JCTR Bonded Joints Paper

8/2/2019 JCTR Bonded Joints Paper

1/32

Published in Journal of Composites Technology and Research, 2002 Page 1

Strength and Fatigue Life Modeling of Bonded Joints in Composite Structure

D. M. Hoyt, Stephen H. Ward and Pierre J. Minguet

ABSTRACT

The aerospace industry lacks a validated, practical analysis method for the strength,durability, and damage tolerance evaluation of composite bonded joints. This paper presents the

results of a combined strength and fracture analysis approach applied to typical bonded jointconfigurations found in rotorcraft composite structures. The analysis uses detailed 2-D non-

linear finite element models of the local bondline. Strength-of-materials failure criteria are usedto predict critical damage initiation loads and locations. A fracture mechanics approach is usedto predict damage growth and failure under static and cyclic loads based on test data for static

fracture toughness (GIc, GIIc) and crack growth rate (da/dN). Results are presented from theapplication of the analysis approach to two joint configurations: 1) a skin-stiffener T-joint and,

2) a bonded repair lap joint. The results demonstrate that the proposed approach can be used topredict critical failure modes, damage initiation loads and locations, crack and/or delaminationstability, static strength, residual strength, and fatigue life. Discussion is also included on how

this approach can be applied in damage tolerance evaluations of composite bonded joints..

INTRODUCTION

Ever increasing aerospace performance requirements make the high strength-to-weightratios and cost efficiency associated with bonded joints attractive. However, bonding cannot befully utilized without validated analytical methods to increase confidence in bonded designs and

to reduce the expensive testing often necessary to certify bonded joints in critical locations.Current standard analysis methods are not capable of predicting all of the complex failure

mechanisms associated with composite bonded joints [1]. Most existing bonded joint analysesdo not include shear deformation of the adherends and cannot account for peel failures at the endof the overlap (Figure 1), which are often a primary cause of joint failure. In addition, they often

truncate the adhesive stress-strain curve to indirectly account for the composite adherend failuremodes not explicitly analyzed. An accurate composite bonded joint analysis method must be

able to predict failure in the adhesive, at the adhesive-adherend interface, within the surface plies

of the laminate itself, at stiffener flange fillets, or at the skin-to-core interface in sandwichstructure, and must also account for nonlinear material behavior.

In addition to being able to predict all critical failure modes and locations, the analysismethod must have the ability to address damage growth and damage tolerance, given the

D. M. Hoyt, NSE Composites, 1101 N Northlake Way #4, Seattle WA 98103Stephen H. Ward, SW Composites, HC68, Box 15G, Taos, NM 87571Pierre J. Minguet, The Boeing Company, MC P38-13, PO Box 16858, Philadelphia, PA 19142


2/32


emphasis now placed on them by aircraft certifying agencies. Many of the failures in compositebonded joints involve delaminations that may grow from small pre-existing flaws or from

damage induced by fatigue loads. Delaminations may also be driven by temperature and/ormoisture induced loading. Recent research indicates that a fracture mechanics approach caneffectively predict quasi-static delamination growth and is best suited to address the issues of

fatigue life, damage tolerance, and the effects of operating environments on composite bonded

joints subjected to cyclic loading [2-3,10-11]. This paper presents the results of a combinedstrength and fracture analysis approach applied to typical bonded joint configurations found inrotorcraft structures.

ANALYSIS APPROACH

The analysis approach presented here overcomes many of the shortcomings of existingmethods and is capable of predicting all critical joint failure modes, as well as tracking damage

growth due to static and fatigue loading. This integrated approach is based on the work ofMinguet, OBrien, and Johnson [2,4-6]. The analysis uses non-linear 2-D FE models (through-the-thickness) of the local bondline together with strength-of-materials failure criteria for the

prediction of critical damage initiation loads and locations, and a fracture mechanics approachfor the prediction of damage growth and failure under static and cyclic loads, Figure 2. All of

the fracture mechanics analysis for crack growth, static strength, and fatigue life is done as "post-processing" based on a single set of FEM results for a series of crack lengths.

Finite Element Modeling

For these analyses, 2D, plane stress, continuum (solid) elements with an 8-noded, bi-quadratic (2nd order), reduced integration formulation (ABAQUS CPS8R elements) are used.

Composite lamina are modeled with linear elastic properties; however, to account for 3D effects,

material properties are entered to achieve a generalized plane strain solution that is betweenclassical plane stress and plane strain assumptions. The difficulty in using 2-D modeling when

representing laminated composites is that, although the laminate may be in a state of plane stress,each lamina is typically not in a state of plane stress. The effect is most marked for angle (e.g.,

+/- 45) plies because of their high in-plane Poissons ratio, while it is small for 0 and 90 plies.The following procedure is an approximation designed to balance accuracy and efficiency with2-D modeling. Starting with the traditional 3-D stress-strain relationships and the traditional

orientations where x,y,z are the laminate axes and 1,2,3 the lamina axes, the two traditionaloptions are:

Plane Strain, where yy = xy = yz = 0 and

Plane Stress, where yy = xy = yz = 0.

The typical choice for 2-D models of laminates where the model is in the thicknessdirection is to use a plane strain approach. A pure plane stress approach would assume that the

laminate in-plane stresses in the laminate y-direction (into the page in a 2-D, through-the-thickness model) are zero. This is clearly not valid since significant stresses in 90 plies resultfrom Poisson strains. On the other hand, using a plane strain approach makes the +/-45 plies

too stiff due to their high Poissons ratio. For this reason, an intermediate generalized planestrain state is used where it is assumed that:


3/32


yy = -Lxx and xy = yz = 0, whereL is the laminate Poissons ratio.

With these assumptions, ply stiffnesses are calculated for each of the ply angles in the laminate.

Adhesives are modeled as non-linear isotropic materials with plastic hardening behavior,to match the true shear stress-strain response. Due to the potentially high plastic strains at the

peak stress locations in the joints, the incorporation of non-linear stress-strain behavior in theadhesive is essential to obtaining an accurate stress representation in areas near the end of abonded joint [7,8]. In order to develop an accurate shear stress-strain curve, the shear stress-

strain behavior is first modeled using the relation developed by Grant [9]:

modulusshearelasticG

stressshearmaximum

toingcorrespondstressshear

stressshear

strainshearelasticmaximum

strainshear

G

where

thenIf

GthenIf

max

ee

e

emax

e

ee

e

==

====

=

=

+

+=


4/32


The interlaminar tension-shear stress interaction criterion is used to predict delaminationof the composite adherends, in laminates with either tape and/or fabric plies. The failure index is

given by:2

IndexFailure

+=

xz

xz

zz

z

SF

where z = through the thickness stress

xz = interlaminar shear in the x-z plane

Fzz = allowable through-thickness strength

Sxz= allowable interlaminar shear strength

The maximum transverse tensile stress failure criterion is used to predict matrix crackingin tape laminates. This failure index has been used successfully in previous research [5] and is

given below:

23

2

3232

axm

maxmax

22

FIndexFailure

+

+

+=

=

where Fmax = max transverse tensile stress in a ply

2 = in-plane transverse principal stress (lamina coordinates)

3 = through the thickness stress (lamina coordinates)

23 = shear in the 2-3 plane (lamina coordinates)

The Von Mises strain failure criterion is used to predict failure in the adhesives. Thefailure index is given by:

maxVMVonMises

SIndexFailure

=

where VonMises = Von Mises equivalent strain

SVMmax = allowable Von Mises strain

Static Strength

An outline of the static strength analysis procedure is shown in Figure 3. The first step inthe static strength analysis is to choose the initial crack size, location, and growth path. Locating

a crack in a critical location simulates either the condition where a crack develops once thedamage initiation load, Pinit, is reached, or the condition where a crack exists due to a

manufacturing or in-service damage event. The selection of an initial crack size should be based


5/32


on many factors, including manufacturing acceptance and/or damage tolerance criteria for thespecific structure.

Next, the location of the crack interface is determined a-priori based on the damageinitiation site and experience with typical crack paths in composite structure. It may benecessary to analyze several crack paths to ensure that the critical path has been identified. The

crack interfaces are modeled along the direction of anticipated crack growth. In bonded jointswith composite adherends, critical crack interfaces can occur between two plies in the adherend,

between the adherend and the adhesive, and within the adhesive. Note that within compositelaminates, it is generally conservative to assume a clean crack path, where the crack tipcontinues along a line between plies or along fibers within a ply during crack growth. Other

matrix cracking, ply bridging, and ply jumping crack behaviors require more energy to propagatethe crack than self-similar crack growth.

Once the crack interface has been selected, duplicate nodes are placed in the FE modelalong the anticipated crack path. A series of runs of the FE model are made for successiveincrements of increasing crack lengths. For each load step in each analysis run, the total strain

energy release rate (SERR, Gtot) is calculated for the crack length from the change in strainenergy in the model between successive crack lengths. At several crack lengths, the Virtual

Crack Closure Technique (VCCT) [12,13] is used to calculate GI, GII, and Gto t, and the mode mix(GII/Gtot). Next, the critical fracture toughness, Gtot,crit is determined for each crack length usingtest data at the appropriate mode mix (GII/Gto t) for that crack length. Then by comparing Gtotfrom the finite element model (calculated at several load steps) to Gtot,crit at a given crack length,the load, Pgrowth, at which the crack is predicted to grow is determined. A residual strength curve

is then plotted as Pgrowth vs. crack length, a, and used to predict static strength and crack stabilityas a function of crack length.

The method of determining the ultimate static strength, Pgrowth,static depends on the shape

of the Pgrowth

versus crack length curve, and on specific criteria, as shown in Figure 4. The Pgrowthvs. a curves can be used to determine residual strength of the joint at any crack length, such as

after the detection of in-service damage. They can also be used in damage tolerance analyses.For example, if the damage tolerance criteria for a given structure states that the joint must carrylimit load in the presence of 0.50 x 0.50 inch damage, the residual strength at a crack length, a =

0.50 inch (Pgrowth,0.50) can be directly compared with the limit load to determine a margin ofsafety.

Fatigue Life

An outline of the fatigue life analysis procedure is given in Figure 5. To predict crackgrowth under cyclic loading, the calculated SERRs as a function of crack length and load level

(Gtot vs. a from FEM) are combined with crack growth rate test data (da/dN vs. Gtot) fromstandard composite or bonded fracture toughness specimens to determine the number of fatigue

cycles required to grow a crack to its critical length. Note that mode mix was not considered inthe fatigue analysis. The use of Gtot (i.e., the difference between the total SERRs at Pmax andPmin) is based on research indicating it to be more important than either GI orGII for cyclicdelamination growth in polymer matrix composites[2,6,14].


6/32


The procedure outlined in Figure 5 is for constant amplitude fatigue loading at a singleload ratio (R-ratio = Pmin/Pmax). First, the total strain energy release rate range is determined as

Gtot = Gtot,max - Gtot,min for each crack increment (a) at a series of load levels. Next, the crackgrowth rate (da/dN) for each crack length and maximum load level is determined from Gtotusing crack-growth-rate test data (da/dN vs. Gto t). The crack growth increment (a) is thendivided by this growth rate to obtain the number of cycles (N) required to progress the crackthat distance under the specified cyclic loading.

Finally, the number of fatigue cycles (N) associated with each increment of crackgrowth are summed from the initial to final crack lengths to determine the number of cycles tofailure, (NPj), at each cyclic load level. The fatigue life (N) of the joint due to loading at that

specific R-ratio can then be determined for any load amplitude from a curve constructed throughthe (NPj, Pmax) data pairs. To address spectrum loading, Pmax vs. N plots are developed from

fatigue test data for various R-ratios and used together with a damage accumulation model (e.g.,Miner's Rule).

Note that if only the onset of fatigue damage is of interest (not crack growth due to cyclic

loading), an alternate approach can be used. That is, the maximum calculated SERR value overthe crack length can be combined with damage onset toughness vs. cycles data (Gonset vs. N) to

predict the number of cycles to damage onset.

APPLICATION OF ANALYSIS

The above analysis approach has been successfully applied to several typical aerospace

configurations, including a T-stiffened skin panel, a single lap joint, a scarf repair joint, and asandwich panel bulkhead attachment. Results from the skin/T-stiffener and single lap joints are

presented here.

Skin/T-Stiffener Model

The skin/T-stiffener joint is shown in Figure 6. This joint configuration represents

integrally stiffened panels used in many current fuselage and wing designs, including integratedbonded designs for stringers, frames, ribs and bulkhead attachments. The skin laminate was

made with IM7/8552 grade 160 carbon fiber tape, the flange used IM7/8552 plain weave (PW)carbon fiber fabric, and the adhesive was FM-300 film. The material properties are given inTables 1 and 2.

Figure 7 shows the model details, including the different ply types and orientations

(material properties), and the element densities relative to the ply and adhesive thicknesses. Theappropriate composite ply properties are entered for each element based on its material andorientation. The properties for a +45 ply and a 45 ply are the same since the model is two-dimensional. In general, one element was used through the thickness of each ply, except in the

region near the flange tip. There, three elements through the thickness were used for theadhesive and for the two plies on either side of the adhesive layer, to more accurately model the

stress gradients in that area.


7/32


Other important areas in the bondline and the adhesive fillet were also modeled in detail.The adhesive filler size, the corner radius of the flange and the thickness of the tip of the tapered

flange are all based upon typical dimensions observed in actual specimens. This level of detailedmodel avoids stress singularities that would be caused by the combination of sharp corners (i.e.,no rounded flange tip or resin pocket) and material property discontinuities. For the tapered

flange, the tip thickness is equal to two plies. The radius at the flange tip corner is chosen equal

to one ply thickness to better represent actual part geometry (perfectly sharp corners are notproduced by typical machining processes). The height of the adhesive fillet extends up two plieson the flange and the slope of the fillet is roughly 45.

The model was run to a maximum load (PFEM) of 50 lbs with geometric and material

nonlinearity enabled. The load-displacement response of the joint is shown in Figure 8.Through-the-thickness normal and shear stress results in the area near the flange tip are shown as

contour plots in Figures 9 and 10, respectively, for the three-point bending loadcase at themaximum applied load. Significant plastic yielding of the adhesive was predicted in a smallregion adjacent to the flange tip as shown in these figures. The contour plots were created

without averaging the nodal results across boundaries between different materials and plies. This

ensures that inappropriate averaging, which can obscure peak stress regions, does not occur.

Skin/T-Stiffener Damage Initiation Analysis

Based on previous research and data from literature [5,15], the following strength valueswere used to calculate the damage initiation failure indices discussed earlier:

Skin interlaminar tension: 3000 psi

Skin interlaminar shear: 5000 psi

Flange interlaminar tension: 3000 psi

Flange interlaminar shear: 5000 psi

Skin transverse (in-plane) tension: 5000 psi

Adhesive Von Mises strain 0.05 in/in

The results are shown in contour form in Figures 11 and 12. The predicted damage

initiation load for each failure index was calculated by interpolation between the nonlinear loadsteps, and is summarized in Figure 13. Damage is first predicted to initiate in the top 45 skinply in the in-plane transverse tension failure mode at a location near the end of the adhesive

fillet. This represents the onset of a matrix crack in the 45 ply. Progressing to higher load, themodel predicts an interlaminar failure in the top 45 skin ply below the end of the flange. This

represents the onset of a delamination; given that the 45 ply is predicted to have a matrix crack,

it is expected that this delamination would start at the matrix crack and propagate along theinterface between the first two skin plies. This delamination propagation behavior is consistent

with test results from similar tests reported in [4,5]. The adhesive is predicted to fail at higherloads than the skin laminate, which is a desirable design condition and consistent with test results

on this type of bonded joint.


8/32


Skin/T-Stiffener Static Strength Analysis

Based on the results of the damage initiation analysis, a crack was introduced into the

model to represent a matrix crack in the skin at the tip of the adhesive followed by crack growthbetween the top two skin plies, as shown in Figure 14. Duplicate nodes were placed in the modelalong the crack then successively released and analyzed for a series of crack lengths. The crack

was 'grown' to a total length of acrit = 0.40 inches, which represents a maximum allowabledamage size based on typical design criteria. The smallest element size along the delamination

was 0.00444 inches.

Total strain energy release rate (Gtot) and mode mix (GII/Gto t) were calculated as afunction of crack length using the fracture mechanics methods described earlier. Since each non-

linear run has several load steps, Gtot can be calculated for each load level and plotted as shownin Figure 15. The mode mix was plotted versus crack length and a curve fit was made as shown

in Figure 16. The curve shows that, as the crack is opened, the amount of mode II fracture (in-plane shear mode) relative to mode I (opening mode) gradually increases.

The mode mix at each chosen crack length (in this case increments of 0.05 inches were

used) is then combined with fracture toughness test data to determine the critical fracturetoughness, Gtot,crit , Figure 17. Gtot,crit represents the amount of strain energy required to advance

the crack an infinitesimal amount. As test data were not available for IM7/8552 during thisstudy, data were estimated based data for similar materials [5,16,17].

The critical fracture toughness values, Gtot,crit s, for each crack length were then combined

with the predicted strain energy release rate, Gtot, from the FEM (Figure 15) to determine theload at which crack growth is predicted. This load, Pgrowth, occurs when Gtot is equal to Gtot,crit .

The values of Pgrowth vs. crack length were then plotted as shown in Figure 18. The staticstrength of the joint, Pgrowth,static, is determined using the procedure outlined in the AnalysisApproach section. In this case, additional load beyond the predicted damage initiation load of

25.6 lbs. is required to advance the crack, as shown in Figure 18. The crack will begin to grow ata load of 43.3 lbs. Since the slope of the Pgrowth vs. a curve is negative, the crack will become

unstable once that load is reached. Therefore, the predicted static strength of the joint,Pgrowth,static, is 43.3 lbs. Note that in this case, static strength is dependent on the chosen initialcrack length. That is, if a larger initial crack size had been chosen, a lower static strength would

be predicted. Also note that only one crack location was modeled to demonstrate feasibility. Fora complete analysis of the skin/T-stiffener joint, crack growth from the other potential damage

initiation sites in the adhesive and the flange laminate, as shown in Figure 13, would beevaluated.

The Pgrowth vs. a curve can also be used to determine the residual strength of the structure

at a given crack length. In this skin/T-stiffener example, suppose in-service damage of 0.40inches was detected. The residual strength could then be determined from the Pgrowth curve

(Pgrowth,0.40 = 0.60 * 50 lbs. = 30 lbs.) and compared with the load requirements and damagegrowth criteria for the structure to determine the disposition.

Skin/T-Stiffener Fatigue Life Analysis

The durability of the skin/T-stiffener joint under fatigue loading was then assessed usingthe methods discussed above in the Analysis Approach section. For the purposes of this study,


9/32


the fatigue crack path was assumed to be the same as the static crack path. Values of Gtot,FEM(= Gtot,max Gtot,min , corresponding to cyclic loads Pmax and Pmin, see Figure 5) for three R-ratios

(0.1, 0.5 and 0.75) were interpolated from the existing FEM load steps.

Next, these values ofGtot,FEM were compared to crack growth rate test data to determinethe predicted crack growth rate at a given crack length for each load level, as shown in Figure 19.

The test data were estimated and assumed to be independent of R-ratio, since data for IM7/8552were not available for this study. The estimated crack growth rate data were combined with thecalculated SERRs to generate a set of S-N type curves for several R-ratios, Figure 20. Forconstant amplitude loading, the Pmax vs. N curve for the corresponding R-ratio can be used to

directly determine the number of cycles to failure. For example, for Pmax = 28.9 lbs (67% ofpredicted ultimate static strength), the cycles to failure at an R-ratio of 0.10 are predicted to be

49,877.

The cycles to failure in this example are based on an arbitrary maximum allowabledamage size of acrit = 0.40 inches. This critical length would typically be determined by criteria

or by residual strength requirements. If a residual strength criterion is used, the Pgrowth curve

from the static strength analysis can be used to determine the critical crack length (acrit) forfatigue life analysis. The structure may be considered failed when the part can no longer carrya given load (e.g., limit load), which is typically higher than the fatigue load. The crack length atwhich the joint falls below the required residual strength (based on the static Pgrowth vs. a curve)

can then be used as acrit .

Skin/T-Stiffener Summary of Predictions

Damage Initiation Load: Pinit = 25.6 lbs

Matrix crack in top skin ply followed by delamination between top two skin plies

Static Strength: Pgrowth,static = 43.3 lbs

Unstable crack growth at crack length = 0.05 inches

Fatigue Life: (assuming joint failure at crack length = 0.40 inches)

Low cycle fatigue, Pmax = 28.9 lbs --> 49,877 cycles

While directly comparable static and fatigue test results for this configuration were notavailable, the predicted damage locations, loads, and cycles to failure are consistent with

previously developed test data from similar specimens [18,19].

Single Lap Joint Model

The single lap joint shown in Figure 21 represents a single-lap-shear flaperon repair.This type of high load transfer joint is critical to the understanding of joint analysis and fatiguebehavior. The two-dimensional (through-the-thickness) finite element model of the joint shown

in Figure 21 was constructed based on a typical tilt-rotor flaperon skin repair joint [20]. The skinlaminate is made with IM6/3501-6 grade 145 carbon fiber tape; the repair laminate uses


10/32


AS4/3501-6 5-harness (5HS) carbon fiber fabric. The adhesive is Magnolia 6363 paste. Thematerial properties are given in Tables 1 and 2.

The joint is axially loaded. One-half of the joint was modeled with symmetry boundaryconditions at the centerline, as shown in Figure 21. An axial load of 3000 lbs was applied to theend of the model. The loading tabs were simulated in the model, and were constrained from

moving in the thickness direction (Y).

Figure 22 shows the model details, including the different ply types and orientations

(material properties), and the element densities relative to the ply and adhesive thicknesses. Theappropriate composite ply properties are entered for each element based on its material andorientation. The joint was modeled with 65F material properties. One element was used

through the thickness of each ply, except for two skin and one repair plies adjacent to theadhesive and for the adhesive layer where three elements through the thickness of each ply were

used. Through-the-thickness normal stress and shear stress results in the area at the end of therepair laminate are shown in Figures 23 and 24 at the maximum applied load (3000 lbs).

Single Lap Joint Damage Initiation Analysis

The same three damage initiation failure criteria were used as for the skin/T-stiffenermodel. Based on data from literature [5,15], the following strength values were used to calculate

the failure indices in the lap joint materials:

Skin interlaminar tension: 3000 psi

Skin interlaminar shear: 5000 psi

Repair interlaminar tension: 4000 psi

Repair interlaminar shear: 6000 psi

Skin transverse (in-plane) tension: 5000 psiAdhesive Von Mises strain 0.05 in/in

Figure 25 shows the adhesive Von Mises strain failure index plotted along the entire

bondline. Higher stresses were observed at the repair laminate termination (left end) than at theskin laminate termination (right end). A survey of all three failure indices at both ends of the

joint indicated that the left end of the joint was more critical in all cases. This is likely becausethe flaperon laminate is thinner and less stiff (smaller percentage of 0 plies) than the repairlaminate, which results in more bending in the flaperon skin. Figures 26 and 27 show failure

index contour plots of the maximum transverse tensile stress criterion at P = 3000 lbs, and theinterlaminar tension-shear stress interaction criterion at P = 2400 lbs, respectively. The thickness

directions of the contour plots are exaggerated by a factor of 3 for clarity. These plots show thatthe critical location is in the 0 ply at the end of the repair adherend.

Damage is predicted to initiate as a delamination between the 0 ply and the 45 ply

above it. For the purposes of the damage growth modeling, it was assumed that a through-the-thickness matrix crack in the two 45 plies above the 0 ply would also occur. This behavior is

consistent with test results from similar tests reported in [20]. A summary of the predicteddamage initiation loads and location is shown in Figure 28. The adhesive is predicted to fail at


11/32


higher loads than the skin laminate, which is a desirable design condition and consistent with testresults on this type of bonded joint.

Single Lap Joint Static Strength Analysis

The predicted damage initiation load and location was used as the starting point for the

fracture mechanics based strength analysis. A crack was placed in the model at the left end ofthe adhesive bondline (end of the repair laminate) and then grown incrementally at theinterface between the top 45 and 0 skin plies to a total length of 1.10 inches, which represents

a maximum allowable damage size based on criteria. Figure 29 shows the deformed model for acrack length of 0.72 inches. The smallest element size along the delamination was 0.00444inches.

The model was then run for each increment of crack growth. As in the skin/T-stiffeneranalysis, the total strain energy release rate (Gtot) and the fracture mode mix (GII/Gtot) were

calculated and plotted as a function of crack length as shown in Figures 30 and 31. Figure 31shows that as the crack opens from 0.05 inches to 0.50 inches, the mode mix shifts from mode Idominated fracture (opening mode) to mode II dominated (in-plane shear mode), then remains

fairly constant as the crack continues to grow to 1.10 inches. From this curve, the mode mix atany crack length can be determined.

The mode mix at each chosen crack length (in this case, increments of 0.15 inches wereused) is then compared to fracture toughness test data to determine the critical fracturetoughness, Gtot,crit (Figure 32). As test data were not available for IM6/3501-6 at 65F,

estimates were based on data for similar materials [5,6,16]. Crack growth is predicted at theload, Pgrowth, at which Gtot is equal to Gtot,crit . Interpolation was used to determine Pgrowth for each

crack length.

The values of Pgrowth vs. crack length were then plotted using the same method as for the

skin/T-stiffener joint. As shown in Figure 33, Pgrowth at the initial crack length (ainit = 0.05inches), is lower than the predicted damage initiation load, Pinit = 1875 lbs. As can be seen in thefigure, Pinit corresponds to a crack length of 0.25 inches. This indicates that as soon as damage

initiates, the crack will grow to this length. After that, additional load is required to continuecrack growth, since the slope of the Pgrowth vs. a curve is still positive in that region. Once themaximum static load (Pgrowth,static = 2028 lbs.) is reached at a = 0.50 inches, the crack becomes

unstable and continues growing to the critical length. Note that in this case, static strength is notdependent on the chosen initial crack length (assuming the chosen initial length is less than 0.50

inches). That is, the same maximum static load will be predicted for any initial crack crack sizebetween 0.05 inches and 0.50 inches, since regardless of the initial length, 2028 lbs will berequired to grow the crack to its critical length. This is in contrast to the skin/T-stiffener

example where, if a larger initial crack size had been chosen, a lower static strength would havebeen predicted (Figure 18).

Single Lap Joint Fatigue Life Analysis

The durability of the single lap joint under fatigue loading was then assessed in the samemanner as for the skin/T-stiffener joint. Again, the fatigue crack path was assumed to be the

same as the static crack path and the calculated change in total strain energy release rate,Gtot,FEM, was compared to crack growth rate test data to determine the predicted crack growth


12/32


rate at a given crack length for each load level, Figure 19. Pmax vs. N curves were thendeveloped for several R-ratios as shown in Figure 34. The dashed lines show the results from the

skin/T-stiffener joint for comparison. For constant amplitude loading, the Pmax vs. N curve forthe corresponding R-ratio can be used to directly determine the number of cycles to failure. Forexample, for Pmax = 1358 lbs (67% of predicted ultimate static strength), the cycles to failure at

an R-ratio of 0.10 are predicted to be 132,569. The cycles to failure in this example are based on

an arbitrary maximum allowable damage size of acrit = 1.10 inches. This critical length wouldtypically be determined by criteria or by residual strength requirements.

Single Lap Joint Summary of Predictions

Damage Initiation Load: Pinit = 1875 lbs

Delamination in 0 tape skin ply will open to 0.25 inches once damage initiates

Static Strength: Pgrowth,static = 2028 lbs

Unstable crack growth at crack length = 0.50 inches

Fatigue Life: (assuming joint failure at crack length = 1.10 inches)

Low cycle fatigue, Pmax = 1358 lbs -->132,569 cycles

While directly comparable test results for this configuration were not available, thepredicted damage locations, loads, and cycles to failure are consistent with similar test data as

reported in Reference 20.

CONCLUSIONS

It has been shown that the analysis approach presented here for composite bonded joints

can be used for predicting critical failure modes, damage initiation loads and locations, staticstrength, residual strength, and fatigue life. The analysis approach was applied to two different

joint configurations. Only a single delamination location was analyzed for each configuration, inorder to demonstrate the analysis approach. For a complete analysis of a given configuration,several potentially critical delamination locations would be evaluated. The fracture mechanics

analysis in particular has demonstrated the ability to:

Predict crack growth stability under static loads

Predict static ultimate strength and critical crack lengths

Predict crack growth under fatigue loads

Accommodate a variety of durability and damage tolerance criteria related to initial flawsizes and critical lengths.

These results have been achieved through the use of basic material fracture toughnessdata, and without reliance on complicated and controversial stress-based failure criteria. This


13/32


analysis approach has the potential to be very useful for damage tolerance analyses of bondedand composite structure by:

Using the shape of P vs. a curve to select critical crack size for residual strength analysis.

Predicting residual strength to compare and validate designs

Predicting crack growth under repeated loads to select inspection methods and intervals.

Substantial material and geometric non-linearity was observed in the modeling, which

indicates that a non-linear analysis is required to properly address the structural behavior. Also,due to the time intensive nature of the post processing of finite element model results,automation of the analysis would be essential for practical applications.

REFERENCES

1. Composite Materials Handbook, Mil-Handbook-17, Volume 3E, Section 5.2, January 1997.

2. Johnson, W.S., et al., Applications of Fracture Mechanics to the Durability of BondedComposite Joints, FAA Final Report DOT/FAA/AR-97/56, 1998.

3. Murri, G.B., OBrien, T.K., Rousseau, C., Fatigue Life Methodology for Tapered

Composite Flexbeam Laminates, NASA Tech Memo 112860, 1997.

4. Minguet, P. J. and OBrien, T. K., Analysis of Skin/Stringer Bond Failure Using a Strain

Energy Release Rate Approach, Proceedings of the Tenth International Conference onComposite Materials (ICCM-X), Vancouver, British Columbia, Canada, August 1995.

5. Minguet, P.J., Analysis of the Strength of the Interface between Frame and Skin in a

Bonded Composite Fuselage Panel, Proceeding of the 38th AIAA Structures, StructuralDynamics and Materials Conference, 1997.

6. Johnson, W.S., Mall, S., A Fracture Mechanics Approach for Designing Adhesively BondedJoints, NASA Tech Memo 85694, September, 1983.

7. Hildebrand, M., The Strength of Adhesive-bonded Joints between Fibre-reinforced Plastics

and Metals, Technical Research Centre of Finland, 1994.

8. Adams, R. D. and Wake, W. C., Structural Adhesive Joints in Engineering, Elsevier Applied

Science Publishers, London, 1984.

9. Grant, P., Analysis of Adhesive Stresses in Bonded Joints, Symposium: Joining in FibreReinf. Plastics, Imperial College, London, I.P.C. Science and Technology Press, 1978, p. 41.

10.Fernlund, G., et al., Fracture Load Predictions for Adhesive Joints, Composites Scienceand Technology, Vol. 51, pp. 587-600, 1994.

11.Charalambides M.N., et al., Strength Prediction of Bonded Joints, 83rd Meeting of theAGARD SMPBolted/Bonded Joints in Polymeric Composites, 1997.


14/32


12.Wang, J.T., Sleight,D.W., Raju,I.S., Martin,R.H., and OBrien,T.K.,Computational Methodsfor Using Shell Elements in Skin Stiffener Disbonding Analysis, NASA CP 3229, 1993.

13.Rybicki, E.F. and Kanninen, M.F., A Finite Element Calculation of Stress Intensity Factorsby a Modified Crack Closure Integral,Engr. Fracture Mechanics, Vol. 9, 1977, pp931-938.

14.Mall, S., Ramamurthy, G., and Rezaizdeh, M. A., Stress Ratio Effect on Cyclic Debondingin Adhesively Bonded Composite Joints, Composite Structures, Vol. 8, 1987, pp. 31-45.

15.Tsai, Stephen W., Composites Design, 3rd Ed, Think Composites, Dayton, OH, 1987.

16.Ilcewicz, L. B., Keary, P. E. and Trostle, J., Interlaminar Fracture Toughness Testing ofComposite Mode I and Mide II DCB Specimens, Polymer Engineering and Science, May1988, Vol. 28, No. 9.

17.Schaff, J.R., Davidson, B.D., Life Prediction Methodology for Composite Structures, Parts Iand II,Journal of Composite Materials, Vol. 31, No. 2/1997.

18.Krueger, Ronald, Cvitkovich, Michael K., O'Brien, T. Kevin and Minguet, Pierre J., "Testingand Analysis of Composite Skin/Stringer Debonding Under Multi-Axial Loading," Journalof Composite Materials, Vol. 34, No. 15/2000.

19.Cvitkovich, M., OBrien, T.K., Minguet, P., Fatigue Debonding Characterization inComposite Skin/ Stringer Configurations, NASA Tech Memo 110331/Army Research Lab

Report 1342, April 1997.

20.Stewart, M., An Experimental Investigation of Composite Bonded and/or Bolted RepairsUsing Single Lap Joint Designs, Bell Helicopter Textron Report 299-100-779, 26 January

1999/PhD. Thesis, University of Texas at Arlington, December 1996.

Table 1: Lamina Material Properties

IM7/8552

Grade 160

Tape

IM6/3501-6

Grade 145

Tape

AS4/3501-6

5HS Fabric

E1 20.7 23.8 9.5 Msi

E2 1.65 1.57 9.5 Msi

E3 1.65 1.57 1.57 Msi12 0.34 0.32 0.0513 0.34 0.32 0.32

23 0.45 0.45 0.32G12 0.65 0.89 0.87 MsiG13 0.65 0.89 0.87 Msi

G23 0.65 0.623 0.87 Msi


15/32


Table 2: Adhesive Material Properties

Adhesive Temperature

G elastic

(psi) 12

E elastic

(psi)

Tau

elastic

(psi)

Tau max

(psi) plastic

FM-300 70F 200000 0.34 536000 4000 5000 0.300Magnolia 6363 -65F 135000 0.34 361800 5800 9820 0.231

Figure 1: Common Failure Sequence for Composite Bonded Joints (Showing AdherendDelamination Due to Peel Stresses in the Joint)

Joint Configuration and Loads

Database Analysis

Joint Configuration

and Loading Input

Joint Geometry

Critical Loads

Fatigue Spectra Materials

Environments

Static Analysis Results

Ultimate load

Crack stability

Fatigue Analysis Results

Cycles to failure

P vs. N

Spectra

Global Loads from

Global FE Model

Sub-element Loads

from non-linear FEM

Material Properties and Criteria

Fracture Mechanics

DamageInitiationAnalysis Results

Initial damage load

Damage mechanism Location

X

Y

Z

VLC

Local Bondline FE Model

Local FE Model w/Crack

X

Y

Z

V2L1

C11

Output Set: Step 1, Inc 5

Deformed(0.315): Total Translation

Strength of Materials

Fracture ToughnessData

Stiffnesses and

nonlinear propertiesStrength Data

Structural DesignCriteria

Fatigue Data

Figure 2: Outline of Bonded Joint Analysis Approach


16/32


Local FEM with Introduced Crack

Strain Energy Release Rates (GI, G

II,

Gtotal

) for Multiple Crack Lengths at

Several Load Increments

Results Combined with Material

Gtotal,critical Data to Obtain G total,criticalvs.

Crack Length Curve

Pgrowth

Values Calculated for Each

Crack Length

Crack

Gtotal

Crack Length, a

a1

a2

a3

a4

a5

a6

a7

Pa1P

a2 Pa3Pa4 P

a5 Pa6 Pa7

IncreasingLoad

+ =

FEM & VCCT Test Data

GII/Gtotal

Crack Length, a

Gtot,crit

G II / Gtotal

Pgrowth

Crack Length, a

a1

a2

a3

a4

a5

a6

a7

Pa1

Pa2P

a3 Pa4

Pa5

Pa6P

a7

(A) Crack Arrest

(B) Unstable Growth

Figure 3: Static Strength Analysis Procedure

- Negative slope means crack is unstable;

once Pgrowth for a init is reached, joint will fail

- Positive slope means additional load

required to grow crack

Pgrowth,static

= Static Strength

(C)

Pgrowth

Crack Length, a

Positive Slope

Pgrowth,static

ainit

acrit

based

on

criteria

(D)

Pgrowth

Crack Length, a

Pgrowth,static

acrit

based

on

criteriaa

init

Crack Arrest

Pgrowth

Crack Length, a

Negative

Slope

Pgrowth,static

ainit

(B)

More Load

Required toGrow Crack

(A)

Pgrowth

Crack Length, a

Negative Slope

Pgrowth,static

ainit

Determination of Pgrowth,staticfor four possible shapes ofload vs. crack length curve

UNSTABLE

STABLE / UNSTABLE

UNSTABLE / STABLESTABLE

Figure 4: Static Strength from Pgrowth Residual Strength Curves


17/32


Pthresh

NP1

NP2 NP3 NP41 Nrunout

NPthresh

P1

P2

P3

P4

Pgrowth

Cycles (N)

Load(P)

a / (da/dN) = N at a i, Pj

da/dN(in/cycle)

Gtot

Crack growth rateat given a

i, P

j

Gtot

at ai, P

jfrom

FEM

Test Data

For Each Load Level, Calculate SERR, Gtotal

, for

Series of Crack Increments, a

Using Material da/dN Data, Calculate CrackGrowth Rate and Divide By a to Obtain Numberof Cycles, N, to Grow Crack bya

Sum Up N From ainit to acritical To Obtain CyclesTo Failure, N

P

Plot NP Results For All Load Increments

Crack Length, aainit acrit

P1

P2

Pj

P4

PFEM

Gtot,max at ai,Pj

ai

P3

Gtot,min at ai,P

j

Gtot = Gtot,max - Gtot,minIn

cre

asin

gLoad

Figure 5: Fatigue Life Analysis Procedure Using Crack Growth Approach


18/32


1 1P/2

1

2

P = 50 lb

Symmetric B.C.

Flange Tip

Frame or stiffener

Flange Tip of flange

SkinBondline

Since Critical Location Known to beFlange TIP, FE Model Incorporates

Skin and Stiffener Flange Only.

Flange: [45/0/45/0/45/0/45/0/45] IM7/8552 Fabric

Skin: [45/-45/90/45/-45/0/-45/45/90/-45/45] IM7/8552 Tape

Adhesive: FM-300 Film

Figure 6: Skin/T-StiffenerFinite Element Model

Figure 7: Skin/T-StiffenerModel Detail at Flange Tip

45 Fabric

0 Fabric

Adhesive

90 Ta e

45 Ta e 2 lies)

0 Ta e3 Elements er Pl in Ti Re ion

Skin Panel

Tee Flan e


19/32


Skin/T-Stiffener Damage Initiation Model

Load vs Deflection at Stiffener Centerline

0

10

20

30

40

50

60

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Displacement at Center of Stiffener (Left End of Half Model) (inch)

AppliedLoad,P

(lbs)

Load-Displacement Curve

Linear Line

Figure 8: Skin/T-StiffenerPredicted Non-Linear Deflection

Figure 9: Skin/T-StiffenerThrough-Thickness Normal Stress

High peel stresses in adhesive

and top skin ply


20/32


Figure 10: Skin/T-StiffenerThrough-Thickness Shear Stress

Figure 11: Skin/T-StiffenerMaximum Transverse Tensile Stress Failure Index

Large plastic strains inadhesive at flange tip

Contours shown for P = 30 lbs

Max Transverse Tensile Stress CriterionMatrix crack in top 45 skin ply predicted

Critical load: P = 25.6 lb.


21/32


Figure 12: Skin/T-StiffenerCFRP Interlaminar Tension-Shear Stress Interaction and

Adhesive Von Mises Strain Failure Indices

F.I. (2), P=36.2 lb.Interlaminar Stress

(Delamination)

F.I. (1), P=25.6 lb.

Max Transverse Tension

(Matrix Crack)

F.I. (3), P=45.4 lb.VonMises Strain

(Adhesive)

Figure 13: Skin/T-StiffenerPredicted Damage Initiation Loads and Locations

Adhesive VonMises Strain Criterion

Adhesive failure predicted

Critical Load: P = 45.4 lbs

CFRP Interlaminar Interaction CriterionDelamination in top skin plies predicted

Critical Load: P = 36.2 lbs

Contours shown for P = 50 lbsFailure index > 1.0 predicts damage initiation


22/32


Matrix crack in skin at tip of

adhesive followed by crack

growth between top two skinplies to a length of 0.40

Figure 14: Skin/T-StiffenerAnalyzed Crack Path

Gtotal versus Crack Length

Crack Between Skin Plies 1 (+45) and 2 (-45)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Crack Length, a (in)

Gtotal(in-lb/in^2

)

Data from FEM

Interpolated points for chosen crack lengths

P/PFEM = 1.0

PFEM = 50 lbs

(ainit) (acrit)

P/PFEM = 0.2

P/PFEM = 0.4

P/PFEM = 0.6

P/PFEM = 0.8

FE model is run to PFEM for a series of crack

lengths as the crack is opened from the

chosen initial crack length (0.05) to the

chosen critical crack length (0.40)

Figure 15: Skin/T-StiffenerStrain Energy Release Rate, (Gtot)FEM vs. Crack Length, a


23/32


Fracture Toughness Mode Mix Ratio (G II/Gtotal)

Crack Between Skin Ply 1 (+45) and Ply 2 (-45)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45


ModeMixRatio(G

II/Gtotal)

Calculated using FEM nodal

data & VCCT

Curve fit showing chosencrack length increments

Mode Mix Ratio shown for

PFEM = 50 lb, the applied

load to the FEM

chosen initial

crack size, ainit

chosen critical crack size,

acrit, based on critieria

Figure 16: Skin/T-StiffenerDetermination of Mode Mix for a Given Crack Length

Critical Fracture Toughness (Gtot,c) versus Mode Mix (GII/Gtot)

for IM7/8552 tape, RT, Estimated Data

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Mode Mix, GII/Gtot

Gtot,c

(in-lb/in2)

** Estimated Data **

100% G I

100% G II

Mode mix for chosen cracklen ths 0.05" < a < 0.40"

GII/Gtot

Gtot,c

Figure 17: Skin/T-StiffenerDetermination of Critical Fracture Toughness (Gtot,crit) fromFracture Toughness Data


24/32


Pgrowth versus Crack Length, a

Crack Between Skin Plies 1 (+45) and 2 (-45)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45


Pgrowth/PFEM

Additional load required to

propagate damage

Pgrowth = PFEM = 50 lbs

Max load at 0.866 --> Pgrowth,static = 43.3 lbs

(ainit) (acrit)

Negative slope indicates unstable crack

growth (i.e., lower load required for

propagation as crack length increases)

Curve can also be used to determine

residual static strength at a given cracklength during fatigue damage growth

(e.g. P residual,0.40 = 0.608 * 50 lbs = 30.4 lbs)

0.512 --> Pinit = 25.6 lbs

(damage initiation load)

Figure 18: Skin/T-StiffenerPredicted Residual Strength - Pgrowth vs. Crack Length, a

Crack Growth Rate (da/dN) vs. Strain Energy Release Rate (Gtot)

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

0.1 1.0 10.0 100.0

Log[Gtot] (in-lb/in^2)

Log[da/dN],(in/cycl

IM6/3501-6, -65 F, CLS, R = 0.1

IM7/8552, RT, CLS, R = 0.1


Gtot from FEM for a givenload level (P) and crack

length (a)

Crack growth rate

(da/dN) for a givenP and a

Figure 19: Determination of Crack Growth Rate from Test Data


25/32


Figure 20: Skin/T-StiffenerPredicted Cycles to Failure vs. Load Level and R-Ratio


26/32


0.50 0.50 0.35 0.50

1.94

Symmetric BCs

Flaperon Skin

[45/-45/0/45/-45/-45/45/-45/45]IM6/3501-6 tape

Repair Laminate

[45/0/0/45]

AS4/3501-6 fabricEnd Tabs

P

P = 3000 lb. (1.5 inch wide specimen)

Adhesive: Magnolia 6363 paste

Figure 21: Single Lap JointFinite Element Model

Figure 22: Single Lap JointModel Detail at End of Repair Laminate

Flaperon Skin

Laminate

Re air


27/32


Figure 23: Single Lap JointThrough-Thickness Normal Stress

Figure 24: Single Lap JointThrough-Thickness Shear Stress

Peel stresses in adhesiveand top skin plies

High shear stress in

adhesive and 0 skin ply


28/32


Flaperon Repair Lap Joint, Axial Load

Static Load Failure Indices in Adhesive

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.9 1.4 1.9 2.4 2.9

X Position

FailureIndex

Load = 3000 lb

Load = 2400 lb

Load = 18200 lb

Load = 1200 lb

Load = 600 lb

Von Mises Strain Criteria (vm_max = 0.05)

(Loads based on 1.5 inch wide specimen)

Adhesive Failure at 3096 lb

Figure 25: Single Lap JointAdhesive Von Mises Strain Failure Indices

Figure 26: Single Lap JointMaximum Transverse Tension Failure Index


Failure index > 1.0 predicts damage initiation

Max Transverse Tensile Stress Criterion

(Y Scale Exaggerated for Clarity)


29/32


Figure 27: Single Lap JointCFRP Interlaminar Tension-Shear Stress InteractionFailure Index

Figure 28: Single Lap JointPredicted Damage Initiation Loads and Locations


Failure index > 1.0 predicts damage initiation

CFRP Interlaminar Interaction Criterion

Delamination in 0 skin ply predicted

Critical Load: P =1875 lbs

P =1875 lbs

Interlaminar Stress

P =3096 lbs


30/32


Figure 29: Single Lap JointModel with Skin Delamination

.

Gtotal versus Crack Length

Crack Between Skin Plies 2 (-45) and Ply 3 (0)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2


Gtotal(in-lb/in

^2)

Data from FEM

Interpolated points for chosen crack lengths

P/PFEM = 1.00

PFEM = 3000 lbs

(ainit) (acrit)

P/PFEM = 0.200

P/PFEM = 0.388

P/PFEM = 0.556

P/PFEM = 0.756

FE model is run to P FEM for a series of crack

lengths as the crack is opened from the

chosen initial crack length (0.05) to the

chosen critical crack length (1.10)

Figure 30: Single Lap JointStrain Energy Release Rate, (Gtot)FEM vs. Crack Length, a

Matrix crack in skin at tip of adhesive followed

by crack growth between skin plies 2 and 3 to

a to a length of 1.10 inches

Deformations and Y-scale exaggerated for clarity


31/32


Fracture Toughness Mode Mix Ratio (G II/Gtotal)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20


ModeMixRatio(GII/Gtotal)

Calculated using FEM nodaldata & VCCT

Curve fit showing chosencrack length increments

chosen initial cracksize, ainit = 0.05"

chosen critical crack size,acrit , based on critieria

Mode Mix Ratio shown forPFEM = 3000 lb, the applied

load to the FEM

Figure 31: Single Lap JointDetermination of Mode Mix for a Given Crack Length

Critical Fracture Toughness (Gtot,c) versus Mode Mix (G II/Gtot )

for IM6/3501-6 tape, -65F, Estimated Data

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Mode Mix, GII/Gtot

Gtot,c

(in-lb/in

2)


100% GI

100% GII

Mode mix for chosen crack

len ths, 0.05" < a < 1.10"

GII/Gtot

Gtot,c

Figure 32: Single Lap JointDetermination of Critical Fracture Toughness (Gtot,crit) fromFracture Toughness Data


32/32

Pgrowth versus Crack Length, a


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20


Pgrowth/PFEM

Pgrowth = PFEM = 3000 lbs

Max load at 0.676 --> Pgrowth,static = 2028 lbs

(ainit) (acrit)

Negative slope indicates unstable crack

growth (i.e., lower load required forpropagation as crack length increases)

0.625 --> Pinit = 1875 lbs

(damage initiation load)

(0.05")

Pgrowth vs. a curve indicates that crack will open

to 0.25" once damage initiates (at Pinit) then

require more load to open to 0.50". The crackwill then become "unstable" as shown.

Figure 33: Single Lap JointPredicted Residual Strength - Pgrowth vs. Crack Length, a

Load Ratio (Pmax / PFEM) vs. Cycles (N)


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.E+00 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12

Log[Cycles, N]

Pmax

/Pgrowth,static

R = 0.75

R = 0.5

R = 0.1

Pgrowth,static = 2028 lbs

P vs. N curves are developed for a

series of R-ratios and used to

address both constant applitude

and spectrum fatigue loading

Figure 34: Single Lap Joint Predicted Cycles to Failure vs Load Level and R Ratio

Documents

JCTR Bonded Joints Paper