Abstract A double cantilever beam specimenloaded with uneven bending moments (DCB-UBM) is proposed for mixed mode fracturemechanics characterisation of adhesive joints, lam-inates and multilayers. A linear elastic fracturemechanics analysis gives the energy release rateand mode mixity analytically for both isotropic andorthotropic materials. By varying the ratio betweenthe two applied moments, the crack tip stress statecan be varied from pure mode I to pure mode IIfor the same specimen geometry. The specimen al-lows stable crack growth. A special test fixture isdeveloped to create uneven bending moments. Asa preliminary example, the DCB-UBM specimenwas used for characterising fracture of adhesivejoints between two laminates of thermoset glassfibre reinforced plastic.
Keywords Delamination · Fracture · Mixedmode · J integral
B. F. Sørensen (B) · K. Jørgensen · R. C. ØstergaardMaterials Research Department, Risø NationalLaboratory, P.O. Box 49, Frederiksborgvej, DK-4000Roskilde, Denmarke-mail: [email protected]
T. K. JacobsenLM Glasfiber, Rolles Møllevej 1, DK-6640Luderskov, Denmark
Nomenclatured distance between measurement pointsh thickness of core layer in sandwich speci-
mens� moment arms spacing between the rollersB width of specimenD position of the neutral axis of a bimaterial
beamE Young’s modulusE1 Young’s modulus of core material in sand-
wich specimensE2 Young’s modulus of beam material in
sandwich specimensE11 Young’s modulus (in the x1-direction) of
the orthotropic specimensE22 Young’s modulus (in the x2-direction) of
the orthotropic specimensG shear modulusG12 shear modulus in the x1–x2 plane of the
orthotropic specimenG energy release rateH beam heightI0 non-dimensional moment of inertiaI1 non-dimensional moment of inertiaJ0 initial value of fracture resistanceJext J integral evaluated along external bound-
ariesJR fracture resistanceKI mode I stress intensity factor
164 B. F. Sørensen et al.
KII mode II stress intensity factorM1 bending moment applied to beam # 1M2 bending moment applied to beam # 2P applied forceR radius of rollerδ∗ end-opening of the bridging zoneη ratio between thickness of core and beam
(η = h/H)θ rotation angle of transverse beamλ non-dimensional orthotropic parameterν Poisson’s ratioρ non-dimensional orthotropic parameterψ phase angle of stress intensity factorE extensometer displacement non-dimensional measure of position of
neutral axis ( = D/h)ext integration path along the external bound-
aries of a specimen� non-dimensional curve-fitting parameter� stiffness ratio of sandwich (� = E1/E2
under plane stress)
1 Introduction
Many modern components and constructions, frommicrochips to ships and large wind turbine bladesare made of materials arranged in layers. Mixedmode cracking is commonly observed in layeredstructures, since they often have weak planes suchas interfaces between individual layers. Examplesof mixed mode failure modes are splitting cracks,delamination of laminates and interface cracks insandwich structures and adhesive joints. Undermixed mode cracking, the failure process zone issubjected to both normal and shear stresses. Ear-lier studies have shown that the interfacial fractureenergy, expressed in terms of the critical energyrelease rate, Gc, can depend on the mode mixity.Usually, the critical energy release rate increaseswhen the amount of tangential crack opening dis-placement (“mode II”) near the crack tip becomeslarger than the normal crack opening displace-ment (“mode I”) (Cao and Evans 1989; Wang andSuo 1990; Thouless 1990; Liechti and Chai 1992).The increase in macroscopic fracture energy withincreasing amount of crack tip sliding has beenattributed to various mechanics, such as crack facecontact by asperities near the crack tip (Evans and
Hutchinson 1989), to differences in the crack tipplasticity (Tvergaard and Hutchinson 1993) and toelectrostatic effects between the crack faces (Liangand Liechti 1995).
A number of Linear Elastic Fracture Mechanics(LEFM) mixed mode and mode II test configura-tions have been developed, such as the asymmet-ric DCB-specimen, End-Notched-Flexure (ENF),Cracked Lap Shear (CLS) and Mixed Mode Bend-ing (MMB) (Williams 1989; Hashemi et al. 1990;Reeder and Crew 1992; Fernlund and Spelt 1994).Each method has its advantages and drawbacks;see e.g. Shivakumar et al. (1998) for a recent over-view of the most used test methods. For instance,the MMB test configurations proposed by Ree-der and Crew (1992) and by Fernlund and Spelt(1994) both have the advantage that the entiremode mixity range from pure mode I to pure modeII can be obtained by the same specimen geometry.However, crack growth may be unstable for somemixed mode and mode II specimen configurations,such as the MMB and the ENF tests (Shivakumaret al. 1998; Ozdis and Carlsson 2000). Further-more, mode II specimens where a transverse forceis transmitted between the crack faces are suscepti-ble to friction between the points where the trans-verse forces are applied (Carlsson and Gillespie1989; Williams 1989; Hashemi et al., 1990) – thefriction coefficient between delaminated surfacescan be as high as 0.4 – 0.6 (Schön 2000). By the useof mode II specimens where a transverse force istransmitted between the crack faces it is thereforedifficult to investigate whether toughening mech-anisms occur and to quantify their toughening ef-fects.
It is desirable to characterise mixed mode crack-ing parameters by specimens that (i) enable testsunder the full range of mode mixity, so that thesame specimen geometry can be used for all modemixities, reducing possible error sources associatedwith processing differences, and (ii) allow stablecrack growth for all mode mixities. A further com-plication is that some composites generate fibrebridging during cracking. This leads to a signifi-cant rise in the fracture resistance (Albertsen et al.1995). In many cases, large scale bridging (LSB)develops, which makes a LEFMs analysis invalid(Bao and Suo 1992). Instead, the analysis of the testspecimen can be based on the J integral (Suo et al.
DCB-specimen loaded with uneven bending moments 165
1992). Then, (iii) it is preferable to use fracturemechanics test specimens for which the J integralcan also be obtained in analytical form under largescale bridging.
The purpose of the present paper is to developa fairly general fracture mechanics test configura-tion for characterising mixed mode crack growth.The proposed test configuration fulfils the require-ments described above, including the capability ofcharacterizing large scale bridging by the J integral.
The paper is organised as follows: First, the basicmechanics of the proposed specimen is presented.Next, we describe the practical implementationof the test method. Finally, we illustrate the testmethod by fracture mechanical characterisation ofan adhesive joint.
2 Description of test configuration
The proposed test configuration, a double canti-lever beam (DCB) specimen loaded with unevenbending moments (DCB-UBM) at the two beams,is shown in Fig. 1a. In the following we analysethe DCB-UBM specimen by the usual assumptionsof a small-scale failure process zone, small strains,small displacements and small rotations. The planestress energy release rate and mode mixity are ob-tained in closed form for linear-elastic isotropicand orthotropic materials. Plain strain is treated inAppendix A.
2MM
M
M +M1 2
M1
M2
M
M
H
H
H
H
H
H
Γext
(a)
(b)
(c)
Fig. 1 The double cantilever beam specimen loaded withuneven bending moments (DCB-UBM) (a), can be ob-tained by superposition of a pure mode I specimen (b) anda pure mode II specimen (c)
2.1 LEFM failure process zone parameters
LEFM is applicable when the failure process zoneis much smaller than the smallest specimen dimen-sion (for DCB-specimens, the beam height, H).Then, for crack propagation along a weak planein a homogeneous material, the singular crack tipstress field is characterised in terms of the energyrelease rate, G, and the mode mixity, ψ , which isdefined as the phase angle of the stress intensityfactors (Hutchinson and Suo 1992).
ψ = tan−1(
KII
KI
)(1)
with KII and KI being the mode II and mode I stressintensity factors, respectively. Equation 1 is validonly for homogeneous solids; a slightly differentdefinition is used for the mode mixity for cracksalong a bimaterial interface (Hutchinson and Suo1992).
An appropriate failure criterion for mixed modecrack propagation along a weak plane is of the form(Jensen 1990; Hutchinson and Suo 1992)
G = Gc(ψ) (2)
where Gc is the critical energy release rate. Equa-tion 2 implies that the critical energy release ratedepends on the mode mixity. Usually, Gc(ψ) in-creases significantly as ψ approaches 90◦.
2.2 J integral and mode mixity
The energy release rate can be calculated by eval-uating the path independent J integral (Rice 1968)along a path following the external boundaries ofthe specimen (Fig. 1a), ext. The only non-zerocontributions come from the beam-ends. When thebeams are longer than a few times the beam height,the stress states at the beam-ends are unaffected bythe stress field around the failure process zone ofthe crack. The beams are then subjected to purebending. Under these conditions, the J integralcalculation for the isotropic DCB-UBM specimenshown in Fig. 1a gives (plane stress)
G = J = 21(M21 + M2
2)− 6M1M2
4B2H3Efor |M1| < M2,
(3)
where M1 and M2 are the two applied moments,B is the specimen width, H is the specimen height
166 B. F. Sørensen et al.
and E is the Young’s modulus. For plane strainthe result should be multiplied by (1-ν2), whereν is the Poisson’s ratio. Since the J integral anal-ysis of the DCB-UBM specimen is valid for bothLEFM and LSB problems, we will subsequentlyuse the symbol J when referring specifically toLSB; otherwise for LEFM problems the symbolG will be used for the energy release rate. Notefrom (3) that the energy release rate of the DCB-UBM is independent of crack length. Thus, if theapplied moments are measured it is not neces-sary to measure the crack length during an exper-iment. Furthermore, for experiments conductedunder displacement control (“fixed grips”) the mo-ments will decrease during cracking, so that the en-ergy release rate will decrease. Then, crack growthshould be stable.
The DCB-UBM specimen (Fig. 1a) can be con-structed by a superposition of a pure mode I speci-men (Fig. 1b) and a pure mode II specimen(Fig. 1c), both analysed by Hutchinson and Suo(1992). Then, ψ can be obtained as
ψ = tan−1
(√3
2M1 + M2
M2 − M1
)|M1| < M2. (4)
Plots of G and ψ as functions of the ratio betweenthe moments are shown in Fig. 2. G and ψ are bothwell-behaving in the sense that no rapid variations,with respect to M1/M2, occur. The practical impli-cation is that it is not necessary to control M1/M2with a high degree of accuracy. Small changes inM1/M2 during an experiment do not change nei-ther G nor ψ significantly.
The analysis can be extended to specimens ofan orthotropic material. It has been shown (Suo1990; Suo et al. 1991) that for any simple connectedplane body of orthotropic material with stressesprescribed at its boundaries, the stress field de-pends on two non-dimensional elastic parameters,λ and ρ, defined by
λ = E22
E11ρ =
√E11E22
2G12− √
ν12ν21, (5)
where E11, E22, ν12, ν21 and G12 are the in-planeengineering constants (Young’s moduli, Poissons’ratios and the shear modulus); the subscripts indi-cate coordinate axis following standard notationof classic composite literature (Lekhinitskii 1981).Restrictions in the elastic properties lead to the
-1.0 -0.5 0.0 0.5 1.00
5
10
15(a)
Nor
mal
ised
Ene
rgy
Rel
ease
Rat
e
JB2 H
3 E11
/M22
Ratio between Moments, M1/M2
(b)
-1.0 -0.5 0.0 0.5 1.00
10
20
30
40
50
60
70
80
90
0.10.5
λ=E22/E11
=1
Mod
e M
ixity
,ψ (
degr
ees)
Ratio between Moments, M1/M2
Fig. 2 Plots of (a) the normalised energy release rate ofthe DCB-UBM specimen as a function of the ratio betweenthe applied moments, and (b) the resulting mode mixity afunction of the ratio between the applied moments
requirements λ > 0 and ρ > −1 (Suo 1990). Typi-cal values are λ = 0.4 and ρ = 2 for unidirectionalglass fibre composites and λ = 0.08 and ρ = 3 fora unidirectional carbon fibre composite with thefibre direction parallel to the x1-axis. For isotropicmaterials, λ = ρ = 1.
Evaluating the J integral along the externalboundaries of the orthotropic DCB-UBM speci-men gives (plane stress)
G = J = 21(M2
1 + M22
) − 6M1M2
4B2H3E11for |M1| < M2
(6)
The mode mixityψ of the DCB-UBM specimencan be determined analytically. A notable resultis that ψ(λ, ρ, M1/M2,) is independent of ρ for theorthotropic DCB-UBM,ψ = ψ(λ, M1/M2). This isan exact result, shown first by Suo (1990); a slightly
DCB-specimen loaded with uneven bending moments 167
different proof is given in Appendix B. With themode mixity of the isotropic specimen given by (4),the mode mixity of the orthotropic specimen canbe obtained by the use of the orthotropy rescalingtechnique (Suo et al. 1991). The result is
ψ(λ, M1/M2) = tan−1
{λ−1/4
√3
21 + M1/M2
1 − M1/M2
},
for |M1| < M2. (7)
The mode mixityψ is shown as a function of M1/M2in Fig. 2b for some values of λ. For fixed M1/M2,ψincreases (i.e., the amount of mode II stress inten-sity increases) with increasing orthotropy (decreas-ing λ for λ < 1).
2.3 Practical implementation of test fixture
The principle of creating different bending mo-ments in the two free beams of the DCB-UBMspecimen is shown schematically in Fig. 3. Forcesof identical magnitude, P, are applied perpendicu-lar to two transverse arms connected to the end ofthe beams of the DCB specimen. The un-crackedend of the specimen is restricted from rotation butcan move freely in the x1-direction. Different mo-ments are obtained if the length of the two moment
2 1
P
P
P
P
Transversearm
DCB-specimen
x2
x1
Fig. 3 Schematics of the proposed loading method; themode mixity is controlled entirely by altering the lengthof one of the moment arms, e.g. �1
arms, �1 and �2, of the transverse arms are uneven
M1 = P�1 and M2 = P�2. (8)
M1 and M2 are defined positive when they act inthe counter clockwise direction, as shown in Fig 1.
It follows from (7) and (8), that with one of thebeam arm fixed, say �2 ≥ 0, the mode mixity canbe changed simply by altering the other momentarm, �1, (the moment arms �1 and �2 are takenpositive when the left force of each moment actsin the x2-direction, as shown in Fig. 3).
Identical forces are obtained by the use of awire arrangement, see Fig. 4. The idea builds uponearlier fixtures for pure mode I testing (Freimanet al. 1973; Sørensen et al. 1996)1. A 1.5 mm thicksteel wire runs from the upper part of a tensiletest machine, mounted at a 2.5 kN load cell (model1210AJ-2.5KN-B, Interface Inc., Scottdale,Arizona, USA), via rollers to one of the trans-verse arms, down to rollers at the lower part ofthe tensile test machine and up again to anotherload cell in a similar manner in the other side ofthe fixture. The rollers at the transverse arms aremounted through holes. This allows easy and welldefined adjustments of the moment arms. Duringan experiment, the force in the wire increases asthe lower part of the test machine moves down-wards. The gravitational forces of the transversearms (made in aluminium) are outbalanced by heli-cal springs. Some additional considerations weremade to minimize errors in the applied momentsas the specimen deforms, see Appendix C.
Typical dimensions of the DCB-UBM specimensare shown in Fig. 5. Specimens can be cut out asrectangular bars from planar plates. Steel parts thatfit into the grips are fixed to each beam by four steelscrews (M5) and an adhesive.
3 A case study: strength of adhesive joints inpolymer matrix composites
We illustrate the usefulness of the DCB-UBM testconfiguration by studying the strength of adhesive
1 Since this work was completed, we became aware of arather similar test apparatus developed by Plausinis andSpelt (1995). Their test apparatus also applies uneven bend-ing moments to a DCB specimen. The two uneven momentsare made by two uneven forces that are connected via wiresto holes in the neutral axis of the beams.
168 B. F. Sørensen et al.
Fig. 4 Photos of the testset-up. Overview over thetest fixture
Laminate
Adhesive
Slip Foil
5
11
4.1
28
49.9
18
30
300
H
Hh
M5
60
Fig. 5 Sketch of the specimen geometry. Steel parts areattached to the laminates by screws and adhesive. Nominaldimensions: H = 8 mm, h = 3 mm and B = 30 mm
joints. The adherends are made of laminates ofplies of a unidirectional thermoset glass fibre rein-forced plastic. The lay-up of the adherend lami-nates was [±45/05]s i.e. almost unidirectional.Since the lay-up is symmetric, the laminates donot posses bending-twisting coupling. The speci-men is not elastically homogenous as presumedin the analysis in Section 2; the stiffness of theadhesive layer differs from the stiffness of the lam-inate. The presence of the adhesive layer is ac-counted for in the calculation of J; the appropriate
J integral solution for the sandwich specimen isgiven in Appendix D.
3.1 Manufacturing and testing of the DCB-UBMspecimens
DCB specimens were processed as follows. First,two plates (300 by 300 mm, thickness approximately8 mm) were made of a glass fibre/polyestercomposite. Details of the fibre and matrix typesare proprietary. The plates were made by hand-lay up of dry fibre bundles, followed by matriximpregnation by vacuum infusion and post-cured.After consolidation, the two plates were bondedtogether by a thermoset adhesive. A thin slip foilwas placed at the one end of the plates to act as apre-crack and ease crack initiation. Spacers wereused to control the thickness of the adhesive layer(3 mm). The adhesive was post-cured. Specimens,30 mm in width, were cut from the sandwich plates.Steel parts were fixed to each beam by four steelscrews (M5) and an epoxy adhesive (Scotch-WeldDP 460 from 3 M, hardened at 40◦C for 2 h). Thespecimen geometry is shown in Fig. 5.
DCB-specimen loaded with uneven bending moments 169
Prior to the measurements, the specimens werepre-cracked. The pre-cracking was conducted byloading the specimen by a near-symmetrical load-ing (M1/M2 = −0.45) until a load drop occurred(The near-symmetric loading was chosen to forcethe crack towards the adhesive-laminate interface— a symmetric loading might have caused crack-ing in the middle of the adhesive layer). A crackextension, typically about 5–10 mm in length, hadoccurred. The crack experienced fibre bridging.However, since the initiation had occurred froma thin inset and not a truly sharp crack tip, theinitiation value is not valid. Therefore, in order toenable the determination of the fracture tough-ness of a sharp un-bridged crack, the crack wasmachined. The cut was made in the adhesive layerby a band saw until 1–2 mm from the crack tip. Thecut was made in the adhesive layer at the laminateinterface, effectively removing most of the bridg-ing fibres. The purpose of this re-notching was tocreate a specimen that had a truly sharp crack tipwith very limited fibre bridging.
Following re-notching, steel pins were mountedin holes drilled in the laminates at the end-of theinserts. The pins were separated by a distance, d,see Fig. 6. An extensometer (Instron, type 2620–602), range ±2.5 mm, was mounted at the pins,in a way that it could rotate freely and therebyrecord the magnitude of the crack opening dis-placement, E, see Fig. 6. It should be noted thatE comprises both the end-opening of the bridg-ing zone, δ∗, and the elastic strain in the spec-imens from x2 = −d/2 to x2 = d/2; however,the latter is assumed to be so small that it can beneglected (Sørensen and Jacobsen 2000), so thatE ≈ δ∗. An LVDT was mounted parallel to thebeam to record the tangential displacements atthe end of the notch, see Fig. 4.
The specimen was loaded monotonically at aconstant displacement rate of 5 mm/min. Data(elapsed time, load, end-opening displacement)were collected at a personal computer using a 16 bitdata acquisition board (PCI-DAS6013 from Mea-surement Computering, MA, USA) and a dataacquisition programme (Labtech Notebook Pro,version 12.1, Laboratory Technology Corporation).Loading was continued until a stationary load level(indicating steady-state fracture resistance)was achieved.
BridgingZone
d
Un-bridgedCrack
Pin
d+∆E
(a)
(b)
Fig. 6 Schematics illustrating how the total crack openingdisplacement is measured by an extensometer mounted attwo pins, initially separated by a distance, d, before cracking(a) and (b) during cracking
3.2 Test results — DCB-UBM specimens
Figure 7 shows pictures of some cracking spec-imens under various mixed mode loadings. Thecrack propagated along the adhesive/laminateinterface towards the beam that was subjected tothe highest moment. After some crack extension(typically around 20 mm), a new crack formed atthe next interface within the laminate, betweenthe 0 and 45◦ plies (delamination). Subsequently,both the interface crack and the delaminationcrack grew simultaneously; usually the delamina-tion crack propagated at the highest rate. The
underlying cause for the formation of the delam-ination crack is not understood. Fibre cross overbridging was observed for both cracking planes,see Fig. 7. The measured fracture resistance of thepure interface crack and the combined interfacecracking/delamination differed for some speci-mens. The failure sequence is shown schemati-cally in Fig. 8.
JR was calculated from Eq. 21 valid for the sand-wich specimen by the use of the elastic data given inTable 1 and � = 0.1 and η = 0.35. Accounting forthe stiffness of the adhesive layer is not very signifi-cant for the specimens evaluated here. Neglectingthe adhesive layer, i.e. calculating JR from the Eq
6 (orthotropic specimen) and using the elastic datagiven in Table 2 (giving λ = 0.3 and ρ = 2), gave avalue approximately 10% lower than the sandwichEq. 21. The nominal mode mixity — also neglect-ing the adhesive layer — was calculated from (7)with λ = 0.3.
Figure 9 shows typical results; the fracture resis-tance, JR, as a function of end-opening, δ∗. The frac-ture resistance increases from a relative lowvalue, corresponding to unbridged cracks, to asteady-state level, as the fibre bridging zone evolves.For experiments conducted at a higher nominalmode mixity ψ , the fracture resistance increasesfaster and attains a higher steady-state level.
DCB-specimen loaded with uneven bending moments 171
Machinednotch
Interfacial cracking
Adhesive layer
Interlaminar cracking(delamination)
Interfacial cracking
45
0
0
(b)
(a)
Fig. 8 Schematics of the cracking modes in the DCB-UBM sandwich specimens. Initially, cracking occurs alongthe adhesive/laminate interface (a), however, a delamina-tion crack develops in the laminate (b). Subsequently, bothcracks propagate, while showing fibre bridging
Since the mode mixity ψ looses its property as aparameter characterising the failure process zoneunder LSB, attention will be focussed at the initia-tion of crack growth of unbridged cracks. The initi-ation fracture resistance, J0, i.e., corresponding toan unbridged crack, is shown in Fig. 10 as a functionof the nominal mode mixity, ψ . Although there issome scatter in the results, it appears that a higherψ gives a higher J0. The scatter may be partiallyattributed to differences in the amount of bridgingfibres following the re-notching. The experimentaldata was fitted (by eye) with the phenomenologicalcriterion proposed by Hutchinson and Suo (1992)
J0(ψ) = J00{1 − tan2[(1 −�)ψ]} (9)
where J00 is the initial fracture resistance at ψ = 0◦
and � is a dimensionless constant. A fitted curveis shown in Fig. 10; upper and lower bounds areshown as dash-dotted lines. The correspondingparameters are listed in Table 3.
4 Discussion
4.1 Drawbacks and advantages of the DCB-UBMtest specimen
The DCB-UBM test configuration proposed in thepresent study was developed for static fracturetests. However, the concept of applying uneven,pure moments may also be useful for studying fa-tigue. Indeed, the apparatus by Plausinis and Spelt(1995), which also uses a wire arrangement, hasbeen used successfully for cyclic fracture tests ofadhesive joints (Dessereault and Spelt 1997). How-ever, our test fixture uses long wires, and we thinkthat this loading method is probably not well suitedfor rapid load changes usually used in cyclic load-ing. Another obvious drawback of our DCB-UBMconfiguration is that a special fixture is required,but this is also the case for other mixed modespecimens (Reeder and Crews 1992; Fernlund andSpelt 1994; Shivakumar et al. 1998). The mixedmode bending methods proposed by Reeder andCrews (1992) only provides stable crack growthunder dominating mode I cracking; under domi-nating mode II, the crack growth can be unsta-ble (Ozdis and Carlsson 2000). An advantage ofthe present method (and the one of Plausinis andSpelt (1995)) is that crack growth is stable overthe entire mode mixity spectrum. Then, for anymode mixity, the fracture energy can be deter-mined from arrested cracks having truly sharp tips.The measurement is therefore not sensitive to de-tails of the crack starter (notch or film insert). Atest is performed rather quickly, since there is no
Table 1 Orthotropic elastic data for the composite lami-nate (thermoset glass fibre reinforced plastic with the stack-ing sequence [+45/− 45/010/− 45/+ 45])
E11(GPa) ν12(−) E22(GPa) ν21(−) G12(GPa)
34 0.27 10 0.08 4
Table 2 Isotropic elastic data for bimaterial specimen.Subscripts indicate material number (material 1: adhesive;material 2: composite)
E1(GPa) ν1(−) E2(GPa) ν2 (−)3 0.35 34 0.27
172 B. F. Sørensen et al.
0.0 0.5 1.0 1.5 2.0 2.5 3.00
500
1000
1500
2000
2500
3000
3500
4000
ψ = 90
ψ = 73
ψ = 24
Fra
ctur
e R
esis
tanc
e, J
R (
J/m
2 )
Total End Opening, δ* (mm)
˚
˚
˚
Fig. 9 Measured fracture resistance, JR, as a function ofthe nominal mode mixity
Table 3 Fitting parameters to J0(ψ)
J00(J/m2) �(−)
Upper bound 450 0.33Best fit 300 0.39Lower bound 125 0.42
need to interrupt the test for crack length mea-surements. The method allows load rate effects tobe investigated. The DCB-UBM is also well suitedfor studying effects of friction between the crackfaces since loading–unloading experiments can beconducted under pure mode II or any mode II-dominated stress state. Unlike mode II test config-urations where the test specimen is loaded withtransverse forces (Carlsson and Gillespie 1989),no contact force must be transmitted between thefracture planes for the DCB-UBM specimen. Forthe DCB-UBM specimen, crack face contact andfrictional interaction can, however, occur if thefracture surfaces are rough. Anyhow, frictional ef-fects are expected to be smaller for the DCB-UBMspecimen than e.g. for an ENF specimen. Finally,the J integral solutions hold true also under LSB.No crack length or compliance measurements areneeded for the calculation of J. The equations forthe fracture resistance (eqs. 3 and 6) are not inval-idated by the development of a large-scale bridg-ing zone (Bao and Suo 1992; Suo et al. 1992). Thisallows the determination of cohesive laws. This willbe pursued elsewhere.
0 10 20 30 40 50 60 70 80 900
500
1000
1500
J0(ψ)= J00 {1+tan2[(1-Λ)ψ]}
J00=300 J/m2
Λ =0.39
Initi
al In
terf
ace
Fra
ctur
e R
esis
tanc
e
J 0 (
J/m
2 )
Mode Mixity, ψ (degrees)
Experiments Best Fit Upper/Lower Bounds
Fig. 10 The initial fracture resistance as a function of thenominal mode mixity. The solid line represents the best fit;dash-dotted are upper and lower bound curves
4.2 Importance of orthotropic elastic parameters
For many well established composite fracturemechanics test configurations, such as the stan-dard DCB specimen loaded with wedge forces andthe end load split specimen, ψ depends on bothλ and ρ (Suo et al. 1991; Bao et al. 1992a). Thus,for those specimens, all five elastic constants E11and E22, ν12, ν21 and G12 must be known before ψcan be determined accurately. For some specimensψ depends only weakly on ρ, so that the depen-dency can approximately be ignored (Bao et al.1992a,b). However, as shown, ψ is exact indepen-dent of ρ for the orthotropic DCB-UBM. Thus,under plane stress neither the energy release ratenor the mode mixity of the orthotropic DCB-UBMspecimen depend on ρ. This has important impli-cation for practical fracture testing of orthotropicDCB-UBM specimens. Then, only two elastic con-stants must be known to determine G and ψ forplane stress: E11 and E22. This reduces the amountof experimental work required for determinationof elastic properties of orthotropic materials.
Summary and conclusions
A mixed mode test configuration, the DCB speci-men loaded with uneven bending moments(DCB-UBM), was proposed for fracture mechan-ics characterisation of mixed mode cracking. Theenergy release rate and mode mixity are given
DCB-specimen loaded with uneven bending moments 173
analytically for specimens made of isotropic andorthotropic materials. A special loading fixture basedon steel wires was developed.
Crack propagation in adhesive joints betweencomposites made of thermoset glass fibre rein-forced plastic was investigated. The initiation frac-ture energy of an unbridged crack, J0, was found toincrease with increasing mode mixity. In the earlystages of cracking, the crack propagated along theadhesive/laminate interface; fibre cross over bridg-ing occurred. In the later stages, a new crack formedat the next interface within the laminate; fibrebridging also occurred here. The fibre bridging re-sulted in a rising crack growth resistance.
Acknowledgements
This work was supported by an EFP2003 fund(No. 1363/03–0006) from the Danish Ministry ofEnvironmental and Energy. We thank Jens Ols-son, Christian H. Madsen and Morten D. Sørensenfor experimental help. We also wish to thank ananonymous referee for making us aware of the testapparatus developed by Plausinis and Spelt (1995).
Appendix A: Plane strain
For plane strain, the engineering orthotropic con-stants E11, E22, ν12 and ν21 should be replaced withE′
11, E′22, E′
12 and v′21, respectively in all equations,
including the definitions of λ and ρ. The planestrain parameters are (Bao et al. 1992)
E′11 = E11
1 − ν13ν31v′
12 = ν12 + ν13ν32
1 − ν13ν31
E′22 = E22
1 − ν23ν32v′
21 = ν21 + ν23ν31
1 − ν23ν32(10)
The shear modulus G12 remains unchanged.
Appendix B: Dependency of mode mixityon λ and ρ
For a crack running parallel to the x1-direction therelationship between the energy release rate, G,and the mode I and II stress intensity factors, KIand KII, can be written as (Suo 1990)
G = nE11
[λ−3/4K2I + λ−1/4K2
II] (11)
where the constant n depends on ρ only,
n =√
1 + ρ
2. (12)
As shown by Suo (1990), the combinationsλ−3/8KI and λ−1/8KII are independent of λ. Fur-thermore, by superposition the stress intensity fac-tors should depend linearly on M1 and M2. Thus,we can write
λ−3/8KI + iλ−1/8KII = (a + ib)M1 + (c + id)M2,
(13)
where i = (−1)1/2 and the four (for the time beingunknown) real constants a, b, c and d (units m−5/2)depend on E11, ρ and H but not on λ, M1 and M2.
The four unknown parameters are determinedby analysing four load cases. First we analyse thetwo load cases M1 �= 0; M2 = 0 and M1 = 0; M2 �=0 separately by inserting (13) into (11) to get G,which is then set equal to G calculated from (6).This gives
|a + ib|2 = |c + id|2 = 214n
1B2H3 (14)
so that (13) can be can written as
λ−3/8KI + iλ−1/8KII =√
212√
n
1BH3/2[−M1eiω + M2eiω+γ ]
. (15)
Here ω and γ are two non-dimensional phase an-gles (within the interval [−90◦, 90◦]) that remainto be determined. On dimensional grounds ω andγ should depend on the dimensionless parametersM1/M2, M1/E11H3, λ and ρ. However, due to lin-earity, ω and γ should not depend on M1/M2 andM1/E11H3. Furthermore, since the left hand sideof (15) is independent of λ, so must the right handside be. Thus, ω and γ should not depend on λ.They can only depend on ρ.
In order to determine γ , we consider a third loadcase, M1 �= M2 �= 0, insert (15) into (13), and putthe result (λ−3/8KI and λ−1/8KII) into (11). Thenwe obtain
G = 214
1B2H3E11
[M2
1 + M22 − 2M1M2 cos γ
].(16)
Setting (16) equal to (6) we find
cos γ = 17
(17)
174 B. F. Sørensen et al.
or γ = 81.78◦. Note that n cancels out so that γ isindependent of ρ.
With γ determined, we proceed to findω. As thefourth loading case we select one particular loadcombination, M2 = −M1 > 0. For this load case,the geometry and loading of the specimen is sym-metric. Therefore, the shear stress σ12 must vanishalong the symmetry line (x2 = 0). Consequently,the mode II stress intensity factor, KII, must bezero. Enforcing this condition to (15) gives
sinω + sin(ω + γ ) = 0. (18)
With γ given by (17) we obtain
tanω = −√
32
(19)
or ω = −40.89◦. It follows from (17) and (19) thatboth γ and ω are independent of ρ and λ. Withλ−3/8KI and λ−1/8KII being independent of λ andρ, the mode mixity ψ can be determined analyti-cally by orthotropic rescaling.
Appendix C: Effects of finite displacement, rota-tion and friction
The analysis of the test specimen was made underthe assumption of small displacements and smallrotations. This requires that the difference in beamrotation and deflections ahead and behind the fail-ure process zone (the path where the J integral isevaluated) must be small. However, the deflectionsand rotations of the complete test specimen canstill be finite. As the specimen is loaded, the twobeams of the DCB-beams deflect and rotate. Thetransverse arms also move and rotate, since theyare fixed to the ends of the DCB-specimen. Bothfinite displacements and finite rotation cause thetrue moments to differ from the nominal moments,Eq. 8. The moments are created by a wire that runsthrough rollers arranged as shown in Fig. 11. Thus,the true moment arm is
� = 2R + s cos θ , (20)
where R is the radius of the rollers, s is the spac-ing between the centres of the rollers, measuredalong the transverse beam arm and θ is the angleof the transverse arms with respect to the x2-axis.
In order to reduce the effect of rotation, the trans-verse arms were angled 10◦ in the direction oppo-site of the rotation that they would undergo duringloading. Thus, the transverse beam (the momentarm) reaches the neutral position (θ = 0◦) after arotation of 10◦. With this design, the error in theapplied moment will always be less than 6% aslong as the beam-ends rotate less than 20◦. Notefrom (20) that decreasing the spacing between therollers s, decreases the error. Thus, in most cases,the error will be significantly smaller than 6%.
In order to minimize the effect of the deflec-tion (in the x2-direction) of the specimen, the ver-tical distance (the x1-direction, Fig. 3) betweenrollers at the upper and lower parts of the testmachine should be maximized. In our set-up, thisdistance exceeds 2 m, so the error in the moment isvanishing.
The friction in the rollers (mounted at ball bear-ings) and wires were measured by pulling the wirealong its direction. The friction was found to in-crease linearly with the applied load. Based onthese experiments, the frictional moment duringtesting was estimated to be less than 3% of theapplied moment, i.e. so small that it could be ne-glected. This was confirmed by experiments; thedifference in the two load signals was insignificant.
Appendix D: DCB-UBM sandwich specimen
Sandwich specimens, e.g. specimens where two skinlayers are joined by a thin core layer that is muchthinner than other relevant specimen dimensions,
P
P
θ
s
Roller
Transversearm
Wire
R
Fig. 11 Geometric relationship between transverse beamrotation angle θ and moment arm �
DCB-specimen loaded with uneven bending moments 175
are often used for characterising interfacial crackgrowth in bimaterial specimens. This type of speci-men is attractive since, if cracking occurs along theinterface so that the core layer remains attached toone of the skin layers, the residual stresses do notcontribute to the energy release rate (Wang andSuo 1990). The strain energy in the adhesive layeris much smaller than that in the beams. Then, (3)and (6) are applicable. However, if the core thick-ness, h, is not much smaller than the thickness ofthe skins, H, the core thickness must be taken intoaccount, as in the following. Assuming that the coreand skin to possess isotropic elastic properties, theJ integral evaluated along the external boundariesof the sandwich specimen gives (plane stress)
G = J = 1B2H3E2
{M2
1
2η3I0+ 6M2
2
− (M1 + M2)2
2η3I1
}, (21)
where H and E2 denote the thickness and theYoung’s modulus, respectively, of the skin layers(material #2). For plane strain, the result shouldbe multiplied by the term 1 − ν2
2 , where ν2 is thePoisson’s ratio of material #2. The non-dimensionalparameters η, I0 and I1 depend on the stiffnessproperties and layer thickness as describedbelow. Equation (21) valid for both small-scalefailure process zone and for LSB problems. Thenon-dimensional parameters η, I0 and I1 used inEq. (21) are defined as follows (Østergaard andSørensen, in preparation):
I0 = 13
1η3 −
η2 + 2
η
+�(
1η2 + 1
n+2 − 2
η−+ 1
3
), (22)
η = hH
, (23)
M1
M2
M1+M
2
HhH
#1#2
#2D
Neutralaxis
Neutralaxis
Fig. 12 Definition of geometry and material numbers forthe DCB-UBM sandwich specimen
and
I1 = 112
(� + 8
η3
12η2 + 6
η
). (24)
The stiffness ratio parameter, �, is defined as(plane strain)
∑= E11 − ν2
2
E21 − ν21
, (25)
where E1 and E2 are the Young’s moduli of mate-rial #1 (the core layer) and #2 (the skins), respec-tively, and ν1 is the Poisson’s ratio of material #1.For plane stress (25) becomes � = E1/E2. Theparameter is a non-dimensional measure of theposition of the neutral axis (D denotes the distancefrom the top of the skin layer to of the neutral axisof the bimaterial beam), see Fig. 12,
= Dh
= 1 + 2�η +�η2
2η(1 +�η). (26)
The parameters I0, η,� and are identical to theones derived by Suo and Hutchinson (1990) in theiranalysis of bimaterial fracture specimens.
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