Integrity of edge-debonded patch on cracked panel

Theoretical and Applied Fracture Mechanics 12 (1989) 121-139

121

INTEGRITY OF EDGE-DEBONDED PATCH ON CRACKED P A N E L G.C. S I H Institute of Fracture and Solid Mechanics, Lehigh Unioersity, Bethlehem, Pennsylvania 18015, USA

and T.B. H O N G

*

Aeronautical Research Laboratory, AIDC/CSIST, Taichung, Taiwan, Republic of China

The structural integrity of reinforced cracked panels with patches that are partially damaged is analyzed by assessing the load carrying capacity and failure instability of the system, Two typical types of path edge debonding are considered; they are referred to as collinear and transverse debonding with respect to the crack plane. The former refers to debonding over a region ahead of only one of the crack tips where the load and geometry are symmetric across the crack plane while the latter is concerned with debonding over a region to the side of the crack where symmetry is no longer preserved across the crack plane. Finite elements are employed to obtain the stresses and strains from which the strain energy densities can be determined for analyzing the failure behavior of the patched panels. The local and global maximum of the minimum strain energy density function, designated by [(dW/dV)~'~]L at L and max [(dW/dV).c~ ]G at G, are found and applied to define failure instability. The distance l between L and G serves as a measure of crack instability; it increases with the debonded area. That is, debonding tends to enhance failure instability by fracture initiating from the existing crack. For approximately the same area of debonding, crack initiation for collinear debonding would be more unstable as compared with transverse debonding for loads directed normal to the crack. Introduced also is a Patch Effectiveness Index (PEI) that serves as a measure of the load carrying capacity of the damaged patch. In this case, transverse debonding is more detrimental than coninear debonding because a more significant reduction in the load transfer path occurs in the former case. In general, both l and PEI would have to be considered for assessing the integrity of the damaged patch.

1. Introduction A l l structures c o n t a i n defects o r flaws t h a t are either i n h e r e n t in the m a t e r i a l o r d e v e l o p e d d u r i n g service. W h e t h e r these i m p e r f e c t i o n s are h a r m f u l o r n o t d e p e n d o n the specific a p p l i c a t i o n . I n high p e r f o r m a n c e structures such as aircraft a n d s p a c e vehicles, even small cracks m u s t b e carefully o b served, p a r t i c u l a r l y in regions o f high m e c h a n i c a l c o n s t r a i n t where a n u n f a v o r a b l e d i s t u r b a n c e can cause the s u d d e n release o f energy b y r a p i d c r a c k p r o p a g a t i o n . Since r e p l a c e m e n t b y d i s c o n n e c t i n g a p o r t i o n o f the structure is n o t always possible, local repair, w h e n a c c o m p l i s h e d successfully, c a n restore d a m a g e d structural c o m p o n e n t s to its original level of strength a n d d u r a b i l i t y . * Formerly, Graduate Student, Department of Mechanical

Engineering and Mechanics, Lehigh University, Bethlehem, Pennsylvania 18015, USA. 016%8442/89/$3.50 © 1989, Elsevier Science Publ:Aaers B.V.

C o n v e n t i o n a l r e p a i r o f c r a c k e d metallic c o m p o n e n t s involves riveting o r b o l t i n g a n d the use of m e c h a n i c a l fasteners. W h i l e the resulting j o i n t can b e d i s a s s e m b l e d subsequently, m e c h a n i c a l a t t a c h m e n t s r e q u i r e m a c h i n i n g o f holes in the a d j o i n i n g m e m b e r s ; this w e a k e n s t h e l o a d c a r r y i n g c a p a b i l ity of the m e m b e r s a n d i n t r o d u c e s a d d i t i o n a l conc e n t r a t i o n of stresses a n d strains at the b e a r i n g surfaces. R e c e n t a d v a n c e s o n c o m p o s i t e m a t e r i a l s a n d adhesive b o n d i n g techniques have d e v e l o p e d m o r e efficient a n d effective r e p a i r techniques which are s u m m a r i z e d in [1]. I n p a r t i c u l a r , the boron-fiber-reinforced plastic (BFRP) patches had c o n t i n u e d success in the r e p a i r o f c r a c k e d aircraft structures. Since 1975, over 300 r e p a i r s have b e e n m a d e o n the H e r c u l e s w i n g p l a n k m a d e o f 7075-T6 a l u m i n u m w h i c h suffered stress c o r r o s i o n cracking. T h e fatigue lives o f t h e M a c c h i l a n d i n g wheel more than doubled after patching. Extension of fatigue lives were also m a d e o n the lower w i n g

122

G.C. Sih, 72B. Hong / Integrity of edge-debonded patch on cracked panel

skin of Mirage III and fuselage bulkhead flange of AT-3 * [2,3]. Bonded BFRP repairs are also more cost-effective because inexpensive and simple molding techniques can be used to patch cracks in curved sections while cracks can also be more easily monitored through the patch. What occurs in practice, however, is that tools or other objects are unintentionally and frequently dropped on the patch reinforcements. This causes partial debonding of the patch and can reduce its load carrying capacity. Such a concern has been the subject of many U.S. Air Force and Navy research projects. The majority of the work, however, were empirical in nature and did not provide any insights into assessing the remaining strength of patches that have been unbonded by different amounts at different locations relative to the crack. One of the objectives of this work is to define a Patch Efficiency Index (PEI) that serves as a measure of loss in load carrying capacity due to debonding. Influence on crack instability will also be analyzed by application of the strain energy density criterion [5-7] which has been employed to explain the fracture behavior of many engineering problems of practical interest in the past. This criterion applies to crack initiation, subcritical growth, and onset of unstable fracture without invoking additional assumptions. To be employed are the local and global maximum of the minimum strain energy density function that determines not only the prospective path of subcritical crack growth but also the degree of crack instability [8-10]. The redistribution of stresses, strains and strain energy densities in the patched panel are obtained from the finite element analysis. The 1/r behavior of the strain energy density field near the crack is preserved by the 1/9 and 4/9 shift of the nodes adjacent to the crack plane. Here, r stands for the distance measured from the crack tip. Since load transfer takes place from the panel to the patch via the adhesive layer in the thickness direction, the patch panel problem is three-dimensional by nature. Sufficient accuracy can be achieved by using an effective modulus for the patch, adhesive and panel. Patch debonding is simulated to occur in regions collinear and transverse to the crack.

* This is the advanced fighter pilot training jet built by the Republic of China in 1982 [4].

Since additional stress singularities are created along the debonded border of the patch, an effective crack border is defined for estimating the local intensification of the strain energy density function which can be an order of magnitude higher than that away from the crack border. Debonding does not only reduce the bearing surface for load transfer but also introduces additional concentration of energy density. Special emphases are placed on determining the oscillation of the strain energy density function whose amplitude decays with increasing distance from the crack tip. These results are displayed graphically and applied to analyzing the load carrying capacity and failure stability of the debonded patch panel.

2. Edge-debonding of patched panel with a center crack Because of the step-wise discontinuity that prevails at the edge where the patch terminates on the panel surface, the local stresses can be many times higher than the average. Aggressive environments, together with unfavorable mechanical loads, can further aggrevate the situation and lead to peeling and delamination. It is not always obvious whether the damaged patch should be completely removed and repaired. Such a decision should rest on a knowledge of the remaining load carrying capacity of the patch and the specific nature of damage. For example, debonding in regions collinear to the crack can differ appreciably from that which occurred in regions transverse to the crack. The development of predictive capability becomes necessary so that unnecessary a n d / o r premature repair can be avoided.

2.1. Collinear and transoerse edge-debonding Two types of edge-debonding will be considered. They are referred to as collinear and transverse debonding as illustrated in Figs. l(a) and l(b), respectively. Let F denote the complete patch contour of length 2~rR. Collinear debonding takes place along the arc ABC in Fig. l(a) creating a debonded area of depth d and a crack border y. The same applies to transverse debonding in Fig. l(b) where DEF is the debonded portion of F. Shown in Figs. 2(a) and 2(b) are cross-sectional

G.C. Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel ,",-

•

123 -

c

Debonded} oreD-~ ~E F

Debondedarea/ B

Patch contour 1"~ (b)Tronsverse debonding

(o) Collinear debonding

Fig. 1. Schematics of collinear and transverse debonding.

views B-B and C - C referred, respectively, to those indicated in Figs. l(a) and l(b). As a result of edge-debonding, new crack borders are created giving rise to stress singularities in addition to those for the original crack in the panel. Hence,

the intensification of the strain energy density state along the periphery of the reentrant comer F - A B C or F - D E F differs from those along the debonded border ~,. Such a distinction must be clearly recognized.

crack borders created by edoe-debonding

Original crock edge

//,--Panel edge

Adhesive

layer

a

New crack borders created _ _

~ by edge-debonding'-~/' // /

I !

'

/

I

Patch

/ ---J~--Original |!crack

~

/

_..--I ~

Adhesive layer

I

R

,

Fig. 2. Cross-sections of collincar and transverse debonding; (a) section B - B in Fig. l(a); (b) section C - C in Fig. l(b).

124

G.C. Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel

~Y

2.2. Effective debonded edge border energy density

cry

Practical application of crack patching must be supplemented not only by mechanical tests on structural details but also by a reliable stress analysis for determining the appropriate combination of the patch material and geometry for a given situation. For aircraft metallic skin structures whose thickness dimension is relatively small in comparison with those in the plane, bending effects can be neglected. Globally speaking, the patch, adhesive and panel behave effectively as a whole; the combined stiffness may be estimated. Suppose that the stress states along "t in Figs. l(a) and l(b) are obtained by considering the effective stiffness Ej* defined as ( g ; / a ) j Ap/a + (Es) j As Ej* = Ap/a + A s ,

j = x. y, z;

(1) in which ( E ; / , ) j stand for the effective stiffness of the patch and adhesive, i.e., ( E p ) j Ap + EaA a

(E~*/a)j=

Ap/a

,

j = x , y, z;

(2)

,:"

,/B"/" or

C or Y

' / ~ - ~- t !

E

z_ Debo o.onded area

Fig. 3. Effective local stress state for element near effective crack border.

If r denotes the distance normal to y at a given point as shown in Fig. 3, the strain energy density function d W / d V in a local element can be expressed in terms of a .strain energy density factor

s [5-71: dW

s

dV

r"

(5)

On the contour 3,, an effective crack length a* may be defined to yield an effective Mode I stress intensity factor k~* = o . * ~

and

Apl a = Ap + A a.

_ _ tangent

(3)

(6)

Knowing that (1 + v)(1 2v)k,~ 2E -

The orthotropic properties of the patch are reflected by the difference in the stiffness modulus in the x- and y-direction, oriented along and normal to crack plane in Fig. 1. The subscripts p, a and s refer to the patch, adhesive and panel, respectively. While the adhesive can affect load transfer in the thickness direction, its influence on the effective modulus in the xy-plane can be neglected. In this respect, it suffices to employ ext. (1) for finding the effective in plane stresses o*, Oy and r ~ everywhere in the patched zone by including the influence of patches on both sides of the panel. The contour y in Fig. 3 may then be regarded as an effective crack border. An effective transverse stress component %* can also be defined in terms of off and o* by invoking the condition of plane strain:

S=

(7)

both eqs. (5) and (6) may be applied to each point on the effective debonded edge or crack border % i.e., (dW)*~ = ( l + v ) ( l - 2 v ) ( ° * * ) ( 5 ) ~ - ~ i ,

(8)

where E* is the effective Young's modulus of the patch, adhesive and panel in the z-direction. For an edge crack * of length a*, the ratio a * / r may be taken approximately as 20 and hence, eq. (6) simplifies to (dW)* =

lO(l+v)(1-2v)(o.*) 2 E*

(9)

Note that since d W / d V in an element away from o.* = v(o~* + ay*), where v = v*.

(4) * For a central crack of length 2a, the ratio a / r is ten (10).

G.C. Sih, TB. Hong / Integrity of edge-debondedpatch on crackedpanel

~

125

4tt

/debonding

Ponel

/-Transverse

t [ I/~; / ; pa~' L ch

U12 Collinear

debondlng h

,

o,

t

t

t

Io- t 1_ ~

b

t

X

1

tit

b

Fig. 4. Plane and thickness view of patched panel; (a) plane view, (b) side view.

the crack is proportional to the local stress square divided by the modulus, the factor ten (10) in eq. (9) represents an order of magnitude increase in the strain energy density function in regions near the effective crack border.

2.3. Problem specification The finite element method will be used to obtain the stresses, strains and strain energy densities in a cracked panel that is reinforced on both

1

sides with partially debonded composite patches. The adhesive layer acts as the interface where the load is transferred from the panel to the patch. Load and Geometry. Depicted in Figs. 4(a) and 4(b) are the plane and side view of a panel 2b wide and 2h high that contains a center crack of length 2 a centered at the origin of a rectangular Cartesian coordinate system (Oxy). A circular path of radius R is bonded to both sides of the panel where debonding can occur in the x- or y-direction as explained earlier. A load of o = 50 ksi is

Table 2 Effective mechanical properties of patch and adhesive

Table Mechanical properties of 7075 A1 panel E × 106 (psi)

G x 106 (psi)

p

(E;/.)x

(E;/.)y

(E;/.)z

G;/.

× 106 (psi)

× 106 (psi)

× 106 (psi)

× 106 (psi)

10.3

3.9

0.32

0.44

3.39

0.44

0.13

,~. 0.249

G.C. Sih, 72B. Hong / Integrity of edge-debonded patch on cracked panel

126

Table 3 Effective mechanical properties of patch, adhesive and panel E* × 106 (psi)

E* ><106 (psi)

E~* × 106 (psi)

G* × 106 (psi)

p*

10.74

13.69

10.74

4.03

0.308

Y 335

347

© applied uniaxially along the y-axis such that stress state is symmetric about the x-axis for collinear debonding and no symmetry prevails if debonding occurs in the transverse direction. The dimensions of the patched panel are such that a = 1.5 in, b = 5.0 in, h = 10 in and R = 3 in. In the thickness direction, tp = 0.005 in, t a = 0.004 in and t s = 0.09 in. Material properties. The panel is made of 7075 aluminum alloy that has a yield strength Oys = 75 ksi and ultimate strength af = 98 ksi. Its Young's modulus and Poisson's ratio are given in Table 1. Given in Tables 2 and 3 are the effective mechanical properties of the patch, adhesive and panel as computed from eqs. (1) and (2). Grid patterns. The patched panel is discretized using 452 nodes and 86 elements as shown in Fig. 5. The circular boundary with nodal points 41, 187 . . . . . 188 around the complete contour represents the outer edge of the patch while the line crack lies within the patch whose tips are located in the two small squares. Enlarged views of the grid pattern for the patch and crack region are given in Figs. 6 and 7, respectively. The exact shapes of the debonded area are shaded; they approximate those shown in Fig. l(a) for collinear debonding and in Fig. l(b) for transverse debonding. Since the surfaces over the debonded area are separated, two sets of nodal points will have to be defined. Those points that overlap are distinguished in Tables 4 and 5. The left and right crack tip correspond to node 24 and 23, respectively. Since the debonded patch geometry is symmetric with respect to the x-axis for collinear d~fl~onding and to the y-axis for transverse debellding, the full grid pattern must be considered

Table 4 Overlapping nodal points for collinear debonding Panel Patch

37 416

39 447

41 448

67 449

68 450

93 451

94 452

323

@

333

@

345

@ /

501

289

277

/

201

/

~

287

299

187 @

202

200

256

@

z44x ''J 2~90~

~

@

@

@ 8@

278

502

548

199

@

~ 28

254

500

® 346

0

X

524 354 Fig. 5. Grid pattern for patched panel. 336

for obtaining the numerical results of both types of debonding.

3. Structural integrity evaluation: strain energy density criterion Failure stability of the patched panel can be analyzed by application of the strain energy density criterion [5-7] which takes into account the effects of all the stresses and strains rather than favoring on a particular component of the stress or strain. The criterion considers energy absorption by an element undergoing shape and volume change. The former can be identified with yielding or permanent deformation while the latter with fracture. Depending on the load type, structure

G.C. Sih, T.B. Hong / Integrity of edge-debondedpatch on crackedpanel

127

231 24~

~89

241

/

@ 16,5

361

42

@ I01

16~

~r

166

102

®

16z

®

3/ /

0 24'~ 252 Fig. 6. Grid pattern for patch.

automatically from the stationary values of the strain energy density function.

Table 5 Overlapping nodal points for transverse debonding Panel

Patch

Panel

Patch

243 221 207 239 235 231

454 456 458 452 450 448

217 203 233 237 241 219 205

447 446 449 451 453 455 457

3.1. Physical meaning of stationary values In general, the stresses and strains in structural components are nonuniform; they vary from one location to another. This gives rise to oscillations of the energy stored in a unit volume of material. It is not difficult to visualiT.e that when distortion dominates, volume change dV is small; this makes the quotient dW/dV a relative maximum, i.e., (dW/dV)max whereas when dilatation dominates or dV being large, dW/dV attains a relative

geometry and material, the proportion of dilatational and distortional energy in an element can vary from location to location and be determined 177

165

I01

163

175

r 3£

178

166

102 Fig. 7. Grid pattern for crack region.

164

176

G.C. Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel

128

minimum or (dW/dV)min. The locations of the prospective sites of yielding and fracture initiation can, therefore, be assumed to coincide with the locations of maximum (dW/dV)max or (dW/ dV) max and maximum of (dW/dV)min or (dW/dV)m~. If the load is made to increase, (dW/dV)m~ will first encounter a threshold that corresponds to yield initiation. This will be followed by (dW/dV)mia~ reaching the threshold (dW/dV)¢ that corresponds to fracture initiation while yield initiation corresponds to (dW/dV)m~ reaching (dW/dV)o. Since (dW/dV)o is, by deftnition, less than (dW/dV)c, yielding will always

occur prior to fracture and at a different location. 3.2. Basic hypotheses Based on the physical explanation given earlier, assumptions can be made to predict the initiation of fracture in patched panels. It can be stated that: (1) Fracture is assumed to initiate at a location that coincides with the maximum of the relative minimum strain energy density, i.e., max

(dW/dV)min.

(2) Fracture initiation occurs when (dW/dV)m~ reaches the critical value (dW/dV)c. The quantity (dW/dV)c can be obtained from the area under the uniaxial true stress and true strain curve: (

= f0'co de,

(10)

where c c is the ultimate strain of the 7075 aluminum panel that contains the crack. In accordance with eq. (5), (dW/dV)~ can be related to the critical strain energy density factor S~ as S~ rc

3.3. Local and global instability The classical linear elastic fracture mechanics approach based on the concept of Kl~ considers only the onset of rapid fracture and does not yield any information of crack stability or instability. The tendency of a crack to arrest or to propagate unstably depends on the combined influence of loading, geometry and material. Such behavior is reflected by the local and global stationary values of the strain energy density function. On physical grounds, it was argued in [7] that at each point in a nontrivial stress a n d / o r strain field, there exists at least one maximum and one minimum of dW/dV. These values are known as the local stationary values [(dW/dV)max]L and [(dW/dV) rain]L such that a new coordinate system is used for each point. When every point in the structure is referred to the same coordinate system, the resulting maxima and minima are known as the global stationary values [(dW/dV)max]G and [(dW/dV)min]G. The distances between the local and global stationary values of dW/dV can serve as a measure of the failure stability of a system by yielding a n d / o r fracture. If the discussion is limited to fracture instability, then only the distance l between [(dW/dV)m~]L, say at L, and [(dW/dV)~] G, say at G, needs to be considered; it is indicative of the degree of crack instability. The notation ( )mia~ stands for the maximum of the many minima. Illustrated in Fig. 8 is a single crack system loaded symmetrically where both L and G would then lie on the same straight line. Crack growth is predicted to initiate from L to G. A system with large l is said to be more unstable as compared with that having a smaller I.

Apptied symmetricIood mox

(11)

in which re corresponds to the critical ligament of material at the onset of rapid fracture. Hence, Sc can be related to the valid ASTM KI~ fracture toughness value by letting S---, Sc and k a Kac/ ~ in eq. (7). Typical values of Sc for many metal alloys can be found in [11]. The radius of the core region r0 is a limiting macroscopic distance as discussed earlier.

ro ~ [ ( d V ) m l n ] G J

!

~--Existing

I

"

crock

Fig. 8. Location of local and global maximum of minimum strain energy density function for symmetric loading.

G.C. Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel

o__

129

condition for stable crack growth must be known if structural reliability is to be assessed accurately. The challenge is to be able to estimate the remaining strength of damaged structural components.

,

4.1. Debonding classification

cc

I

I,

/ - Patch

I

I -al

Distance ahead of crack

Fig. 9. Concept of crack instability index.

3.4. Stability of crack patching Depending on the combined influence of load, geometry and material, failure by fracture can either be confined locally to the crack tip region or extended beyond the patch into the panel once fracture initiation has occurred. This is illustrated schematically in Fig. 9. Aside from the discontinuity of dW/dV at the patch edge, the top curve shows that G, the location of [(dW/dV)~]G, is outside the patch while the lower curve shows that G lies inside the patch. For a given patch thickness tp, l can be longer or smaller than R - a depending on the type of debonding. The location of L or [(dW/dV)mi~ max]L Occurs at ro which will be taken as 10 -2 in in the present analysis. Once L and G are known, l can be obtained to assess the influence of edge debonding on the failure instability of patched panels.

4. Discussion of results

Failure of engineering structures can occur slowly over a period of time or suddenly without warning. The transition from subcritical damage to catastrophic fracture depends on several factors that involve loading, structure geometry and material type. Precise knowledge of the terminal

Analytical prediction of the failure behavior of patch panels that are damaged at different locations by different amounts becomes necessary for the development of fracture control procedures. Referring to Fig. 6, collinear debonding occurs over element no. 71 and 81 while transverse debonding occurs over element no. 27 and 47. A detailed account of the geometric discontinuities arising from debonding has already been discussed in relation to Figs. l(a), l(b), 2(a) and 2(b). The different cases to be analyzed are summarized in Table 6. The area of debonding increases with the depth d. In collinear debonding, the distance between the crack tip and debonded edge 7 in Fig. l(a) decreases with increasing d. Transverse debonding occurs in regions above the crack with ~, approaching the top crack surface as d is increased. These two types of edge debonding will exert different effects on crack instability.

4.2. Crack patching instability Crack patching instability will be determined by computing for the strain energy density function along the prospective path of crack growth. In the case of collinear debonding that occurs symmetrically with reference to the crack plane, the crack growth path is straight ahead along the Table 6 Classification of collinear and transverse debonding Case No.

Debonding depth d (in)

Debonded area (in2)

Collinear debonding CO C1 C2 C3

0 0.5 0.9 1.2

0 0.965 1.605 1.875

Transverse debonding TO T1 T2 T3 T4

0 1.0 1.5 2.0 2.4

0 3.816 5.316 6.816 8.016

G.C. Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel

130 14

Table 7 Strain energy density across panel width and y = 10 in. for Case CO

l

[3

x

c (3

~ 9

Crack-~ 7

G at node 31

o

I

i

2 3 Distancex (in)

i

4

i

5

Fig. 10. Oscillation of energy density for patch without debonding Case CO.

line y = 10 in in Fig. 5. The direction of crack initiation is no longer obvious for transverse debonding and must be found by application of the strain energy density criterion [5-7]. Collinear debonding. For coUinear debonding, it suffices to compute dW/dV along the line y = 10 in where nodes 48, 42 . . . . . 47 are situated as indicated in Fig. 5. The condition of plane strain is assumed such that dW

dV

l+v

E

x

dW/dV

(in)

(psi)

48 42 36 34 32 24 1 23 31 33 35 351 37 39 41

0.(30(0000 2.0000000 3.0000000 3.1366667 3.2733333 3.5000000 5.0000000 6.5000000 6.7266667 6.8633333 7.0000000 7.3333333 7.6666667 7.8333333 8.0000000

0.1100453 0.0775261 0.0799074 0.0812673 0.0777485 0.9302381 0.0115008 0.9302381 0.0777485 0.0812673 0.0799074 0.0789412 0.0778085 0.0773376 0.0775261

/ ~atchedge

tip I

Node No.

[(1-11) ( °2 + °.v2) - 21P°x°Y "l- 2 "r2y]

(12) is computed from the corresponding strains using the material properties E and v for the aluminum panel in Table 1. As a base line comparison, Fig. 10 plots dW/dV against the distance x at y = 10 in for Case CO which represents a perfectly bonded patch as defined in Table 6. Note that dW/dV attains the highest value near the crack tip. At a distance r0 = 0.01 in, [(dW/dg)~x]L = 51.82 x 10 -2 psi is obtained while [(dW/dV)~]o= 7.7775 × 10 -2 psi occurs at G which corresponds

to node 31. The value of (dW/dV)mm = 7.7338 × 10 -2 psi at the patch edge F is not the maximum; it is less than [(dW/dV)~,.~]~. Refer to the numerical values given in Table 7. A crack instability index value of l = 0.210 in is thus obtained. It is confined near the crack tip. Once debonding occurs, the potential of crack initiation increases. This is indicated by the increase in 1 as illustrated in Fig. 11 and the numerical results of dW/dV in Table 8 for Case C1. While [(dW/dV)~.]L = 51.95 × 10 -2 psi remains locally near the crack

Table 8 Strain energy density across panel width at y = 10 in. for Case C1 Node No.

x

dW/dV

(in)

(psi)

48 42 36 34 32 24 1 23 31 33 35 351 37 39 41

0.0000000 2.0000000 3.0000000 3.1366667 3.2733333 3.5000000 5.0000000 6.5000000 6.7266667 6.8633333 7.0000000 7.3333333 7.6666667 7.8333333 8.0000000

0.1099691 0.0775221 0.0799359 0.0813042 0.0777885 0.9308934 0.0115378 0.9321579 0.0784687 0.0822079 0.0811683 0.0804788 0.0814993 0.0828354 0.0846324

G.C. Sih, T.B. Hong / Integrity of edge-debondedpatch on cracked panel 14

/ - Effective crock border 7"

13

14

131

Effective crack border y

13

I

r0

! ?_

~ " 12

'0

I !

x

'0

I

x

-o :>,

IO

:=

I

~

oJ

._c o

~- 9

y- 9

L z tnode39

-G at node 37

8

I Crack-,x

tip \

7

/

03

o

i

I

;

~ Patchedge

Crack--~

/ - Patch edge ! F

F

tip\ ,

7

5

r

0

I

Distance x (in)

2

3

i

i

4

5

Distance x(in)

Fig. 11. Oscillation of energy density for collinear debonding Case C1.

Fig. 12. Oscillation of energy density for collinear debonding Case C2.

tip, three global (dW/dV)mi~ are found in Table 8; they are: 7.8469 x 10 -2 psi, 8.0479 × 10 -2 psi and 8.1499 x 10 -2 psi occurring at nodes 31, 351 and 37, respectively. The location of maximum (dW/dV)~cm or [(dW/dV)~]o = 8.1499 × 10 -2 psi locates G. This gives 1 = 1.170 in which is the distance between x =r0 and node 37. Additional debonding further increases 1. Oscillation pattern of dW/dV for Case C2 is exhibited in Fig. 12. The location of [(dW/dV)~..]o = 8.3382 x 10 -2 psi has now shifted to node 39 which lies just inside the debonded patch edge F. This gives l = 1.190 in. A weaker minimum of dW/dV -- 7.1640 × 10 -2 psi occurred at node 31 which can be seen from the data in Table 9. For Case C3, G has moved to node 41 that coincides with the debonded edge of the patch. The corresponding variations of d W / d V along the prospective crack growth path is shown in Fig. 13 with [(dW/dV)~]o = 8.2739 x 10 -2 psi which is larger than the (dW/dV)r~ of 7.2596 X 10 -2 psi at node 31. Table 10 gives the values of dW/dV for all x at y = 10 in. A complete description of the global dW/dV minima for Case C3 is displayed graphically in Fig. 14. Two of the

four dW/dV minima occurred to the left side of the crack tip and two to the fight side. The maximum global (dW/dV)mi~ is at node 41. Summarized in Table 11 are the crack instability index

Table 9 Strain energy density across panel width at y = 10 in. for Case C2 Node No.

x

dW/dV

(in)

(psi)

48 42 36 34 32 24 1 23 31 33 35 351 37 39 41

0.0000000 2.0000000 3.0000000 3.1366667 3.2733333 3.5000000 5.0000000 6.5000000 6.6266667 6.6633333 6.7000000 6.9666667 7.4000000 7.7000000 8.0000000

0.1100358 0.0775459 0.0799569 0.0813256 0.0778123 0.9312817 0.0115793 0.9616300 0.0716403 0.0794913 0.0820217 0,0847007 0.0839328 0,0833818 0,0833842

G. C Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel

132 14

/

13

l

r0

,4I

Effective crack border y

,3]

~

--

- ~ -

Center

line

.

.'12 ~0

0

x >

x >

"1o

g

If,

o~

-lo

:-, I0 .

o

.

.

.

.

.

.

.

.

.

g ._¢

._= 8 ~9

o 03

I 9 3 at node4 I,I

G at node 41

. . . . . .

Crack-, tip\

J

/ ~atch edge

J

)

I

3

edge

'

,

4

5

-5

Distance x (in) Fig. 13. Oscillation of energy density for collinear debonding Case C3.

l and the corresponding maximum of the local and global (dW/dV)mi~ for collinear debonding of the patch.

Table 10 Strain energy density across panel width at y = 10 in. for Case C3 Node No.

x

dW/dV

(in)

(psi)

48 42 36 34 32 24 1 23 31 33 35 351 37 39 41

0.0000000 2.0000000 3.0000000 3.1366667 3.2733333 3.5000000 5.0000000 6.5000000 6.6266667 6.6633333 6.7000000 6.7666667 7.2000000 7.6000000 8.0000000

0.1100739 0.0775538 0.0799614 0.0813302 0.0778202 0.9312945 0.0115994 0.9599313 0.0725963 0.0816074 0.0846575 0.0897654 0.0851831 0.0830337 0.0827394

-I

-3

I

3

-5

Distance x(in)

Fig. 14. Variation of strain energy density with distance along crack plane for Case C3.

Since [(dl'V//dV)~max.]L in all cases are much higher than [(dW/dV)~]~, fracture initiation, when it occurs, will start from the vicinity of the crack tip such that it tends to propagate from L to G. Crack tends to become more unstable as l increases with increasing cotlinear debonding area, a result that is not unexpected. Transverse debonding. Because of the lack of symmetry across the y-axis, the plane on which fracture will initiate must be found by obtaining the angle 0 at which dW/dV acquires a local

Table 11 Crack instability data for collinear debonding d W rasx

d W max

× 1 0 - 2 (psi)

× 1 0 - 2 (psi)

51.82 51.95 51.48 51.87

7.7775 8.1499 8.3382 8.2739

(in) CO C1 C2 C3

0.21 1.17 1.19 1.49

G.C. Sih, T.B. Hang / Integrity of edge-debondedpatch on crackedpanel

the debonded area is relatively small in comparison with the patch, the small amount of anti-symmetry about the y-axis has negligible influence on the direction of crack initiation. The coordinates of G are found to be x G = 6.72 in and Yo = 10.0 in with G lying on the x-axis. Refer to Figs. 4 or 5 for the origin of the xy-coordinate system which coincides with the lower left corner of the panel. This given a value of l = 0.22 in which is only slightly larger than that of the Case CO. Fracture is predicted to initiate from L with O0 = 0 o where [(dW/dV)~max. ]L is an order of magnitude higher than [(dW/dV)~]G. As debonding is increased to Case T2, a slight increase in x R = 6.75 in is detected while YR remains approximately at 10 in with G lying on the x-axis. Only a slight increase of l = 0.25 in is detected. The results in Table 12 reveal that significant increase in l occurs for cases T3 and T4. The angle 00 now differs significantly from zero. Crack instability is also seen to increase with increasing transverse debonding. In general, it is desirable to localize crack initiation by minimizing l. This can be accomplished by increasing the patch thickness parameter tp = 0.005 in which corresponds to using only one ply of the b o r o n / e p o x y patch. It is not uncommon to have a six-ply patch with tp = 0.03 in in which case, both 1 and dW/dF near the crack can be lowered.

Load direction ebonded area

[ F a i l u r e path

I

~--Patch edger

Fig. 15. Unsyrnmetric crack initation for transverse debonding.

minimum,

i.e.,

a(dW/dV) a0

=0,

for

r = r 0 ; 0 = 0 0.

133

(13)

This determines the position L as shown in Fig. 15. The position G at which [(dW/dV)~.~]G occurs must be found by observing the maximum of minimum of dW/dV at points in the upper halfplane. Potential path of failure from L to G may deviate from the 00 plane if the transverse debonding is appreciable. To be analyzed are the Cases T1, T 2 , . . . , T 4 inclusive. The Case TO is the same as CO and will not be elaborated. Without going into details, the results given in Table 12 refer to those in the aluminum panel. They are obtained by using the material properties of the panel and not the effective properties for finding the strains in the patched panel. The coordinates of G ( x o, YG) and values of [(dW/dV)~'.]o are obtained from the constant strain energy density contours. For the case of T1,

~.3. Patch effectiveness index The load carrying capacity of a debonded patch can be best described by the Patch Effectiveness Index or PEI. It is indicative of the increase in the local strain energy density as the patch edge debonded into a crack-like border. Refer to the notations adopted in Figs. 1 and 2 for collinear and transverse debonding.

Table 12 Crack instability data for transverse debonding No.

oo (degree)

[(dwT] ~ rain L

TO T1 T2 T3 T4

0* 0o 00 35 o to 36 * 35 o to 36 *

51.82 51.13 51.13 63.01 76.15

(Xo, Yo)

[ / d W ' ~ max]

l

(in)

[~,J - ~/= ~ o

(in)

×10 -2 (psi)

x I O - 2 (psi) (6.50, (6.72, (6.75, (7.70, (7.70,

10.00) 10.00) 10.00) 10.55) 10.55)

7.78 7.90 8.00 8.10 8.10

0.21 0.22 0.25 1.25 1.25

134

G. C. Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel

Table 13 Strain energy densities on comer nodes of undebonded patch boundary F

Table 15 Effective strain energy densities on corner nodes of effective crack border 7 for Case C2

Node No.

x (in)

y (in)

dW/dV (psi)

Node No.

x (in)

y (in)

(d W / d V ) * (psi)

42 189 243 231 241 187 41 188 242 232 244 190

2.0000000 2.2656000 3.5000000 5.0000000 6.5000000 7.7344000 8.0000000 7.7344000 6.5000000 5.0000000 3.5000000 2.2656000

10.0000000 11.2344000 12.5980000 13.0000000 12.5980000 11.2344000 10.0000000 8.7656000 7.4020000 7.0000000 7.4020000 8.7656000

0.1030321 0.1169941 0.1445479 0.1359486 0.1445479 0.1296374 0.1030321 0.1295913 0.1446806 0.1361696 0.1446806 0.1169750

187 397 353 398 188

7.7344000 7.1000000 7.1000000 7.1000000 7.7344000

11.2344000 10.6000000 10.0000000 9.4000000 8.7656000

0.2220438 0.3220008 0.1978553 0.3220008 0.2218955

Table 16 Effective strain energy densities on c o m e r nodes of effective crack border "r for Case C3

Collinear debonding. In this case, (PEI)c is defined as

x

7

+ ~ (dW/dV)i i= 1

(14) F-ABC

Nodal points on F, ~, and F - A B C in eq. (14) can be found in Fig. 6. On the patch periphery F, there are a total of 36 nodes such as 41, 187, etc. The corresponding strain energy densities along the undebonded patch border can be computed using plane stress condition

2 -~x + -~y -

y (in)

(dW/dV) * (psi)

187 397 353 398 188

7.7344000 6.8000000 6.8000000 6.8000000 7.7344000

11.2344000 10.3000000 10.0000000 9.7000000 8.7656000

0.2261401 0.3023992 0.2229095 0.3023992 0.2259905

E y . . . . . Vxy w i t h Pxy = Pyx = p * a n d a x y = G * . R e f e r to Table 13 for their numerical data which

-1

dV

x (in)

The effective mechanical properties E * , E y .... , v* in Table 3 are used in eq. (15) instead of E x,

~ (dW/dg i=1

Node No.

-~y + - ~

apply to cases CO and TO. The 13 nodes on "t for collinear debonding correspond to those on the periphery of elements 71 and 81 in Fig. 6 except for nodes 37 and 39. Nodes 187 and 188 are considered as part of the effective crack border in Fig. 3; they correspond, respectively, to points A and C in Fig. l(a). The effective strain energy density ( d W / d V ) * (i = 1, 2 . . . . . 13) are computed by using eq. (9). Tables 14, 15 and 16 give, respectively, the numerical values for Cases C1, C2 and

o~oy+ -~y . (15)

Table 14 Effective strain energy densities on c o m e r nodes of effective crack border 2, for Case C1 Node No.

x (in)

y (in)

(dW/dV) * (psi)

187 397 353 398 188

7.7344000 7.5000000 7.5000000 7.5000000 7.7344000

11.2344000 11.0000000 10.0000000 9.0000000 8.7656000

0.3178775 0.2399621 0.2673697 0.2399364 0.3176705

Table 17 Strain energy densities on c o m e r nodes of undebonded portion of patch F-ABC for Case C1 Node No.

x (in)

y (in)

dW/dV (psi)

42 189 243 231 241 242 232 244 190

2.0000000 2.2656000 3.5000000 5.0000000 6.5000000 6.5~ 5.0000000 3.5000000 2.2656000

10.0000000 11.2344000 12.5980000 13.0000000 12.5980000 7.4020000 7.0000000 7.4020000 8.7656000

0.1030268 0.1169996 0.1445910 0.1359480 0.1444918 0.1446239 0.1361751 0.1447239 0.1169801

G.C Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel

135

Table 18 Strain energy densities on corner nodes of undebonded portion of patch F - A B C for Case C2

Table 20 Effective strain energy densities on comer nodes of effective crack border 3' for Case T1

Node No.

x (in)

y (in)

dW/dV (psi)

Node No.

x (in)

y (in)

(dW/dV) * (psi)

42 189 243 231 241 242 232 244 190

2.0000000 2.2656000 3.5000000 5.0000000 6.5000000 6.5000000 5.0000000 3.5000000 2.2656000

10.0000000 11.2344000 12.5980000 13.0000000 12.5980000 7.4020000 7.0000000 7.4020000 8.7656000

0.1030587 0.1170142 0.1445635 0.1358069 0.1447319 0.1448648 0.1360339 0.1446962 0.1169948

243 387 375 385 241

3.5000000 3.5000000 5.0000000 6.5000000 6.5000000

12.5980000 12.0000000 12.0000000 12.0000000 12.5980000

0.3873195 0.3886998 0.2368600 0.3886998 0.3873195

Table 19 Strain energy densities on comer nodes of undebonded portion of patch F - A B C for Case C3 Node

x

y

No.

(in)

(in)

dW/dV

(psi)

42 189 243 231 241 242 232 244 190

2.000012(0 2.2656000 3.5000000 5.0000000 6.5000000 6.5000000 5.0000000 3.5000000 2.2656000

10.0000000 11.2344000 12.5980000 13.0000000 12.5980000 7.4020000 7.0000000 7.4020000 8.7656000

0.1030694 0.1170127 0.1445496 0.1356783 0.1449842 0.1451173 0.1359053 0.1446826 0.1169935

C3. The debonded portion of the patch involves not only the boundary ABC which contains the five nodes between 187 and 188, but also the two nodes 37 and 39 shown in Fig. 6 which are to be excluded. This gives a total of 29 nodes on F - A B C and the corresponding comer node strain energy densities are given in Tables 17 to 19 inclusive. Transverse debonding. In the same way, a patch effectiveness index (PEI)t can be defined for transverse debonding:

(dW/dV),

(PEI)t = ~Li=I

1

(dW/dV)r

Iv -1

(16) Li=l

Table 21 Effective strain energy densities on corner nodes of effective crack border Y for Case T2 Node No.

x (in)

y (in)

(dW/dV) * (psi)

243 387 375 385 241

3.5000000 3.5000000 5.0000000 6.5000000 6.5000000

12.5980000 11.5000000 11.5000000 11.5000000 12.5980000

0.4089636 0.4270710 0.2409197 0.4270710 0.4089636

Table 22 Effective strain energy densities on corner nodes of effective crack border -/ for Case T3

r

t Li=1

same as CO, no discussion is necessary. According to Fig. 6, the 13 nodes on the effective crack border 3, for cases T1, T2 . . . . . T4 are those on the periphery of dements 27 and 47 including nodes 241 and 243 but excluding those on the outer edge between 241 and 243. The corresponding effective comer node strain energy densities (dW/dV)* computed from eq. (9) can be found in Tables 20 to 23 inclusive. Again, it is understood that the debonded boundary F - D E F excludes nodes 203, 217 and the ones between 241 and 243 on F. This gives a total of 29 nodes and the corresponding comer node strain energy densities are given in Tables 24 to 27 inclusive. Summary of results. Inserting the values of (dW/dV)~ on F(i= 1, 2 . . . . . 36) in Table 13; (dW/dV)* on "r (i = 1, 2 . . . . . 13) in Tables 14 to

F- DEF )

S i n c e the case of undebonded patch TO is the

Node No.

x (m)

y (m)

(dW/dV)* ~si)

243 387 375 385 241

3.5000000 3.5000000 5.0000000 6.5000000 6.5000000

12.5980000 11.0000000 11.00000120 11.0000000 12.5980000

0.4317626 0.4507174 0.2343238 0.4507174 0.4317626

G.C. Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel

136

Table 23 Effective strain energy densities on c o m e r nodes of effective crack border T for Case T4

Table 26 Strain energy densities on c o m e r nodes of undebonded portion of patch F - D E F for Case T3

Node No.

x (in)

y (in)

(dW/dV)* (psi)

Node No.

x (in)

y (in)

d W/d V (psi)

243 387 375 385 241

3.5000000 3.5000000 5.0000000 6.5000000 6.5000000

12.5980000 10.6000000 10.6000000 10.6000000 12.5980000

0.4481116 0.4650570 0.2289545 0.4650570 0.4481116

42 189 187 41 188 242 232 244 190

2.0000000 2.2656000 7.7344000 8.0000000 7.7344000 6.5000000 5.0000000 3.5000000 2.2656000

10.0000000 11.2344000 11.2344000 10.0000000 8.7656000 7.4020000 7.0000000 7.4020000 8.7656000

0.1065010 0.1200004 0.1200004 0.1065010 0.1192081 0.1435513 0.1332489 0.1435513 0.1192081

Table 24 Strain energy densities on c o m e r nodes of undebonded portion of patch F - D E F for Case T1

Node No.

x (in)

y (in)

dW/dV (psi)

42 189 187 41 188 242 232 244 190

2.0000000 2.2656000 7.7344000 8.0000000 7.7344000 6.5000000 5.0000000 3.5000000 2.2656000

10.0000000 11.2344000 11.2344000 10.0000000 8.7656000 7.4020000 7.0000000 7.4020000 8.7656000

0.1045291 0.1194658 0.1194658 0.1045291 0.1177444 0.1440677 0.1349693 0.1440677 0.1177444

Table 27 Strain energy densities on c o m e r nodes of undebonded portion of patch ] ? - D E F for Case T4

Node No.

x (in)

y (in)

dW/dV (psi)

42 189 187 41 188 242 232 244 190

2.0000000 2.2656000 7.7344000 8.0000000 7.7344000 6.5000000 5.0000000 3.5000000 2.2656000

10.0030t)00 11.2344000 11.2344000 10.0000000 8.7656000 7.4020000 7.0000000 7.4020000 8.7656000

0.1070725 0.1204377 0.1204377 0.1070725 0.1199801 0.1434531 0.1324210 0.1434531 0.1199801

Table 25 Strain energy densities on comer nodes of undebonded portion of patch F - D E F for Case T2

Node No.

x (in)

y (in)

dW/dV (psi)

42 189 187 41 188 242 232 244 190

2.0000000 2.2656000 7.7344000 8.0000000 7.7344000 6.5000000 5.0000000 3.5000000 2.2656000

10.0000000 11.2344000 11.2344000 10.0000000 8.7656000 7.4020000 7.0000000 7.4020000 8.7656000

0.1055463 0.1197131 0.1197131 0.1055463 0.1183888 0.1437756 0.1341651 0.1437756 0.1183888

16; and (dW/dV)i o n / ' - A B C (i = 1, 2 . . . . . 29) in Tables 17 to 19 into eq. (14), (PEI)c for collinear debonding can be computed. Their values are given in Table 28. N o t e that the patch effectiveness tends to decrease with increasing area of debonding. Such a parameter is useful for ranking the effectiveness of damaged patch and it includes all the combined effects of loading, geometry, material and the type of damage.

Table 28 Patch effectiveness index for collinear debonding Case No.

d (in)

Debonding area (in 2)

(PEI)c (%)

CO C1 C2 C3

0 0.5 0.9 1.2

0 0.965 1.605 1.875

100 86.6 85.1 84.0

Following the same procedure, (dW/dV)i on F ( i = 1, 2 . . . . . 3) in Table 13; (dW/dV)* on y (i = 1, 2 . . . . . 13) in Tables 20 to 23; and (dW/dV)i on F (i = 1, 2 . . . . . 9) in Tables 24 to 27 can be put into eq. (16) to yield (PEI)t for transverse debonding. The results are given in Table 29. A comparison of (PEI)t in Table 29 with (PEI)c in Table 29 shows that transverse debonding in much more detrimental than collinear debonding. The patch effectiveness index (PEI)t is much smaller than (PEI)c for approximately the same area of debonding. The location of the damaged area in

G.C. Sih, T.B. Hong / Integrity of edge-debonded patch on cracked panel Table 29 Patch effectiveness index for transverse debonding Case No.

d (ill)

Debonding area (ill2 )

(PEI) t (~)

TO T1 T2 T3 T4

0 1.0 1.5 2.0 2.4

0 3.816 5.316 6.816 8.016

100 80.8 73.6 67.1 62.6

relation to the crack can greatly influence the load carrying capacity of the patch. This is clearly demonstrated by the results in Tables 28 and 29.

5. Conduding remarks Analyzed in this work is the effect of patch debonding on the remaining strength of reinforced panels that contain initial cracks. The influence of the patch thickness, adhesive layer and panel thickness are accounted for by applying the effective stiffness concept in conjunction with the finite element procedure. Two types of edge-debonding are considered; they correspond to debonding in regions coUinear and transverse to the crack whose plane is assumed to be normal to the applied load. When a portion of the patch is detached from the panel, free surfaces are created giving rise to a new crack border. This crack lies in a plane normal to that in the panel. An effective patch/ adhesive/panel medium is defined for calculating the local intensification of the strain energy density function. Among the important contributions are the concept of crack instability parameter and patch effectiveness index. The distance 1 between the local and global maximum of the minimum strain energy density function gives an indication of failure instability by fracture. Instability tends to increase with l. For about the same size of patch damage, collinear debonding leads to more unstable fracture than transverse debonding. The patch effectiveness index PEI measures the load carrying capacity of the damaged path. In this case, transverse debonding is less favorable as compared with collinear debonding. In general, both 1 and PEI must be considered for assessing the integrity of the damaged patch. Although crack patching analyses have been published in many previous publications that can

137

be found in [1], the majority of the works were concerned with determining the displacements and stresses in the structure rather than developing predictive capability. In this respect, the major contributions of the present investigation can be summarized as follows: • The local and global maximum of the minimum strain energy density function are obtained to define crack patching instability for collinear and transverse edge-debonding. Geometric and material parameters can thus be optimized for specified loading conditions such that fracture can be stabilized and localized in the vicinity of the original crack. The means for monitoring crack growth can then be developed to assure safe service life of the reinforcement. • Using the undamaged patch as the base line, a patch effectiveness index is defined so that different damage areas and locations can be ranked in terms of a single parameter for comparison. This additional information can be used to determine the remaining strength of damaged reinforcement. Even though useful information has been obtained without resulting to a three-dimensional analysis, additional refinements can be made to improve on the failure prediction of reinforced panels with damaged patches. Some of the items for future considerations can be mentioned in passing.

5.1. Plastic deformation Since the local stress, strain and energy density in the immediate vicinity of the crack tip can be extremely high, plastic or permanent deformation cannot be avoided. The incremental theory of plasticity can be incorporated into the finite element formulation where the energy dissipated by plasticity must now be distinguished from that used to create new crack surfaces. The locations of yielding or permanent deformation can be determined from a knowledge of the maximum of the maximum strain energy density as discussed earlier. Applications of the strain energy density criterion to include plastic deformation can be found in [12-14]. Corrections for change in the local strain rates and strain rate history can also be made by modifying the classical theory of plasticity where the constitutive relations for the elements near the crack will no longer be the same

138

G. C. Sih, 72B. Hong / Integrity of edge-debonded patch on cracked panel

as those far away. The details on this can be f o u n d in [15]. 5.2. Directional interaction o f loading and debonding

Service loading can, in general, change direction in relation to the location of edge-bonding and crack. It is no longer obvious that the fibers in the b o r o n / e p o x y patch should be directed normal to the crack plane. I n practice, a quasi-isotropic composite patch is p r o b a b l y preferred so as to account for all possible orientations of the load. Information of the remaining strength of the patched panel for different position of edge-debonding would be useful for making decisions on maintenance and repair. 5.3. Subcritical c r a c k growth

The capability for predicting subcritical crack growth in patched panels subjected to m o n o t o n i c and cyclic loadings prior to the onset of unstable fracture is needed for establishing inspection procedures. This can be accomplished b y employing the strain energy density criterion [16-18]. I n addition to the conditions stated earlier for determining fracture initiation, it can be further assumed that crack growth segments follow constant S j / r j , where j = 1, 2 . . . . . n. That is, once crack m o t i o n is initiated by reaching ( d W / d V ) c , the following prevails

-dgc or

rl 07)

ra ,

in which

Sl < S 2 < ... < S j < ...
... < r j <

... < r e ,

(18)

correspond to increasing rate of crack growth leading to unstable fracture and S l > S 2 > ... > S j > . . . > S a , r l > r 2 > ... > ~ > ... > r a ,

(19)

correspond to decreasing rate of crack growth resulting in crack arrest. Depending on the interaction of load, geometry and material, the crack

failure behavior m a y involve the combination of the conditions in eqs. (18) and (19). N o t e that thickness effect, plastic deformation and arbitrary load and d e b o n d i n g direction can all be included in the analysis of subcritical crack growth.

Acknowledgement The authors would like to acknowledge Mr. Y.D. Lee for the time he spent on modifying the c o m p u t e r p r o g r a m used in this work.

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