1)G-E(0°);(2)B-E(45°);h1/h 2=1/4 oCR a

qo /(ya)l/2 for Comparison L/r CR

s1= o. 01 s1= o. 10 s1= o.50 s1= 0.20 (Chapter III) s2=-0.0l f32=-0.50 s2=-0.50 s2=-0.80

0.067 278. 1 280.9 280.5 281.8 298. 5

o. 100 157.8 158. 3 158. 6 158. 9 164.3

o. 133 98.9 99.2 99.2 99.3 101.9

0.167 71. 9 72.0 72. 1 72. 1 73.5

0.200 59.5 59.6 59. 6 59. 6 60. 4

0.233 54.3 54.4 54.4 54.4 54.7

0.267 52.9 52.9 52.9 52.9 52.9

0.300 53. 1 53. 1 53. 1 53. 1 53. 1

0.333 54.0 54.0 54.0 54.0 53.9

0.367 54.8 54.8 54.8 54.8 54.7

0.400 55.4 55.4 55.4 55.4 55.4

0.433 55 .. 8 55.8 55.8 55.8 55.8

0.467 55.9 55.9 55.9 55.9 55.9

0.500 56.0 56.0 56.0 56.0 56.0

0.533 55.9 55.9 55.9 55.9 55.9

0.567 55.9 55.9 55.9 55.9 55.9

0.600 55.9 55.9 55.9 55.9 55.9

0.667 55.8 55.8 55.8 55.8 55.8

0.833 55.8 55.8 55.8 55.8 55.8

1. 000 55.8 55.8 55.8 55.8 55.8

70

Tab 1 e 13

Effect of a1 and s2 on q /(y )112:(l)B-E(0°);(2)B-Al(45°);h1/h2=1/4 OCR a

qo /(ya) 1/2 for Comoarison L/r CR

s1= 0.01 s1= o. 10 a1= o.50 s1= 0.20 (Chapter III) a2=-0. 01 s2=-0.50 s2=-0.50 s2=-0.80

0.067 394. 7 395. 3 395. 6 395. 7 402.0

o. 100 216.7 216.7 216.7 216.7 217.4

o. 133 143. 0 143.0 143.0 143.0 142. 2

0. 167 114. 5 114. 4 114. 4 114. 4 113. l

0.200 105.2 105. l l 05. l 105. l 104.0

0.233 104.2 104. 2 104. 2 104.2 103.7

0.267 106.0 106.0 106.0 106. 0 106.0

o. 300 108.0 108.0 108.0 108. 0 108.2

0.333 109.3 109. 3 109.3 109.3 l 09. 5

0.367 109. 9 l 09. 9 l 09. 9 l 09. 9 110. 0

0.400 110. 0 110. 0 110.0 110.0 11 o. l 0.433 110. 0 110. 0 110. 0 110.0 110.0

0.467 l 09. 9 l 09. 9 l 09. 9 109.9 l 09. 9

0.500 109.8 109. 8 109.8 l 09. 8 109.8

0.533 109. 8 l 09. 8 109.8 109.8 109. 8

0.567 l 09. 8 109. 8 l 09. 8 109. 8 109.8

0.600 109.8 109. 8 109.8 109.8 109. 8

0.667 109.8 109.8 109. 8 l 09. 8 109.8

0.833 109.8 109.8 109.8 109.8 l 09. 8

1. 000 109. 8 l 09. 8 109.8 109.8 109.8

500

400

~~ 300

200

71

CRACK ARREST REGION LAYERS

(I) GRAPHITE - EPOXY ( 0°)

( 2) BORON - EPOXY ( 45°)

h 1 /h 2 =I

0 /31 = 0.01 ; /32 = - 0.01

~ /31 = f32 = 0

IOOL_~__;t__~__J,~~~~:=::::::::L~~..-1....___J

o.o 0.2 0.4 Q.6 o.s 1.0

L/r

FIGURE 19: EFFECT OF ~· Bz AND CONTINUITY IN Nxz ON qoCR:

(1) G-E(0°); (2) B-E(45°); h1/h 2=1

900

800

700

600

Jll~ > 500

400

300

200

7" {_

LAYERS

(I) BORON-EPOXY ( 0°)

( 2) BORON - ALUMINUM ( 45 °)

h 1 /h 2 = I

0 P 1 = 0.2 i f3 2 = - o.s 6 '31 = '32 = 0

100L-~~.....L~~--L.~~~J.._~~_._~~-"--

o.o 0.2 0.4 0.6 0.8 1.0

L/r

FIGURE 20: EFFECT OF Bl' s2 AND CONTINUITY IN Nxz ON qoCR:

(1) B-E(0°); (2) B-Al(45°); h1/h2=1

;1~

500

400

300

200

I.;

73

CRACK ARREST REGION LAYERS

(I ) GRAPHITE - EPOXY ( O 0 )

( 2) BORON -EPOXY ( 45°)

h 1 I h 2 = 4

0 /31 = 0.01 i /3 2 = - 0.01

[!) /31 = !32 = 0

100L-~~-'-~~-L.~--=:::~~::::::::::__i___~~..L__J

o.o 0.2 0.4 0.6 0.8 1.0

L/r

FIGURE 21: EFFECT OF 51, B2 AND CONTINUITY IN Nxz ON qo : CR

(1) G-E(0°); (2) B-E(45°); h1/h2=4 )

74

1800

1300

800

LAYERS

(I) BORON - ALUMINUM ( 0°)

(2) GRAPHITE-EPOXY (45°)

0 f31 =0.5 i /32=-0.5

A f31 = f32 = 0

300L-~~-'-~~--L~~~...1..-~~~~~~-'---'

o.o 0.2 0.4 0.6 0.8 1.0

LI r

FIGURE 22: EFFECT OF Bl' B2 AND CONTINUITY IN Nxz ON qoCR:

(1) B-Al(0°); (2) G-E(45°); h1/h 2=4

75

400

300

200

LAYERS

(I) BORON- EPOXY ( 0°)

( 2) GRAPH IT - EPOXY ( 45°)

0 /31 = 0.01 i /32 = -0.01

A/31 =/32=0

100'--~~.....L.~~--!~~~...J......~~-'-~~~...__--

o.o 0.2 0.4 0.6 0.8 1.0

L/r


(l) B-E(0°); (2) G-E(45°); h1/h 2=1

600

500

400

300

200

0.0

7(. ~·

I .. 1>l I CRACK I ARREST I REGION I I I I I

0.2

LAYERS

(I) BORON-EPOXY (0°)

( 2) GRAPHITE - EPOXY ( 45°)

h 1 I h 2 = 4

0 /31 = 0.01 i /32 = - 0.01

t:. f31 = /32 = 0

0.4 0.6 o.a

L/r

1.0


(l) B-E(0°); (2) G-E(45°); h1/h 2=4

Ji~

77

,_ p

900 CRACK

700

500

300

ARREST REGION

LAYERS

( I) BORON - EPOXY ( 0°)

( 2) BORON- ALUMINUM ( 45°)

h1 /h2=4

0 !31=0.1 i !32 =-0.5

~ /31 = !32 = 0

IOOL-------L-------L------1-------'-----__.____, 0.0 0.2 0.4 Q.6 a.a 1.0

L/r


(1) B-E(0°); (2) B-Al(45°); h1/h 2=4

78

agreement with a ratio of 1. The inclusion of shear force continuity

across the crack tip alters the delamination behaviour drastically

when the thickness ratio is 4. Such changes are also seen in some

cylinders with a ratio of 1. The formulation in Chapter 3 yielded a

solution that had two stable regions, and, in some cylinders with a

thickness ratio of 4, a "crack arrest" region. The present analysis,

by accounting for shear force continuity and arbitrary crack tip geo-

metry, revises such an abnormal behaviour into an unstable-stable-neu-

tral crack growth pattern (Figures 19 to 25). The behaviour is affected

predominantly when the half crack length is small and the thickness ra-

tio of the cylinder is larger than one. The present analysis gives a

truer picture of the delamination process as it represents the real prob-

lem in a more realistic fashion. Moreover, it establishes the fact that

the queer predictions of Chapter 3 are a result of poor modelling. Ex-

perimental verification of these results are called for to help support

the present analysis.

CHAPTER V. EFFECT OF MIDPLANE AXIAL AND SHEAR STRAINS

The governing equation for a layered cylindrical shell was obtained

in Chapter 2 by combining the compatibility and equilibrium equations

through the many assumptions that were made regarding the shell geometry,

the crack geometry and the loading. In addition to the simplifications

that resulted through the axisymmetry in the problem, the midplane axial

and shear strains were considered negligible compared to the circumfer-

ential strain. This led to an expression for the laminate inplane

stress resultant in the circumferential direction, U , in terms of ,XX

the hoop strain w/r and the change in curvature w xx (Equation 6). The '

shell compatibility and equilibrium equations were combined for the axi-

symmetric problem to yield Equation ( 5) :

2 l b21

( dll b21 l (92) - -) \'! + --\•/ - - u = -q a22 , xxxx r a22 ,xx r ,XX z

When Equation (6) was substituted into the above equation, the fourth

order ordinary differential equation governing the radial displacement

in a cylindrical shell with a loading q2 was obtained (Equation 7). This

was the shell equation used in Chapters 2, 3 and 4.

In the present chapter the effect of the inclusion of the midplane

axial and shear strains on the shell analyses carried out in the earlier

chapters is studied. When the axial strain and the shear strain of the

reference plane are assumed to be of the same order as the circumferen-

tial strain of that surface, U is a function of all the three strains ,xx [8]. To obtain an expression for U solely in terms of the hoop strain, ,xx w/r, and the change in curvature, w , the axial and shear strains of ,xx the reference plane must be expressed as functions of w/r and w As ,xx

79

80

a result of the axisymmetry in the model, u, 00 and u,xo are zero.

These give a measure of the axial stress resultant and the shear stress

resultant of the reference plane. Expressing u, 00 and u ,X0 in terms of

'ti ,xx and all the reference plane strains [8] the following equations

are obtained:

Bl l "' = 0 ,XX (93)

B 14w = 0 ,xx (94)

where u and v are the displacements in the axial and circumferential

directions. The midplane stress resultant in the circumferential direc-

tion, u,xx' may likewise be expressed as a function of the reference

plane strains and the change in curvature w [8]: ,xx U = A12u + A22w/r + A24v - s12w ,XX ,X ,X ,XX (95)

Equations (93) and (94) are solved simultaneously to obtain u ,x and v as functions of the hoop strain w/r and the change in the curva-,x ture w These expressions, when substituted into Eauation (95), yield ,xx a relationship between U and w/r and w that is different from Equa-, xx 'xx tion (6). On replacing u,xx in Equation (92) by this new expression, the

governing equation for the cylindrical shell, accounting for the axial

and shear strains in the reference plane, is of the form shown below: 2

b21 d 11 (dll

b21 Y5)w,xx -

4 ( 96) - -)w + (-- -2- y?":d 11 /r = -q a22 ,xxxx a22r z r 2 2 where yl = (A44 - A14;A11 )r /d 11 ( 97)

y 2 = (Bl4 - B11A14/All)r/dll (98)

y (Al2Al 41 All

2 (99) = A24)r /dll 3 y = B11 /(A11 r) (A14IA11 h2IY1 ( l 00) 4

81

Y5 = A12IA11 + (A14IA11h3IY1 ( l 01 )

y = (\Al2 - 812/r'+ A24Y2/Yl)r2/dil ( l 02) 6 2 ( 103) Y7 = (A22 - A12Y5 + A24Y3/yl)r /dll

The shell equation (96) is similar in form to Equation (7) used in

the earlier chapters. The inclusion of the midplane axial and shear

strains has changed the coefficients premultiplying the radial displace-

ment wand its second and fourth order derivatives in Eauation (7).

This affects the radial displacement solutions of the three shell ele-

ments considered in the earlier analyses. The present investigation

aims at determining quantitatively the effect of this change in the

shell displacement solutions on the delamination behaviour of the lay-

ered cylinder. Except for the change in the shell equation, the remain-

der of the analysis remains the same in each of the former problems.

The earlier solutions are modified by the changes in the particular solu-

tions and in the roots of the homogeneous equations for the three shells.

The modified shell equations are used in the analyses of the previous

chapter and the variation of the critical loading value is found as a

function of the initial half crack length. Results are obtained for

either type of loading. The shell is made more flexible due to the

inclusion of the reference plane axial and shear strains. This is

manifested in all the results as a decrease in the critical loading

value. The magnitude of the drop in the value depends on the geometric

and material properties of the plies that constitute the cylinder. The

type of loading alters the region where this effect is predominant.

Figures (26) and (27) show the effect of the axial and shear strains

of the midplane on PcR for a boron-aluminum (0°), graphite-eroxy (45°)

82

cylinder and a boron-epoxy (0°), ~raphite-epoxy (45°) cylinder of unit

thickness ratio. A considerably large decrease in PcR is seen in both

the cylinders. The graphite-epoxy (0°), boron-epoxy (45°) cylinder of

unit thickness ratio shows a small drop in PcR due to the inclusion of

the additional strains (Table 14). A cylinder made of boron-epoxy (0°)

and boron-aluminum (45°) plies shows negligible changes in PcR values

for three different thickness ratios (Tables 15 to 17).

The results for a layered cylinder loaded on the inner surface show

a different behaviour. Three layer combinations with a thickness ratio

of one were analyzed: boron-aluminum (0°) and graphite-e~oxy (45°),

boron-epoxy (0°) and graphite-epoxy (45°), and graphite-epoxy (0°) and

boron-epoxy (45°). In this case, a large drop in the critical load

occurs only when the initial crack length is small (Fiqures 28 to 30).

As L increases the drop in q0 cr is negligible. Anisotropic layered cylin-

ders of boron-epoxy (0°) and boron-aluminum (45°) plies, of different

thickness ratios, show practically no change in the critical load for all

values of the initial debond lengths (Tables 18 to 20).

83

800

LAYERS

( I ) BORON - ALUMINUM ( 0°) 700 ( 2) GRAPHITE - EPOXY ( 45°)

h1 /h2 = I

600 13, = 0.01 i /32 = - 0.01

0 U,x AND v,. IGNORED

0 U,x AND V,x INCLUDED

500

c'.31~ 400

300

200

100

o--~~_....._~~__.~~~......._~~_._~--~...._--

o.o 0.2 0.4 0.6 0.8 1.0

L/ r

FIGURE 26: EFFECT OF MIDPLANE AXIAL AND SHEAR STRAINS ON PcR:

(l) B-Al(0°); (2) G-E(45°); h1/h 2=1

84

800

LAYERS

( I ) B 0 R 0 N - EPOXY ( 0 ° ) 700 ( 2) GRAPHITE - EPOXY (45°)

h, I h = I 2

13, :: 0.01 f32 :z - 0.01 soo I

0 U •x AND V,x IGNORED

0 U,x AND V,x INCLUDED

500

~1~ 400

300

200

100

O"--~~-'-~~~'--~~-'-~~~..__~~~~

o.o 0.2 0.4 0.6 0.8 l.O

L/r

FIGURE 27: EFFECT OF MIDPLANE AXIAL AND SHEAR STRAINS ON PcR: (1) B-E(0°); (2) G-E(45°); h1/h 2=1

Table 14

Effect of midolane axial and shear strains on PcR:(l)G-E(0°); (2)B-E(45°);h~/h2=1

PcR/(ya)l/2 for L/r

s1= 0.01 Comparison s1= 0.20 Comparison s2=-0.0l (Chap. IV) s2=-0.80 (Chap. IV)

0.067 852.5 865.5 869. 1 901. 6

o. 100 386.9 398. 9 401. 8 426.5

0. 133 226.4 242.4 239. 8 266.2

o. 167 155.5 173.9 167. 4 194.0

0.200 119. 7 137. 9 130. 2 153.9

0.233 100.0 115. 7 108.7 127.3

0.267 89. 1 101. 6 95.8 109.3

0.300 82.6 92.4 87.4 97.2

0.333 78.9 86.6 81. 8 89.3

0.367 76.9 83.3 78.4 84.6

0. 400 75.9 81. 7 76.5 82.2

0.433 75.8 81. 3 75.7 81. 2

0.467 76.0 81. 6 75.6 81. 2

0.500 76.5 82.2 76.0 81. 7

0.533 77.0 82.9 76.5 82.5

0.567 77. 5 83.6 77. 1 83.2

0.600 78.0 84.2 77.6 83.9

0.667 78.6 85.0 78.4 84.8

0.833 78.8 85.3 78.8 85.3

1.000 78.7 85.2 78.7 85.2

86

Table 15

Effect of midplane axial and shear strains on PcR:(l)B-E(0°); (2)B-A1(45°);h 1/h 2=1/4

pCR/(ya)l/2 for L/r

B1= 0.01 Comparison s,= 0.20 Comparison s2=-0. 01 (Chap. IV) s2=-0.80 (Chap. IV)

0.067 474.7 475.0 474.3 474.7

0. 100 222.8 222.8 222.6 222.6

o. 133 142.4 142.4 142.2 142.3

0. 167 112. 8 113. 0 112. 7 112. 9

0.200 103.3 103. 6 103. 3 103.5

0.233 102. 3 102. 6 102.3 102.6

0.267 l 04. 1 l 04. 5 l 04. l 104.5

0.300 l 06. l 106.5 106. l 106. 5

0.333 107.5 107.8 107.5 107.8

o. 367 108. l 108.4 108. l l 08. 4

0.400 108.2 108.5 108. 2 108.5

0.433 108. l 108.5 108. l 108.5

0.467 108. 1 108.4 l 08. l 108.4

0.500 108.0 l 08. 3 l 08. 0 108.3

0.533 107. 9 108.3 l 07. 9 108.3

0.567 107. 9 108.3 107. 9 108.3

0.600 l 07. 9 108.3 107. 9 l 08. 3

0.667 107. 9 l 08. 3 107.9 108.3

0.833 107. 9 108.3 107.9 108.3

l. 000 107.9 108.3 107. 9 l 08. 3

87

Table 16

Effect of midplane axial and shear strains on PcR:(l)B-E(0°); (2)B-Al(45°);h 1Jh2=1

!( ,112 f PcR Ya' or L/r

s1= 0.01 Comparison s1= 0.20 Comparison s2=-0.0l (Chap. IV) s2=-0.80 (Chap. IV)

0.067 1265.4 1267.4 1297.6 1302. 8

o. 100 582.7 583.6 611. 8 613.6

0. 133 354.0 355.9 377.9 380.6

o. 167 255.4 257.4 274.4 277. 0

0.200 206.7 208.5 220.5 222.6

0.233 180.3 181. 7 188. 9 190.4

0.267 166.3 167.5 171. 0 172. 1

o. 300 159. 4 160.4 161. 4 162.3

0.333 156. 8 157.7 157. 2 158. 1

o. 367 156.8 157.7 156.4 157.3

0.400 158. 1 159.0 157. 3 158. 3

0.433 159. 7 160.7 159. 0 159. 9

0.467 161. 3 162.3 160.6 161. 6

0.500 162.5 163.5 162.0 163.0

0.533 163.3 164.3 163. 0 164. 0

0.567 163.7 164.8 163.6 164. 6

0.600 163. 9 165. 0 163.8 164.9

0.667 163. 9 165. 0 163.9 165. 0

0.833 163.7 164.7 163.6 164.7

1.000 163.7 164.7 163.6 164. 6

88

Table 17

Effect of midplane axial and shear strains on PcR:(l)B-E(0°); (2)B-Al(45°);h 1/h2=4

PcR/(ya)l/2 for L/r

s1= 0.01 Comparison s1= 0.20 Comparison s2=-0. 01 (Chap. IV) s2=-0.80 (Chap. IV)

0.067 469. 3 472. 5 170. 2 105. 8

0. 100 277. 0 282.7 287.9 294. 3

o. 133 234.6 241.2 252.2 260.0

o. 167 221.4 227. 1 248. 1 256.0

0.200 210. l 214.6 243.0 249.4

0.233 199.0 202.6 232.7 238. 1

0.267 190.2 193.4 221.3 225.9

0.300 184. l 186.9 21o.1 214.0

0.333 180. l 182. 8 200.0 203.3

0.367 177. 8 180. 4 191. 4 194.3

0.400 176. 8 179. 3 184.8 187.5

0.433 176. 6 179. l 180. 3 182.8

0.467 176. 9 179. 4 177. 7 180. 1

0.500 177. 5 180. 1 176.4 178. 9

0.533 178.2 180. 8 176.2 178. 7

0.567 178.8 181. 4 176.5 179. 1

0.600 179. 3 182.0 177. 1 179. 7

0.667 180.0 182.7 178.4 181 . 1

0.833 180.3 182.9 180.0 182.6

1.000 180. 1 182.8 179. 9 182. 6

89

550

500

450

LAYERS

(I) BORON -ALUMINUM ( 0°)

(2) GRAPHITE - EPOXY ( 45°)

h1 I h2 = I

13, = 0.01 i '32 = - 0.01

O u,x AND v,x IGNORED

D u,x AND v,x INCLUDED

400'-------L-------1-------'------~------..._--

o.o 0.2 0.4 0.6 0.8 1.0

L/r

FIGURE 28: EFFECT OF MIDPLANE AXIAL AND SHEAR STRAINS ON qo :

CR (l) B-Al(0°); (2) G-E(45°); h1/h2=1

90

300 LAYERS

( I) BORON - EPOXY ( 0 ° ) ( 2) GRAPHITE - EPOXY ( 45°)

h1 I h2 = I

13, = o. 2 i f3 2 = - 0.8

o u,x AND v,x IGNORED

o u,x AND v,x INCLUDED

250

200

150 L.,.._ __ _J_ ___ ,L._ __ __,_ ___ ~ __ __._ _ _.

o.o 0.2 0.4 0.6 0.8 1.0

L/r

FIGURE 29: EFFECT OF MIDPLANE AXIAL AND SHEAR STRAINS ON q :

OCR

(1) B-E(0°}; (2) G-E(45°); h1/h2=1

91

300 '

200

100

LAYERS

(I) GRAPHITE - EPOXY ( 0°) (2) BORON - EPOXY ( 45°)

h1 I h2 = I /31 = 0.01 i /32 = -0.01

o u, 11 AND v,11 IGNORED

o u,K AND v,.,. INCLUDED

o~~~_.__~~--"'~~~-'-~~_._~~~-----

o.o 0.2 0.4 0.6 0.8 1.0

L/r

FIGURE 30: EFFECT OF MIDPLANE AXIAL AND SHEAR STRAINS ON q :

°CR (1) G-E(0°); (2) B-E(45°); h1/h2=1

92

Table 18

Effect of midplane axial and shear strains on q : (1 )8-E(0°); (2)B-Al(45°);h1/h 2=1 OCR

1/2 q I ( y ) for L/r

oCR a

Bl= 0.01 Comparison :31= 0.20 Comparison s2=-0.0l (Chap. IV) s2=-0.80 (Chao. IV)

0.067 495.3 511. 4 506.6 522.7

o. 100 408.4 416.2 422.0 430.0

0. 133 314.5 317.5 330.5 334.0

o. 167 249.6 251. 3 265. 1 267.2

0.200 210. 9 212. 1 223.2 224.6

0.233 188.0 188.8 196. 2 197.0

0.267 175.4 175. 9 180.0 180. 4

0.300 169. 1 169. 4 171. 0 171. 3

0.333 166.7 166.9 167. 1 167.3

0.367 166.7 167.0 166.3 166.5

0.400 167.9 168.2 167.2 167.5

0.433 169.5 169. 8 168.8 169. l

0.467 171. 0 171. 3 170.3 170.7

0.500 172. 1 172. 4 171. 6 172. 0

0.533 172. 8 173.2 172. 6 172. 9

0.567 173. 3 173. 6 173. 1 173.5

0.600 173.5 173. 8 173.4 173.7

0.667 173. 5 173. 8 173.5 173. 8

0.833 173. 2 173.6 173. 3 173.6

1. 000 173. 2 173. 6 173. 2 173.5

92

Table 19

Effect of midplane axial and shear strains on q :(1)8-E(0°); (2)8-Al (45°) ;h/h2=1/4 OCR

1/2 qo /(ya) for L/r CR

s1= 0.01 Comparison s1= 0.20 Comoarison 62=-0.0l (Chap. IV) s2=-0.80 (Chap. IV)

0.067 389.2 394. 7 390.5 395. 7

0. 100 216. l 216.7 216.2 216.7

0. 133 143.0 143. 0 142.9 143. 0

0. 167 114. 3 114. 5 114. 2 114.4

0.200 105. 0 105.2 104.9 105. 1

0.233 104.0 104.2 103.9 104. 2

0.267 105. 8 106.0 105.8 106.0

0.300 107.8 108.0 107.8 108.0

0.333 109. 1 109. 3 109. 1 109.3

0.367 109.7 109.9 109.7 109.9

0.400 109. 8 110. 0 109.8 110. 0

0.433 109.7 11o.0 109. 7 110. 0

0.467 109.7 109.9 109.7 109.9

0.500 109.6 109.8 l 09. 6 109.8

0.533 109.6 109.8 109.5 109. 8

0.567 109.5 109.8 109. 5 109. 8

0.600 109.5 109. 8 109.5 109.8

0.667 109. 5 109. 8 109. 5 . 109. 8

0.833 109. 6 109. 8 109.6 109.8

1. 000 109. 6 109.8 109.6 109.8

94

Table 20

Effect of midplane axial and shear strains on q :(l)B-E(0°); (2)B-Al(45°);h1/h 2=4 OCR

1/2 q I (y ) for L/r

oCR a

s1= 0.01 Comparison s1= 0.20 Comparison s2=-0. 01 (Chap. IV) s2=-0.80 (Chap. IV)

0.067 333.9 338.6

0. 100 276.6 278.6 285.4 287.2

o. 133 254.3 256.8 265.4 268.2

o. 167 246.6 248.9 263.7 266.8

0.200 239. 7 241.3 261. 5 263.9

0.233 232.4 233.4 255.7 257.6

0.267 226.2 226.9 248.7 250. 1

0.300 221. 6 222.2 241.3 242.3

0.333 218.7 219. 1 234.2 234.8

0.367 217.0 217.3 227.9 228.3

0.400 216.3 216.6 222.7 223. 1

0.433 216.2 216.5 219.2 219.4

0.467 216.5 216.8 217.0 217.2

0.500 217.0 217.3 216.0 216.2

0.533 217.6 217.9 215.8 216. 1

0.567 218. 1 218.5 216.0 216.4

0.600 218.5 218.9 216.6 216.9

0.667 219.0 219.4 217.6 218. 1

0.833 219.2 219.6 218.9 219.3

1. 000 219. 1 219.5 218.8 219.2

CHAPTER VI. CONCLUSIONS

An initial crack is assumed to be present in the adhesive layer at

the interface between two plies in a layered anisotropic cylinder. For

different values of this debonded length the amount of the applied

loading that causes the delamination to propagate is determined. Two

problems are considered simultaneously; one with the driving force as

the pressure on the crack surface, and the other with a loading on the

inner cylindrical surface. Thin shell assumptions are made to reduce

the equations to a tractable form, and further simplifications are

introduced through axisymmetric crack geometry and loading. The analy-

sis can easily accommodate the effects of the adhesive layer though the

results presented in this dissertation assume these to be negligible.

A very simple model is initially analyzed to obtain closed-form

solutions for the critical delamination loads. An infinite cylinder

with a sharp, flat crack tip is assumed. While intuitively reasonable

results were obtained for the cylinder with a pressure on the crack

surface, some interestingly intriguing phenomena were seen to take place

in cylinders with a loading on the inner surface. The thickness ratio

of the plies plays an important role in this delamination behaviour. A

"crack arrest" region, where the delamination would not propagate for

any real value of the applied loading on the inner cylindrical surface,

was observed for a large thickness ratio (h 1/h2>>1).

In order to predict the above behaviour, the model is refined to

accommodate any general crack tip geometry. In addition to this, shear

continuity across the crack tip, not imposed in the initial model, is

95

96

also enforced. The results obtained in this case establish the fact

that the abnormal behaviour of certain cylinders analyzed earlier is due

to the violation of the shear force continuity across the crack tip. It

is also observed that the slopes of the three shell elements at the

crack tip, indeed, are very nearly zero, justifying the assumption of a

sharp, flat crack in the simpler model. The present analysis makes it

advisable to resort to numerical techniques to solve for the radial

displacements in the cylinder. A finite cylinder would make this impera-

tive.

The model is further modified to include the effects of the mid-

plane axial and shear strains, which were considered negligible in the

earlier models compared to the midplane circumferential strain. The

flexibility of the cylinder is increased by providing for these extra

strains and the critical delamination loads are decreased. The reduc-

tion in the magnitude of these critical loads depends on the geometric

and material properties of the plies, and the type of applied loading.

The drop in q is observed only in the region where the crack length 0 cR

is small, when the loading is applied on the inner cylindrical surface.

The critical pressure on the crack surface decreases considerably for

all values of the initial crack length, and this is quite large for

large crack lengths. Certain cylinders show negligible effect on the

critical loads due to the inclusion of the reference plane axial and

shear strains.

Based upon judgment, it appears that the last model represents the

real delamination behaviour in a realistic fashion. Experimental in-

vestigations are called for to compare the predicted delamination behav-

iour and establish the validity of the model considered here.

REFERENCES

l. Stambler, I., "Bright Future Forecast for Composites in Aerospace", Interavia, December 1972.

2. Ambartsumyan, S. A., Theory of Anisotropic Shells, NASA TT F-118, May 1964.

3. Ambartsumyan, S. A., Theory of Anisotropic Plates, Technomic Publishing Co., Westport, Conn., 1970.

4. Ashton, J. E., Halpin, J. C., and Petit, P. H., Primer on Composite Materials: Analysis, Technomic Publishing Co., Westport, Conn., 1969.

5. Ashton, J. E., and Whitney, J. M., Theory of Laminated Plate~, Technomic Publishing Co., Westport, Conn., 1970.

6. ~omposite Materials, a Multivolume Treatise (8 Volumes), edited by Broutman, L. J., and Krock, R.H., Academic Press, New York, 1974.

7. ~odern__i:om_posite Materials, edited by Brout~an, L. J., and Krock, R. H., Addison-Wesley, Reading, Mass., 1967.

8. Calcote, L. R., Th~AnaJ...:l~~ of Laminated Cornposite Structures, Van Nostrand Reinhold, New York, 1969.

9. Composite Engineeri_Q_g Laminates, edited by Dietz, A.G. H., M.I.T. Press, Cambridge, Mass., 1969.

10. Hearmon, R. F. S., Applied Anisotropic Elasticity, Oxford University Press, Oxford, 1961.

11. Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, 1963.

12. Lekhnitskii, S. G., Anisotropic Plates, Gordon and Breach, New York, 1968.

13. McCullough, R. L., Concepts of Fiber-Resin Composites, Marcel Dekker, New York, 1971 .

14. Scala, E., Composite Materials for Combined Functions, Hayden Book co., Rochelle--Par~N~~-r9T:f ______________ _

15. Spencer, A. J. M., Deformations of Fiber-Reinforced Materials, Oxford University Press, Oxford, 1972.

16. ~omposite Materials vJorkshop, edited by Tsai, S. \~., Halpin, J. C., and Pagano, N. J., Technomic Publishing Co., Westport, Conn., 1968.

97

98

17. Mechanics of Composite Materials, Proceedings of the Fifth Symposium on Naval Structural Mechanics, 1967, edited by Wendt, F. W., Liebowitz, H., and Perrone, N., Pergamon, New York, 1970.

18. Jones, R. M., Mechanics of Composite Materials, Scripta Book Co., Wash. D. C., 1975.

19. Proceedings of the Sixth St. Louis Symposium on Composite Materials in Engineering Design, Washington University, St. Louis, 1972.

20. Sinclair, P. M., 11 Composites: Designers \~ait and Contemplate 11 ,

Industrial research, October 1969, p. 59.

21. Composites Recast, Air Force Materials Lab., Wright-Patterson AFB, Ohio, 1972.

22. Rayburn, C. L., and Westburg, P. W., 11 Shrouds for Space Payloads", Space/Aeronautics, Vol. 47, February 1967, p. 66.

23. Griffith, A. A., 11 The Phenomena of Rupture and Flow in Solids", Philosophical Transactions, Roy. Soc. (London) Series A, Vol. 221, 1920, pp. 163-198.

24. In.,iin, G. R., 1'Fracture Dynamics 11 , Fracturing of Metals, Am. Soc. of Metals, Cleveland, 1948, pp. 147-166.

25. OrO\\lan, E., "Energy, Criteria of Fracture 11 , l·ielding Research Supple-ment, Vo 1 . 2 0 , 1 9 5 5 , p. 1 5 7 5 .

26. Neuber, 11 Kerbspannungslehre 11 , Engl. Trans. Edwards Bros., Ann Arbor, 1958.

27. Neuber, 11 Fracture Mechanics 11 , Structural Mechanics, Pergamon Press, New York, 1960.

28. Kuhn, 11 The Prediction of Notch and Crack Strength Under Static and Fatigue Loading 11 , SAE Paper 843 C, May, 1964.

29. Kuhn, "Unified Notch Strength Analysis for Wrought Aluminum Alloys", NASA TN D-1259, 1962.

30. Barenblatt, G. I., 11 Mathematical Theory of Equilibrium Cracks in Brittle Fracture 11 , Adv. Appl. Mech., Vol. 7, 1962, p. 55.

31. Fracture, An Advanced Treatise, Vols. I to VI, edited by Liebowitz, H., Academic Press, New York, 1968.

32. Fracture of Structural Materials, Tetelman, A. S., and McEvily, Jr., A. J., John Wiley & Sons, 1967.

33. Method of Analysis and Solutions to Crack Problems, edited by Sih, G. C., Wolters-Nordhoff Publishing Co., 1972.

99

34. Fracture Mechanics, Lecture Notes by Smith, C. \·J., V.P.I.& S.U., Blacksburg, Virginia.

35. A Short Course in Fracture r1echanics, Paris, P. C., Univ. of Washington Press, 1960.

36. Fracture of Solids, edited by Drucker, and Gilman, Interscience, New York, 1963.

37. Sih, G. C., et al., "Fracture Mechanics Studies of Laminate Systems", Air Force T eciln i cal Report AFML-TR-70-112, Pa rt I II , \.Jri ght-Pa t terson, 1972.

38. Lauraitis, K., "Failure Modes and Strength of Angle-Ply Laminates", Univ. of Illinois TAM Report No. 345, 1971.

39. Almond, E. A. et al., "The Fracture of Pressurized Laminated Cylin-ders", Journalofthe Iron and Steel Institute, October, 1969.

40. Sih, G. C., and Chen, E. P., "Fracture Analysis of Unidirectional and Angle-Ply Composites'', Lehigh University, Technical Report NADC-TR-73-1 for the Naval Air Development Center, Warminster, Pennsylvania, January 1973.

41. Sih, G. C. et al., "Fracture Mechanics for Fibrous Composites", Ameri-can Society-:ro-r-Testing and Materials, Special Technical Publication, 1973.

42. Hilton, P. D., and Sih, G. C., 11 A Sandwiched Layer of Dissimilar Material Weakened by Crack-Like Imperfections'', Proceedings of the 5th Southeastern Conference on Theoretical and Applied Mechanics, Vol. 5, 1972, p. 949.

43. Paris, P., and Erdogan, R., 11 A Critical Analysis of Crack Propagation Laws'', Transactions of ASME, Journal of Basic Engineering, December 1963.

44. Donaldson, D.R., and Anderson, W. E., "Crack Propagation Behaviour of Some Airframe Materials'', Proceedings of the Crack Propagation Symposium, Cranfield, Vol. 2, September 1961.

45. Yaffe, E. H., "The Moving Griffith Crack", Philosophical Magazine, Ser. 7, Vol. 42, 1951, pp. 739-750.

46. Cotterell, B., "On the Nature of Moving Cracks", Journal of Applied Meehan i cs, 1964.

47. Proceedings of the Crack Propagation Symposium, Vols. I and II, Cranfield, 1961.

48. Erdogan, F., "Crack Propagation Theories", NASA CR-901, December 1967.

100

49. Irwin, G. R., "Fracture Mechanics", Structural Mechanics, Pergamon Press, New York, 1960, p. 557.

50. Yang, C. T., "A Study of the Law of Crack Propagation", Transactions of ASME, Journal of Basic Engineering, September 1967.

51. Bueckner, H. F., "The Propagation of Cracks and Energy of Elastic Deformation", Transactions of ASME, Journal of Applied Mechanics, 1958.

52. Craggs, J. vJ., "On the Propagation of a Crack in Elastic-Brittle Material", Journal of Mechanics and Physics of Solids, Vol. 8, 1960.

53. Zak, A. R., and Williams, M. L., "Crack Point Stress Singularities at a Bimaterial Interface", GALCIT SM 42-1, California Institute of Technology, January 1962.

54. Sih, G., and Rice, J., "The Bending of Plates of Dissimilar Materials with Cracks'', Transactions of ASME, Journal of Applied Mechanics, Vol. 31 ' 1964.

55. Ang, D. D., and vii 11 i ams, M. L., "Combined Stresses in an Orthotropi c Plate Having a Finite Crack", Journal of Applied Mechanics, 1961.

56. Sneddon, I. N., "Crack Problems in Mathematical Theory of Elasticity", Report No. ERD-126/l, North Carolina State College, May 1961.

57. Sechler, E. E., and Williams, M. L., "The Critical Crack Length in Pressurized Cylinders", GALCIT 96, CIT, Final Report, September 1959.

58. vlu, E. M., and Reuter, R. C., "Crack Extension in Fiber-glass Rein-forced Plastics'', University of Illinois TAM Report No. 275, 1965.

59. Paris, P. C., "The Mechanics of Fracture Propagation and Solutions to Fracture Arrest Problems'', Document No. D2-2195, Boeing Co., 1957.

60. Sanders, Jr., J. L., "On the Griffith-Irwin Fracture Theory", Transac-tions of ASME, Journal of Applied Mechanics, 1968.

61. Erdogan, F., and Ratwani, M., "Fatigue and Fracture of Cylindrical Shells Containing a Circumferential Crack", International Journal of Fracture Mechanics, Vol. 6, No. 4, December 1970.

62. Folias, E. S., "An Axial Crack in a Pressurized Cylindrical Shell", Int. J. Fracture Mechanics, Vol. l, 1965, p. 104.

63. Fol ias, E. S., 11 A Circumferential Crack in a Pressurized Cylinder", Int. J. Fracture Mechanics, Vol. 3, 1967, p. 1.

64. Copley, L. G., and Sanders, Jr., J. L., "A Longitudinal Crack in a Cylindrical Shell Under Internal Pressure'', Int. J. Fracture Mecha-nics, Vol. 5, 1969, p. 117.

101

65. Duncan, M. E., and Sanders, Jr., J. L., "The Effect of Circumferential Stiffener on the Stress in a Pressurized Cylindrical Shell with a Longitudinal Crack", Int. J. Fracture Mechanics, Vol. 5, 1969, p. 133.

66. Hahn, G. T., Serrate, M., and Rosenfield, A. R., "Criteria for Crack Extension in Cylindrical Pressure Vessels'', Int. J. Fracture Mechanics, Vol. 5, 1969, p. 187.

.. 67. Erdogan, F., Ratwani, M., and Yuceoglu, U., "On the Effect of Orthotropy

in a Cracked Cylindrical She11", Int. J. of Fracture, Vol. 10, 1974, p. 369.

68. Yuceoglu, U., and Erdogan, F., "A Cylindrical Shell with an Axial Crack Under Skew-Symmetric Loading", Int. J. Solids Structures, Vol. 9, 1973.

69. Williams, J. G., and Ewing, P. 0., "Crack Propagation in Plates.and Shells Subjected to Bending and Direct Loading", Proceedings of the 2nd Int. Conference on Fracture, Brighton, April 1969.

70. Folias, E. S., "On the Theory of Fracture of Curved Sheets", Engineer-ing Fracture Mechanics, Vol. 2, 1970.

71. Steverding, B., and Nieberlein, V., "Fracture of Pressurized Cylindri-cal Shells", Engineering Fracture Mechanics, Vol. 6, 1974.

72. Anderson, R. B., and Sullivan, T. L., "Fracture Mechanics of Through-Cracked Cylindrical Pressure Vessels", NACA TN 0-3252, 1966.

73. Weiss, V., and Yukawa, S., "Critical Appraisal of Fracture Mechanics", ASTM STP No. 381, 1964.

74. Sih, G. C., "Some Basic Problems in Fracture Mechanics and New Concepts", Journal of Engineering Fracture Mechanics, 1973.

75. l~illiams, M. L., "Stress Singularities, Adhesion and Fracture", Proceed-ings of the 5th U.S. National Congress of Applied Mechanics, 1966.

76. England, A.H., "A Crack Between Dissimilar Media", J. Applied Mechanics, Vol. 32, No. 2, June 1965.

77. Griffith, A. A., "The Theory of Rupture", Proc. of First Int. Congress of Applied Mechanics, Delft, 1924, pp. 55-63.

78. Erdogan, F., "Stress Distribution in Bonded Dissimilar Materials with Cracks", J. Applied Mechanics, Vol. 32, tlo. 2, June 1965.

79. ~~ill iams, M. L., "On the Stress Distribution at the Base of a Stationary Crack", J. Applied Mechanics, Vol. 24, 110. l, March 1957.

80. l~illiams, M. L., "The Prediction of Critical Debonding Length for a Case-Bonded Sol id Propellant Rocket Motor", Fourth Meeting of ICRPG Working Group on Mechanical Behaviour, San Diego, California, November 1965.

102

81. ~Jilliams, M. L., "The Relation of Continuum Mechanics to Adhesive Fracture", Journal of Adhesion, Vol. 4, 1972.

82. Williams, M. L., Journal of Applied Polymer Science, Vol. 13, 1969, p. 29.

83. Uilliams, rL L., "The Fracture Threshold for an Adhesive Interlay-er", Journal of Applied Polymer Science, Vol. 14, -1970, p. 1121.

84. Kulkarni, S. V., and Frederick, D., "Propagation of Delamination in a Layered Cylindrical Shell'', Int. J. Fracture Mechanics, Vol. 9, 1973.

85. Adams, D. F., "Elastoplastic Crack Propagation in a Transversely Loaded Unidirectional Composite'', Journal of Composite Materials, January 1974.

86. Anderson, G. P., Devries, K. L., and Williams, M. L., "The Peel Test in Experimental Adhesive Fracture Mechanics", presented at SESA Spring meeting, May 1974.

87. Rice, J. R., and Johnson, M.A., "The Role of Large Crack Tip Geo-metry Changes in Plane Strain Fracture", US AEC Tech. Report No. 38, Brown University, September 1969.

88. Sendeckyj, G. P., "Debonding of Rigid Curvilinear Inclusions in Longitudinal Shear Deformation", Air Force Flight Dynamics Labora-tory, AFFDL-TM-72-3-FBC, 1972.

89. Sih, G. C., Hilton, P. D., and Hei, R. P., "Exploratory Development of Fracture Mechanics of Composite Systems", Air Force Materials Lab., June 1970.

90. Swedlow, J. L., and Williams, M. L., "A Review of Recent Investiga-tions into Fracture Initiation at GALCIT", Office of Aerospace Re-search, U.S. Air Force, ARL 64-175, October 1964.

91. Hoover, W. R., and A 11 red, R. E., "The Effect of Crack Length and Bond Strength on the Delamination Process in Bosic-Al Composites", J. Composite Materials, Vol. 8, January 1974.

92. Chen, E. P., and Sih, G. C., "Interfacial Delamination of a Layered Composite Under Anti-Plane Strain", J. Composite Materials, Vol. 5, Jan. 1971.

93. Erdogan, F., and Cook, T. S., "Antiplane Shear Crack Terminating at and Going Through a Bimaterial Interface", Int. J. of Fracture, Vol. 10, June 1974.

94. \fright, M.A., and Iannuzzi, F. A., "The Application of the Princi-ples of Linear Elastic Fracture Mechanics to Unidirectional Fiber Reinforced Composite Materials", J. Composite Materials, Vol. 7, Oct. 1973.

103

95. Arin, K., "An Orthotropic Laminate Composite Containing a Layer with a Crack", Lehigh University, Jan. 1975.

96. Zweben, C., "Fracture Mechanics and Composite Materials: A Critical Analysis", Materials Sciences Corp., April 1972; Also ASTM STP 521, 1973.

97. Cruse, T. A., and Osias, J. R., "Exploratory Development on Fracture Mechanics of Composite Materials'', Carnegie Mellon Uni-versity, April 1974.

98. Sih, G. C., and Chen, E. P., "Fracture Analysis of Unidirectional Composites'', J. Composite Materials, Vol. 7, April 1973.

99. Zweben, C., "Failure Analysis of Unidirectional Fiber Composites Under Combined Axial Tension and Shear'', J. Mech. Phys. Solids, Vol. 22, 1974.

100. Erdogan, F., "Fracture Problems in Composite Materials", Symp. on Fatigue at the School of Engineering and Appl. Sc., George Wash. Univ., May 1972.

101. Waddoups, M. E., Elsenmann, J. R., and Kaminski, B. E., "Macro-scopic Fracture Mechanics of Advanced Composite Materials", J. Composite Materials, Vol. 5, Oct. 1971.

102. Marston, T. U., Atkins, A.G., and Felbeck, D. K., "Interfacial Fracture Energy and the Toughness of Composites", J. Materials Science, Vol. 9, 1974.

103. Rice, J. R., "An Examination of the Fracture Mechanics Energy Balance From the Point of View of Continuum Mechanics'', Brown Univ., May 1965.

104. Sneddon, I. N., "Two Lectures Concerning Pressurized Cracks", North Carolina State Univ., Appl. Math., Research Group, Feb. 1969.

105. Swedloi,-1, J. L., and Ritter, M.A., "Toward Assessing the Effects of Crack Front Curvature (CFC)", NASA Report SM-58, April 1971.

106. Erdogan, F., and Ratwani, M., "Fracture Initiation and Propaga-tion in a Cylindrical Shell Containing an Initial Surface Flaw", Nuclear Engrg. and Design, Vol. 27, 1974.

107. Rati,-1ani, M., Erdogan, F., and Irwin, G. R., "Fracture Propagation in a Cylindrical Shell Containing an Initial Flaw", Lehigh Univ., Aug. 1974.

108. Trantina, G. C., "Fracture Mechanics Approach to Adhesive Joints", J. Composite Materials, Vol. 6, April 1972.

104

109. Fracture Mechanics of Aircraft Structures, edited by Liebowitz, H., Advisory Group for Aerospace R&D, NATO, Jan. 1974.

110. Sih, G. C., and Liebovlitz, H., "On the Griffith Energy Criterion for Brittle Fracture", Int. J. Solids Structures, Vol. 3, 1967.

111. Mossakovskii, V. I., and Rybka, M. T., "An Attempt to Construct a Theory of Fracture for Brittle Materi a 1 s Based on Griffith's Criterion", PMM, Vol. 29, 1965.

112. Thresher, R. W., and Smith, F. 111., "The Partially Closed Griffith Crack", Int. J. Fracture, Vol. 9, No. l, March 1973.

113. Malyshev, B. M., and Salganik, R. L., "The Strength of Adhesive Joints Using the Theory of Cracks", Int. J. Fracture Mechanics, Vol. l, 1965, p. 114.

114. Dannenburg, H., "Measurement of Adhesion by a Blister Method", J. Appl. Polymer Science, Vol. 5, 1961, p. 125.

115. Kulkarni, S. V., "Mechanics of Delaminations in Layered Cylindrical Shells", Ph.D. Dissertation, V.P.I.& S.U., 1973.

116. Rosen, B. W., and Kulkarni, S. V., "Failure ~:echanisms in Composite Materials", Materials Sciences Corporation, Presented at the Society of Engrg. Science 10th Annual Meeting.

117. Pian, T. H. H., Tong, P., and Luk, C.H., "Elastic Crack Analysis by a Finite Element Hybrid Method", MIT, supported by the Air Force Office of Scientific Research under Contract F44620-67-C-0019.

118. Kobayashi, A. S., Maiden, 0. E., Simon, B. J., and Iida, S., "Ap-plication of the Method of Finite Element Analysis to Two-Dimen-sional Problems in Fracture Mechanics'', University of Washington, Department of Mechanical Engineering, ONR Contract Nour - 477(39), TR No. 5, October 1968.

119. Chan, S. K., Tuba, I. S., and Wilson, vJ. K., "On the Finite Element Method in Linear Fracture Mechanics'', Engineering Fracture Mechanics, 1970, Vol. 2, pp. 1-17.

120. Byskov, E., "The Calculation of Stress Intensity Factors Using the Finite Element Method with Cracked Elements", Int. J. Fracture Mechanics, Vol. 6, No. 2, June 1970.

121. Anderson, G. P., Ruggles, V. L., and Stibor, G. S., "Use of Finite Element Computer Programs in Fracture Mechanics", Int. J. Fracture Mechanics, Vol. 7, No. l, March 1971.

122. Oglesby, J. J., and Lomacky, 0., "An Evaluation of Finite Element Methods for the Computation of Elastic Stress Intensity Factors", Naval Ship Research and Development Center, RD Report 3751, Dec. 1971 .

123. Thin Elastic Shells, Kraus, H., John Wiley & Sons, New York, 1967 . ..

124. Stresses in Shells, second edition, Flugge, W., Springer-Verlag, New York, 1973.

125. J_heory of Plates and Shells, second edition, Timoshenko, S., and Woinowsky-Krieger, S., McGraw Hill, New York, 1959.

126. General Theory of Shells and its Application to Engineering, Vlasov, V. Z., NASA Technical Translation, NASA-TT-F-99, 1964.

0 C'\

c c

l\PP~'!DI X :\

C PRORLE~ 1: PRESSU~E CN CRCCK SU~F~C::; FLAT CRACK TIP; MIDPLANE AXIAL & SHEA? C STRAINS IG~JRED

c c

797

701

CC~PLEX RM21 9 R~22,R~L,R~2,KWJ,~~4

UWPL::X R!HR,RM2R 9 RM3R,R1111tR, ('<'.PLX, CS'.PT 0 D I '-1 E: ;'~ S I Cl N iJ ( 6 , 6 > , T < 2 J , Q < 3 , 3 I , :: 1 1 ( 2 l , c?. 2 ( 2 l , G 1 2 ( 2 ) , G NU 1 2 ( 2 l , l.G\:U2U2),THFTA(2l ,H(J),Qt:3(3,3l

D I '-~:: "; S I 0 N A I J ( 3 • 3 l , 3 I J { 3 , 3 l , D I J ( J , 3 ) , '.)MAL :3 { 3 , 3 ) , S ~.;A L A { 3 , 3 l , 1 S ; " ~-. L 3 T ( 3 , 3 l 9 R i{ { 3 , 3 l , S IJ. ,":. L 0 { 3 , 3 l , L ( 3 l , '.1 ''1 ( 3 ) DI.'-~'.:'•SIIJ"-1 XXX{3) ,YYY(3) ,ZZZl3l OI~~NSIU~ PAK(3),PAP(3) UI~~NSION CRKL(5JJ JI~2NSION TITLE!40J R~A0{5,797lTITL>::

!=Q~V. H { 20:\/+) ~·.R. I Tc ( o , 7 9 7 l TI Tu: R::A0(5, 7·Jl )RAD, ICRKL FJ~M~T(Fl0.0,15)

P.t 60( 5, 2001 l (CRKL( I), I=l, I~KKL l 2JJl FOD~4T(l0F8.0J

c I s H::: LL = 1 ' 2 ' 3 c [RR Es pc N D T 0 T r c s c Lu r I c ~J s F IJ R T H E f~ E G I iTJ s I 1 ' I 2 ' I I RF. s p E c T I v EL y DC 1 ·"" '.: 1 I SHE L L = 1 , 3

C N IS THE NUMBER CF LAMINAE Rr:6.D(5,l0J)N

1 J C F J ~ ·~ t. T ( I 3 J C I;~ l T I !', L I l i: u ( 6 v 6) W H IC H I S THE ;~ t3 r) .'·: !\. T ;:. IX

DJ l 2 I= l v 6 DC 12 J=l,6

12 l)(l,Jl=J.JL)

C THICKNESS OF LAMINATE IS NAMED DEPTh C T('~l IS Tl-1E THlCKNtSS CJF THE i" 1 TH L~'Alr\11

Rt= ;'.l D ( 5 , l) l) ( T ( ~\) , M = l , ~·J ) 101 FJRMAT(l0F8.0l

OEPTH=',}.CO D CJ 3 M = l, f~

3 DZ.: P TI-<= DE n TH+ T{ M l C Q ~AT~IX UF EACb L~MINA

DU 10 I=l,"J R~A0(5,102)Ell(l},E22(IJ ,Gl2{Il ,~~Ul2( !),THETA(! l

l)2 FO~MAT(3E2~.7 1 2F6.1) G\U2l(I)=GNU12( I>*E22{!)/~ll(!J G'\JU2=G'\JU12 (I l *GNU2 l (I) :) { l , l I = E 11 ( I ) I ( l • 0 0 - G r·J U 2 ) Q ( 2 ~ 2 J = c 2 2 C i l I ( l • 0 0- GN U 2 l Q ( 1 , 2 l = f 11 ( I ) '~ CN U 2 l { I l I ( l • C.• J - G :~ u Z ) Q(3,3l =Gl2C I)

C Tri FT,\ +Vt: FRIJI-'. GLOGAL TO LA;'l li\/A PF: I\~( I P.t1L I F l T H c T A { I l • E Q • 9 0 • J ) Gr~ T 0 5 :) A ;\! G L = ( T H E T ;\ ( I ) I i a 0 • J a ) :?< 3 0 14 l ') '.) 2 7 C=f:D S( A.\JGL) S = S I i~ ( ANG L J GL~ T ~ 6 'J

5 .: c = ;; • ;_) ') S=l. 1j0

oO C2.=C~q,2

S2 = s~, :x 2 C3=C'~C2

S3=S'~~2 C4=C:·,.:C3 S4=S <S3

C THE 03 ~ATRIX OF EACH LA~INA

AXES

0 ...._,

(..) 3 ( 1 ' l ) = Q { 1 ' 1 ) :t: c 4 + 2 • 0 * ( Q ( l ' 2 ) + 2 • o ) "'"l) ( 3 ' 3 ) ) * s 2 *' c 2 + Q ( 2 ' 2 ) * s 4 QP(2,2l=Q(l, ll*S4+2.JC~C0Cl,2)+2.~2~Q(3,3l l•S2*C2+Q(2,2)*C4 Q~(i,2)={0( l,l)+Q{2,2)-4.~~*0(3,~ll~SZ*C2+Q(l,2)*(54+C4l

Q 8 ( 1 ' 3 ) = + {(J( l ,l ) -Q ( 1 ' 2 ) - 2 • i.) '" (J (.1 ' 3 ) p s * c 3 + ( Q { l ' 2) -() ( 2 ' 2 ) + 2 • 0"' H)( 3 , 3 I l * S3 *C o c; u d > = + < Q c i, u -o < t, n - 2. c" c CJ .3 i i * s 3 * c +' o < i 9? J -Q < 2 , z > + 2 • o *

1'~ ( 3 ,3 ) ) * s ..... c 3 Q B ( .J , 3 l = ( Q ( 1 , 1 ) + G ( 2 , 2 l - 2 • · J • ~ •< C ( l , 2 l - 2 • :; C >:< Q ( 3 , 3 l ) * S 2. * C 2

i+Q(3,3)*(S4+C4l Qe<Z, l )=QB( i,2) Q8C3dl=~ac1,J> QS(3 ,2l =QB(2,3) IF( 1-l >15,llt,15

C H ( l ) I S D I ST AN CE FR 0 M M I 0 P L A r\ E HJ T w~ 8 8 T T () M Cl F T HE ! ' T H L t\ "l PJ A 14 HC i l =-DEPTH/2 .OO 15 Hll+U=HCil+T(l)

Dl=HCI)-H(l+l) D2=(H( I)**Z-d(I+l)**Z)/2.-JO 03 = ( H ( I l * * 3-H I+.!. l '~ '~ J) I 3. v ~

c TH:: SI:.Ji~S JF 01,02,D3 ARE CHAi\i,;rc '\S TH: PLY LAYLIP IS DIFFERENT FR0."1 C THAT I~ MY M.S. PRCGRA~

01=-l)l 02 =-02 03=-D.3 no 25 K=l,3 DO 2'5 J=l,3 DCK,Jl=D(K,J)+8l~Q~(K,J)

JJ=J+3 D ( K , J J ) = 11 ( K t J J ) + 02 * ~ 3 ( K, J ) KK=Kt3

25 0(KK,JJ)=D(KK,JJ)+03~Q8(K,JJ OC' 3J K=l ,.3

0 .-o

DC 3J .J=4,6 30 D(J,Kl=O{K,J) l J co~n PWE

hR IT E ( 6, 1 l 0 ) N 110 FCR~AT(///lCX, 1 LA~INATE CONSTA~1·s 1 112x, 1 NU~QE~ OF LAYEqS= 1 ,I3)

inR n;:c 6 ,211 > 211 FO~~AT(///2X, 1 l~PUT VARIA2LES [F ElCH LAMI~A'//)

DO 201 I=l,l\ nR I T:: ( 6 , 11 1 ) T ( I } 'E 11 { I ) ' :: 2 2 ( r ) ' 1; l 2 ( I ) , CN u 1 2 i I ) , T f-{ f: TA ( I )

111 FGR"1AT(2X,F6.4,2X,3E20.7,2X,F6.~,zx,F6.2/) 201 CO·\iTI~U~

i~R I TE ( 6, 2 12 ) 212 FO~~AT(///2X, 1 THE ABD MATRIX 15 1 ///l

i'i R IT t< .:i , 1 12 ) ( ( D ( t<., J ) , J = 1 , 6 } , K = l , 6 l 112 FORM.H('..J' 9 6'=.2~.8//)

DO 501 I=l,3 OCJ 502 J=l,3 AIJ( I,J}=D(I,JJ i31J( !,J)=f)( I,J+3) DIJC I,Jl=D( 1+3,J+3 l

5 J 2 C Ci'H I '-J U E 5 0 1 C G ~-J T I NU E

PAK( ISHELL)=BIJ( 1, 2) PAP( ISHELU=OIJ(l,l) DC 5J4 I=l,3 DO :,,:,4- J=l, 3

5 J 4 S ~· ."'-. L A ( I , J ) = f'. I J ( ! , J ) CALL MINV<SMALA,J,CET,L,M~,9)

CALL GMPRO(S~ALA,a!J,SMALB,3,3, 3,S,9,9) DO SJ7 I=l,3 DO 5J7 J=l,3

5:) 7 S "1 Al J T ( I , J l = S MJ~ UH J ,I )

___, 0 •o

CALL GMPRO(S~ALBT,BIJ,RR,3,3,J,0,S,9) D:J 5 'J 8 I = l , 3 DO :3J8 J= l d

503 S\11\l')( I ,J)=R~( I,J)-OLJ( I, J l C EVALUATI~N OF TH~ CONSTANT CQ~~FICf~NTS IN ThE SHfLL EQUATIJ~S

PC = S ·,14 L D ( l , l ) - S ~ l~ LB ( 2, l ) * * 21 S '•11\ l A ( 2 , 2 ) c Ri\O IS Hii.:: 1~NER ?.AL:rus OF Tli'.: CYLil\JD[:K

~~ R I T f: ( 6 , 7 i l ) R A D 7 1 l F 0 R r'-1 ti T ( I / 9 R ,\ C = 1 , E 1 5 • 7 )

3 C = ( l • :) IR AD l :i:: ( S i\1 A Lf3 ( 2 , 1 ) I S tit, L ,'; ( 2 , 2 l + S I J ( 1 , 2 ) ) CC=-lIJ(2,2)/RlC**2 XXX( ISHELL)=CC wR I T t ( b , 5 l u )

510 FOR~AT(///5X,' THE CG~FFICIE~TS A,A,C I~ THE SHELL EOUDTIONS'l WRIT[{6,511)PC,BC,CC

511 FJR~~T(//3(5X,E22.7))

IFIBC.LE.l.OE-04l8C=J.O Disc~ec*BC-4.0#PC*CC

IF(D!SC.LT.O.CJGU TO bOl i-IRI T:;:(6 ,602)

6 <J 2 F C R '-L\ T ( I I ' T H E F 0 0 T S M ~< * 2 :, ;.<. f'. R ;_ 1-' L 1 l RMllR=-BC/(2.J*PCJ+SQRT(DISCl/{?.O*PC> RM22~=-3Cl<2.o~PC)-SJRT(CISC)/(2.0~Pc>

RA l)S 1 = 1\ n S ( RM 2 1 R ) RAt\S2=AbS(RM22R) I F ( R. ··12 1 R • GE • 8 • 0 ) ::.: ,Vj 1 R = CM PL X ( Sr: R T { L\ t, 2 S l ) , 0 • :j } IF(G~21P.LT.J.J)R~lR=C~PLX(S.),~~~T(P~8SlJ}

R ,v 2 ? = - R ;.11 R I F- ( R '-12 2 P .• GE • 0 • 0 ) RM 3 R, = C ~.{ ? l X ( '.:. 0 ~ T ( ;1 /\ n S 2 ) , 0 • C ) I r: ( ,< ··12 2 r\ • L T • :; • I] ) K :'! 3 K = c i'! p L x ( c • ·.~ 1 s \.Ji; T { " /'. es 2 ) ) R 1'-\ t+ ;~ = - ,;.z ,"1 3 R h rz I T c ( u , 6 0 3 ) R M 1 R , R ; 12 '< , P. \1 3 ~ , P '·' t+ f<

I=>

G':J Fl 1JO1 6Jl R~2l=CMPLX(-R(/(2.0*PC),+SCRT<-rISC)/C2.0~PC))

R~22=CMPLX(-BC/(2.0=PC) 1 -SOkT(-D!SC)/(2.0~PC)) RM l =-C '.)(m T ( RM2 l l ~~~2=-K."11

RM3=CSGRT(~M22l

R~.'.t= -f'.; .. 13 WqlT~(6,603)R~l,PM2,R~3,R~4

603 FORMAT(//' Tht Four< ROOTS OF TH[ HC:•L.iGENECUS PA~f OF THE SOLUTIU''1 l 4RE I //(2E2C. 71))

Y Y Y ( I S Ii i: L U =A B S ( R E J\ L C KM l I ) l Z Z ( I S H EL U =ABS ( A 1 M -~ ~ ( P :•11 ) )

lOJ l C!J1'H Ii'JUf Cil=XXX(2) SI2=XXX(l} CII=XXX(3) 4A=YYY(2) Af3=YYY(l) .1C=YYY(3) AD=lZZ(J) WRITE(6,606)AA,A6,AC,AD

606 FOP.~,1Afl// 1 Tt-<E ClJNSTANT fXPfJ~'i!:'lTS l'-J THE Dt:FLECYIOf\i SOLUTIQ\lS FO·~

lRCGIJNS Ilri2,Il ~~SPECTIV~LY ~RE 1 //4E20.7//)

C HL IS Tiff HALF CRACK LEl\GTH on 3J01 IHL=l,IC~Kl HL=C.~KL( IHL) Al =A;~'"'lil BL=Ab*HL SA=CQS(nl) SR=CCSH(BL) ~C=S INC BU SD=SI·m(SL)

SH= c.::;s < 4U SI:::CCSH(AL) S J = S If'; { ~ L ) SK= S I \ft ( i\ L) t-11'. =S;''~sf., H~= SC:'< sn liE=Sf\'"')C HF=SC*Sf3 HG= SrPS I HH=SIPSK rlI=SJ''Sl HJ=SJ*SK RJ1=2.J*(HG-HJ* (HH-hl )/(rH+Hl)) R J 2 = 2. '-'* ( H A-H B* (HE -HF ) I ( HE +Hr: ) ) R J :> = 4 • 0 *A A * A,'\.* P .I\ P { 2 ) "' { HJ +I- G""' ( hf-< - h l l I ( fl H + H I ) l R J 4 =It • -:) * A 8 * :. 3 * p A D ( 1 ) ~ ( H 3 + r A ,., ( Hr~ - h F ) I ( ~i [ + H F ) ) R Ki' l = r l K ( 2 ) * R J l/ ~ :'. 0 + iU 3 2KP2=PAK{l)*RJ2/RACt~J4

'<. K P 3 = P l\ K { 3 l n, t:i. D + ( J\ C * ,'\. C + A D "' ,r_, D } '< ' A 1-l ( 3 ) RKP4=~KPL/PJL+PKP2/RJ2-RKP3

~ K P 5 = ( P !\ K ( 2 ) I R A C - R. K P 1 I P. J 1 ) I C 1 l ~KP6=lR~P2/RJ2-PAK(l)/KAC)/CI2

C21QJ={RKP5+RKP6)/RKP4 C9QO=!C21QC-l.~/CI2)/RJ2

c l lJ •.J = < c 2 i w , J + r • -~j 1 c r 1 l nu l P J R. l = 2 • :J"' C 1 Q J * ( 4 • 0"' H rP H I '~HG- 2 • C1 "'HJ "' ( I- i-i '" H H - H 1 "'H I ) ) I ( H H + H I ) * ,,~ 2- l • 0 0 'J

l/Cil P J ?, 2 = 2 • } * ( H H * H H + f- I '* H I l I { { I- H + 1-i I ) '-' !:. ,\ l P FU = 2 • i) ,;,: C 9 Q 0 * ( 2 • 0 '" H Q '~ ( H r; * h :: - f-< F -·, H 1~ l -4 • 0 "' H A * H E "' H F ) I ( H E + HF ) "'" * 2- 1 • C 0 0

l /CI 2 P JR 4 = - 2 • G :*'{ r. E ~:tff +HF r.- HF ) I ( A 8 ,., ( h:: +r1 !- i ) CPl=RJ3~HJ/(A~*P~P(2}*(HH+Hill

N

CP2=~J4*HB/(A8*PAPCll*(HE+HFll

C P 3 = 2 • U ., A A * ( 4 • 0 "' ~ A * A A "" P :\ ? ( 2 l ;c ( f-j f ! 'i'.f 1 H + I-'. l " H I l - H G * R J J ) I ( H H + H r ) C P-=t = 2 • (.} '~ t\ [3 ~' ( 't . J"" ;.'\ G ,~ 1'. B ~ P ~ P ( 1 ) '' ( !-i E * I- c + f-' r- *HF ) - HJ\ *P. J 4 ) I ( h E t HF ) CP6=-(CP3*~Jl-RJ3*CPl)/CCil*RJl*~?)

CP7=(CP4*~J2-RJ4~CPl)/(CI2~RJl~*ll

CP5=-CI1*CP6+Cl2~CP7

OC2l=CCP6tCP7-C21J0=CP5)/~KP4

DC9=(DC21-CP2*C900)/RJ2 DCl=<0C21-CPl*ClQO)/RJl lUBYL=PJRl+OCl*PJR2+PJR3+CC9*PJ~4

IF(ZU~YL.LE.O.G)GO rn 30Jl C GAMAt'.1 IS Hit SPECIFIC f\DHE:SIVE F?:~STU~E C:NEPGY PJ LH/IN. c PCR rs THl: CRITIC.:\L HJT~RNAL PPESSUP.i:: rr~ THE CYLI~JOER AT \~HICH

C DELAMIN>~TIOi-..J PR:JPAGAfcS C PPP=PCR/(GA~AAl**J.5

c c

PPP=S•~RT(2.J/ZUf3YL)

·,~P. I T f ( b , 7 ;) 2 l H L, PPP 7J2 FOR~~T(//' THF HALF CR~CK L~N~Trl !S'//EZ0.7// 1 TH~ CRITICAL INTL.

lPRESSIJK_;: Il'J T~f CYLPWER, Ii'~ risr, /'.<T viHICH DEU'ir•lHJATION PQ.OPAGATES z, JIVIDED BY'/' THE SCARE ~CCT OF T~~ SPECIFIC ADHESIVE FkACTURE 3ENERGY (IN La./!N.) IS'//E2U.7)

3.JOl CGNTHUE STtJP aw

C SAMPLE 8ATA SET c r '-'

E~FECT llF ANlSGT~QPY; PRESSURE GN C~~CK SU~FACE

(l) :B-ECO) (2):G-El45J 3.Q 30

w

(). l 0.9 l.9

l 0.05

l 0 •. '.) 'j

2

0. l ') 1.-J 2.J

(J • 2 1.1 2.2

·~ • 3 .:~ C' ,:; J ·~"J E + 0 6

G • l S 5 ~ D :; ::; E +- C 8

a.cs o.1s 0.3JJO)OJE+-G8 0.1350JOOE+08

l). 2 5 l. 2 2.4

0.3 1. 3 2.6

0 • 2 rn :· (', r ~~ +- :_· 7

c. -, Q :_i J .... : -~~· ~ :: + :-j (1

O. 2 700.:JuJE+J 7 O. 7~00COJ;:+uu

0 o It

1.4 2oH

0.5 1 • 5 J.-)

u.o 1. 6 3.5

J.7 1.7 4.0

0.9300QQJE+06 J.21 0.00

~.83JQJ00E+06 0.21 45.JO

u.93JOJOJE+06 0.21 o.oo 0.8300000E+06 0.21 45.00

J.8 l. 8 5.'.)

__, __, ~

Ul

c !'.. P P c '.\ 1) I X P c C PRCULE:--: 2: l_J,'\Dlf\G CN l'Ji\it:R. CYlli\l.>:IC:\L SURFACE; FLH CR/<CK TIP; nDPLA"JE C hXI~L ~ SHEAP STRAINS IGNORED c c

CJMPLEX RM21,R~22,R~l,RM2,RM3,~~4 CC~PLEX RMLR,RM2~,R~3~,R~4R,C~PLX,CSCRT

0 0 I Mt= :·! S I 0 N 0 ( 6 , 6 ) , T ( 2 ) , Q ( 3 , 3 ) , :: l 1 ( . ~ ) , c 2 2 ( 2 ) , G l 2 ( 2 ) , G N lJ l 2 ( 2 ) , l GN U 2 1 ( 2 ) , THE TA ( 2 ) , 1-l ( 3) , Q 13 ( 3 , 3 l

D I :·'. E "JS I lJ N A I J ( ::> , 3 ) , f3 I J ( 3 , j ) , 0 I J { j , J ) , SM AL E ( 3 , J l , Sf.~ AL A ( 3 , 3 ) , 1 S :\' ;;, l r~ T ( 3 , 3 ) , RR ( 3 , 3 ) , S M 1\ L D ( J , 3 ) , L ( 3 ) , :.; :-i ( 3 )

0 I :1 E N S IO N X X X ( 3 ) , Y Y Y ( 3 ) , l l l ( 3 ) D i "1 c :-1 S I 0 N P A K ( 3 ) , P t\ P (3 ) JI~ENSICN CRKL(50) OI~~~SIJN TITLrC4J) RE.'.'\C(5,797 HITL':

7 9 7 F fJ K \l ·\ T ( 2 :) A 4) WR!Tc(o,797)TITLE R E ;.\ j; ( 5 , 7 0 1 ) R A 0 , I C q K L

7~1 fOq~~T(FlC.:,lS)

RE:\D(5,2u0ll ((t;_K.L( I} ,I=l ,ICt:>KL) 20)1 F0RMAT(l0F8.0)

c l s H E L L = 1 ' 2 ' 3 c c R k E s r GJ c T () T J-l !: s c L u T r u, s F 0 R T H E R c G I l) ~'! s I L , I 2 ' I I R E s p ::: c T I v E L y DO lJOl ISHELL=l,3

C N I S T H l: ~,; U e-1 R E ;:{ 0 r- L A i-1 I N ~ t: R E A D ( 5 , l ;J 0 )f~

100 FC'<·i/\,T(I3) C INITitLIZE 0(6,6) WHICH IS THE A~C ~!TPIX

on 12 1 = i, 6 on i2 J-=l,6

12 D(I,J)=0.00

c THICK~~ss JF LAPI~ATE IS NAMEC D~DT~

c T(M) IS THc THICKillF.SS OF THE :V 1 Tr Lt::JU-/\ '<. E :1 L ( '..> , l J 1 ) ( T ( :1• l , M = l , r-.i )

101 FC~~AT(l0F9.0)

D E P T H = ':. • C rj D ~ 3 .··1 = l , i'J

3 OEPTH=Di:PTH+T(.,i) C Q MAT•UX UF EACh LA't!NA

DO 10 I=l,N RE A 0 ( 'j, l 0 2 I E 11 ( I ) , r: 2. 2 ( I ) , G 12 ( I l , G0.i U 1 2 { I ) , THE TA ( I )

1~2 FORMAT(JE2J.7,2F6.J) GNU2l<Il=GNUl2( Il*E22(1)/Ell<Il GNtJ2=GNU12( I l*•3':U21 (I) (J ( 1 , l) = f. l lC I> I ( l • 0 0-GN U 2 l Q(2 ,2 > =E22 < r >I< 1.oo-1;Nu2 > QC 1, 2 l =Ell ( I l >'.<GNU2 1 (I> I ( l. 0>: -G:JJ2 l (J(3,3)=Gl2( l)

C THETA +V~ FROM GLCdAL TO LAMINA P~!\CIPAL AXES IFCTHETi\(I) .EQ.CJO.O)GQ TO 5.:; AN~L=CTHETA(l)/l80.0J)*3.141~927

C= CiJS l ANGL l S=SIN(ANGU GO TO 60

SC C=G • .:,::i S= l •. )Q

6C C2 =C *~'2 S2=S'°'~'2 C3=C*C2 S3=s~~s2

C4=C..-lcC3 S4 =S *S 3

C THE QB ~ATRIX OF EACH LA~INA

O"I

CJ 13 ( 1 , l ) = Q ( l , l.) * C 4 + 2 • 0 * { Q ( l , 2 H· 2 • CJ 0 "· 1 ( 3 , 3 ) ) * S 2 * C 2 + Q ( 2 , 2 ) * S 4 () p ( 2 '2. ) = Q ( l ' l ) ·~ s 't + 2 • J 0 * ( () ( l , ~ ) T 2 • 0:; :.c lJ ( 3 , 3) ) * s 2 *C 2 t- Q ( 2 '2 ) ~ c 4 Q ~ ( l , 2 ) = ( (J ( l ' l ) + ·~ ( 2. , 2 } -4 • J 0 ,.. i) ( 3 ' 3 ) ) * s 2 * c 2 +Q ( l , 2 ) * ( s 4 + c 4 ) QP(l,~>=+(Q(L,Ll-Q(L,2)-2.G•((3,3))*S~C3+(Q(l,ll-Q(2,2)+2.0~

lQU,31 )i;i:S3*C Qr.; ( 2 , 3 ) = + ( Q ( l , l ) -Q ( 1 , 2 ) - 2 • C :!: (J ( :~ , 3 ) ) "' S 3 >,'< C + ( Q ( l t 2 ) -Q ( 2 9 2 ) + 2 • <: :-!:

lQ( 3, 3) l"'S*CJ <J [~ ( 3 ,3 ) = ( Q ( l ' l ) + (~ ( 2 , 2 ) - 2 • 0 Cl 'I'<.: ( l ' 2 j - 2 • .) Cl * (J{ 3 , 3 ) ) * s 2 * c 2

l t-0 C 3 , 3 l * C 54 t-C 4 ) (Ji3(2,U =Qf3( l,2) Qd ;,,i )=QrHl,3) l.)f3(j,2)=(JF3(2,3) IF( I-l) 15,l4,l5

c H(I) IS DISTANCE FR.OM j-1lDPLANE rn T·ff GUTTOM Of THE I'TH L!\MINA 14 H( I )=-O!:PTH/2.JJ l 5 H ( l + l ) =rl { I ) + T ( r )

Dl=H{ I )-H( I+l) 02 = ( H ( I > * * Z -H ( I+ l) 'I":(< 2.) I 2.. 'J J D3 = ( H C I > * >!:: 3 -h C I + 1 ) * * 3 l I 3 • 0 0

C TrlE SIG~S 1F Dl,02,03 ~RE CHANGED 4$ THE PLY LAYUP IS DIFFERENT F~O~

c TH!.;T !IJ ~w i.:.s. PRCGRAM Di=-Dl D2=-D2 03=-J3 DCJ 2~ K=l,3 DO 25 J=l,3 D(K,J)=O(K,J)+Dl*OB(K,J) JJ=J+3 D(K,JJ)=O(K,JJl+DZ*QB(K,J) KK=K+3

25 0(KK,JJl=DIKK,JJ)+D3*Q8(K,JJ DO 3J '<=1,3

~

DC 30 J=lt,o JC r.LJ,Kl=O(l<,J) 10 cc~nr:--JuE

v-.P. I T [ ( 6 , 11,J ) N llC FC~M~T(///lJX,'LAMI~ATE CCNST~~TS'//lX,'NU~BER OF LAYfRS=',13)

't;~, I TE ( 6 , 2 11 ) 2 1 1 F o [) ~.q\ T u 1 12 x , 1 11'; Pu T v r~ F r f\ ~ L c s ,Jr c.. :. c H LA :Vi I ~JA • 1 1 >

DD 2Jl I=l,N ~" ~ I T £'. ( 6 s 1 l l } T ( I ) ' E 1 l { I ) ' E 2 2 ( I ) ' G 1 2 ( I } ' er,: u l 2 ( I ) ' THE T {\ ( I )

111 FORM4T(2X,F6.4,2X,3E2Ua7,2X,~L.4 1 2X,~h.2/)

201 CC1f!TINUE Y.kIT=C6'1212l

212 F0°~~T(///2X, 1 THE 430 MATRIX [~ 1 ///l

~iR I Tc ( 6, 112) ( ( 0 ( K, J ) , J = l , 6 l , i< = l , 0 l 112 FOR,Y\.'\T( 1 J 1 ,6E2i.i.P//)

00 5 J L I= 1, 3 Du 502 J=l,3 AIJ( 1,Jl=D(I,Jl B I J ( I , .l ) = D ( I , J -t- 3 ) OIJ( I,J)=O( 1+3,J+3)

so2 cc:n uH.:E 501 CONTI ~:UE

PAK{ ISHELU=BIJ( 1,2} PAP( ISHELU=DIJ(Ltl) DD 5 .J 1t I = 1 , 3 DfJ 5J4 J=l,3

5 J 4 S ~~ f\, L 6. ( i , J ) = /'. I J ( I , J l CALL M[NV(SMALA,3,CEf,L,NV,q) C /I LL r; :'1 PP u ( SM AL A , R I J , SM AL 3 , 3 , 3 , 1 , 'J , 9 , ') l oc 5J7 l=l,3 DO 507 J=l,3

5G7 SM~LGT(I,Jl=S~ALR(J,Il

co

c ALL G ,\1 p RD { s....., AL e r ' B I j 'RR 'j ' _) ' 3 ' (~' s' r..; ) f) G 'J .. rn I = l , 3 OU JIJ~ J=l '3

5 ::> 8 S ~·, 1'. L ! l ( I , J ) = q R { I , J ) - G I J ( I , J l C E VA LU 4 T i L ~ i~ 0 F THE CC NS T A I\ T C 0 E F F I C I !~ r, T S I 1'J T HE SHELL c Q U ,~ T I 0 N S

PC= S M f.. L J ( l , 1 ) - S ~,.I\ LB ( 2 , l l '""' 2 I S ·1 ". L t" { 2. , 2 l c RAD rs TH'.: INi--JER ~l~OIUS OF THE CYLU:c:::P

\rJ R l T :-: ( b , 7 l l ) R A. D Hl FUR"l1\T(f/' RAO=• rE15.7J

F3 C = ( l • JI R 1\ 0) * ( SM AL B ( 2 , l ) IS t>' !\ L .\ ( Z , 2 l + ~~ I J I l , 2 ) ) CC=-~IJ!2,2)/RA0~~2

XXX( I SH[ll l=CC WRIH:(6,51J)

510 FiJR''i,\T(///5X, 1 THE UJEFFICH~:IS t1,t),C IN THE SHELL EQUATllJNS 1 )

WRITEC6,511)PC,BC,CC 51 J F 0 R :~ .f, T ( I I 3 ( 5 X , c 2 0 • 7 ) )

I F ( i3 C • L r:: • l • () E- . A l B C = ,-1 • C 01sc~~C3 BC-4.o~rc*CC

IF(CISC.LT.O.O)GO T~ b0l i~ K l T :;: ( 6 , 6 J 2 )

602 FORM~T(!/' Tl-<E F:OOTS ~',**2 t.RC ::<::AL') RF 21 R = - P,C I ( 2. C* PC) +SORT (!)IS Cl I { 2. ~. '·' P': ) PM22R=-GC/{2.0*PC}-SURT(OISC)/(2.a~rcJ

H A ;5 S 1 = :\ H S ( R ~~ 2 1 R ) R ·'\ i3 S 2 = .\ 3 S { ~ M2 2 ;:\ ) I F { R ·.: 2 ll~ • G E • J • J l P ,v l. P. = C !•'. PL X { S r n, 7' ( P :~ E S l l , G • •J ) lF(~~21R.LT.O.J)RMik=CMPLX(O.~,SQ~T(~~RSl))

R r-12. R = - ~ '~ l R I F ( :::> '· 12 2 R • G E. 0 • 0 ) ~ ~13 R = U i P L X { S C ~ T ( ? !\ 0 S 2 ) i 0 • 0 ) I F ( R >l 2 2 ~ o l T o 0 • 0 ) ·~I~ 3 ?. = C. M P L ,( l '. . ; , S i) :~ T ( ~ ,\ G S 2 ) ) R.'•ViR =-F--1"13R WRIT~(6,603lRM1R,~M2~,PM3R,~~4~

l.O

GCJ T'.J l 1~!) l 6Jl RM2l=C~PLX(-8C/(2.0*PCJ,+SQ~T(-CI5Cl/(2.C*PC)I

~ .'·1 ;~ 2 = :::. :·1 :l L X ( - '3 C I ( .? • G * PC ) , - S Q .{ T ( - i) I S C: l I ( 2 • C ~~ P C ) l ~ ~'. l = :: S ~KT ( RM 2 l ) R/v'2=-R'11 R:-'3=CS•JRT(~M22)

R. II, !t = - IZ "1 3 kR!TE(6,603)RMl,QM2,RM3,R~4

603 FOR',.,,\ T ('/I' THE FOUR ROCT S CF LARE '//(2E20.7/))

YYY ( [$H~LL) =~BS ( P:AL ( RMl) l l Z l { I S :-E: L L I = A !3 S ( .U "1 AG ( R M 1 ) )

l J 0 l C 0 ;'~ TI ~ W E CI2=XXX(ll Cll=XXX(j) At1=YYY ( 2) AR=YYY(ll AC-=YYY(3) AD=lZl(3) ~RIT~(6,606)A~,A3,AC,~D

THi: i-.c:.:cc;Er--JEOUS Pt.RT fJF Tl-E SOLUTIG:'•l

606 FCR:·\.'ITC// 1 THE: C:J/\JST.'.INT ::xP:-:r--.c:·ns I~: THE D!::FLECTIQ:-.J S 1JLUTIONS F:JR 1 R E G I 'J N S I l 9 I 2 7 I I R E S PE C T I V E L Y ~ i'. E ' I I 1t i:: 2 (J • 7 I I )

C Hl IS THE HALF CP1">.CK LF~GTH no 3·::}1 IHL=l,ICRKL HL=CRKL( IHLl A L = ·~ .". ~' li L BL=At3"'hl S/\=C.JS (BL) sa=CJSH(~LJ

SC=SINCRU SO=SPJrl(3Ll SH =C 1JS (AU

N 0

SI=CCSH(AL) SJ=Slr'J(All SK=Sif\:H(Al) HA=SA~=SB

HS =SC *SD 1~~ =SA'~ SO HF=SC*Sl3 HG= Srl""· SI HH=SH "=SK HI=SJ~SI

HJ=SJ~'SK

R J l = 2 • ·) * ( H G - HJ* ( H H - H I }/ ( H H + H I ) ) RJ2=2.0*(HA-HP*(HE-HF)/(HE+Hf)) R J J = 4 • ) ,... A A >r- A 1\ * P .6. P ( 2 ) >,<( Ii J + r G * ( h H-1-: I ) I ( H H + H I ) ) RJ4=4.0*AB*AB*PtP(lJ~(HB+~~*(r~-~FJ/(HE+HF)J

RK?l=PAK(2)*~Jl/RAD+RJ3

RKP2=PAK(l)*RJ2/RAO+RJ4 RKP3=PA~(3)/RADt(AC~AC+AC~AD)*PAP(3)

KKD4=RKPl/RJl+RKP2/PJ2-RKP3 RKP~=PAK(3)/RAO-?KPl/RJl-RKP2/~J2

RKP6=q_J4/P..J2 C21Q~=lRKP5/Cil+RKP6/CI2J/RKP4

C 9 tJ 0 = ( C 2 l 1J ih l • J I C I I - l • 0 I C I 2 ) I i< J 2 ClQJ=(C21QG+l.O/CII)/RJl CPl=RJ3*HJ/(AA*PAP(2)*(Hh+Hl) I CP2=~J4*He/(AB*PA?(l)*(HE+~Fll

C P ::, = Z • ') •• A A * ( 4 • Cl ::: A A* A 1\ *PA P ( 2 ) * ( H H ><fif • + r I * H I ) - HG * R J 3 ) I ( H H + H I ) CP4=2.J*A6~(4.0~AB*AB*PAPlll*CHE~Hf+H~*HFl-HA*RJ4)/(HE+HF)

CP6=(CP4•kJ2-RJ4=CP2l/RJ2*~2

CPj=(CP3~RJl-RJ3~CPll/RJ1**2+CP~

DC 21 = ( CP 6/ C I 2- ( C 21Q0 +l • JI CI I ) "'CP ~ ) I~ KP 4 OC0=(0C2l-CP2*C9QC)/~J2

....... N .......

ZUOYL~l.O/Cl2+2.J•C9Q0*(4.0*HA*H~a~F-2.0*lHE*rlE-HF*~F)*HB)/(HE+HF)

l * ,,, 2 ~- 2. :) '~ ( HE* Ht: t- Hi:* liF I *DC 9 I ( A ? .. , ( rl C +Hf l l + 2 • •: *AC * 0 C 2 l / ( .\C * AC+ A 0 * I\ D ) llJBYL=-LUJ3YL IF(ZU3YL.LE.J.0)G0 TC 3JQl

C GAMA/.., IS THE SPECIFIC /\DHESIVE fF:.\'.:TUf~E :::N~R.GY P.J L8/IN. C PCP. IS THE CRifICAL UHt~NAL Pi\f:SSU~~ Ir·J THE CYLINJER AT i·1H[CH C DELA~!NATICN PROPAGATES C PPP=PC~/lGAMAhl**C.5

PPP=SQRT(2.0/ZUBYL) ~RI TC( 6, 7G2 )I-fl, PPP

712 FOR~AT(//' THE HALF CRACK LENGTH IS'//~2C.7//' THE CRITICAL INTL. lPR::SSLHE IN THE CYLINlJE:R, Ii'I PSI, t.T l~HICH O~LAMINATIO\I PROPAGATES 2, DlVIlJ~D BY'/' THE SQARE RCDT Jr T~E SPECIFIC AOHESIV~ FRACTUR~ JEN~RGY <IN LB./IN.> IS'//~2C.7)

c c

3001 CGNT It\J:JE SflP i: !\! J

C SAMPLE U1\TA SET c c ~FFECT OF A~ISOfROPY; PRESSU~E O~ IN~~R SURFACE OF CYLINDER

3.0 30 0.1 8a9 l.9

l 0. 08

l c.02

c. l 5 l , \ . "' 2.0

0.2 1.1 2.2

1). 3 .:.JJ:.J0•)~+08

0 .2 5 1. ~ 2.4

0.3 l.3 2.6

J. 2 7,:)Q'.)CJ[ +-J7

0.4 l. /t 2.3

J.5 0.6 1. 5 1. 6 3.0 3.5

0.q3oooooc:+o6

0.1 1.7 4.0

0.21 o.ou

o.a l • 8 5.0

_, N N

123

(:' f'.:•

0 n

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Q

l{'\ ()

l{'\

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r.; N

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0 +

+ + U

J U

.! l!J

·:...1 0

(~·; ~,

~

0 .-.

(') ~;

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'.'.) 0

I!'\ r<i

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+ Ll !

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er, cc

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ro ''\! 0 0

c f' pp i:;-.! [ ! x c c C PROBLE:~ .3: PRESSURl ON CR,\CK SU~F.\Ci-; Gr.f';:::RAL CRACK TIP GEOMETRY; SHEi\~ FOR.CE C C 0 NT I N U I T Y /\ C R 0 S S CR AC K T I P ; :"-1 lf) P L. .. ;,,; :: ti X I !, L E. S H E A R S IR A I N S I GNU R E D c c

CO~PLEX RM21,R~22,RM1,RM2,R~J,R~4

CO~PL~X RM1R,PM2R,RM3R,~~4R,C~PLX,CSORT

JOINENSION 0(6,6) ,T(2),Q(3,3),~ll(2),E22(2),Gl2(2),GNU12(2), lGNU2L(2),THETA(2),H(3J,Q8(3,3)

D I ME \IS I lli-J A I J ( 3 , 3 ) , B I J ( 3 , 3 ) , D I J ( 3 , :: ) , S !-1 AL B ( 3 , 3 J , S i"i AL !\ ( 3 , 3 J , 1 S ~Al f3 T ( 3 ,3 I , RR ( 3 , 3 ) , S to' AL D ( 3 , 3 ) , l( 3 ) , ""...,, ( 3 ) OIME'~SION XXX(3) ,YYY(3) ,lZZ(3) DIME~SIUN PAK(3),PAP(J)

~ DIMENSION OBK(6,6),CVK(6),CVP(61 DIME~SION CRKLC50l or~ENSIO~ TITLE(4~)

REA0(5,797>TITLE 797 FOR."'1AT(20~\4)

WR1Tc(6,797JTITLE RE~u(5,777lBETAl,BETA2

777 FOR~AT(2Fl0.0) WRITt(o,73liRETAl,RETA2

7 3 l F 0 RM AT (/ ' a ET A l =' t F 6 • 3 ' 5 x' ' l~ •: T .:iz = I ' F6 • 3 ) RlAD(5,7~l)RAD,I~RKL

7Jl FORMAT(FlO.O,I5J Rf M)( 5 , 2 0 01 I (CR KL< I ) , I= l , IC". KL )

20)1 FORMAT(l0F8.0) C ISH::LL=l,L,3 CORRESPOi'JC TO THE SCLUTICl'.S FCk THE REGIONS Il,I2,II RESPECTIVELY

O~ lJOl ISHELL=l,3 C N IS THE NU~BEq OF LA~I~AE

READ(5, lOOlN

100 FO;.{M!-T( I3) C I \J I T I J\ LI l t:: rJ ( 6 , 6 ) "m IC H I S THE l', rm '-~ ;\ tr• I X

0 i_l L~ I = l , 6 DC 12 J=l,6

l 2 D ( I , J l = : ... 'J 0 c THICK~~ESS •JF LA'.'-IIt!.Hc IS r~/IMEO CcPT.--i C T(Ml IS T~E THICKNESS CF THE M'l~ L~~I~A

R t .\ D ( 5 , 1 C l l ( T ( M ) , M = l , N ) 101 FlJRMAT(lJF8.0l

DEPTH=O.C'O on j ;·i=l,N

3 O~PTH=D~PTH+T(M)

C W ~1 A rn I X J F EACH LA .\l li'-J A DQ l J I::: 1, !-J R E .~ !J ( S , l J 2 ) E 11 ( I } , E 2 2 ( I ) , G l 2 ( I l , G :-.: U l 2 ( ! ) , Tr t TA l I )

102 F0RM~T(3[20.7,2Fo.Cl Gi;U21Cil=G/\iU12( ll"'1=22iil/Ell Cl) G td l 2 = (; i\i IJ l 2 ( I ) * G N IJ 2 i ( I ) Q ( l , l l = f. l 1 ( f ) I ( 1 • ;) 0 - G N U 2 ) 0(2 ,2) =E'22( Il/ (l .J0-GNU2) \~ ( 1 , 2 l = t: 11 ( I , '·' 1~ N U 2 l ( I ) I ( 1 • C: C - G t J U 7 l Ql3,3l=Gl2( IJ

C THETA •V~ FROM GLCBAL TC LAMINA PP!~CIPAL AXES IF(TH'.::Tl\(l).EQ.9').C)GO TO 5·; A i'J:..; L = ( TH E T A { I ) I l8 0 • J'J ) ::c 3 • l 41 ':5 'J 2 7 C=CfJS(ANGL) S=SIN(At\GL) GU TO (iO

50 C=·'.;.C)') S= 1 • G :

6 0 c 2 =C: '~ ,~:::.

S2=S".<*2

N <.n

C3=C*C2 53=5'~52 C4=C~~c.3

54=S"~S3

C THE QS MATRIX OF EACH LAMINA Q 3 ( l , l ) = Q ( l , l ) '"' C 4 + 2 • :J ~ ( Q ( l , 2 ) +- 2. J J ,. I) ( 3 , ,j ) ) .\< S 2. 1i' C 2 + () ( 2 , 2 ) * S 4 Q 8 ( 2 T 2 ) = Q ( l ' l ) * s 4 + 2 • J 0 * ( Q ( l , 2 ) + :2 • 0 1J * c ( 3 ' 3 ) J *S 2 * c 2 + \) ( 2 ' 2 ) * c 4 'J Li l l , 2 I = ( Q < l , l > + CJ I 2 , 2 J - 't • rJ c~ -:: (J < 3 , 3 l l ¥ S 2 >'< C 2 + Q ( l , 2 > >:< < S 4 + C 4 > Q !:3 ( l '3 ) = + ( q ( l ' l ) -l) ( l '2 J - 2 • 'j ''" 0 ( 3 ' 3 ) ) >'.': s ~· c 3 ... ( (,) { l '2) -Q ( 2 '2 ) + 2 • 0 *

lQ(3,3) l*S3'i<C Q8(2,3)=+(C(l,ll-Q(l,2)-2.C*~(3,3) >~S3*C+(Q{l,2J-Q(2,2)+2.0*

lQ( 3,3) l*S*C3 Q i3 ( 3 , 3 ) = ( lJ ( l , l ) t :'~ ( 2, 2 )- 2 • J 0 * Q ( l , 2) - 2 • 'JO* Q ( 3 , 3 ) ) * S 2 *C 2

l+Q(3,3)*($4+C4J QR(2 ,1) =Q8(l,2) QJ:1(3rl)=QlHl,.3) Q3(3,2l=QB(2,3) IFCI-Ul5,14,15

C H(I) IS DISTANCE FROM MIDPLANE TD THE 80TTOM OF THE I 1 TH LA~INA 14 H(l)=-DEPTH/2.00 15 HCI+ll=HCIJ+T<Il

Ol=H(l)-H(l+l) 02=(!1( I )**2-H( I+l l**2 J/2 .00 D3=PH I )**3-H( I+ll**3}/3.00

C THE SIG~S OF Dl,02,03 ARE CHANGEC ~S T~E PLY LAYUP IS DIFFERENT FROM C THAT 11\J iW M.S. PRCGRAM

Dl=-Dl 02=-:)2 OJ =-03 DfJ 2 5 '<.= l , 3 DO 2 5 J= l, 3 O(K,Jl=D(K,J)+Dl*QBC~,J)

I'\.)

O'l

JJ=J+3 0{K,JJ)=D(K,JJ}+D2*QB(K,J} KK=K+3

25 0(KK,JJ)=O(KK,JJ)+D3*QB(K,J) DO 30 K=l,3 DO 3J .J=4,6

30 O{J,K)=D{K,J) 10 CONTINUE

WRITE(6,110)N llG FORMAT(///lOX,•LAMINATE CONSTA~TS 1 /12x, 1 NUMBER OF LAYERS= 1 ,13)

WRITE ( 6, 21 ll 211 FORMAT(///2X, 1 1NPUT VARIABLES CF EACH LAMINA'//)

DO 2 0 l I=l, N WRITi:(6,lll>Tl I),Ell(I),E22( I),Gl2( I),GNU12(I),THETA( I)

111 FORMAT(2X,F6.4,2X,3E20.7,2X,F6.4,2X,F6.2/) 201 CONTINUE

WRIT'.:(6,212) 212 FORM~T{///2X, 1 THE ABD MATRIX IS'///)

WRITE ( 6, 112) ( ( 0 ( K, J), J= l , 6) , K = l , 6) 112 FORMAT('0 1 ,6E20.8//)

DO 5Gl l=l,3 DQ 502 J=l,3 A I J ( I , J ) =D ( I , J ) BIJ( I,J)=O{ I,J+3) DIJ (l, J) =D( 1+3,J+3)

502 CONTINUE 501 CONTINUE

PAK( ISHELU=BIJ(l,2) PAP{ISHELL)=DIJ(l,L) DO 5'.)4 l=L,3 DO 504 J=l,3

504 SMALA(l,J}=AIJ{I,J)

N -....J

CALL MINV{SMALA,3,CET,L,~M,9) CALL GMPRD(SMALA,BIJ,SMALB,3,3,~,y,9,9)

DO 5J7 I=l,3 DO 507 J=l,3

507 SMALBT(I,J)=S~ALB(J,I)

ClLL GMPRD(SMALBT,BIJ,RR,3,3,3,9,9,9) DO 5):3 l=l,3 DO 51J8 J=l,3

508 SMALD( I,J)=RR(I,Jl-CIJ(l,J) C EVALUATION OF THE CONSTANT C8EFFICIENTS IN THE SHELL EQUATIONS

PC=SMALD(l,l)-SMALB(2,1)**2/SMALA{2,2) C RAD IS THE INNER RACILIS OF THE CYLiNCER

wRITE(6,7ll)RAD 711 FORMATl// 1 RAD=' ,El5.7)

BC=(l.0/RAD)*lSMALB(2,l)/SMALA(2,2l+BIJ(l,2)) CC=-AIJ(2 9 2)/RAD**2 XXXC ISHELU=CC WRITE{6,510)

5 l 0 F 0 R '.'-1 A T {/ / / 5 X , 1 T 1-i E C 0 E FF I C I U; T S A , E 1 C I N T H E S HE L L EC U A T I 0 N S 1 I WKITE{6,51l)PC,BC,CC

511 FORMAT(//3(5X,E20.7)) IFCBC.LE.l.OE-04lBC=O.O DISC=3C*BC-4.0*PC*CC IFCOISC.LToO.O)GC TO 6Cl WRIT':(6,602)

602 FORMATC// 1 THE ROOTS M**2 ARE REAL'} RM 2 l ~ = - B C I ( 2 • Cl * P C l + S QR T C 0 I S C ) I ( 2 • -o "'' P C ) RM22R=-BC/(2.0*PCl-SQRT(OISCl/(2.0*?Cl i<.ABS1=ABSCRM21R) RABS2=ABSCRM22R} IF CRM2lR.GE.O.O)R.MlR=CMPLX(SQR.T(RAE\Sl),O .0) 1FtRM21R.LT.0.0)RM1R=CMPLX(O.J,5QRT!~ABSll)

N co

RM2R=-R"'llR lF(RM22R.GE.Q.O}RM3R=CMPLX(SQRT(RhES2),0.0) IF(~M22R.LT.OoOIRM3R=CMPLX(O.O,S0RTCRABS2))

RM4R=-RM3R WRITE(6,6J3)RM1R,PM2R,RM3R,R~4~

GO TO 1001 601 RM2l=CMPLX(-3C/(2.0*PC),+SQRT(-ClSC)/(2.o~PC))

RM22=CMPLX(-BC/(2.0*PC),-SQRT(-DISC)/(2.0*PC)) RMl=CSQRT(RM21) RM2=-RMl RM3=CSQRT<RM22) RM4= -RM3 W~ITE(6,603)RM1,RM2,RM3,RM4

603 FORMAT(//' THE FOUR ROOTS OF THE HCMGGENEOUS PART OF THE SOLUTION lARE 1 //(2E20.7/))

YYYC !SHELL)=ABSCREAL<RMl)) lllC ISHt:LU=ABSIAIMAG(RMl))

1001 CONTINUE Cll=XXXC2) CI2=XXX(l) CII=XXX(3) AA=YYY(2) AB=YYY(l} AC=YYY(3) AD=ZZZ(3} wRITEC6,606)AA,AB,AC,AD

606 fOR~Af(// 1 THE CONSTANT EXPO~ENTS IN THE DEFLECTION SOLUTIONS FOR lREGIONS Il,12,II RESPECTIVELY ARl 1 //4~20.7//)

C HL IS THE HALF CRACK LENGTH DO 3uOl IHL=l,ICRKL HL=CRKL·( IHU AL=AA""HL

__, N \0

Bl =ABr;;HL sr.=CUS(cL> SB =C CS H (BL) SC= SIN (BL) SD=S u~HCi3L) s:~=CCJS (All SI=CflSH(AL) SJ=SI!\l(AL) SK=S I f'.:H (ALI Ht.=s~=·sa

1-'.B=SC,...SD HE:=SA*SD HF=SC"'53 rG=SH"'SI HH=SW•SK HI =SJ:,;•SI HJ=SJ~·SK

IF( C IllL/2>:t2) uEfJ .IHL lGO TO 6?"1 HHJ\= HA 1-!HB= r~ rlHE=HE: HHF=HF HHG=HG HHH=HH HHI=Hl HHJ=HJ

059 XXA=•2.J*HG*PAK(2)/RAC+4.C*AA~AA*HJuOfP(2) XX3=-2.0*HJ*PAK(2)/R~0+4.0*~A*A~$HG~PLP(2)

XXC=+2.0•HA*PAK(l)/DAD+4.0~AB~AB*~B*PAP(l)

XX0=-2. )*l-H3*PAK ( l) /h.AD+4. J*AP~AP~'ht:~rAP ( l) XXE=(PAK(ll/C!2-PAK(2}/Cll}/~AC

XXF=PAK(3)/RAD-PAP{3)*(AC*AC-AD~AO)

w 0

XXG=-2.0*AC*AO~PAP())

YYA=•2.)*Ah*(HH-~l)*PAK(2)/PAn•~.oft~~~*3*(1-H+Hl)*PAP(2}

Y Y :1 :: - 2 • rJ *A ;:i. ¥ ( H H i-1-< i ) *?AK ( 2 } Ir;.;. AD+ 4 • J ,, L~ :(< ,., 3 *- ( 1-1 H- H I ) *PAP ( 2 l y YC = + 2 • (""AB* ( HE- r F , * p AK ( 1 ) IR r '.) + 4. (, .... iJ p * ~' 3 ~ ( I- E +r F ) *PAP ( l ~

Y Y D = - l • ;) * A B.;.: ( h E + 1c F } "' P A K ( l ) I 'r< t~ •_H t1 • G "' /, [; "' >:< 3 * ( H E- H F } * P A P ( l ) YYE=AC*(-PAK(3)/RAD-(3.0*AO*fD-~c~:c>~PtP(3})

YYF=A~*(-PAK(3}/RAD+C3.0*AC¥hC-h0"ACl~PAP(J))

DO 707 J=l,6 CVK(J)=O.O DO 7 :Y/ I= l, 6

707 DBK ( I, J) =u. 0 DBKC l, 1)=2.0*HG DSK(3, l}=HH-rlI DBK ( 5, l I =XXA CBI\( 6, l )=YYA DBK ( l, 2) =-2 .O':<HJ OBK(J,2)=-HH-Hl 08t<(5,n=XX8 DBK(b,2)=YYf3 0[3K( 2,3) =2.•j*HA 08K(4,3)=HE-Hr DBK(5,3)=XXC DBi(( 6, 3) =YYC DBK(2,4l=-2.J*HB CBK ( tt, 4) =-HE-HF Of3K( 5' It) =XXD D BK ( 6 , tt) = Y Y D ORK( 1, 5) =-1.·J DBK(.2, '.5}=-1.0 DBKC3,5l=BETAl~AC/(2.0*AAl

OBK(4,5)=BETA2*AC/C2oC*AB) Cf'\K( S, 5)=-XXF

__, w __,

CBK(b,5}=-YYE DBK(3,6)=8ETALuAC/(2.0*AA) DBK( 't ,6) =B Ef A2* fd)/ (2 ,O*AB) 08K(:.. ,6) =-XXG CP.KC6,o)=-YYF CVK( 1) =1.0/Cll c v K ( 2) =-1 I 1.J/ c I 2 CVK(5)=-XXE CALL SIMQ(CBK,CVK,6,KS,36) If(( IHL/2*2),Ei.1.IHUGO TCJ 7:..~9

D 0 7 :} 8 I I = l , 6 7 0 8 CV P < i I ) =CV K ( I I )

GO TCJ 300 l 709 CU~O=CVP(l)

C2 1~0=C VP( 2) C9iJO=CVP(3) Cl·JQJ=CVPl4) C 2 LC) :j = C VP ( 5 ) c 2 2 .Jt:"J = c v p ( 6 ) DHL=CP.KLC IHU-CRKL ( IHL-1) DCl=(CVK(ll-CVP(l) )/UHL DC2=(CVK(2)-CVP(2) I/OHL DC9=(CVKl3)-CVP(~))/DHL

DC1".)=(CVK(4)-CVP(4) )/OHL OC2l=CCVK(5)-CVP(5))/0HL OC22=CCVK(6)-CVP(6))/DHL z u oYL =2. o "'HHG* c lQJ+ < rHHYHH I > :::sc l / M.-2. O*HHJ *C 2C,)0 + < HHH-HH I> "*DC 21 AA-

l 2. 0*HHA*C9QO- C Hr E +~HF l *OC 9/ A~+2.~*HhG*ClOQO+(HHF-HHE>*OClO/AB-1.01 2Cil-l.iJ/CI2

IFCZUilYL.LE.O.O)GO TJ 3001 C GAMAA IS THE SPECIFIC AOH~SIVE FrACTU~E ENERGY IN L3/IN. C PCR IS THE CRITICAL INTE~NAL PRESSURE IN THE CYLINDEQ AT WHICH

w N

C DELAMIN~TIC~ PROPAGATES C PPP=PCR/CGAMAA)•*0.5

PPP=SQ~TC2.a1zuqYLl

WRITE(6,7J2)CRKLCIHL-l),PPP 702 F01"1.':.T(//' THF: HALF CRACK L::;-,.;;n-1 IS 1 //E20.7//' Tl-'1:: C~ITICAL INTL.

lP~ESSURE Hl THE CYLirmER, IN PSI. :'IT :!HICH OtLA·Hi\Jt.TION PROPAGATES 2, OIVIJt:D BY'/' THE SQArtL: RCGT Lif- THC: SPfC!FIC AnHESIVE FRACTURE 3cl\!ERGY (IN LB./IN.) IS 1 //c20.7)

c c

3 GO l Cll1'JT I "·I UE STCP END

C SA~PLE DATA SET c ,-.....

EFFECT OF GETAl & BlTA2. I 1) : B- E ( :J ) ( 2 ) : G- E ( 4 5 )

PRfSSU~~ G~ C0 ACK SURFACE

0. 2 -0. 8 3. c 50 :J.2 0.1 1.2 l. 7 2.5

l 0.05

::; • 2.;: 1 0.7J5 1.2(''.) l.1'J5 2.505

J.3 0.8 1.3 1.8 2.75

0 .3 COCJOOf. +C8 l

0.05

2 0 .3 C!:JUOOOE +08

C.301 \~·. 4 0.80~ 0.9 1.305 1.4 i.uu5 l. q 20755 3.0

0. 2 700;J ;}11:: + CJ1

0 • 2 7 a 0 0 C CF + ·~, 7

,_·. 4 -. l (;. 5 :>.501 c. 9:.J 5 l ,'") ov 1.005 l. 4;· 5 l. 5 l • 5C' 5 l.qJ5 2.0 2.005 : . • ·'.)'J 5 3.5 3.5C5

0.93J0')00E:+06

::., • 93::E:.O.::·UE +06

0.6 l 0 l l. 6 2.25 4.0

0.21 (). ')0

(J. 21 0. C'•O

'). 60 I. l. l 0 5 1. 605 2.255 4. 01

w w

00

0

0

• •

00

............ N

N

. ~

() 0

-0 ..0 0

0

+ ... l!J

Li.I

00

0

0

00

c .. : 0 :.-_, 0 '.<"\rt'\ O

' CJ'

' .. ") 0

,..._ r-.

(.)

... + t:• t; I

!') co

(...

~:l ,....

0 0

0

,._ ,._ N

N

00

cc 00

00

+

+ U

J U

J 0

·:::> '--' 0 C>

Q

r:; (") '.f'\

c~, o

() ,·.-, r .. )

• :-<"\

('() 0

• c,

0

134

w (.,.,

C APPENUIX U c C PROBLEM 4: LGAOl~G C~ INNER CYLINCRIC~L S1JPFACE; G~NERAL CR~CK TIP GEO~ETRY; C SliEAR FO~CE CONTINUITY ACROSS CRACK TIP;MIDPLANE AXIAL & S~E4R STRAINS IGNORED c c

CO~PL[X RM21,~M22,kML,RM2,RM3,RM4

COMPLEX RM1~,~M2R,RM3R,R~4R,C~PLX,CSC~T 0 9 I ME NS I 0 N C ( 6 , 6 ) , T ( 2 l , Q ( 3 , 3 l , E l l < 2 ) , ': 2 2 < 2 ) , G 12 ( 2 l , GNU l 2 ( 2 ) , lG~IU2U2l ,THETA(2l ,!-1(3) ,Q~(3,3)

D I ,~ E ~~ S I 0 N A I J ( 3 r J J , ;3 I J ( 3 , 3 ) , C ! J ( ~-\ , 5 ) , SM f'., L i3 ( 3 , 3 ) , S MAL A ( 3 , 3 J , l S ~1 t. L o T ( 3 , 3 ) , RR ( 3 , 3 ) , S ~·AL 0 { 3 , 3 ) , L ( 3 ) , ·'i '-\ ( 3 ) DI1~t:NSIUN XXX(J) ,YYY(3) ,ZZZ(3} 0 P':I: l'~S IO~I PAK U l, PAP (3) Dll'-'Er:SID:'l CRKL(5Jl DIME~SION C9K{6,6),CVK(6),CVPl6) Dl~ENSION TITLE(40) qEAD(5,197)TITU:

7 9 7 F 0 RM 1H ( 2 0 A 4} ~RIT~(6,797)TITLE

RE~0(5,777lBETAl,BETA2

777 FORM~T(2Fl0.0l hRIT~(6,73l)~ETAl,BETA2

731 F01-:MAT(! 1 13ETAl= 1 ,F6.3,SX,' iE:TA£·:= 1 ,F6.J) KEACC5,70l)RAO,ICRKL

701 FOR~AT(Fl0.0 1 151 READ(5,2'JJ1) (CRKL( I ),I=l ,ICPl<L)

2001 FORMtT(lOF8.0) c lSHELL-=l,2,3 cnRRESPOND TO TH: SCL'JT!Cl\S :=oR THE REGIONS Il, 12, II RESPECTIVELY

DO 1001 ISHt:LL=l,3 C N IS THE NU~G~R OF LA~IN4E

REl\0(?, UJOJN

100 FORM!1T( I3) C I ~JI TI A. L I Zc D ( 6 , 6 l >·;H l CH I S TH:: 1\ t, J 'A~ T 11 IX

on 12 I=t,6 DC 12 J=l,o

l2 0( I, JI :=·J .•JO C TH I C K '.\ E S S ;~ F L A:~ Ii~ iH E I S N AM c C D E 0 T H C T ( M ) I S TH::'. TH I CK W: S S CF THE :_., 'TH L fl ·I I i'\ ,.'\

RE /'\D ( 5, lO l) ( T ( M} ,M= l, N) 101 FORMAT(l0F8.0)

OEPTH='J .·JO DO 3 M=l,N

3 DEPTH=~=PTH+T(M) C Q MATRIX OF EACH LAMINA

DO lJ I=l,N RE~D(5,102lEll( I),E22(l),~12CI),GNU12CI)pJHETA(I)

102 FOR~AT{3E2J.7,2F6.0) GN U2 l ( I l =GNU 12 ( I >'" ::z 2 ( I ) I>.: 11 ( I l GNU2=GNU12(l)*GNU21(1) Q ( l, l l = E 11 ( I l I ( 1. 0 0-GN U2) Q(2,2l=r:22( I>/ Cl .OO-GNU2 l Q ( 1 , 2 ) = E 11 ( I >''•' G N U2 l ( I ) I ( 1 • ;) C. - G'JU 2 l Q(3,3l=G12Cll

C THETA +-VE F!<OT,. GLCBAL TG LAi"1P~A PRHJCIP1\L AXES I F C T <-I E T A ( I l ~ E Q • g 0 • 0 l G l: T 0 5 ·~' A •+; L = { T r-l J:: T 1\ ( I ) I l 8 0 • 0 0 ) * 3 • l'tl 5 9 2 7 C = C \J S ( l\t·l G L ) S=Sfr\{ANGL) GO T J 6 u

50 C=G. •}] S=l.)0

60 CZ=C,,.""2 S2=S~"'2

~

w Q)

C3=C,•C2 53 = S':' s~ C4=C: '"'( J S4=s·~s3

C THE QB ~ATRIX CF EhCH l.~~INA

Q p, ( 1 ' 1 ) = Q ( 1 ' l ) * c 4 + 2 • ::_.tr. ( Q ( l ' 2 ) .. 2 • :~ j ': '.: ( ) ' 3 ) ) * s 2 * c -~ + Q ( 2 ' 2 ) '" s 4 Q G ( 2 , 2 ) =CH l , l) * S 4 + 2 • 0 G * ( Q ( l , 2 ) + 2 • 0 :: '~ !JC 3 , 3 ) H• S 2 * C 2 + Q { 2 , 2 ) * C 4 QB ( l .2. ) = C CH l , l l -t C ( 2 , 2 ) -t• • J 0 * (.; ( 3, 3 l ) * S <'.'. * C 2 +Q ( l , 2 ) * ( S 4 +C 4 ) Q e. ( 1 ' 3 l = + ( Q ( l ' l } -Q ( l ' 2 ) - 2 • (; :i: (J ( 3 ' 3 ) ) .~ s ,., c 3 + ( Q ( l ' 2 } -~1 ( 2 f 2 ) + 2 • 0 *

lQ{.3,Jl ):ii:S3*C Q l3 ( 2 t 3 } = + ( (J ( l ' l ) -Q ( l , 2 ) - 2 • 0 ~( . .J ( 3' 3 ) ) ;;: s 3 *C + ( Q ( l ' 2 ) -(~ { 2 ' 2 ) + 2 D I),. ..

1Q(3,3l }:::S~C3 Ql3(3 ,3) =(0( l,l) +QC2 ,2 l-2.00*0(1,2 l-2 .JO*Q( 3,3) l*S2*C2

1+0(3,3l*(S4+C4l fJB(2,i)=QEHl,2) QB{3,l)=QB(l,3l QB ( 3, 2) =0:,:!3( 2 ,3 l IF(!-U 15,14,t5

C l-l{l) IS JISTAl\JCE FR.Of>~ MIDPLANE TC1 Hit BOTTUM l!F THE I'TH LA~1I~A 14 H(l)=-O~PTH/2.JO . l 5 H ( I + l ) =H { I ) + T ( I )

Dl=H( I )-H( l+l) 02=( H( I )**2-H( I+U~*L> 12.oc 03 = ( H C I ) * * 3- H ( I rl ) * ~ 3 ) I 3 • G C

c THE SIG~S GF 01,02,03 A~E CHANGED rs 7H[ PLY LAYUP IS DIFFERENT FROM C THAT IN ~y M.S. PRCGRA~

01=-Dl 02=-D2 03=-03 DO 2) K= 1, 3 DO 2 5 J= l, 3 O(K,J) =O(K,J)+Ol=QB(K,J)

--' w '-I

JJ=Jt-3 J(K,JJ)=D(K,JJ)+D2*0B(K,J) KK=K+::>

25 0(KK,JJ)=D(KK,JJl+D3*QR(K,J) DO 30 K=l,3 D~ 3() J=4,6

30 O(J,Kl=O(K,J) 10 CGl\TUWF.

V.Q.IH:(6,llJ)N l l 0 F 0 RM AT ( I I I l 0 X , ' L /I M I f\\ AT E C 0 N S T '\ i ~ T S 1 I I 2 X , 1 MJ M 13 E R Q F L A Y [ q S = 1 , l 3 )

\-iR I TE ( 6, 211 ) 211 FORMAT(///2X,'INPUT VARIABLES CF ~tC~ LA~!NA 1 //)

DC 2.Jl I=l,N i'i R I T t ( 6 , 111 ) T ( l ) , ~ l l ( I ) , E 2 2 ( I ) , G ~ 2 ( I l , GNU 12 ( I I , TH f: TA ( I )

111 FOR~~T(2X,F6.4,2X,3E20.7,2X,F6.4,2X,Fb.2/l 201 cc~JT INUE

\otR I T ~ ( 6 , 2 l 2 ) 212 FORMAT{///2X, 1 THE ABO MATRIX IS 1 ///}

WRIT t { 6, 112) ( ( D ( K, J l, J= i , 6) , K = 1, 6 ) 11 2 F 0 RM.!\ T ( 1 ;.1 1 , 6 E 2 0 • 8 / I )

00 5·)1 !=1,3 DO 502 J-=1,3 AIJ(l,J)=O(!,J) !3IJ( I ,J)=O( I,J+3) CIJC I,J)=0(1+3,J+3)

502 CDNTif·~UE:

501 CONTINUE PAK( I~HELLl=BIJ(l,2) PAP(!SHELL)=OIJ(l,ll DO 504 I=l ,3 OJ 5~.A J=l,3

5-J4 SMAL:I( I,JJ=AIJ( I,J}

w co

CALL MINV(S~ALA,3,CET,L,M~,9)

CALL GMPRD(SMALA,BIJ,S~ALB,3,3,3,9,9,91

DO 5J7 I=l,3 DC 'JJ7 J=l,3

5J7 SMhLBT(J,J)=S~ALG(J 1 1)

C~LL GMPRD(SMALBT,EIJ,RR,3,3,3,q,9,9) DO 5U8 I=l,3 DO 5J8 J=l,3

5 0 8 SM AL;) ( I , J l =RR ( I , J) - DI J ( I , J ) C EVALUATIO~ OF THE CONSTANT COEffICI~NTS IN TH~ SrlELL EQUATIONS

PC = S !~AL D ( l , l ) - SM/\ U.3 ( 2, l ) M• 21 S "11\ L A ( 2 , 2 l C RAD IS THE INNER RADIUS OF THE CYLIND~F

1-.R I T '.: ( 6 , 71 l I R.A D 711 FORMl\T(// 1 R!\D=',El5.7)

BC=(l.O/RAC)~(SMALBC2,l)/S~ALA(2,2l+AIJ(l,2))

CC=-~1J(2,21/RAD**2 XXX (I SH ELLl =CC \"iP. I Tc ( 6, 5 l J I

510 FCRMAT(///5X, 1 THE COEFFICIE~fS L,G,C IN THE SHELL EQUATIONS') WRITE(6,5lllPC,BC,CC

511 FO~~AT(//3(~X,~2J.7)) IFCHC.LE.l.OE-04)~C=J~O DISC=3C*BC-4.0*PC*CC IFCDISC.LT.0.0)GO TO 601 WPITc(6,6J2)

602 FORMAT(//' T~E ROOTS ~**2 ARf ~E~L')

Kr_,, 2 l i~ = - H CI ( 2 • 0 :;c PC) + S .:JR T ( 0 IS C l I ( 2. • r.· '' i-> C. } Pfll:22R=-8C/ (2.0*PCl-SlJRT (01 SC l/ (2 .O"."P': l RA t} S 1 =AB S ( !{ M 2 l R l RA8S2=flBS(RM22R) IFCR~21R.GE.O.O)RMlR=CMPLX(SQRT(~ABSllt0•0)

lF(R~2lR.LT.O.O)~MlR=CMPLX(0.~,SQRT(RABSl))

w l..O

R~2R=-RM1R

IF(Q~22R.GE.0.~)~M3~=CMPLX(SQRTlPAGS2l,O.'~I

IF (PM22K.LT.J.O)RM3R=C~PLX(J.O,SO~T(P~2S2)} R .~14 ?. = - R ~~ 3 R WR!TE(6,603)RM1R,RM2R,RM3R,R~4~

G'.J T~ 1001 6Jl RM2l=C~PLXC-9C/(2.G~PC),+SCRT(-DISCl/(2.0~PC))

RM22=C~PLX(-BC/(l.Q$PCJp-SQRTl-DISCl/(2.0*PC)}

RMl=c;rJ~H(i~M21)

RI·~ 2=-P :'-i 1 R:'13=CSlJRT CRM22) RM4=-R.M3 WRITE(6,603)RM1,RM2,RM3,RM4

603 FORMAT(//' TbE FOUR ROOTS OF THl ~c~nGENECUS PART OF THE SOLUTION lARE '//(2f20.7/))

YYY(lSHF.LL)=AeS(REAL(RMll) llL (I SH ELLI =/\BS ( ,\ IM:\G( Rt'-11))

1001 CUNTUWE Cl2=XXX(lJ CII=XXX(3) AA=YYY(2) AB=YYY{l) AC=YYY(3) AD=LZZC3l WRITE(b,606)AA,A3,AC,AD

606 F0~~-11\T(//' THE CJNSTANT F.XPQ;";f:'.'-iTS P'l THE DEFLECTION SOLUTim~s FOR lR~GICNS Il,I2,II RESPECTIVELY ARE'//4~20.7//I

C HL IS TH~ ~!ALF CRACK LENGTH DO 3001 IHL=lvlCRKL HL=C1KLCIHL) AL=AA*HL l3L=Ao*HL

.f-:> C>

SA=CC:SCGL) S!'.\=CJSH(Hll SC=S!~Htll)

SD=SI~HCBU

SH=C!JS(t\L) SI=ClJSH(AL) SJ=SIN(~L)

SK=SI~H(Al.)

HA=SA*S'3 HH=SC*SfJ HE=Sl~*SO

HF=SC*S3 IFC(IHL/2'l<2}.EQ.IHL)G0 TO 659 HH/\= HA HH.3 =HB HHE=HE t-HF=HF

659 HG=Srlor.SI HH=Srl,,.SK Hl=SJ~~sI

HJ=S J~<SK XXA=•2.0*HG*PAK(2)/RAD+4.0*~A~~A*HJ*PAP(2)

XX3=-2.0*HJ*PAK(21/RA0+4.C*AA~AA•HG*PAP(2)

XXC=+2.0*HA•PAK(ll/RAD+4.0*A8*A~*hP*P~P(l)

XXD=-2J:*Hf3*PAK(1 l /RAr)+4.ll*AR~~.3*Hfp;:P,W ( l) XX~=PAK(l}/(RAC•Cl2)

X X F = P A'< ( 3 ) IR A D - P t. P ( 3 ) ~' ( A C * A C. - A L)" A C ) xx:;=-2. 'J:i;tA(:l<AO><PflP ( 3) XXrl=Pt\K{3)/(RAD*CI I) Y Y /1 = t 2 • 0 *A i\ * ( H H- H I } >r PAK ( 2 I I ~- .'\ ~);- 't • !: :..: .L f, * * 3 * ( h H • H l ) *PAP ( 2 I YYB=-2.0*A~~(HH+Hll*PAK(2)/R~O+~.JcfA**3*(1-'H-Hil*PAP(2)

Y Y C = + 2 • 0 *AB* ( HE - HF ) *PAK ( l ) I R. AC • 't • ·:J ::-: A B ~ * 3 * ( r E +HF ) * P AP ( l )

__, .+:>

y y [):::: -2 • G >'.:AB:;< ( Hi: +HF ) * p AK ( l ) I K II D + lt • ,'J "'J\ r "":;c 3 * ( r E - HF , *P A p ( 1 } YY~=~Cn(-PAK(3)/RAO-C3oO~~D*An-~C~AC)~PAP(3))

YYF=hD~(-PAK(3)/RAD+(3.J•AC*AC-~O*~Dl~PLP(3))

DfJ 7J7 J=l,6 CVK{ .J) =O.O DJ r:_.7 I=l ,6

707 0!3K(I,Jl=O.O CBK{ 1, l l=2 .O*HG OPK( 3, 1) =HH-HI DBK(5,l)=XXA DBK(6,l)=YYA ORK( L,2)=-2.0*HJ CGK(3,2)=-HH-Hl 0!3KC 5,2)=XXB Dt3K(6,2)=YYB on K ( 2 ,3 ) =2 • J *HA OBK( 4,3)=Hc-Hf-DGK(5, 3)=XXC DBK( o,3 )=YYC DGK( 2, 1t)=-2 .O*HA DBKl'+,tt) =-hE-hF Df3K('.:>,4l=XXD OBK(6,4l=YYO DSK(l,5}=-1.Q OSK( 2, '>l=-1.0 OBK(3,5l=3ETAl*AC/(2.0*AA) OBK(4,5)=BETAZ•AC/(2.J*A~)

DB'<.(5,5)=-XXF Cl3K( 6, 5.)=-YYE Du:<.( 3,6) =BETAl~AC/ (2 .O*AA) DBK(4,6)=BETA2~t0/(2.0~Ae>

OBK(5,6)=-XXG

..j:::> N

f"JP.K(6 9 6) =-YYF CVK(ll=l.O/CII CVK'2) =1 .O/CI I-LIJ/CIZ CVi<.( 5) =XXH-XXE CALL SI~Q(OBK,CVK,6,KS,36)

IF((lHL/2*2).EO.IHLlGO TO 7C9 DC 7Jd II=l,6

70 8 C VP ( Ii l =C VK ( I I) GC T'J YJ:Jl

70CJ CUJv=CVPll l C2<'J]=CVP(21 C9rVi=CVP( 3) ClOQ.~=CVP('tl

C2UL=CVP(5} C2ZOJ=CVP{6l OHL =CF',KU IHL )-CRKL ( IHL-1 I DC I.= l CV K ( 1 ) -CV P ( 1) l I 0 H L iJC2=!CVK(2l-CVP(2) )/OHL DC '; = ( C V K ( 3 l - CV P ( 3 l l I D H L DC10=lCVKl4l-CVP(4) )/DHL DC21=CCVK(5)-CVP(5) )/OHL DC22= CCVK(6)-CVP(6) UDHL Z U r2 Y L = i • 0 I C I 2 + 2 • 0 * H H A :i;c C 9 t') 0 +l' C 9 ~' I H1 [ + ~ HF ) I 1\ B - 2 • 0 * H H B *C l 0 Q 0 +DC l 0 *

llHH~-HHFl/AG+lAC*DC21-AD*OC22}/lAC~AC+AD*AO)

zu1:YL=-lUl~YL

IFIZU3YL.LE.0.0)G0 TO 30Jl C GAMA~ I c; THE SPECIFIC ACHES IVE Fi.l.:\CTllf c r:NERGY IN LR/IN. C PC R I S TH t C P. I T I CA L I NT ERNA L PR .~ S 'S 'J ~ ~= l '~ T H E CY L I "J 0 ER AT ,,J H I C H C 0 EL A:~ Ii~ A T ION P R GP A GA T E S C PPP=PCR/{GAMA!l**C.5

PPP=SORT ( 2. 01 ZUi3YL l \~~ l T t ( 6, 7 0 2 l CR KL ( I H L -1 l , PP?

~ w

702 FO~M4T(//' THE hALF CRACK U::f\.:;~H IS 1 //f:20.7//' THE CRITICAL INTL. l p ~ ;:: s s u K :: ! N T I-' E c y L [ ~ [) E R ' p; p s I ' /• r ~· H I c H D E LA M I NA Tl m~ p ~ (J p AG A TE ') 2, UIVIDED RY'/' THE S~ARE RCCT 0f T~~ SPfCIFIC AC~ESIVE FRACTURE

c c

3 U~t: ;~c; Y (IN LR. I H~. ) I$ 1 I I E2 0. 7l 3 0 0 l C 0 ~'! T I :\J U E

STOP END

C SAMPLE DATA SET c c EFF~CT OF B~TAl & BET~2. PRESSURC CN IN~ER CYL. SURFACE ( l ) : 13 - fl U 0) ( 2 i : G- t ( 4 5 )

0.2 -0.B 3.C 50 0.2 J.2'Jl 0.3 o .. 7 l. 2 l. 7 2.s

l o.ca

l ·J. DZ

2 (,. 0 8

·~. 7 ') 5 l • 205 l. 7<15 2.:.J]:.J

o.a 1.3 108 2. 75

:.L32J:JOOOE+J8

a.1.35Jo~-:t:+oa

] • ·::2 0 • ·.:1 2 J '.) ) CJ E ·H) 8 C.l.S~OOCGt+OB

0.301 0.4 0.805 G.9 l.JOS l • 1t l. 805 1.9 2.755 3.C

O. lHOCC'C~:::: ... un

O. 7JOOJJJ~:t-J6

'.). ta·:woac;'::rCB u.1000~0ei:::+c6

C.401 '~·•gr, 5 1c 1t'J5 1.9·5 3. C05

0.5 0.501 0.6 l 0 c 1.005 l.l 1.5 1.505 1. 6 2.0 2 o0:}5 2.25 3.5 3.5C5 4.0

o.9soooooE+07 0.23 a.oo

0.8300000E+06 0.21 45.00

0.950000uE+07 0.23 0.00 0.83000C0E+06 a.21 45.00

0.601 l. l 'J5 l. 605 2. 2 55 4.01

~- i

--I _.,. _.,.

-+::> U1

c c

;\P?::ND!X ~

C PROHLE.,1 '.:>: PRESSU~E Ch CRACK SlPFAC=; (E·H:KAL CRACK TIP GErn-'IETK.Y; SHEAR fOf~CE

C CJNTU~L;ITY ;\C!<.lSS CRACK TIP; M!r)i)l_1\".JE: .'1Xl/\L £. SHE,\R STRAINS lNCLUO[:J c c

CC~PL[X RM21,RM22,RM1,RM2,RM3,R~4

C fJ i" PL t X R :·1 l R 9 RM 2 R , R fl 3 R 7 R r' 4 R, CV? l. X , CS Q P T 0 DI :-t,f r~ SI 0 N 0 ( 6 , 6 ) , T ( 2 l , (J ( 3 , 3 l , :: 1 l ( 2 ) , E 2 2 ( 2 l 9 G 12 ( 2 ) , GNU 12 ( 2 ) , 1G:'-:U2 1 { 2 ) , THE TA ( 2 l , H ( 3 ) , QB ( 3 , 3 )

D I '1 c f·; S I lJ N A I J ( J , 3 l , f3 I J ( 3 , 3 ) , n I J ( 5 , 3 I , SM A L B ( 3 , ~ ) , S !-1 A L A { 3 , 3 ) , 1 SM AL r1 T { 3 , 3 ) , RR { 3 , 3 ) , S t": Al 0 ( 3 , 3 l , L ( J ) , I~.,, ( 3 )

D I ~-1 E i\l S I iJ N X X X ( 3 l , Y Y Y ( 3 I , l l l ( 3 l DIMENSION PAK(3l,PAP(3) Dl!'-'Ef~SION DBK(6,6),CVK(6),CVf1(bl DI~E~SION CRKL(5QJ CI~ENSICN TITLE{40) R.Ei\0{ S, 797) TITLE

797 FO:<.MAT(2'.)A4) hRITE(69797lTITLE RE 110 ( 5, 777lBET~\1 , BET A2

777 FORMAT(2F10.0) WR IT f ( 6, 7 31 l 8 ET A 1, f3 ET A 2

7 3 1 F 0 R ['-\ !>. T ( I I B E T A 1 = I ' F 6 • 3 9 5 x ' I f"l E T /l 2 = ' ' F 6 • J ) RE~0(5,70l)qAO,ICRKL

701 FOPMATlFl0.0,15) RE [In ( :> I 2 0 :n ) { c K Kl ( I ) 1 I = l ? I c r: KL )

2001 FORMAT{10f8.0) C !SrlELL=l,2,3 CGFRESPCNC TG THE SCLUiiO.S FOR THE REGFJNS Il,I2,II RESPECTIVELY

D G 1 0 :) 1 I S H i: L L = l ,:; C N IS THE ;\lUMBER OF LL.,'-q,.;A;:

RE/~D{:.>,LC'j)N

l 0 0 r- J ~ '·LH ( I 3 ) C I N I T I l'l. L I l E i) ( 6 , 6 ) ~ H I C H I S T H E A ~ ;; .~ •"- T P I X

DD lL. I=l,6 DC 12 J=l,6

12 DCI,J)=C.uC! C TH IC K >-i c SS 'Jr- LAM l !\u\T E IS f\ Mi EC Dr P TH C T{M) IS TH~ THICKNESS OF THE M1 TH l~MI~A

REl\[){5,101) ( T(M) ,,v=l ,N) 101 FUR~AT(10F8.0)

DEDTH=0.00 DO 3 ;":= l, ~~

3 DEPTH=DEPTH+f(MJ C Q MATRIX CF EACH LA~INA

DO l J I= l , N R [ .!!.. L) ( 5 , l 0 2 ) E 11 ( I ) , E2 2 ( I ) , G 12 ( I ) • G NL' 12 ( I ) , THETA ( I )

102 FOR~AT(3E2J.7,2Fb.0) G!~UZUIJ=G:~L1 12( Il*E22{1)/Ell (Il G :...i U 2 = G ~-J U 1 2 ( I ) * G i\ U 2 l < I J Q C l , l l = E 11 ( I ) I ( 1 • ·2 G - GNU 2 ) Q ( 2, n = E22 (I) I ( l .JJ-GNU2 J Q ( l , 2 I = c l 1( I >* G '·W 2 l ( I ) I { l • 0 0- r; ~ l :.J 2 ) Q(3,3l=Gl2( I)

C TH ET I\ +- V f FR 0 M G LC 5 ~ L T C L AM I NA F' ~ Ir.,.: I r /, L .AX ES IF ( TH'.: T :\ ( I J • E Q • q J. D ) G 0 T 0 5c.; ANGL=(fHcTACil/l8Q.OJ)~3.141~~27

C=CGS C ,\NGL l S= SPH A~GL) GlJ TJ 60

5C C=J.JJ S=l.J~

60 CZ=C•M2 S2=S**2

_, .j:o> O'l

C3=C>:•C2 S3-=S*S2 C4=C*C3 S 1~=S*S3

C THE QB MATRIX Of EACH L~MINA QG(l,l)=Q(l,l)*C4+2.0*(Q(l,2)~2.00*0(3,3))*SZ*CZ+Q(2,2)*S4

Q?(2,Z)=Q(l,ll*S4+2.00*(Q(l,2l+2.0l*C(3,3)J*S2*C2+Q(2,Zl*C4 QB < l , 2) = ( Q ( l , 1 > + Q ( 2 , 2 )-4 • O O ~'<J U ,3 J ) * S 2 *C 2 +Q < l , 2 ) :;:: < S 4+C 4) QB(l,3)=+(Q(l,l)-Q(l,2)-2.0*Q(J,3))*S~C3+{Q(l,2)-Q(2,2)+2.0*

lQ(3,3) )*S3*C QB(2,3)=+(Q(l,l)-Q(l,2}-2.J•Q(3,3) >~S3*C+CQ(l,2)-Q(2,2)+2.0*

1 Q ( 3 , 3 ) ) * S *C 3 Q6(3,3)={Q(l,l)+Q(2,2)-2.CO*Q(l,2)-2.00*Q(3,3))*SZ*C2

1 +Q < 3 .3 > * C S 4 +C 4 ) QB(2,l)=QB(l,2) QB(3,U=QB(l,3} QfH3,2)=QB(2,3) IF( I-1) 15, 14, 15

C H(l) IS DISTANCE FROM MIDPLA~E TO THE BOTTOM OF THE I'TH LAMINA 14 HCI>=-DEPTH/2.00 15 H(l+ll=H(l)+T(l)

D l = H ( I ) -H ( I+ l ) 02=(H(l)**2-H(I+l)**2J/2.~0 03=(H(Il**3-H(I+1)**3)/3.00

C THE SIGNS OF 01,02,03 ARE CHANGED 4S THE PLY LAYUP IS DIFFERENT FROM C THAT IN MY M.S. PROGRA~

01 =-Dl 02=-02 03=-03 DO 25 t<.=1,3 DO 25 J=l,3 O(K,J)=O(K,J)+Ol*OB(K,JJ

--' +:> -...J

JJ=J+3 0(K,JJl=D(K,JJ)rD2*QR(K,J) KK=l<.+j

25 DCK~,JJ)=D{KK,JJ)rD3*QB(K,J) Du 3') K=l,3 D~ 3J J=4,6

30 O(J,i".)=O(K,J) l 0 CU f\JT l:'W E

hRIT~(6,ll0)N

110 FCkMAT(///lOX,'LAMINATE CGNSTANT3'//2X, 1 NU~8ER OF LAYERS=',13) W~IT:::lo,211)

211 HlRMAT(///2X, 1 rnPUT VARI.1\BLES CF ::.4C.H LAMINA'//} DO 2 H I= 1, N w•ur::C6tlll)T( I),Ell(l),E:22( I),G12( I),Gf\JU12( !),THETA( I)

111 FOR~AT(2X,F6.4,2X,3E20.7,2X,F6.4,2X,F6.2/) 2 0 1 C CJIH I "! U E

w:~ ITE< 6,212) 212 F0'-{~~!\T(///2X, 1 THf: 1\g[) MATRIX IS'///)

WRIT c:( 6, 112) { ( D ( K, J ) , J= l , 6 ) v K = l , .S) 112 FORMAT( 1 0 1 ,6E20.8//)

00 5 .:, 1 I= l ,3 JO 5CJ2 J=l,3 A I J { I , J) =D C I , J ) BlJ( I,J)=D(I,J+JJ QI J( I ,.J)=DC 1+3, J+3)

502 CGf\JTINUE 501 CC1NT INUE

PAK{ lSH~LU=BlJ(l,2) PAP( ISHELL)=DIJ(l,l) DIJ 5J4 I=l,3 D!J ~'.)4 J=l ,3

504 SMALA(l,J)=AIJI I,J)

--' +:> co

CALL MINV(S~AL:\,3,0ET,L,MM,S)

CALL GNPRO(SMALA,BIJ,SMALR,3,J,3,9 1 9,9) DD 5 J 7 I = l , 3 or 5J7 J=l,3

507 s:··,·~L·H(l,J)=StJALB(J, [) C.'~LL GMPRO(S~Aun,i3IJ,RR,3,::J,J,'1,9,S)

DC 5J8 I=l,3 DO 5J8 J=l,3

508 S~·'.'\LD( I,J)=RR( I ,J)-OlJ( I,JJ C EV~LU~TIDN OF THE CCNSTANT COEFFICIENTS IN THE SHELL ~QUATIONS

G Mi J l = ( :\ I J ( 3 ,3 ) - A { J ( l , 3 ) >'.< '* 2 I A LJ ( l , l ) ) '" R A C * R A CI S M i\ L 0 ( l , l l GA '-1 J 2 = ( BI J ( l , J ) - B I J ( l , l ) *A I J ( l , 3 ) It I J ( 1 , l l ) *RAD I S Vi I\ L D { l , l ) G /, .·1 J 3 = ( A I J ( l , 2 ) *A I J ( l , 3 > I A I J ( 1 , l ) - td J ( 2 , 3 ) ) :t: R. AD~ RAD I SM A L D ( 1 , l ) GAtJJ4=BIJ(l,l)/AIJ(l,l)/RAC-AIJ{l,3)/AIJ(l,l)*GAMJ2/GAMJl GA;~ J 5 = 1\ I J ( 1 , 2) I A l J ( l , l ) +A I J C l , ·3 ) / l\ I J ( l , 1 ) * G AM J 3/ Gil. M J 1 Gh~J6=(GAMJ4*AlJ(l~2)-BJJ(l,2)/~hO+~lJ(2,3)*GAMJ2/GAMJll*RAD*~AO/

lSM~LC( 1, l) G~MJ7=(AIJ(2,2)-AIJ(l,2~*GA~J~t~lJ(2,3l*Gt~J3/GA~Jl)*RAO*RAD/

l S\1,H D ( l , l ) PC=SMALO(l,l)-SMALB(2,ll**2/S~ALA(2,2)

C R A 0 I S T H E INN ER R A D I US 0 F T H E CY L L'iD £.. R \.<RITE(6,7ll)RAD

711 FCRMAT(// 1 RAC=' ,El5o7) 3C=CSMALB(2,l)/SMALA(2,2)-SMALJ(l,l)*GAMJ6/RAD)/~AD

CC=-GAMJ7*S~ALD(l,l)/RAD**4

XXX <I SH ::LU =CC wQIL:C6,510)

510 FGR:·\AT(///5X 7 ' Tl-1E CUEFFICIEl\TS t:..,P.,C IN THE SHELL EQUATIONS'} ~sITE(o,5ll)PC,HC,CC

511 FCRMAT(//3(5X,E2C.7)) IF(8C.LE.l.OE-04)RC=Oo0 DISC=GC*8C-4.0*PC*CC

.+::> \.0

IF(OISC.LT.8.0lGU TC 601 WP, I TE ( 6, 6 JZ )

&02 FORM~T(// 1 ThE ROOTS M*•Z A~f R~AL'l R~21R=-BC/(2.0•PCl+SQ~T(CISC)/(2.0~PC}

RM22R=-RC/(2.C~PCl-SQRTCOISCl/C2.~~rci

R.l.~~S l=ABS ( RM2 lK.) RA[3S2=1\BS < R~J.22R. l IF(R~21R.GE.G.ClR~lR=CMPLX(SCRT(~AESl},0.L)

IF(R~21R.LT.0.0l~MlR=CMPLX(O.O,SQRT(RAHSl))

RM2R=-RM1R IF(~~22R.GE.0.2)RM3R=CMPLX(SQPT(RAPS2l 1 0.0) 1F{~M22q.LT.O.OJRM3R=CMPLXCU.0,SCRT(PA8S2ll

~M4R=-R:'-13R

WRITE(6,6)3)R~LR,RM2R,RM3R,R~4R

GO T'..J lD.Jl 6Jl RM2l=CMPLX(-BC/(2.J*PC),+SQ~T(-D SC)/{2.0•PC))

R~22=CMPLX(-BC/(2.J*PC),-SQPTC-D SC)/{2.0*PC>l RMl=CSQRT(RM21) RMZ=-RMl RM3=CSQ~T(RM22)

RMt1-=-P.M3 WRITf(6,6S3)~MlvRM2,R~3,?~4

603 FORMAT(//' THE FCUR ROOTS OF TM~ ~CM~GENEOUS PART OF THE SOLUTION LARE '//C2E2J.7/)) YYY(ISHELLl=AGSCREAL(R~l))

lll (I SH [LU =ABS (AI MAG( PMl)) lOJl CONTINliE

Cil=XXXCZJ CI2=XXX( 1) CII=XXX(3) AA=YYY(2) AB=YYYC 11

Ul 0

AC=YYY(3) flD=ZZl{3) WRI TE(6,606) AA,.'1.B,AC,AD

606 FORMAT(//' THE CONSTANT EXPC~ENTS IN THE DEFLECTION SOLUTIONS FOR lREGlONS 11,IZ,ll RESPECTIVELY ARE'//4E20.7//)

C HL IS THE HALF CRACK LENGTH DO 30Cl IHL=l,ICRKL Hl=CRKL( !HU AL=AA*HL BL=AB*HL SA=COS(BL) SB=COSH( BU SC=SIN(BU SO=SINH(Bl) SH=CQS(AL) SI =CCSH (AU SJ=SIN(Al) SK=SINH(AU H~=SA*SB HB=SC:t.:Sl) HE= Si~ :;c SD HF=SC*SB HG=SH*SI HH=SH>XSK Hl=SJ*SI HJ=SJ*SK IF(( IHL/2*2).EQ.IHLIGO HHA=HA ~Hd=H3

HHE=HE 1--HF=HF HHG=rlG

TO 6S9

tT1 _.

HHH=HH f-HI=HI HI-I J =HJ

6 5 9 X XA = + 2 • C *HG* PAK l 2 I I~ AD+ 4 • :) */\A"" A;\* HJ* PAP ( 2 ) XX8=-2.0*HJ*PAK(2)/RAD+4.0*AA*A~*HG*PAP(2) XXC= +-2 .O*HA*PAK( 1) /RAD+4 .O*A8*AP>l<hP.*PAP( 1) XXD=-2.C*HR*PAK(l)/~AD+4.0•AB*AB*HA4PAP( 1) XXE=lPA~(l)/CI2-PAKl2)/Cil)/R~O XXF=PAK(3)/RAO-PAPl3)*lAC*AC-Au~~O) XXG=-2.0*AC*AD*PAP(J) YYA=+2.0*AA*(HH-HI)*PAK(2)/RA0•4.0*AA**3*(hH+Hl)*PAP(2) YYB=-2.0*AA*(HH+Hll*PAKl2l/RAD+4.C*AA**3*(H~-Hl)*PAP(2)

YYC=+2.~*~&*lHE-HF)*P~K(i)/~A0+4.0*AB**3*lHE+HF)*PAP(l) YY0=-2 .O*AB*( HE+HF) *PAt<.( l )/R.L\.0+1• .O~A8**3*( HE-HF )*PAP ( l) YYE=AC~l-PAK{3)/~AD-(3.0*AD*AC-AC*AC)*PAP(3)J

YYF=AD*C-PAK(3)/RA0+(3.0*AC*AC-AD*AC)*PAP(3)) 00 7·J7 J-=l,6 CVK(J)=O.O DO 707 1=116

707 DBK( I,J)=J.O OBK(l,1)=2.0*HG DBK(J,l)=HH-HI DBK(5,l)=XXA DBK(6, U=YYA DBK( l,2)=-2.0*HJ 08K(3,2)=-HH-HI 05K(5,2)=XXB 06K(6,2)=YYB DBKC2,3)=2.0*HA DBK(4,3)=HE-HF CBK(5,3l=XXC DBK ( 6 ,3) =YYC

U1 N

CAK(2,4)=-2.0*HO 0 9 K ( lt , 4 ) = - H E - HF DB '< ( :) , 1t ) = X XD CAt<C6,4)=YYD OBKl 1,5)=-l.J DRKCZ,5)=-1.0 OBKC3,5)=BETAl*AC/(2.0*AA) DBK(4,5)=BETA2~AC/(2.0*AB)

DB!<.(5,5)=-XXF DBK(6,5)=-YYE DBKC3,o)=BETAl*AD/C2.0*AA) DBK(4,6)=B=TA2*A0/(2.0*AB) 08K(5,6)=-XXG CBK(6,6)=-YYF CVK(l)=l.O/Cll CVK(2l=-l.O/CI2 CVK(5)=-XXE CALL SIMQ(OBK,CVK,6,KS,36) IF((iHL/2*2).EQ.!HL)GO TO 709 DO 7'J8 II=l,6

708 CVP(lll=CVK(III GO TO 3:)1; 1

70CJ Cl;JO=CVP( ll C2QO=CVP(2) C9r)'.:'.i=C VP ( 3) Cl010=CVP(4) C21QO=CVPC5) C22QC=CVP(6) DHL=CRKL(IHL)-CRKLCIHL-l) DCl=<CVK(l)-CVPCl) )/OHL DC 2 = ( CV K ( 2 ) -CV P ( 2 ) ) I 0 H L DC9=(CVK(3J-CVP(3) )/OHL

<J1 w

DC10=(CVK(4)-CVP(4))/DHL DC 21 = l CV K ( 5) -CV P ( 5 l l I OHL DC22=(CVK(6)-CVP(6) )/OHL ZUSYL=2.0*HHG*ClQO•(HHH+HHI>*DCl/AA-2.0*HHJ*C2Q0+(HHH-HHil~ocz/AA-

12. ')*HH A* C9QO- ( HH E+ HH F) *DC 9 /A 8 +2 .O "'HH t3 *C 10 QO • ( HHF- HH E) *DC l J/ AB-1. 0 I 2CI1-l.O/CI2

IF(ZUBYL.LE.O.OIGO TG 3801 C GAMAA IS THE SPECIFIC ADHfS IVE FRACTURE ENERGY IN LB/ IN. C PCR IS THE CRITICAL INTERNAL PRESSURE IN THE CYLINDER AT WHICH C DELAMINATION PROPAGATES C PPP=PCR/{GA~AAl**0.5

PPP=SQRT(Z.0/ZUBYL) WRITE(6,702)CPKUIHL-l) ,PPP

702 FORMAT(//' THE ~ALF CRACK LENGTb IS'//E2C.7//' THE CRITICAL INTL. lPRESSURE IN THE CYLINDER, ll\l PSI, AT WHICH OELAMINATION PROPAGATES 2, CIVICEO BY'/' THE SQARE RCCT aF TbE SPECIFIC ADHESIVE FRACTURE 3ENERGY (IN LB./IN.) IS 1 //E20.7l

c c

3001 CONTINUE ST.JP END

C SAMPLE .D~TA SET c c

cFFECT OF HETAl & BET~2. ( lJ : B- E ( J ) ( 2) : G- E ( 4 5 }

0.2 -o.a 3.0 50 ·J. 2 ) • 20 l u.3 0.301 Oe7 0.705 a.a 0.805 1. 2 1.205 1.3 1.305

PRESStJRf. CN CRACK SURFACE

0.4 .J.401 Q,.5 o.501 0.6 C.9 G.905 1.0 l.005 1.1 1.4 l.4•j5 1.5 le505 1.6

0.601 l. l 05 1.60'>

__.. U1 ~

1. 7 1. 7'.15 1.8 1. 805 1.9 2.5 2.505 2.75 2.755 3.J

1 a.cs

0. 3 C00000~+08 0.2 700000~+0-, 1

1J. cs O .3 :J0000E+'.>8 o.21coccct:+-c1

2 o.i:5 :; • c 5

•J .30·J0000t:+08 0.270G02GE+U7 0 • 3 J :j Q ') 0 0 E + 0 8 0 .2. 700'.:iJJE+O 7

1 • <)<_: 5 2.0 2 .oo 5 3.005 3.5 3.505

J.9300000E+06

0.93JOOOOE+06

0.9300000E+06 0. 'J3 OOOOOE +06

2.25 4.0

0.21

0.21

0.21 0.21

o.oo

o.oo

;).00 o.oo

2.255 4.01

U1 U1

(..Tl ())

c ~PP'.:NOIX F c C PRURLEM 6: LOAOI~G CN INNER CYLINC~ICAL SURFACE; GCNERAL CRACK TIP GEOMETRY; C SHEAR FORCE CCNTINUITY ~T CRACK TIP; ~iDPLANE AXIAL & SHEAR STRAINS INCLUD~D

c c

CC~PLEX RM21,R~22,R~l,RM2,R~3,q~4

CO~PL~X RM1R,RM2R,R~JR,R~4R,CMPLX,CSQRT

0 0 I ·l E NS I ON 0 ( 6 , 6 l , T C 2 ) , Q ( 3 , 3 ) , E l 1. ( 2 ) , E 2 2 ( 2 ) , G 1 2 ( 2 l , GNU 12 C 2 ) , l GI'~ lJ 2 1 ( 2 ) , TH E TA ( 2 ) , H ( 3 l , Q 8 ( 3 , 3 )

D I Mc t'\ S I ON A I J ( 3 , 3) , 13 I J ( 3 , 3 ) , DI J ( 3 , :J l , SM AL 6 ( 3 , 3) , SM AL A ( 3 , 3 ) , l SM .AL r3 T ( 3 , 3 ) , RR ( 3 , 3) , SM AL D ( 3 , 3 ) , L ( 3 } , M l"1 ( 3 )

D IM ENS I ON XX X ( 3 ) , Y Y Y ( 3 ) , l Z l ( 3 l D l i·'. E I'~ S I fJ N P ~ K ( 3 ) , D t\ P (3 ) DI~E~SICN CRKLl50) OIMtNSION DGK(6,6) ,CVK(6) ,CVP(6) DI~ENSION TITLE(40) REACC5,797)TITLE

797 F:JRMAT(20A'+) 1-J R I T E ( 6 , 7 9 7) T IT L E READ(5,777)BETAL,BETA2

177 FORMAT(2F10.0) hRIT~(6,731)5ETAl 7 BETA2

731 FQRM/\T(/ 1 BETlil= 1 ,Fo.3,5X, 1 f3Eft'12= 1 ,F6.3) REAC(5,70l)RAO,ICRKL

701 FC;:<.MAT(FL0.0,!5) REaD(5,20Cll(C~KL(l),I=l,ICRKL)

2001 FORMAT{l0F8.0) C ISHELL=L,2,3 CORRESPONO TO THE SOLUTIONS FOR THE REGIONS Il,12,II RESPECTIVELY

o~J l 1Ji) l IS HELL= l ,3 C N IS THE NU~BEP CF LA~I~AE

R E ,\ D ( 5 , l 0 0 l N

lGC FOR'"1/\T(I3) C INITIALILE 0(6,6) WHICH IS THE ABD M~TRIX

DJ 12 I=l,6 DO 12 J=l,6

12 0(1,.J)=O.OO C THICKNESS OF LAMINATE IS NAMED DEPTH C T(M) IS THE THICKNESS GF THE M'TH LAMINA

RE00(5,10l)(T(~J,M=l,N)

101 FCRMAT(l0F8.0) DEPTH=0.00 DO 3 M= l, N

3 DEPTH=DEPTH+T(M) C Q MATRIX OF EACh LA~INA

DO l 0 I= 1, N REA0(5,102)Ell( I),E22(1),Gl2(J),GNU12ll),T~ETA(l)

102 FORMAT(3E20.7,~F6.0) G!\J U 21 ( I ) =GNU 12 ( I ) * E 2 2 ( I ) IE 11 ( I ) GNU2=GNU12(1)•GNU2l(I) Q ( l, l) =Ell (I)/ ( l .OO-GNU2) Ql2,2J=E22(1)/(l.OC-GNU2) Q( l.2)=ElU I>*GNU21 CIJ/(l .. 00-GtWZ) Q( 3,3)=Gl2( Il

C THETA +VE FROM GLOBAL TC LAMINA PklNCIPAL AXES IF( THETA( I) .EQ.90.0JGO TO 50 ANGL=(THETA(l)/130 .. 0J)*3.1415927 C=COS ( ANGU S=Sll\(ANGL) GO TD 60

50 C=J.00 S= l. JO

60 C2=C*"'2 S2=S*'>:•2

U1

"""

C3 =C ""C 2 SJ=s~~sz

C4 =C ;~C3 S4= s~: S3

C THE QB MATRIX OF EACH LAMINA Q3(1,l)=~Cl,l)*C4+2.0*(Q(l,2)t2.GO*Q(3,3l )*S2*C2+Q(2,2)*S4 Q B ( 2 , 2 ) = Q ( l , l >* S 4 + 2 • 'J 0 * ( \~ ( 1 , 2 ) + 2 • 'J; ¥ Q ( 3 , J l ) * S 2 >:< C 2 + Q ( 2 , 2 l * C 4 Q5(1,2)~(Q(l,l)+Q(2,2)-4.00ll0(3,3l)*S2*C2+QCl,2)*{S4+C4)

QBC1,3)=+(Q(l,ll-QC1,2l-2.0~Q(3,3ll*S*C3+(Q(l,2)-Q(2,2)+2.0*

lQ(J,3) l*S3*C Qb{2,3)=+(Q(l,l)-Q(l,2)-2oO*QC3,3))*S3*C+CQll,2)-Q(2,2)+2.0*

l Q ( 3, 3) ) * S""' C3 QB ( 3 , 3 ) -= ( Q ( l , l ) + Q ( 2 , 2 ) -2 • 0 0 * Q ( l , 2 l -2 • 0 0 * Q ( 3 , 3) ) *S 2 * C 2

l+Q(3,3)*(S4+C4) QfH2,ll=QB(l,2 > QF3(3,l)=QBC1,3) Q8(3,2l=QBC2,3) IF( I-1) 15, 14, 15

C HCI) IS DISTANCE FRO~ MIOPLA~E TC THE PGTTOM OF THE l'TH LAMINA 14 H(l)=-DEPTH/2.00 15 H(I+l)=H(l)+T(l)

01 =HC I )-H( l+l) 02=( H( I )**2-H( I+U*,,~2) 12.CJO D3=(H( I J::.<*3-H( l+l )**3)/3 .OO

c THE SIGNS OF 01,02,03 ~RE CbANGED AS THE PLY LAYUP IS DIFFERENT FROM C THAT IN MY M.S. PRCGRAM

Dl=-01 02=-02 03=-03 DO 25 K=l,3 DO 25 J=l,3 O(K,J)=O(K,J)+Dl*OB(K,J)

t.T1 co

JJ=J+3 0(K,JJ)=O(K,JJ)+02*QB(K,J) KK=K+3

25 0(KK,JJ)=D(KK,JJ)+03*QB(K,J) DO 3'J K= l, 3 00 3J J=4,6

30 0( J,K)=D(K,J) lC COIJTINUE

WR. IT E { b, 110) N 110 FORM,H(J//lOX,' LA:'1H,ATE CONSTA.HS 1 //2X, 'NUMBER OF LAYERS= 1 , 13)

WRITE(6,2ll> 211 FORMATC///2X, 1 INPUT VARIABLES Cf E1-\CI-' LAMINA'//)

00 2 J l I= l, N WRIT~(6,lll)T(I),Ell(l),E22(!),Gl2(1),GNU12(!),THETACI)

111 FORMlTC2X,F6.4,2X,3E20.7,2X,F6.4,2X,F6.2/) 201 CQ;\JT[NUE

WRITE(6,212) 212 FCR~4T(///2X, 1 THE A6D MATRIX IS'///)

wR I TE ( 6, 112 l ( ( D ( K, J) , J = 1 , 6) , K = l , 6 } 112 FLlRMAT{'0 1 ,6E20.8//)

DO 501 I=l,3 DO 5)2 J=l ,3 AJJ(!,J)=O(J,J) BI JC I, J)=D( I ,Jt-3 l DIJ( I,Jl=D(l+3,Ji·3)

502 CO~! TI i'lUE 501 CCNT I~JU!:

PAK(ISHELL)=BIJ(l,2) PAP(ISHELLJ=OlJ(l,l) DO 5 ,)4 I= l ,3 DO 504 J=l,J

5 Q4 SM /l LA ( I , J) =A I J ( I , J )

U1 l.O

CALL ~INVCSMALA,3,CET,L,~M,9)

C A L L G M P K. D ( S M A L A , t3 I J , S (.-! A L B , 3 , 3 , 3 , q , 9 , 9 ) 00 5 0 7 I= l, 3 00 507 J=l,3

5 () 7 S M ,\ L 8 T ( I , J ) = S IV A L B C J , I ) C~LL GMPRO(S~ALBT,BIJ,RR,3 7 3,3,9,9,9) DO 5J8 I=l,3 DO 503 J=l ,3

508 SMALD{I,J)=RRCI,J)-DIJ(I,J) C EVALUATION OF THE CONSTANT COEFFICIENTS IN THE SHELL EQUATIONS

GA .'-1 J l = ( A I J ( 3 , 3 ) - A I J ( l , 3 ) * ~,. 2 I L\ I J C l , 1 ) ) t1R AD* RAD I SM AL D ( 1, l ) GA ''1 J 2 = (B I J (1 , 3 ) - 8 I J ( l , l ) ..x A I J ( 1 , 3 l I A I J ( l , l ) ) *RA 0 ISM AL D ( l , 1 ) GAMJ3=(A!J(l,2)*AIJ(l,3)/AIJ(l,i)-AIJ(2,J))*RAD*RAD/SMALO{l,l) GAMJ4=BIJ(l,l)/AIJ(L,l)/~AD-AIJ(l,3)/AIJ(l,l)*GAMJ2/GAMJ1

GAMJ5=AIJ(l,2)/AIJ(l,lJ+AIJ(l,3l/~IJ(L,ll*GAMJ3/GAMJl

GA :-1J6 = (GA M J 4':C A I J ( l , 2 ) - !:l I J ( l , 2 ) Ii~ A 0 +A I J { 2, 3) * GM1J 21 GA M J 1 ) *RAD *RAO I lSM:\LDll,l} GAMJ7=(AIJ(2,2)-41J(l92)*GA~J5~AIJ{~,JJ*GAMJ3/GAMJl)*RAD*RAD/

lSMt.LDll,1) PC=S~ALD(l,l)-S~AL8(2,1)~*2/SMALA(2,2)

C RAD IS THE INNER RADIUS CF T~E CYLINCE~

hR I TE ( 6, 711 ) RAO 711 FORMAT(//' RAD=',El5.7)

BC= ( S ~~ ALB l 2 , l ) IS :-.'Al A ( 2 , 2 ) - S /'-~AL C ( 1 , l ) *GA M J 6 IR A IJ ) IR AD CC=-GAMJ7~S~ALD(l,l)/RAD**4

XXX( I SrlELL l =CC wRITEC6,5l0)

51C FORMAT(///5X, 1 THE COEFFICIE~TS A,R,C IN THE SHELL EQUATIO~S')

WRITE(6,5ll) PC, BC, CC 511 FOR~AT(//3(5X,E2G.7})

IFCBC.LE.l.JE-J4)8C=O.O DISC=BC~BC-4.0~PC*CC

O"I 0

IF(O!SC.LT.O.O)GO TO 601 \-IR I TE ( 6, 602)

602 F~R~AT(//' ThE ROOTS ~**2 ARE REAL'J RM21R=-BC/(2.0*PCJ+SQRT(O!SC)/12.J•PC) R~22R=-CC/l2.C*PC)-SQRT(DISC)/(2.0*PC)

RAB5l=Al3SlRM21R) KA8SZ=A5S(RM22RJ IF(RM21P.GE.O.O)RM1R=CMPLX(SQRT(RABS1),0.0) lf(RM21P.LT.0.0)RM1R=CMPLX(O.O,SQRTIRABS1JJ RM2K=-Rr-'.1R IF(RM22R.GE.O.OJRM3R=CMPLX(SQRTCRABS2J,O.OJ IF(RM22R.LT.O.O)RM3R=C~PLX(Oo0,SQRT(RABS2J)

RM4R=-R?.13R ~RITE(6,603JRM1R,RM2R,RM3R,RM4R

GO TO 1001 601 RM2l=CMPLX(-BC/(2.G*PC},+SCRTC-OISCl/(2.0*PC))

R~22=CMPLX(-BC/(2.0*PC),-SQRT(-01SC)/(2.0*PCJ)

RM l=C SQRT( RM2 l J RM2=-RM1 RM3=CSQRT(RM22) RY4= -RM3 WRITEC6,603)RMl,RM2,RM3,RM4

603 FORMAT(//' T~E FOUR ROOTS OF THE HOMOGENEOUS PART OF THE SOLUTION lARE '//(2E~0.7/J)

YYYCISHELLJ=ABS(REAL(RMl}) llll ISHELL)=ABS(.411\1AGlRMl))

180 l CONT PlUE Cl2=XXXll) Cll=XXX(3) AA=YYY(2) 1'J.8=YYY ( l) ."4.C=YYY( 3 I

_, O'I _,

AO=ZLZ(3) hRITf:(6,606)AA,AG,AC,AD

b06 FORMAT(//' ThE CCNSTANT EXPONENTS IN THE CEFLECTIJN SOLUTIONS FOR l~EGIONS IL,12,II RESPECTIVELY APE 1 //4E20.7//J

C HL IS THE HALF CRACK LENGTH D 0 3 0 ·J l I H L = l r I C R Kl HL=CRKL( IHL) AL=/\A*Hl [3L:::AR ... hL SA=CCS (BU SB=COSH(BL) SC= S I ~~ ( BL J SO=S I NH (BU SH=CCS(AL) SI=COSH(AL) SJ=Sli\l(AL) SK=SINH(!-.L) HA=SA*SB HB=SC*SD HE=Sl\ *SD rF=SC*SB IF((!HL/2*2).EQ.IHL)GO TO 659 HHA=HA HHB=HB HHE=HE 1--HF=HF

659 HG=SH~<SI

HH=SH*SK HI=SJ*SI HJ= SJ* SK XXA=+2.0*HG*PAK(2)/RAD+4.0*AA*AA*HJ•PAP(2) XXB=-2.0*HJ*PA~(2)/RA0+4.C*AA*AA*HG*PAP(2)

O"I N

XXC=+2.0$HA*P~K(l)/~AD+4.J*AB*AS*HB*PAP(l)

XXD=-2.·J*HB>!<PAK( l)/~AD+'toC*AB*.~E~~HA*PAP( l) XXE=PAK(l)/(RAC*CI2) XXF=PAK(3)/RAD-PAP{3)*(AC*AC-DC~ACl

XXG=-2.0*AC~AD*PAP(3)

XXH=PAKl3)/(RAD*CII) YYA=+2.J*AA*(HH-Hl)*PAK(2)/RAD+4.GaAA~•J•(~H+HI)*PAP(2)

YYB=-2.0*AA*(~H+Hil*PAK(2)/RA0+4.0*AA**3*(~~-Hl)*PAPl2) YYC=+2.0*AR*CHE-HF)*PAK(l)/RAD+4.0~nB**3*(HE+HF)*PAP(l) YY0=-2.0*AB*(HE+hF)*PAK{l)/RAD+~.J*AH**3*(HE-HF}~PAP(l) YYE=AC*(-PAK(3l/RA0-(3.0*AC*AG-hC•ACl*PAP(3)) YYF=h0*(-PAK(3)/RA0+(3.0*AC*AC-~D*AD)*PAP(3))

00 7fJ7 J=l,6 C V K ( J ) =O • 0 DO rJ 7 I =l , 6

707 DBK(l,J)=OeO D3K( 1,1)=2.0*HG DnKL.:3, l>=HH-HI OBK(5,l)=XXA DBK(6,l)=YYA CBK{ lr2l=-2.0*HJ DBK(3,2)=-HH-HI DSK(5,2)=XXB DBK(6,2)=YYB DBK(2,3}=2.0*HA DBK(4,3)=Hf.-HF D8K(5,3)=XXC OBK(6,3)=YYC DBK(2,4)=-2.0*HB DBK( 1+, 1+) =-HE-HF DBK(:),4)=XXD DBK(6,4)=YYD

_,

°' w

08!<.( i, 5)=-l.O DBK(Z,:>l=-l.Q DBK(3,5)=RETAl*AC/(2.0*AA) OBKC4,5l=BETA2*AC/l2.0*ABl 0 8 K ( 5, 5 ) =-·XX F Di3K(6,5)=-YYE OBKC3,6)=8ETAl*AO/l2.C*AA) OBKC4,6)=aETA2*A0/(2.0•AB) DBK( 5 ,6 )=-XXG DBK(6,6)=-YYF CVK( ll=l.O/CII CVK(2)=l.O/CII-l.u/Cl2 CVK(5)=XXH-XXE CALL SIMQ(OBK,CVK,6,KS,36) IFC ( IHL/2*2) .EQ.IHL)GO TO 709 DO 7Ji3 II=l,6

708 CVPC I I) =CVK( I I> GO TO 30C.l

709 Cl\JO=CVP( l) C2 QO =CV Pl 2 ) C9QO=CVP(3) C1UQO=CVP(4) C21QO=CVP(5) C22QC=CVPC6l DHL=CRKLCIHLl-CRKLCIHL-l) DCl=CCVK(l )-CVP(l) )/OHL OC2=CCVK(2)-CVPC2) )/OHL DC9=(CVK(3)-CVP(3))/DHL DClO=lCVK{4)-CVPC4))/DHL DC2l=CCVKC5l-CVP{5))/0HL OC22=CCVK(6)-CVP(6))/DHL ZUSYL=l.0/CI2+2.0*HHA•C9QO+OC9~lHHE+HHF)/AB-2.0*HHB*ClOQO+OClO*

....... O'I ~

l(HHE-HHF)/AB+(AC•OC21-AD*DC22)/(AC*AC+AD*ACJ ZUfWL=-ZUBYL IF(ZUBYL.LE.0.0)GO TO 3001

C GAl-1AA IS THE SPECIFIC ADHESIVE FR.\CTUf<E cf\ffPGY IN Ll3/IN. C PCR IS THE CRITICAL INTER~Al PRESSURE JN THE CYLINDER AT WHICH C OELA~l~AT!GN P~OPAGATES

C PPP=PC~/(GAMAA)**0.5 PPP=SQRT(2.0/ZUBYL) wRITE(6,702lCRKL(lHL-l),PPP

702 FORMAT(//' THE HALF CRACI< LEf\GTH IS 1 //E20.7// 1 THE CRITICAL IMTL. lPRESSURE IN THE CYLINDER, IN PST, AT WHICH OELAMINATION PROPAGATES 2, CIVICED BY'/' THE SQARE RCOT OF THE SPECIFIC ADHESIVE FRACTURE 3ENERGY (IN LB./ IN.) IS 1 //E20.7)

c c

3001 CONTINUE STOP END

C SAMPLE DATA SET c c

EFFECT OF BETAl & BETA2. PRESSURE C~ INNER CYL. SURFACE (l):B-AUO) (2J:G-E(45)

0.01 -0.01 3.0 50 0.2 J.201 0.3 0.301 0.4 0.401 0.5 0.501 0.1 J.1cs a.a o.acs (;. 9 f • 9,' 5 1.0 l 0 0•".15 1.2 1.2~5 1.3 l.3C5 1.4 1. 4'.)5 1.5 l.505 1.7 1.705 1.8 1.805 1 • c; l. 9:)5 2.0 2.005 2.5 2.5·:)5 2.75 2.755 3.0 3. G=:' 5 3.5 3 .50 5

l 0.05

0.6 1. 1 1.6 2.2s 4.0

0.6Jl 1.105 1.605 2.255 4. 0 l

CTI U1

166

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The vita has been removed from the scanned document

PROPAGATION OF DELAMINATION IN

LAYERED ANISOTROPIC CYLINDERS

by

Ramaswamy L. Ramkumar

(ABSTRACT)

The propagation of delamination in a layered anisotropic cylinder

is studied under different loading conditions. Initially, a closed-form

solution is obtained for an infinitely long cylinder by treating it as a

thin shell, with an axisymmetric crack in the adhesive layer between two

laminae, and an axisymmetric loading acting as the driving force. The

axial and shear strains of the reference plane are considered negligible

compared to its circumferential strain. The cylinder, after delamination,

is modelled as three shell elements joined at the crack tip. A sharp,

flat crack tip is assumed, and the slopes of the three shell elements at

the crack tip are set to zero. The radial displacements of the three

regions, obtained through the imposition of the proper boundary and

matching conditions, are substituted into Clapeyron's theorem to obtain

the total strain energy in the cylinder. The change in this strain

energy for a unit change in the crack length is used in the Griffith

energy balance criterion to evaluate the critical loading at which the

delamination propagates. The critical loading is studied as a function

of the half crack length. Due to the assumption of a flat crack tip,

only the moment continuity across the crack tip is imposed. Normal

shear continuity is not enforced and this leads to interesting results

in a few cylinders. 11 Cra.ck arrest 11 regions, or ranges of crack length

within which no real value of the loading could make the delamination

propagate, are observed in some cases. Two different loading conditions

are treated: one with a pressure acting on the crack surface, and the

other with a loading on the inner cylindrical surface. "Crack arrest"

phenomenon occurs in the latter problem for certain layer thickness ra-

tios.

As the next step, a more general crack tip geometry is assumed and

shear force continuity across the crack tip is enforced. This model

establishes the fact that the abnormal delamination behaviour in the

previous model, like the 11 crack arrest", is due to the absence of normal

shear continuity across the crack tip. It also shows that the crack tip

assumption made in the previous formulation - a flat, sharp crack - is

valid. When the crack tip geometry is generalized the radial displace-

ments in the cylinder are obtained numerically. Four different crack

tip geometries are assumed.

The axial and shear strains in the reference plane are included

in a subsequent model to determine their effect on the delamination behav-

iour. A general observation is that the critical value of the loading is

lowered for any initial crack length. The magnitude of the drop depends

on the geometric and material properties of the plies that constitute the

layered cylinder. When the loading is applied on the inner cylindrical

surface, the effect is predominant for very small crack lengths. On the

contrary, the decrease in the critical value of the pressure acting on

the crack surface is considerable for large crack lengths.

LD5655.V856_1976.R35.pdf - VTechWorks

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