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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
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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:
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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
==
====
=
=
+
+=
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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
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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].
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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.
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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.
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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,
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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
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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
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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
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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
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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.
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Published in Journal of Composites Technology and Research, 2002 Page 14
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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
Crack Length, a (in)
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
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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
Crack Length, a (in)
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
** Estimated Data **
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
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Figure 20: Skin/T-StiffenerPredicted Cycles to Failure vs. Load Level and R-Ratio
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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
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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
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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
Contours shown for P = 3000 lbs
Failure index > 1.0 predicts damage initiation
Max Transverse Tensile Stress Criterion
(Y Scale Exaggerated for Clarity)
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Figure 27: Single Lap JointCFRP Interlaminar Tension-Shear Stress InteractionFailure Index
Figure 28: Single Lap JointPredicted Damage Initiation Loads and Locations
Contours shown for P = 2400 lbs
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
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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
Crack Length, a (in)
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
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Fracture Toughness Mode Mix Ratio (G II/Gtotal)
Crack Between Skin Plies 2 (-45) and Ply 3 (0)
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
Crack Length, a (in)
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)
** Estimated Data **
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
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Pgrowth versus Crack Length, a
Crack Between Skin Plies 2 (-45) and Ply 3 (0)
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
Crack Length, a (in)
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)
Crack Between Skin Plies 2 (-45) and Ply 3 (0)
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