Adhesively bonded disk under compressive diametrical load

International Journal of Solids and Structures 0 0 0 (2018) 1–15

ContentslistsavailableatScienceDirect

International

Journal

of

Solids

and

Structures

journalhomepage:www.elsevier.com/locate/ijsolstr

Adhesively

bonded

disk

under

compressive

diametrical

load

E.

Radi

∗

,

E.

Dragoni,

A.

Spaggiari

Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, via G. Amendola 2, 42122, Reggio Emilia, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 16 November 2017 Revised 30 March 2018 Available online xxx Keywords:

Airy stress function Bipolar coordinates Plane elasticity Singular integral equation Perturbation analysis Adhesive bonding Mode mixity Collocation method

a

b

s

t

r

a

c

t

Aclosed-formfull-fieldsolutionispresentedforstressesanddisplacementinacirculardiskcontaininga diametricaladhesivethinlayerinducedbytwooppositecompressiveloadsactingalonganarbitrary di-ametricaldirection.Forthesakeofsimplicity,theadhesivelayeristreatedasatangentialdisplacement discontinuitybetweenthe two disk halves.The problemis splitinto symmetricand skew-symmetric loadingconditions.Nocontributionisexpectedfromtheadhesivelayerforthesymmetricproblem.For theskew-symmetricloading condition,ageneralintegralsolution inbipolarcoordinateshas been as-sumedfortheAirystressfunctionintheformofaFouriersinetransform.Theimpositionoftheboundary conditionsthenallowsustoreducetheproblemtoaFredholmintegralequationofthefirstkinddefined onthehalf-lineorequivalentlytoasingularintegro-differentialequationdefinedonaboundedinterval. Apreliminaryasymptoticanalysisofthestressanddisplacementfieldsattheedgesoftheadhesivethin layershows thatthestress fieldisfinitetherein, butthe rotationdisplays alogarithmic singularity.A numericalsolutionofthesingularintegro-differentialequationisthenprovidedbyassumingapower se-riesexpansionfortheshearstress,whosecoefficientsaredeterminedbyusingacollocationmethod.An approximateclosed-formsolutionisalsoderivedbyexploitingaperturbationmethodthatassumesthe ratiobetweentheshearmodulusofthediskmaterialandtheshearstiffnessoftheadhesivethinlayer assmallparameter.Theshearstressdistributionalongthethinlayerturnsouttobemoreandmore uni-formastheadhesiveshearstiffnessdecreases.Inordertovalidatetheanalyticalresults,FEinvestigations andalsoexperimentalresultsobtainedbyusingDigitalImageCorrelation(DIC)techniquesarepresented forvaryingloadingorientationandmaterialparameters.

© 2018ElsevierLtd.Allrightsreserved.

1. Introduction

Adhesivebondjointsfindanincreasingnumberofapplications in aircraft,marineand civilstructures eveniftheir strength can-not beaccuratelypredictedby theoreticalmodeling,duetostress singularity that mayarise at their edges. The occurringof stress singularity isa consequenceof theassumed linearelastic behav-iorofthematerialsand,evenifnotrealistic,itmayhowevercause initiation and propagation of cracks and debonding. Therefore, a strong requestcomes fromstructural engineersanddesignersfor thedefinitionofareliableexperimentalmethodologythatenables onetocharacterizetheadhesivebondingstrengthpropertiesupto failure,undervariousloadingandphysicalconditions.

Adhesive research over theyears has strivedto identify spec-imen designs of bonded joints where shear and tensile stresses occur without stress concentration at the edges, due to mate-rials elastic mismatch. Testing the adhesive in thin film pro-vides more realistic information than retrieving its properties from tests on bulk specimen, due to several reasons. For thin

∗ _{Corresponding author.}

E-mail address: eradi@unimore.it (E. Radi).

film, indeed,the chemical effect of the adhesive-adherend inter-face plays a significant role, as suggestedby Krogh et al. (2015) . Moreover, for test on bulk the curing conditions may differ be-cause of uncontrolled exothermic reactions in massive form, as shown by Adnan and Sun (2008) , and the stress triaxiality pro-moted in-situ by the adherends cannot take place, as stated by Wang and Rose (1997) . Therefore, the mechanical and chemi-cal properties of the interphase region differ from the bulk. All bond properties are affected by the thickness of the adhesive layer, butthe influence on Young’s modulus is especially signifi-cant,Peretz (1978) .Intechnicalliteraturewecantracesome stan-dardspecimenthatleadtonon-singularstressdistributioninthin film adherends, such as:thick adherend shear test, prepared ac-cording to the ASTM D5656-10 , the napkin ring torsion test, as showninthe ASTM E229-97 .Thistest specimenwasanalyzedby Grant (1987) then by Castagnetti et al (2011) and improved by Spaggiari et al. (2012,2013) and the solid rod butt-joint in tor-sion, proposed by Adams (1977) . Moreover, there exist other in-teresting options which exploit rigid adherendsbonded together suchas:theArcantest,proposed byArcan et al. (1987) ,the butt-bonded notched beam under four-point antisymmetric bending, studiedbyWycherley et al. (1990) andIosipescu (1967) and,more https://doi.org/10.1016/j.ijsolstr.2018.05.021

0020-7683/© 2018 Elsevier Ltd. All rights reserved.

2 E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15 recently,the three-point bendtest ofa sandwich beammadeby

bonding two thick flats on top of each other, as suggested by Moussiaux et al.(1987), He et al. (2001) andBrinson et al. (1995) . Thesepromisingmodelswerestudiedanalyticallyandnumerically by Dragoni and Brinson (2015) andvalidated experimentally us-ingDigital ImageCorrelation(DIC) by Spaggiari et al. (2016) .The bondedBrazilian disc is a very simple test coupon, easier to be manufactured than the previous ones. It has been proposed by Martin et al. (2012) for evaluating the shear strength of epoxy joinedceramics.Moreover,unlikethenapkinringortubular spec-imens,itdoesnotneedatorsionaltestingmachineand,compared tothethreeorfourpointbendingspecimens,ithasnoneedofad hocfixtures.

The bonded Brazilian disk can be simply compressed with a universaltestingmachineanditisabletocreatethedesiredstress field in theadhesive undertest, with a variabledegree ofmode mixity, and without a strong edge effect at the corners as al-ready observed by Martin et al. (2012) . Due to its simple speci-menpreparationandexperimentalperformance,theBraziliantest isawidely usedtestforthedeterminationofthetensilestrength andfracturetoughnessofbrittlematerialslikerocksandconcrete (Berenbaum and Brodie, 1958; Awaji and Sato, 1978; Dong et al., 2004 ).Inthestandard test,ahomogeneous thindiskis diametri-callycompressedandthat loadingcausesan almost uniform ten-silestress perpendicular to the loading direction, which yields a tensile failure. This problem has attracted the interest of many researchers(Kourkoulis et al., 2012, 2013; Markides et al., 2011 ), whoapproacheditby usingthemethodofcomplex potentialsof planeelasticity(Muskhelishvili, 1954 )orthemostgeneralsolution ofthe biharmonic equation in polarcoordinates (Mitchell, 1965 ). An exhaustive depictionof the problem can be found in the re-view paperon theBrazilian test and its generalization by Li and Wong (2013) . A numberof generalizationsof the test to bonded bimaterialdisksorringshasbeenalso proposed intheliterature inordertodeterminethetoughnessoftheinterfaceunderpureor mixed-modeconditions(Wong and Suo, 1990; Soares and Tianxi, 1998; Banks-Sills and Schwartz; 2002; Budzik and Jensen, 2014 ). Theseinvestigations provideonlynumericalresults orasymptotic analysesofthecracktipfieldsinabondedbimaterialdiskorring performedby usingthe linear elastictheory of interface fracture mechanics. However, no full-field analytical solutions have been suppliedfor a disk with an adhesive thinlayer subject to com-pressiveloadalonganarbitrarydirection,whichinducesboth nor-malandshearstressalongthebondingjoint.Suchaconfiguration mayindeedbeusedforthecharacterizationoftheadhesive bond-ing strength proprieties. Therefore, an accurate evaluation ofthe stressdistributionalongtheadhesivethinlayerundermixed load-ingconditionsisaprerequisitetoassurethestructuralintegrityof manybondedcomponents.

In the presentwork, we propose the useof theBrazilian test ona bonded diskfor the characterization ofadhesion properties ofthe adhesive bonding. The main advantage of thistest is that anycombinationofshearandnormalloadingcanbeachievedby appropriate choice of the bonding inclination angle withrespect totheloadingdirection.Tothisaim,we firstinvestigatethe two-dimensionalproblemofthestressanddisplacementdistributionin thetwo diskhalves,bytreating theadhesive thinlayerasa tan-gentialdisplacementdiscontinuitybetweenthetwohalves.A sim-ilarassumptionwasextensivelyadoptedintheconventional analy-sisofbondedjoints(Klabring, 1991; Koutosov, 2007 ).Forthesake ofsimplicity,onlyshear compliance oftheadhesive layeris con-sidered,leavingthe contributionofnormal compliancefor future analysis.Namely, the adhesive joint is represented asan imper-fectelasticinterfacewhoseshearstressisassumedtodepend lin-earlyonthetangentialdisplacementdiscontinuityacrossthethin layer(Mishuris, 1997,1999,2001) ,whereas aperfectbondbetween

the two semi-disks is considered for the normal displacement. Undertheseassumptions,theshearstressalongthebondlineis fi-nite,beingproportionaltothetangentialdisplacement discontinu-ity.Then,thestressanddisplacementfieldsneartherightwedges atthebondingedgessubjecttouniformshearloadingononeside andvanishing traction onthe other side are non-singular,as ob-servedby England (1971) andlater byBarber (2010) .However, an unbounded rotation is expectedat the corner. Unfortunately, the Mitchellsolutionforabiharmonicfunctioninpolarcoordinatesis notgeneralenoughtoincludethesolutionofthepresentproblem duetothenon-classicalboundaryconditionalongthecommon di-ameterandtotheboundary layerwhichmayariseatboth edges of the adhesive bonding. The methodof complex potentials also requires additional terms to properly model the boundary layer which may arise at the bonding edges. In the present investiga-tionwe insteadadoptbipolarcoordinatesto describethe geome-tryoftheproblem. Themostgeneralformofabiharmonic stress function in bipolar coordinates was found out by Jeffery (1921) . TheapproachwasthengeneralizedbyLing (1948) ,whosolvedthe problemoftwooverlappingholesinaninfiniteplateundergeneral far-fieldstresses by introducinga generalintegral solution ofthe biharmonic equation in bipolar coordinates.Then, he determined theparameters involvedin thesolutionfromthe givenboundary conditions with the aid of Fourier transforms. The use of bipo-lar coordinates has been proved to be effective for investigating mechanicalproblemswhosegeometriesaredefinedbytwocircles ortwocirculararches,e.g. byusingtheJefferysolution,Radi and Strozzi (2009) solved theproblem ofa circulardisk containinga sliding eccentric circular inclusion; Radi (2011) presented an an-alytic solution for stresses induced in an infiniteplate with two unequal circularholes by remote uniform loadings andarbitrary holepressures andcalculatedthecorresponding stress concentra-tionfactors;Lanzoni et al. (2017) investigatedtheeffectivethermal propertiesofcompositesreinforcedwithfiberswhosecrosssection isdefinedbytwocirculararches.

Theproblemissplitintosymmetricandskew-symmetric load-ingconditions,thusreducingthestudytoaquarterofdisk.Here, we focused only on the skew-symmetric loading conditions be-cause the symmetric problem is trivial if the adhesive layer is modeled by an imperfect interface displaying only elastic shear complianceandnonormaldisplacementdiscontinuityisadmitted betweenthe two halves.In thiscase, indeed,the solution ofthe symmetric problem coincides with the solution alreadyavailable foranintegerdisk(Dong et al., 2004 ),sincenotangential displace-mentdiscontinuityoccursbetweenthetwohalvesunder symmet-ricloadingconditions.Therefore,thepresentapproachisnot lim-ited tothe skew-symmetricloading conditionbutitprovides the solutionforthegeneralloadingcase.Inparticular,theshearstress distributionalongthebondlineinthegeneralcasecoincideswith the solution for the skew-symmetric loading condition provided here.The solutionisobtainedherebyexploitingtheintegral rep-resentation of the Airy stress function in bipolar coordinates in theformofa Fouriersine orcosinetransform. Theimpositionof the boundary conditions onthe tractions along the circular con-tour,thesymmetryandbondingconditionsalongtheadhesivethin layer then allows to reduce the problem to a Fredholm integral equationofthefirstkinddefinedonthehalf-line.Aftera prelimi-naryasymptoticinvestigationofthestressanddisplacementfields atthebonding edges, whichturnout to be finitetherein,an ap-proximate solutionisprovided byreducing theproblemtoa sin-gular integro-differential equation definedon abounded interval, andassuming a power seriesexpansion for the shear stress dis-tribution,whose coefficientsare found by meansof a collocation method.Anotherapproximateclosed-formsolutionisalsoderived by using a perturbation methodthat assumes the ratiobetween the shear modulus of the disk material and the shear stiffness Pleasecitethisarticleas:E.Radietal., Adhesivelybondeddiskundercompressivediametricalload, InternationalJournalofSolidsand

E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15 3

Fig. 1. Elastic disk adhesively bonded along a diameter and subject to compressive load P acting along an arbitrary direction defined by the angle θ0 (a). The problem is split into the sum of symmetric (b) and skew-symmetric (c) loading conditions .

Fig. 2. Semicircular elastic disk elastically constrained in the tangential direction along its straight boundary under skew-symmetric loading condition .

ofthe adhesivelayer assmallparameter.In ordertovalidate the analytical results, FE investigations and also experimental results obtained by using Digital Image Correlation (DIC) techniques are presented for varying loading orientation angle and material pa-rameters.

2. Problemdescriptioninbipolarcoordinates

TheproblemconsideredhereconsistsinadiskofradiusRwith an adhesive thin layer along a diameter (Fig. 1 (a)), under plane strain or plane stress loading conditions. The disk is loaded by two opposite loads P acting along an arbitrarydiametrical direc-tion. The problem is split into the sumof symmetric and skew-symmetricloadingconditions,asdepictedinFig. 1 (b)andFig. 1 (c), respectively, thus reducing the analysis to a semidisk bonded to the adhesivethinlayer(Fig. 2 ).A Cartesiancoordinatesystem(0,

x, y) is introduced withthe y-axis directioncoinciding withthe bondline.Let

θ

0denotetheanglebetweenthedirectionofloading andthex-axis.FollowingJeffery (1921) ,useismadeofthebipolar coordinates(

α

,

β

)withthepoleslocatedon they-axisaty=±R. TherelationsbetweenCartesianandbipolarcoordinatesthenwrite x=_coshR

_α

sin

β

− cos

β

, y=

Rsinh

α

cosh

α

− cos

β

. (1)

Eliminationof

α

fromequations(1) thenprovidestheequation ofafamilyofcirclespassingforthetwopoleswithcentersonthe

xaxisatdistanceRcot

β

fromtheoriginandradiusR/sin

β

,namely

(

x− Rcot

β)

2_+y2₌

₍

_R/sin

_β)

2_{. (2)}

In particular, the circular and straight boundaries of the semidisk are defined by the coordinates

β

=

π

/2 and

β

=

π

, re-spectively(Fig. 2 ). Thepolarangle

θ

around thecircledefinedby thecorrespondingvalueof

α

isassociatedwiththebipolar coordi-nate

β

bythefollowingrelations:

sin

θ

=y R =

sinh

α

cosh

α

− cos

β

, cos

θ

=

x R =

sin

β

cosh

α

− cos

β

. (3) For an intact disk with thickness b, the normal and shear stressesalongadiametricaldirectionwhosenormalisinclinedof anangle

θ

0withrespecttothecompressiveloadsP(Fig. 1 (a))are givenbyDong et al. (20 04) andSadd (20 05) andreportedhere:

σ

x = P

π

Rb



1− 4cos2

_θ

0 1+2y2_{− y}4

₍

_{1− 2cos2}

θ

0

)



1+2y2_cos2

θ

0+y4



2



,

τ

xy = 2P

π

Rb



1− y2



2



_1+y2



_sin2

_θ

0



1+y4_+2y2_cos2

θ

0



2 , (4)

wherethenormalstress

σ

x ispositiveiftensile.

ThestressdistributionsprovidedbyEq. (4) areplottedinFig. 3 forvarious values ofthe loading angle

θ

0. We chose to plot the stressesfor

θ

0=30°,60°,70°,80°,namelywithdecreasinginterval as

θ

0increases,becausethestressdistributionbecomesmore sen-sitiveto thevariation of

θ

0 as

θ

0 approaches90°.From Eq.(4.2) itcanbecalculatedthatfor

θ

0≤ 52.2° themaximumoftheshear stressisattainedatthediskcenter(y=0)wherethenormalstress iscompressive,whereasfor

θ

0>52.2° thepeakoftheshearstress increasesanditslocationmovestowardtheouterborder(y_=±R). Correspondingly,the(peeling)normalstressbecomestensileatthe diskcenterandattainsacompressivepeaknearthebondingedges. As

θ

0 approaches90° theshearstress(4.2) tendstovanishalong the loading line,whereas the normal stress (4.1) approaches the uniformtensilevalue P/

π

Rb.The latterresulthas beenexploited forindirecttensiontesting.Similartrendsareexpectedalsoforan adhesively bonded disk,as observed by the numerical investiga-tionsperformedbyMartin et al. (2012) onaepoxy-jointedceramic disk.

In general, for an adhesively bonded disk under symmetric loading conditions (Fig. 1 (b)) shear stress may occur along the bondlinein order to generatethe same tangentialstrain both in thediskandintheadhesivelayer(Lanzoni and Radi, 2009 ). How-ever, ifthe adhesive layer is modelled asan imperfect shear in-terface then under symmetric loading conditions (Fig. 1 (b)) the normal stress along the adhesive bonding coincides with (4.1), whereastheshearstressvanishestherein.Conversely,under skew-symmetricloadingconditions(Fig. 1 (c))thenormalstressvanishes,

4 E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15

Fig. 3. Normal stress (a) and shear stress (b) distributions along the diameter of an intact disk whose normal is inclined of the angle θ0 with respect to the loading direction. whereas theshear stress coincides withthe solutionof the

gen-eralproblem(Fig. 1 (a)). Therefore,in thefollowingwe will focus onlyontheproblemofskew-symmetricloadingofadiskwithan imperfect shear interface (Fig. 2 ), which has never been investi-gatedpreviously.

2.1.Stressfields

TheproblemisformulatedusinganAirystressfunction

χ

(x,y),

whichsatisfiesthebiharmonicequation

χ

=0.Inbipolar coor-dinates,thisconditionbecomes(Jeffery, 1921 ):



∂

4

∂

β

4 +2

∂

4

∂

α

2

∂

β

2+

∂

4

∂

α

4+2

∂

2

∂

β

2− 2

∂

2

∂

α

2+1



χ

J =0, (5) whereJ= R/(cosh

α

− cos

β

).

Beingtheexpectedstressdistributionskew-symmetricwith re-spectto thex-axis, the stress function

χ

can be assumedas the mostgeneralintegralsolutionofthebiharmonicequationin bipo-larcoordinates (5) thatcontainsonlyoddtermsin

α

.Anintegral solutionofEq. (4) thatis oddin

α

then takesthefollowingform

χ

J = _∞ 0 F

(β

)

sins

α

ds, (6) where F

(

β)

= P

π

Rb

{

[f1

(

s

)

sin

β

+f2

(

s

)

cos

β

]sinhs

β

+[f3

(

s

)

sin

β

+f4

(

s

)

cos

β

]coshs

β}

. (7) Thecorrespondingin-planestresscomponentsaregivenby:

σ

α =[

(

cosh

α

−cos

β)

_∂

∂

_β

22−sinh

α ∂

∂α

−sin

β ∂

∂β

+cosh

α

]

χ

J,

σ

β =[

(

cosh

α

−cos

β)

∂

2

∂

α

2−sinh

α ∂

∂α

−sin

β ∂

∂β

+cos

β

]

χ

J,

τ

βα =−

(

cosh

α

− cos

β)

∂

2

∂

β ∂

α

χ

J. (8)

The introductionofthestressfunction(6) inEq. (8) then pro-videsthefollowingintegralrelationsforthestresscomponentsin bipolarcoordinates:

σ

α =

_∞

0

{

[F

(β)

cosh

α

− F

(β)

sin

β

+ F

(β)(

cosh

α

− cos

β)

]

sins

α

− sF

(β

)

sinh

α

coss

α}

ds,

σ

β =

 _∞

0

{

[F

(β

)

cos

β

− s2

₍

_cosh

_α

_{− cos}

_β)

_F

_(β)

_{− F}

_(β)

_sin

_β

_] sins

α

− sF

(β

)

sinh

α

coss

α}

ds,

τ

βα=−(cosh

α

− cos

β)

 _∞ 0 sF

(β)

coss

α

ds. (9) where F

(

β)

=

_π

P

Rb

{

[

(

f1+s f4

)

cos

β

−

(

f2− s f3

)

sin

β

]sinhs

β+

+[

(

f3− s f2

)

cos

β

−

(

f4− s f1

)

sin

β

]coshs

β}

, F

(

β)

=

_π

P_Rb

2s f3+



s2_{− 1}



_f 2



cos

β

−

2s f4−



s2_{− 1}



_f 1



sin

β



sinhs

β

+ P

π

Rb

2s f1+



s2_{− 1}



_f 4



cos

β

−

2s f2−



s2− 1



f3



sin

β



coshs

β

, (10)

according to (7) . The traction boundary conditions on the outer semicircleat

β

₌

π

/2thenrequire:

τ

αβ

(α

,

π

/2

)

=0,

σ

β

(α

,

π

/2

)

=p

(α)

, (11) where p(

α

) is the normal stress along the curved boundary at

β

=

π

/2 consisting in a concentrated radial force P/2 applied at

θ

=

θ

0.ByintroducingtheDiracdeltafunction

δ

itwrites

p

(

α)

=− P 2Rb

δ(

θ

−

θ

0

)

=− P 2Rb

δ(

α

−

α

0

)

cosh

α

0= =− P

π

Rbcosh

α

0 _∞ 0 sins

α

0sins

α

ds, (12)

being sin

θ

₌tanh

α

for

β

₌

π

/2, asit follows fromEq. (3) , and thus

cosh

α

0=1/cos

θ

0, sinh

α

0=tan

θ

0. (13)

Alongtheadhesivethinlayer,namelyat

β

=

π

,wemustrequire vanishingofnormalstress duetotheskew-symmetrypropertyof the problemand proportionality betweenshear stress andradial displacementdueto theelasticshear complianceofthe adhesive thinlayer,namely

σ

β

(α

,

π

)

=0, uα

(α

,

π

)

+₂h_G

τ

αβ

(α

,

π

)

=0, (14)

E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15 5 where G and h are the shear modulus and the thickness of the

adhesive layer,respectively. The caseofan intactdiskis thus re-covered asG tends to infinity or as h becomesvanishing small. ForG₌

μ

,thestressdistributionalongthebondlineatdistanceh/2 fromthemeanlineoftheadhesivelayerisinsteadrecovered.

2.2. Displacementfields

AccordingtoJeffery (1921) ,thedisplacementsu_α andu_β along the directions orthogonal to the curves

α

=const and

β

=const (Fig. 2 )aregivenby

u_α = R 2

μ

J

(

1− 2

ν

¯

)

∂χ

∂α

−

∂

∂β

Q



, u_β = R 2

μ

J

(

1− 2

ν

¯

)

∂χ

∂β

+

∂

Q

∂α



, (15)

respectively,where

μ

istheelasticshearmodulusand

v

=



_v

_planestrain

v

/

(

1+

v

)

planestress (16)

being

ν

the Poisson ratio of the disk material. The function Q

is determined from the Airy stress function

χ

by the condition (Jeffery, 1921 ) Q J =

(

1−

ν)

 



_∂

2

∂α

2−

∂

2

∂β

2−1



χ

J d

α

d

β

= =−(1−ν

)

 

_

s2+1



F

(

β

)

+F

(

β)



sins

α

d

α

d

β

. (17) Introductionof(10) in(17) thenyields

Q J = P

π

Rb



2

(

1−

ν

)

∞ 0

[

(

f2cos

β

+f1sin

β)

coshs

β

+

(

f4cos

β

+f3sin

β)

sinhs

β

]coss

α

ds+Dsin

β



, (18) wherethelasttermdefinesanarbitraryrigidbodymotion.In par-ticular, the radial displacement along the adhesive thin layer at

β

=

π

followsfrom(15.1)and(18) as

2

μ

R uα

(

α

,

π

)

=−

 _∞

0



ν

scoss

α

+

(

1−2

ν

)

sinh₁₊

α

_coshsin

_α

s

α



F

(

π

)

+

(

1−

ν)

F

(

π

)

+F

(

π

)



coss

α

s



ds+ PD

π

Rb. (19)

3. Impositionoftheboundaryconditions

The functionsfi(s), fori=2,3, 4,introduced inEq. (7) can be

determinedintermsoff1(s)byimposingtheboundaryconditions

(11) ,(12) and(14) at

β

=

π

/2and

β

=

π

.Inparticular,from condi-tions(11) and(12) oneobtains

F

(π

/2

)

=0, (20)

 _∞

0

s[scosh

α

sins

α

+ sinh

α

coss

α

]F

(

π

/2

)

ds=

= P

2Rb cosh

α

0

δ(

α

−

α

0

)

. (21)

Bytakingthederivativewithrespectto

α

ofEq. (21)

 _∞ 0 s

(

s2₊₁

₎

_F

(π

_/2

₎

_coss

α

_ds= P 2Rb cosh

α

0

δ



_(α

₋

_α

₀

₎

cosh

α

, (22)

and,then,theinverseFouriercosinetransformofEq. (22) ,one ob-tains F

(

π

/2

)

= P

π

Rb cosh

α

0 s



s2₊₁



 _∞ 0

δ



₍

_α

₋

_α

₀

₎

cosh

α

coss

α

d

α

= P

π

Rb

ssins

α

0+tanh

α

0 coss

α

0

s



s2₊₁



. (23)

Moreover,theintroductionofthestressanddisplacement com-ponents(9) and(19) intotheboundaryconditions(14) yields

F

(π

)

=0, (24) 2

ε

(

cosh

α

+1

)

 _∞ 0 sF

(

π

)

coss

α

ds+  _∞ 0 F

(

π

)

coss

α

s ds= P

π

Rb D 1−

ν

, (25)

wherethefollowingparameterhasbeenintroduced

ε

= h

2R

(

1− ¯

ν)

μ

G. (26)

Letusdefine

q

(

s

)

=ssins

α

0+tanh

α

0coss

α

0 s

(

1+s2

)

=− cosh

α

0 s

(

1+s2

)

∂

∂

α

0



_coss

_α

₀ cosh

α

0



, (27) then, by considering the general expression (7) of function F(s) and its derivatives (10) , the boundary conditions (20) , (23) and (24) thenyield f2

(

s

)

=−2scoths

π

[q

(

s

)

sinh s

π

2 +f1

(

s

)

], f3

(

s

)

=q

(

s

)

sech s

π

2 − f1

(

s

)

tanh s

π

2 , f4

(

s

)

=2s[q

(

s

)

sinh s

π

2 +f1

(

s

)

]. (28)

TheintroductionofEqs. (28) into(10) thenprovides F

(

p

)

=

_π

P Rb 2 sinhs

π





s2− sinh2s

π

2



f1

(

s

)

+



s2_{− coshs}

_π



_sinhs

π

2 q

(

s

)



F

(

π

)

=−

_π

P Rb2s



f1

(

s

)

+2q

(

s

)

sinh s

π

2



. (29)

Finally,theintroductionofEqs. (29) into(25) yieldsthe follow-ingintegralequationfortheunknownfunctionf1(s):

(

cosh

α

+1

)

 _∞ 0 2s sinhs

π

[

(

s 2_{− sinh}2s

π

2

)

f1

(

s

)

+

(

s2_{− coshs}

_π

₎

_q

₍

_s

₎

_sinhs

π

2 ]coss

α

ds −1

ε

_∞ 0 [f1

(

s

)

+2q

(

s

)

sinh s

π

2 ]coss

α

ds= D 2

ε

(

1− ¯

ν

)

, (30) for0≤

α

≤ ∞.Inorder tosimplifythe previous integralequation, letusdefinethefunction

t

(

s

)

= f1

(

s

)

+2q

(

s

)

sinh s

π

2 +

D

2

(

1−

ν)

δ(

s

)

, (31)

then,theintroductionof(31) into(30) providesthefollowing Fred-holmintegral equationofthefirstkindfortheunknownfunction

t(s) _∞ 0



2s

(

cosh

α

+1

)

sinhs

π



(

s2_−sinh2s

π

2

)

t

(

s

)

−(1+s 2

₎

_q

₍

_s

₎

_sinhs

π

2



−t

(

_ε

s

)



coss

α

ds=0,

6 E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15 _∞ 0

2s sinhs

π



s2_{− sinh}2s

π

2



− 1

(

1+cosh

α)

ε



t

(

s

)

coss

α

ds= =G

(

α)

, for0≤

α

≤ ∞, (32)

where, by using (27) and according to Gradshtein and Ryzhik (1965) ,therighthandtermis

G

(

α)

=

_∞

0

s sins

α

0+ tanh

α

0coss

α

0

cosh

(

s

π

/2

)

coss

α

ds=

=4cosh

α

sinh2

α

0cosh

α

0

(

cosh2

α

+cosh2

α

0

)

2 = =−cosh

α

0

∂

∂α

0 2cosh

α

cosh

α

0 cosh2

α

+cosh2

α

0. (33) The integral equation (32)will be solved in Section 6 by us-ingaperturbation approachthatassumes

ε

asasmallparameter (Kanwal, 1971 ).Inthefollowing,weintroducesomefurther trans-formationsinordertoreducetheintegralequation(32)toamore familiarformthatallowusforitsnumericaltreatment.Tothisaim, letusdenotewithT(

α

)theFouriercosinetransformoft(s) t

(

s

)

=

_∞

0

T

(

α)

coss

α

d

α

, (34)

thenbyintroducing(34) inEq. (32) andchangingtheorderof in-tegrationoneobtains

_∞ 0 T

(

z

)

dz  _∞ 0



_2s sinhs

π



s2_{− sinh}2s

π

2



−

₍

1 1+cosh

α)

ε



coss

α

cosszds=G

(α)

, (35)

for 0_≤

α

_{≤ ∞}. By using contour integration and calculus of residues,onecanevaluatetheinnerdefiniteintegralin(35) asthe sumoftheresiduesoftheintegratingfunctionintheupper half-planeofthecomplexvariablesmultipliedby

π

i,namely

 _∞ 0 2scoss

α

cossz sinhs

π



s2_{− sinh}2s

π

2



ds = = 2cosh

(

α

− z

)

− 1 [1+cosh

(

α

− z

)

]sinh2

(

α

− z

)

+ 2cosh

(

α

+z

)

− 1 [1+cosh

(

α

+z

)

]sinh2

(

α

+z

)

. (36)

Moreover, theinversion oftheFouriercosinetransformofthe Diracdeltafunction

δ

(

α

− z)yields

_∞

0

coss

α

cosszds=

π

2

δ(α

− z

)

. (37)

Byusingtheresults(36) and(37) ,thenEq. (35) becomes

_∞ −∞ [2cosh

(

α

−z

)

−1]T

(

z

)

dz [1+cosh

(

α

−z

)

]sinh2

(

α

−z

)

−

π

2

ε

T

(

α)

1+cosh

α

=G

(

α)

. (38) IntegrationofEq. (38) withrespectto

α

thenprovides

_∞ −∞ cosech

(α

− z) 1+cosh

(α

− z)T

(

z

)

dz−

π

2

ε

 _T

_(α)

_d

_α

1+cosh

α

= 2sinh

α

sinh

α

0 cosh2

α

+cosh2

α

0+ arctan



sinh

α

cosh

α

0



tanh

α

0, (39)

for_−∞≤

α

_{≤ ∞}.Let us now introduce the following transforma-tions

z=log1− t

1+t,

α

=log 1+y

1− y, (40)

then, considering that T(y)₌T(_−y) due to symmetry condition, Eq. (39) becomes



1− y2



2

π

p.v.  1 −1



1− t2



_T

₍

_t

₎

(

t− y

)

(

1− ty

)

3dt− 1

ε

 T

(

y

)

dy=

Fig. 4. Normalized angular variation of the leading order stress field in polar coor- dinates near the semidisk corner.

= 2y



1− y2



_sinh

_α

0 4y2₊



_{1− y}2



2_cosh2

α

0 +tanh

α

0arctan 2y



1− y2



_cosh

α

0 , (41)

for− 1≤ y≤ 1, where p.v. denotes the Cauchy principal value of theintegral.Thesingularintegro-differentialequation(42) willbe solvedfortheunknownfunctionT(y)inSection 5 byusingan ap-proximate procedure basedon a power seriesexpansion andthe collocation methods (Erdogan et al., 1973; Monegato, 1987; Badr, 2001 ).

Theshearstressalongtheadhesivethinlayerfollowsfrom(9.3) for

β

₌

π

,byusing(27) ,(29) ,(31) ,(33) and(34) ,as

τ

αβ

(

α

,

π

)

=

_π

P_Rb

(

1+cosh

α)



G

(

α)

−  _∞ 0 2s sinhs

π



s2_{− sinh}2s

π

2



t

(

s

)

coss

α

ds



=−1

_ε

_π

P_Rb T

(

α)

. (42)

4. Asymptoticanalysisnearthecornerofthesemicircle

A preliminary asymptotic analysis of the stress and displace-ment fields near a right-angle elastic wedge with one face free of tractions and theother one subjectto the imperfect interface conditions betweenthe shear stress andthe radial displacement (Mishuris 1999, 2001) isperformedinordertoinvestigatethe be-haviorofthesefieldsatthebondingedges,namelyaty=±R.The resultswillthenbeusefulforthepropertreatmentoftheintegral equation(32)thatwillbeperformedinSection 5 and,in particu-lar,forthedefinitionofapowerseriesexpansion forthefunction

T(y).

Withrespect toa polar coordinatesystem(0, r,

θ

) centredat thecorner ofthewedge (Fig. 4 ),the consideredboundary condi-tionswrite:

σ

θθ

(

0

)

=0,

τ

rθ

(

0

)

=0

σ

θθ

(

π

/2

)

=0,

τ

rθ

(

π

/2

)

+Kur

(

π

/2

)

=0. (43)

E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15 7

Fig. 5. Normalized shear stress distribution along the adhesive thin layer for different values of the relative stiffness parameter εand four different loading orientation angle

θ0 .

A similarstress analysisfor aquarterof plane subjectto uni-formshear loadingon oneface can befound inSection. 11.1.1 of Barber (2010) .

Finite values of radial displacement and shear stress are ex-pectedattheright-anglecorner.Therefore,theleadingorderterms ofthebiharmonicAirystressfunctioninpolarcoordinatescanbe assumedintheform

χ

(

r,

θ)

=r2

(

c0+c1

θ

+c2cos2

θ

+c3sin2

θ)

+r3

(

d0cos

θ

+d2cos3

θ

)

+o

(

r3

)

, asr→0. (44)

The corresponding stress and displacement components near thewedgecorneraregivenby(Williams, 1952 ):

σ

θθ=

∂

2

χ

∂

r2,

σ

rr= 1 r

∂χ

∂

r + 1 r2

∂

2

χ

∂

θ

2,

τ

rθ=−

_∂

∂

_r



1 r

∂χ

∂θ



, (45) 2

μ

ur=−

∂χ

∂

r +

(

1− ¯

ν

)

r

∂ψ

∂θ

, 2

μ

uθ=−1_r

∂χ

_∂θ

+

(

1− ¯

ν

)

r2

∂ψ

_∂

_r, (46) wherethedisplacementfunction

ψ

mustbeanharmonicfunction obeyingthefollowingcondition

∂

∂

r



r

∂ψ

∂θ



=

χ

, (47) namely:

ψ(

r,

θ)

=b

(

cos

θ

)

/r+2c1

(

θ

2− ln2r

)

+4c0

θ

+4rd0sin

θ

+o

(

r

)

, asr→0. (48)

Then,theconstantsci anddi (i=0,1,2,3) canbe determined

8 E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15

Fig. 6. Shear stress distribution along the adhesive thin layer obtained by the perturbation method for small values of the relative stiffness parameter ε, for two different loading orientation angle θ0 .

Fig. 7. Specimen considered in the FE analysis. seriesexpansionsoftheAirystressfunction

χ(

r,

θ

)

= r2 4



π

sin2

θ

+sin2

θ

− 2

θ



+

π

r3 32

ε

(

cos

θ

− cos3

θ

)

+o



r3



_{, (49)}

anddisplacementfunction

ψ

(

r,

θ)

=2

ε

r cos

θ

+ln 2_r+

π

2

θ

−

θ

2₊

π

r 8

ε

sin

θ

+o

(

r

)

. (50) Correspondingly, the asymptotic stress field in the elastichalf planetakesthefollowingform

σ

θθ= 12

(π

sin

2

_θ

_+sin2

_θ

_{− 2}

_θ)

₊3πr

4ε cos

θ

sin

2

_θ

_+o

₍

_r

₎

_,

σ

rr= 12

(π

cos2

θ

− sin2

θ

− 2

θ

)

+16πrε

(

cos

θ

+3cos3

θ

)

+o

(

r

)

,

τ

rθ=12

(

2sin

θ

−

π

cos

θ

)

sin

θ

−8πεrsin

θ(

1+3cos2

θ)

+o

(

r

)

.

(51)

where the normalization condition

τ

r_θ=1 for

θ

=

π

/2 and r=0

hasbeenintroduced,asusualintheasymptoticanalysisofwedge andcrackproblems.Thenormalizedangularvariationofthe lead-ingorderstresscomponents(51) areplottedinFig. 4 .There,itcan be notedthat the stress field is finite at the corner and has no logarithmicsingularity under the boundary conditions(43) , even iftheprincipleofshearstress reciprocityisnot metatthe right-anglecorner(Barber, 2010 ).

Accordingto(51) theshearstressattheedgesoftheadhesive layer,namelyfor

θ

=

π

/2,behavesas

τ

rθ

(

r

)

=

(

1+

π

r

4

ε

)

τ

rθ

(

0

)

+o

(

r

)

. (52)

The displacementfield within thehalf plane corresponding to theAirystressfunction(49) (seeBarber (2010) ,Tables8.Iand9.I) reads: ur=−sin

θ

K + r 8

μ

[

(π

− 4

θ)(

1− 2

ν

¯

)

+

π

cos2

θ

− 2sin2

θ

] −₃₂

π

_με

r2

(

1+2

ν

¯ − 3cos2

θ

)

cos

θ

+o

(

r2

₎

_, u_θ =−cos

θ

K + r 8

μ

[8

(

1− ¯

ν

)

lnr−

π

sin2

θ

− 2cos2

θ

] +

π

r2 32

με

(

1− 2

ν

¯ − 3cos2

θ

)

sin

θ

+o

(

r 2

₎

_{. (53)}

Therefore, the displacement field is also finite at the corner. However, the rigid rotation Wr_θ turns out to be singular asln r

as r→0 dueto the termr ln r occurring in u_θ, as observed by England (1971) andBarber (2010) ,namely

Wrθ= 1 2r

∂

∂

r

(

ruθ

)−

∂

ur

∂θ



= 1 4

μ

[3− 4

ν

¯+4

(

1− ¯

ν)

lnr]+o

(

lnr

)

. (54)

5. Numericalsolutionofthesingularintegro-differential equation

The kernelofthe integral equation is almost generaland sin-gularasstendstoinfinity,andthusnoexactclosedformsolution Pleasecitethisarticleas:E.Radietal., Adhesivelybondeddiskundercompressivediametricalload, InternationalJournalofSolidsand

E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15 9

Fig. 8. Finite element mesh, loading (red arrows) and constraints (green arrows). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

forthefunctiont(s)canbeexpected;itcanbecalculatedonly nu-merically.AsshowninSection 4 ,theshearstressisfoundtobe fi-niteatthebondingedges.Moreover,itissymmetricallydistributed withrespecttothex-axis,asexpectedforskew-symmetricloading conditions.Then,accordingto(42) wecanassumearegularpower seriesexpansionfortheevenfunctionT(y),namely:

T

(

y

)

=∞

n=0

cny2n. (55)

Therefore,thesingularintegro-differentialEq. (41) providesthe followingrelation ∞  n=0 cn

Hn

(

y

)

π

− 1

ε

y2n+1 2n+1



= 2y



1− y2



_sinh

α

0 4y2₊



_{1− y}2



2_cosh2

α

₀+ +tanh

α

0arctan 2y



1−y2



_cosh

α

0 , (56) for−1≤ y≤ 1,where Hn

(

y

)

=



1− y2



2 p.v.  1 −1



1− t2



_t2n

(

t− y

)

(

1− ty

)

3 dt=2n 1− y2 y + +2y2n_ln1+y 1− y +y





y2_{,1,1/2− n}



₊ +[

(

1− 2n

)

ny3₊



_1+4n2



_y − n y

(

1+2n

)







y2_,1,1/2+n



_{, (57)}

where



is the Lerch transcendent function Gradshtein and Ryzhik, 1965 ). Notethat the conditionsHn(±1)=0and Hn(0)=0

followfrom((57) .

Inordertoevaluate theunknown coefficientscn ofthe power

seriesexpansion (55) fromEq. (56) we consider a finite number

N+1 of termsin the seriesexpansion. Then, by adopting a collo-cationmethodweevaluate Eq. (56) atN+1pointschosenasthe Gausspointswithintheinterval[0,1],namelyat

yk=sin



k+1 N+1

π

2



, fork=0,1,...,N, (58) thusobtainingalinearsystemofN+1equationsfortheunknowns

cn (n=0,1,…,N).

OncethefunctionT(y)hasbeencalculated,theshearstress dis-tributionalongtheadhesivebondingfollowsfrom(42) .Theresults fortheshearstressdistributionobtainedforN=30areplottedin Fig. 5 forvariousvaluesoftherelativestiffnessparameter

ε

rang-ing between0 and 5and forfour loading orientation angles

θ

0, namely30°,60°,70° and80° Fromtheseplotsitcanbe observed that,as

ε

tends to0,the sheardistribution approachesthat pro-videdin (4.2) and plottedin Fig. 3 (b)for an intact disk,namely foraninfinitestiffnessoftheadhesivethinlayer.Inthiscase,the shearstressisvanishing atbothjointedges andattains its maxi-mumvalueatthediskcenterforasmallangle

θ

0oratincreasing distanceto thecentreforlarge loading angle

θ

0,thus recovering theresultsofDong et al. (2004) plottedinFig. 3 (b).Astheshear stiffnessoftheinterfacedecreases,namelyforincreasingvaluesof

ε

,theshear stressdistribution along theinterface becomesmore andmore uniform. Note that the area under the curves in each figureis constant andcoincides withtheoverall loadcomponent alongthebondingdirection,namely

R

0

τ

αβ

(α

,

π

)

dy= P

10 E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15

Fig. 9. Analytical (solid line) and FEM (markers) results for the shear stress distribution along the bondline for the same elastic modulus of the disk E = 206 GPa and four different elastic moduli of the adhesive layer.

Fig. 10. Close-up of the broken specimen after the experimental test. A polyurethane upper layer was used to avoid excessive concentrated loads.

As the loading inclination angle

θ

0 increases the shear stress along thebonded jointattains a larger peak closer tothe edges. Inparticular,for

θ

0=90° and

ε

=0 theshear stressbecomes un-bounded therein. However, for

ε

> 0 the shear stress displays a

smoothpeakandnon-singularbehaviorattheedges ofthe bond-ingline,whereitapproachestheasymptoticprediction(52) .Note alsothatthetangentialdisplacementalongthebondlineturnsout tobeproportionaltotheshearstress,accordingtocondition(14.2).

6. Perturbationapproachforsmall

ε

A closed form approximate solution of the integral equation (32) canbe derivedby usingaperturbation methodassuggested inChapter11ofthebookofKanwal (1971) iftheparameter

ε

de-finedin(26) issufficientlysmall,namelyifthestiffnessofthe elas-ticspringisrelativelylarge.Inthiscase,thelowestordertermfor

ε

= 0correspondstoaninfiniteshearstiffnessoftheadhesivethin layerandthus it coincideswiththe solutionof theBrazilian test foran intactelastic disk(Dong et al., 2004 ), whereas thehigher ordercontributionscanbe calculatedbyassuming apowerseries expansionin

ε

ofthefunctiont(s),namely

t

(

s

)

=∞

n=1

ε

n_t

n

(

s

)

. (60)

Byintroducingtheseriesexpansions(60) intheintegral equa-tion(32) andequatingequalpowers of

ε

,theproblemcanbe re-ducedtothefollowingsystemofintegralequationswiththesame kernel: _∞ 0 t1

(

s

)

coss

α

ds=−G

(α)

, (61) _∞ 0 tn+1

(

s

)

coss

α

ds=

(

1+cosh

α)

Gn

(

α)

,

(

n≥ 1

)

(62)

E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15 11

Fig. 11. Force-displacement curve for the bonded disk ( θ0 = 60 °). for0≤

α

≤ ∞,where Gn

(α)

= _∞ 0 2s sinhs

π



s2− sinh2s

π

2



tn

(

s

)

coss

α

ds. (63)

Thefunctionstn(s), forn≥ 1,canbeexplicitlyobtainedby

tak-ingtheinverseFouriercosinetransformsof(61) and(62) ,byusing theresult(33) andthedefinition(63) ,namely

t1

(

s

)

= 4

π

cosh

α

0

_∂

∂

_α

0 _∞ 0

(

1+cosh

α)

cosh

α

cosh2

α

+cosh2

α

0 coss

α

d

α

= =cosh

α

0

_∂

∂

_α

0

coss

α

0 cosh

α

0cosh

(

s

π

/2

)

− sinh

α

0sins

α

0 cosh

α

0sinh

(

s

π

/2

)



, (64) tn+1

(

s

)

=

_π

2 _∞ 0

(

1+cosh

α)

Gn

(α)

coss

α

d

α

.

(

n≥ 1

)

(65)

Finally, the shear stress along the adhesive bonding follows from(42) ,byusing(60) and(61) ,as

τ

αβ(N)

(

α

,

π

)

=

_π

P_Rb

(

1+cosh

α)



G

(

α)

− N  n=1

ε

n_G n

(

α)



, (66) where N is the numberof terms considered in the power series expansion(60) .ThefunctionG1(

α

),t2(s)andG2(

α

)havebeen cal-culatedby using contourintegration andcalculus of residues ac-cordingto their definition(63) and(65) . Forthesake of concise-ness, theirexpressions andderivations havebeenreportedinthe Appendix.

The results obtained for N₌2, namely up to terms of order

ε

2 _{inthe expressionof theshear stress field (66) , havebeen} re-portedin Fig. 6 for some smallvaluesofthe parameter

ε

. Itcan beobservedthatfor

ε

= 0theshearstressdistributionrecovers ex-actlythesolutionfoundbyDong et al. (2004) .Inthiscaseindeed, by using (33) , Eq. (66) provides the result (4.2). As

ε

increases

theshearstressdistributionbecomesmoreuniform.However, the three-termsperturbedsolutionstartslosingitsvalidityalreadyfor smallvaluesof

ε

,namelyfor

ε

₌0.15inthecase

θ

0=30° andfor

ε

=0.10inthecase

θ

0=60°,thus denotingthata largernumber oftermsisrequiredforobtaininganaccurateapproximationwhen theparameter

ε

islarger.Moreover,anunrealisticshearstress con-centrationispredictedatthebonding edges(y=±R)asthevalue of

ε

increases.

7. Verificationbythefiniteelementmethod

A Finite Element analysis of a disk with radius R=50mm bonded along a diameter with a h=0.1mm thin adhesive layer has been performed by using the LUSAS Finite Element system. Meshgeometryandloadingare showninFigs. 7 and8 . Thedisk halves were discretized with quadratic 2D plane-strain/stress el-ements with out of plane width b₌1mm. Two elements were placedthroughthethicknessoftheadhesivelayerwhileeach ad-herendwas partitionedin differentregions with a finermesh in proximity of theadhesive layer. The two regions near the bond-linewereuniformwithoutspacegrading,resultinginquasi-square elements with side of 0.1mm for the adherends and rectangu-larelements withsides 0.1×0.05mm for the adhesive, asshown inFig. 8 .The total numberofelements(adherends₊adhesive)is 184,454(8-nodedquadrilaterals), with2000elementsinadhesive layer(2 elements throughthe thickness). We decided toadopta quite refined mesh since we were interested in getting a fairly precise resultson the bondline,especially aty=±R.The materi-alswere assumedelastic withtheconstitutive propertiesofsteel for the semidisks (E₌206GPa,

ν

₌0.3) andvarious elastic prop-erties for the adhesive layer, in order to verify the effect of the elastic mismatch (E0=206GPa,

ν

0=0.3, E1=2060MPa,

ν

1=0.3,

E2=206MPa,

ν

2=0.3, E3=20.6MPa,

ν

3=0.3). Bothadhesiveand adherends are modelles as linear elastic and isotropic materials and no failure criterion is adopted. The purely elastic response,

12 E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15

Fig. 12. Digital image correlation of the bonded Brazilian disk at point 11,735 N (a), at 13,201 N (b) and the onset of failure at 14,427 N (c).

consistent withthe assumptions ofthe theoretical model,makes theresultsapplicabletobrittleadhesives.

Comparedto Fig. 7 the modelin Fig. 8 is rotated inorder to placethebondlinealong theverticaly-axisandtheexternalload isdecomposedaccordingtotheverticalandhorizontaldirections. ThecentrenodeOoftheadhesivelayerwasconstrained horizon-tallytoeliminateanyrigid-bodymotions(seeFig. 8 ).

The results of FEM analyses are reported in terms of shear stressalongtheadhesivebondline.Sincetheanalyticalframework considered a perfect bond in normal direction and the normal strains within the adhesive thin layer have beenneglected, then thenormalstressalongthebondlineareexpectedtocoincidewith thoseoccurringinan intactdisk(14) .Futurework willbe aimed atestimatingthestressfield incaseofan elasticbondinnormal direction.

Theshearstressissymmetricallydistributedwithrespecttothe centerofthebondlineandfourcurvesarereportedinFig. 9 forthe fourconfigurationsanalyzed.ItisimmediatetoverifythattheFE predictions (single markers) for all the considered configurations coincidewiththeanalyticalresults(solidlines)obtainedfromthe analyticalapproachdevelopedinSection 5 .Some discrepancyhas been found for the finite elements placed at both edges of the bonding line, due to the infinite rigid rotation predicted by the asymptoticanalysisinSection 4 ,whichmaycausesomenumerical instabilityfortheFEcode.

8. Experimentalresultsfromdigitalimagecorrelation

TheDigitalImageCorrelation(DIC)wasusedtovalidate exper-imentally the theoretical and numerical analyses of the bonded disk. DIC is a powerful optical-numerical method which mea-suresfull-field surfacedisplacements by meansof anon-invasive contactless techniques. Several works about the application of DIC techniques in adhesive bonded joints analysis can be traced in technical literature. Comer et al. (2013) and Kumar et al. (2013a,b ) exploited them to analyze composite joints and kiss-ing bonds. Guo et al. (2011) used them to study the adhesive interface. The review work of Hild and Roux (2006) shows the main experimental techniquesabout this methods and the work ofPan et al. (2009) helpsinchoosingthesubset sizeandthe re-gionofinterest.TheDICsystemisabletoassessthefull displace-ment field by using a contactless optical method, whichexploits a highresolution camera anda coherentlight source. The speci-menispaintedusingarandomnetofdots,calledspeckles,which areusedbythesoftwaretocorrelatetheimagesandtodetermine the displacementfield. The camera records theimage of the un-loaded specimenandthen theloading procedure starts, synchro-nized with the CCD acquisition. The system records a set of de-formedimagesastheloadincreasesandcomparesthemwiththe reference configuration. Since the initial position of each subset is known, and its final position can be estimated, it is possible to compute the displacement fields either in the plane (2D DIC, one camera needed) or inthe space, by usingtwo camerasand recovering the out of plane displacement by a triangulating pro-cedure (3DDIC, twoor morecamerasneeded).The experimental testwerecarriedout byusingaDICSoftwarebyDantech Dynam-icscalledIstra4D,inthe2Dconfiguration,sincenomeasureofthe outofplanedisplacementisneeded,asstatedbyPan et al. (2009) . Thecamera isa5MPCCDwitha C-mountlenswith50mmfocal length. The experimental set-up is madeof two flat plates used tocompress thespeckled Brazilian bondeddisk.The speckle pat-ternswereacquiredby randomlysprayingblackandwhitepaints onthetwo flatsemi-disksurfaces,accordingto Pan et al. (2008) . An image captured at the end of the experimental procedure is reportedin Fig. 10 . Thetest rig implementsthe conceptual load-ingconfigurationreportedinFig. 7 foraloadinginclinationangle Pleasecitethisarticleas:E.Radietal., Adhesivelybondeddiskundercompressivediametricalload, InternationalJournalofSolidsand

E. Radi et al. / International Journal of Solids and Structures 0 0 0 (2018) 1–15 13

Fig. 13. Shear strain distribution along the bondline for the three acquisition points of Fig. 12 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

θ

0=60°,in order to demonstratethe feasibility of the analytical proceduredeveloped.Apolyurethanerigidsheetisinsertedonthe topplateinordertogiveasmoothercompressiontothediskand itisvisibleinFig. 10 inyellowandtoensurethecorrectposition ofthespecimen,preventingpossibleslips.Wedecidedtoadd this layer only on the top to be able to verifyif the steel-steel con-tactonthebottompartcouldcauseproblemsduringtesting.This additionalelasticitydoesnot affectthe maximumfailure forceof thejointbutonly thestrokemeasured bythe universalmachine, whichisnotusedsincethefullfieldacquisitionbytheDIC appara-tusisavailable.Thespecimenismadebytwomildsteeladherends (E=206GPa)withthicknessb=10mm,andradiusR=25mm.The two adherendsare bondedwith the high strength epoxy Loctite Hysol 3422 (Henkel, 2003 ), which ensures fast curing and good material strength.The adhesive elastic modulus,accordingto the producer datasheet, is1298MPa,witha nominal tensilestrength of28.6MPaandanelongationatbreakof3.3%.Allpropertieswere retrievedfromtechnicaldatasheetsprovidedbytheproducerand comply withthe ASTMD882 Standard.Some other experimental tests Carbas et al. (2013) reveal that thisadhesive tends to have brittlefailurewhencuredatroomtemperature.Thetestswere car-riedout onaMTSBionix858universal machineequippedwitha 25kNloadcellatquasistaticloadingrateof1mm/min.The force-displacement curveswere sampled at102.5Hzwhile theDIC ac-quisitionwassetat1Hz,inorderto limitthe amountofdatato process.

The force-displacement curve derived from the experimental set-up in the previous section (Fig. 10 ) is reported in Fig. 11 . The curve shows a monotonic non-linear behaviour up to fail-ure at 14,427N. The non-linearity is due to the soft layer of polyurethane needed to avoid misplacement of the specimen and excessive contact pressure. As soon as the first crack in the adhesive bondline occursan immediate abruptfailure ofthe joint takes place. The crack was easily detectable using the DIC equipment.

The primaryoutput ofthedigitalimagecorrelation isthe dis-placementfull field,fromwhichtheDIC softwareisable to com-putealltheother interestingquantities,suchastheshearstrains, throughsimpledifferentiation.Fig. 12 showstheengineeringshear strain map for three salient points highlighted in Fig. 11 . The strains, expressed in millistrain, are calculated using a subset of

17× 17pixel, following the work of Pan et al. (2008) in addi-tionto theDantech Dynamics manual andusinga reference sys-temconsistent withthebondline,visible inFig. 12 (c). Theframe reported in Fig. 12 shows a loading state where it is possible to see the increasing shear strains in the bondline,which reach the maximum at the onset of the crack, shown in Fig. 12 (c), where it is clearly visible an almost uniform and regular strain distribution in the adhesive layer. The shear stress distribution along the bondline at the three points analyzed is reported in Fig. 13 . The first part of the bondline (around 7mm) is not re-ported since the correlation is prevented by the deformation of the PU sheet. After the frame showed in Fig. 12 (c), the correla-tion islost, since the cracked part ofthe image cannot be com-puted due to the excessive displacement. The average value of the shear strain along the bondline is 33.5 mstrain, (blue solid line in Fig. 13 ), consistent with the producer datasheet. The ex-perimentalresultsdemonstratethepracticalfeasibilityofthe pro-posedtestintermsofeasinessofmanufacture andsimpleset-up. ThebondedBrazilian diskis veryeasy tobe obtainedwith mini-mummachiningofabar,anditissimplycompressedbetweenflat plateswith a universal testing machine, withno need of special fixtures.

9. Conclusions

Asimpletestforthecharacterizationofadhesivelybondedjoint has been proposed here, as a generalization of the widely used Braziliantestforhomogeneousbrittlespecimens.Duetothe sim-plegeometryofthetest,mixed-modeloadingconditionsalongthe adhesive layer can be easily achieved by properly selecting the loadinginclinationangle.Tothisaim,analyticalandnumerical so-lutions for the stress and displacement fields in a bonded elas-ticdiskwithadiametricaladhesive thinlayer,subjected to com-pression along an arbitrary diametrical direction, have been de-rivedherein. In particular, theshear stress distribution along the adhesive thinlayer as well asthe stress anddisplacement fields withintheelasticdiskhavebeencalculatedforvariousloading ori-entationangles.Then,theanalyticalresultshavebeensuccessfully comparedwithFEnumericalpredictions.Finally,theDICtechnique wasalsoemployed inorder todemonstrate the feasibilityof the proposedtest.