FRACURE ANALYSIS OF ADHESIVE JOINS USING INRINSIC COHESIVE ZONE MODELS M. Alano a*, F. Furiuele a, A. Leonardi a, C. Maletta a, G. H. Paulino b a Department o Mechanical Enineerin, Universit o Calabria, 8736 Arcavacata di Rende, Ital, e-mail: alano@unical.it b Department o Civil and Environmental Enineerin, Universit o Illinois at Urbana-Champain, Newmark Laborator, 5 N. Mathews Avenue, Urbana, IL 68-35, U.S.A. ABSRAC In this paper Cohesive Zone Model (CZM) concepts are applied in order to stud mode I racture in a pre-acked bonded Double Cantilever Beam (DCB) specimen. A cohesive surace element is implemented in a Finite Element commercial code usin intrinsic cohesive zone models: eponential, bilinear and trapezoidal traction-separation laws. he sensitivit o cohesive zone parameters in predictin the overall mechanical response is eamined, then the load displacement curves obtained with the dierent CZMs are compared and some interestin eatures concernin the prediction o damae onset in adhesive joints are illustrated. Finall, cohesive parameters are identiied comparin numericall simulated load-displacement curves with eperimental data retrieved rom literature.. INRODUCION he use o adhesive joints in automotive, aerospace, biomedical, and mioelectronics industries is widespread; however, inaccurate joint abrication or inappropriate curin ma cause the presence o bubbles, dust particles or un-bonded areas in the bond line. As a consequence the establishment o reliable interit assessment methodoloies and testin procedures is needed in order to tackle racture events in adhesive joints. A valuable approach rom this point o view is represented b the Linear Elastic Fracture Mechanics (LEFM) [-3]. So ar, both the classic one parameter models o elastic and plastic racture have been successull applied but the present some limitations. he require pre-eistin ack like deects or notch and, thereore, ack initiation cannot be treated directl. In addition, the nelect a detailed desiption o what happens in the racture process zone (FPZ) because, provided the FPZ is small compared to the other specimen dimensions, the lump it all into the ack tip. However, a detailed desiption o the FPZ is essential, especiall to understand racture mechanisms and to desin suitable modiications o the material (e.. touhenin b reinorcement in polmeric structural adhesive [4]). he standard model used to desibe the ack tip process zone assumes bonds stretchin orthoonal to the ack suraces until the break at a characteristic stress level. hus, the sinular reion introduced rom LEFM can be replaced b a lateral reion over which non-linear phenomena occur. his model is oten mentioned as the Cohesive Zone Model (CZM) and it can be traced back to the works o Dudale and Barenblatt [5,6]. Accordin to CZM the entire racture process is lumped into the ack line and is characterized b a cohesive law that relates traction and displacement jumps aoss cohesive suraces (-). Unlike racture mechanics based strateies, CZM can be used or the analsis o ack initiation and rowth that, indeed, are obtained as a natural part o the solution without an a priori or ad hoc assumptions. So ar, CZM has been successull applied to model racture in metals, conete, polmers and unctionall raded materials (FGMs) [7-]. Cohesive zone approaches dier in the wa b which cohesive surace elements are inserted in the initial eometr. In the intrinsic approach [] cohesive elements are introduced between volumetric elements rom the beinnin o the analsis as a network o cohesive suraces; on the other hand, in
the etrinsic approach, [3] a cohesive element is introduced in the mesh onl ater the correspondin interace is predicted to have started to ail, i.e. the are inserted adaptivel. In this paper CZM concepts have been applied in order to stud mode I racture in a pre-acked bonded Double Cantilever Beam (DCB) specimen. A cohesive surace element has been implemented in the Finite Element commercial code ABAQUS [4] usin intrinsic cohesive zone models: eponential, bilinear and trapezoidal laws. heoretical and numerical aspects o the CZM are eplained. In particular, details reardin the computation o the orce vector and the tanent stiness matri are presented. he sensitivit o cohesive zone parameters in predictin the overall mechanical response is eamined and then cohesive parameters are identiied comparin numericall simulated load-displacement curves with eperimental data retrieved rom literature [].. GENERAL HEORY AND NUMERICAL ASPECS OF COHESIVE ZONE MODEL (CZM). CZM concepts Let consider the domain Ω showed in Fi..a which is divided into two parts, i.e. Ω and Ω, b a material discontinuit, c. his last deines the interace between the domains Ω and Ω and represents an internal surace not et separated. Presibed tractions, i, are imposed on the boundar. he stress ield, σ ij, in Ω, nelectin bod orces, is related to the eternal loadin and to the closin tractions, i, in the material discontinuit throuh the ollowin equilibrium equations σij = j σ n = ij j i σ n = r ij j i σ n = ij j i in on on on Ω = Ω U Ω u c () where n j is the outward normal, while r i are the reaction orces on the boundar u. Cohesive zone model is, as previousl said, one o the most commonl used tools to model material discontinuities. In the simplest and most usual ormulation o CZM, the whole bod volume remains elastic while the nonlinearit is embedded in the cohesive law which dictates the boundar conditions alon the ack line, c (Fi. ). he pick stress on the cohesive law is the cohesive strenth, σ, o the material while the area under the curve is the cohesive racture ener, G c. As a consequence, racture process can be summarized as illustrated in Fi..b: at irst a linear elastic material response prevails (I), as the load ineases the ack initiates (=σ ) (II) and then, overned b the non linear cohesive law (sotenin curve), it evolves rom initiation to complete ailure (III) with the appearance o new traction ree ack suraces, c and + c (= ) (IV). Fi. - Cohesive Zone Model concepts Fi. - Intrinsic versus etrinsic CZM hereore, the continuum should be characterized b two constitutive laws: a linear stress-strain relation or the bulk material and a cohesive surace relation (cohesive law) that allows ack spontaneous initiation and rowth. It was shown in [5] that the shape o the traction-separation laws plas an important role in the maoscopic mechanical response o the sstem, thereore, it is
important to properl select the shape o the sotenin curve. he development o cohesive zone models in FEM ramework requires bulk inite elements, or modellin the stae () (Fi..b), bordered b cohesive surace elements or the remainin three staes. From this point o view there are basicall two cohesive zone approaches that dier in the wa b which cohesive surace elements are inserted in the initial eometr. In the intrinsic approach [] cohesive elements are introduced between volumetric elements rom the beinnin o the analsis as a network o cohesive suraces. Cohesive tractions inease rom zero to a ailure point that is represented b the cohesive strenth, at which the reach a maimum beore the raduall deease back to zero ollowin the post peak sotenin behaviour (Fi. ). his active network o cohesive suraces introduces a compliance due to the initial slope o the cohesive law, i.e. the penalt stiness, K. his drawback could be addressed ineasin K, however, rom a numerical standpoint, it is not recommendable to use elements with an etremel sti initial response because instabilities in the solution procedure arise [6]. Another noteworth CZM reported in literature is the etrinsic model proposed in [3], which eliminates the artiicial compliance tpical o the intrinsic models mentioned above. In particular, in the etrinsic approach, a cohesive element is introduced adaptivel in the mesh onl ater the correspondin interace is predicted to have started to ail; as a consequence, it is possible to adopt a cohesive law with a ver hih initial stiness like that reported in Fi.. In the present paper, the racture behaviour o adhesive joints is analzed usin intrinsic cohesive zone model. For this particular problem cohesive surace elements are introduced alon a predeined racture path, i.e. the bond line; thereore their number is reduced and this, in turn, ields a lower compliance..3 raction separation laws eamined in the paper In this paper three o the most representative traction-separation laws proposed in literature have been analzed and compared in order to investiate the racture behaviour o adhesive joints: bilinear, trapezoidal and eponential cohesive laws (Fi. 3). In particular, onl pure-mode I decohesion problems are considered so that the attention has been ocused on traction-displacement relationships relatin the normal openin displacement to the dual traction component. In all the traction separation laws presented below, onl positive values o are considered, or neative values, a hih penalt stiness is assumed in order to avoid ack aces interpenetration (Fi. 3). In addition, onl monotonic loadin processes, without local unloadin, will be studied in the paper. his assumption strictl makes the laws thermodnamicall not consistent (i.e. irreversibilit is not accounted or) however it will not aect the results o the comparative analses. In what ollows, the tractionseparation laws analzed in the paper are briel summarized. he cohesive traction accordin to the eponential model is derived trouh cohesive potential ener φ [] φ =σ ep ep() + ; φ = = σ ep (a, b) where is the itical normal displacement and σ, as mentioned above, is the cohesive strenth o the material. As ack ace displacements inease, the traction ineases, reaches a maimum, and then decas monotonicall. he traction interated to complete separation ields the racture ener release rate G c ( ) = σ ep (3) he analtical epression o the bilinear law [7] is iven as ollows = σ K ( ) /( ) > = γ < (4) where K is the penalt stiness o the cohesive zone model that can be adjusted selectin proper values o σ and γ (Fi. 3). he normal work o separation is then iven b G c = σ / (5) he analtical epression o the trapezoidal traction-separation law is iven as ollows [8]
= σ ( K σ ) /( ) < > = λ < = λ (6) the normal work o separation is iven b Gc = σ ( + λ λ ) / (7) As it can be seen, or this particular (-) curve, the overnin cohesive parameters are the cohesive racture ener (G c ), the peak stress (σ ), the itical openin displacement ( ) and the actors λ and λ whose values dictate the shape o =(). Fi. 3 Cohesive laws eamined Fi. 4 - Cohesive zone element.3 Finite element implementation he development o cohesive zone models in FEM ramework requires bulk inite elements, or modellin the stae () (Fi..b), bordered b cohesive surace elements or the remainin three staes: () ack initiation, (3) ack evolution and (4) complete ailure. In this work, our node cohesive zone elements (CZE) with two interation points have been implemented within the commercial FE code ABAQUS [4] usin the user element (UEL) capabilit. A schematic representation o a CZE is iven in Fi. 4. A CZE is made up o two linear line elements (cohesive suraces) that connect the aces o adjacent elements durin the racture process. he two suraces initiall lie toether in the unstressed deormation state (zero thickness) and, subsequentl, separate as the adjacent elements deorm. In particular, the surace constitutive relations presibe the evolution o tractions () enerated aoss ack aces as a unction o displacements jump () at the nodes o the element. hus cohesive elements do not represent an phsical material, but desibe the cohesive orces when material elements are bein pulled apart. he insertion o cohesive surace elements brides linear elastic and racture behaviour allowin or spontaneous ack propaation. In order to carr out the iterations o the method [4], the contribution o cohesive elements to the tanent stiness matri as well as to the orce vector is acquired rom the numerical implementation o the CZM. he implementation o a eneral cohesive element is eplained in what ollows. he line cohesive element has eiht derees o reedom. In particular, the nodal displacements vector in the lobal coordinate sstem is iven as: () () () () (3 ) (3 ) ( 4 ) ( 4 ) { u u u u u u u u } u = (8) in which the order ollows tpical ABAQUS conventions. Crack aces openin or cohesive elements is deined as the dierence between top and bottom nodes thereb leadin to the ollowin deinition in terms o displacements o paired nodes:
δ δ δ δ (,4) (,3) (,3) (,4) u u = u u () ( ) ( ) () u u u u ( 4) (3) (3) ( 4) = Lu = u (9) with [L] bein an operator matri. From the nodal positions, the ack ace openin is interpolated to the Gauss interation points b means o standard shape unctions ξ (,4 ) ~ δ δ = = (,3 ) = δ ξ + ξ δ + ξ δ δ (,3 ) (,4 ) NLu () where ξ is the natural coordinate and N is the matri o shape unctions. Since the constitutive relations are based on tractions and displacements in the local coordinate sstem a transormation rom lobal to local coordinate is needed or the cohesive element. Let [R] deines the orthoonal transormation matri rom lobal (,) reerence rame to element speciic local coordinate sstem. hen the relative displacement vector, or an uncoupled cohesive law, is iven as: = = RNLu Bu () he relative displacements o the element aces eate normal and shear displacements, which in turn enerate element stresses dependin on the constitutive equations o the material. he relationship between tractions and displacements is iven b: = ; t t = n n (a, b) with the last term bein the Jacobian stiness matri while the subsipts t and n denote the tanential and normal directions. he constitutive relationships adopted herein have been presented in the previous section and the are independent o the element ormulation. he stiness matri or cohesive elements can be obtained b minimizin the total amount o potential ener: Π = U + W = da u (3) where is the eternal traction vector; ater some manipulations it ollows so that the cohesive element stiness matri is iven as B BdA u = A (4) K = B BdA = B B w d (5) A where w is the element width. he contribution o cohesive elements to the lobal orce vector is deined in a variational settin usin the principle o virtual work δπ IN = δπ EX δ da = δu F COH (6)
with δ Π and δ Π bein the internal and eternal virtual work respectivel. Ater some IN EX manipulations the equivalent riht hand side nodal orce vector or cohesive elements, F COH, is iven as ollows: B da = w B dc A F = COH (7) c 3. SIMULAION OF FRACURE IN DCB CONFIGURAION he specimen analzed herein is the DCB studied in []. A schematic representation is reported in Fi. 5. he beam has a lenth (L) o mm, a substrate thickness (h) o 5 mm, and a width (B) o 3 mm. he mechanical pre-notch (a ) etends 4 mm rom the let to the riht ede o the beam. he substrate material is aluminium (E s =7 GPa and ν s =.33) and the adhesive is epo (E a =.3 GPa and ν a =.35). he bond line thickness (h a ) is equal to.3 mm. Eternal loadin is imposed under displacement control. Fi. 5 - Schematic representation o the specimen he measured racture touhness reported in [] is G c = 55 N/m. he cohesive strenth, that is a material parameter, is not easil measurable, thereore estimated values are usuall adopted [9]. Likewise, in this paper the cohesive strenth will be selected usin a trial and error procedure until a match between numerical and eperimental P-δ curves will be obtained. he eneral purpose commercial code ABAQUS has been adopted in order to model the specimen: zero thickness cohesive surace elements were used or the cohesive zone and plane strain our nodes isoparametric elements (CPS4) or the surroundin bulk material. Fi. 6 shows mesh details or the reions where cohesive elements are inserted. Fi. 6 - Details o the FE model in the reion were cohesive elements are inserted Fi. 7 - otal dissipated racture ener In order to properl represent tractions in the cohesive zone, spatial disetization alon the ack propaation path is essential and enouh elements (three or more) should be inserted in the FPZ. With coarser mesh, the shape o the interace law ma have a non neliible inluence on the simulated load-displacement curve; however, this problem can be tackled b means o suitable mesh reinements []. In the paper, dierent cohesive element sizes, have been assessed. o illustrate that the element size chosen or the numerical analses is objective and not sensitive to arteacts o the numerical solution, a lobal quantit, i.e. total dissipated racture ener, is evaluated or each element size and then compared. As it can be seen rom Fi. 7, with element size ranin rom. to mm, the racture ener dissipated is almost the same. hereore, usin an element lenth o mm
the proper mechanics o ener dissipation and ack propaation are ensured. Subsequentl, a sensitivit analsis to cohesive racture parameters has been perormed. he inluence o the simulated mechanical responses to cohesive parameters, i.e. material strenth (σ ) and cohesive racture ener (G c ) has been assessed. Fi. 8 illustrates the sensitivit o P versus the openin displacement δ curve to dierent racture eneries and cohesive strenths or eponential and bilinear cohesive laws. (a) (b) (c) Fi. 8 - Sensitivit to cohesive strenth and racture ener. Fi.8 clearl shows that when the cohesive strenth, σ, is held constant (Fi. 8.a-b) as the racture ener ineases the area under the curve (lobal racture ener) and the peak load ineases. On the other hand, when the cohesive racture ener is held ied (Fi. 8.c-d) as the itical strenth ineases, the peak load is ineased while the lobal racture ener is almost constant. he authors obtained similar results or the trapezoidal law [], but, or the sake o brevit, these results are not reported herein. he load-displacement curves predicted b the cohesive laws usin similar racture parameters are now compared (Fi. 9). As it can be seen, a ood areement between the cohesive laws is observed in the linear elastic rane and in the post-peak reions (damae propaation) o the P-δ curves. However, sliht dierences in the maimum load reion (damae onset) are observed, in particular the trapezoidal law overestimates the peak load. his can be addressed as ollows; the trapezoidal law produces a dierent interacial stress proile with respect to the other ones. hese in turn can reatl aect the simulated P-δ curve i the FPZ is lare, i.e. or ver sti bulk materials []. he parameters o the cohesive zone model are then tuned in order to match the eperimental results reported in []. hese last are calibrated b ittin present numerical results to eperimental indins in order to account or the dierences between eperiments and numerical simulations. he simulated load displacement curves alon with the correspondin cohesive parameters that ive minimum deviations between simulations and eperiments are reported in Fi.. As the tanent stiness or the eponential model starts to deease rom the ver beinnin, the initial slope o the correspondin load-displacement curve is aected b this inherent artiicial compliance. When bilinear and trapezoidal models are used this compliance is reduced. It is worth notin that the cohesive strenth adopted or the eponential model is hiher than that o the bilinear one; this is a drawback o the eponential model that does not allow to control the penalt stiness, K, without aectin the cohesive strenth (K ineases i σ ineases). On the contrar, in the linear CZMs, K can be (d)
adjusted ensurin a sti connection between cohesive and bulk elements and thus a proper representation o the undamaed state, without aectin the cohesive strenth. However, the lowest value o the cohesive strenth amon those that allow to obtain the best it with eperimental data is that o the trapezoidal law; the dierences with respect eponential and bilinear laws are -5 % and -4 %, respectivel. Indeed, the trapezoidal law overestimates the load or damae onset and then lower values o the cohesive strenth must be selected in order to match the eperimental results. Fi. 9 - Comparison amon the P-δ curves keepin cohesive parameters as constants Fi. - Eperimental versus numerical P-δ curve 4. SUMMARY AND CONCLUSIONS In this paper CZM concepts have been applied in order to stud mode I racture in a pre-acked bonded Double Cantilever Beam (DCB) specimen. A cohesive surace element has been implemented in a inite element commercial code usin intrinsic cohesive zone models: eponential, bilinear and trapezoidal laws. Basicall, a ood areement amon numerical and eperimental results has been observed and some interestin eatures were reported. In particular, it has been shown that both the linear cohesive zone models are appropriate or modellin the undamaed state o acked adhesive joints: the can control the pre-peak slope o the traction-separation law allowin to reduce compliance. However, the trapezoidal law overestimates the load or damae onset and then lower values o the cohesive strenth must be selected in order to match the eperimental results. his problem was addressed to the dierent interacial stress proile produced b the trapezoidal law which, in turn, can reatl aect the simulated P-δ curve or larer FPZ. REFERENCES [] Kinloch A.J.: Adhesion and Adhesives Science and echnolo, London: Chap. & Hall, 986. [] Pirondi A., Nicoletto G. Proc. o the Italian Group o Fracture, Bari,. (in Italian) [3] Alano M., Furiuele F., Maletta C. Ke En Mat, 6; 35:49-5. [4] Williams J.G.: Fracture Mechanics o Polmers, NY: Halsted Press, John Wile & Sons, 984. [5] Dudale D.S. J Mech Phs Solids, 96; 8:-4. [6] Barenblatt G.I. Adv Appl Mech, 96; 7:55-9. [7] Li W., Siemund. En Fract Mech, ; 69:73-93. [8] Roesler J.R., Paulino G.H., Park K., Gaedicke C. Cem & Con Comp, 7; 9:3-3. [9] Son S.H., Paulino G.H., Buttlar W.G. ASCE J En Mech, 6; 3:5-3. [] Son S.H., Paulino G.H., Buttlar W.G. En Fract Mech, 6; 73:89-848. [] Zhan Z., Paulino G.H. Int J Plasticit, 6; :95-54. [] Needleman A. ASME J Appl Mech, 987; 54:55-53. [3] Camacho G.., Ortiz M. Int J o Solids and Structures, 996; 33:899-938. [4] ABAQUS User s Manual, v6.5-. Pawtucket, USA, Hibbit, HKS Inc;. [5] Chandra N., Li H., Shet C., Ghonem H. Int J o Solids and Structures, ; 39:87-855. [6] Alano M., echnical Report, Universit o Illinois at Urbana-Champain, 6. [7] Geubelle P.H., Balor J.S. Compos Part B, 998; 9:589-6. [8] veraard V., Hutchinson J. W. J Mech Phs Solids, 99: 4:377-397. [9] Alano G., Crisield M.A. Int J Numer Meth En, : 5:7-736. [] Alano G. Compos Science and ech, 6; 66:73-73. [] Alano M., Furiuele F., Leonardi A., Maletta C., Paulino G. H. Ke En Mat, 7; 348:3-6.