Shear-lag analysis of notched laminates with interlaminar debonding

Engineering Frorrure Rmtcd in the U.S.A.

Mechunia

Vol.

22. No.

6. pp.

1013-1029.

1985 e

0013-7944&Q $3.00 + .Oo 1983 Pergamon Press Ltd.

SHEAR-LAG ANALYSIS OF NOTCHED LAMINATES WITH INTERLAMINAR DEBONDINGt Department

A. K. KAW and J. G. GOREE of Mechanical Engineering, Clemson University,

Clemson, SC 2%31, U.S.A.

Abstract-This study considers a method of analysis for predicting the fracture behavior of a notched, unidirectional lamina in the presence of surface constraint layers with debonding between the unidirectional ply and the constraint layers. Two particular cases are presented, the first being a debonded zone of finite width with no longitudinal damage in the unidirectional ply. This solution is then extended to include longitudinal matrix yielding and splitting in the unidirectional ply at the crack tip. The analysis is based on a materials modeling approach using the classical shear-lag assumption to describe the shear transfer between fibers. The fracture behavior of the laminate is studied as a function of initial crack length, the relative physical and geometric properties of the constraint plies and the unidirectional lamina, and width of the debonded zone. The results indicate that debonding can reduce the maximum fiber stress at the crack tip on the order of ten percent. This effect is maximum for a debond width of two or three fiber spacings and is independent of the initial crack length. As the debond width grows beyond this point, the maximum stress increases. For widths of about ten fiber spacings or more, the maximum fiber stress is larger than for the fully bonded case. In the presence of longitudinal matrix damage the same general behavior is found; however, the location of the maximum fiber stress is quite complex. In some cases with large matrix damage and a high constraint ratio, the maximum fiber stress can occur at the end of the debonded zone away from the crack tip.

INTRODUCTION

follows a sequence of papers by Goree et al.[l-81, all of which have been concerned with developing fairly simple mathematical models to represent fracture and crack growth in notched composite laminates. Various forms of crack-tip damage were considered in these studies, with the primary aim being to predict, from a mechanistic point of view, the fracture behavior of notched laminates. Several of these investigations involved experimental studies[7] and [8], both to verify some of the models and to observe the form of the damage growth in particular laminates. The results of [7] indicated that for a unidirectional ductile-matrix composite (boron/aluminum), a shear-lag model which accounted for large scale longitudinal matrix yielding and a small region of stable transverse fiber breaks at the crack tip could accurately represent the fracture process. For a unidirectional brittle-matrix composite (graphite/epoxy), it was found in [8] that the shear-lag model represented the dominant part of the longitudinal splitting at the crack tip but that it was not sufficiently general to account for split initiation and early split growth. The initiation of longitudinal matrix splitting and the early stable portion of the growth (on the order of 10% of the initial transverse crack length) was shown in [8] to be due to transverse tensile stresses in the matrix at the crack tip. It was noted in [8] that these transverse stresses are not given accurately by the shear-lag model and that a more complete solution would be required to describe the early phase of the damage growth. For a unidirectional composite the shear-lag model can be shown to give a good representation of the axial fiber stress as well as the matrix shear stress. For situations in which these stresses control the fracture behavior one can anticipate good results. A more detailed discussion of the accuracy of the shear-lag approximation is now being considered by the second author of this paper. In the present problem, the quantity of most interest is the axial stress in the’fibers near a notch as a function of laminate properties and disbond size. These above studies indicate that a shear-lag based analysis does give a valid representation for this fiber stress. Some of the first work in modeling a unidirectional composite using the shear-lag approximation was done by Hedgepeth[9], where no damage other than an initial transverse notch was considered. The study was extended by Hedgepeth and Van Dyke[lO] for the special case THIS INVESTIGATION

t Supported by the Materials Division, NASA-Langley, 1013

Grant NSG-1297.

1014

A. K. KAW and J. G. GOREE

of one broken fiber with longitudinal splitting. Goree and Gross[l] used Fourier transforms to modify the solution to account for an arbitrary number of broken fibers as well as for longitudinal matrix damage to include both yielding and splitting initiating at the notch tip between the last broken fiber and the first unbroken fiber. Dharani, Jones and Goree[4] added constraint layers to the main lamina to represent either a misalignment of fibers in a multi-ply unidirectional laminate, or for the presence of angle plies which give support to the unidirectional ply. The constraint layer was taken as being fully bonded to the unidirectional lamina at all times in [4]. The analysis presented here extends the constraint layer problem of [4] to include the effect of debonding between the notched unidirectional lamina and the surface constraint plies. The laminate is modeled as a two dimensional region of a unidirectional lamina with symmetrically located surface constraint layers whose fibers make an angle theta with the unidirectional ply (Fig. 1). In the vicinity of a notch in the laminate the broken fibers exert longitudinal shear stresses in the matrix which are transferred to the nearest unbroken fibers. The shear-lag assumption[4, 91 is used for this shear transfer between fibers in the unidirectional lamina. Debonding between plies in the vicinity of the crack is considered to be of finite width and extend to infinity in the longitudinal direction. The extreme fibers of the debond zone are assumed to be attached to each other across the debonded region by a spring of stiffness depending on the constraint layer properties and on the width of the debonded zone. That is, the layer debonds but is still connected to either side of the region and carries load due to the longitudinal displacement of the extreme fibers of the debonded zone. This can be more clearly seen by referring to Fig. 1. As an initial investigation, the basic mechanism of crack growth is limited to a model containing broken fibers only. Subsequently, a model is developed to account for additional longitudinal damage parallel to fibers in the monolayer (Fig. 2). Splitting and yielding of the matrix is assumed to initiate at the notch tip and to progress longitudinally between the last broken fiber and the first unbroken fiber. There are three different zones in the model: (i) unidirectional ply with bonded constraint layers, fiber numbers (0) to (N - 1) and (M + 1) to (=>; (ii) unidirectional ply with debonded constraint layers, fiber numbers (N + 1) to (M - 1); and (iii) intermediate fibers (N) and (M). An equilibrium equation is written for each fiber using the basic stress-displacement relations given by Hooke’s law and the shear-lag assumption. The stresses and displacements are determined as a function of number of broken fibers, constraint layer parameters and debonded size. The results are compared to the corresponding fully bonded cases Of 141.

f

kdebonded Fig. 1. Unidirectional

&,(opplied

stress)

4 zone

lamina with broken fibers, surface constraint layers and debonding.

Shear-lag analysis of notched laminates

t

debonding CJ- (opplied

debonded

Fig. 2. Unidirectional

1015

stress 1

zone

lamina with broken fibers, longitudinal matrix damage, surface constraint layers and debonding.

FORMULATION a. Two dimensional shear-lag model with broken fibers, surface constraint layers and debonding A unidirectional array of parallel fibers with surface constraint layers, debonding and an arbitrary number of broken fibers is shown in Fig. 1. Debonding is assumed to exist from the last broken fiber (L) to an unbroken fiber (M) of the unidirectional lamina. The constraint layers are assumed to be placed symmetrically about the unidirectional lamina to give a laminate with no bending. The broken fibers are assumed to occur along the x-axis and, since the loading is symmetric, only the first quadrant of the laminate is considered in the analysis. The basic analysis and assumptions are the same as in [4], however, in order to indicate clearly the modifications needed to account for surface debonding, it is necessary to repeat some of the formulation. The fibers are taken to be of much greater stiffness and strength than the matrix and the longitudinal load is therefore assumed to be carried by the fibers only. Load is transferred between fibers by shear stresses as given by the classical shear-lag assumption. The axial fiber stress a,(y) and the matrix shear stresses, T,,(Y) and 7:,(y), are then given by the simple relations dz,,, u,(Y)

=

EF

d)

(1)

,

L’n -

I

(Y)l,

vn-I(Y where V,(Y)

=

Ej= =

axial displacement

of the fiber (n) at the location (.v),

Young’s modulus of the fiber,

t=

thickness

of the unidirectional

t’ =

thickness

of the constraint

ply,

plies.

(2) (3)

A. K. KAW and J. G. GOREE

1016

The stiffnesses G&h and Gb/h’ must account for the interaction between tibers[l, 2, 13, 141. GM and CL are typically not the shear moduli for the “neat” matrix nor are h and h’ necessarily fiber center-line distances. The ratios G,dh and Gtlh’ are equivalent stiffnesses and are assumed to be material constants depending on the fiber and matrix properties, the fiber volume fraction, orientation of plies, but not on the size of the damage region. Only for large spacing can GM and h be expected to approach the “neat” matrix and center-line values. For example, in [2, 14, 151, it is shown that the shear stress becomes larger as the fiber spacing decreases, that is, 0(1/d/d) for rigid fibers where “d” is the minimum distance between the fibers. Local failure may occur at critical points through the thickness in advance of laminate splitting which would give an apparent shear stiffness considerably different from that of the matrix alone. By the virtue of the shear-lag assumption the longitudin~ and transverse equilibrium equations become uncoupled and the longitudinal displacement and stress in the fibers as well as the matrix shear stress can be obtained without solving the transverse equilibrium equations. Therefore, only the equilibrium equations in the longitudinal direction will be considered in the following discussion. The debonded fibers, (N) and (M), are considered to be connected by springs due to the presence of the angle-plies of the constraint layers. The springs are assumed to have a linear force-displacement relation and the stiffness (k) per unit area for a particular laminate then decreases proportional to the length of the spring. With reference to the free-body diagrams (Figs. 3 through 5) of the elements for different ranges of fibers, the equilibrium equations are

forn=0,1,2

,...,

N-l,M+l,..

., AF

-

forn=N+l,N+2

,...,

f

dw + (T ln+l -

dy

T

In) = 0

(5)

M-2,M-I,where AF = area of fiber. IY

r

cT,(y+Ay) (y +Ay)l,\

I,

~(v+Ay)l,+,, ’

Fig. 3. Free-body

T(Y)&+,

diagram of a typical element of the fully debonded I) to (M - 1)).

zone (fibers No. (N +

1017

Shear-lag analysis of notched laminates

‘(y+&y)I

”

r&+A&+ /

qfY)l” NOTE: Diagram shown as above for clarity only. In the actual laminate, the constraint layers are symmetric with a layer of thickness t’/2 placed on each side of the unidirectional ply. Fig. 4. Free-body

diagram of a typical element of the fully bonded zone (fibers No. (0) to (N - I) and (M + 1) to fsll.

NOTE: Diagram shown as above for clarity only. In the actual laminate, the constraint layers are symmetric with a layer of thickness t’i2 placed on each side of the unidirectional ply. Fig. 5. Free-body of a typical element of the intermediate

zone (fiber No. (N)).

A. K. KAW and J. G. GOREE

1018

For fiber N

For fiber M AF d‘m --

f dy

+

(++I

- TIM) -

Using the stress-displacement relations (I), (2) and (3), in the above equ~~~b~umequations, the following set of d~eren~e-d~erent~~ equations is obtained: (8) forn=0,1,2

,...)

N--1,&f+

--

GMt

for n =iv+

l,N+2,...,

AFEF~ d2vN -+ Giwt dy=

(vN+,

I,...,

+ (vne1 - 2V, + vn-1) = 0

dy2

M-&M-

(9)

~.F~r~ber~

- 2flN + VN- 1) + CJ&M - WV)- CR(ZIN- “UN-I) = 0.

(IO)

For fiber N AFE& -G,wt

d’~,+g

dy=

+ (w+,

- 2% + W--I) -

-

CRZCW

-v&$)-O,

(11)

where k t’ CR2 = -{G&h) t ’ c R

= --(Gidh’) fG,dhl

t’ f

The ~~ns~r~~t layer provides additions Io~g~tudina~stiffness to the un~dire~t~ona~ply, the effect being given by the constant CR. The debonding effect is represented by the second constraint ratio CRZ. To match the differential equations for the bonded case[4], CR2 must reduce to CR when 1. The only varying parameter for CRZin a particular laminate is the width of the M--N= debond zone. fM - IV). Since C Rz represents a linear spring and loses its stiffness (k) per unit area ~ropo~iona~ to its length then CRZCCI&M - N). Hence,

c R2

=

CR p M-N’

(13)

Noting the coefficient of the second derivative in eqns (8) through fj2), the following changes of variables, as suggested in [41, are made:

y=

- =EF$$, gt? = uxu,,

(14)

1019

Shear-lag analysis of notched laminates

where ua = applied remote

stress,

q, C,, and V&z) are nondimensional. Substituting

eqn (14) into eqns (1) and (2), u,

dvn

(15)

a% ’

=

(16) The resulting equations

in nondimensional 9

for n = 0, 1,2, . . . , N -

+ (1 + CR) (Vn+l - 2v,

drl*

+ (Vn+*

M -

1,N+2,...,

d*V, drl*

+ V,_,)

= 0

(17)

1, M + 1, . . . )

d2V, for n =N+

form are

2,M

+ (V/v+1 - 2VN + v,_,>

- 2v,

(18)

+ V*_*) = 0

- l.ForfiberN

+ C&V,

- V,) - C,(V,

- C&VM

- V,) - C,(V,+,

- V,_,)

= 0,

(19)

= 0.

(20)

For fiber M d*V, h*

+ (VkfII

- 2vA4 + v,_,>

- V,)

These equations can be written as follows, where the left-hand side is the same in each equation, dzv,

+ (1 + C,) (Vn+* - 2v,

dv2 forn=0,1,2

,...,

N-

l,M+

d* V, d$

l,N+2,...

,M -

+ (1 + CR) (VNI-I -

= 0

(21)

I,...,

y$ +(1 + CR) (Vn+1 forn=N+

+ v,_,>

2vN

2v,

2, M -

+

+ v,-I)

= CR(Vn-+I - 2v,

+ V,_,)

(22)

1. For fiber N

VN-1)

=

c,(vN+,

-

VN)

-

c,,(v,

-

(23)

VN).

For fiber M d’ VM -+ drl*

(1 + C,)(V,+,

These difference-differential

- ~V,W+ V,-,) equations

= - CR(V, - V,+,-_I) + c~z(V,+, - V,).

may be reduced to differential

equations

(24)

as in [4]

A. K. KAW and J. G. GOREE

1020

by introducing a new function %?y, 8) defined as

Vo(rl> EC Rq, 8) = ---j-+ c V,(q) cos(n@)

f2-5)

?Z=l

from which

Making use of orthogonality of the circular functions, eqns (2 1)through (24) are then written as one equation, valid for all 3 and n, -

- 2(1 + C,) [l - cos(@)]~ cos(n0) d6 c

Gt(n) cosfl0) cos(n61

where G!(q) = - CR(V~GN(TJ)

=

-

I

CR(~‘N-I

-

(‘XT)) = C,Q(V~ - v,-1) Go(q) = - ; [CR(Vi

VN)

-

- Vo) -

+

l=N+

for

2Vt + VI-I)

-

cR2fvM

CRZ(VM

-

CRZ(bf

l,...,M-

1,

vN> 5

(29) (30)

VN)r

-

(281

vO)l*

(31)

The equation is of the form f

c

F(q, 0) cos(nt3)d0 = 0

for all n and q,

(32)

and as F(q, 6) is even valued in 0, if the integral is to vanish for ah 11,the function F(Q 8) then must be zero. The single equation specifying &n, 6) is then d’v -d’n

s’V = -

$ G,(q) cos(&,, /= N

(331

where fj2 = 2(1 + CR) [I - cos@)]< The solution to the problem of vanishing stresses and displacements at infinity and uniform axial compression on the crack surface will now be sought. The complete solution will be obtained by adding the rest&s corresponding to uniform axial stress and no broken fibers to the solution of the following problem. The appropriate boundary conditions are: V,,(q) = 0,

and

dJ’,>(rl)_ o drl

dvn(q) = 5,,(q) = drl V,,h) = 0

1

as q -+ x for all fibers, at

3 = 0 for all broken fibers,

at q =

0 for all unbroken fibers,

(36)

1021

Shear-fag analysis of notched Iaminates

complete solution to eqn (24), satisfying vanishing stresses and displacement at infinity is given by The

where the function A(8) is yet unknown. The remaining two boundary conditions give

dVm(O) 2

--=-,-[-&A@,+[{

dq

eosh(6t) I5 N C&(r)cos(l0) df

a~

>

cos(ne)

I

de = - I

(38)

for all broken fibers, and sinh(&) 5

G&t) cos(E0) dt

cos(n0) >

I-N

1

d0

= 0

(39)

for all unbroken fibers. Equation (37) can be solved exactly by taking A(6) - 8 f= sinh(6t) 5

0

I-N

G!(t) cosf&) dt = m$OB, cos(m0).

Eljrn~~at~~gA(8) from eqns (38) and (40), the stress boundary condition (35) reduces to x

M epcst) 2

s0

G,(t) cos(i@) dt

cos(n0) de. = -

I =

I=N

(41) From eqns (37) and (401,A@) can be efiminated to obtain Vfq, 60 in terms of B, and Gl(t). RecaUing the relation between V(q, 8) and V,(T), an expression can be obtained for the longitudinal fiber displacements as

1

I

+10 = D(“8”

‘) s G!(t) cos(Kt) dl} cos(n8) de, i-w

(42)

where

The ~o~git~di~al fiber stress is obtained by differentiating eqn (42) with respect to q and is

where p=l

P E-

1

for

t5~,

for

f>q,

Equations (28). (29), (30), (41) and (42) can be solved for the unknowns & and Gt(t).

A. K. KAW and J. G. GOREE

1022

b. Limit case of an infinitely wide debonded

zone

This is an extension of the model developed previously, such that the width of the debonded zone is now assumed to extend to infinity. Since by eqn (12)

cR2

k --- (GM/h)

t’ t ’

and 1 M-N’

kx-

then

CR2 + 0

for a debond zone of infinite width. Physically, the spring between fibers at the extremities of the debonded zone has no stiffness as it has an infinite length. Also, for fibers (M) and (M - I) far from the crack tip V,+, = V,_ , . Then using CR2 = 0

and

If,,, = VM_,

eqns (21), (22), (23), (24) reduce to

d?V, +(1 + drl’ forn=0,1,2

,...,

N-

l,M+

= N + 1, N + 2,. . . ,M

Following the same technique

- 2,M

(45)

- 2v,, + VII_,) = 0

(46)

- l.ForfiberN

, - 2VN + VN-_I) = -

CR(VN+l

as before, the single differential d2v 2drl

= 0

and

+ (V,,-I

d+

d2V, + (V,, dT2

- 2v, + V,_,)

l,...,

d’ forn

c,)(v,+,

-

(47)

v,).

equation to be solved is then

E2 v = - 5 G,(n) cos(lf3), I=0

(48)

where

G/(q) = CR(V/+I

- 2v/ + V/-I).

(49)

For fiber (0) Go(q) = cR(vO

The boundary

conditions

-

VI).

(50)

(34), (35), (36) yield

- 6 2 B, cosbz~) m=O

I +

e-@‘) g

Gl(t) cos(le)

dt

cos(n8) de = - 1

(51)

l=O

for n = 0, 1, 2, 3, . . . , L. Equations (49), (50), (51), (52) can be solved for the unknowns B, and G,(t).

The longi-

1023

Shear-lag analysis of notched laminates

tudinal displacement

is given by

V,(q)

5c

=

{

i

emcsq)

n D(6’8” while the longitudinal

B, cos(mf3)

m=O

‘) ; Gl(t) cos(f0) drj cos(n8) de, I=0

(52)

fiber stress is given by

5

-6

e-(h)

B,

cogme)

m=O

+

S(?J + 1)

F)

-P

2

5

Gr(r)

cor(lej} c0she) de,

(53)

I=0

where

for r’q,

p=l p=

-1

for

c. Two dimensional.shear-lag model with broken fibers, surface constraint layers and debonding

t>q. longitudinal matrix splitting and yielding,

The solution developed in the first section will now be extended to include longitudinal splitting and yielding of the matrix as shown in Fig. 2. All the previous assumptions are assumed valid and it is only necessary to account for the additional damage. It is assumed that splitting and yielding of the matrix initiates at the notch tip and progresses longitudinally between the last broken fiber and the first unbroken fiber. The matrix is assumed to be elastic-perfectly plastic. The last broken fiber is considered to be the first debonded fiber. All the equations remain the same as (4), (5), (6) and (7) for all fibers except for fibers (L) and (L + 1). The equilibrium equation for fiber (N) is - il,y)

-

7’ I,q f

= 0

(54)

when y 5 I,, for fiber (N + I) (55) when y 5 I,, ?. = matrix yield stress, II = length of longitudinal

matrix damage at the crack tip,

f2 = length of longitudinal

matrix split at the crack tip.

The above equations, on introduction for fiber (IV), when y I II AFEFd’w -t

dy2

-

To(4(

-

1,)

-

GM -p”

of the stress-displacement

-

Z’N_

1)

+

yi,

-

ZJN)

relations (I), (2), (3), become,

-

G.i4 $VN

I -

UN_,)!

=

0,

(56)

A. K. KAW and J. G. GOREE

1024

and for fiber (N + I), when y 5 II AFEF d2vN+ I -

Equations

-

t

GM + h

(UN+2

-

vN+l)

+

70

(y

-

12)

(57)

=

0.

UN)

+

70

(y

-

12)~

(58)

-

TO (y

-

b).

(59)

dy2

(56) and (57) are rewritten as

AFEFh d2vN + (1 + c,) Gj~t dy2

(UN+, - 2uN + UN-,) = + (1 + CR> (UN+, -

AFEFh d2vN+ 1 + (1 + CR) (UN+2 G& dy2 = - (1 + CR)

2uN+l

$-

-

(1’Nll

UN)

-

CRZ(VM

-

UN)

t)N)

+

C,42’N+2

-

~‘,A’+,)

(60)

12 =

The resulting overall nondimensional

s

+ (1 +

dn’ fern

=0,1,2

,.,,,

d2v, dn2

+ (1 + c,)

N-

equilibrium equations are

l,M+

(61)

CR) (Vn+, - 2v,, + V,,_,) = 0 l,....

Forfiber(N),whenrt~a

(VA’+, - 2v,,l + VN_1) = (1 + CR)

(VNtl

-

VN)

-

CR,(VM

-

VN)

+

50

(q

-

p>.

(62)

For fiber (N), when q > cx d2 v, dn2

- 2vN + vj,-,)

+ (1 + CR) (vN+,

= CR(VN+l -

vN)

-

~,,(V_M

-

VN).

(63)

For fiber (N + l), when n I a

d’vN+,

+

(1

+

CR)

(VN-2

-

2vN+,

f

VN)

dr12 =

-

(1

+

CR)

(vN+I

-

For n = N + 2, N + 3, . . . , M - 1,andN d2V, + (1 +

CR)

(Vn+l

-

VN)

+

cR(vN+,

-

vN+I)

-

;io(q

-

p>.

(64

+ l,whenn>cr

2v, + v,-,)

= CR(Vn+, - 2v,, + v,-,).

(65)

h2

The differential equation to be solved is the same as eqn (33), where G,(n) must now account for the yield zone and is given by

I015

Shear-lag analysis of notched laminates

GA(T)) = - (1 +

- VA,) + C,& vM - v,)

= -C&V&‘+, %+I($

=

G/(n) =

(1

+

CR(V/+l

-

cR)(vN+,

-

for ns&,

I -V,)+CRZ(V~-VN)-?O(~-_)

cR)(vN+

2v/

+

vN)

-

cR(vN+,

for n > -

vN+l)

+

(Y ,

%(q

(66)

-

p>

for

q5a9

V/-I)

(67)

(68)

for 1 = N + 2, . . . , A4 - 1 and N + 1 for n > (Y.The expressions for G,(n) as given by eqns (66)-(68) must match at n = (Y(that is, at end of the yield zone) for continuity of displacements. This requirement gives Vd4

(6%

- v.N+l(a) = 50.

Equations (41), (42), (66), (67), (68) and (69) can be solved for the unknowns B, and G,(t). The expressions for displacements and stresses remain as given by eqns (42) and (44). The equations to be solved are expressed explicitly in terms of the displacement function Gl( t) and the constants B,, giving rise to a set of Fredholm integral equations of the second kind defined within a semi-infinite interval of integration. The equations are of similar form to those of 141. The solution technique makes use of a method by Reiz[ 171, which reduces the set to a system of algebraic equations for the constants B, and explicit values of the displacement function G,(r) at specific quadrature points. Once these values are known, the longitudinal fiber displacements and stresses can be computed. RESULTS a. Debonding with no longitudinal matrix damage in the unidirectional p!\ The effect of the width of the debonded zone was of particular significance in this study. Results are given in Fig. 6 for various numbers of broken fibers with a constant constraint ratio. Debonding was assumed to start at the last broken fiber and extend longitudinally to infinity (Fig. 2). The critical fiber is defined as the fiber which has the maximum stress. For this problem the critical fiber was the first unbroken fiber at the crack tip (n = 0). The stress in the critical crack-tip fiber decreased initially for a small debonded zone, but subsequently increased with an increase in the width of the debonded zone. In fact, the stress concentration in the limit case of an infinitely wide debonded zone, was more than that of the bonded case (Table 1). One result of particular significance is that the maximum decrease in the stress in the critical fiber occurs for a small debonded zone and is essentially independent of the initial crack length. Figure 6 shows that a debonded zone width of about two fibers spacings results in the largest decrease in the maximum fiber stress for 5, 7, 9 and 21 broken fibers. Debonding acts like a constraint (CR,) between the last broken fiber (N) and last debonded fiber(M), resulting in a redistribution of stresses in the vicinity of the crack tip, hence decreasing the stress in the critical fiber. A higher constraint ratio resulted in a larger drop in critical stresses as shown in Fig. 7 but gave higher critical stresses in the limit case (Fig. 8).

Table 1. Maximum stress concentration

vs width of debond zone

Number of broken fibers = 7 First debonded fiber (NJ = 3 Contraint ratio (CR) = 0.5 Width of debonded zone (A4 - N) fibers 1

2 3 4 x

Maximum stress concentration K

2.5461 2.3258 2.3392 2.3601 2.5813

1026

A. K. KAW and J. G. GOREE

5 debonded zone starts from last broken fiber. crock length = no. of broken fibers x fiber spacing.

v

2 I

9 bray

b $

7 broken fibers /

‘111

5 broken fibers

4 b -

ehh

-

I I 2 3 4 a2 debonded zone width, (M-N 1 fibers

Fig. 6. Maximum fiber stress as a function of debonded zone width and crack length for a constraint ratio of 0.5.

4 debonded lost

tqt?

zone

broken

storts

from

fiber.

Z

,

2 I

2 debonded

3 zone width, (M-N)

fibers.

Fig. 7. Maximum fiber stress as a function of debonded zone width and constraint seven broken fibers.

ratio for

1027

Shear-lag analysis of notched laminates

4-

infinite

debonded

0

zone width.

3 number [crack

7

5 of broken

length

= fiber

9

fibers,

II

n

spacing x n]

Fig. 8. Maximum fiber stress as a function of crack length and constraint ratio.

b. Debonding

with longitudinal matrix damage

in the unidirectional

ply

The effects of debonding accompanied by longitudinal matrix splitting and yielding at the crack tip are indicated in Figs. 9 and 10 where some typical results were obtained for seven broken fibers. A two/one split strain to yield strain condition is assumed. This ratio was selected for comparison with the results of [4] and is approximately equal to that for brittle epoxy. A debonded zone of two fiber widths starting at the last broken fiber is assumed. The maximum fiber stress, normalized by a laminate constant

To = To

J

EFht -

)

GMAF

is plotted against the normalized applied stress. Figures 9 and 10 give results for CR = 0.5 and 1.0, respectively. The results are plotted for a monolayer having four different combinations of constraint and/or damage as given below. For case (i) no matrix damage corresponds to a fully elastic matrix. The remaining three cases have an elastic-perfectly plastic matrix with a stress-strain behavior as shown on the inserts of Figs. 9 and 10. Transverse notch 6)

Constraint layer

Matrix damage

Debonding

X

(ii)

X

(iii)

X

X

X

(iv)

X

X

X

X X

A. K. KAW and J. G. GOREE

1028

fC,= 0.5, with d~onding)

3 To

2

(net

R= 0, inelastic

section)

matrix 1

2 SJ yield l Split Fig. 9. Maximum fiber stress as a function of applied stress for a constraint seven broken fibers.

3-

ratio of 0.5 and

Cose (iii) --_(CR= l.O,no debonding;‘. (C,_+=l.O,with

l

debonding)

split

Fig. 10. Maximum fiber stress as a function of applied stress for a constraint ratio of I .O and seven broken fibers.

Shear-lag analysis of

notched laminates

1029

For case (ii), the unidirectional lamina with no constraint, it was found in [l] that once the split forms the critically stressed fiber unloads and the split length becomes unbounded under 5-10% further increase in applied stress. The fracture behavior then reduces to that of an unnotched laminate with the net-section fracture stress being independent of the initial crack length. The experimental study of this split growth as reported in [S] points out that this predicted behavior is indeed reasonably accurate if the early slow growth region (about one-tenth the length of the initial transverse crack size) is ignored. That is, by split initiation one refers to the beginning of the rapid growth region due to shear rather than the early stable growth due to transverse matrix normal stresses. For cases (iii) and (iv), the critically stressed fiber does not reduce to a net section state but continues to carry additional load after splitting with increasing applied load. However, in the presence of debonding, the maximum fiber stress is relieved. Hence, in terms of load carrying capacity, the worst case is (i), where the monolayer has no damage other than an initial transverse notch, while the best case is (ii), where the notched monolayer is not constrained but has longitudinal matrix damage. Cases (iii) and (iv) lie between the above models where debonding allows the larger load carrying capacity. For all fully bonded constraint layer cases, the maximum fiber stress occurs in the first unbroken fiber at the end of the split (y = II) for no (or low) constraint ratio[ I] and at the notch tip (y = 0) for high constraint ratios[l6]. The same behavior occurs for debonded cases, but in case of high constraint ratios and high values of alpha (a) and beta (p), the maximum fiber stress occurs in the last debonded fiber at y = 0. This shows that under the above conditions the first unbroken fiber is highly relieved of stresses and can result in discontinuous damage of the fibers. REFERENCES [II J. G. Goree and R. S. Gross, Analysisof a unidirectionalcompositecontainingbroken fibers and matrix damage. Engng Fracture

Mech.

13, 395-578

(1979).

r21 J. G. Goree and R. S. Gross, Stresses in a three-dimensional Engng Fracture

Mech.

13, 395-405

unidirectional

composite containing broken fibers.

(1980).

[31 L. R. Dharani and J. G. Goree, Analysis of a hybrid, unidirectional laminate with damage. IUTAM

Symposium

on Mechanics

of Composite

Proceedings

ofrhe

Materials (August 1982).

141 L. R. Dharani, W. F. Jones and J. G. Goree, Mathematical modeling of damage in unidirectional composites. Engng Fracture Mech. 17, 555-573 (1983). [Sl L. R. Dharani and J. G. Goree, Analysis of a hybrid, unidirectional buffer strip laminate. Proceedings ofthe 2nd International Conference on Composire Structures, Paiseley, Scotland (September 1983). [61 L. R. Dharani and J. G. Goree, Analysis of a unidirectional, symmetric buffer strip laminates with damage. Engng Fracture Mech., in press. [71 W. F. Jones and J. G. Goree, Experimental determination of internal damage growth in unidirectional boron/ aluminum composite laminates. Mech. Composire Material 58, 171-178 (1983). @I J. M. Wolla and J. G. Goree, Longitudinal splitting in unidirectional composites; analysis and experiments. 21st Annual Meeting of the Sot. of Engineering Sci., Blacksburg, VA (October 1984). r91 J. M. Hedgepeth, Stress concentration in filamentary structures. NASA TN D-822 (1961). 1101 J. M. Hedgepeth and Van Dyke, Stress concentration from single filament failures in composite materials. Textile Res. 39, 613-626

(1969).

1111 C. Zweben, Fracture mechanics and composite materials: A critical analysis. ASTM, pp. 65-97 (1973). WI P. Kuhn, Stresses in Aircraft and Shell Structures. McGraw-Hill, New York (1962). [I31 S. B. Batdorf, Measurement of local stress distribution in damaged composites using an electric analogue. Adv. Aerospace

Struct. Materials,

71-74 (1982).

[I41 B. Budiansky and G. F. Carrier, High shear stresses in stiff fiber composites. MECH-51, Division of Appl. Sci., Harvard University (March 1984). [ISI J. G. Goree and E. B. Wilson, Jr., Transverse shear loading in an elastic matrix containing two elastic circular cylindrical inclusions. J. Appl. Mech. 34, 51 l-513 (1967). [I61 J. G. Goree, L. R. Dharani and W. F. Jones, Mathematical modeling in unidirectional composite. NASA CR3453 (1981). [I71 A. Riez, On the numerical solutions of certain types of integral equations. Ark. Math. Fysik 29, l-21 (1943). (Receil,ed

6 Nol,ember

1984)