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Influence of free-edge delaminations on the extension-twist coupling of elastically tailored composite laminates
Ma&l.
Comput.
Modelling Vol. 19, No. 314, pp. 8S107, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/94 $6.00 + 0.00
08957177(94)E0025-I
Influence of Free-Edge Delaminations on the Extension-Twist Coupling of Elastically Tailored Composite Laminates* J. LI School of Textile and Fiber Engineering, Georgia Institute of Technology Atlanta, GA 30332-0295, U.S.A. E. A. ARMANIOS School of Aerospace Engineering, Georgia Institute of Technology Atlanta, GA 30332-0150, U.S.A. Abstract-A
closedform solution is developed in order to analyze the influence of freeedge delaminations on elastically tailored composite laminates. The analysis is based upon a shear deformation theory and a sublaminate approach. The solution is applied to predict the influence of free-edge delamination on the behavior of a class of laminates exhibiting extension-twist coupling. Comparison of analytical predictions with test data from laminates with mid-plane and off mid-plane free edge delaminations is performed. The results indicate that free-edge delamination can have a significant influence on extension-twist coupling.
1. INTRODUCTION The potential application of unsymmetrical composite laminates has been brought into focus through the use of elastic tailoring which had its first application in aerospace industry. Advancement in the manufacturing processes also made possible the production of unsymmetrical laminates with tailored properties. However, the use of unsymmetrical laminates in structural design depends on the full understanding of their behavior under various loading conditions and their damage tolerance characteristics. Composite laminates exhibit large interlaminar stresses at interfaces near the laminate free edges. It is well-documented [l-8] that delamination caused by interlaminar stresses arise at the free edges of symmetric laminate under uniform extension. Symmetric laminates subjected to bending [9-141, torsion [14-211, combined extension and bending [ll] and combined extension, bending and torsion [13, 19-211, have also been considered. However, limited work has been devoted to unsymmetrical laminates. Li [22] and Armanios and Li [23] developed a simple analytical model to predict the interlaminar stresses and the total strain energy release rate for delaminated unsymmetrical laminates under extension. While symmetric configurations are simple to analyze and possess unique design and manufacturing advantages, their damage modes often alter their initial symmetry. An example is the case of a laminate subjected to unsymmetrical loading such as bending. Free edge delamination is expected to initiate on one side of the symmetry plane creating an unsymmetrical effect. Understanding and predicting the influence of free-edge delamination on the overall behavior of the laminates will provide quantitative measures of the extent of the damage to ensure their damage tolerance. *This was supported by the Army Research Office under Grant DAAH04-93-G-0002 Excellence in Rotorcraft Technology. This support is gratefully acknowledged.
ss part of the Center of
J. LI
90
Figure
E. A. ARMANIOS
AND
1. Laminate
configuration
2. ANALYTICAL
and loading.
MODEL
Consider a laminate with an arbitrary lay-up subjected to combined extension, bending and torsion as shown in Figure 1. The laminate is assumed to be initially flat. Unsymmetrical lay-ups induce warping, bending or twisting as a result of curing stresses. Appropriate changes in design can lead to hydrothermally stable laminates. Such a design strategy is outlined in Winckler [24] in order to produce a class of unsymmetrical laminates with extension-twist coupling. These laminates do not exhibit changes in bending and torsional curvatures with variations in temperature or moisture content, while keeping their extension-twist coupling when subjected to mechanical loads. The generic laminate shown in Figure 1 is subjected to combined extension Fx, torsion MT and bending moment MB. For a homogeneous anisotropic elastic body bounded by a cylindrical surface, a generalized plane deformation [25] exists when the stress tensor does not vary along the generator z. The displacement field in the absence of rigid body modes can be written as u(z,y,t)=Eox+~xZ+mxy+U(y,z),
(1)
v(x,y,z)=-;n1x2+Cxz+v(y,z),
(2)
zv(x,y,z)
1 = -~Kx~-cxy+W(y,z),
(3)
where u, v, and w denote displacements relative to the x, y, and z axes, respectively. Axial extension is denoted by ~0, while K and ~1 are bending curvatures in the x-t plane and x-y plane, respectively. The angle of rotation per unit length about the x-axis, or twist, is denoted by C. The laminate is modeled as consisting of groups of plies or sublaminates. Each sublaminate is treated as a homogeneous anisotropic body bounded by a cylindrical surface and subjected to interfacial stresses. A generic sublaminate, subjected to interfacial stresses on the top and bottom surfaces and stress resultants along the edges, is shown in Figure 2. The thickness to width ratio of each sublaminate is much smaller than unity. Therefore, the bending curvature ~1 is negligible. Further simplification is achieved by considering transverse shear deformation only in the expression of the displacement components V(y, t), V(y, z) and W(y, Z) associated with the reference surface. The displacement field in equation (1) through equation (3) takes the form u(x,y,x)=Eox++x(Z+6)+U(Y)+tP,(Y),
v(x, Y, z) = V(Y) + z t&(Y) w(s,y,z)
1 = -SK”2
+
c x (2 + a
- Cz (Y + P) + W(Y)*
(4 (5) (6)
Free-Edge
Delaminations
91
Y, ” -
Figure 2. Sublaminate
notation
and sign convention.
The arbitrary constants 6 and p are to be determined from continuity of displacements between the sublaminates and from the overall boundary conditions. Shear deformation is recognized through the rotations PZ and & The corresponding strains are
The strain components associated with the reference surface are denoted by superscriptO. These are defined as E!&=&c+K6, IE, =
8 YY=v
K,
KY = r,z
Y& = u,, + c4
,Y,
ZY =
PY,Y, -I$!,
= Pa, + yy,
=
Pz
-
cc;+
Pz,y
+ c,
(8)
P),
where partial differentiation is denoted by a comma. The constitutive relationships can be written in terms of resultant forces and moments and associated strains and curvatures as follows
-AH
A12
-416
Bll
B12
B16 B26
A12
A22
A26
B12
B22
A16
A26
A66
B16
B26
B66
&I
B12
B16
Dll
D12
D16
B12
B22
B26
012
022
D26
- Bl6
B26
B66
D16
D26
D66
(9)
(10) For a sublaminate of thickness h, the stiffness coefficients in equations (9) and (10) are defined as (&j,
&j,
Dij) = /‘;,;I&(W2)d~,
where Qij are the transformed reduced stiffnesses [26].
(11)
J. Lr AND E. A. AFLMANIOS
92
By substituting equation (8) into equations (9) and (lo), the constitutive relationships can be written as
(12)
(13) where -All
[Xl=
All6
+ &I
A166 + &6
Al2
A126
+ &2
A266 + B26
A16
A166 + &6
A666 + B66
&l
&la
B166 + D16
B12
B12fi + D12
B266 + D26
&sS
B66S + D66
_ B16
+ Dll
+ D16
(14
06) The sublaminate equilibrium equations are obtained from the virtual work principle as N zy,y + t,, - tr, = 0,
(17)
N?/,y + t,, - tl, = 0,
(18)
Qy,y+~u -m = 0,
w-9
M q,,y -
Qz + ; (tuz+ h,) = 0,
(20)
M y,y -
Q1/+ 2 (hq,+ h,) = 0,
h
(21)
where the interlaminar shear and peel stresses at the sublaminate upper and lower surfaces are denoted by t,,, tuy , p, and tl,, tly, pi, respectively, as shown in Figure 2. At the sublaminate boundaries, y = constant, the following should be specified N,, or u,
3. SUBLAMINATE
NY or V,
Qvor W
ANALYSIS
J&, orP,,
and
OF DELAMINATED
Mv or BY
(22)
LAMINATES
For a symmetric laminate under antisymmetric loading, such as bending or torsion, or an unsymmetrical laminate under extension, free edge delaminations may develop in the upper or lower half of the laminate as shown in Figure 3. One half of the laminate cross section can be modeled using four sublaminates as shown in Figure 4. Also appearing in Figure 4 is the sublaminate numbering scheme and the coordinate systems. Sublaminates 2 and 3 represent the group of plies above and below the delamination, respectively, while Sublaminates 1 and 0 represent the same group of plies in the untracked portion. The governing equations (4)-(22) developed in the previous section will be applied to each sublaminate.
Free-Edge Delaminations
Figure 3. Delaminated
unsymmetrical
Figure 4. Sublaminate
3.1. Kinematic
93
laminate under extension.
model and coordinate
systems
Relations
To suppressthe rigid body modes, it is assumed that the center of the laminate is fixed. That is, at X = Y = Z = 0 u=lJ=w=o, (23) VJ = WJ = VJ - w,y = 0. The continuity of displacements at the interface between Sublaminates 1 and 0 can be written as
(x4,$) (X,Y.-$) vo (XJ,$) (X7%-$) wo (X7%$) =Wl (X7%-$), uo
=ul
=vl
(24)
where hr and ho denote the thicknesses of Sublaminates 1 and 0, respectively. The subscripts associated with the variables denote the respective sublaminate. This convention is adopted throughout the following sections. Substitute equations (4) through (6) into equation (24) to get
(25)
J. LI AND E. A. ARMANIOS
94
From the displacement continuity conditions between Sublaminates 1 and 2, and 0 and 3 and the overall conditions in equation (23), the constants S and p for each sublaminate can be determined as
3.2. Equilibrium
Equations
The responses associated with Sublaminates 0 and 1 are coupled through the interface continuity conditions expressed in equation (25). The upper surface of Sublaminate 1 and the bottom surface of Sublaminate 0 are stress free. The shear and peel stresses at the interface between Sublaminate 0 and 1 will be denoted by t,, t,, and p, respectively. Apply the equilibrium equations (17)-(21)
to Sublaminates 0 and 1 to get &/l
+ &go
&I
+ &/o
= 0,
(27)
=
(28)
0,
&,I + Qyo = 0, hl + +%,l,u - Qzl = 0,
&yl,y
M WO,Y -- “2”NZYO,Y
(2% (30)
-
&to = 0,
(33)
-
&VI = 0,
(32)
ho M Yod -- 2 Nvod/ - QDo = 0.
(33)
M YbY +
$N,l,,
For Sublaminate 2, the upper and lower surfaces are stress free. The equilibrium equations reduce to &,2
=
NzY2
Qg2 = 0,
(34)
M xy2,y - Qs2 = 0.
(35)
=
Mar2 =
Similarly, the equilibrium equations for Sublaminate 3 can be written as NV3 =
3.3.
Resultant
Forces
Nzy3
Qy3 = 0,
(36)
M q,3,y - Q+3 = 0.
(37)
=
My3
=
and Moments
From equation (27) through equation (29), the mid-plane displacement functions of Sublaminate 1 are related to the rotations of Sublaminates 1 and 0 through the following relations
(38)
and wl,,=wo,y=-cy-
A45 A44
where
A:5 ---A44
A405 A:4
Ai4
A44
A44
A44
(3%
Free-Edge Delaminations
[T] = [ ;;;
[I [ ji
=
;;Y2
1[ -l
Bk -
ho o TA26
Bi2
_
!$;,
B6 _ 66
hoA 2
Bl
_
!?A6
412
B1’2 + By2
+ A:,&+
A16
B;6
+ A;,&
+ @‘6
95
66
26
A:,So + A:,&,
2
!.?A0 2 22
BO26 _
2 26 !!!?A6
22
26
1
BO _
’
(41)
Bi6 + A;,61 + Ai6bo + A;,61 + Az6S0 I ’ (42)
Bi6
+
BA6
+ Bi6
The superscript associated with the stiffness coefficients denotes the respective sublaminate. The over bar denotes the following summation (43) The resultant forces and moments in Sublaminates 1 and 0 are given
0 0+
by
w =VI “,” +[El]My}, c {NO} = [-lo] i
LEOIMl/l
(44)
(45)
7
IQ) = [W 02 - {d . C. (Y+ a - b),
(46)
where
{N’} = (hl
j&l
M,,l)T,
(47)
Nq/o Me0 M,o l&/o >T,
(48)
&yl
{NO} = ( Nzo &o
a1
qll
{&I = (OS1 Qzo &,I
Q~o)~,
-41 A:,& + B:, 4.41 + B:6
1
‘A:2
(4% A:6
42
A:,61
+ Bt2
A:,61 + Bi6
42
Ai
A:6
A:661
+ @6
A&& + B&
A;6
Ai6
B,‘,
B:,h
+ &
&,61
Bi2
Bi6
B212
Bi6
B1’2 B:,61 + Di2 mB:6
@651
+ D:6
+ @6
-
Bi661 + Dfs Bij661
+
D&_
_B216
41 &So + B!‘, 46~0+ @I3 42
-490
+ By2
4660
+ @6
A!6
4660
+ BY6
A90
+ B&
-
B616
-412
A!6
42
46
A16
A16
B!,
B&So + DTI
B1),60 + 05
B?2
B?6
B102
By260
+ Dy2
Bz6&, + Di6
@2
B:6
BY6
BY6SO
+ @6
@660
D&
~~26
B:6
-B1’6
0
B&3 0
+
Bi2
0
42
42
B:,
0
A:6
A026
Bi6
0
B:,
0
A;6
A:6
D:6
0
D12
0
BT2
@2
D;,
0
D;,
0
By6
B:6
o
Ob6
o
_ B:6
B:6
.Oi6
-
(50)
ITI
1
(52)
J. LI AND E. A. ARMANIOS
96
-+A!,
By6 - $A!,
-?A!,
-$A% h
BO 26 _ !!?A0 2 26
--
By2 - $An,
hl 2 A 022
Bi2
-
922 ho
0
Bi6
-
TA26 h0
0
‘X2
-
[SO] = -2 h1B6 16
-T L
hl
Do16 _
0
B 26
0 -- hl 2 B66
ho@ 2
16
-- ;@2
Do26 -hoBO 2
26
-Yj-
Do _ 66
Al
_
55
h”B6 2 66
hl
072
0 B 22
hl -TB26
($d2
--
ho 0 +312
-
Do22 _ %!BO 2 22
0
Do _ 26
ho@ 2
A&A!5
A44 At5
_
A:6
Ai6
A86
BY2
B202
B?6
Bt6
Bi6
Bi6
--
cA_!5J2
--
PI
7
(53)
A&A&
-444
A44
A!5Ai4
A&A:4
A44
A44
-444
[k”] =
A16
26
Ai!A&
-444
AE2
&4Ai4 A44
--
(54)
-4:4A404 A44
Ai4At4 -
SYM
A44 (q}T
=
(A:,
-
2A:,
At5 - gA;,
Ai
- kA;,
Ai
- 2A&)
.
(55)
The resultant forces and moments associated with Sublaminates 2 and 3 can be expressed as
(56)
(58)
(59) where
Matrix [T’] and vector {c’}
[P] =
AA B:, B:6
;;i$;i;
{ ml}T
=
(Nz2
M22
KY2
),
(60)
{ fi”}T
=
( Nz3
Mz3
Mq,3
1.
(61)
are defined as
;!;$;;]
-
[$
;;
;;]
[@I,
(62)
Free-Edge
Delaminations
97
where [ii’]=
[$
$
31’[::
@I’)=
[;
;i;;;
3
$]-‘{
zig],
;].
Similarly, matrices [TO] and [RO] and vectors {p} and {II’} are obtained from equation through equation (65) by replacing superscript 1 by 0 and 61 by 60, respectively. 3.4. Boundary
(64)
(65)
(62)
Conditions
The continuity conditions between Sublaminates 1 and 2, and 0 and 3, and the boundary conditions at the free ends of Sublaminates 2 and 3 can be written as NV1 lyl=o &I
lvl=o
(67)
~l/ll,l=o
=
07
(68)
hoI,,=
=
07
(69)
Mwllyl=o
=
M4/l=o
for the
7
(70)
= 0,
(71)
~3cYolY0=o = ~~Y31yo=0 7
(72)
ay31yo=-a
Solution
(66)
07
= 0,
Ky21yl=-a
3.5.
=
= 0.
(73)
Rotations
Substitute the force and moment resultants in equation (44) through equation (46) into equation (30) through equation (33) to obtain a set of ordinary differential equations in terms of the and using equation (66) through rotations of Sublaminates 1 and 0. Solving these equations equation
(69) to determine
the integration {P) = PI [e-“1
constants
yields
{GI + I.q) C (Y + a - b) 9
(74)
where (75) Matrix [eeSg] is a 4 by 4 diagonal matrix of characteristic roots s, where s2 are the eigenvalues. The ith diagonal element corresponding to the ith characteristic root, si, is e-3aY. When si equals zero 7emsiY is replaced by -(y-b+a). Matrix [a] is 4 by 4 whose columns, 4, are the eigenvectors. The eigenvalues and eigenvectors are determined from (76)
(77)
J. LI AND E. A. ARMANIOS
98
The vector {G}T
= (Gi Gz Gs G 4 ) in equation (74) represents four integration constants and
can be expressed as (73) where [G] is given by
[G] =
[I1s
(79)
[~]-‘[M]-‘([L]+(M]{~}(OO1)).
Matrix [i] in equation (79) is diagonal. Its z‘th element is the inverse of the ith characteristic root. For a zero root the corresponding diagonal element equals 1. Matrices [L] and [M] are given as
(81) Matrix
[&I in equation (75) is a diagonal matrix whose elements are obtained from
h
lI&
The rotation @sZ of Sublaminate
(82)
lk4 I 2 can be expressed as
,f3zZ= { sinh(s,y) where the characteristic
= [aIT [k”] [@]a
lks
cosh(s,y)
y + a - b} [H]
,
(33)
root sC is given by
s, =
(84)
Matrix [H] is defined as [HI=
[;:
;;
;:I,
(35)
where
(86) H22 = H12
coth (~,a> + ksch 4
(s,a) , csch
(~,a)
,
99
Free-Edge Delaminations
[W’ (PI + WI {a (070, 1)).
[@‘I = [I’]
The remaining
displacement
functions
associated
with Sublaminate
(87) 2 can be related
to fls, by
(88)
Similarly, COSh(Sdy) ?~+a-
P3t = {Sinh(S&)
(89)
b
and
where sd is obtained by replacing superscript 1 by 0 in equation (84). Matrix [I] is similar to matrix [H] defined in equation (85). Its elements are obtained by replacing H by I, s, by sd and superscript 1 by 0 in equations (86) and (87). 3.6.
Loading Conditions
The laminate is assumed to be subjected to an extensional force Fx , bending moment kl~ and twisting moment &fT at its ends. These are expressed in terms of the sublaminate forces and moments
by
J
b-a
Fx = 2
(&1+~zo)&+2
J0
0
wz2
+X3)4/,
(91)
--a
b-a .
J
MB=2
0
(Mzl+Mzo + 61~,1+
G~Iv,~) dy
0
+2
CM22
+ Mz3
+ 61Nz2
(92)
+60Nz3)dy,
I -a
it&=2
J
b-a I"zyl
0
+2
’ s -a
WY2
+ MZYO + + KY3
Equation (93) can be simplified conditions to get
MT
=4
J
-
6lN~yl + ~o~~~cI - (Qzl + Qzo)(y + a -
b)] dy (93)
(912 + Q=~)(Y -t- a - b)] dy.
by using the sublaminate
equilibrium
equations
and boundary
0
b-a
[Mz,i + Mz,o + &&,i
+ ~o&,o]
Substitute the force and moment tions (91), (92) and (94) to get
dy + 4
Wzy2
+ KY31
dy.
(94)
s -a
0
resultants
in equations
(44), (45), (56) and (58) into equa-
(95)
100
J. LI AND E. A. ARMANIOS
Matrix [cr] represents the stiffness of the delaminated laminate. Its elements cyij are, in general, unsymmetric and are determined from
IQ1=
2 i$O Pii
+ %)
2 ks (%
+ Q)
2 *kc (J% + %)
2i$0 (G
+ &>
2 $c PA
+ %>
2$s (G
4 2 F;, + pjl L
! 2
+
4 6 F& + F&
4 & Fgi2+ p&
i=O
+ %a)
i=O
i=o
0
0
0
& (F;Z3 + pj3) i$o 4 (F!I + Q1) 2 i$o 4 (F&t+ Fj2) 2 i$o 4 2 &F,i,
4 ‘$ &F;,
4 2 6iF&
i=O
i=O
(96)
I.
i=O
The parameters in equation (96) are elements of the following matrices [F1l =
([Yll+ [r’l(3 (0 0 1,) (b- a)- [Ye] PI [Cl[Cl3
q PI [Cl[@ PO1= ( ho1+ [tOI(3 (0 0 1,)(b- a)- [rO] 1
[q
[?I a + {P} (4 [HI
=
(97) (98) (99)
9
[FO]= [TO] a + {CO”) (Wd) PI,
(100)
where [C] is a diagonal matrix whose elements are defined as
for si = 0, for si # 0,
17
(101) (102)
and (w’) = { sinh(s,a)
1 - cosh(s,a)
a} ,
(103)
(w”) = { sinh(sda)
1 - cosh(sda)
a}.
(104)
For a laminate subjected to an extensional force FX only, the induced bending curvature IE and twist C can be obtained as (105) The induced bending curvature and twist are functions of delamination length a. The ratio of twice the twist over the extension force, defined as the extension-twist coupling parameter and denoted by &d,
can be written as
QP Pl6d
=
(106)
F;;2b.
The extension-twist coupling parameter pi&, corresponding to a laminate with no delamination can be derived using Classical Lamination Theory (CLT), its expression is given by [22]: plsu
= -%
where
WT
= -
NZ wB _
pll
-
/412wB
p33p21
-
p23p31
p22p33
-
b23p32
-
/113wT’
’
(107)
WW
Free-Edge Delamination5
101
wT = p22p31 - p32p21
(109)
CL221133 - p23p32 ’
A22
A26
A26
A66
B26
B26
022
B22
Similarly,
for a laminate
subjected
to a bending
B22
moment
n/ie,
(111) and for a laminate
subjected
to a twisting
MT
moment
(112) The extension-twist tions,
coupling
Pisd for a laminate Ic, = 0,
can be written
subjected
to the following
boundary
condi(113)
MT=$'Fx
as p16d =
2c
J'xl2b
=
-
4qa31
- lla11) (114)
alla33
- m3a31'
where IJ is a coefficient of proportionality which represents the torsional associated with the test conditions. The corresponding CLT expression for the coupling can be written as 2Psi - +11 Pl6u
=
%1W33
-%-Jl3~31'
restraining
moment
(115)
4. APPLICATION The solution developed in Section 3 will be used to analyze a class of unsymmetrical laminates with [-e, (90 - f3)z, -8, 8, (0 - 90)2, 01~ lay-up. This class of laminates is designed to exhibit extension-twist coupling with no initial warping due to curing stresses. The laminate geometry and material properties are given in Table 4. It is seen from equation (105) that the induced twist and bending curvature are expressed in terms of the stiffness oij, which is dependent on the delamination length and its location. The induced twist C normalized by se/H is plotted against the delamination length a, in Figure 4, for a laminate with a mid-plane delamination and 0 = 20°. The ply thickness is denoted by H in the figure. For zero delamination length, the induced twist can be obtained from 1 ure 4 indicates that the extension-twist coupling Classical Lamination Theory (CLT) [22]. F’g decreases as the delamination length increases. Further investigation of the effect of free-edge delamination on the extension-twist coupling is provided through the variations of the extension-twist coupling parameter &d. Free-edge delamination is assumed at one interface at a time starting from the bottom interface or Interface 1 between 0 and (0 - 90) up to the top interface or Interface 7 between (90 - 0) and -8. The angle 8 varies from 10” to 80” at 10“ intervals. Parameter ,&sd normalized by flisU is shown in Figure 6 through 4. A delamination length of 15 ply thickness is assumed for all cases. The numerical values used to plot Figure 6 through 4 are provided in Table 4 for convenience. The values of&, MCM 19-3/4-H
J. LI AND E. A. ARMANIOS
102
0.22 -
Table 1. Geometry and material prop erties of AS4-3502 graphite/epoxy.
Err=113.31
GPa (16.44 Msi)
E22=E33=9.73
GPa (1.41 Msi)
Grs=Grs=5.66
GPa (0.82 Msi)
&s=3.40
GPa (0.50 Msi)
0.16 -
vrs=vrs=O.25 vss=O.42
0.14
Ply thickness H=0.14 mm (0.0055 in) Semi-width
0
.
'_I.
5
' 15
IO
b=68H.
-
20
-
'. 25
' 30
-
' 35
am
Figure 5. Extension-twist coupling as a function of mid-plane free-edge delamination length for 0 = 20°.
Table 2. Normalized extension-twist
Table 3. Extension-twist
e P16u
coupling E
(a=15H).
coupling parameter prsU (Deg./N).
10
20
30
40
.00291
.00401
.00367
.00164
50 -.00164
60 -.00367
70 -.00401
80 -.00291
are given in Table 4. The symmetrical response shown in Figure 6 through 4 and Table 4 about Interface 4 is a result of the applied uniform extensional loading. Figure 6 through 4 indicate that the influence of free-edge delamination on the extension-twist coupling depends on the interface location and the lay-up angle 8. For the case of 8 = 30”, the extension-twist coupling decreases irrepective of the interface in which free-edge delamination occurs. For B = 40°, the induced twist changes directions when the delamination is placed at Interface 2 or 6. For 0 = 50”, the extension-twist coupling can increase by as much as 350%.
5. COMPARISON
WITH
TEST DATA
The analytical predictions are compared to test data for [-30/602/-30/30/-60~/30]~ laminates with mid-plane and off mid-plane delaminations. The delaminations were created by a 3.175 mm (0.125in) Teflon FEP film placed along the length of the laminates at two locations. The first, placed at the mid-plane free edges (Interface 4), while the second, symmetrically placed at 60/60 and -6O/ - 60 interfaces (Interfaces 2 and 5). A schematic of these damaged laminates appears in Figure 14. The delamination was symmetrically stacked in the second configuration, in
Free-Edge
I
2
Delaminations
3
4
5
103
6
7
Interface Number Figure
6. Influence
of free-edge
I
delamination
2
3
on extension-twist
4
5
6
coupling
for 0 = 10’.
coupling
for 0 = 20°.
coupling
for 0 = 30°.
coupling
for 8 = 40’.
7
Interface Number Figure
7. Influence
of free-edge
delamination
on extension-twist
1.0
5
0.9
e z 0.8
0.7 I
2
3
4
5
6
7
Interface Number Figure
8. Influence
of free-edge
delamination
on extension-twist
1.0 2
‘0
a a
?!
0.5
0.0
4.5
4
-1.0
I
2
3
4
5
6
7
Interface Number Figure
9. Influence
of free-edge
delamination
on extension-twist
J. LI
104
I
2
AND
E. A. ARMANIOS
3
4
5
6
7
Interface Number Figure
10. Influence of free-edge delamination
I
2
3
on extension-twist
4
5
6
coupling
for 0 = 50”.
coupling
for fJ = 60’.
coupling
for 0 = 70°.
coupling
for 0 = 80’.
7
Interface Number Figure
11. Influence
of free-edge delamination
on extension-twist
1.5
5
1.0
Q 5! m 0.5
0.0 1
2
3
4
5
6
7
Interface Number Figure
12. Influence
of free-edge delamination
1
2
3
on extension-twist
4
5
6
7
Interface Number Figure
13. Influence
of free-edge delamination
on extension-twist
Free-Edge Delaminations
a
I-I
-W-l+ -_
105
2b
--_
(a) Mid-plane damage
(b) Off mid-plane damage Figure 14.
order to avoid warping due to curing stresses. The laminates were fabricated using AS413502 graphite/epoxy material whose properties are given in Table 5. A custom loading transducer was used in the test [27]. The transducer was designed to allow freedom of twist at the end of the specimen while undergoing axial load. It was found however, that the transducer causes a torsional restraining moment which was proportional to the applied extensional force. The equivalent loading condition for the portion of the test specimen away from the ends can be represented by equation (113). The coefficient of proportionality for the torsional restraining moment, J!Jis fO.04331 mm (0.001705 in) for all the test cases considered. For the class of laminates with [-0, (90 - e),, -19, 6, (6 - 9O)s, e]~, layup ?I, is negative for 0 5 8 I 45” and positive for 45’ 5 0 5 90’. The comparison between analytical predictions and test results is shown in Table 5. For mid-plane delamination, the analytical prediction shows a reduction of 18.0% which is 2.7% lower than the test data. However, for off-mid plane delsmination the analysis predicts 27.5% larger reduction than the test result. For the case of off-mid plane delamination, the tested specimens included an additional delamination symmetrically placed along each free edge as shown in Figure 14(b). However, due to the antisymmetrical distribution of the interlaminar normal stresses about the mid-plane, one delamination front on each free-edge will be subjected to tensile normal stresses while the other to compressive normal stresses. The contact problem associated with the latter, is not considered in the analysis. This contributes to the relatively larger discrepancy of extension-twist reduction between analytical prediction and test data for the off mid-plane case. Table 4. Material properties for extension test specimens. &1=126.1GPa
Table 5. Percentage of coupling reduction.
(18.3Msi)
&2=&,3=11.6
GPa (1.68 Msi)
G12=G13=5.45
GPa (0.79 Msi)
C&3=3.27 GPa (0.47 Msi) V12=Vls=O.31 V23
=0.53
Ply thickness H=0.13 mm (0.005 in) Semi-width
b=90H.
6. CONCLUSION An analytical model for the free edge delamination analysis of unsymmetrical laminates is developed. The model is applied to the study of the influence of free edge delamination on the extension-twist coupling in a [-0, (90 - e)s, -8, 8, (0 - 9O)s, e]T class of laminates made of graphite/epoxy material. It is found that the extension-twist coupling is influenced by the extent of the delamination. Moreover, free edge delamination may decrease or increase the coupling depending on the location of the delamination and the lay-up angle, 8, of the laminate. Test
106
J. LI AND E. A. ARMANIOS
data confirm the coupling reduction trend predicted by the analysis for the cases of a laminate with mid-plane and off-mid plane delamination and are in excellent agreement with the analytical prediction for the case of mid-plane delamination.
REFERENCES 1. 2. 3. 4. 5.
6. 7. 8. 9. 10. 11.
12.
13.
14. 15. 16. 17.
18. 19.
20.
21.
22. 23.
24.
N.A. Puppo and H.A. Evensen, Interlaminar shear in laminated composites, J. Compos. Mater. 4, 204-220 (1970). R.B. Pipes and N.J. Pagano, Interlaminar stresses in composites under axial extension, J. Compos. Mater. 4, 538-548 (1970). A.S.D. Wang and F.W. Crossman, Some new results on edge effect in symmetric composite laminates, J. Compos. Mater. 11, 92-106 (1977). N.J. Pagano, Free edge stress field in composite laminates, Int. J. Solids & Struct. 14, 401 (1978). T.K. O’Brien, Characterization of delamination onset and growth in a composite laminate, Damage in Composite Materials, ASTM STP 775, (Edited by K.L. Reifsnider), pp. 140-167, American Society for Testing and Materials, Philadelphia, (1982). J.M. Whitney, Stress analysis of a Mode 1 edge delamination specimen for composite materials, AZAA Journal 24 (7), 1163-1168 (July 1986). E.A. Armanios and L.W. Rehfield, Sublaminate analysis of the interlaminar fracture in composites: Part I-Analytical model, Journal of Composites Technology & Research 11 (4), 135-146 (Winter 1989). E.A. Armanios and L.W. Rehfield, Sublaminate analysis of the interlaminar fracture in composites: Part II-Applications, Journal of Composites Technology & Research 11 (4), 147-153 (Winter 1989). N.J. Salamon, A finite-difference program for stresses in anisotropic, layered plates in bending, NASA TN D-8059, National Aeronautics and Space Administration, (1975). N.J. Salamon, Interlaminar stresses in a layered composite laminate in bending, Fibre Science and Technology 11, 305-317 (1978). E.A. Armanios and L.W. Rehfield, Interlaminar fracture analysis of composite laminates under bending and combined bending and extension, Composite Materials: Testing and Design (Eighth Conference), ASTM STP 972, (Edited by J.D. Whitcomb), pp. 81-94, American Society for Testing and Materials, Philadelphia, (1988). L. Ye and B.X. Yang, Interlaminar stress analysis in symmetric composite laminates under bending, Proceedings of International Conference on Computational Mechanics, May 25-29, 1986, (Edited by G. Yagawa and S.N. Atluri), Volume 1, pp. IV-236-IV-242, Springer-Verlag, Tokyo, (1986). W.S. Chan and 0.0. Ochoa, An integrated finite element model of edge-delamination analysis for laminates due to tension, bending, and torsion loads, Proceedings of the AIAA/ASME/AHS 28th Structures, Structural Dynamics, and Material (SDM) Conference, 1987, AIAA Paper No. 87-0704CP, Part 1, pp. 27-35. P.L.N. Murthy and C.C. Chamis, Free-edge delamination: Laminate width and loading conditions effects, Journal of Composites Technology & Research 11 (l), 15-22 (Spring 1989). W.S. Chan and 0.0. Ochoa, Assessment of free-edge delamination due to torsion, Proceedings of the Second Technical Conference, pp. 469-478, American Society for Composites, Lancaster, (1987). R.D. Kurtz and J.M. Whitney, Torsion of laminates containing orthotropic layers, Proceedings of the Third Technical Conference, pp. 115-124, American Society for Composites, Lancaster, (1988). J. Li and E.A. Armanios, Analysis of unidirectional and cross-ply laminates under torsion loading, mcture Mechanics: Twenty-Second Symposium, Volume II, ASTM STP 1131, (Edited by S.N. Atluri, J.C. Newman, Jr., IS. Raju and J.S. Epstein), pp. 421-435, American Society for Testing and Materials, Philadelphia, (1992). E.A. Armanios and J. Li, Interlaminar stress predictions for laminated composites under bending, torsion and their combined effect, Composite Engineering 1 (3), 277-291 (1991). W.-L. Yin, Free-edge effects in laminates under extension, bending and twisting, Part I: A stress function approach, Proceedings of the AIAA/ASME/AHS 32th Structures, Structural Dynamics, and Material(SDM) Conference, 1991, AIAA Paper No. 91-0959-CP, Part 2, pp. 985-995. W.-L. Yin, Free-edge effects in laminates under extension, bending and twisting, Part II: Sublaminate/Layer modeling and analysis, Proceedings of the AIAA/ASME/AHS 33 th Structures, Structural Dynamics, and Material(SDM) Conference, 1992, AIAA Paper No. 92-2228-CP, Part 1, pp. 48-58. J. Li, Free-edge interlaminar stress analysis in laminated composites under combined loading, Proceedings of the 33rd Struckres, Structural Dynamics, and Materials (SDM) Conference, April 13-15, 1992, Dallas, Texas, pp. 698-706. J. Li, Interlaminar fracture analysis of laminated composites under combined loading, Ph.D. Thesis, pp. 63-104, 119-123, School of Aerospace Engineering, Georgia Institute of Technology, (1992). E.A. Armanios and J. Li, Interlaminar fracture analysis of unsymmetrical laminates, Composite Materials: Fatigue and Fmcture, Fourth Volume, ASTM STP 1156, (Edited by W.W. Stinchcomb and N.E. Ashbaugh), pp. 341360, American Society for Testing and Materials, Philadelphia, (1993). S.I. Winckler, Hygrothermally curvature stable laminates with tension-torsion coupling, Journal of the American Helicopter Society 31 (7), 56-58 (July 1985).
Free-Edge Delaminations 25. 26. 27.
107
S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, pp. 103-108, Holden-Day, Inc., San Francisco, (1963). J.R. Vinson and R.L. Sierakowski, The Behavior of Structures Composed of Composite Materials, pp. 46-47, Martinus Nijhoff Publishers, (1986). E.A. Armanios, D. Hooke, M.P. Kamat, D. Palmer and J. Li, Design and testing of composite laminates for Testing and Design, ASTM optimum extension-twist coupling, 11th Symposium on Composite Materials: STP 1206, (Edited by E. Camponeshi), American Society for Testing and Materials, Philadelphia (to appear).
Comput.
Modelling Vol. 19, No. 314, pp. 8S107, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/94 $6.00 + 0.00
08957177(94)E0025-I
Influence of Free-Edge Delaminations on the Extension-Twist Coupling of Elastically Tailored Composite Laminates* J. LI School of Textile and Fiber Engineering, Georgia Institute of Technology Atlanta, GA 30332-0295, U.S.A. E. A. ARMANIOS School of Aerospace Engineering, Georgia Institute of Technology Atlanta, GA 30332-0150, U.S.A. Abstract-A
closedform solution is developed in order to analyze the influence of freeedge delaminations on elastically tailored composite laminates. The analysis is based upon a shear deformation theory and a sublaminate approach. The solution is applied to predict the influence of free-edge delamination on the behavior of a class of laminates exhibiting extension-twist coupling. Comparison of analytical predictions with test data from laminates with mid-plane and off mid-plane free edge delaminations is performed. The results indicate that free-edge delamination can have a significant influence on extension-twist coupling.
1. INTRODUCTION The potential application of unsymmetrical composite laminates has been brought into focus through the use of elastic tailoring which had its first application in aerospace industry. Advancement in the manufacturing processes also made possible the production of unsymmetrical laminates with tailored properties. However, the use of unsymmetrical laminates in structural design depends on the full understanding of their behavior under various loading conditions and their damage tolerance characteristics. Composite laminates exhibit large interlaminar stresses at interfaces near the laminate free edges. It is well-documented [l-8] that delamination caused by interlaminar stresses arise at the free edges of symmetric laminate under uniform extension. Symmetric laminates subjected to bending [9-141, torsion [14-211, combined extension and bending [ll] and combined extension, bending and torsion [13, 19-211, have also been considered. However, limited work has been devoted to unsymmetrical laminates. Li [22] and Armanios and Li [23] developed a simple analytical model to predict the interlaminar stresses and the total strain energy release rate for delaminated unsymmetrical laminates under extension. While symmetric configurations are simple to analyze and possess unique design and manufacturing advantages, their damage modes often alter their initial symmetry. An example is the case of a laminate subjected to unsymmetrical loading such as bending. Free edge delamination is expected to initiate on one side of the symmetry plane creating an unsymmetrical effect. Understanding and predicting the influence of free-edge delamination on the overall behavior of the laminates will provide quantitative measures of the extent of the damage to ensure their damage tolerance. *This was supported by the Army Research Office under Grant DAAH04-93-G-0002 Excellence in Rotorcraft Technology. This support is gratefully acknowledged.
ss part of the Center of
J. LI
90
Figure
E. A. ARMANIOS
AND
1. Laminate
configuration
2. ANALYTICAL
and loading.
MODEL
Consider a laminate with an arbitrary lay-up subjected to combined extension, bending and torsion as shown in Figure 1. The laminate is assumed to be initially flat. Unsymmetrical lay-ups induce warping, bending or twisting as a result of curing stresses. Appropriate changes in design can lead to hydrothermally stable laminates. Such a design strategy is outlined in Winckler [24] in order to produce a class of unsymmetrical laminates with extension-twist coupling. These laminates do not exhibit changes in bending and torsional curvatures with variations in temperature or moisture content, while keeping their extension-twist coupling when subjected to mechanical loads. The generic laminate shown in Figure 1 is subjected to combined extension Fx, torsion MT and bending moment MB. For a homogeneous anisotropic elastic body bounded by a cylindrical surface, a generalized plane deformation [25] exists when the stress tensor does not vary along the generator z. The displacement field in the absence of rigid body modes can be written as u(z,y,t)=Eox+~xZ+mxy+U(y,z),
(1)
v(x,y,z)=-;n1x2+Cxz+v(y,z),
(2)
zv(x,y,z)
1 = -~Kx~-cxy+W(y,z),
(3)
where u, v, and w denote displacements relative to the x, y, and z axes, respectively. Axial extension is denoted by ~0, while K and ~1 are bending curvatures in the x-t plane and x-y plane, respectively. The angle of rotation per unit length about the x-axis, or twist, is denoted by C. The laminate is modeled as consisting of groups of plies or sublaminates. Each sublaminate is treated as a homogeneous anisotropic body bounded by a cylindrical surface and subjected to interfacial stresses. A generic sublaminate, subjected to interfacial stresses on the top and bottom surfaces and stress resultants along the edges, is shown in Figure 2. The thickness to width ratio of each sublaminate is much smaller than unity. Therefore, the bending curvature ~1 is negligible. Further simplification is achieved by considering transverse shear deformation only in the expression of the displacement components V(y, t), V(y, z) and W(y, Z) associated with the reference surface. The displacement field in equation (1) through equation (3) takes the form u(x,y,x)=Eox++x(Z+6)+U(Y)+tP,(Y),
v(x, Y, z) = V(Y) + z t&(Y) w(s,y,z)
1 = -SK”2
+
c x (2 + a
- Cz (Y + P) + W(Y)*
(4 (5) (6)
Free-Edge
Delaminations
91
Y, ” -
Figure 2. Sublaminate
notation
and sign convention.
The arbitrary constants 6 and p are to be determined from continuity of displacements between the sublaminates and from the overall boundary conditions. Shear deformation is recognized through the rotations PZ and & The corresponding strains are
The strain components associated with the reference surface are denoted by superscriptO. These are defined as E!&=&c+K6, IE, =
8 YY=v
K,
KY = r,z
Y& = u,, + c4
,Y,
ZY =
PY,Y, -I$!,
= Pa, + yy,
=
Pz
-
cc;+
Pz,y
+ c,
(8)
P),
where partial differentiation is denoted by a comma. The constitutive relationships can be written in terms of resultant forces and moments and associated strains and curvatures as follows
-AH
A12
-416
Bll
B12
B16 B26
A12
A22
A26
B12
B22
A16
A26
A66
B16
B26
B66
&I
B12
B16
Dll
D12
D16
B12
B22
B26
012
022
D26
- Bl6
B26
B66
D16
D26
D66
(9)
(10) For a sublaminate of thickness h, the stiffness coefficients in equations (9) and (10) are defined as (&j,
&j,
Dij) = /‘;,;I&(W2)d~,
where Qij are the transformed reduced stiffnesses [26].
(11)
J. Lr AND E. A. AFLMANIOS
92
By substituting equation (8) into equations (9) and (lo), the constitutive relationships can be written as
(12)
(13) where -All
[Xl=
All6
+ &I
A166 + &6
Al2
A126
+ &2
A266 + B26
A16
A166 + &6
A666 + B66
&l
&la
B166 + D16
B12
B12fi + D12
B266 + D26
&sS
B66S + D66
_ B16
+ Dll
+ D16
(14
06) The sublaminate equilibrium equations are obtained from the virtual work principle as N zy,y + t,, - tr, = 0,
(17)
N?/,y + t,, - tl, = 0,
(18)
Qy,y+~u -m = 0,
w-9
M q,,y -
Qz + ; (tuz+ h,) = 0,
(20)
M y,y -
Q1/+ 2 (hq,+ h,) = 0,
h
(21)
where the interlaminar shear and peel stresses at the sublaminate upper and lower surfaces are denoted by t,,, tuy , p, and tl,, tly, pi, respectively, as shown in Figure 2. At the sublaminate boundaries, y = constant, the following should be specified N,, or u,
3. SUBLAMINATE
NY or V,
Qvor W
ANALYSIS
J&, orP,,
and
OF DELAMINATED
Mv or BY
(22)
LAMINATES
For a symmetric laminate under antisymmetric loading, such as bending or torsion, or an unsymmetrical laminate under extension, free edge delaminations may develop in the upper or lower half of the laminate as shown in Figure 3. One half of the laminate cross section can be modeled using four sublaminates as shown in Figure 4. Also appearing in Figure 4 is the sublaminate numbering scheme and the coordinate systems. Sublaminates 2 and 3 represent the group of plies above and below the delamination, respectively, while Sublaminates 1 and 0 represent the same group of plies in the untracked portion. The governing equations (4)-(22) developed in the previous section will be applied to each sublaminate.
Free-Edge Delaminations
Figure 3. Delaminated
unsymmetrical
Figure 4. Sublaminate
3.1. Kinematic
93
laminate under extension.
model and coordinate
systems
Relations
To suppressthe rigid body modes, it is assumed that the center of the laminate is fixed. That is, at X = Y = Z = 0 u=lJ=w=o, (23) VJ = WJ = VJ - w,y = 0. The continuity of displacements at the interface between Sublaminates 1 and 0 can be written as
(x4,$) (X,Y.-$) vo (XJ,$) (X7%-$) wo (X7%$) =Wl (X7%-$), uo
=ul
=vl
(24)
where hr and ho denote the thicknesses of Sublaminates 1 and 0, respectively. The subscripts associated with the variables denote the respective sublaminate. This convention is adopted throughout the following sections. Substitute equations (4) through (6) into equation (24) to get
(25)
J. LI AND E. A. ARMANIOS
94
From the displacement continuity conditions between Sublaminates 1 and 2, and 0 and 3 and the overall conditions in equation (23), the constants S and p for each sublaminate can be determined as
3.2. Equilibrium
Equations
The responses associated with Sublaminates 0 and 1 are coupled through the interface continuity conditions expressed in equation (25). The upper surface of Sublaminate 1 and the bottom surface of Sublaminate 0 are stress free. The shear and peel stresses at the interface between Sublaminate 0 and 1 will be denoted by t,, t,, and p, respectively. Apply the equilibrium equations (17)-(21)
to Sublaminates 0 and 1 to get &/l
+ &go
&I
+ &/o
= 0,
(27)
=
(28)
0,
&,I + Qyo = 0, hl + +%,l,u - Qzl = 0,
&yl,y
M WO,Y -- “2”NZYO,Y
(2% (30)
-
&to = 0,
(33)
-
&VI = 0,
(32)
ho M Yod -- 2 Nvod/ - QDo = 0.
(33)
M YbY +
$N,l,,
For Sublaminate 2, the upper and lower surfaces are stress free. The equilibrium equations reduce to &,2
=
NzY2
Qg2 = 0,
(34)
M xy2,y - Qs2 = 0.
(35)
=
Mar2 =
Similarly, the equilibrium equations for Sublaminate 3 can be written as NV3 =
3.3.
Resultant
Forces
Nzy3
Qy3 = 0,
(36)
M q,3,y - Q+3 = 0.
(37)
=
My3
=
and Moments
From equation (27) through equation (29), the mid-plane displacement functions of Sublaminate 1 are related to the rotations of Sublaminates 1 and 0 through the following relations
(38)
and wl,,=wo,y=-cy-
A45 A44
where
A:5 ---A44
A405 A:4
Ai4
A44
A44
A44
(3%
Free-Edge Delaminations
[T] = [ ;;;
[I [ ji
=
;;Y2
1[ -l
Bk -
ho o TA26
Bi2
_
!$;,
B6 _ 66
hoA 2
Bl
_
!?A6
412
B1’2 + By2
+ A:,&+
A16
B;6
+ A;,&
+ @‘6
95
66
26
A:,So + A:,&,
2
!.?A0 2 22
BO26 _
2 26 !!!?A6
22
26
1
BO _
’
(41)
Bi6 + A;,61 + Ai6bo + A;,61 + Az6S0 I ’ (42)
Bi6
+
BA6
+ Bi6
The superscript associated with the stiffness coefficients denotes the respective sublaminate. The over bar denotes the following summation (43) The resultant forces and moments in Sublaminates 1 and 0 are given
0 0+
by
w =VI “,” +[El]My}, c {NO} = [-lo] i
LEOIMl/l
(44)
(45)
7
IQ) = [W 02 - {d . C. (Y+ a - b),
(46)
where
{N’} = (hl
j&l
M,,l)T,
(47)
Nq/o Me0 M,o l&/o >T,
(48)
&yl
{NO} = ( Nzo &o
a1
qll
{&I = (OS1 Qzo &,I
Q~o)~,
-41 A:,& + B:, 4.41 + B:6
1
‘A:2
(4% A:6
42
A:,61
+ Bt2
A:,61 + Bi6
42
Ai
A:6
A:661
+ @6
A&& + B&
A;6
Ai6
B,‘,
B:,h
+ &
&,61
Bi2
Bi6
B212
Bi6
B1’2 B:,61 + Di2 mB:6
@651
+ D:6
+ @6
-
Bi661 + Dfs Bij661
+
D&_
_B216
41 &So + B!‘, 46~0+ @I3 42
-490
+ By2
4660
+ @6
A!6
4660
+ BY6
A90
+ B&
-
B616
-412
A!6
42
46
A16
A16
B!,
B&So + DTI
B1),60 + 05
B?2
B?6
B102
By260
+ Dy2
Bz6&, + Di6
@2
B:6
BY6
BY6SO
+ @6
@660
D&
~~26
B:6
-B1’6
0
B&3 0
+
Bi2
0
42
42
B:,
0
A:6
A026
Bi6
0
B:,
0
A;6
A:6
D:6
0
D12
0
BT2
@2
D;,
0
D;,
0
By6
B:6
o
Ob6
o
_ B:6
B:6
.Oi6
-
(50)
ITI
1
(52)
J. LI AND E. A. ARMANIOS
96
-+A!,
By6 - $A!,
-?A!,
-$A% h
BO 26 _ !!?A0 2 26
--
By2 - $An,
hl 2 A 022
Bi2
-
922 ho
0
Bi6
-
TA26 h0
0
‘X2
-
[SO] = -2 h1B6 16
-T L
hl
Do16 _
0
B 26
0 -- hl 2 B66
ho@ 2
16
-- ;@2
Do26 -hoBO 2
26
-Yj-
Do _ 66
Al
_
55
h”B6 2 66
hl
072
0 B 22
hl -TB26
($d2
--
ho 0 +312
-
Do22 _ %!BO 2 22
0
Do _ 26
ho@ 2
A&A!5
A44 At5
_
A:6
Ai6
A86
BY2
B202
B?6
Bt6
Bi6
Bi6
--
cA_!5J2
--
PI
7
(53)
A&A&
-444
A44
A!5Ai4
A&A:4
A44
A44
-444
[k”] =
A16
26
Ai!A&
-444
AE2
&4Ai4 A44
--
(54)
-4:4A404 A44
Ai4At4 -
SYM
A44 (q}T
=
(A:,
-
2A:,
At5 - gA;,
Ai
- kA;,
Ai
- 2A&)
.
(55)
The resultant forces and moments associated with Sublaminates 2 and 3 can be expressed as
(56)
(58)
(59) where
Matrix [T’] and vector {c’}
[P] =
AA B:, B:6
;;i$;i;
{ ml}T
=
(Nz2
M22
KY2
),
(60)
{ fi”}T
=
( Nz3
Mz3
Mq,3
1.
(61)
are defined as
;!;$;;]
-
[$
;;
;;]
[@I,
(62)
Free-Edge
Delaminations
97
where [ii’]=
[$
$
31’[::
@I’)=
[;
;i;;;
3
$]-‘{
zig],
;].
Similarly, matrices [TO] and [RO] and vectors {p} and {II’} are obtained from equation through equation (65) by replacing superscript 1 by 0 and 61 by 60, respectively. 3.4. Boundary
(64)
(65)
(62)
Conditions
The continuity conditions between Sublaminates 1 and 2, and 0 and 3, and the boundary conditions at the free ends of Sublaminates 2 and 3 can be written as NV1 lyl=o &I
lvl=o
(67)
~l/ll,l=o
=
07
(68)
hoI,,=
=
07
(69)
Mwllyl=o
=
M4/l=o
for the
7
(70)
= 0,
(71)
~3cYolY0=o = ~~Y31yo=0 7
(72)
ay31yo=-a
Solution
(66)
07
= 0,
Ky21yl=-a
3.5.
=
= 0.
(73)
Rotations
Substitute the force and moment resultants in equation (44) through equation (46) into equation (30) through equation (33) to obtain a set of ordinary differential equations in terms of the and using equation (66) through rotations of Sublaminates 1 and 0. Solving these equations equation
(69) to determine
the integration {P) = PI [e-“1
constants
yields
{GI + I.q) C (Y + a - b) 9
(74)
where (75) Matrix [eeSg] is a 4 by 4 diagonal matrix of characteristic roots s, where s2 are the eigenvalues. The ith diagonal element corresponding to the ith characteristic root, si, is e-3aY. When si equals zero 7emsiY is replaced by -(y-b+a). Matrix [a] is 4 by 4 whose columns, 4, are the eigenvectors. The eigenvalues and eigenvectors are determined from (76)
(77)
J. LI AND E. A. ARMANIOS
98
The vector {G}T
= (Gi Gz Gs G 4 ) in equation (74) represents four integration constants and
can be expressed as (73) where [G] is given by
[G] =
[I1s
(79)
[~]-‘[M]-‘([L]+(M]{~}(OO1)).
Matrix [i] in equation (79) is diagonal. Its z‘th element is the inverse of the ith characteristic root. For a zero root the corresponding diagonal element equals 1. Matrices [L] and [M] are given as
(81) Matrix
[&I in equation (75) is a diagonal matrix whose elements are obtained from
h
lI&
The rotation @sZ of Sublaminate
(82)
lk4 I 2 can be expressed as
,f3zZ= { sinh(s,y) where the characteristic
= [aIT [k”] [@]a
lks
cosh(s,y)
y + a - b} [H]
,
(33)
root sC is given by
s, =
(84)
Matrix [H] is defined as [HI=
[;:
;;
;:I,
(35)
where
(86) H22 = H12
coth (~,a> + ksch 4
(s,a) , csch
(~,a)
,
99
Free-Edge Delaminations
[W’ (PI + WI {a (070, 1)).
[@‘I = [I’]
The remaining
displacement
functions
associated
with Sublaminate
(87) 2 can be related
to fls, by
(88)
Similarly, COSh(Sdy) ?~+a-
P3t = {Sinh(S&)
(89)
b
and
where sd is obtained by replacing superscript 1 by 0 in equation (84). Matrix [I] is similar to matrix [H] defined in equation (85). Its elements are obtained by replacing H by I, s, by sd and superscript 1 by 0 in equations (86) and (87). 3.6.
Loading Conditions
The laminate is assumed to be subjected to an extensional force Fx , bending moment kl~ and twisting moment &fT at its ends. These are expressed in terms of the sublaminate forces and moments
by
J
b-a
Fx = 2
(&1+~zo)&+2
J0
0
wz2
+X3)4/,
(91)
--a
b-a .
J
MB=2
0
(Mzl+Mzo + 61~,1+
G~Iv,~) dy
0
+2
CM22
+ Mz3
+ 61Nz2
(92)
+60Nz3)dy,
I -a
it&=2
J
b-a I"zyl
0
+2
’ s -a
WY2
+ MZYO + + KY3
Equation (93) can be simplified conditions to get
MT
=4
J
-
6lN~yl + ~o~~~cI - (Qzl + Qzo)(y + a -
b)] dy (93)
(912 + Q=~)(Y -t- a - b)] dy.
by using the sublaminate
equilibrium
equations
and boundary
0
b-a
[Mz,i + Mz,o + &&,i
+ ~o&,o]
Substitute the force and moment tions (91), (92) and (94) to get
dy + 4
Wzy2
+ KY31
dy.
(94)
s -a
0
resultants
in equations
(44), (45), (56) and (58) into equa-
(95)
100
J. LI AND E. A. ARMANIOS
Matrix [cr] represents the stiffness of the delaminated laminate. Its elements cyij are, in general, unsymmetric and are determined from
IQ1=
2 i$O Pii
+ %)
2 ks (%
+ Q)
2 *kc (J% + %)
2i$0 (G
+ &>
2 $c PA
+ %>
2$s (G
4 2 F;, + pjl L
! 2
+
4 6 F& + F&
4 & Fgi2+ p&
i=O
+ %a)
i=O
i=o
0
0
0
& (F;Z3 + pj3) i$o 4 (F!I + Q1) 2 i$o 4 (F&t+ Fj2) 2 i$o 4 2 &F,i,
4 ‘$ &F;,
4 2 6iF&
i=O
i=O
(96)
I.
i=O
The parameters in equation (96) are elements of the following matrices [F1l =
([Yll+ [r’l(3 (0 0 1,) (b- a)- [Ye] PI [Cl[Cl3
q PI [Cl[@ PO1= ( ho1+ [tOI(3 (0 0 1,)(b- a)- [rO] 1
[q
[?I a + {P} (4 [HI
=
(97) (98) (99)
9
[FO]= [TO] a + {CO”) (Wd) PI,
(100)
where [C] is a diagonal matrix whose elements are defined as
for si = 0, for si # 0,
17
(101) (102)
and (w’) = { sinh(s,a)
1 - cosh(s,a)
a} ,
(103)
(w”) = { sinh(sda)
1 - cosh(sda)
a}.
(104)
For a laminate subjected to an extensional force FX only, the induced bending curvature IE and twist C can be obtained as (105) The induced bending curvature and twist are functions of delamination length a. The ratio of twice the twist over the extension force, defined as the extension-twist coupling parameter and denoted by &d,
can be written as
QP Pl6d
=
(106)
F;;2b.
The extension-twist coupling parameter pi&, corresponding to a laminate with no delamination can be derived using Classical Lamination Theory (CLT), its expression is given by [22]: plsu
= -%
where
WT
= -
NZ wB _
pll
-
/412wB
p33p21
-
p23p31
p22p33
-
b23p32
-
/113wT’
’
(107)
WW
Free-Edge Delamination5
101
wT = p22p31 - p32p21
(109)
CL221133 - p23p32 ’
A22
A26
A26
A66
B26
B26
022
B22
Similarly,
for a laminate
subjected
to a bending
B22
moment
n/ie,
(111) and for a laminate
subjected
to a twisting
MT
moment
(112) The extension-twist tions,
coupling
Pisd for a laminate Ic, = 0,
can be written
subjected
to the following
boundary
condi(113)
MT=$'Fx
as p16d =
2c
J'xl2b
=
-
4qa31
- lla11) (114)
alla33
- m3a31'
where IJ is a coefficient of proportionality which represents the torsional associated with the test conditions. The corresponding CLT expression for the coupling can be written as 2Psi - +11 Pl6u
=
%1W33
-%-Jl3~31'
restraining
moment
(115)
4. APPLICATION The solution developed in Section 3 will be used to analyze a class of unsymmetrical laminates with [-e, (90 - f3)z, -8, 8, (0 - 90)2, 01~ lay-up. This class of laminates is designed to exhibit extension-twist coupling with no initial warping due to curing stresses. The laminate geometry and material properties are given in Table 4. It is seen from equation (105) that the induced twist and bending curvature are expressed in terms of the stiffness oij, which is dependent on the delamination length and its location. The induced twist C normalized by se/H is plotted against the delamination length a, in Figure 4, for a laminate with a mid-plane delamination and 0 = 20°. The ply thickness is denoted by H in the figure. For zero delamination length, the induced twist can be obtained from 1 ure 4 indicates that the extension-twist coupling Classical Lamination Theory (CLT) [22]. F’g decreases as the delamination length increases. Further investigation of the effect of free-edge delamination on the extension-twist coupling is provided through the variations of the extension-twist coupling parameter &d. Free-edge delamination is assumed at one interface at a time starting from the bottom interface or Interface 1 between 0 and (0 - 90) up to the top interface or Interface 7 between (90 - 0) and -8. The angle 8 varies from 10” to 80” at 10“ intervals. Parameter ,&sd normalized by flisU is shown in Figure 6 through 4. A delamination length of 15 ply thickness is assumed for all cases. The numerical values used to plot Figure 6 through 4 are provided in Table 4 for convenience. The values of&, MCM 19-3/4-H
J. LI AND E. A. ARMANIOS
102
0.22 -
Table 1. Geometry and material prop erties of AS4-3502 graphite/epoxy.
Err=113.31
GPa (16.44 Msi)
E22=E33=9.73
GPa (1.41 Msi)
Grs=Grs=5.66
GPa (0.82 Msi)
&s=3.40
GPa (0.50 Msi)
0.16 -
vrs=vrs=O.25 vss=O.42
0.14
Ply thickness H=0.14 mm (0.0055 in) Semi-width
0
.
'_I.
5
' 15
IO
b=68H.
-
20
-
'. 25
' 30
-
' 35
am
Figure 5. Extension-twist coupling as a function of mid-plane free-edge delamination length for 0 = 20°.
Table 2. Normalized extension-twist
Table 3. Extension-twist
e P16u
coupling E
(a=15H).
coupling parameter prsU (Deg./N).
10
20
30
40
.00291
.00401
.00367
.00164
50 -.00164
60 -.00367
70 -.00401
80 -.00291
are given in Table 4. The symmetrical response shown in Figure 6 through 4 and Table 4 about Interface 4 is a result of the applied uniform extensional loading. Figure 6 through 4 indicate that the influence of free-edge delamination on the extension-twist coupling depends on the interface location and the lay-up angle 8. For the case of 8 = 30”, the extension-twist coupling decreases irrepective of the interface in which free-edge delamination occurs. For B = 40°, the induced twist changes directions when the delamination is placed at Interface 2 or 6. For 0 = 50”, the extension-twist coupling can increase by as much as 350%.
5. COMPARISON
WITH
TEST DATA
The analytical predictions are compared to test data for [-30/602/-30/30/-60~/30]~ laminates with mid-plane and off mid-plane delaminations. The delaminations were created by a 3.175 mm (0.125in) Teflon FEP film placed along the length of the laminates at two locations. The first, placed at the mid-plane free edges (Interface 4), while the second, symmetrically placed at 60/60 and -6O/ - 60 interfaces (Interfaces 2 and 5). A schematic of these damaged laminates appears in Figure 14. The delamination was symmetrically stacked in the second configuration, in
Free-Edge
I
2
Delaminations
3
4
5
103
6
7
Interface Number Figure
6. Influence
of free-edge
I
delamination
2
3
on extension-twist
4
5
6
coupling
for 0 = 10’.
coupling
for 0 = 20°.
coupling
for 0 = 30°.
coupling
for 8 = 40’.
7
Interface Number Figure
7. Influence
of free-edge
delamination
on extension-twist
1.0
5
0.9
e z 0.8
0.7 I
2
3
4
5
6
7
Interface Number Figure
8. Influence
of free-edge
delamination
on extension-twist
1.0 2
‘0
a a
?!
0.5
0.0
4.5
4
-1.0
I
2
3
4
5
6
7
Interface Number Figure
9. Influence
of free-edge
delamination
on extension-twist
J. LI
104
I
2
AND
E. A. ARMANIOS
3
4
5
6
7
Interface Number Figure
10. Influence of free-edge delamination
I
2
3
on extension-twist
4
5
6
coupling
for 0 = 50”.
coupling
for fJ = 60’.
coupling
for 0 = 70°.
coupling
for 0 = 80’.
7
Interface Number Figure
11. Influence
of free-edge delamination
on extension-twist
1.5
5
1.0
Q 5! m 0.5
0.0 1
2
3
4
5
6
7
Interface Number Figure
12. Influence
of free-edge delamination
1
2
3
on extension-twist
4
5
6
7
Interface Number Figure
13. Influence
of free-edge delamination
on extension-twist
Free-Edge Delaminations
a
I-I
-W-l+ -_
105
2b
--_
(a) Mid-plane damage
(b) Off mid-plane damage Figure 14.
order to avoid warping due to curing stresses. The laminates were fabricated using AS413502 graphite/epoxy material whose properties are given in Table 5. A custom loading transducer was used in the test [27]. The transducer was designed to allow freedom of twist at the end of the specimen while undergoing axial load. It was found however, that the transducer causes a torsional restraining moment which was proportional to the applied extensional force. The equivalent loading condition for the portion of the test specimen away from the ends can be represented by equation (113). The coefficient of proportionality for the torsional restraining moment, J!Jis fO.04331 mm (0.001705 in) for all the test cases considered. For the class of laminates with [-0, (90 - e),, -19, 6, (6 - 9O)s, e]~, layup ?I, is negative for 0 5 8 I 45” and positive for 45’ 5 0 5 90’. The comparison between analytical predictions and test results is shown in Table 5. For mid-plane delamination, the analytical prediction shows a reduction of 18.0% which is 2.7% lower than the test data. However, for off-mid plane delsmination the analysis predicts 27.5% larger reduction than the test result. For the case of off-mid plane delamination, the tested specimens included an additional delamination symmetrically placed along each free edge as shown in Figure 14(b). However, due to the antisymmetrical distribution of the interlaminar normal stresses about the mid-plane, one delamination front on each free-edge will be subjected to tensile normal stresses while the other to compressive normal stresses. The contact problem associated with the latter, is not considered in the analysis. This contributes to the relatively larger discrepancy of extension-twist reduction between analytical prediction and test data for the off mid-plane case. Table 4. Material properties for extension test specimens. &1=126.1GPa
Table 5. Percentage of coupling reduction.
(18.3Msi)
&2=&,3=11.6
GPa (1.68 Msi)
G12=G13=5.45
GPa (0.79 Msi)
C&3=3.27 GPa (0.47 Msi) V12=Vls=O.31 V23
=0.53
Ply thickness H=0.13 mm (0.005 in) Semi-width
b=90H.
6. CONCLUSION An analytical model for the free edge delamination analysis of unsymmetrical laminates is developed. The model is applied to the study of the influence of free edge delamination on the extension-twist coupling in a [-0, (90 - e)s, -8, 8, (0 - 9O)s, e]T class of laminates made of graphite/epoxy material. It is found that the extension-twist coupling is influenced by the extent of the delamination. Moreover, free edge delamination may decrease or increase the coupling depending on the location of the delamination and the lay-up angle, 8, of the laminate. Test
106
J. LI AND E. A. ARMANIOS
data confirm the coupling reduction trend predicted by the analysis for the cases of a laminate with mid-plane and off-mid plane delamination and are in excellent agreement with the analytical prediction for the case of mid-plane delamination.
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6. 7. 8. 9. 10. 11.
12.
13.
14. 15. 16. 17.
18. 19.
20.
21.
22. 23.
24.
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S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, pp. 103-108, Holden-Day, Inc., San Francisco, (1963). J.R. Vinson and R.L. Sierakowski, The Behavior of Structures Composed of Composite Materials, pp. 46-47, Martinus Nijhoff Publishers, (1986). E.A. Armanios, D. Hooke, M.P. Kamat, D. Palmer and J. Li, Design and testing of composite laminates for Testing and Design, ASTM optimum extension-twist coupling, 11th Symposium on Composite Materials: STP 1206, (Edited by E. Camponeshi), American Society for Testing and Materials, Philadelphia (to appear).