SUBLAMINATE-ANALYSIS OF INTERLAMINAR FRACTURE IN COMPOSITES•*

**

Erian A. Armanios and Lawrence W. Rehfield

School of Aerospace Engineering

Georgia Institute of Technology

Atlanta, Georgia 30332 USA

(404) 894-3067

Abstract

A simple analysis method based upon a transverse shear deformation theory

and a sublaminate approaSh is utilized to analyze a Mixed-Mode edge delamination

specimen. The analysis provides closed form expressions for the interlaminar

shear stresses ahead of the crack, the total energy release rate, and the eiTergy

release rate components. The parameters controlling the behavior are identified.

The effect of specimen stacking sequence and delamination interface on the strain

energy release rate components is investigated. Results are compared with a

finite element simulation for reference. The simple nature of the method makes

it suitable for preliminary design analyses which require a large number of

configurations to be evaluated quickly and economically.

* This work was sponsored 6y the NASA Langley Research Center under Grant

NAG-1-558.

** Research Engineer and Professor, respectively.
: Introduction

Del ami nations along the free edges of laminates subjected to tensile

loading have been observed during testing and service. The presence of

delamination, initiated by interlaminar stresses, causes redistribution of

the stresses among plies in a laminate and, therefore, usually results in a

reduction of stiffness and strength. The edge delamination (ED) test has

been proposed by Pagano and Pipes to characterize the interlaminar peel

• • 2 strength of laminated composite materials. O'Brien extended the scope of

the test., to investigate delamination onset and growth in graphite/epoxy

laminates unsder uniform extension. . A simple expression was also developed

for the total energy release rate. The energy release rate components G.,

Gj, and G,.,. associated with the opening, shearing and tearing modes,

respectively, were estimated based on a finite element simulation and the

2 3 crack-closure method ' .

A similar approach was used to study delaminations around an open hole

4 in composite laminates . Discrete locations around the hole boundary were

modeled as straight edges, with the ply orientations rotated by an

appropriate angle. Delamination was found to be governed by the percentage

of Mode I for a given geometry under static loading for the graphite/epoxy

material systems under consideration.

Whitney and Knight developed an ED specimen which produces Mode I

behavior. The analysis was based on classical laminated plate theory and

v . 3

continuity of displacements, force resultants, and moment resultants

between the cracked and uncracked regions of the plate were not satisfied.

In addition, such an approach precludes any reasonable determination of the

effect of specimen geometry on GT.
Recently Whitney developed a higher order laminated plate theory which

includes transverse shear deformation and a thickness-stretch mode to

analyze a Mode I ED specimen. The effect of specimen geometry on strain

energy release rate was also investigated for 12 ply laminates of the class

[8/-e2/e/902]s and for the class [03/903Js.

It is the purpose of the present work to develop a simple model for

the analysis of Mixed-Mode ED specimens. Such a model provides closed-form

estimates of G,, GJJ and Gj,,. and hence allows one to establish appropriate

7 8 failure criteria ' for del ami nation.

Preliminary Remarks

Consider the ED specimen shown in Figure 1 subjected to a uniform

strain e - e. Due to symmetry one quarter of the laminate is analyzed as

X

shown in Figure 2. The response is only a function of Y and Z. The

laminate is divided into sublaminates with thicknesses h, and h and local

coordinate sys.tems z,, y.. and z , y . The crack length is .denoted by a.

Sublaminates 1 and 2, and 0 and 3 represent the groups of plies above and

below the interface along which delamination occurs, respectively.

In order to provide an accurate estimate of interlaminar stresses, a

higher-order theory should be considered since classical laminated plate

theory predicts zero interlaminar stresses. A shear deformation theory can

be used for this purpose. This theory provides a good estimate for

interlaminar shear stresses T and T . However, the interlaminar peel

stress a is not accurate. The reason for that is the absence of

thickness strain.
From symmetry, the transverse displacement w is zero at 2=0, hence the

prescription of i w at the middle plane Z=0 fixes w everywhere. In this

case, .the vertical shearing force resultant at both ends cannot be

prescribed and the distribution of the peel stress will not be correct.

Inspite of this simplification, reliable energy release rate components can

be estimated based on interlaminar shear stresses. G. is evaluated as GT -

(Gjj+Gjjj), where the total energy release rate is denoted by G,-.

In the present formulation thickness strain is neglected and

consequently considerable simplification in the analysis is achieved.

Another source of simplification in the present approach is due to the

modeling of the structure as sublaminates—group of plies that are

conveniently treated as laminated units. This approach can be applied with

confidence if the characteristic length of the response is large compared

9

.to the individual sublaminate thickness.

Overview of the Analytical Solution

In the .following sections a step by step procedure is provided for the

solution of the ED specimen. Intermediate results are also provided. The

governing equations are derived in Appendix I. Expression for the

interlaminar stresses, total energy release rate and energy release rate

components are given in Equations (34), (40) and (42)-(43), respectively.

The parameters associated with these equations are provided explicitly in

terms of the stiffness coefficients in Appendix II.

A solution based on Classical Lamination Theory is given in Appendix

III. This solution represents the behavior in the interior of the

laminate. Application of the present analysis and comparison with a finite
element simulation for 63 test cases is presented under the section

entitled Results and Discussion.

The reader interested in results and comparison can refer directly to

the "Results and Discussion" section on page 28.

Analysis

Assume the following displacement field within each sublaminate:

u = XE + U(y) + zpx(y)

v = V(y) + zpy(y) ' (1)

w = W(y)

where u,v, and w denote displacements relative to the x, y, and z axes,

respectively, and e is a uniform axial strain. Coordinates y and z are

local coordinates as shown in Figure 2. The present formulation recognizes

shear deformation through the rotations B and 8 . The corresponding

strains are

Exx

"yy

Yxy ~ 'y

V = Py

 + W'y

(2)

The variables associated with sublaminates 0 through 3 will be written with

subscripts 0 through 3, respectively.
From symmetry

w0(y,-ho/2) = 0

or WQ = 0

(3)

From continuity of displacements at the interface between sublaminate (1)

and (0)

. uQ(y, h0/2) = u^y.-h^Z)

vQ(y, ho/2) = u1(y,-h1/2)

wo(y, ho/2) = w1(y,-h1/2)

(4)

Substitute from Equation (1) into (4) to get

hi ho

Uo = Ul - T Plx - i pox

h

1

-

 Vl - T

W

l =

 W0 = °

(5)

Governing Equations

The governing equations for each sublaminate are derived in Appendix I

using a virtual work approach. These equations are written below for

convenience.

6
Constitutive Relationship

Nx

N

y

N

xy

X

My

V,

=

All A12 A16 B12 B16

A12 A22 A26 B22 B26

A16 A26 A66 B26 B66

Bll B12 B16 D12 D16

B12 B22 B26 D22 D26

B16 B26 B66 D26 D66

e

%

s

py,y

Px,y

V =

A A H

44 45

_

A45 A55_

Equilibrium Equations

xy,y

y,y

y.y

(6)

Mxy,y * Q

My,y -

 Qy

mx =

my

 =

(7)

where n , n , q, m and m are defined in Equation (1-9) of Appendix I.

The equilibrium equations can be written in terms of kinematic

variables by substitution from Equation (6) into Equation (7). The result

is
A66Lyy

A26Lyy A22Lyy

B66Lyy B26Lyy

B26Lyy B22Lyy

SYM

<

D66Lyy ' A55>

(

D26Lyy ' A45> (D22Lyy ' A44>

L

yy =

 d

Ly = d/dy

U "

V

w

p

x

_

p

y

= -

n

X

n

y

-q

m

x

m

y

(8)

Solution Methodology

The above collection of equations are to be applied1 to individual

plies of a laminate or to groups of plies... sublaminates. The solution

steps are summarized in the following:

1. Divide the laminate into sublaminates according to geometry and

loading condition. The sublaminate length is selected such that

within the sublaminate the geometry and loading are continuous as

is commonly done in engineering analysis of simple structures.

2. The displacements, resultant forces and moments, and interlaminar

stresses in each sublaminate are governed by the equilibrium

Equation(7), the constitutive relations (6) and the

displacement distributions (1). Write these equations for each

sublaminate in the analysis model.

3; Apply interlaminar continuity conditions and enforce traction or

displacement conditions at the extreme upper and lower surfaces

of the laminate.
4. Solve the system of coupled ordinary differential equations for

the element variables.

5. Enforce the boundary conditions at constant values of y, the

0

laminate "'sections, as well as continuity requirements between

sublaminate ends in order to find the values of the arbitrary

constants resulting from the solution in step 4.

6. Determine interlaminar stresses, resultant forces and moments

displacement distribution and energy release rate.

Application

 %to the ED Specimen

The ED configuration is divided into- four sublaminates as shown in

Figure 2. The response associated with sublaminates 1 and 0 is coupled

through the continuity conditions at their common interface. Hence, the

variables associated with both sublaminates are to be solved

simultaneously. The situation is different with sublaminates 2 and 3

where the continuity conditions are relaxed due to the presence of the

crack. Therefore, the variables associated with these sublaminates are not

coupled.

The solution procedure for sublaminates 1 and 0—the uncracked portion

of the laminate—is presented first, followed by the cracked portion

represented by sublaminates 2 and 3.

Uncracked Region of the Laminate:

(i) Sublaminate 1

The upper surface of this sublaminate is stress free. Denote the

shear and peel stress at the bottom surface by t , t and p, respectively,

: X y

The transverse displacement VL, is zero from Equation (5). Hence, the

equilibrium equation in terms of the displacement variable takes the form

9
A66

 Lyy

26

 Lyy

0

B

26\y

' A I R I A26 yy

 b66 Lyy

A22 Lyy B26 Lyy

B

26\y

B

* |

— _ L.

22 yy

t

A

44\

6

 Lyy"A45 *

2

lLyy-A441}

Ul

v

1

"ix

_ V

= -

'-*

_t

P

2~

h

(9)

\

(ii) Sublaminate 0

From symmetry condition at the sublaminate bottom surface the shear

stresses t. and t, are zero. From reciprocity of stresses at the interface

between sublaminates 0 and 1, the interlaminar stresses at the upper surface

of siiblaminate 0 are it , t and p. Denote the peel stress at the sublaminate

bottom surface by p... Hence the equlibrium equation takes the form

MA °L A- °l 66 Lyy M26 Lyy

AM °L A °L 26

 Lyy M22 Lyy

R °l R °l B66

 Lyy 26 Lyy

B26°Lyy B22°Lyy

R °l R °l B66

 Lyy b26 Lyy

B26°Lyy B22°Lyy

-AM °L -A °L 45

 Ly

 A44

 Ly

0 0 0 0

6

 Lyy"A55 } (D26 Lyy"A45 )

6°Lyy-A45°) (D22°Lyy-A44°^

Uo

Vo

POX

Q

°

y_

= -

»x

ly

Pj'P

ho

2~

 tx

h

0

(10)

10
form:

Equations (9),. (10) and (5) can be combined to yield the following

<

FllLyyV>

F2iLyy i

F41Lyy

SYM

(F22Lyy"A44 *

F32Lyy (F33Lyy"A55 \

(F42Lyy"A45 } F43Lyy (F44Lyy"A55 }

p

0 V

p

= 0

(11)

The parameters in Equation (11) are defined in Equation (II-l) of Appendix

s

II.

Assume an exponential solution of the form

'

 plx'

* * * *, ,

)e

sy

(12)

Substitute from Equation (12) into Equation (11) to get the following

characteristic equation

E0 =

(13)

Coefficients £„ through EQ are defined in Equation (II-5) of Appendix II.

Parameter Ea represents the determinant of the coefficients of L in matrix

equation (11) while EQ is the determinant when L is set to zero. Parameter EQ

depends solely on the stiffness coefficients A-^, A,.,- and A^^ for both

sublaminates while E0 is predominantly influenced by the bending and coupling

o

coefficients D.. and B... Hence its numerical value can be orders of magnitude

I J ' J

smaller than the remaining coefficients. This fact results in the presence of a

boundary zone in the response.

11
Typical values of a nondimensional form of the coefficients Eg through

EO is given in fable II for three laminates made of T300/5208

graphite/epoxy material. The nondimensional form of the coefficients is

obtained by making the following substitution

s= sb (14)

Where b is the laminate semi-width. The arrows in the layups of Table II

indicate..the interfaces containing delaminations. The material properties

and geometry^ appear in Table I.

The characteristic roots controlling the behavior are determined from

Equation (13), which has a closed-form solution. However when A.g, B.g,

D.g (j = 1,2) and A45 are neglected, Equation (11) takes an uncoupled form

and consequently the characteristic equation (13) can be factorized into

two biquadratic equations. The stiffness coefficients A.g, B.g and D.g

represent coupling interaction while A.,- depends predominantly on the shear

modulii G-. and G?3. For a composite material where these shear modulii

are approximately the same, A45 can be neglected.

The uncoupled form of the characteristic equation is

^ ~^ ~^

 + ^ ^ + ^ ^

= 0

(15)

12
The first, bracket in Equation (15) control V and p behavior while the

second U and p . The absolute values of the roots in Equation (15) will be A

denoted by s .., s „>

 s i and s 2' These can De regarded as a good

approximation for the roots of the coupled equation (13). A comparison

between the coupled and uncoupled roots for a typical laminate is provided

in Table III. These roots are found to be real for the material system and

layup used. The predictions of the uncoupled equation (15) are in good

approximation with the coupled equation (13). Also, the larger

characteristic roots s .. and s , correspond to bending behavior while the

smaller s ^ and s ~ control the membrane behavior. This is shown in the

following section.

Membrane behavior can be modeled by setting

Mx = My = Mxy = 0 (16)

in the equilibrium equations for sublaminates 1 and 0. The characteristic

equation (13) reduces to

The membrane parameters F... and Fp2m are defined in Equation (II-9)

of Appendix II. they depend on A^. and B^. coefficients. The

characteristic roots predicted by Equation (17) are included in Table III.

By comparison with the roots of Equation (15), bending behavior is more

localized than the membrane behavior as the characteristic roots

controlling bending are larger. This fact is expected since Classical

Lamination Theory (CLT), which predicts membrane-type behavior, prevails

in the interior of the laminate.

13
Since the laminate width is large compared to its thickness and to the

crack length, the response in sublaminates 1 and 0 is predominantly

decaying from the crack tip and, therefore, only the roots with a negative

sign will be considered in this solution.

Cracked Region of the Laminate:

Sublaminate 2

This sublaminate represents the upper group of plies in the cracked

portion of the laminate. Since there is no restriction on the transverse

displacement W, boundary conditions on Q can be specified.

The upper and lower surfaces in this sublaminate are stress-free and

atvy,=-a there is a' free edge. The equilibrium equation (7) reduces to

N = -N = Q = • M = 0

2 -2 2 2

M - Q = 0

Xy2,y X2

(18)

By substituting these conditions into the constitutive relations (6), to a

single differential equation in terms of p? obtained.

Cd22 + D26lcd32> P2x,yy 55 A 1

H44 -I

(19)

Parameters Cd12, Cd22 and Cd-2 are defined in Equation (11-10) of Appendix

II. Solution of Equation (18) leads to

scyl "scyl

P2x

 = Hl e H2

e

where . 0 < y- < - a

14
s =

55 44

1/2

(20)

Since the crack length 'a' can be small, positive as' well as negative

signs of the root s have been considered. The arbitrary constants H. and

Hp are determined from the boundary condition at

•Y! = -a Mxv (-a) = 0 (21)

i .

 xy2

and continuity conditions at y, = 0 with sublaminate 1.

The displacements at the bottom surface of sublaminate 2 are

= v.(0) + (Cdn

= u.(0)

where

* v.(0) = v2(0,Ch1/2)

u.(0) = U

H2(l-e

-e50 l) + H2(l-e

 C

(22)

The linear terms in y. in Equation (22) represent the displacement

when 3= 0.

15
(ii) Sublaminate 3

The upper surface of this sublaminate is stress-free, while the

interlaminar shear stresses on the lower surface are zero from symmetry

conditions. Moreover, there is a free edge at y = -a. The equilibrium

equation (7) reduces to

= N = 0

Xy3 "y3

M

xy3,y

- Qv = 0

M - - Q = 0 y3,y v"y3

(23)

By substituting these conditions into the constitutive relations (6),

two coupled differential equations in p~ and p~ is obtained

J

22Lyy"

 A44

f\

_

J

aeV

 A

45

J

2eV

 A45°

Q

J

66Lyy"

 A55_

[

P

3y

L

P

3x_

= 0

(24)

Parameters J22 , J2g and Jg6 in Equation (24) are defined in Equation

(11-12) of Appendix II.

Solve the differential equations for p~ and P3 .

Ss2yo "

ss2yo

ssy -ss^y.

where

+ I2 e -

1 u)I + n, do e '

£ O

u

 + I4e

 u)

-a < y0 < o

16
n. = - (J22ss2 -A44°)/ (J26sSj

2 - A45°) (25)

The constants I. through I, are found from the boundary conditions at

y = -a . M (-a) = Mxv (-a) = 0 (26) o y3 xy3

and continuity conditions at y =0 with sublaminate 0.

The roots ss. and sSp in Equation (25) are found by solving the

characteristic equation resulting from equation (24).

The'displacements at the upper surface of sublaminate 3 are given by

v3(yo, ho/2) = v.(o) + wdn eyo + (wd12+ ^ + n^d^) [I^e -

1 °- !)•

"

sslyo ho Ss2yo

+ I2 (e

 l ° - 1)] + (wd12 + -f + n2wd13) [I3 (e ^ °- 1)

-ss2y

+ I4 (e

 d °- 1)]

u3(yQ, ho/2) = u.(6) * wd21 eyo * [wd22 + r^ (wd23. + ^) ] [I^e

 1 ° - 1)

"

SSy h SSy

I2 (e - 1)] + [wd22 + n2 (wd23 + -)] [I3(e

+ I4 (e * ° - I)] (27)

Parameters wd..., wd.,2, wd.,3, wd,,., wd22, and wd23 are defined in Equation

(11-12) of Appendix II.

In order to determine the energy release rate components by the

virtual crack-closure method , the relative displacements at the crack

surface as well as the interlaminar stresses at the crack tip are needed.

From Equations (22) and (27) the relative displacements are

17
AV = v- (y , h /2) - v?(y - I-

•50 0 C 1 j .

h. h

= (wd^ - Cd,, + 75- Cd0 1 ) ye +(wd1 0 + —x + ru wd,,)

J.J. II c. ol it c 1 lo

ss^ -ss.y

[Ij_ (e

 L - 1) + 12 (e L - 1)]

h ss9y ~ss9y

+ (wd12 + ^ + n2 wd13) [I3(e ^ - 1) + I4(e * - 1)]

o

 h1

 Sry

 "Sr

y

+ (t:d19 - ~ Cd-,) [H. (1 - e c ) + H, (1 - e c )] 1*1 c. 6£ 1 ^

AU -•••Li3(y , h /2) - Upfy^ - h,/2) .

s

h SSly

= (wd01 - Cd01)ye + [wd00 + n, (wd00 + 75—)] [I,(e - 1)

£l CL tLc. J. co c. 1

+ I2 (e

 1 - 1)]

+ [ wd22 +-n2 (wd23 + 2^) [I3 (e

 2 - 1) + I4 (e 2 - 1)]

h, s y -s y

+ (Cd22 - -i-) [H^l - e ) + H2 (1 - e • )]

-a < y < 0

(28)

The linear terms in Equation (28) represent the relative displacements

when the shear deformations p? , p3 and P3 are neglected. The remaining

terms are exponential, and their effect oh the predictions are depicted in

Figures 4 and 5, for two typical laminates. The dotted lines denoted by

All, and AV, in the figures correspond to the linear contribution, while the

solid lines All and AV include the exponential terms. The crack length 'a1

in figures is 10 percent of the laminate semi-width and the applied strain

is 1000 micro in/in: The simple linear distribution represents a good

18
approximation for the test cases considered. Due to the simplicity and

accuracy of the linear displacement distribution, the energy release rate

computations will be based on the linear displacement contribution only. A

discussion of the effect of the exponential terms on the calculation of the

total energy release rate will be provided later.

Interlaminar Stresses

Denote the absolute values of the roots in Equation (13) by s. (j=

J

1,4). The response can be written in the form

-s.y

~

siy

,

=

 6*

 e

-s.y

j = 1,4; o < y < b

(29)

Summation over the range of index j is implied in Equation (29).

Parameters a., to., Y., v., and u. are defined in Equation (11-14) of

J J O J J

Appendix II. The arbitrary constants of integration associated with the

displacements are C and C .

19
Substitute from Equation (29) into the constitutive relationship for

sublaminates 1. and 0 to get

N*11

\

V

\

Mxy

=

A

 1 A * A 1

11 12 16

1 1 1

HA A A 12 H22 H26

A * A * A 1

H16 26 H66

1 1 1

B12 B22 B26

B R R - - D0/_ D/-/- 16 26 66

e

Cv

_

Cu_ - - .

Nx

lj

Nylj

Nxyxj

Myij

MXYIJ

q^r

AX,1

A

 1 A l

H44 H45

A 1 A 'I H45

 rt55

~i"

"j

-s,y

G.e

 J

J

AA ° A ° A ° ll

 H12 A16

AH ° A ° A ° 12

 M22 26

AH ° A ° A ° 16

 H26 H66

B12 B22 B26

B16 B26 B66

M

"S.y

"v =

A

44

 A45

AH ° A ° 45

 H55 ";"

(30)

20
The parameters associated with Equation (30) are defined in Equation

(11-15) of Appendix II.

There are twelve arbitrary constants of integration namely: G. (j=

J

1,4), C , C , Hp H2 and I- through I,. These can be found from the

boundary condition at the free edge of sublaminates 2 and 3. expressed in

Equations (21) and (26) and from the • continuity conditions between

sublaminates 1 and 2, and 0 and 3 at y = 0. These are:

N (0) = N (0) = 0

V

0) = vo) =

Mv/ (0) = Mv (0) = 0

yl y2

N (0) = N (0) = 0

y

o

 y3

Nxy0(0) =

Mxy2(0) =

(0) = M (0)

(31)

21
Boundary .conditions on Q cannot be specified in sublaminates 1 and 0

as a result of neglecting the transverse normal strain in the assumed

displacement function. Equations (21), (26) and (31) represent 14 boundary

conditions for twelve constants. However, the last two continuity

conditions at the interface between sublaminates 0 and 3 in Equation (31)

lead to

44° V0) + A45° POX(°> =

 A44°

that is,

Qv (0) = Q (0) .

yo y3 (32)

which cannot be specified. Therefore, the conditions p (0) = P3x(0) and

3Q (0) .= 3- (0) cannot be prescribed for consistency.

Substitute for the resultant forces and moments in sublaminates 0

through 3 into the first nine equations in (31) and solve for the arbitrary

constants. As C and C are easily obtained, their expressions are listed

below for convenience.

Cv = (k26 k!6 - k66k12> £/D

Cu = (k26k!2 - k22 k!6) e/D

where

k!2 " A12

 + A12

k!6 = A161 + A16°

Parameters koc, kcc, k00 and D are defined in Equations (II-3) and

£O DO C.C

(H-4).

22
The resul.tant axial forces and moments in Equation (30) consist of two

parts: a constant term and an exponentially decaying term. In the

interior of the laminate the second term is negligible and the response is

controlled by the first term. Since CLT is retrieved in the interior of

the laminate, the first term can be recognized as the CLT prediction. This

is shown in Appendix III where a CLT solution is derived for the ED

specimen. This'' solution is simple and can be derived on a ply-by-ply

basis. Moreover, the sign of the resultant axial force N determines the

sign of the interlaminar peel stress. A compressive peel stress tend to

retard delamination at a given interface.

Interlaminar stresses at the interface between sublaminates 1 and 0

are found from equilibrium.

' \i

 SJ

The distribution of the peel stress p is not in equilibrium since the

boundary condition on the shear force Q.,(0) cannot be prescribed.

rb

yl

f Pdy = f Q dy = Q (b) - Q (0)

JQu j u Q 1 y-p l j y- L i y-i

 (35)

Qw (b) = 0 as Qw is an exponentially decaying function, however Qw(0)

0.

\ .. \**/ — w« X»/ • ** **l I ^/%1^^/ltivllWIVAII ^ «MW ^f*^J • • • y iwtll^*wi**tl j I1*,*iv*rw*ri "l/'

23
A comparison of the interlaminar shear stress distribution t along

the interface between sublaminates 1 and 0 appears in Figure 6 for three

laminates. The applied uniform strain is 1000 micro in/in.

Energy Release Rate

The total energy release rate can be determined by considering the

work done by external forces . For a uniform applied strain e = e f\

throughout the laminate this can be written as

r _ -e dP

\

 GT - 2~ da"

where P is the axial applied force and is given by

r(b-a) ro

P = (N- + N ) dy + (N + N )dy

Jo xl xo

 J-a

 X2 X3 (37)

The axial forces N and N in sublaminate 1 and 0 respectively are

xl xo

given in Equation (30) while the axial forces in sublaminate 2 and 3 are

given by

N

x2 = <Ai;

NX3 = <A

li

+ AT H + A C(\ + R

1

 M12 ^11 H16

 LQ21

 D

L°

 + A12° Wdll + A16° Wd21)e +

+

 (J2

12

 Cd31)e + JlS

c

(Hle

ss.

(J2 + ni J3)sSl (il6

 J

ss?y

+ n2 J3) ss2 (I3e * -

- H2e ;

,y -ss.

L

- I 2 e '

-ss?y

I4e )

)

L

y

(38)

24
where

V^

 + A12* Cd12 + C Cd22 + B12 Cd32

J2 = B12° + A12° Wd12 + A16° Wd22

J3 =B1 6°+A1 3° wd13 +A16° wd23 (39)

Substitute from Equations (30) and (38) into Equation (36) and using

Equation (37) to get

GT -i £K12 C

v

 + K16Cu ' Cdn + A16 Cd21 + B

wdn + A16° wd21) e

12 Cd31)e

J

 + GR

(40)

where GD is an exponential function of the crack length. For a crack

K

length larger than a few ply thicknesses, the contribution of GR is

negligible. This situation is depicted in Figure 7 where a normalized

total energy release rate for a typical laminate is plotted against the

r

crack length divided by ply thickness. The laminate thickness is denoted

by t in the figure and the ply thickness is h. In this case, GT reaches

the constant value predicted by the first term in Equation (40) for a value

of crack length larger than four ply thickness. Also appearing in Figure 7

is a comparison with a finite element solution presented in Figure 6 of

Reference 4.

The total energy release rate is a global parameter which does not

depend strongly on the local details at the crack tip. This is the reason

why relatively simple modeling approaches yield adequate predictions for

the total energy release rate.

25
This finding checks the results of References 4 and 8 where simple

>

closed-form expressions for GT are in good agreement with predictions based

on refined theories or on finite element simulation.

Energy Release Rate Components

• o • ' ' .

Using the virtual crack-closure method G,, and G,^ are given by

- -Gn = Lim • t (5-r)Av(r)dr

A

1. 6->0

 26 J0

 y

1 f6

GUT = Lim =± t (6-r)Au(r)dr 111 ^n

 25 -Jn.

 x

(41)

where 6 is a virtual crack step size. Unless a singularly exists in the

stress field, Equation (41) yields the trivial result (GTT= GTT T = 0) when

9

the limit as 6 tends to zero is determined. Consequently, a sufficiently

large finite step size is essential to get an answer when using models that

do not exhibit singular behavior.

Subsitute from Equation (34) for the interlaminar stresses in term of

force resultants into Equation (41) and use the linear terms in Equation

(28) for the relative displacements to get

V

Gin = \

26
where

h

G l

v

 = wdir cdn + T

0

u

 = Wd

 21'

 Cd

 21

I rA

A

 Vdy

J

0

 xyl (42)

The finite crack step size is denoted by A. The resultant forces N ,

and N . exhibit a boundary layer behavior. This is shown in Figures 8 and

9 where the resultant force distributions are plotted along the laminate

width. In the interior a constant value corresponding to the CLT

prediction is reached.

The average resultant force values F and F depends on the selected

value of the crack step A. Recommended ad hoc values as a percentage of

the initial .crack length have been suggested in the finite element

representation of the crack-closure method. However, within the boundary

layer region N - and N « have steep gradients and consequently a small

variation in the selection of A leads to large variations in F and F .

Since N ^ and N . distributions at the crack tip are controlled by the

boundary layer decay length, the crack step size A should be selected based

on the boundary layer length rather than a percentage of the initial crack

length.

Mode I energy release rate is found from

Gj = 6T - (Gn + GIH) (43)

27
Gj. and G,... are given in Equation (42) and G, in Equation (40).

The energy release rate components predicted by this approach show a

good correlation with finite element solutions when the crack step size A

is selected as

§3)]2

 (44)

where s. (j=l,4) are the characteristic roots in increasing, magnitude

J

nondimensionalized by the ply thickness h. The distinct roots s. control

* J

the decay length associated with different physical variables. It has been

found empirically that the combination above gives very good correlation

with finite element simulations for over sixty cases that have been

compared. Although the functional form is not simply identified with a

boundary length it 'nevertheless contains the proper information.

For a general layup the crack size A is influenced by all four roots.

However, for laminates where Mode III is negligible the crack step size is

influenced by the characteristic roots s^ and s, that control V and 6

behavior. For this situation, the following crack step size expression

provides good correlation with the finite element solution:

•£= 2r6 (s4 i2)

2 (45)

Results and Discussion

An extensive comparison between the energy release rate components

predicted by the present approach and a quasi 3-D finite element solution

has been performed. The results appears in Tables IV through X and in

Figures 10 through 28. The finite element results in Tables V through X

8

 28
are related to. the work of Reference 4 and were provided by the authors.

The properties and geometry used appear in Table I. The applied uniform

strain is one micro in/in.

In Table IV a comparison is provided for six laminates. The first

three layups were reported in Reference 3. The 6,,,. component in these

layups are negligible. The remaining layups represent cases where G,, Gj,

and GJJ, components are all finite.

The laminates presented in Tables V-X are rotated stacking sequences

for two quasi-isotropic layups. The first is a [0/90/+45] and the second

is a [45/90/-45/0] . The results of the first laminate appear in Tables

V-VII while in Tables VIII-X the results of the second laminate are

provided. These laminates were used in Reference 4 to investigate,

delaminations around an open hole. The strain energy release rate

distribution around the hole boundary, for delaminations growing in a

prescribed interface, was calculated by assuming that each circumferencial

portion acts as a straight edge subjected to an .appropriate uniform

circumferencial strain. Hence, at each circumferencial angle 0 the

laminate reflects a new stacking sequence where the load is applied in the

0-direction tangent to the hole. The angular position corresponding to

each stacking sequence is provided in Tables V-X.

The results in Table V are plotted in Figures 10-12. In Figures 10

and 11 the percentages of G,, and G^r are plotted against the angular

position, 0, around a hole in the first laminate. The finite element

results are shown in solid lines while the results of the present analysis

appear in dotted lines. A comparison of GT is shown in Figure 12.

Similarly, the results in Table VI and VII are plotted in Figures 13-16 and

29
17-19, respectively. The results of the second laminate are" provided in

Tables VIII-X. The results in Table VIII are shown in Figures 20-22.

Similarly the results of Tables IX and X are plotted in Figures 23-25 and

26-28. In Figures 22, 25 and 28 as well as in Tables VIII-X the values of

4

GT from the simple expression derived by O'Brien are included for

comparison. The plots in Figures 26-28 have been discontinued at G = -45°

as compressive peel stress occurs at the 45/90 interface. 'The sign of the

> o '

peel stress at a given interface can be determined from a simple

membrane-type model. This is shown in Appendix III. Under compressive

peel stressxthe crack surfaces tend to close and a special modeling

approach should be used. One possible approach was proposed in Reference 9

in connection with a double cracked-lap-shear specimen tested under

compression loading.

Conclusion

Interlaminar stresses and energy release rates are estimated for the

ED test using a shear-type deformation theory and a sublaminate approach.

The predictions are obtained in closed form and the parameters controlling

the behavior are identified. The governing equations are derived using a

virtual work approach. Due to the absence of transverse strain the

interlaminar peel stress distribution is not in equilibrium. The

interlaminar shear stresses, however, show reliable trends. The energy

release rate components Gn and GT T I are estimated based on the

interlaminar shear stresses and relative displacements using the virtual

crack closure method. The total energy release, GT, is determined from the

30
rate of change-of the work done by the external forces with crack length.

Then G, is obtained as the difference between GT and (G,, + GJTJ) .

An extensive comparison between the energy release rates predicted by

the present approach and a quasi 3-D finite element solution for over sixty

test cases, is performed. The agreement is good. The methodology outlined

in this work "is simple and the results are generated using a Hewlett

Packard 9845B desktop computer.

ACKNOWLEDGEMENTS

N.

The authors acknowledge the benefit of discussions and suggestions

with Dr. Kevin O'Brien. The extensive comparison presented in this work

for over sixty test cases was based on the finite element results of Dr.

I.S. Raju.

31
Appendix I

Derivation of the Governing Equations

In this Appendix the governing equations for the sublaminate shown in Figure

3 are derived using the principle of virtual work.

Consider a sublaminate... a single ply or group of plies conveniently

treated as laminated units of thickness h. The origin of a cartesian coordinate

system is located within the central plane (x-y) with the z-axis being normal to

this plane, the material of each ply is assumed to possess a plane of elastic

symmetry parallel

 sto xy as shown in Figure 3.

Stress and moment resultants,

(Nx' V V

• V V

h/2

V

 = I (

V V Txz' V)

-h/2

dz

h/2

<°

-h/2

v Vzdz

Because of the existence of a plane of elastic symmetry, the constitutive

relations are given by

°x

ay

°z

_

 T*y_

cn

C12 C22 SYM

rL r r 13

 L23 33

r r r r

_ 16 26

 L36 L66_

ex

ey

ez

Yxy

xz

SYM

XZ

32

d-2)
where C.. are components of the anisotropic stiffness matrix and Y , Y

and YV, are engineering shear strains xz

The displacements are assumed to be of the form

u = U(x,y) + ZB (y) A

v = V(y) + zP

w = W(y) (1-3)

where u,.v and w are the displacement components in the x, y and z

directions,

 s respectively. Equation (1-3) in conjunction with the

strain-displacement relations of classical theory of elasticity leads to

the following kinematic relations

exx

 = U'x

£yy = S + zpy,y

Yxy ~ 'y

 Z x,y

Yxz ~ ^x

YT = B + W, (1-4) yz

 py 'y

Substitute Equation (1-4) into Equation (1-2) and put the results into

Equation (1-1). This yields the following constitutive

relations:

33
N

x

"y

N

xy

M

x

\

M

xy

X

HA A A R

ll

 M12 rt!6 Bll

A

12

 A22 A26 B12

A

16

 A26 A66 B16

o R R R n

" 11

 b!2 B16 Ull

B

12

 B22 B26 D12

B

16

 B26 B66 D16

Q

y _

 A44 A45

_

Q

x J L A«5 'A55_

B

12

 B16

B

22

 B26

B

26

 B66

D

12

 D16

,

D22 D26

D

26

 D66

' Vv

;

 Jx

U

'x

%

s

0

p

y,y

px 1

where

I - \

h/2

C.. (1, z, z")dz

' »J

-h/2

(1-5)

The usual' coupling between bending and extensional modes for a laminate of

arbitrary construction occurs in Equation (1-5) through the stiffness terms

V

The variation of the strain energy due to virtual displacements 6u, 6v

and 6w is

6V =

(1-6)

34
where 6e , 6e , 6e , 6y , 6y and 6y are the strains associated with

the virtual displacements. Substitute from 'Equation (1-3) and integrate

through the thickness using Equation (1-1) to get

60 = [N 511, + N 6V, + N 6U, +0 66 +0 (68 + 6W, ) + M 66 I y v \f \t y\/\ / v v »/ v \/ v v y j A J" «y J" jr " *» Jr jr Jr AAj A

d-7)

x

The variation of the work done by the external forces and by the

surface fractions is

<SW = j (n 6U + n 6V + q6W + m 6B + m 63 ) dA

J A A y xxy y

6°n

 + ns

 5°s

where a bar denotes values on the boundary, n and s are coordinates normal

and tangential to the edge, and

nx

 =

ny

 = ^y ~

q = P2 - PI

where n and n can be regarded as effective distributed axial forces, m A y x

and m effective distributed moments and q an effective lateral pressure.

35
From the principle of virtual work the equations of equilibrium and

boundary conditions are determined from the Euler equations and boundary

conditions of the variational equation.

6V = 6W . (1-10)

Substitution of Equations (1-7) and (1-8) into Equation (1-10) leads the

following equations of equilibrium:

N + N + n = 0

x,x xy,y x

N + N + n = 0

xy,x V,y y

Qv + 0 + q = 0

x,x

 vy,y

 M

M

x,x+ Mxy,y - <>x

 + mx = °

Vx+My,y -

 Qy + "V = ° (1-11)

and one member of the following five products must be prescribed on the

sublaminate edges

NnV NnsUs' >Vn> Mns PS and <>n W ^^ .

For the ED specimen under uniform extension, U(x,y) in Equation (1-3)

is given by

U(x,y) = xe + U*(y) (1-13)

and the response is a function of y and z coordinates only. For this case

the equilibrium equations (1-11) take the form

36
V. * "x =

= 0

y.y

My,y -

qy

 + nly = 0

(1-14)

X

Substitution of the constitutive relations in Equation (1-5) into Equation

A66Lyy

A26Lyy

0

B66Lyy

B26Lyy

A22Lyy

0

B26Lyy

B22Lyy

"

A44Lyy

-A/icL 45 y

"

A44Ly

SYM

(D22L

yy-A44)

U*

V

w

p

X

Jy_

= -

n

X

n

y

-q

m

X

m

y

where the operators

= d2/dy2

Ly = d/dy

and for the boundary conditions at y = constant prescribe

Nxy or U*, Ny or V, Qy or W, My or py and Mxy or PX

(1-15)

d-16)

(1-17)

37
: Appendix II

Definition of Parameters

In this appendix the parameters in Equations (11) - (30) are defined

in terms of the sublaminates thickness and A,., B,. and D-. coefficients.

' J ' J ' J

Sublaminates 1 and 0:

The F.. parameters (i=l-3, j=l-4) in Equation (11) are defined as

F -n 1 +-iR 1 -l- h \i + h ii rll " U22

 x 2 B22

 M22 Vll R26 Ull

F21

 = h22

 V12 - h26 U12

1

 hl 1

F31

 = D26 + 2~~

 B26

 + h22

 V13

 + h26

 U13

F41

 = h22

 V14

 + h26

 U14 •

F =n° Qft' o + r w +r u i-22

 U22 2 B22 C22 V12 L26 U12

F32

 = h26

 V12

 + h66

 U12

F43

 = h26

 V14

 + h66

 Ul4

F33 =

 DM

 + ^e + h26

 Vi3 hee Ui3

F42

 = D26 " T B26 + C22

 V

14

 + C26

 U14

F44 = D66 " 2~ B66 + C26 V14 + C66 U14 ' (II-l)

38
Parameters u,. and v,. (i= 1-4) in Equation (II-l) relate the

displacements IL and V, to the rotations through the following equation

yy

yy

vll Lyy v!2Lyy v!3Lyy v!4Lyy

ullLyy u!2Lyy U13Lyy u!4Lyy lx

ox

where

(H-2)

12

13

14

= (K26 h26 - K66 h22)/D + hl/2

= (K26C26- K66 C22}/ D + ho/2

= (K h -

 K 66

C66 - K66 C26)/D

U13

?'

U14

D =

hl/2

ho/2

(H-3)

39
and

hn = B + A 22

 B22 2 A22

1

 hl 1

h26

 = B26 + 2~

 A26

1

 hl 1

h66

 = B66 + F A

66

= 8 '° + — A °22' 2 H22

J> A 2

 M26

^ A l

2

 M66

K - A + A '^OO *5*!? *3 0'

= AM + A 26

 M26 •

K66

The coefficients Eg through £Q in Equation (13) are defined as

E8 =

 F11W1 - F31X1 + F2!

 Yl -

 F4!Z1

E6 = F11W2 - A44X Wl - F

31

 X

2

 +

 \5l Xl

 + F21Y2 ' F

41 h

F = FA ° A ° - fA °}21 fF F - F 2} +1

L - fF A° + F A°

4 L«44 "55 ^45 ; J ^nr33

 r3^ T vr22 H55 44 H

44

-

 2F42 A45° ) Vll** - F

31A45^ ' *U

 W2 +

 *K

 X2 + F21 Y3 ' F

41

 Z3

F = -TA ° A ° - fA °^1 FF A ^ + F A * - ?F A ^1

2

 LM44 M55 IM45 ' J LhllM55 r33 44 ^r31M45 J

- FLrtA * A ^ - fA *^1 fF A ° + F A ° - ?F A °)

44

 M55 AH45 ; J l 22M55 h44A44 ^42 45 }

F = FA ° A ° - fA °121 FA

 l A X - fA

 1^2

t 1 O LA44 A55 ^A45 j J LA44 a55 tA45 J J

40
where

W

l

 = F33 ^F

22

 F44 F42 * F32 F44 + 2F43 F42 " F43 F22

WW = F (FA°+FA°-2 F A °} - A ^ (F F - F } 2 ^33

 ir22H

55

 r44H

44 ^42

 H45 ' H55 {fZ2 44 42 ;

+ F 2 A°-2FFA°+F 2 A °

32 55 ^r43r32H45 43 44:

2

X

l

 = F31 ^F

22F44

 F42 * F32 (F21 F44 " F41F

42) + F43^F21F42 " F41F22)

X = -F - If A °+F A °-? F A °^- A ^ IF F -F

 2)

2

 h31 J 22 55 r44rt44 ^h

42H45

 ; M45 Ir22h

44 42

 ;

+ F fFA°-FA°^-FfFA°- F A 0}

32

 l 21 55 r41H

45

 ; r43ir21 45 41 M44 ;

Y

l

 = F31 ^F

32F44 " F43F42^ " F33^F21F44 " F41F42^

 + F43 (F21F

43 " F41F32)

v = ~ A ^f F F- F F }- F f F A °- F A ° )

2

 H45 U32 44 r43r

42; 31

 U32H

55 43 45

 ;

+ A55 (F21F

44 " F41F42^

 + F33 (F21A

55 " F41A45 *

Y = A ^FA °- F A °^- A ( F A °- F A ° ^ Y3

 M45 lh32M55 r43H45 ; M55 ir21H55 r41H45 ;

Zl = F31^F32F42 " F43F22) " F33(F21F42~ F41F22) + F32(F21F43~ F41F32^

Z2 = F31(F43A44 " F32A45 ^ ~ A45 (F32F42 " F43F22) + A55 (F21F42 " F41F22)

-F IF A °-F' A °^

33^41rt44

 r21H45 '

7 =-A ^ (F A °-F A °}+ A ^ (F A °-F A °1

^3

 M45 l 43 44 32H45 ; 55 ir41H44 r21 45 ;

(H-5)

41
when the coefficients A.g, B.g, D.g (j=l,2) and A45 are neglected

we have

V13

 = V14 = Ull = U12

and

31 ?< " F32

 = F42= (H-6)

for this.case Equation (II-2) is

i,yy

ji,yy

vllLyy Vi2Lyy

0 0 u!3Lyy

 u!4Lyy

and Equation (11) takes the uncoupled form

(H-7)

<

FllLyy

F01L

21 yy

o

o

(F22Lyy "A44

o

o

<

F33Lyy

SYM

(F44Lyy

5iy

oy

5lx

ox

= 0

(H-8)

42
By neglecting bending effects in sub!ami nates 1 and 0, the rotations

p and p can be expressed in terms of p.. and p, and the characteristic

oy ox ±y ix

equation reduces to

= 0

where

Tllm ' 2 22 ll 12 22

'1 A55

(H-9)

Parameters F.. and represent a membrane behavior and therefore

do not depend on the bending coefficients D...

Sublaminate 2:

The elements of the matrix Cd in Equation (19) are defined as

-1

[Cd] 3x2

"22

J22

66

'26

SYM

'22

12 26

(11-10)

Displacements IL and V_ and rotation p? in sublaminate 2 are related

to the applied uniform strain e and the rotation p? through the matrix Cd

by .

43
"2,y

J2,y

32y,yJ

[Cd]

2x,y

(II-H)

Sublaminate 3:

Parameters ^ J?6

 and ^66 TM ^clliat^on (24) are defined by

J22 = D22° + B22° Wd12 + B26° wd22

J26 = D26° + B22° Wd13 + B26° wd23

J66 = D66° + B26° Wdi3 + B66° wd23

where

[wd]

2x3

-1

AH ° A ° 22

 M26

AH ° A ° 26 66

Al

Al

o R o R o

, B22 B26

0 p 0 p 0

»

 B26 B66

(11-12)

Displacements U- and V- in sublaminate 3 are related to the applied

uniform strain e and the rotations p- and P3 through the matrix wd by

[wd]

3y,y

53x,y

(H-13)

44
Interlaminar Stresses:

Parameters a., ty. and Y- in Equation (29) are defined as

J J J

(lhF s - A ^ 31Sj

 A45 >

22 ^i ~ 44 42 ^j ~ 45

aj

 V13

-1

<

F

3lY -

F21SJ

Uj

 = aj

 U13

(11-14)

The parameters associated with the resultant force and moment distributions in

sublaminates 1 and 0, Equation (30). are given by

9

=

A12 A16. B12 B16

MA A R R 22 H2g D22 t326

A A B B 2o DO 2o DO

B22 B26 D22 D26

B26 B66 D26 D66

1-

45
Nxojl

%j

Nxyoj

Myoj

xyoj

=

AH °12

A22

u

H26

RB ° 22-.

RB ° 26

•

»lS

*26°

u

H66

RB ° 26

X

RB ° 66

h

(0 0 _ 0 A °\

{012 2 rt!2 '

h

(o 0 0 . (K

(B22 2 A22

 }

d

 h

( a ° _ _!£ o o . A u\

(*26 2 26 '

h

If) 0 0R 0>

(D22 2 B26

 }

h

ID ° _ ° R °^ (D26 2 B22

 }

h

((bB ° - — A ° !6 2 M16

h

IB ° - -° A ° ^26 2 26

o

 ho o (R ° ° fl °

^

B66 2 M66

h

/n o _ o R o

(D26 2 B26

h

/n o OR o

(D66 2 B66

hi

11 - -i A ° 2 A12

h.

- -± A ° 2

 H22

hl o - — A 2

 H26

hi

> 1 R o 1 2 B26

hi _ _1 D o

2

 B26

hi

- ^ A ° 2

 H16

hi

- -± A ° 2

 H26

hl o - — A 2

 H66

h. _ 1 R o

2

 B26

hi1 R o

2

 B66

(11-15)

46
Appendix III

Classical Lamination Theory (CLT) Solution

In the interior of the laminate CLT prevails that is, a membrane

solution controls the behavior. The resultant forces N and N are easily

determined. In the following a ply-by-ply model is constructed.

The constitutive relationship for each ply for a membrance behavior is

given by.

Nx

NY

Nxy

=

HA A A ll H12 rt!6

A12 A22 A26

_

A16 A26 A66 _

e

ey

_ V •{III-D

Superscript k denotes the kth ply. According to the present formulation

(IH-2)

YX =U(y),y

Since e and y are functions of y only, therefore, from continuity of

displacements at the interfaces between the kth arid (k+l)th plies we have

u(y, )

v(y,

k+1 -hk+1

= uk+1 (y, -\-}

(IH-3)

Substitute for the displacements from Equation (1) into Equation (III-3)

and differentiate w.r.t. y to get

47
- e -

 ey

From symmetry, consider one

of forces get

quarter of the laminate. From equilibrium

1 .;*;•„'••

k=l k=l

Substitute from Equation (1II-1) into (III-5) to get

S22

 ey

 + S26

 Yx

 + S12 £ = °

S26

 ey

 + S66

 Y

x

 + S

16

where

" A k - t = 1,2 m = 2 and 6 (HM)>

s^m " /. im

k=l

Solving for . and ,„ fr. Equation (.11-6) to get

y

- ''66*12

 SS)£/ S

= (S26S12 * S

22S16)£/S

2 (IH-7)

5

 =

 S22 S66 " (S26}
Substituting from Equation (III-7) into Equation (III-l) to get

By comparing the expressions for C and C in Equation (33) with Equation

(III-7), we find that for sublaminates 1 and 0

and

Therefore the first term in Equation (30) for N , N and N in sublaminate

x y xy

1 and 0 is the CLT prediction.

From equilibrium considerations the sign of the resultant axial force

k N in each ply determines the sign of the interlaminar peel stress. This

is illustrated for a [0/45/90/-45] with a delamination at the 0/45 interface. With reference to Figure 26, this layup corresponds to a

[45/90/-45/0] laminate rotated at an angle 0 = -45 . T S

{ For the properties shown in Table I and using Equations (III-7) and

(III-8) the resultant forces N and N in each ply can be determined.

These are shown in Table XI.

Consider the free body diagram of the top ply shown in Figure 29. From

equilibrium of forces in the vertical direction the peel stress

distribution should reverse its sign such that its resultant is zero.

Furthermore, the peel stress at the crack tip should be compressive in

order to balance the moment of the compressive resultant force N .

49
References

1. Pagano, N.J. and Pipes, R.B., "Some Observations on the Interlaminar

Strength of Composite Laminates," International Journal of Mechanical

Sciences. Vol. 15, 1973, pp 679-688.

2. O'Brien, T.K., "Characterization of Del ami nation Onset and Growth in a

Composite Laminate," Damage in Composite Materials, ASTM STP 775, K.L.

Reifsnider, Editor, American Society fo Testing and Materials,

Philadelphia, 1982, pp 140-167.

3. O'Brien, T.K., "Mixed-Mode Strain-Energy-Release Rate Effects on Edge

Delamination of Composites," Effects of Defects in Composite Materials, ASTM

STP 836, American Society for Testing and Materials, Philadelphia, 1984,

125-142. "•-, •-•

X

4. O'Brien, T.K. and Raju, I.S., "Strain-Energy-Release Rate Analysis of

Delamination Around on Open Hole in Composite Laminates, "AIAA Paper 84-0961,

presented at the 25th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and

Materials Conference, May 14-16, 1984, Palm Springs, California.

5. Whitney, J.M. and Knight, Marvin, "A Modified Free-Edge Delamination

Specimen," Delamination and Debonding of Materials, ASTM STP 876, W.S.

Johnson, Ed., American Society for Testing and Materials, Philadelphia, 1985,

pp. 298-314.

6. Whitney, J.M., "Stress Analysis of a Mode I Edge Delamination Specimen

for Composite Materials," AIAA Paper • 85-0611, presented at the 26th

AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference,

April 15-17, 1985, Orlando, Florida.

7. Johnson, W.R. and Mangalgiri, P.:D., "Influence of the resin on

Interlaminar Mixed-Mode Fracture," NASA TM 87571, July, 1985, Langley

Research Center, Hampton, Virginia. .

8. Russell, A.J. .and Street, K.N., "Moisture and Temperature Effects on the

Mixed-Mode Delamination Fracture of Unidirectional Graphite/Epoxy,

"Delamination and Debonding of Materials, ASTM STP 876, W.S. Johnson, Ed.,

American Society for Testing . and Materials, Philadelphia, 1985, pp.

349-370.

9. Armanios, E.A. "New Methods of Sublaminate Analysis for Composite

Structures and Applications to Fracture Processes," Ph.D. Thesis, Georgia

Institute of Technology, March 1985.

50
10. Irwin, G.R., "Fracture," Handbuk der Physik, Vol. V, Springer-Verlag,

(1958), pp. 551.

11. Rybicki, E.F. and Kanninen, M.F., "A Finite Element Calculation of Stress

Intensity Factors by a Modified Crack Closure Integral," Eng. Fract. Mech.,

Vol"9, 1977, pp. 931-938.

51
Table I. Material Properties and Geometry of the ED Specimen

EU = 134 Gpa

E22 = 10.2 GPa

G12 = 5.52 GPa

v12 =0.3

G31 = G23 = 3.1 GPa

Ply thickness (h) = 0.14 x 10"3 m

x

Semi-Width (b) = 140h

Crack length (a) = 6h

Table II. Comparison of Characteristic Equation Coefficients

Laminate

[15^60/-75/-30]s

[-35/55^10/-80]s

[40/-50/85^5]s

-9

E8x 10

0.143

2.282

0.095

-9

E6x 10

-6.940 x 103

-6.049 x 104

0

-6.389 x 10"5

-9

E4 x 10

9.089 x 107

4.614 x 108

p

1.071 x 10°

-9

E2 x 10

-3.132 x 1011

-1.155 x 1012

1 1

-4.096 x 1011

-9

EQX 10

3.193 x 1014

5.676 x 1014

1 A

3.193 x 101

Table III. Comparison of the Characteristic Roots for a [35/80>^-55/-10]s Layup

Model

Coupled

Uncoupled

Membrane

\

5547

5538

\

4286

3737

%

2714

2427

2802

%

2438

2108

2162

52
z.1— 1

CO

hz.

UJ

z.

0.

CL :

s:

oCJ . —E ^

O

UJ S3-

CO i-H

UJ II

1

UJ _O

oe

»»

>- JC

o 10

C£.

UJ II

LLJ ITJ

u_

O CO

Z UJ

o 2:

LO •-<

1-1 CJ

0£ UJ

eC CL.

•<D_ CO

O 0

O UJ

>

i— i

O

I —

f*

n

UJ

1— Li_

i— i

»— i

0 1-

^—.

UJ

CO

UJ

oe

Q.

^"

UJ

u_

{

0

1 — 1

[ (

ID

UJ

CO

LU

o:

Q-

.

^"

LU

u_

1 —

^LU

CO

LU

o:

Q.

LU

CJ

LU

"^

cr

LU

CO

CD

s ~^*

1—4

v>*

0

^£f^

0

CD

CM

•

0

o•

o

t-H

CO

•

CO

I— 1

0

o

CJ1

in•

10

CO

(/>

I— 1

CD

en

**s^-

o^^

in

CO

t

*x^

in

ro

CO 1 — 1

CD

CD

0

•

O

o»

CM

"*

in•

CD

<=!-

CD

00

in

in•

en

in

</) i — i

o

en"*^(-

in

CO

i

^x^

CD

in

ro• *

O r0 .-I

00

o in

• •

O in

CO

o ^-

• •

CO 0

^ .

r- en . •

CO ^~

o en

CM IO

CM r-l

CO CO

• •

«D en

CM

(0

I i </) in i — i •— i

0 1

en \ ^<<- m

m !—•

co \«-

1 O

^s^ IO

in i

CO "*».

•^ o

o ro

i i i i

00

en

CO

vo•

0^0-

•

CM

en.

i-H

in

^

in.

CO

r— |

V}

1 1

0

•-ii

"•x^

in

m

i "^s^fo

00

in

ro

i — i

r—

^

<£>

f^

•

to»

.

I— 1

t-H

CO

(^-,__!

vo

o

CM

in.

00

•-H

I/}

1 1 o

00

"X^

0

r- 1

*x^(-

m

in•^^

in

CO

i

i — i

53
-

.(_)

re

c

o

'.Jj

re

.,_

£

re

CD «O

0 1

o

CD T-t

O)

-o n

UJ '.

U)

1

0) >

•— JC

0 0

T~ r^*

T-H

sre n

3

O

S- JT

•r- IO

<_J

II

+-> re

•i — ^— •»

s

CO

CD O

-»-> re

re <+-

E i-

••- o>

C [ 1

re c

— i t— i

</> in

i — i *^

in i

+ 1 £

S

**x. •+*

0

01

^^

o

1 1

>

O)

r—

O

re

i—

s:

U1—

0 I—

*-H ^Z

UJ

CO

LJJ

C£

Q.

1 1 i

CD

"^

i— t

i — i

C3 I—

•z.

UJ

CO

UJ

cc

Q2:

H- ui

es u_

i — i

eT* s

UJ

CO

UJ

cc.

Q.

s:

UI

1- u.

es

I-H

CD h—^y

O 1 1 1

^ CO

o:

Q_

UI

CJ

z.

UJ

ID

f~y

UI

CO

z

1— (

NX

o

^j^

1—CO

cc o

^C I-H

_J hcu co

•z. o

•=C QCO

cn

CM

CO

CO

•

CM

CO

CO

CO

^~

T— )

CO

•

t— 1

^J~

10

en

•

CM

CO

O

•

1—1

in

to

CO

m

in

«D

rm^

OT i — i

m

i

v^.^-

in

oo

0

in

i

-v.

O

^~

I — I

O

o

in

r-H

t— 1

in

^jO)

•

^j-

<*

*=»•

CM

p^

CO

o

•

p*»

IO

in

cj-

•

CO

CO

CO

•

CM

r-H

t— 1

•

^fCM

cn

in

o

CO

V)

i — i

o

i-H

|

x^$-

0

00

in

mi

•N^

in

CO

i — i

o

in

m

T-H

m

ID

C7>

CM

•

10

CD

OO

CO

00

in

oo

•

IO

r^

T-H

CM

•

in

<T>

T-H

•

c^-

00

•

CD

1 — 1

r01

oo

T-H

(/I

I—I

in

T-H

1

•*x^fm

i~-

0

10

i

"v^

o

CO

1 — 1

0

o

10

in

01

to

CO

f^.

•

10

,3-

f~-

CO

OO

CM

f^.

•

CM

OO

10

00

•

P"^.

i-H

00

IO

CD

.

CO

f^^

•=*•

o

T-H

(/

1 !

CD

CMi

•-<<•

CD

>~-

in

UD

1

•i —n

CM

i i

0

in

10

54
01

c

o

•T—

C

•1s—

, — ~

0)

Q

0) (S3

O) 1

TJ 0

UJ . r-t

1 'I I

0) N

i — UJ

o

S- 0

(1$ e^"

t—* «-H

3

U II

&~

••- JQ

-c .c

4J VO

S ii

CD IT)

-4J

n}

C 0)

•i — o

E: CO ^

<T) (4—

— J £-

to o>

1 -) 4_3

VT> C

^J" 1— 1

+ 1 in

NN| ^~

«» "**•••.

0 0 Ch <y>

^Sj.

O

i — i

•

*-M

:>

0)

r^

-Q

*TJ

s:UJ

u_

I- t—

UJ

vo oo

o uj

T-H OC

UJ

1 1

1—

CD

h—t

^ 1—

CD "Z.

UJ

UJ

ce.

CL.

•SL

u_

1 — CD

t— 1

t t 1

CD Z:

UJ

oo

UJ

ce

Q.

s:UJ

Uh-

^^ h~"

•— " Z.

CD UJ

00

UJ

f>^

Q.

UJ

Q_)

^^UJ

"^

cr

UJ

oo

o

i — i

*^s

0

•a:

1—

oo

o; o

^C *~~* —1 1— ~~^ ft_H

CD OO

Z. O

•=t Q.

rco.

VD

T— 1

•

VO

CO

•3-

0

o

vo

vo

CO

CO

o

CM

CM

1

.—1

O

•

I— 1

o

CO

t— I

CO

CO

•

CO

f— 1

V)

1 — I in

voi

*">^

in

CM

^•s^-

o

rCO

CM

i

i — i

o

0

1 —

1

vo

CO,

CO

1— 1

r—{

•

vo

00

CO

,-^

en

^f

CM

^

CO

CO

T— I

CO

1

f^

rH

,

. O

o

CO

in

en

m

•

CM

(/}

l — i

0

r**

1 v^

O

CM

"v^-

m

vo

in

OJ

i

i — i

o

in

vo

i

t-jvo.

in

CM

^~

•

in

vo

CM

iC —O

vo

r-l

1-1

CO

vo

CO

CO

CO

p^

•

CO

^CO

,_!

1— 1

, — 1

•— t

•

in

.-H

(/)

r— i in

p~.i

'*^Si in

t— 4

*

S

N^(~

CO

VO

CO .

co

1

1 1

0

0

VO

1

r^

CM

.

"*

CO

»— 1

•

"*•.

^H

l~-

^

vo

in

vo

vo

vo

vo

vo

i— i

T- 1

enr^..

^>

t— <

,_,.

vo

CO

CM

vo

in•

CO

I— 1

V)

i — i

0

CO

**»^

o

i-H

^•^tm

in^«^_

in

co

i

i i

o

in

ini

0

r-.

CM

O

vo•

CM

CO

o

CO

o

in

CO

CO

CO

vo

CO

vo

t— i

•

CM

^-

co

CO

CO

c^-o

•

^C-M

(/)

1 — I in

COi

*^^

m

^••x^-

o

in^x^

CO

^J-1

1 1

0

0

in

i

in

en.

i— i

CO

CO

I— 1

vo

*Jo

1

o

o

o

0

in

vo

CO

i— 1.

CM

1^.

0

1 —

in

CO

p^

CO

•

p^

CM

(/)

l — 1

CO

en

"^s^

0

%V<l

f~

in

^~

-x^

in^ji

i — t

o

in^j-i
4-

)

fCEO(BE'i(00) 0Q) — . 0) 10 T3 1 LU O

cH

10

) 1 o w

IE--.

S- _E

fO CO-

3 iH OS- 1

• I —0 .

0

_c *>

40 c~

•r- 10

S

II

O)

4-> fO

m -—

E

•r- CU

E 0 (0 ro —1 "*- S- O) r— i -M in E

<d- .— i

+ 1 0

CTl

0 0 cn01 — 1t 1— 1 *— 1 ^>Q) r—-QfQ| —

g-

LU

1 |U—

co

100 I

—LUCOLUCUQ. 1 1 i u_

1r^

^s^

«-> t— •-• Z. i— i LU O COLUo; Q.

til

O i ^ r—0« h

-

LUCOLUQ: Q. SI LULu

h-

co

->^

•-H I—

CO 2:

LUCO1 , 1 L&J oeQ. LUyLU^3f"YLUCOCO1— 1 NX0^£i—CO

•z.

C£. O

^C *~H

—1 1—

^ 1 — I co co

•z. o <: Q.

o

^t-

^H<oCM•

iH

•=J-

CT1 cnCMLO• to

CT> ^-coCDCO0• oCMCM.OoCOCD 1 — I 0LO|

•*x^ O

^J-

•*^10COinii — i oinCOi

,3-

f~~

co

tH^>• CO10ocncnCMin• incn^inCDCD• oo^~.oCOo^C

O

in

ini

^s^ inCO

•x^oCO

^v^f-

o

rH11 — 1 ooCO1

CO

CO

LOCOCO^-^CDCOcniOcnCOcnr-o

._] COcn• oIOCOo100int

o

o(0i

-•^0CO

^x^ inr-

"^^f-

in

tHii — i oini

CO010

tHLO• inCO**10cn10CO

tHcni —cn

iHinCO• i— i cn . in-•

tHcnCM<oC

O

intoi

"v^ inCM

••x^ Or-

**x^(-

oCMI1 l Oo1

COotocoin• inini— i ^cn^>CMcnCO10COco^-0• COcn• CMcor-r-1

0

o

r-»

x^CDCM

*x^ m10 • x^f

-

10

CM1

i — i Oin10i

01

•*LOO

rH• incnCM

tHcn

tHCM• to00incoLOinin• ^~

p^. co• COinCM01

-co

LO

r^*1 x^10

tH

*«^_ O10

^^jr-

oCOii i oo101

IOin

•*' oCO•

^J-

CO

CT1 r-co^-0COCOin01^^-co• tor-o* ^> CMIOOCO oCO

"^^0

tH

*v^ inin

"*^^f-

incoii — i oinini

10COCOC1 —M• COIOLO^COto

iH• oCOCMOl o

iH10CM• COCMIO• ^- CO10

tH

tHC

O

1 — 1 in00i^\ in

*^_ 0in

*«Njf- o^fil r O0ini

0

tHCM

t^- t— 1• CMCOCOt— l COcoCMCO

r-.

CO^f CO

tH

f^*.

CO• cn

^J-

^o.*t

cr^ CO

rH

iHC

O

1 1 0cn

""X^ CD

"^v.

LO^" >*s

<(~

LO^"1 1 1 OLO^*1
UJ

1 — 1

cd

f

3

O

5:

fl \— Ul

^ C5 | L

o

i-H

1—

UJ

UJ

C£.

4-> CL

m

E

o

4-> 1— UJ

fO O u_

•I— 1 — 1

E •-•

i— CD HCO 2:

Q

i , 1 LLJ

oo

CD UJ

-O XO CL

1 t-H

CD

II *5~

O UJ

:n w u.

i

s- cu

fO _C """x

r— 0 «-•

3 ^" 1 1 )

0 t-l C3 2!

C 1 i 1 >— LJ-J

•r- II tO

C_> UJ

.a a:

JZ CL

4->

'i io

CD 11 21

4_) 1 1 1

m m u _

E •> — •

•r- (— E 0> CD

m o "-v.

— 1 *T< . ** <o *— ^ |— ~

4- & -Z.

CO S- UJ

1 l CD - OO

O 4-*

 : 1 1 1

\ E - o:

in »— i CL

1 O

o in

01 ^~ 1 1 1

\ 1 0 in ~z

<3- ul

1 l ^3

cr

UJ

H-l 00

> z.

1 — 1

cu :*;

i— O

-Q <=C

m HI— 00

C£ O

^C T-H

_J h-

"~i r— 1

O OO

Z 0

=C CL

p^.

oo

o

in

oo

t-H

CO

00

,_,

,-H

10

co

01

CO

«5f

t— • 01

00

00

,_,

i

c~~>

0

0

oo

IO

CM

1in —

00

C/l

i — i

in

i

'••x^to

in

i

in

00

•*v^

o

T^-

1 1

0

in

oo

i

oo

in

CO

^*3-

CO

CO

•3-

CO

01

01

00

01

co

01

01

01

^CM

t— 1

1

o

o

o

oo

CM

CM

^o

o

(/]

I—I

o

t— 1

1

>^$-

m

m

o

00

m

co

i — i

o

o

CO

1

CM

i-H

in

0

o

in

T-H

o>

,3-

f00

01

01

CO

«*

01

01

01

o

I— 1

1

o

0

0

,_,

oo

t-H

^

in

o

(/!

i — i

in

i— i

i

*x^(-

o

10

in

r-

*^_

o

CO

1 1

0

in

i —i

oo

in

00

&

m

in

in

in

01

i— i

01

01

,3.

*»•

oo

01

^.

•*•

o

1

I— 1

"*'

o

m

00

i-H

c^-

T-H

T-H

(/]

1 1

0

00

1

**"sc(-

m

10

o

r-

~x^

in

Osl

i — i

0

o

1

01

in

oo

i —

in

10

m

m

in

00

oo

01

t-H

oo

io

01

00

01

0

P-^

*

T-H

fr0

T-H

f»

T-H

CO

1 — i

m

CM

1

"x^-

o

p«^

1

in

10

*\

o

CM

l i

0

in

<o

i

t-H

CO

in

|^_

oo

in

o

t-H

in

CO

o

to

O)

00

CO

,3.

01

10

•*

CO

01

•*

CO

i-H

in

o

01

i-H

00

l/l

1 — 1

O

CO

1

**x^(-

in

r^«.

i

o

«0

"**x^

in

T-H

i — i

o

0

i

oo

,3.

IO

^J.

o

CO

,3-

o

oo

00

01

5-Jm

o

01

^00

rm

00

10

ID

in

o

i-H

^o

oo

10

i — i

in

CO

i

"Xsjfo

oo

1

in

in

•••^

o

t-H

O

in

in

i

T^-

"*

CO

CO

CO

o

co

CO

o

00

10

00

00

^.j.

00

in

^

00

T-H

oo

00

i— i

•-H

in

CO

i— i

m

10

CO

CO

i — i

o

i^-

i

^^tm

oo

i

o

in

^x.

in

i i

o

CD

in

i

o

>=*•

00

0

00

T-H

co

00

o

00

10

r--

^_,

r^

CO

f00

t-H

Ol

i-H

T-H

T-H

00

T-H

OO

10

53-

oo

t-H

00

(/]

( — 1

LT>

^J-

|

"*"X5(-

o

01

in

T^f-

"\_

O

1 — 1

o

in

^ji

oo

10

t-H

OO

VO

t— 1

(0

in

i— i

r-^

CD

oo

in

ri —

CO

in

oo

CO

.3.

CM

r^

oo

t— i

00

in

in

ri— i

IO

CO

^-,-

oo

CO

1 — 1

o

in

i

^xjtm

oo

CD

^~>

vx

m

i

i — i

0

o

T^~

I

in oo

CM 00

T-H T-H

in r^

oo oo

T-H T-H

o oo

oo oo

t-H T-H

00 CO

01 10

o 01

co

«3- CO

T-H ^J"

m *a00

oo o in 01

T-H 0

CM T-H

00 CD

01 0

^- o

T-H

T^- |-^

<* m

r^ 01

^ r^

«a- r— 01 in

01 in

in 01

CO CO

i — i r— i

in o

in vo

i i "^vjf- *^<$-

o in

oo rin o

CO CO

*X^ ^s^

CD in

t— 1 t-H

1 1

1 — 1 1 — 1

0 0

m o

CO CO

i i

57
n)

o

-4J

to

c

•t—

Q ^s

d) VD

O) 1

-o o

UJ i-H

0)

r— U)

O

3:

f- O

ro •*

i — i-H

O II

i.

•i- JQ

O

±> 10

<l> to

±> • —

(0

C 0)

•r- O

E to

to M-

_l S1 — I C

0 •-•

in in

i i

\ "^

01 Ol

vn

i — i

X

<u

(0

1—

UJ

h-H

O

<J ^.

VO UJ

I— 1

tUJ

UJ

a;

Q.

21

u_

1— h1-1 CO

i— i UJ

i-H DC

czj o.

s:

u.

1—

tl UJ

i— i UJ

H-t CJC

CU Q.

U.

1—

1— UJ

(•3 CO

I-H ce.

CD 0.

UJSTACKIN

G SEQUEf

ANGULA

R

POSITIO

N J

CO

o

CM

VD

01

i-H

00

00

I— 1

CM

in

CM

00

CO

VD

CO

CM

CO

CO

in

CM

i— 1

CO

VO

r~-

i — i

in

o

in

00

i

o

in

0

in

00

cor--vDcooiocovor — cocOi-HC D

'si- co vo co. <a-*d-oovoco-*oo.- i

CMcota-LnvDvoin^cocMCMCMC M

CMCMinvoinr-ooooocMoococ o

^ CO ^3" «£f O O ^" •& i-H CO CO ^ «^fCMCO«3-LnVOVOin*3-COCMCMCMC M

CMOCDOOVOOOi-HVOOOCOOIOC D

COCMCOCMCOCOCOCMOCMOOCn O

CMco^-ininininta-cocM^H^HC M

in in I-H o inor-.CMO>ininoio o

«d-r^r~-cococMOOoococooiov o

o»cooiCMcoinvooop~cOi-HCMi n

vooooooioioioioioiooinc M

**-CMvoooovb'd-vQr~in*d-aiC M

oooimcMOOvocMcocoi-i i — voo i

^oi'd-ininr^-i — iin«3-vOi-HOC M

VD(-~-OOOOOOOOO1O1Oir~-COi- H

r-»K*-VOf--COCMCD^-CMCOCMO1C M

cooor-vovoinr~-cnvDi-Hovor ~

eOI--CMi-Hi-li-HOOOi-HCOCOi- H

CM i i I-H co m VD

O<=J-i-HCOVOOOOOCOCOOO O

Or-HO1Ol^-OOOO«3-«3-COi n

^-p-.i-iooooooi-ii- i oo' r~

CM CM in. VO VD

O O CM - «d-CMCMOOCOCOCOi-iCOCMr- l

i-H">j-Lnoinco«tf-i-Hr~cocDCMv o

r--oor--vo*3-cocMCMCMincD<7>c M

•-H i-H CO

VO«l-CM^l-sa-'*lO<d-COCMOOCMi- H

i-HOimr>-r-~cor--vovD'*p--cni n

i-HCMCOCOCOCMOO<^-inCMVOOO 1

i-H i— 1 i-H i— 1 i— 1 i-H i— 1 CM CM

COtO(/)(/)WI/ ) CO- W

</> i — i o in CD in o in i — i vt \ — i in o

i — iini-Hi-HCMCMcocoo i — lOinv o

0 1 i i i i i i<3-inin i i

inoinoinoino ^ i "^^oi n

«i-ininvovDr — r~oom v^moor^ -

1 1 1 1 1 1 1 1 00 O OO ^<- ^~#-

oincDinomoi n "^je- "-><«- CD co co

oioocor--i--vovoinoin«d-\ \

inoinoinoino vv . >v iini-(i- i

^^-cococvicMi — ii-tino i i i

oooooooooooo o

omoLnoinoino.inoini n

oioooor^-r~-vDvomin^-'^-co' *

i i i i i i i i i i i i
z.

UJ

1 — 1

oc

_

o

^- •

CO' UJ

,-H

1—

Z.

LU

-P CO

(rt 1 1 i

C Q.

O

(0

E S

• r- 1 1 i

E 1— LL.

ro cs

'co -~- I-H

Q ID i— i

1 •— 1—

CO O CD Z. .

Oft i-H 1 1 1

T3 C/)

LU II LU

1 '"•. U) . Q.

O J=

1C 0

«* s:

i- t-H 1 1 i

to u.

— II I—

3 C3

0 J3 -V.

i. •-<

U -G CD Z

IO •> iii

f~ J^

-M II LU

•r- CC.

2 « QL

4-> 0)

to u

c m s :

•f* *1 Ll J

fO CD

— 1 •!-> 1—

I/) 1— 1 1— «

i — i C5 1—

CD O Z.

\ cn LU

m \ co

^J* tO 1 1 1

i <*• of

~\ o_

o

m

«* UJ .

I — I O"^

LU

ID

X CDUJ

O) CO

r^

-Q O

tS -"*'*'

L . .

J-- t-^

\l£

0

^t

J—

^ Z.

CC O

•a: •-•

-I H-

~"S f—i

tn co

z. o

«=c a.

CM

CD

t-H

CM

i-H

t-H

O

«-t

t-H

rr-~

o

00

CO

o

CO

00

CD

0

1^

cn

CO

o

t-H

CO

CM

CM

oo

CM

co

(/)

1 — 1

Lf)

•^v^

0

1

*^^

LO

00

•*x^fo

LO

1 — 1

o

LO

00

o

<*

CM

o

CM

CM

^

t-H

CM

o

I— 1

CM

00

co

i-H

to

CO

CO

0

<3-

co

01

o

t-H

I— 1

cn

CO

O)

oo

CM

vt

1 — 1

CD

>«^

tn

^*

1

1

o

^4

-NScn

*^~

LO

1 1

O

CD

cn

^>t

^-

CO

01

co

CO

CM

CO

r— 1

r-~

^.jco

in

CD

CO

CO

01

CD

T— 1

,-J.

I— I

cn

CD

CM

=*

o

CO

CM

(fl

1 — 1

LO

^v^

o

in

1

v^

LO

CO

0

1 1

o

LO

00

i

oo

^-

IO

*

o

CO

,-J.

CO

vo

r^

CO

T-H

CM

CD

01

CO

t-H

t-H.

1

O1

co

in

in

CO

O)

CO

CM

(/)

i — i

O

,— 1

1

•^^

in

in

i

x^

o

CO

in

CO

i i

0

o

oo

1

t-H

CO

in

rCM

LO

cn

o

in

CO

**•

0

cn

^>

t-H

CM

cn

^J"

in

OJ

i

LO

CO

c3-

CM

,-H

CM

'

i-H

O

CO

(/)

1 — 1

LO

I — 1

1

*s^^

0

co

1

*v*.

m

r-

*v^$-

o

CO

i — i

o

LO

r— - i

01

i —

LO

i— 1

1 —

LO

CO

in

LO

in

CD

CO

cn

CO

r~

CO

cn

i— i

"-i

CO

i

P^

«-«

CO

co

o

CD

o1—

CO

(/)

r— i

0

CM

1

>^

in

r*^.

i

^^

CD

f*^.

^xj(-

LO

CM

1 — 1

O

CD

p^.

1

CO

rin

rin

t-H

in

in

CO

in

cn

00

o

in

cn

cn

0

CO

i

cn

i-H

m

r-~

CM

0

CO

in

i — i

in

CM

**^

0

i —

x^

LO

••x^fo

CM

I — i

O

in

CO

i

CM

m

01

01

*

oo

CO

<3-

OO

rcn

co

,-H

co

cn

o

f^-

CM

i

0

o

r-l

CM

CM

in

•3-

00

CM

V)

1— 1

o

CO

••••^

LO

i —1

"x^

o

CD

**s^-

m

i — i

o

0

CO

1

CO

in

CO

r-

•5t

CO

i— i

•aCO

LO

CD

cn

cn

CM

o

(V.

cn

P^

«-«

CM

1

i-H

O

CM

i— 1

CO

co

LO

CM

</>

i — i

LO

CO

1

**^

o

oo

i

"N^.

in

LO

v<$-

o

t— i

i — i

o

in

m

i

r-. rco oo

0 0

LO CM

CM *3-

t-H i-H

CD CD

CM CM

t-H t-H

CO 0

O CD

0 0

O O

,-H r-H

O CO

CO t-H

r-. r01 cn

CD r**

CD r-

•-H O i i

co r~-

0 0

0 0

oo m

cn t-n

t-H CD

CM r~-

co rCM CM

w </)

r— T i — i

o o •3- in

i i

LO LO

CO CO

-^ o

O •=3-

cn ~^<«- v

vj«- in

in i i — i i — i

o o

0 0 in •*

i t

CO CO

in rco in

in 01

oo <*•

co m

r-l -» CO

•a- co

co io o t-i in

0 00

o cn

T—t

co

O1 t-H

in «tf01 cn

•-H in

^3 i~^

O ^H

1

cn m

co cn

0 0

o r-.

CO CO

0 0 i

co m

co cn

CO slV) (/)

i — i i — i

in o

cn co

i i

CD in

oo rtn o

CO CO

o m

i— 1 i-H

1 1 1 — 1

0 0

cn o

co ro

t i

59
Table XI. Resultant Forces for a [0>^45/90/-45] Laminate

Ply Angle Ny (N/m) " Nxy (N/m)

0 -0.008 0

45 2.671 3.031

90 -5.335 0

-45 2.671 -3.031

60
OUTER

SUBLAMINAtE

INNER

SUBLAMINATES

FIGURE 1. THE ED SPECIMEN

61
N .

N\\\\\\\\\\\

CO2LUh-co>- CO

III I-QCLOOOQZQ. CCOCOLUQLU<_l CDCO

CNJ

LUDCD

62
N

UJH-CODCOoDC

LLJ ZHIoccoLLZoKZHI>OOZgCOQZ<COZoi-COLUQC=>OLL

63
00

D

X

ro

CM

o

CXJ

SlN3H3GVndSIQ

o

LU LU

IE fI- <

GO _J

hZ co

LU i—.

21 O

LU 0>

U \

< in

_i ro

Q. I

CO \-

i—i in Q ro

\

LU O

_i cr

LU O

OH LL

LL LU

O CJ

Z LL

O (X.

GO ID

M CO

Q_ CJ

o o:

CJ CJ

LU

cv

D

CD

f-H

LL

64
u

O

/

CM O 7

lN3W33V"ldSia

in

DX

o

u < I 21- <

C/) w

I— r— . 2 O U 00 ^ I UJ \ O O _o_ in ui in M \ Q inK> LJ i< _u o CK LL L_ U O O< 2 LL O Q: co :D M C/)

ct: < ^

Q.-Q

X < o o: o o inuo: Z) o

(—I U_

65
14.0

£ 12.0

CO

3

 10

-o

cc

co

cc

<

UJ

I

CO

8.0

6.0

DC

£ 4.0

2.0

0.0

(b-a)

1 F35/-35/0/901

L

 I s

2 f35/0/-35/9ol

I

s

3 [o/35/-35/9oJ1

0.90 0.92 0.94 0.96 0.98

FIGURE 6. COMPARISON OF INTERLAMINAR SHEAR STRESS

66
o

\

Zujinuio:CL

\

LUOzUJQ:uU_LUQ:

o

w

i—i

IDL

OO

000rv£) ID \D -ro

LUCOLU11 LUOHI CDOHLU~7LUI Zh- 1 OH\-COQLUMOI— i ii —M<LUQU_0ZOhHI—CDZIDLu<c

CO

CO

LUZCDii.i — i Q_DOQLUQhHI— i

r^\

- CM o

O

CO

OH' LU O h-OH UJOHIDCD

\—\ U_

LUM

I—I CO

67
o

x

X X

00

in

_Q

\

X

o

HH

tZ)

DO

I—I

o:

GOi—i

Q

LU

CJ

o:

o

U_

LU

I—

M

_l

LO

I

LO

O

OJ

o

ro

Z)

GO

UJ

cc

LO cu

(W/N) S33dOJ

o

— o

00

LU

Cd

68
o

oo

in

CM

o

in

i

in o

(W/N) SBGdOJ lNVl~inS3d

_Q

\

X

O

CD

LU

t-

<

_l

00 00

LU

O

o:

o

u_

I—

CO

Ld

o

\

o

\-

m

in

in

ro

LJ

o:

ID

CD

I-H

LL

69
oo

I11 • 111I 2

:

LULL

1—ZLUGOLUQ: Q_

4

I—I CD \ A\

\

*Lo00 o o 19

CD o

CXJ

U

inoininoin

o

LU_JOIQZ=) Oo: ohHCOoQ_

ex. _Z) CDZ

^ QJ

s-1- L<L rv

 a

^ LU

_Z

) occMCJ n: LJI—inxoo\ooUJ ccLL

in

ino

t—i i—_LJQLUCDQLU

70
h

-

10

oo

oCO 31

oO

OOJ

oLO\DO10

O

LU_lOIOOorQ

I—I I—

I—I COOQ.o:<_iID

_oorZ)CJo:Muz:LUh-_Jeo ID\OO\O

LUa<u_ccLUI—IDIDOLUaLUCDQLU

LUCD • M

LJL

71
10

O 00 v£> ^f

' ^ goi

— vi

CXJ C

O

O

in

LU

_J

O

Q

Z

13

O

u

_J

o

o:

_

Z)

o

o:

M

o

n:

u

OJ

LJ

CC

Z)

CD

hH

Lu

LU

CJ

<

LL

o:

LU

in

^ti

\

in

"

in

in

o

h- 1

ht— 1

00

0

Q_

ct

<^^*r.

_J

Z)

0

-z.<r

^

 h yH

< t

-« z

ro ^ -in < i

5 LU

\ °

§ LU

> QLU

72
LU

OO

LJ Q

.Lj_ Q * <«LUCOLU

LO^tI O00 O O

If).1 LOLO

OOCO

Or^I O

LUOQZIDOo:zo

0 LU

1 O<

•Q: LL < cc _l LU Z) h- o -z. o in

"tf

n: + H \ M O LU K h- <. M O z: M < h- I <-> CO t ^ o 1 ' a01<£1 z> LO CD in Si) Z 1 < CO 1 — 1 LO

-xl-

+1\O

CT

1 \Oi — i I— I X<_JLUQLUCDQLU LUo:

.Z)

CDi—i LL

i, 7

3
Oo

o00

O

oCM

O

IT)

^d-I Oinin10oor-i LU_ion:a

-z. ~DOo:Z)CD

LU_JOIEZ)CJO

•j:\—i—i 3?LU

(—(

0h- 1 I—1— 1

(Jl 0CL

X

/5CO I — 1 - in-H\OO\O

1,

LUDCZ)U_

LU

I_L LUH2MinoOLUQLUCDQLU

74
oo

LUCD

21 UJ

LU o: u_ a. o00

o

1

OO

C\J

Lc

o

in^IOIDI inini - o^oi inor>i L

U_JoIEQoo: o

(7) OCLo: CDz:

LU_JOIor<3or3LUH<w

i—i in\ooinLUo: IDO

i—i

LL_

UO<

LJ_

ccuo

I—I _

LiJQUCDQLU

0 !N30d3d

75
O

00

OJ

o

IDOini LOLOI o^DI LOOr-i LU_JoIQOQ:oMh-M

(/)OCL_Z)

LU_JO31o:IDOo:HHOUJCO

I—1 LO^t+1\OO\O

1. UJonZ)CDhHU_

LUCJ<LLo:UJLO\Oo_UJQLJCDQLU

76
I

D I—

i

oo

o00

o^0o

-sh

0

C\J

 C e

OinIDino10Oinr^i o00I

If) 00I. U

J

_JoIEQZIDOo: oMh-oQ. Ct IDCD

UJ_) o1

 UJ

o: u

LJ_

OaI— i o

ot<

_J <

in < ^t- —I +1 LU \ Q0> UJ o U

0 !N33d3d

UJ a. IDCD

I—I U_

77
in LU

1 "-

1

1 —

z 1

LU . I

CO 1 f

X LU '

LU QT |

^ °- 1

o <

4 f

•!

x

l T

Ml

& \ \

M1

1

i

i *T

i

i

i

+ <i

i ^j

i

i

0

- ID

I

ID

- ID

I

O

- <vO

1

ID

> ^)

i

1

O

> ri

ID

-

I

i

i ° \\ °P i

i

i i t i i i ID

4 oo

o o o o o 0 '

O 00 ^0 Kl- CM ^

J-O !N30d3d

LU

_l

O

I

Q

Z)

0

o:

<

z

o

1— 1

1—

CO

o

Q_

o:

<

_i

^)

CJ

o

ICULA

R

o:

i— i

CJ

IE

1—

M

3:

LU

[—

<

Z

f— i

;z

<

_j

0)

1 — 1

ID__ i,

^vT

I.I

T!

\

o

rr\

\j • \

\

0

i — i

•

00

LU

. on

ID

CJ

LL

LU"ERFA

C

1 —

o

&

\

o

11

<

z

o

1 — 1

1—<

~7

'

\ — 1 1 1

•^~

->_

_l

LU

Q

LU

CJ

Q

LU

78
i

n

o

00CM

1

9 g

oi

O

oGOiin00i L

U_JOor

U

oini inini o

v£>

I in

<£>i or-i inri \i LU_]OiQZIDOCt<^*"f» Zo1 — 1 1 I 11 —M

c/) oQ_DC •-. <_l^}n

<_i—_, -)* CJC£I— I CJ31f-h- 1 ^LU[—<Zf— ) X<_Jwi — i in^f+i\o

<

LL. DCLUih-ZM0

CT^ \O1—<Zoh- 1 h-<Zf— H

^!"' X<_JLUQ

O

UJ

UJ

LUo: IDLL

79
4

(

1—

~z_

LU

00

X LU • LU o:

LL CL

• 4

1 t 1 1

2 o o o

0

 00 v£ ^t

o

• *- ro

/

 7 1 ' / / // - 1

 /

1 /

rfu ^

\\

\\

•

VV

\

(fa

^

\

\

\ -

\

\

t

\

\

in

i

o

i

in

•^t

!

O

- in

i

in

L ID

r

o

- ^o

1

in

4- <O

] ' 1

I °

t ^

} in

M |1 1 ^

&

 Jo

t °°

I

|,io

o o ' C\J ^

LU

_i

o

I

Q

Z

13

0

C£

<

£

—,

ov

 -*

I — I

I—

h- 1

CO

o

Q_

o:

<c

1

ID

CD

<

~-

o:

0

LL

LU

1—

<

o:

LU

00

<

LU

_J

LU

>-

CD

CY

LJ

Z

LU

LL

O

Z

0

00

1-H

o:

CL

^~

o

CD

O

CM

LU

U

CDi i

LU

O

LL

o:

LU

(-

Z

h- 1

o

\

LO

•^jI

1—

<

Q

LU

\—

<(

Z

i — i

^

<

1

LU

Q

(0

i — i

O

\

ID

<v^-

1

O

0^

IDt

i i

LL

80
_oIEQZZ>oo:oMI—M

(7)OQ.OH_Z>CD

o:o u

LL. 0< U LL h- o: <o:uCO< oin uo:CD < DCU Q Z LU U h-<

LL. ~Z. O M21

-z. < O _J CO LU M Q CY< M CL r-i 2: o O \ U LO

^f-I • \ — O OJ 0^\LJ ID C£ ^J- Z)OMU_

81
Z

Z LJ

UJ HH 00 OH LU CO

LULL Q_ O

Oroi inroi oLOOIDIDI O.I IDI OI IDoooID00

LU_JOQZIDO

Cr: OMI—I— I 00oCLo:_IDCDZ

o:O L

U

LL O<

LU LL

h- Q:

< LU

LU \—\ 00< O LU \ _l ID LU ^t o: i >- t— CD < crLU Q z LU LU h-<

I_L Z O M2 z < O _J

(7) LU

M Q <CL2:OCJ MoI

CXI O

CM O\U ID CDMLL

82
1 oo

(

i i -aXi

0 00

00 vO ^J-

/i 0

 c

CM

 e

in00ooin00

LU_)OiQ

~Z. Z)OQ:oMCOoCLo:_Z>CD

LU_JoIT_Z)oo:f— i CJn:

LL)

I—O\ID\OCh\inCMLdccZ)CDMLL

UO<LLo:LUI—10

<sf

I \oo_LUQLUCDQLU

83
00

oCO

O^0

—v~*t o

•^t

^ w

0

 c

OJ

 e

oroi LOro• i oI LOI OLOI LOLOI OI LOO.LOO00I LO00OLOGO

LUOIEQZO^o:ooooo_o:_JIDCD

LU_lomo:_Z)occt— I oLUo\LOo0^\LO

CXJ UJo:Z)CDMLL

LUO<LLo:LULOXo.o_UQLUCDQLU

84
Z 11

1

LUco Q: U QQ

LJ Q: -

LL Q. O

O

oK). 1 IDK)I O

<sfl ID^fI OIDI IDIDI O^0I ID^0I Or-i IDr-i o00I ID00I OID00

U_loIQZIDOo:ocooCLo:

U_lomoQ:i— i on:uO\ID\OLD

00OJ

O

IDOJLUC£IDCD

LUCJ<LLCZLUID^fl \Oo_LUQLUCDQLU

9

01 X

85
LULL

h-ZLUCOLUo:CL

,oo,o,o

v£>

iO^1 JH O

91

O

LO00

LU_ioIEazIDOOCOoQ_Q:_IDCD

Q:o u LL CJ< LU LL I- Q: < LU a: h- LU

C/) <

o

LU \ _l LO LU ^t- cr i >- h- CD < o: • LU Q Z LU LU I-< LL Z O MH Z < O _l CO LU M Q o:< w X O o \ O LO^I • \ vO O CM 0>\ LU LO OH ^t CD

l—I

LL

86
LJ

LUCOLUo:Q_

o o

!N33d3d

oinOo

o

LJ_lO31QZZ>DQ:OCOoCLo:CDZ<

OHo L

J

LL CJ LJ LL \— C£ < LJ OH h- LJ M CO< O LJ O _J \ LJ in ct: ^j- >- h- CD < LJ Q Z LJ LJ I— LL Z O M Z < O _J CO LJ M Q

ct: CL r-

!

:z o o \ o ini r^ o CM &\ LJ in OH *& IDCDMLL

87
oroi IDroi oID^tI OIDI IDIDI OvOI IDSOI OIDo00I ID00OChIDCO

LU_lOIEQ2Z)OonoCOoQ_On_Z)CD

Q:O L

U

•|_L CJ< U LL h- o: < LU o: H

~z.

LjJ HH

CO< o U Ch _l \ UJ ID o: -sh >- t- CD < o:U Q Z- LU UJ h-< LL Z O M. O _l CO LJ M Q cc<Q_X O O \w CJ ID

^J-I • \

00 O

01 0^\

LU ID

on ^J-

Z)

CDMU_

88
ox LLJ H- II

Tcc JD

Oqd

tJZ

0)

<If)^-I *^OO) In0. Oh- UJ T. h- DCOQOCDHI HI DC

Li. c»oqo O) CMg

LJL

89