Composites Science and Technology 70 (2010) 1386–1394

Contents lists available at ScienceDirect

Composites Science and Technology

journal homepage: www.elsevier .com/ locate /compsci tech

Fracture characterization of sandwich structures interfaces under mode I loading

D.A. Ramantani *, M.F.S.F. de Moura, R.D.S.G. Campilho, A.T. MarquesDepartment of Mechanical Engineering and Industrial Management, Faculty of Engineering of Porto University, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal

a r t i c l e i n f o

Article history:Received 8 March 2010Accepted 17 April 2010Available online 24 April 2010

Keywords:A. SandwichB. Fracture toughnessC. Finite element analysis (FEA)E. Welding/JoiningCohesive damage model

0266-3538/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compscitech.2010.04.018

* Corresponding author. Tel.: +351 918087241; faxE-mail address: dimitraramantani@hotmail.com (D

a b s t r a c t

The accurate prediction of failure of sandwich structures using cohesive mixed-mode damage modelsdepends on the accurate characterization of the cohesive laws under pure mode loading. In this work,a numerical and experimental study on the asymmetric double cantilever beam (DCB) sandwich speci-men is presented with the objective to characterize the debonding fracture between the face sheet andthe core under pure mode I. A data reduction method based on beam theory was formulated in such away to incorporate the complex damaging phenomena of the debonding due to the material and geomet-ric asymmetry of the specimen, via the consideration of an equivalent crack length (ae). ExperimentalDCB tests were performed and the proposed methodology was followed to obtain the debonding fractureenergy (GIc). The experimental tests were numerically simulated and a cohesive damage model wasemployed to reproduce crack propagation. An inverse method was followed to obtain the local cohesivestrength (ru,I) based on the fitting of the numerical and experimental load–displacement curves. With thevalue of fracture energy and cohesive strength defined, the cohesive law for interface mode I fracture ischaracterized. Good agreement between the numerical and the experimental R-curves validates the accu-racy of the proposed data reduction procedure.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The sandwich structural concept, where two thin and strongface sheets are bonded to a thick and low density core, offers supe-rior specific flexural stiffness and strength and has received wideacceptance in weight critical structures such as airplane parts, shiphulls and wind turbine blades [1]. However, it is well known thatthe performance of sandwich structures depends primarily onthe integrity of the bond between the face sheet and the core[2–4]. The prediction of this type of damage has attracted muchinterest since such an attainment would increase the confidenceon the design of sandwich structures and subsequently expandtheir field of application.

Within the context of developing numerical methodologies tosimulate delamination as well as failure of adhesive bonds inmonolithic composite components (laminates), cohesive damagemodels have attracted much interest due to their well establishedadvantages compared to the stress based and fracture mechanicsmethods [5,6]. This methodology, including a mixed-mode formu-lation, has been extensively used in the aforementioned cases andhas provided results in excellent agreement with experimental andanalytical ones [7–9]. However, the application of cohesive damagemodelling in sandwich structures for the simulation of debondingfailure is still limited [10–13]. The good performance of cohesive

ll rights reserved.

: +351 225081584..A. Ramantani).

damage models depends primarily on the accurate determinationof the parameters characterizing the cohesive law, i.e. the fracturetoughness and local strength, Gc and ru, respectively. In mixed-mode formulations this requisite is translated to characterizationof the cohesive parameters of the material or interface under puremode loading, i.e., opening I, shearing II and tearing III. Therefore, itis of great importance the choice of the test method and datareduction scheme to predict the fracture energy. In the case ofsandwich structure, the bi-material interface between the stiff facesheet and the compliant core, as well as the geometric asymmetryof the sandwich fracture specimen, induce complex damaging phe-nomena, such as mode mixity at the crack tip (crack kinking initi-ation), crack tip and tab rotation, as well as crack tip stretching,which can make the application of common test methods and datareduction schemes problematic. Several researchers have investi-gated the fracture debonding of sandwich materials under modeI, analytically, numerically and experimentally, because the energyrequired to induce this failure mode is smaller than the shearingand tearing. In these studies, the double cantilever beam (DCB)[14,15], tilted sandwich debond (TSD) [16–20], single cantileverbeam (SCB) [21] sandwich specimen, and also modified versionsof them [22–24] have been analysed. Parameters like angle of theTSD specimen, face sheet and core thickness, core density andcrack length have been studied in order to meet the requirementsof pure mode I fracture [15,16,18,24]. Regarding the data reductionmethods commonly applied, they can be summarised incompliance calibration, area method and modified beam theory

D.A. Ramantani et al. / Composites Science and Technology 70 (2010) 1386–1394 1387

which includes modifications accounting for the effect of the coreon the global deformability of the specimen and local deformabi-lity of the crack tip [14,16,20–26]. Avilès and Carlsson [14], pre-sented an analysis of the compliance and energy release rate ofthe double cantilever beam (DCB) sandwich specimen followingtwo approaches. In the first, the arms of the DCB specimen wereconsidered as rigidly built-in (fixed) at the debond tip, and in thesecond an elastic foundation allowing for displacements of the de-bond tip was assumed. In the ‘‘elastic foundation analysis” (EFA),the upper arm of the specimen is considered partially supportedby an elastic foundation (the core is characterized by an exten-sional modulus j) rather than rigidly built-in [25]. Compliancewas determined for both approaches and the results were com-pared to finite element analysis (FEA) and to experimentally deter-mined values. It was shown that a foundation effect from the coreincreases the DCB specimen compliance quite substantially. Themodified beam theory (MBT) was followed for the determinationof fracture energy according to Shivakumar et al. [26]. The EFAand FEA results were in remarkable agreement, though lower thanthe experimental ones while the built-in beam analysis resultswere lower to a greater extent compared to the experimental. Itwas suggested that a way to incorporate the foundation effect inthe beam analysis is to consider an effective crack length by forcingcompliance equivalence between the EFA and the built-in analysis(aeff = a + D). An analytical expression for D was obtained and ver-ified against experimental values. Other works reported in refer-ences [17,25,26] emphasize the necessity to account for beamroot rotation at the crack tip since it was verified that this issuedrastically influences the specimen compliance.

In this paper, the asymmetric DCB sandwich test is studiednumerically and experimentally. A data reduction scheme basedon beam theory and specimen compliance is presented. The meth-od is based on crack equivalent concept and its formulation allowsincorporating all the complex damaging phenomena of the deb-onding due to the bi-material interface and geometric asymmetryof the specimen. Problems like measurement of the parameter Daccounting for crack tip rotation and deflection and also the cracklength monitoring during propagation are overcome. Experimentaltests on sandwich beams comprised of carbon/epoxy face sheetsover medium density PMI foam were conducted and the proposedmethod was used to calculate the fracture energy. Numerical sim-ulations of the DCB tests were performed with the use of ABAQUS�

software. A cohesive damage model was employed to simulatecrack propagation. This model is implemented in the numericalcode via interface elements placed between the face sheet andthe core. An inverse method was followed to obtain the remainingcohesive parameter (ru,I) based on fitting the numerical P–d curveswith the experimental ones. It was verified that the numericalR-curves agree with the experimental ones, thus validating theproposed data reduction scheme. The cohesive law characterizingdebonding between the face sheet and the core in the sandwichmaterial under mode I was obtained.

2. Compliance-based beam method

As already mentioned in the previous paragraph, the compli-ance calibration method, area method and the corrected beam the-ory are the data reduction schemes typically used to measure thefracture energy under mode I loading. However, the applicationof these methods depends on the rigorous measurement of thecrack length during propagation which can become a difficult task,depending on the specimen deformation and the natural appear-ance of the core material in which the crack grows (e.g. low densityfoams). In most cases, the crack length measurement is performedwith the use of an optical device, such as the optical microscope

placed in one or both sides of the specimen, or by stopping the ma-chine after crack propagation and identifying the crack tip[16,22,26]. Alternatively to crack length measurement, the relationbetween the compliance (C) and crack length (a) can be deter-mined by separate tests independently of the fracture test[25,27]. However, this method also presents disadvantages. First,it is more expensive and time consuming since several specimensmust be fabricated and tested to establish the C = f(a) relation. Sec-ond, the normal variability between different specimens consti-tutes an additional source of errors which, as will be shown, doesnot exist if the proposed procedure is followed. Third, the compli-ance calibration using several different initial crack lengths is usu-ally performed considering low levels of load that do not induceany damage. This procedure does not accurately describe the frac-ture phenomenon since in reality fracture takes place with a pro-cess zone ahead of the crack tip. This region can be nonnegligible (depends on the material) and influences the specimencompliance.

To overcome the aforementioned difficulties, a data reductionscheme based on the beam theory and measured specimen compli-ance named compliance-based beam method (CBBM) is proposed.In the CBBM method the compliance equation for the asymmetricsandwich DCB specimen is formulated following beam theory. Thetwo beams, the debonded upper face sheet (A) and the core andlower face sheet (B) in Figs. 1 and 2, named arms of the DCB sand-wich specimen, are considered fixed to the intact part of the spec-imen (cantilever beams) and the total strain energy is obtained asthe sum of the strain energy of each arm considering bending andshearing deformations

U ¼ UA þ UB

¼Z a

0

M2A

2DAdxþ

Z a

0

Z hs=2

�hs=2

s2A

2Gsbdydx

" #

þZ a

0

M2B

2DBdxþ

Z a

0

Z 0

� hc2þhs�l1ð Þ

s2BsðlowerÞ

2GsþZ l1�hc

2

0

s2BsðupperÞ

2Gs

"

þZ l1þhc

2

l1�hc2

s2Bc

2Gc

!bdydx

#ð1Þ

where Mi = Px (0 6 x 6 a) and Di (i = A, B) are the bending momentand stiffness of each arm respectively, l1 ¼ EshsðhcþhsÞ

2ðEshsþEchcÞ is the distance

of the neutral axis of the arm B from the neutral axis of the foam(Fig. 2),

sA ¼32

Pbhs

1� 4y21

h2s

!; �hs

26 y2 6

hs

2ð2Þ

is the equation giving the shear stresses through the thickness ofthe arm A and

sBc ¼PEc

2DB

hc

2þ l1

� �2

� y22

" #; l1 �

hc

2

� �6 y2 6 l1 þ

hc

2

� �ð3Þ

sBsðupperÞ ¼P

2DBEs l1 �

hc

2

� �2

� y22

!þ 2Echcl1

" #;

0 6 y2 6 l1 �hc

2

� �ð4Þ

sBsðlowerÞ ¼P

2DBEs l1 �

hc

2

� �2

þ 2Echcl1

!1� y2

2

hs þ hc2 � l2

1

� �0@

1A;

� hc

2þ hs � l1

� �6 y2 6 0 ð5Þ

-5

0

5

10

15

20

25

30

0 0.05 0.1 0.15 0.2

τB (analytical)

τB (numerical)

τB [MPa]

y 2 [

mm

]

Fig. 3. Analytical and numerical shear stress distribution along the thickness of thearm B.

Fig. 2. Debonded region of the asymmetric DCB specimen and definition of x–y coordinate systems for the beam analysis.

A

B Rohacell 71 IG (Ec, Gc)

Carbon/Epoxy 0o (Es, Gs)

L=240 mm

a =80 mm

b=30 mm

hs=1.5mm

hc=25mm

C

x y

z o

Fig. 1. Geometry and dimensions of the DCB sandwich specimen.

1388 D.A. Ramantani et al. / Composites Science and Technology 70 (2010) 1386–1394

are the equations giving the shear stresses along the thickness ofthe arm B. Specifically, Eq. (3) characterises the shear stressesdeveloped within the foam, and Eqs. (4) and (5) the shear stressesdeveloped within the part of the face sheet above the neutral axisand below the neutral axis, respectively (Fig. 2). The subscripts sand c designate the face sheet and the core, respectively. P is thetransverse load in each arm, Ei, Gi, i = s,c are the elastic propertiesof the components and b is the width. Eqs. (3) and (4) were obtainedfollowing sandwich beam theory as described in [28] by consideringa symmetric sandwich beam comprising of a composite plate with2 l1 � hc

2

� �thickness between two parts of foam with hc thickness

(Fig. 2). Eq. (5) was obtained considering that the shear stress distri-bution is characterized by a quadratic polynomial sBs(lower) =asy2

2+bs. The parameters as and bs were obtained by satisfying thecondition of null stresses at the bottom of the face sheety2 ¼ � hc

2 þ hs � l1� �� �

and stress continuity between the upper andlower parts of the face sheet (y2 = 0). The stress field given by theanalytical equations was validated via comparison with the one ob-tained from a numerical model of the arm B as a single cantileverbeam in bending. The model was built in ABAQUS� software withthe use of 8-node plane stress elements (CPS8) and the shear stres-ses were obtained at the integration points along the thickness ofthe beam. The good accuracy of the numerical and analytical stressfields, presented in Fig. 3, validates the analytical equations.

From Eq. (1) and by applying the Castigliano theorem, the dis-placement d is obtained

d ¼ @U@P¼ 1

3DA þ DB

DADB

� �Pa3 þ 12

10bhsGsþ F

� �Pa ð6Þ

where

F¼ b

4D2B

E2c

Gc

115

l1þhc

2

� �4 23hc

2�7l1

� �þ l1�

hc

2

� �3 23

l1þhc

2

� �3 "(

�15

l1�hc

2

� �2!#þ 1

Gsl1�

hc

2

� �8Es

15l1�

hc

2

� �2 "

�ðEs l1�hc

2

� �2

þ5Echcl1

!þ 2Echcl1ð Þ2

!#

þ 1Gs

815

Es l1�hc

2

� �2

þ2Echcl1

!2hc

2þhs� l1

� �24

359=; ð7Þ

D.A. Ramantani et al. / Composites Science and Technology 70 (2010) 1386–1394 1389

In the performed beam analysis, the compliance (C = d/P) givenby Eq. (6) implies zero rotation and displacement at the crack tipdue to the assumption that the arms are fixed to the intact partof the specimen. However, in reality, the two arms are elasticallyconnected to the intact part resulting in crack tip rotation anddeflection. Additionally, the higher bending stiffness of the lowerarm (B) leads to a slight rotation of the specimen at large openingdisplacements. The combined effects of all deformations result inan increase of the specimen compliance. It is suggested that a con-venient way to incorporate this phenomenon in the beam analysisis by considering an equivalent crack length, ae. The ae is calculatedfrom Eq. (6) by setting the compliance C equal to the specimencompliance registered during the test (C = d/P) where a is replacedby ae. The equivalent crack is considered to be the sum of threeparameters, ae = ao + D + DaFPZ. The parameter ao is the initial cracklength measured before the initiation of the test, the parameter Dstands for all the aforementioned deformation modes and theparameter DaFPZ for an additional phenomenon, the developmentof the fracture process zone (FPZ) ahead of the crack tip duringdamage propagation. To this matter it has to be noted that the en-ergy dissipated at the fracture process zone developed ahead of thecrack tip due to formation of micro-cracks (Fig. 4a and b) can belarge and for this reason it is important to be included in the se-lected data reduction scheme [29,30]. The solution of Eq. (6) canbe found using the mathematical software Matlab� (see AppendixA). The CBBM method presents two great experimental advanta-ges. First, the parameter D is not necessary to be defined experi-mentally [26] since it is indirectly incorporated in Eq. (6) via theconsideration of the measured specimen compliance which isinfluenced by the value of D. If necessary, this parameter can bedefined by setting Eq. (6) equal to the initial compliance C0 and ob-tain ae = a0 + D (DaFPZ = 0 at low levels of loading since no damage

(a)

(b)

micro-cracks

5 mm

Fig. 4. The asymmetric DCB sandwich test and (a) sub-interface crack prop

is present), and consequently the parameter D. Second, using thepresented methodology, the difficulty of crack length monitoringduring the test is overcome. This is an important advantage rela-tively to the classical data reduction schemes because the cracklength ae becomes a calculated value rather than a measured one.

The strain energy release rate in mode I can now be obtainedfrom the Irwin–Kies equation

GI ¼P2

2bdCda

ð8Þ

leading to

GI ¼P2

2bDA þ DB

DADB

� �a2

e þ12

10bhsGsþ F

� �� ð9Þ

This procedure provides an entire R-curve (GI = f(ae)), thus lead-ing to a clear identification of the fracture energy at the plateau.This will be detailed in the Section 3.1.

3. Experimental work

For the experimental work, a sandwich material comprising ofcarbon/epoxy face sheets over medium density PMI foam was se-lected. The geometry and the dimensions of the DCB specimenare presented graphically in Fig. 1. A thick core specimen was cho-sen in order to promote mode I crack propagation. It is demon-strated in many studies [15,20,22], that for thick sandwichspecimens with low density foam and low fracture toughness (asit is expected in this case since the PMI foam material is very brit-tle) the crack has the tendency to propagate inside the core, veryclose and parallel to the interface indicating mode I fracture. How-ever, in order to assure mode I loading conditions at the crack tip, a

micro-kink

(c)

5 mm

agation, (b) micro-cracks at the crack tip, (c) micro-kink phenomena.

Table 1CFRP components elastic orthotropic properties for a unidirectional ply [7] and elastic properties of the PMI foam core material.

Face sheets (carbon–epoxy 00) Core (Rohacell 71 IG, PMI Foama)

E1 = 109.0 (GPa) m12 = 0.34 G12 = 4.32 (GPa) E = 92 (MPa)E2 = 8.82 (GPa) m13 = 0.34 G13 = 4.32 (GPa) G = 29 (MPa)E3 = 8.82 (GPa) m23 = 0.38 G23 = 3.20 (GPa)

a Data provided by the manufacturer.

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0

0.04

0.08

0.12

0.16

0.2

0.24

85 95 105 115 125 135 145 155 165 175

P [

N]

δ [mm]

GI[N

/mm

]

ae [mm]

(a)

(b)

Fig. 5. Representative experimental P–d curve (a) and corresponding R-curve (b).

1390 D.A. Ramantani et al. / Composites Science and Technology 70 (2010) 1386–1394

numerical analysis of the DCB specimen was performed based oncohesive element modelling where the mode mixity at the cracktip was extracted. It was verified that the percentage of mode IIin the total energy is very low, near 3%, which confirms the domi-nance of mode I loading. The analysis is presented in detail inSection 4.2.

The DCB test specimens were cut from a sandwich panel[300 mm � 300 mm] fabricated at the Faculty of Engineering of Por-to University (FEUP). The sandwich panel was fabricated by adhe-sive bonding of the two face sheets to the core with the use of apress system. The face sheets consisted on unidirectional laminatesof CFRP, produced by hand lay-up and cured in a hot plates press.The laminates, with dimensions of 400 mm � 400 mm, were fabri-cated following a 0� lay-up (x direction in Fig. 1) of 10 plies of car-bon/epoxy pre-preg (SEAL� Texipreg HS 160 RM) with 0.15 mm ofply thickness. Curing was performed inside a press for 1 h at 140 �Cunder 4 bar pressure. The core material consisted on PMI Rohacell71IG foam with 0.075 g/cc density. The foam was supplied in bulkand it was cut into sheets of 300 mm � 300 mm � 25 mm (±2 mm)with the use of a water-cooled tile saw machine with a diamond-coated blade. The foam after cut was dried inside an oven for 2 hunder 50 �C. The properties of the composite (orthotropic proper-ties for a unidirectional ply aligned in the x direction) and corematerial are given in Table 1. For the adhesive bonding of the facesheets to the core, a ductile epoxy adhesive, Araldite� 2015, wasused whose elastic properties, E and G moduli, were measuredexperimentally in bulk tensile and TAST tests, respectively(E = 1850 MPa, G = 680 MPa) [31,32]. The adhesive bonding proce-dure involved abrading of the face sheets bonding surfaces with180 grit sandpaper and cleaning thoroughly with acetone. Thefoam was cleaned with compressed air. Bonding was performed in-side a press at ambient temperature. Each face sheet was bondedseparately under 1 bar pressure. The pre-crack was formulatedby placing between one of the composite plates and the core a thinTeflon film (0.025 mm). Four hours under the aforementionedpressure were sufficient for the adhesive to fully wet the bondingsurfaces and consolidate to its final thickness (�0.3 mm). The pan-els were left at room temperature for 1 week before machining toassure complete curing of the adhesive. The specimens were cutfrom the sandwich panel with respect to the dimensions indicatedin Fig. 1 using a water-cooled tile saw with a diamond-coatedblade and left to dry inside an oven for 2 h under 50 �C. White paintwas applied along the interface and a scale was glued underneaththe interface to facilitate the track of the crack growth during thetest. To accomplish pure vertical loading without introducing mo-ments and axial force, aluminium piano hinges were bonded to thetop and bottom face sheet of the DCB sandwich specimen at theend of the pre-cracked region with a two-component adhesive,Araldite� 420 (E = 4000 MPa, v = 0.3). This adhesive was chosendue to its high strength and toughness [8]. The pre-crack in eachspecimen was measured before testing with an optical microscopein both sides of the specimen. The crack length was reported as theaverage of the two recordings. It has to be noted that several at-tempts to extend the initial pre-crack before testing were not suc-cessful, since micro-kink phenomenon took place leading to crackdeviation from its initial plane. This means that fracture energy atcrack initiation will be affected by pre-crack blunting. Seven DCB

tests were carried out on an Instron� 8801 hydraulic testing ma-chine equipped with a 1 KN load cell, under displacement control(4 mm/min) and room temperature. The load–displacement (P–d)curve was registered during the test. Pictures were recorded duringthe specimens testing with 10 s intervals using a 10 MPixel digitalcamera in order to follow the deformation phenomena as well asthe propagation of crack.

3.1. Experimental results

A typical load–displacement curve for the DCB tests is shown inFig. 5a. From the curve it is clear that crack propagation occurred inan unstable manner throughout the test. The load increases line-arly prior to crack initiation and then drops suddenly. Carefulobservation of the crack propagation revealed that the crack ini-tially ‘‘micro-kinked” into the core over a short length (�0.9 mm)(Fig. 4c) and then propagated inside the core, parallel and veryclose to the interface (sub-interface crack between 0.5 and1.5 mm bellow the interface) (Fig. 4a and c). The crack distancepropagation, which was monitored with the help of the scale at-tached on the specimen, varied between 5 and 15 mm. However,the exact identification of the crack tip was carried out with diffi-culties owing to the porous nature of the core ‘‘hiding” the crack tip

D.A. Ramantani et al. / Composites Science and Technology 70 (2010) 1386–1394 1391

and also due to the formation of the FPZ ahead of the crack tip hav-ing multiple micro-cracks which were visible in the white paintedpart (Fig. 4b).

From the experimental P–d curves the respective R-curves wereobtained with the use of the CBBM method. In order to apply themethod, the P–d curves were ‘‘smoothed” by considering only thevalues of the load at peaks, since the subsequent drops correspondto sudden crack propagation. A representative R-curve is presentedin Fig. 5b. A high initial value of GI was obtained which is clearlyaffected by crack starting. In fact, the procedure used to performthe pre-crack and the micro-kink phenomenon both contributeto spurious additional energy consumption at crack initiation. Thisaspect emphasizes the importance of getting an entire R-curve,where the fracture energy corresponding to self-similar crackgrowth and not influenced by spurious effects can be straightfor-wardly measured from the plateau. Consequently, the critical frac-ture energy in mode I for each test was evaluated from the plateauof the respective R-curve, which corresponds to an almost lineartrajectory of the crack, thus not being so affected by the mixed-mode loading characteristic of micro-kink phenomenon.

4. Numerical analysis

In the numerical part of the work, the DCB sandwich test spec-imens were represented numerically with the use of ABAQUS�

software in combination with a cohesive mixed-mode damagemodel able to simulate crack propagation. The cohesive damagemodel was implemented in the numerical code via interface finiteelements which were placed inside the core, parallel and slightlybelow the face sheet–core interface. The interface elements allowthe separation of the material and therefore the reproduction ofthe crack path. The objective of the simulations was to performan inverse method based on fitting the numerical with the exper-imental P–d curves in order to obtain the cohesive strength, ru,I,under mode I loading and consequently determine both the cohe-sive parameters of the pure mode I law, for the specific materialstudied.

4.1. Cohesive mixed-mode damage model

The cohesive mixed-mode damage model used in this numeri-cal work is based on a linear constitutive relationship (Fig. 6) be-tween stresses (r) and relative displacements (d) betweenhomologous points of the interface elements having zero thickness.

σu,i

σum,i

σ i

δom,i δo,i

δum,i δu,i

δ i

Pure mode model

Mixed mode model

Gic i = I, II

Gi i = I, II

Fig. 6. The linear softening law for pure and mixed-mode cohesive damage model.

In the pure-mode model (i = I or i = II), the constitutive equationcan be defined in two different ways. Before damage starts to grow

r ¼ Dd ð10Þ

where D is the diagonal matrix containing the penalty parameter(d) and d the relative displacements between homologous pointsin modes I and II. The values of the penalty parameter must be quitehigh in order to hold together and prevent interpenetration of theelement faces. Following a considerable number of numericalsimulations [33,34], it was found that d = 106 N/mm3 produced con-verged results and avoided numerical problems during the non-linear procedure. After the peak stress is reached (ru,i) a gradualsoftening process between the stress and the relative displacementis observed, defined as

r ¼ ðI� EÞDd ð11Þ

where I is the identity matrix, E a diagonal matrix containing thedamage parameter

ei ¼du;i di � do;i� �

di du;i � do;i� � ð12Þ

and do,i is the displacement corresponding to the onset of damageand di the current relative displacement in mode i. In pure modeloading the strength along other directions is abruptly cancelled.The maximum relative displacement du,i for which complete failureoccurs, is obtained by equating the area under the softening curveto the respective critical fracture energy

Gc;i ¼12ru;idu;i ð13Þ

In a general case, failure is more likely to occur under a mixed-mode (I + II) loading. Therefore, a formulation for interface ele-ments should include a mixed-mode damage model, which, in thiscase, is an extension of the pure-mode model described above(Fig. 6). Damage initiation is predicted by using the quadratic stresscriterion

rIru;I

� �2þ rII

ru;II

� �2¼ 1 if rI > 0

rII ¼ ru;II if r 6 0ð14Þ

where ru,I and ru,II represent the ultimate normal and shear stres-ses, respectively, and it is assumed that normal compressive stressdo not induce damage. Defining an equivalent mixed-modedisplacement

de ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

I þ d2II

qð15Þ

and a mixed-mode ratio

b ¼ dII

dIð16Þ

and considering Eq. (10), Eq. (14) can be written as

dom;I

do;I

� �2

þ dom;II

do;II

� �2

¼ 1 ð17Þ

where dom,i (i = I, II) are the relative displacements at damage initi-ation, corresponding to critical interface stresses rum,i. CombiningEqs. (15)–(17) the value of the equivalent mixed-mode displace-ment leading to damage initiation (dom) can be easily obtained:

dom ¼do;Ido;II

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

o;II þ b2d2o;I

q ð18Þ

The mixed-mode damage propagation is simulated consideringthe linear fracture energetic criterion

GI

GIcþ GII

GIIc¼ 1 ð19Þ

1392 D.A. Ramantani et al. / Composites Science and Technology 70 (2010) 1386–1394

The released energy in each mode at failure can be obtainedfrom the area of the minor triangle (Fig. 6)

Gi ¼12rum;idum;i ð20Þ

being dum,i (i = I, II), the relative displacement in each mode forwhich complete failure occurs. Considering Eq. 10, 13, 15, 16, 19,and 20, the mixed-mode relative displacement leading to total fail-ure (dum) can be obtained

dum ¼2ð1þ b2Þ

ddom

1GIcþ b2

GIIc

" #ð21Þ

The values of de, dom and dum are introduced into Eq. (12), in-stead of di, do,i and du,i, respectively, thus establishing the damageparameter under mixed-mode. The damage parameter is intro-duced in Eq. (11), thus simulating damage propagation undermixed-mode loading. It should be noted that the mixed-modemodel is general, in the sense that it can be applied under any com-bination of modes, even for the pure mode cases. A more detaileddescription of the method can be found in reference [6].

4.2. Inverse method

Each of the DCB specimens was modelled numerically with re-spect to its specific dimensions. Plane stress 8-node quadrilateral

δ, P

Fig. 7. Deformed shape of the DCB sandwich specimen duri

Elemen

=80mm

σi

τi

Fig. 8. Schematic representation of the crack tip i

elements with reduced integration (CPS8) available in ABAQUS�

library were used to build the mesh. The face sheets were modelledas orthotropic material and the foam as isotropic. Zero thicknessinterface elements with 6 nodes, compatible with the 8-node solidelements, were placed inside the core, parallel and slightly under-neath (0.125 mm) the face sheet–core interface. The positioning ofthe interface elements inside the core was chosen to representmore accurately the crack path observed experimentally. Detailsof the mesh and boundary conditions can be seen in Fig. 7. A re-fined mesh was used in the damage propagation region consistingof 0.125 mm rectangular elements. The lower edge node of the arm(arm B) was clamped and a vertical displacement was applied tothe upper one (arm A). The C part of the specimen (uncrackedextremity in Fig. 1) is free to move which allows the specimen torotate and align the crack tip under an almost pure mode I loading(Fig. 7). Geometrical non-linear analysis was considered due topronounced deformation of the specimens which is expected atlarge opening displacements.

In order to assure that for the specific DCB sandwich test geom-etry (Fig. 1) mode I loading conditions prevail at crack initiation, aVCCT analysis was performed based on cohesive element model-ling. The interface elements placed along the crack plane allowobtaining the stresses and relative displacements in the vicinityof the crack tip (at the nodal points in Fig. 8). Thus, instead of nodalforces, VCCT was applied using the nodal stresses obtained fromthe interface elements. The values of G can then be calculated by

z

y x

ng propagation and details of the mesh at the crack tip.

ts width (b =30mm)

=0.125mm

upper face sheet

core

n 2D for the application of the VCCT method.

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35

σu,I=2.8MPa

σu,I=1.0MPa

σu,I=6.0MPa

testP

[N

]

δ [mm]

Fig. 9. Influence of ru,I on the P–d curve.

Table 2Cohesive parameters for mode I fracture.

# GIc (N/mm) ru,I (MPa)

1 0.133 2.82 0.119 2.83 0.134 2.84 0.131 2.85 0.120 2.66 0.127 2.67 0.146 3.2

Average 0.130 2.8St. dev. (%) 7.11 7.14

0.00

0.04

0.08

0.12

0.16

0.20

0.24

80 100 120 140 160 180

Experimental Numerical

GI[N

/mm

]

ae [mm]

Fig. 10. Experimental and numerical CBBM R-curves.

D.A. Ramantani et al. / Composites Science and Technology 70 (2010) 1386–1394 1393

the product of the relative displacements at the opened point(nodes l1 and l2) with the stresses at the closed point (node i)(Fig. 8) according to the following equations

GI ¼12riDul; ð22Þ

GII ¼12siDv l ð23Þ

where Dul, Dvl, represent the relative displacements at node l in thex and y direction respectively, and ri and si the normal and shearstresses at node i. It was verified that the percentage of GII relativelyto Gtot = GI + GII is about 3% which verifies the clear predominance ofmode I conditions for this geometry.

In each numerical model representing one of the experimentalDCB tests, the value of fracture energy, GIc, obtained from the pla-teau of the respective R-curve, was used as an input in the cohesivedamage model. For the remaining parameter, the cohesivestrength, ru,I, a typical value close to the tensile strength of thecore material, was chosen [11,12,17,22]. The numerical P–d curves,were compared with the respective experimental ones and the va-lue of ru,I was slightly reduced or increased until a good accuracybetween the two curves was obtained. It was noticed that thenumerical value of maximum load, Pmax, is lower than the experi-mental one because the initial crack in the experimental DCB sand-wich specimen was not extended before testing, which lead to aninitially blunt crack tip. This value was not taken into considerationin the calculations of fracture energy since it is not representativeto natural crack propagation (non self-similar crack propagation inthe beginning). A parametric study on ru,I revealed that small vari-ations on the value of this parameter does not greatly influence theP–d curve (Fig. 9).

The results of the experimental and numerical analysis are sum-marised in Table 2. The average values of the cohesive parameters,fracture energy, GIc and cohesive strength, ru,I represent the aver-age cohesive law characterizing face sheet–core crack growth un-der mode I loading, for the analysed sandwich.

5. Validation of the CBBM method

In order to verify the adequacy of the CBBM method to measureaccurately the fracture energy under mode I loading (GIc), one ofthe DCB experimental tests was simulated numerically and theCBBM method was applied to confirm if the experimental R-curvewill be reproduced (equal plateau values). A 2D numerical analysis(explained in previous paragraph) was followed where the cohe-sive parameters defined with the inverse method for the specifictest, via fitting the experimental with the numerical P–d curve,were used as inputted parameters in the cohesive damage model.The CBBM method was applied and the R-curve was obtained. InFig. 10 the numerical R-curve is compared with the respectiveexperimental one. Their good agreement in the plateau region val-idates the proposed methodology.

6. Concluding remarks

In this work a numerical and experimental study on the asym-metric double cantilever beam (DCB) sandwich specimen is pre-sented. The objective is to obtain the mode I cohesive lawcharacterizing the debonding fracture between the face sheet andthe core via the determination of the cohesive parameters, fractureenergy, GIc and cohesive strength, ru,I. A data reduction method ispresented based on beam theory and specimen compliance (C = d/P),named compliance-based beam method (CBBM). According tothis method, an analytical expression for the specimen complianceis defined based on beam theory and considering an equivalentcrack length ae, which includes ao + D + DaFPZ. The value of equiv-alent crack length, ae, is calculated from the analytical expressionof compliance by considering the compliance value registered dur-ing the test. With the equivalent crack length concept all the com-plex damaging phenomena of the bi-material interface andgeometric asymmetry of the specimen, which have a significantinfluence on the value of the compliance, are incorporated in theanalysis. The parameter D accounts for the local and global defor-mation mode effects and the parameter DaFPZ for the fracture pro-cess zone (FPZ) developed ahead of the crack tip. The application ofCBBM method presents two great advantages. First, the parameterD is not necessary to be defined experimentally since it is indi-rectly incorporated in the compliance equation and secondly, usingthe presented methodology, the problem of crack length measure-ment during propagation, resulting from difficulties to identify thecrack tip, is overcome. DCB tests were performed and the CBBMmethod was applied to obtain the mode I fracture energy, GIc fromthe plateau value of the R-curve. It is necessary to note the impor-tance of obtaining an entire R-curve in order to identify the frac-ture energy corresponding to self-similar crack growth and not

1394 D.A. Ramantani et al. / Composites Science and Technology 70 (2010) 1386–1394

influenced by spurious effects such as an initial micro-kink phe-nomenon. Each test was simulated numerically with the use of acohesive mixed-mode damage model allowing modelling crackpropagation. An inverse method was followed based on fittingthe numerical with the experimental P–d curves by altering the va-lue of the cohesive strength, ru,I. With the fracture energy andcohesive strength, GIc and ru,I respectively, being defined, the cohe-sive law for mode I fracture of debonding between the face sheetand the core is characterized. In order to verify the ability of theproposed methodology to reproduce the plateau value correspond-ing to the fracture energy, GIc, one experimental test was numeri-cally simulated with the cohesive parameters determinedexperimentally using the CBBM. The good agreement on the pla-teau values between the numerical and experimental R-curves val-idates the proposed methodology.

Acknowledgements

The first author would like to thank the EU Marie Curie Re-search Training Network MOMENTUM ‘‘Multidisciplinary Researchand Training on Composite Materials Application in Transport Modes”(MRTN-CT-2005-019198) for supporting the work here presented.The First author would also like to thank Mr. Paulo Nóvoa and Mr.Nelson Pereira of INEGI Institute (Institute of Mechanical Engineer-ing and Industrial Management, Porto, Portugal) for their help withthe experimental part of the work.

Appendix A

Eq. (6) can be expressed as,

u1a3e þu2ae þu3 ¼ 0 ðA:1Þ

where

u1 ¼13

DA þ DB

DADB

� �; u2 ¼

1210bhsGs

þ F and u3 ¼ �dP¼ �C

ðA:2Þ

Using the Matlab� software and considering only the real solu-tion, we obtain

ae ¼1

6u1A� 2u2

AðA:3Þ

where

A ¼ �108u3 þ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

4u22 þ 27u2

3u1

u1

� �s !u2

1

!13

ðA:4Þ

References

[1] Zenkert D. The handbook of sandwich construction. EMAS: UK; 1997.[2] Berggreen C, Simonsen BC. Non-uniform compressive strength of debonded

sandwich panels-II. Fracture mechanics investigation. J Sandw Struct Mater2005;7(6):483–517.

[3] Avilès F, Carlsson LA. Experimental study of debonded sandwich panels undercompressive loading. J Sandw Struct Mater 2006;8(1):7–31.

[4] Luundsgaard-Larsen C, Berggreen C, Nokkenvend A. Residual strength of in-plane loaded debonded sandwich panels-Experimental investigation.Sandwich Structures 7: advancing with Sandw Struct Mater. In: Proceedings

of the 7th international conference on sandwich structures, Aalborg, 29–31August 2005, Springer Netherlands; 2005. p. 451–60.

[5] de Moura MFSF, Gonçalves JPM, Chousal JAG, Campilho RDSG. Cohesive andcontinuum mixed-mode damage models applied to the simulation of themechanical behaviour of bonded joints. Int J Adhes Adhes 2008;28(5):419–26.

[6] de Morais AB, de Moura MFSF, Gonçalves JPM, Camanho PP. Analysis of crackpropagation in double cantilever beam tests of multidirectional laminates.Mech Mater 2003;35(7):641–52.

[7] Campilho RDSG, de Moura MFSF, Dominques JJMS. Modelling single anddouble-lap repairs on composite materials. Compos Sci Technol2005;65(13):1948–58.

[8] Campilho RDSG, de Moura MFSF, Domingues JJMS. Using a cohesive damagemodel to predict the tensile behaviour of CFRP single-strap repairs. Int J SolidsStruct 2008;45(5):1497–512.

[9] Campilho RDSG, de Moura MFSF, Pinto AMG, Morais JJL, Domingues JJMS.Modelling the tensile fracture behaviour of CFRP scarf repairs. Composites PartB – Eng 2009;40(2):149–57.

[10] Han TS, Ural A, Chen CS, Zehnder AT, Ingraffea AR, Billington SL. Delaminationbuckling and propagation analysis of honeycomb panels using a cohesiveelement approachi. Int J Fract 2002;115:101–23.

[11] El-Sayed S, Sridharan S. Performance of a cohesive layer model in theprediction of interfacial crack growth in sandwich beams. J Sandw StructMater 2002;4:31–48.

[12] El-Sayed S, Sridharan S. Cohesive layer models for predicting delaminationgrowth and crack kinking in sandwich structures. Int J Fract 2002;117:63–84.

[13] Ramantani DA, Campilho RDSG, de Moura MFSF, Marques AT. Stress andfailure analysis of repaired sandwich composite beams using a cohesivedamage model. J Sandw Struct Mater. doi: 10.1177/1099636208104520.

[14] Avilès F, Carlsson LA. Analysis of the DCB sandwich for debondcharacterization. Eng Fract Mech 2008;75:153–68.

[15] Carlsson LA, Matteson RC, Avilès F, Loup DC. Crack path in foam cored DCBsandwich fracture specimens. Compos Sci Technol 2005;65:2612–21.

[16] Li X, Carlsson LA. The tilted sandwich debond (TSD) specimen for face/coreinterface fracture characterization. J Sandw Struct Mater 1999;1:60–75.

[17] Li X, Carlsson LA. Elastic foundation analysis of tilted sandwich debond (TSD)specimen. J Sandw Struct Mater 2000;2:3–32.

[18] Li X, Carlsson LA. Fracture mechanics analysis of tilted sandwich debond (TSD)specimen. J Compos Mater 2001;35:2145–68.

[19] Viana GM, Carlsson LA. Mode mixity and crack tip yield zones in TSD sandwichspecimens with PVC foam core. J Sandw Struct Mater 2002;4:141–55.

[20] Viana GM, Carlsson LA. Influences of foam density and core thickness ondebond toughness of sandwich specimens with PVC foam core. J Sandw StructMater 2003;5:103–17.

[21] Cantwell WJ, Davies P. A test technique for assessing core-skin adhesion incomposite sandwich structures. J Mater Sci Lett 1994;13:203–5.

[22] Shivakumar K, Smith SA. In situ fracture toughness testing of core materials insandwich panels. J Compos Mater 2004;38:655–68.

[23] Cantwell WJ, Scudamore R, Ratcliffe J, Davies P. Interfacial fracture in sandwichlaminates. Compos Sci Technol 1999;59:2079–85.

[24] Mc Garva LD, Astrom BT. Experimental investigation of compression mouldingof Glass/PA12-PMI foam core sandwich components. Composites Part – A1999;30:1171–85.

[25] Ozdil F, Carlsson LA. Beam analysis of angle-ply laminate DCB specimens.Compos Sci Technol 1999;59:305–15.

[26] Shivakumar K, Chen H, Smith SA. An evaluation of data reduction methods foropening mode fracture toughness of sandwich structures. J Sandw StructMater 2005;7:77–90.

[27] Berkowitz CK, Johnson WS. Fracture and fatigue tests and analysis ofcomposite sandwich structure. J Compos Mater 2005;39:1417–31.

[28] Allen HG. Analysis and design of structural sandwich panels. Oxford:Pergamon Press; 1969.

[29] de Moura MFSF. Numerical simulation of the ENF test for the mode II fracturecharacterization of bonded joints. J Adhes Sci Technol 2006;20:182–9.

[30] Viana GM, Carlsson LA. Mode mixity and crack tip yield zones in TSD sandwichspecimens with PVC foam core. J Sandw Struct Mater 2002;4:141–55.

[31] Lee KY, Kong BS. Theoretical and experimental studies for the failure criterionof adhesively bonded joints. J Adhes Sci Technol 2000;14:817–32.

[32] Marques EAS, da Silva LFM. Joint strength optimization of adhesively bondedpatches. J Adhes 2008;84:917–36.

[33] de Moura MFSF, Gonçalves JPM, Marques AT, de Castro PT. Modellingcompression failure after low velocity impact on laminated compositesusing interface elements. J Compos Mater 1997;31:1462–79.

[34] Gonçalves JPM, de Moura MFSF, de Castro PMST, Maques AT. Interface elementincluding point-to-surface constraints for three dimensional problemswith damage propagation. Eng Comput: Int J Comput Aid Eng Soft 2000;17:28–47.