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Bulging of cracked panel in extension due to unbalanced patching
Theoretical and Applied Fracture Mechanics 19 (1993) 13-27 Elsevier
13
Bulging of cracked panel in extension due to unbalanced patching R.C. Chu Aviation and Space Industry Development Administration 1, 333 Keelung Rd., Sec. 1, Taipei, Taiwan, ROC
Y.S. Lin Institute of Applied Mathematics, National Chung-Hsing University, Taichung, Taiwan, ROC
When a cracked panel is patched only on one side, there is a tendency for the reinforced panel to bulge even if the load is applied in the plane. The Mindlin plate theory is applied to examine such an effect. Considered are variations of transverse shear across the adhesive and transverse normal stresses through the sheet and patch. The formulation makes use of the variational principle for determining the constitutive relations. Finite element is employed to discretize the system of equations and to solve for the displacements of cracked panels with unbalanced patching. Stress intensity factor solutions and stresses are obtained and compared well with available numerical results and experimental data.
1. Introduction Defects in aircraft structures are known to exist from manufacturing a n d / o r overload in service. Many of them are hidden and become detectable only when they are sufficiently large in locations conducible to inspection. Aged aircrafts are particularly vulnerable to structure failure by fatigue crack growth. Recent advances in repair technology have made possible to restore the structural integrity of aircrafts that otherwise would be retired. The discipline of fracture mechanics plays a central role in the development of this technology. It is based on the concept of redirecting the load away from the cracked region. A reduction in the local stress intensity or energy level would prevent the crack from further growth. Conventional repair method for cracked aircraft structures involve welding, drilling holes at the crack tips and attaching metal reinforcing patches with rivets or bolts. These techniques can be problematic as they can introduce additional stress concentration and corrosion if the electrochemical properties of the metal patch are not properly matched with those of the aircraft structure. Moreover, the fatigue strength of the metal patch is limited. Recent advances in the adhesive bonding of fibrous composites suggest the possibility of developing more efficient repair techniques [1-8]. In particular, the boron-fiber-reinforced plastic (BFRP) patches have had much success in repairing cracked aircraft components. Since 1975, over 300 repairs have been made on the Hercules wing plank made of 7075-T6 aluminum which suffered stress corrosion cracking. The fatigue lives of the Macchi landing wheel were more than doubled after patching. Successful extension of fatigue lives was also made on the lower wing skin of Mirage III and fuselage bulkhead flange of AT-3 [7]. Bonded BFRP repairs are also most cost-effective because simple molding technique can be used to patch cracks in curved sections while cracks can also be more easily monitored through the patch. The shear spring element were employed [2,3] to analyze cracks in adhesively bonded panels. Assumed in [4-7] are the linear variations of the adhesive shear stress across the thickness. Partially debonded patches were also analyzed [8] whereby a patch effectiveness index was defined to rank the load reduction capacity of the reinforced structure. Up to now, all of the analyses considered symmetric 1 Also at National Chung-HsingUniversity, Taichung, Taiwan, Republic of China. 0167-8442/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
14
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
patch across the thickness so that out-of-plane effect is ignored. In service, patch is administered only on one side of the panel and there is the tendency for the reinforced structure to bulge even if the sheets are thin a n d / o r the loads are applied in the plane. The influence of such an effect on patching is the subject of this investigation. In what follows, a finite element procedure is developed to analyze the bulging of cracked panel reinforced on one side. Mindlin's plate theory will be used to account for variations of the adhesive transverse shear and transverse normal stresses across the panel thickness. The variational principle is applied to determine the constitutive relations in the adhesive layers. Results are discussed and compared with those obtained by other investigators.
2. Preliminaries Consider a composite patch of thickness hp bonded to an elastic sheet of thickness h s by an adhesive layer of thickness h a as shown in Fig. 1. The coordinate x~3k) is directed along the thickness such that (x l, x 2) coincide with the plane of the panel and adhesive, respectively. The superscript k could stand for p, s and a which refer, respectively, to the panel, sheet and adhesive.
2.1. Patch and sheet Let u~k), %(k) and ~/~k) denote for the displacements, strains and stresses respectively for k = p and s. Given are the in-plane stress and strain relations:
{
orl,~(k)
[ e u ~(k'
or227
= [D](S<){ e22 )
0"12J
t 2e12 ]
'C13/C33 3s -I- C23/0C33/ or(k)
(1)
such that e ~ ) = ½[u~kf, + U~k )],,,j
(2)
The material matrix [D] (k) is given by
[-Cll Cl2
0 [
G2
C66 ]
[D] Lo
0
hp
Patch
ha
Adhesive
+
hs
(3)
o
Panel
• x I .x2
lxta)
• Xl,X 2
l x~s)
• Xl,X 2
Fig. 1. Coordinate systemfor patch, adhesive and panel.
R.C. Chu, Y.S. Lin / Bulging of crackedpanel in extension
15
where C}~) are related to the elastic constants. For the patch, [D]
=
(k),
~(k) = ~33
,/.13//C55
~ C33 ]
.(t,) _ ell k C33
.(k) + t:22 ~ C33
(4)
2.2. Adhesive A linear displacement field is assumed in the adhesive: 1
,i = ~a [u~P)(x,, X2, - h p / 2 ) -
u~S)(x,, X2, hs/2)]
(5)
Note that E1 and e 2 are the shear strain and e 3 stands for the transverse normal strain. The strain-stress relations in the adhesive are:
T( a ) = a(J)E}a),
0"3(a) = G(3)e(3a)
(6)
in which z) a) are the shear stresses and a3~) is the transverse normal stress in the adhesive while G
Mindlin's plate theory
According to the plate bending theory of Mindlin, the in-plane displacements displacement u~3k) for k = p, s are given by
u~k)(xl, X2, x(3k)) = uj(k)( xI, X2) + x(3k)w(k)( xI, X2),
u~.k) and out-of-plane
u(k)(Xl, X2, x(3k)) = u(3k'( xI, X2)
(7) where U/k) and w) k) are the displacements of the mid-surface of the patch for k = p and sheet for k = s. A transverse shear stress and transverse normal stress field distribution through the thickness of the sheet and patch which satisfies the boundary condition are taken to be in the patch
-(P)[ ,.- X2, x(3P)) .3jI.~I,
2hoe j
1
]/+~) a'3
3
-- __x(P) 3 _
2 3 ~33rr(P)/~Xl~ X2 ' x(P)) = 4v3 1,-,-(a)[9 _ 6_X(p) / _ _ x ( P ) I ] [~ 'Ihp 3 thp 3]J
(8)
and in the sheet 3--~O(S)[1 ~-(s)/x 3jl~ l' X2' x(S') .--_ 2hs~.i [ _
"2 2 __x
1 1 hp 3 ] ] + , [J1 hs 3 ] + --X(3 hs s) -- -~ 2
33 ~, 1,
,_
,
1 2 r~s)[a/__x(S)t
"l- hsX(3S'--
3
]
(9)
hs 3 ] J
where Q~k) are shear force in the sheet and patch:
Q~k) =
f hkf2
~(k.) dx~3k)
J_hk/2 "31
(10)
16
R. C. Chu, Y.S. Lin / Bulging of cracked panel in extension
3. Governing equations
Constitutive relations for the adhesive can be developed by application of the variational principle for the boundary value problem at hand. For a one-sided patch, it can be shown that
(11)
2
The quantities Ctk’ with j = 1, 2 stand for J Clk’= Cg),
C$“) = ~45”’
(12)
The cross-sectional areas for k = p, s are denoted by kk) and A occupies the mid-plane in which (x,, x2) are defined. Tractions $‘) are applied on the portion of A denoted by A* with outward unit normal vector n. Application of eqs. (1) to (10) and Gauss’s theorem, eq. (11) yields the equilibrium equations, boundary conditions and constitutive equations that include appropriate shear corrections. 3.1. Equilibrium equations Let Iv;“’ and IV);’ denote, respectively, the stress resultants and moments: jvf)
=
jhk”
&)
dX$“‘,
Mj/? =
j;;‘;2uhk’x”
”
-h,/2
dXik’
(13)
k
The equilibrium equations can thus be written as
(14)
3.2. Boundary conditions Boundary conditions that are consistent with the constraint of the problem are 6U.W = 0: I
k!k’ = N!kjn. 1 lJ
I’
aW!k)
I
=
0.
.
fi!k)
I
=
M(k)n. I/
J
(15)
in which &‘=
jhx’2 f(k) dX$k’, -h,/2
’
&W = j-;:/:L@Wx$k) dX$k’ k
(16)
R.C. Chu, Y.S. Lin / Bulging of crackedpanel in extension
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3.3. Constitutive equations Recall from eqs. (7) that U/k) and wff ) refer to the mid-surface of the sheet and panel; they depend only on (xl, Xz). The variational formulation yields the relations between O Ck) and aUC3P)/~xjor U(p) 3,i. They are given by Q}p)=
[5_h
C(p) +
1 h2/_/(j)]
--1 h h /-/(J)[u,(s)
U (s)]
+ ~hpHU)[ U / p ) _ ½hpw~p)_ ~.(s) _ ½hswff)] Q)s)=
w(p) --I- U (p)] --I--[5hsC(S) _}_i-~o~s--1 h2/4-(j)]] [ w ( s ) + U(S)13,j]
l_~hphsH(J)[
+ ~h~H
(17)
U / s ) - 7"'~"i l h ~"'~)11
Moreover, T(a) and or3~a) take the forms "r(a) =
lhpH(')([w~ p)+ U3tp)]
0r(a) = hpH(3) (
+ hs [w{S)+ hp t Y
1+ 3.Yl
12 [u/p)- lhpw) p)- u/s)- 'h ,,,
~
1 [/...,(p)/l_l[(p ) ± h W(p)'~ -4- t"(P)/1/'[(P) ---1/1 u,(p)~l ~,avl,1 -- lO p 1,1 / --'-'23 ~ 2"-'2,2 -- 10')p"2,211
~'~["'13
(18)
"'33
-[- C(~)
["-'13 1,2~..'1,1 - lO"s"l,1]
"-'23 1,2'.-~2,2 + ~',sW2,2]]
Introduced in eqs. (17) and (18) are the following contractions: 1 1 hp 1 hs ha H
1
13 hp
n (3)
13 h s
ha
35 "-33t~(P)4- 35 C3(~ )+ - - G(3)
(19)
Recall that G (j) and G (3) are defined in eqs. (6). Once ~jo(k), ~.~a) and or3
W
involving the parameters hk, H (3) and
Mi~k)=M
(20)
expressions can be found in Appendix A.
4. Finite element formulation
The total strain energy of an element of the repaired structure can be written as
JA
dXl
(21)
18
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension 13
1
11
9
5
Fig. 2. Schematicof isoparametricelement. The strain tensor {~} are defined in terms of the displacement gradients: = [
mo~ u ~ + m ~ w~~' + u ~3 , 1
~
w~, . , ~
w ~ + w~
w~~ + mo~ u ( ~ _ ±~ .,~.~- - up~ ~'3,2 ~ 2'~p'" 1
--
,1,
,.,.,
~r~srVl
U ( p ) - 12 ~" p'~ , A2, ( P ) - U2(s) -- 2'*sty lla ,,6,(S) U 2,2~ (s) 2 ~ U 3 ( P ) - U3(S), UI(S?, ,
u~,sg + v~??, ,.,,"~ . , c . ,.,.~ + .,.~ w?~ + u 3,1, ~s~ " 1 , 1 , " 2 , 2 , rVl,2 "2,1~
w~s) + U3,2
(22)
while the stress tensor {¢} in terms of stress resultants and moments: {~r}T = [N(~'), N2(~), Nl(~'), M~ '), ""22a'~(0),M w),2, Q~P), Q(2°),
~?,, ~a~, ~.~, X:~', n~', N,% M~', M ~ , M ~ ), Q]~), Q~2~)]
(23)
The displacement vector {~} for the repair structure which is defined in terms of [U/(k)]~ and [w~k)]~ in /-direction (j = 1. . . . . n), with n = 16 being the number of nodes for the sheet and patch, as illustrated in Fig. 2. That is, {~}T= [(Uw))T, (U~))T ' (Ujp))T, (U(~))~,..., (U(sp))T ' (U(~))T]
(24)
such that ,,
,,
2 , , , (v3,%,
(2s)
The matrix [B] relates eqs. (22) and (24) as {~} = [B]{~}
(26)
Refer to Appendix B for the specific expression of [B] which is a 19 × 80 matrix. It follows from eqs. (17), (18) and (20) that {o-} = [De]{e}
(27)
with [D e] being a 19 × 19 matrix which can be found in Appendix C. Without going into details, the surface integral in eq. (21) can be expressed in the form fA{E}T{or} dx I dx 2 = {(~}TtKe]{(~}
(28)
the matrix [K e] provided that
[g e] = f-,'_ l i l-'_ l 1[ B]T[ De][ n] det[ J] d~ d.
(29)
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
19
The Jacobian matrix is [J] which is used to transform the physical coordinates (xl, x 2) to the curvilinear coordinates (~, -q) that take the values of + 1. Such details can be found in [10].
5. Discussion of results
Three different arrangements of crack patching will be examined. In all cases, the patch will be bonded only to one side of the panel. Asymmetry across the mid-plane of the panel will cause the patched panel to bulge even if the load is applied in the original plane of (x 1, x2).
5.1. Edge crack Consider a rectangular sheet made of aluminum with dimensions of 12 in x 4 in x 0.1 in. It contains an edge crack of length a = 0.59 in as shown in Fig. 3. The region near the crack is reinforced by bonding a semi-circular patch with radius b = 2.3 in and thickness hp = 0.035 in. The patch is made from a g r a p h i t e / e p o x y laminate whose fiber direction is perpendicular to the crack. The adhesive has thickness h a = 0.009 in, Young's modulus E a = 0.261 x 103 ksi and Poisson's ratio/'a 0.4. Anisotropic properties given by =
E l l = 2 1 . 4 × 103 ksi,
E22 = E33 = 1 . 3 2 × 10 3 ksi
G12 = G13 = 0 . 6 9 × 10 3 ksi,
G23 = 0 . 2 9 X 10 3 ksi,
(30) /'12 = 0 . 2 9
The Young's modulus and Poisson's ratio of the sheet are, respectively, E s = 10.3 X 103 ksi and/'~ = 0.32. It is stretched uniformly in direction normal to the crack with an applied stress of tr = 20 ksi. Obtained are the normal strain ell in the x~-direction at eight (8) different locations as indicated in Fig. 3. Their values, summarized in Table 1, are compared with those found in [5] and from experiments. The effect of bulging is considered to be important as the present results are in closer agreement with the experimental data for unbalanced patching than those in [5] that did not consider such an effect.
5.2. Reinforcement exposing the crack ends A rectangular sheet of photoelastic material with dimensions of 13.38 in x 2.95 in x 0.12 in contains a center crack of length 2a = 1 in as shown in Fig. 4. Bonded to the center region of the panel is a patch strip with thickness hp = 0.096 in, width w = 0.236 in and length 2Lp ranging from 0.787 in to 4.40 in. The patch is made from a glass/epoxy laminate with the following elastic properties: E l l = 2 × 10 3 ksi,
E22 = E33 = 0 . 5 1 X 10 3 ksi,
G12 = G13 = G23 = 0 . 1 8 7 × 10 3 ksi,
(31)
/'12 = 0 . 3 3
x2
2.36 in ~----~t/--GREP Area
2.36 in
1 -26 I / "I 3-1- -~,s/ I-
12 in
"l
Fig. 3. Edged cracked sheet reinforced by semi-circular patch.
xl
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
20
Table 1 Normal strain ell at eight different locations in patched panel No.
Coordinate (in)
Strain etl × 10 6 (in/in) Present
Experiment
Reference [5]
1 2 3 4 5* 6 7 8*
(2.56,2.00) (0,3.35) (0,0.12) (1.57,0.12) (1.57,0.12) (0,1.57) (0.98,0.98) (0.98,0.98)
2217 1497 - 18 708 1563 917 988 1674
2042 1646 97 674 1588 898 961 1722
1905 1790 2517 1081 1106 1268 1264 1266
* On the back side.
x2
°
l
° --
xl
n
I " ~ 2L p ~ " I
I~
13.38
-I
in
Fig. 4. Central cracked sheet reinforced by patch strip.
1.2
1.0
0.8
C•7-Experiment [10]
@
Present method
0.6
D Finite element [5]
0.4
0.2
0
. . . .
i
,
,
,
,
I
1
i
,
,
2
,
I
,
,
3
,
i
I
,
,
,
,
4
I
S
Lp/a
Fig. 5. Normalized stress intensity factor versus patch length ratio.
G
G
x2 Location B 7
l x1
~
- 6 in
20 in
'1
Location A
I"
25 in
"1
.1
Fig. 6. Central cracked sheet reinforced by rectangular patch.
21
R.C. Chu, Y.S, Lin / Bulging of cracked panel in extension 20
IS
\
Finite element [6]
~
$
1
2
3,
4
5
hp x 10"2 (in)
Fig. 7. Variation of maximum normalizedpatch stress with patch thickness. The fiber direction is placed normal to the crack. Note that both crack ends are exposed because the patch is narrower than the crack length. The thickness of the adhesive is h a - - 4 × 10 -4 in while the Young's modulus and Poisson's ratio of the sheet are E s = 0.45 X 103 ksi and vs --- 0.36. Let K~p) and K~°) denote, respectively, the Mode I stress intensity factor with and without the patch. The ratio K~P)/K~°) is plotted against the normalized patch length Lp/a in Fig. 5. The experimental data in [11] are also given by the open circles. Again, the present results including correction for unbalanced patching agreed much better with experiments in contrast to those obtained by finite element calculations [5] without taking into account the effect of bulging.
5.3. Finite crack with complete patching The center crack of length 2a = 1.5 in in Fig. 6 is patched completely by a reinforcement sheet of 6 in x 2 in. Dimensions of the aluminum panel are 25 in × 20 in x 0.09 in. The patch is made from a unidirectional b o r o n / e p o x y laminate such that the reinforced fiber direction is perpendicular to the crack. The thickness of patch is an n-ply laminate, each ply of laminate is 5 x 10-3 in. The material properties of patch are El] = 30.2 x 10 3 ksi,
E22 = E33 = 3.692 X 103 ksi,
Gl2 = G13 = 1.05 x 103 ksi,
G23 = 0.716 x 103 ksi,
(32) V12 = 0 . 1 6 7 7
Both the sheet and adhesive are assumed to be isotropic and homogeneous. The adhesive is h a = 4 x 10 -3 in thick; it has a Young's modulus E a -- 392 ksi and Poisson's ratio va -- 0.4. The Young's modulus and Poisson's ratio of the aluminum are E s - 10.3 x 103 ksi and vs = 0.32. Displayed in Fig. 7 is the decay maximum normal stress ratio ,T(p)/,, •" 1 1 / v with increasing patch thickness hp. The two upper curves obtained from [6,8] gave higher values than the present results. An overestimate of "--(P) would result if bulging were not considered. Vll Plotted in Fig. 8 is the ratio ~'~a)/o" in the adhesive layer as a function of the patch thickness hp. The data refer to the region marked A near the crack center as shown in Fig. 6. As compared with the results in [6,8], the same trend is obtained with the present results being the lowest. Near the path edge region marked B in Fig. 6, ~'~a)/tr increased with increasing hp. This applied to the results in [6,8] as well except that they attained higher values. The transverse normal stress tr~)/tr increased with hp at both locations A and B. This is exhibited by the two curves in Fig. 10. Finally, the ratio K~P)/K~°) is computed, the
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
22
1.4
i
1.2 ! Finite element [6] 1.0
i
0.8
0.6
D " ~ ' ~ ~
0.4
hod
D
-~
0.2 Of . . . . 0
i , 1
,
,
L i i 2
,
i
,
i , 3
i
h ,
,
,
,
i
~1
i 5
hp x 10-2 (in)
Fig. 8. Normalized transverse shear stress versus patch thickness in the adhesive.
1,0
0.8
0.6
t~
Spring element [8] 0,4
,.,t IOl 0.2
0
i
i
L
0
i
i
i
i
i
1
,
i
. . . .
i
2
. . . .
i
3
. . . .
i
4
5
hp x 10"2 (in) Fig. 9. Normalized transverse shear stress versus patch thickness in the adhesive.
0.5
0.4
0.3
t~ t~
0.2
0.1
1
2
3
4
5
hp x 10"2 (in) Fig. 10. Normalized transverse normal stress versus patch thickness in the adhesive
23
R. C. Chu, YS. Lin / Bulging of cracked panel in extension
bpring
o-
0
“‘I
element
1 “,““““““““’2
IO] 3
hp x 10.’
4
5
(in)
Fig. 11. Normalized stress intensity factor versus patch thickness.
numerical values of which are shown in Fig. 11 as h, is increased. Although the trend is the same as the two lower curves obtained from the data in [6,8], the deviation attributed to bulging is significant.
6. Conclusions Presented in this work is a finite element formulation of the problem of crack patching by application of the plate bending theory of Mindlin. Particular attention is given to the bulging cracked panel when it is patched only on one side even if the load is applied in the plane and the thickness of the patch and sheet are thin. A reduction in the efficiency of patching results with increasing out-of-plane deflection. Such a conclusion is validated by comparing the present findings with experimental data and results that do not consider the effect of unbalanced patching.
Appendix A: Resultant
stress and moment
relations
Outlined below are the expressions of TV:!) and i14jji”)in terms of the mid-plane displacement gradients L$tF)and w,$) for k = p, s while the superscript a refers to the adhesive. The subscripts j = 1, 2 refer to the mid-plane x1.x2 and 3 to the direction xg along the thickness. Introducing the contraction r.k’ = H(3)h,h,A\!‘$!’ lJ
J
(3%
with $9J = q”‘/q:’
(34)
the stress resultants in the panel are
(35)
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
24
N(2p)= [~ l ' ~ p "P' 1( P2 )
F P P ] UJ I ( , ~ ) -I- [ h p C ( 2 p) + 1 F 2 ~ P ] /'-'2,2 r(p) + !4--12
-- 2 ~ h p [ P P P , , , ( P ) . . L P P P , , , ( P ) ] Jr- 1 [ / - , p s / T ( s ) j_ r, ps/T(s)] --12 rVl,1 -- • 22 v'2,2,1 ~ [ ~ 21 '-'1,1 -- " 22 ~2,2,1
+ ~6"s]/,I.[r'psu,(s), 21 ,r 1,1 Jr --22r'PS' ],v2,2.1 ~'(s)
Jr 2"p''I/"/-./(3)}l(P)[ I T ( PI.'-'3 ),,2
-- U3(s)]
N(p> = / t', p~-f " (66 p ) l [l TL-~ ( p1,2 ) + U(,~)]
(36) (37)
and in the sheet are
N[[)
1 [ r P s u ( P ~ 4-/-'psTT(P)I !h { F'PSu,(p) -4- p p s u , ( p ) ] = ~ t l] ],1 -- " 21 " 2 , 2 '1 -- 2o"p,1--11 "'1,1 -- " 21 "'2,2,1 q-[h
F'(s) -{- 1 I'SS]
"s"" ' 1
4--11,1Ul(S?
l /-'SS] T/'(S) [hsC~)+'~-12,1"-'2,2
+
It, [ r, ss,~,(s) + - -pss. (s)l 12w2,21
+~'%[--11'"1,1
1 [ F, pslT(p ) ..~/-,pslT(p)] N2(~ ) = 4 [ - - 1 2 Wl,1 - " 22 'J2,2 I -
_lh l-4(3):ds)[tt(p) 1 t'-'3
Jr 2 " s . . . .
~ . rr,s~,,(s)
U3(S)]
(38)
I h [ /.-,psu,(p ) 4. F, psu,(p)] 2 o " p [ - - 1 2 '1"1,1 - " 22 " 2 , 2 J
+[hsC{S2)+~_12lU~)l+ lr sl + 26'%[~t 12'"1,1 + "
--
+
1 r'ss] 22j U(s) 2,2
r s ~ . (s)] 1/i l../(3)~(s)l'TT(p ) 22W2,2] + 2 ' ) s . . . . 2 ["3
U3(S)] -
(39)
-
N(~) -- - z, r.(s)t.(~) + V(~)] " s ' 6 6 [ t'a 1,2 2,1,1
(40)
Similarly, the moments in the panel are found: M(p)
11 =
-- l h [ /"PP/T(P) 4-/"PPl/(P)] 1 / , 2 [ 1/, f-,(p) 1/-,pp]..,,(p) 20'~pL--11 "1,1 - - - - 1 2 '-'2,2,1 + 2'~p|'6'~pX-'ll + 50--11 J " l , 1 1 / , 2 [ 1_/, t~(p) 1 F ' p P ] u,(P) + 2 " p [ 6 " p ~ - ' 1 2 + 50--12 ] ' v 2 , 2 --
~hp[
/"PS/'T(S) ..[- r ' p s / / ( s ) ] 11 'Jl,1 --12t"2,2,1
1/, /, [ F, psu,(s) Jr r~pst4,(s) ] + 1/~2I..I(3)]L(P)[/'T(p) lOO"p'~s[~ll "1,1 --12"2,2] l6r~p xx ~1 [I-'3 g(p)
U(S)]
(41)
= _ ~ o h p [ PPPTT(P) .a_ F'PPlT(P)] 1 / , 2 [ 1/t t-~(p) 4- l _ / ~ p p ] , , , ( p ) --12 '-'1,1 - - " 22 ~2,2,1 + 2'~Pt6'~pX-'12 -- 50--12 ] " 1 , 1 1 / , 2 1 1 / , f~(p)_L I T ' p P ] u , ( P ) 1 / , [ r , psl/-(s) ppsi/(s) l + 2"*p[ 6 " p ' - ' 2 2 -- 50" 22 ] " 2 , 2 -- 2-6',p[ L 21 '-'1,1 + - 22 ~2,21
rrps.,(~)21 "1,1
~"p"s['l/,/,
+ "F'PSu'(S) ] 2 2"v2,2]] -I- "~-6t, p ,t'2 , ] l-/(3)l(P)[ U 32( P[ ) , v . -- U3(s)]
M~p) = ~1t ~~,3c,(p)[w(p) w(p) ] p " 6 6 [ 1,2 + 2,1,1
(42)
(43)
Those in the sheet are given by g[~)
=
__1h [ F p s u ( p ) + p p s l l ( p ) l 20 s [ 11 1,1 " 21 " 2 , 2 1
-
-
1 +~h~[F~[U(]~)]+r~tr(ol + --12v2,2.1
1 /, h [ p p s , , , ( p ) + p p s u , ( p ) ] lO0"p"s,1--11 'Vl,1 " 21 " ' 2 , 2 1 1/4211/,, f'(S) 1 /-'ss]. (s) ~ s [ 6 ~ s ~ " l l + 5"0--11,1Wl,1
1/,2[ 1/,, f , ts) 1 / - s s l , , , ( s ) a- ] / , 2 L r ( 3 ) ~ ( s ) l / r ( p ) U3(S)] + 2 " s L6"'s~"12 + -5"6--12 ] '"2,2 -- 1-6'~s ~ "~1 ["-'3 --
M(2~2) =
½h[FP~U(p) + F'pslT(p)] L 12 1,1 " 22'-'2,2,1
_ 1__/1 /, [ r, ps,,,(p) + r'psu,(p)] 1 0 0 " p " s [ - - 1 2 "Vl,1 " 22 "v2,2.1
+½hs[F~U(~+rssm(s) 1 " 22"2,2'1 +
1 / . 2 1 1 / . f'(S)_l_ 1 /-'SS]u,(S ) 2~S ['6t~S"12 -- ~ - - 1 2 ] '"1,1
1/.,21" 1/,, /"'(s) 1 r, ss] ,,,(s) 1 /,21../(3)It(s)[ iT(p ) U3(S)] + 2"'~s [ 6 ~ s ~ ' 2 2 + 5"6" 22,1 "v2,2 + "1"6"s ~'. '"2 L " 3 --
M~) = ~1/,3r~(s)[~,(s) w(S)] " s ~ 6 6 ,1 "" 1,2 + 2,1,1
(44)
(45) (46)
Appendix B: Stress-displacement matrix
The matrix [B] in eq. (26) is given by [B]
= [bl,
b 2,
b 3. . . . .
b8]19x8 o
(47)
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
25
in which -E 1
0 ] (48) E1
19x10
The quantities E (j -- 1, 2, 3) stand for
aN,
0
0
0
0
Ox 1
oN,
0
0
0
~)x2
aN,
ON,
0X 2
0X 1
0
0
0
0
0
aN;
0
0
0x 1 E 1 --~
(49)
a~ 0
0
0
0 ax 2
a~ 0
0
o
~x 2
~)x 1
ON,. 0
0
N,.
0
0
N,
ax 1
0
oN,
0
ax 2
E2=
8x5
0
0
0
Ni
0
0
0
0
N,.
0
E3 =
-N/ 0
(50)
- 5hpN 0
0
1 - ~-h~N,
0 -N/
0 0
3×5
1° 1
- ~h~N,. 0
(51)
3)<5
In eqs. (48) to (51), Nj are known as the shape functions [10].
Appendix
C: S t r e s s - s t r a i n
matrix
The relations between the stress tensor {or} and strain tensor {~} in eq. (27) are determined by the material matrix:
[D e ] =
-d1
0
d6
d7
0
0
d2
0
0
d9
dT
0
d3
d8
0
aT
o
d~ d,
0
0
d9T
0
0
d5
(52) 19× 19
R.C. Chu, Y.S. Lin / Bulging o f cracked panel in extension
26
The parameters d~, d 2 , . . . , d 9 are given by + I F~IP
hpClf )
0
-- I
-- 1
pp
t, f'(P) ~_ 'tp*-'22 -- _l4*F'PP 22
0
_
-- I
pp
0
hpC~ )
h ~-,(p) _,. ! F ' P P p~12 " 4--12 dl=
0 --
pp 2ohpFll 1 pp 20hpF12
hpC}~ ) + ~Fl°ff
~hprlPl ~
_1
pp 2ohpFl2
-
_ _l
pp 2ohpffl2
0
½hpF~P
0
0 + ~OFl~p] !~,2 [ $~, t-(p) 2#*p[.6,.p~12 + ~0/~1~p]
P'pl.g"p~ll
0
0
5_/, (~(p) ..~ 1 / , 1 2 / 4 0 ) 6tt p~'55 1~4, ~p **
0
0
5h t"(P)+ 6'*p~44
0
lhpH(1)
0
0
(53)
0
0
0
2ohpr~12
l h 2 [ lh f'(P)
~, r~(P) 12.* p'.-~66
d2=
2ohpFll
r~(p) 2,,p16,,p~12 + ~ r ~ 2 p ] 1/.2[ 1/,, f,(p)
±,,2r!a
0 lhpH(1) 1 /42/_/(2) "i'~'" p**
0
0 ~hpH
0 0 lhpH(2)
H (1)
0
0
H (2)
(2)
(55)
d 3 = [ H (3)]
h F(S)+ lrss
- h /'~'(s)..~ 1 p s s '*s~ll 4--11
1 1, rss
0
h C(S)+ ~rss
s~12 4--12 h /--,(s)+ 1y, ss s'.-q2 4 - 22
0
-- 2"0'" s--11 1/4 F s s - ~'%--12
0
0
/4 r(s) )~s~66
0
s~12
d4=
4"--12
1/, --
~t,s_ll 12
0
1 /.3(~(s)
I4(3)
~hs + 1 2 [ 1/4 (~(s) Shs g " s " " 12 + 1 2 [ 1/4 /",(s) 6 ' ~ s " - " 11
1 /4 y, ss ~'*s--12 1/4 p s s 2"0' % * 22
0 5"0--11J
1 y'ss]
1/,2[ 1/, /-,(s) -.I- 1 y, ss] ~ * * s [ 6 ' t s " 1 2 -- ~ - - 1 2 ]
1 --12 /'~SS]J 5"0
1/,2[ 1/4 /",(s) --I- 1 p s s ] 2 ' i s [6*%*'22 -- ~ * 22]
(56)
,IT 6 h
0
1/1 y, ss -- ~-6,%.t 22
--
-- ~t%a
d 6 = p.p..
1/, y'ss ~'6,~s--12
F, ss
1/,I p s s
[-
(54)
0
0
~hsC(sS5) + hZsn(1)
0
0
5/4 (-,(s) -.I- 1 h2/../(2) g " s " ~ 4 4 -- l - ~ ' * s * *
(57)
(58)
0
- ½hpA(lp' _
d7= -
~_/4
),(P)
5.~p~.2
1p p s --11 1 F.ps 4" 21
1F, PS 4"~12 I F , PS 4"* 22
0
0
lhprpS
1/, -- ~"
y.ps
d 8 = 2hsH(3)[A(~ ), A~), 0, g'%"1 -0 0 0 0 lhpH(1) 0 d9=
l~h s
½hsr
0 0
~hsFff( 0
~oh,Fff2~ 0
1 /1 /,I y . p s -- "i-~,~p,%--ll 1 /, /, y, ps -- ~ * + p * ~ s * 21
1 /.I /, F.ps -- T~,~p,~s--12 1 /.. /t y ' p s -- T~r~p*+s * 22
0
p*12
-~hpF~ ~
- ½hpFff
o
0 '
0
0
0
H (1)
0
0
0
H (2)
g'%"2 I
~ 2 h p H (2)
s
½hsr
s
(59)
(60)
(61)
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
27
Acknowledgement This research was supported by the National Science Council, Republic of China, under Contract N S C 81-0401-E-005-501.
References [1] A.A. Baker and R. Jones, eds., Bonded Repair of the Aircraft Structures, Engineering Application of Fracture Mechanics, Vol. 7, G.C. Sih, ed. (Martinus Nijhoff: Dordrecht, The Netherlands, 1988). [2] M.M. Ratwani, Analysis of cracked adhesively bonded laminated structures, AIAA 17, 988-994 (1979). [3] A.S. Kuo, A two-dimensional shear spring element, AIAA 22, 1460-1464 (1984). [4] R.A. Mitchell, R.M. Wooley and D.J. Chivirut, Analysis of composite reinforced cut-outs and cracks, AIAA 13, 744-749 (1975). [5] R. Jones and R.J. Callinan, Finite element analysis of patched crack, J. Struct. Mech. 7, 107-130 (1970). [6] R. Jones and R.J. Callinan, A design study in crack patching, Fibre. Sci. Tech. 14, 99-111 (1981). [7] R.C. Chu, C.L. Ong, S. Shen and T.C. Ko, Composite materials bonding technology, AIDC-ARL Technical Report No. 76C-013, 1984. [8] R.C. Chu and T.C. Ko, Isoparametric shear spring element applied to crack patching and instability, Theoretical and Applied Fracture Mechanics 11, 93-102 (1989). [9] G.C. Sih and T.B. Hong, Integrity of edge-debonded patch on cracked panel, Theoretical and Applied Fracture Mechanics 12, 121-139 (1989). [10] O.C. Zienkiewicz, The Finite Element Method, 3rd ed. (McGraw-Hill: New York, 1977). [11] R. Chandra and A. Subramanian, Stress-intensity factors in plates with a partially patched central crack, Experimental Mechanics 29, 1-5 (1989).
13
Bulging of cracked panel in extension due to unbalanced patching R.C. Chu Aviation and Space Industry Development Administration 1, 333 Keelung Rd., Sec. 1, Taipei, Taiwan, ROC
Y.S. Lin Institute of Applied Mathematics, National Chung-Hsing University, Taichung, Taiwan, ROC
When a cracked panel is patched only on one side, there is a tendency for the reinforced panel to bulge even if the load is applied in the plane. The Mindlin plate theory is applied to examine such an effect. Considered are variations of transverse shear across the adhesive and transverse normal stresses through the sheet and patch. The formulation makes use of the variational principle for determining the constitutive relations. Finite element is employed to discretize the system of equations and to solve for the displacements of cracked panels with unbalanced patching. Stress intensity factor solutions and stresses are obtained and compared well with available numerical results and experimental data.
1. Introduction Defects in aircraft structures are known to exist from manufacturing a n d / o r overload in service. Many of them are hidden and become detectable only when they are sufficiently large in locations conducible to inspection. Aged aircrafts are particularly vulnerable to structure failure by fatigue crack growth. Recent advances in repair technology have made possible to restore the structural integrity of aircrafts that otherwise would be retired. The discipline of fracture mechanics plays a central role in the development of this technology. It is based on the concept of redirecting the load away from the cracked region. A reduction in the local stress intensity or energy level would prevent the crack from further growth. Conventional repair method for cracked aircraft structures involve welding, drilling holes at the crack tips and attaching metal reinforcing patches with rivets or bolts. These techniques can be problematic as they can introduce additional stress concentration and corrosion if the electrochemical properties of the metal patch are not properly matched with those of the aircraft structure. Moreover, the fatigue strength of the metal patch is limited. Recent advances in the adhesive bonding of fibrous composites suggest the possibility of developing more efficient repair techniques [1-8]. In particular, the boron-fiber-reinforced plastic (BFRP) patches have had much success in repairing cracked aircraft components. Since 1975, over 300 repairs have been made on the Hercules wing plank made of 7075-T6 aluminum which suffered stress corrosion cracking. The fatigue lives of the Macchi landing wheel were more than doubled after patching. Successful extension of fatigue lives was also made on the lower wing skin of Mirage III and fuselage bulkhead flange of AT-3 [7]. Bonded BFRP repairs are also most cost-effective because simple molding technique can be used to patch cracks in curved sections while cracks can also be more easily monitored through the patch. The shear spring element were employed [2,3] to analyze cracks in adhesively bonded panels. Assumed in [4-7] are the linear variations of the adhesive shear stress across the thickness. Partially debonded patches were also analyzed [8] whereby a patch effectiveness index was defined to rank the load reduction capacity of the reinforced structure. Up to now, all of the analyses considered symmetric 1 Also at National Chung-HsingUniversity, Taichung, Taiwan, Republic of China. 0167-8442/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
14
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
patch across the thickness so that out-of-plane effect is ignored. In service, patch is administered only on one side of the panel and there is the tendency for the reinforced structure to bulge even if the sheets are thin a n d / o r the loads are applied in the plane. The influence of such an effect on patching is the subject of this investigation. In what follows, a finite element procedure is developed to analyze the bulging of cracked panel reinforced on one side. Mindlin's plate theory will be used to account for variations of the adhesive transverse shear and transverse normal stresses across the panel thickness. The variational principle is applied to determine the constitutive relations in the adhesive layers. Results are discussed and compared with those obtained by other investigators.
2. Preliminaries Consider a composite patch of thickness hp bonded to an elastic sheet of thickness h s by an adhesive layer of thickness h a as shown in Fig. 1. The coordinate x~3k) is directed along the thickness such that (x l, x 2) coincide with the plane of the panel and adhesive, respectively. The superscript k could stand for p, s and a which refer, respectively, to the panel, sheet and adhesive.
2.1. Patch and sheet Let u~k), %(k) and ~/~k) denote for the displacements, strains and stresses respectively for k = p and s. Given are the in-plane stress and strain relations:
{
orl,~(k)
[ e u ~(k'
or227
= [D](S<){ e22 )
0"12J
t 2e12 ]
'C13/C33 3s -I- C23/0C33/ or(k)
(1)
such that e ~ ) = ½[u~kf, + U~k )],,,j
(2)
The material matrix [D] (k) is given by
[-Cll Cl2
0 [
G2
C66 ]
[D] Lo
0
hp
Patch
ha
Adhesive
+
hs
(3)
o
Panel
• x I .x2
lxta)
• Xl,X 2
l x~s)
• Xl,X 2
Fig. 1. Coordinate systemfor patch, adhesive and panel.
R.C. Chu, Y.S. Lin / Bulging of crackedpanel in extension
15
where C}~) are related to the elastic constants. For the patch, [D]
=
(k),
~(k) = ~33
,/.13//C55
~ C33 ]
.(t,) _ ell k C33
.(k) + t:22 ~ C33
(4)
2.2. Adhesive A linear displacement field is assumed in the adhesive: 1
,i = ~a [u~P)(x,, X2, - h p / 2 ) -
u~S)(x,, X2, hs/2)]
(5)
Note that E1 and e 2 are the shear strain and e 3 stands for the transverse normal strain. The strain-stress relations in the adhesive are:
T( a ) = a(J)E}a),
0"3(a) = G(3)e(3a)
(6)
in which z) a) are the shear stresses and a3~) is the transverse normal stress in the adhesive while G
Mindlin's plate theory
According to the plate bending theory of Mindlin, the in-plane displacements displacement u~3k) for k = p, s are given by
u~k)(xl, X2, x(3k)) = uj(k)( xI, X2) + x(3k)w(k)( xI, X2),
u~.k) and out-of-plane
u(k)(Xl, X2, x(3k)) = u(3k'( xI, X2)
(7) where U/k) and w) k) are the displacements of the mid-surface of the patch for k = p and sheet for k = s. A transverse shear stress and transverse normal stress field distribution through the thickness of the sheet and patch which satisfies the boundary condition are taken to be in the patch
-(P)[ ,.- X2, x(3P)) .3jI.~I,
2hoe j
1
]/+~) a'3
3
-- __x(P) 3 _
2 3 ~33rr(P)/~Xl~ X2 ' x(P)) = 4v3 1,-,-(a)[9 _ 6_X(p) / _ _ x ( P ) I ] [~ 'Ihp 3 thp 3]J
(8)
and in the sheet 3--~O(S)[1 ~-(s)/x 3jl~ l' X2' x(S') .--_ 2hs~.i [ _
"2 2 __x
1 1 hp 3 ] ] + , [J1 hs 3 ] + --X(3 hs s) -- -~ 2
33 ~, 1,
,_
,
1 2 r~s)[a/__x(S)t
"l- hsX(3S'--
3
]
(9)
hs 3 ] J
where Q~k) are shear force in the sheet and patch:
Q~k) =
f hkf2
~(k.) dx~3k)
J_hk/2 "31
(10)
16
R. C. Chu, Y.S. Lin / Bulging of cracked panel in extension
3. Governing equations
Constitutive relations for the adhesive can be developed by application of the variational principle for the boundary value problem at hand. For a one-sided patch, it can be shown that
(11)
2
The quantities Ctk’ with j = 1, 2 stand for J Clk’= Cg),
C$“) = ~45”’
(12)
The cross-sectional areas for k = p, s are denoted by kk) and A occupies the mid-plane in which (x,, x2) are defined. Tractions $‘) are applied on the portion of A denoted by A* with outward unit normal vector n. Application of eqs. (1) to (10) and Gauss’s theorem, eq. (11) yields the equilibrium equations, boundary conditions and constitutive equations that include appropriate shear corrections. 3.1. Equilibrium equations Let Iv;“’ and IV);’ denote, respectively, the stress resultants and moments: jvf)
=
jhk”
&)
dX$“‘,
Mj/? =
j;;‘;2uhk’x”
”
-h,/2
dXik’
(13)
k
The equilibrium equations can thus be written as
(14)
3.2. Boundary conditions Boundary conditions that are consistent with the constraint of the problem are 6U.W = 0: I
k!k’ = N!kjn. 1 lJ
I’
aW!k)
I
=
0.
.
fi!k)
I
=
M(k)n. I/
J
(15)
in which &‘=
jhx’2 f(k) dX$k’, -h,/2
’
&W = j-;:/:L@Wx$k) dX$k’ k
(16)
R.C. Chu, Y.S. Lin / Bulging of crackedpanel in extension
17
3.3. Constitutive equations Recall from eqs. (7) that U/k) and wff ) refer to the mid-surface of the sheet and panel; they depend only on (xl, Xz). The variational formulation yields the relations between O Ck) and aUC3P)/~xjor U(p) 3,i. They are given by Q}p)=
[5_h
C(p) +
1 h2/_/(j)]
--1 h h /-/(J)[u,(s)
U (s)]
+ ~hpHU)[ U / p ) _ ½hpw~p)_ ~.(s) _ ½hswff)] Q)s)=
w(p) --I- U (p)] --I--[5hsC(S) _}_i-~o~s--1 h2/4-(j)]] [ w ( s ) + U(S)13,j]
l_~hphsH(J)[
+ ~h~H
(17)
U / s ) - 7"'~"i l h ~"'~)11
Moreover, T(a) and or3~a) take the forms "r(a) =
lhpH(')([w~ p)+ U3tp)]
0r(a) = hpH(3) (
+ hs [w{S)+ hp t Y
1+ 3.Yl
12 [u/p)- lhpw) p)- u/s)- 'h ,,,
~
1 [/...,(p)/l_l[(p ) ± h W(p)'~ -4- t"(P)/1/'[(P) ---1/1 u,(p)~l ~,avl,1 -- lO p 1,1 / --'-'23 ~ 2"-'2,2 -- 10')p"2,211
~'~["'13
(18)
"'33
-[- C(~)
["-'13 1,2~..'1,1 - lO"s"l,1]
"-'23 1,2'.-~2,2 + ~',sW2,2]]
Introduced in eqs. (17) and (18) are the following contractions: 1 1 hp 1 hs ha H
1
13 hp
n (3)
13 h s
ha
35 "-33t~(P)4- 35 C3(~ )+ - - G(3)
(19)
Recall that G (j) and G (3) are defined in eqs. (6). Once ~jo(k), ~.~a) and or3
W
involving the parameters hk, H (3) and
Mi~k)=M
(20)
expressions can be found in Appendix A.
4. Finite element formulation
The total strain energy of an element of the repaired structure can be written as
JA
dXl
(21)
18
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension 13
1
11
9
5
Fig. 2. Schematicof isoparametricelement. The strain tensor {~} are defined in terms of the displacement gradients: = [
mo~ u ~ + m ~ w~~' + u ~3 , 1
~
w~, . , ~
w ~ + w~
w~~ + mo~ u ( ~ _ ±~ .,~.~- - up~ ~'3,2 ~ 2'~p'" 1
--
,1,
,.,.,
~r~srVl
U ( p ) - 12 ~" p'~ , A2, ( P ) - U2(s) -- 2'*sty lla ,,6,(S) U 2,2~ (s) 2 ~ U 3 ( P ) - U3(S), UI(S?, ,
u~,sg + v~??, ,.,,"~ . , c . ,.,.~ + .,.~ w?~ + u 3,1, ~s~ " 1 , 1 , " 2 , 2 , rVl,2 "2,1~
w~s) + U3,2
(22)
while the stress tensor {¢} in terms of stress resultants and moments: {~r}T = [N(~'), N2(~), Nl(~'), M~ '), ""22a'~(0),M w),2, Q~P), Q(2°),
~?,, ~a~, ~.~, X:~', n~', N,% M~', M ~ , M ~ ), Q]~), Q~2~)]
(23)
The displacement vector {~} for the repair structure which is defined in terms of [U/(k)]~ and [w~k)]~ in /-direction (j = 1. . . . . n), with n = 16 being the number of nodes for the sheet and patch, as illustrated in Fig. 2. That is, {~}T= [(Uw))T, (U~))T ' (Ujp))T, (U(~))~,..., (U(sp))T ' (U(~))T]
(24)
such that ,,
,,
2 , , , (v3,%,
(2s)
The matrix [B] relates eqs. (22) and (24) as {~} = [B]{~}
(26)
Refer to Appendix B for the specific expression of [B] which is a 19 × 80 matrix. It follows from eqs. (17), (18) and (20) that {o-} = [De]{e}
(27)
with [D e] being a 19 × 19 matrix which can be found in Appendix C. Without going into details, the surface integral in eq. (21) can be expressed in the form fA{E}T{or} dx I dx 2 = {(~}TtKe]{(~}
(28)
the matrix [K e] provided that
[g e] = f-,'_ l i l-'_ l 1[ B]T[ De][ n] det[ J] d~ d.
(29)
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
19
The Jacobian matrix is [J] which is used to transform the physical coordinates (xl, x 2) to the curvilinear coordinates (~, -q) that take the values of + 1. Such details can be found in [10].
5. Discussion of results
Three different arrangements of crack patching will be examined. In all cases, the patch will be bonded only to one side of the panel. Asymmetry across the mid-plane of the panel will cause the patched panel to bulge even if the load is applied in the original plane of (x 1, x2).
5.1. Edge crack Consider a rectangular sheet made of aluminum with dimensions of 12 in x 4 in x 0.1 in. It contains an edge crack of length a = 0.59 in as shown in Fig. 3. The region near the crack is reinforced by bonding a semi-circular patch with radius b = 2.3 in and thickness hp = 0.035 in. The patch is made from a g r a p h i t e / e p o x y laminate whose fiber direction is perpendicular to the crack. The adhesive has thickness h a = 0.009 in, Young's modulus E a = 0.261 x 103 ksi and Poisson's ratio/'a 0.4. Anisotropic properties given by =
E l l = 2 1 . 4 × 103 ksi,
E22 = E33 = 1 . 3 2 × 10 3 ksi
G12 = G13 = 0 . 6 9 × 10 3 ksi,
G23 = 0 . 2 9 X 10 3 ksi,
(30) /'12 = 0 . 2 9
The Young's modulus and Poisson's ratio of the sheet are, respectively, E s = 10.3 X 103 ksi and/'~ = 0.32. It is stretched uniformly in direction normal to the crack with an applied stress of tr = 20 ksi. Obtained are the normal strain ell in the x~-direction at eight (8) different locations as indicated in Fig. 3. Their values, summarized in Table 1, are compared with those found in [5] and from experiments. The effect of bulging is considered to be important as the present results are in closer agreement with the experimental data for unbalanced patching than those in [5] that did not consider such an effect.
5.2. Reinforcement exposing the crack ends A rectangular sheet of photoelastic material with dimensions of 13.38 in x 2.95 in x 0.12 in contains a center crack of length 2a = 1 in as shown in Fig. 4. Bonded to the center region of the panel is a patch strip with thickness hp = 0.096 in, width w = 0.236 in and length 2Lp ranging from 0.787 in to 4.40 in. The patch is made from a glass/epoxy laminate with the following elastic properties: E l l = 2 × 10 3 ksi,
E22 = E33 = 0 . 5 1 X 10 3 ksi,
G12 = G13 = G23 = 0 . 1 8 7 × 10 3 ksi,
(31)
/'12 = 0 . 3 3
x2
2.36 in ~----~t/--GREP Area
2.36 in
1 -26 I / "I 3-1- -~,s/ I-
12 in
"l
Fig. 3. Edged cracked sheet reinforced by semi-circular patch.
xl
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
20
Table 1 Normal strain ell at eight different locations in patched panel No.
Coordinate (in)
Strain etl × 10 6 (in/in) Present
Experiment
Reference [5]
1 2 3 4 5* 6 7 8*
(2.56,2.00) (0,3.35) (0,0.12) (1.57,0.12) (1.57,0.12) (0,1.57) (0.98,0.98) (0.98,0.98)
2217 1497 - 18 708 1563 917 988 1674
2042 1646 97 674 1588 898 961 1722
1905 1790 2517 1081 1106 1268 1264 1266
* On the back side.
x2
°
l
° --
xl
n
I " ~ 2L p ~ " I
I~
13.38
-I
in
Fig. 4. Central cracked sheet reinforced by patch strip.
1.2
1.0
0.8
C•7-Experiment [10]
@
Present method
0.6
D Finite element [5]
0.4
0.2
0
. . . .
i
,
,
,
,
I
1
i
,
,
2
,
I
,
,
3
,
i
I
,
,
,
,
4
I
S
Lp/a
Fig. 5. Normalized stress intensity factor versus patch length ratio.
G
G
x2 Location B 7
l x1
~
- 6 in
20 in
'1
Location A
I"
25 in
"1
.1
Fig. 6. Central cracked sheet reinforced by rectangular patch.
21
R.C. Chu, Y.S, Lin / Bulging of cracked panel in extension 20
IS
\
Finite element [6]
~
$
1
2
3,
4
5
hp x 10"2 (in)
Fig. 7. Variation of maximum normalizedpatch stress with patch thickness. The fiber direction is placed normal to the crack. Note that both crack ends are exposed because the patch is narrower than the crack length. The thickness of the adhesive is h a - - 4 × 10 -4 in while the Young's modulus and Poisson's ratio of the sheet are E s = 0.45 X 103 ksi and vs --- 0.36. Let K~p) and K~°) denote, respectively, the Mode I stress intensity factor with and without the patch. The ratio K~P)/K~°) is plotted against the normalized patch length Lp/a in Fig. 5. The experimental data in [11] are also given by the open circles. Again, the present results including correction for unbalanced patching agreed much better with experiments in contrast to those obtained by finite element calculations [5] without taking into account the effect of bulging.
5.3. Finite crack with complete patching The center crack of length 2a = 1.5 in in Fig. 6 is patched completely by a reinforcement sheet of 6 in x 2 in. Dimensions of the aluminum panel are 25 in × 20 in x 0.09 in. The patch is made from a unidirectional b o r o n / e p o x y laminate such that the reinforced fiber direction is perpendicular to the crack. The thickness of patch is an n-ply laminate, each ply of laminate is 5 x 10-3 in. The material properties of patch are El] = 30.2 x 10 3 ksi,
E22 = E33 = 3.692 X 103 ksi,
Gl2 = G13 = 1.05 x 103 ksi,
G23 = 0.716 x 103 ksi,
(32) V12 = 0 . 1 6 7 7
Both the sheet and adhesive are assumed to be isotropic and homogeneous. The adhesive is h a = 4 x 10 -3 in thick; it has a Young's modulus E a -- 392 ksi and Poisson's ratio va -- 0.4. The Young's modulus and Poisson's ratio of the aluminum are E s - 10.3 x 103 ksi and vs = 0.32. Displayed in Fig. 7 is the decay maximum normal stress ratio ,T(p)/,, •" 1 1 / v with increasing patch thickness hp. The two upper curves obtained from [6,8] gave higher values than the present results. An overestimate of "--(P) would result if bulging were not considered. Vll Plotted in Fig. 8 is the ratio ~'~a)/o" in the adhesive layer as a function of the patch thickness hp. The data refer to the region marked A near the crack center as shown in Fig. 6. As compared with the results in [6,8], the same trend is obtained with the present results being the lowest. Near the path edge region marked B in Fig. 6, ~'~a)/tr increased with increasing hp. This applied to the results in [6,8] as well except that they attained higher values. The transverse normal stress tr~)/tr increased with hp at both locations A and B. This is exhibited by the two curves in Fig. 10. Finally, the ratio K~P)/K~°) is computed, the
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
22
1.4
i
1.2 ! Finite element [6] 1.0
i
0.8
0.6
D " ~ ' ~ ~
0.4
hod
D
-~
0.2 Of . . . . 0
i , 1
,
,
L i i 2
,
i
,
i , 3
i
h ,
,
,
,
i
~1
i 5
hp x 10-2 (in)
Fig. 8. Normalized transverse shear stress versus patch thickness in the adhesive.
1,0
0.8
0.6
t~
Spring element [8] 0,4
,.,t IOl 0.2
0
i
i
L
0
i
i
i
i
i
1
,
i
. . . .
i
2
. . . .
i
3
. . . .
i
4
5
hp x 10"2 (in) Fig. 9. Normalized transverse shear stress versus patch thickness in the adhesive.
0.5
0.4
0.3
t~ t~
0.2
0.1
1
2
3
4
5
hp x 10"2 (in) Fig. 10. Normalized transverse normal stress versus patch thickness in the adhesive
23
R. C. Chu, YS. Lin / Bulging of cracked panel in extension
bpring
o-
0
“‘I
element
1 “,““““““““’2
IO] 3
hp x 10.’
4
5
(in)
Fig. 11. Normalized stress intensity factor versus patch thickness.
numerical values of which are shown in Fig. 11 as h, is increased. Although the trend is the same as the two lower curves obtained from the data in [6,8], the deviation attributed to bulging is significant.
6. Conclusions Presented in this work is a finite element formulation of the problem of crack patching by application of the plate bending theory of Mindlin. Particular attention is given to the bulging cracked panel when it is patched only on one side even if the load is applied in the plane and the thickness of the patch and sheet are thin. A reduction in the efficiency of patching results with increasing out-of-plane deflection. Such a conclusion is validated by comparing the present findings with experimental data and results that do not consider the effect of unbalanced patching.
Appendix A: Resultant
stress and moment
relations
Outlined below are the expressions of TV:!) and i14jji”)in terms of the mid-plane displacement gradients L$tF)and w,$) for k = p, s while the superscript a refers to the adhesive. The subscripts j = 1, 2 refer to the mid-plane x1.x2 and 3 to the direction xg along the thickness. Introducing the contraction r.k’ = H(3)h,h,A\!‘$!’ lJ
J
(3%
with $9J = q”‘/q:’
(34)
the stress resultants in the panel are
(35)
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
24
N(2p)= [~ l ' ~ p "P' 1( P2 )
F P P ] UJ I ( , ~ ) -I- [ h p C ( 2 p) + 1 F 2 ~ P ] /'-'2,2 r(p) + !4--12
-- 2 ~ h p [ P P P , , , ( P ) . . L P P P , , , ( P ) ] Jr- 1 [ / - , p s / T ( s ) j_ r, ps/T(s)] --12 rVl,1 -- • 22 v'2,2,1 ~ [ ~ 21 '-'1,1 -- " 22 ~2,2,1
+ ~6"s]/,I.[r'psu,(s), 21 ,r 1,1 Jr --22r'PS' ],v2,2.1 ~'(s)
Jr 2"p''I/"/-./(3)}l(P)[ I T ( PI.'-'3 ),,2
-- U3(s)]
N(p> = / t', p~-f " (66 p ) l [l TL-~ ( p1,2 ) + U(,~)]
(36) (37)
and in the sheet are
N[[)
1 [ r P s u ( P ~ 4-/-'psTT(P)I !h { F'PSu,(p) -4- p p s u , ( p ) ] = ~ t l] ],1 -- " 21 " 2 , 2 '1 -- 2o"p,1--11 "'1,1 -- " 21 "'2,2,1 q-[h
F'(s) -{- 1 I'SS]
"s"" ' 1
4--11,1Ul(S?
l /-'SS] T/'(S) [hsC~)+'~-12,1"-'2,2
+
It, [ r, ss,~,(s) + - -pss. (s)l 12w2,21
+~'%[--11'"1,1
1 [ F, pslT(p ) ..~/-,pslT(p)] N2(~ ) = 4 [ - - 1 2 Wl,1 - " 22 'J2,2 I -
_lh l-4(3):ds)[tt(p) 1 t'-'3
Jr 2 " s . . . .
~ . rr,s~,,(s)
U3(S)]
(38)
I h [ /.-,psu,(p ) 4. F, psu,(p)] 2 o " p [ - - 1 2 '1"1,1 - " 22 " 2 , 2 J
+[hsC{S2)+~_12lU~)l+ lr sl + 26'%[~t 12'"1,1 + "
--
+
1 r'ss] 22j U(s) 2,2
r s ~ . (s)] 1/i l../(3)~(s)l'TT(p ) 22W2,2] + 2 ' ) s . . . . 2 ["3
U3(S)] -
(39)
-
N(~) -- - z, r.(s)t.(~) + V(~)] " s ' 6 6 [ t'a 1,2 2,1,1
(40)
Similarly, the moments in the panel are found: M(p)
11 =
-- l h [ /"PP/T(P) 4-/"PPl/(P)] 1 / , 2 [ 1/, f-,(p) 1/-,pp]..,,(p) 20'~pL--11 "1,1 - - - - 1 2 '-'2,2,1 + 2'~p|'6'~pX-'ll + 50--11 J " l , 1 1 / , 2 [ 1_/, t~(p) 1 F ' p P ] u,(P) + 2 " p [ 6 " p ~ - ' 1 2 + 50--12 ] ' v 2 , 2 --
~hp[
/"PS/'T(S) ..[- r ' p s / / ( s ) ] 11 'Jl,1 --12t"2,2,1
1/, /, [ F, psu,(s) Jr r~pst4,(s) ] + 1/~2I..I(3)]L(P)[/'T(p) lOO"p'~s[~ll "1,1 --12"2,2] l6r~p xx ~1 [I-'3 g(p)
U(S)]
(41)
= _ ~ o h p [ PPPTT(P) .a_ F'PPlT(P)] 1 / , 2 [ 1/t t-~(p) 4- l _ / ~ p p ] , , , ( p ) --12 '-'1,1 - - " 22 ~2,2,1 + 2'~Pt6'~pX-'12 -- 50--12 ] " 1 , 1 1 / , 2 1 1 / , f~(p)_L I T ' p P ] u , ( P ) 1 / , [ r , psl/-(s) ppsi/(s) l + 2"*p[ 6 " p ' - ' 2 2 -- 50" 22 ] " 2 , 2 -- 2-6',p[ L 21 '-'1,1 + - 22 ~2,21
rrps.,(~)21 "1,1
~"p"s['l/,/,
+ "F'PSu'(S) ] 2 2"v2,2]] -I- "~-6t, p ,t'2 , ] l-/(3)l(P)[ U 32( P[ ) , v . -- U3(s)]
M~p) = ~1t ~~,3c,(p)[w(p) w(p) ] p " 6 6 [ 1,2 + 2,1,1
(42)
(43)
Those in the sheet are given by g[~)
=
__1h [ F p s u ( p ) + p p s l l ( p ) l 20 s [ 11 1,1 " 21 " 2 , 2 1
-
-
1 +~h~[F~[U(]~)]+r~tr(ol + --12v2,2.1
1 /, h [ p p s , , , ( p ) + p p s u , ( p ) ] lO0"p"s,1--11 'Vl,1 " 21 " ' 2 , 2 1 1/4211/,, f'(S) 1 /-'ss]. (s) ~ s [ 6 ~ s ~ " l l + 5"0--11,1Wl,1
1/,2[ 1/,, f , ts) 1 / - s s l , , , ( s ) a- ] / , 2 L r ( 3 ) ~ ( s ) l / r ( p ) U3(S)] + 2 " s L6"'s~"12 + -5"6--12 ] '"2,2 -- 1-6'~s ~ "~1 ["-'3 --
M(2~2) =
½h[FP~U(p) + F'pslT(p)] L 12 1,1 " 22'-'2,2,1
_ 1__/1 /, [ r, ps,,,(p) + r'psu,(p)] 1 0 0 " p " s [ - - 1 2 "Vl,1 " 22 "v2,2.1
+½hs[F~U(~+rssm(s) 1 " 22"2,2'1 +
1 / . 2 1 1 / . f'(S)_l_ 1 /-'SS]u,(S ) 2~S ['6t~S"12 -- ~ - - 1 2 ] '"1,1
1/.,21" 1/,, /"'(s) 1 r, ss] ,,,(s) 1 /,21../(3)It(s)[ iT(p ) U3(S)] + 2"'~s [ 6 ~ s ~ ' 2 2 + 5"6" 22,1 "v2,2 + "1"6"s ~'. '"2 L " 3 --
M~) = ~1/,3r~(s)[~,(s) w(S)] " s ~ 6 6 ,1 "" 1,2 + 2,1,1
(44)
(45) (46)
Appendix B: Stress-displacement matrix
The matrix [B] in eq. (26) is given by [B]
= [bl,
b 2,
b 3. . . . .
b8]19x8 o
(47)
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
25
in which -E 1
0 ] (48) E1
19x10
The quantities E (j -- 1, 2, 3) stand for
aN,
0
0
0
0
Ox 1
oN,
0
0
0
~)x2
aN,
ON,
0X 2
0X 1
0
0
0
0
0
aN;
0
0
0x 1 E 1 --~
(49)
a~ 0
0
0
0 ax 2
a~ 0
0
o
~x 2
~)x 1
ON,. 0
0
N,.
0
0
N,
ax 1
0
oN,
0
ax 2
E2=
8x5
0
0
0
Ni
0
0
0
0
N,.
0
E3 =
-N/ 0
(50)
- 5hpN 0
0
1 - ~-h~N,
0 -N/
0 0
3×5
1° 1
- ~h~N,. 0
(51)
3)<5
In eqs. (48) to (51), Nj are known as the shape functions [10].
Appendix
C: S t r e s s - s t r a i n
matrix
The relations between the stress tensor {or} and strain tensor {~} in eq. (27) are determined by the material matrix:
[D e ] =
-d1
0
d6
d7
0
0
d2
0
0
d9
dT
0
d3
d8
0
aT
o
d~ d,
0
0
d9T
0
0
d5
(52) 19× 19
R.C. Chu, Y.S. Lin / Bulging o f cracked panel in extension
26
The parameters d~, d 2 , . . . , d 9 are given by + I F~IP
hpClf )
0
-- I
-- 1
pp
t, f'(P) ~_ 'tp*-'22 -- _l4*F'PP 22
0
_
-- I
pp
0
hpC~ )
h ~-,(p) _,. ! F ' P P p~12 " 4--12 dl=
0 --
pp 2ohpFll 1 pp 20hpF12
hpC}~ ) + ~Fl°ff
~hprlPl ~
_1
pp 2ohpFl2
-
_ _l
pp 2ohpffl2
0
½hpF~P
0
0 + ~OFl~p] !~,2 [ $~, t-(p) 2#*p[.6,.p~12 + ~0/~1~p]
P'pl.g"p~ll
0
0
5_/, (~(p) ..~ 1 / , 1 2 / 4 0 ) 6tt p~'55 1~4, ~p **
0
0
5h t"(P)+ 6'*p~44
0
lhpH(1)
0
0
(53)
0
0
0
2ohpr~12
l h 2 [ lh f'(P)
~, r~(P) 12.* p'.-~66
d2=
2ohpFll
r~(p) 2,,p16,,p~12 + ~ r ~ 2 p ] 1/.2[ 1/,, f,(p)
±,,2r!a
0 lhpH(1) 1 /42/_/(2) "i'~'" p**
0
0 ~hpH
0 0 lhpH(2)
H (1)
0
0
H (2)
(2)
(55)
d 3 = [ H (3)]
h F(S)+ lrss
- h /'~'(s)..~ 1 p s s '*s~ll 4--11
1 1, rss
0
h C(S)+ ~rss
s~12 4--12 h /--,(s)+ 1y, ss s'.-q2 4 - 22
0
-- 2"0'" s--11 1/4 F s s - ~'%--12
0
0
/4 r(s) )~s~66
0
s~12
d4=
4"--12
1/, --
~t,s_ll 12
0
1 /.3(~(s)
I4(3)
~hs + 1 2 [ 1/4 (~(s) Shs g " s " " 12 + 1 2 [ 1/4 /",(s) 6 ' ~ s " - " 11
1 /4 y, ss ~'*s--12 1/4 p s s 2"0' % * 22
0 5"0--11J
1 y'ss]
1/,2[ 1/, /-,(s) -.I- 1 y, ss] ~ * * s [ 6 ' t s " 1 2 -- ~ - - 1 2 ]
1 --12 /'~SS]J 5"0
1/,2[ 1/4 /",(s) --I- 1 p s s ] 2 ' i s [6*%*'22 -- ~ * 22]
(56)
,IT 6 h
0
1/1 y, ss -- ~-6,%.t 22
--
-- ~t%a
d 6 = p.p..
1/, y'ss ~'6,~s--12
F, ss
1/,I p s s
[-
(54)
0
0
~hsC(sS5) + hZsn(1)
0
0
5/4 (-,(s) -.I- 1 h2/../(2) g " s " ~ 4 4 -- l - ~ ' * s * *
(57)
(58)
0
- ½hpA(lp' _
d7= -
~_/4
),(P)
5.~p~.2
1p p s --11 1 F.ps 4" 21
1F, PS 4"~12 I F , PS 4"* 22
0
0
lhprpS
1/, -- ~"
y.ps
d 8 = 2hsH(3)[A(~ ), A~), 0, g'%"1 -0 0 0 0 lhpH(1) 0 d9=
l~h s
½hsr
0 0
~hsFff( 0
~oh,Fff2~ 0
1 /1 /,I y . p s -- "i-~,~p,%--ll 1 /, /, y, ps -- ~ * + p * ~ s * 21
1 /.I /, F.ps -- T~,~p,~s--12 1 /.. /t y ' p s -- T~r~p*+s * 22
0
p*12
-~hpF~ ~
- ½hpFff
o
0 '
0
0
0
H (1)
0
0
0
H (2)
g'%"2 I
~ 2 h p H (2)
s
½hsr
s
(59)
(60)
(61)
R.C. Chu, Y.S. Lin / Bulging of cracked panel in extension
27
Acknowledgement This research was supported by the National Science Council, Republic of China, under Contract N S C 81-0401-E-005-501.
References [1] A.A. Baker and R. Jones, eds., Bonded Repair of the Aircraft Structures, Engineering Application of Fracture Mechanics, Vol. 7, G.C. Sih, ed. (Martinus Nijhoff: Dordrecht, The Netherlands, 1988). [2] M.M. Ratwani, Analysis of cracked adhesively bonded laminated structures, AIAA 17, 988-994 (1979). [3] A.S. Kuo, A two-dimensional shear spring element, AIAA 22, 1460-1464 (1984). [4] R.A. Mitchell, R.M. Wooley and D.J. Chivirut, Analysis of composite reinforced cut-outs and cracks, AIAA 13, 744-749 (1975). [5] R. Jones and R.J. Callinan, Finite element analysis of patched crack, J. Struct. Mech. 7, 107-130 (1970). [6] R. Jones and R.J. Callinan, A design study in crack patching, Fibre. Sci. Tech. 14, 99-111 (1981). [7] R.C. Chu, C.L. Ong, S. Shen and T.C. Ko, Composite materials bonding technology, AIDC-ARL Technical Report No. 76C-013, 1984. [8] R.C. Chu and T.C. Ko, Isoparametric shear spring element applied to crack patching and instability, Theoretical and Applied Fracture Mechanics 11, 93-102 (1989). [9] G.C. Sih and T.B. Hong, Integrity of edge-debonded patch on cracked panel, Theoretical and Applied Fracture Mechanics 12, 121-139 (1989). [10] O.C. Zienkiewicz, The Finite Element Method, 3rd ed. (McGraw-Hill: New York, 1977). [11] R. Chandra and A. Subramanian, Stress-intensity factors in plates with a partially patched central crack, Experimental Mechanics 29, 1-5 (1989).