Reliability Analysis of Bonded Joints with Variations in Adhesive Thickness

Transcription

1 Relablty Analyss of Bonded Jonts wth Varatons n Adhesve Thckness Jean-Dens MATHIAS* IRSTEA, Laboratore d Ingénere pour les Systèmes Complexes, Campus des Cézeaux, 24 Avenue des Landas BP 50085, Aubère Cedex, France Maurce LEMAIRE Clermont Unversté, Insttut Franças de Mécanque Avancée, EA 3867,Laboratore de Mécanque et Ingéneres, Campus de Clermont-Ferrand les Cézeaux - BP265, AUBIERE Cedex, France Short Ttle: Relablty analyss of bonded jont *Correspondng author: emal:jean-dens.mathas@rstea.fr, tel: +33(0) , fax : 33.(0)

2 ABSTRACT Bonded jonts are used n several ndustral applcatons as a surrogate of more expensve repars, but ther relablty must be ascertaned. Falure n a bonded jont manly occurs n the adhesve due to stress concentratons that drectly depend on the adhesve thckness. In practce, t s dffcult to ensure a good accuracy of the fnal adhesve thckness, leadng to uncertanty to ts spatal varablty. Ths uncertanty greatly nfluences the strength of the bonded jont. Ths work deals wth one of the man key-ssues n bonded jonts: the nfluence of the spatal varatons n the adhesve thckness on the relablty of the jont; an excessve shear stress level caused by the adhesve thckness varatons may lead to falure. Ths paper provdes relablty analyss by consderng the adhesve thckness as a stochastc feld. The expermental thckness feld s obtaned so as to dentfy the stochastc parameters. These parameters are then ntroduced n a structural relablty model to evaluate the falure probablty. Results show the nfluence of adhesve thckness uncertanty on bonded jont falure. KEYWORDS: bonded jont, falure, shear stress, adhesve thckness, stochastc feld, relablty

3 1 Introducton Bonded composte patches are used as structural repars n several felds such as cvl engneerng for damaged concrete structures [1] or aeronautcs for components whch exhbt damages, defects or mpacts [2]. Another ndustral applcaton conssts n usng patches for the preventon of damage appearance. The servce lfe of such renforced structures s expected to be ncreased and expensve repars or replacements are also expected to be avoded. However, t s well known that shear stress peaks near the free edges of the composte patches may cause falure of the jont. Indeed, some studes have shown that 53% of sgnfcant defects on bonded structures proceed from adhesve bond falures [3]. The shear-lag model was the frst model whch enabled the calculaton of stress dstrbuton n the adhesve, and consequently the predcton of bonded jont falure [4]. Ths model was then refned to account for some addtonal factors such as spew fllet [5], large deflectons [6] or possble elastc plastc response of the adhesve [7,8]. Some other nonlnear models have been developed, for nstance to take nto account the vscoelastcty of the adhesve [9] or the nfluence of the load level on the parameters of the Burgers model [10]. It s well known that the adhesve thckness greatly nfluences the shear stress dstrbuton. Several studes have shown the nfluence of ths parameter. For example, cohesve zone models have been used to analyze the nfluence of the adhesve thckness [11]. A comparatve numercal and expermental study has also been done for alumnum sngle-lap jonts bonded wth alumnum powder flled epoxy adhesve [12]. The type of falure accordng to the adhesve thckness has been also studed [13]. It s commonly admtted that adhesve thckness presents some uncertan varatons owng to the jont fabrcaton process. These varatons can sgnfcantly ncrease the shear stress peak near the free edge of the jont. Relablty methods [14] can take nto account ths varablty by consderng the adhesve thckness as a stochastc feld. Therefore, t enables the assessment of the falure probablty of the bonded jont. Lnear or quadratc approxmatons of the lmt state (characterzng the bonded jont falure) or response surface approxmatons can be used for the calculaton of ths falure probablty avodng a too hgh number of smulatons such as classcal Monte Carlo smulatons [14] [15]. It has been successfully appled n several studes, especally n mechancal applcatons [16] [17]. An expermental nvestgaton of adhesve thckness varatons s frst addressed n ths paper. For ths purpose, a three-dmensonal measurng machne s used n order to obtan the adhesve thckness feld of bonded jont. The stochastc parameters of the adhesve thckness feld are then dentfed by a stochastc decomposton. Afterwards, these parameters are ntroduced n a structural relablty model to calculate falure probabltes and normalzed senstvtes wth respect to the mean [14]. Fnally, the relablty model s used to calculate the safety coeffcent for a target falure probablty. 2. Expermental assessment of the adhesve 2.1 Expermental set-up Sngle-lap specmens have been prepared by the ndustral partner, AIA of Clermont- Ferrand (French Mnstry of Defence). Composte patches were bonded to alumnum (see Fgure 1). The substrate was made of alumnum 2024 T3 wth a thckness 4 mm. The composte patch was made of Hexcel carbon prepreg system 914 T300 (epoxy/carbon). Four - 3 -

4 undrectonal ples were used. The total thckness of the patch was 0.5 mm. The dmensons of the composte patch were equal to LxL wth L=70 mm. The adhesve used was a Redux 312 adhesve. The ntal settng of the adhesve thckness durng the jont fabrcaton process was 0.15 mm. Mechancal propertes of each materal are reported n Table 1. As mentoned above, the adhesve thckness may present some uncertan varatons owng to the jont fabrcaton process. Before dentfyng the stochastc parameters, the adhesve thckness feld has to be measured. For ths purpose, a three-dmensonal measurng machne (MMT TEMPO MCA 10 from TRI-MESURES, France) was used here to obtan the thckness feld of the adhesve wth respect to the x- and y-drectons. The resoluton was 5 µm. Measurements were made on both faces of the specmen. The ptch u (the dstance between two measures) here was 1.1 mm whch corresponds to 64 ptches for the specmen length n the x-drecton. 37 ptches were used for the specmen wdth n the y-drecton. The path of the feeler of the three-dmensonal measurng machne s represented n Fgure 1. The total thckness t tot (x,y) was then obtaned. The composte thckness t comp (x,y) and the alumnum thckness t alu (x,y) were assumed to be constant. The adhesve thckness t adh (x,y) can be determned as follows: t adh tot alu comp ( x, y) t ( x, y) - t ( x, y) - t ( x, y) (1) Note that varatons n thcknesses of the alumnum plate t alu (x,y) and of the composte t comp (x,y) were neglgble n comparson wth the adhesve thckness varatons. 2.2 Feld decomposton The adhesve thckness t adh (x,y) s now consdered as a stochastc feld t adh (x,y, ω ) n whch are random ponts n space. In the followng, we consder a one-dmensonal mechancal model based on the Volkersen s model [4] (see Secton 4.1) n the x-drecton. Ths model has some lmtatons and more complex and accurate models could be used. However, ths model s commonly used n the bonded jont communty. Therefore, for the sake of smplcty and to show the feasblty of the current approach, ths model s chosen here. The x-drecton corresponds to the fber drecton of the composte patch and to the drecton of the loadng (see secton 4.1). Ths leads to consder the stochastc propertes of the stochastc feld only along the x-drecton. In ths case, we consder the stochastc adhesve thckness feld along the y-drecton t adh (x,y, ω ) as 37 expermental realzatons named replcas (37 ptches were used for the specmen wdth n the y-drecton) wth 64 ponts per replca (n the x-drecton). Each adhesve thckness replca depends only on the x- a drecton and s denoted as t ( x, ω ) We consder these expermental replcas a t ( x, ω ) 1 37 as realzatons of the stochastc feld t a ( x, ω) that we want to characterze. The thckness feld of the adhesve t a ( x, ω) presents two types of varatons. The frst varaton s global and can be represented by a mean determnstc feld t mean (x) whch can be modelled by a polynomal functon. The second varaton s modelled wth a stochastc feld Y ( x, ω), whch leads to: t a mean ( x,ω) t ( x) Y ( x, ω) (2) - 4 -

5 For the dentfcaton procedure of t mean (x) and Y ( x, ω), we use the expermental replcas. For each replca, we have the same decomposton: t a ( mean x,ω ) t ( x) Y ( x, ω ) (3) For each replca, the am s to extract a trajectory Y ( x, ω ) that has a mean equal to 0 (see Fgure 2). Decomposton of the adhesve thckness s represented n Fgure 3. Each replca a t ( x, ω ) s smoothed leadng to the mean determnstc feld t mean (x). Resduals are then obtaned by subtractng the mean determnstc feld from the full thckness feld and are consdered as the trajectory Y ( x, ), wth a mean equal to 0. We can see that the mean determnstc feld t mean (x) decreases near the free edge of the patch. Ths s due to the jont fabrcaton process n ths case. Ths determnstc decrease n the adhesve thckness leads to a stress concentraton wthn the adhesve near the free edge. Uncertan varatons may ncrease ths stress concentraton. The am then s to determne the stochastc parameters of the stochastc feld Y ( x, ω) from replca trajectores Y ( x, ) and to ntroduce them n a relablty model. 3. Identfcaton of the Y ( x, ω) stochastc feld parameters 3.1 Assumptons Before begnnng the dentfcaton, some assumptons regardng Y ( x, ω) must be made. Y ( x, ω) s supposed to be a statonary second-order process. As explaned above, expermental data are consdered as realzatons of Y ( x, ω). For each replca, stochastc parameters have to be dentfed. The ptch between two abscssae x j, x j+1 s denoted as u j. The parameters m, σ, R u ) and C u ) can be calculated as: Y 2 Y Y ( k Y ( k R Y N 1 my y( x j ) N j1 N σy y( x j ) my N j1 N k 1 ( uk ) y( x j uk ) y( x j ) N k j1 2 C ( u ) R ( u ) m Y k Y k Y (4) Usng Equatons (4), the stochastc parameters can be dentfed from the expermental measurements (see Secton 2). Based on the assumptons descrbed above, the trajectory process s totally defned by the mean m Y, the standard devaton Y and the autocorrelaton functon C Y u ). The mean m Y and the standard devaton Y are calculated from (4). The ( k - 5 -

6 autocorrelaton functon C Y ( u k ) has to be modelled. For ths purpose, a negatve exponental functon f cor s commonly used: C Y u cor ( u ) f ( u) e (5) k where u s the dstance between two measurement ponts of the bond and s a length parameter. 3.2 Identfcaton of Y ( x, ω) parameters Y The expermental feld s developed frst followng (3) leadng to trajectory felds x, ω ) wth a mean equal to 0. The decomposton s represented n Fgure 3. The ( dentfcaton procedure s appled to trajectory felds Y x, ω ) n order to dentfy the ( mean m Y, the standard devaton Y and λ. For the dentfcaton of the λ -value, a least squares-based method s used. As explaned above, we consder the stochastc feld as 37 replcas. The stochastc parameters of Y ( x, ω) are calculated as follows: my 1 σy 36 1 λ λ σ Y m Y 0 m m m (6) The rato between the standard devaton and the mean value of the dentfed parameters characterzes the varablty of the results. Ths rato, the coeffcent of varaton (c.o.v.), s chosen to control the result qualty. C.o.v. of the standard devaton (respectvely of ) s equal to 0.2 (respectvely 0.3). Note that we have found smlar results n the y-drecton. Ths sgnfes that the stochastc feld Y(x,y,) has sotropc propertes. The dentfcaton of the mean m Y and the standard devaton Y s very well known. However, the dentfcaton stablty of the value has to be valdated. Especally, we have to check that a c.o.v. equal to 0.3 does not lead to a sgnfcant error n the -estmate. In order to check the -dentfcaton process stablty, a senstvty analyss s done. -dentfcaton process stablty To valdate the -dentfcaton process stablty, we buld frst a synthetc sample data set generated from Y ( x, ω) as t s dentfed: 64x37 ponts (leadng to 37 replcas of 64 ponts), a ptch equal to 1.1 mm, L=70 mm. The dentfed stochastc parameters are also used n the 6 sample data: m 0 m and σ m. However, for the -value, we generate sample Y Y - 6 -

7 data wth 20 dfferent values of unformly dstrbuted n the range from 0.3 mm to 6 mm. As explaned above, for a gven -value, c.o.v. of the dentfed parameter characterzes the varablty n the results and the result qualty. The c.o.v. drectly depends on 3 parameters: the length L, the ptch u and the -estmate. Therefore, a senstvty analyss of the c.o.v. s done wth respect to the rato L and the rato L u. The calculated c.o.v. of the -estmate s plotted accordng to these ratos n Fgure 4. An acceptable zone s defned n the fgure. It corresponds to an error less than 15% n the value. Ths constrant corresponds to a c.o.v. less than 0.6. Pont A represents the expermental characterstcs of our problem (u=1.1 mm, L=70 mm, =1.39 mm). The c.o.v. of the -estmate calculated from Equaton (6) s equal to 0.3 whch satsfes the crteron. Now the stochastc parameters of the adhesve thckness feld have been dentfed and valdated by the stablty of the -dentfcaton process. They are now ntegrated n a relablty model. 4. A combned mechancal and relablty model 4.1 Mechancal model Some classcal models have been developed under some smple assumptons by Volkersen [4] or Hart-Smth [7] for nstance. More complex models can be used n order to model non-lnear behavor. In ths frst approach, t has been decded to choose a commonly used model developed by Volkersen [4] n order to hghlght the feasblty of the proposed approach. The dfferental equatons whch govern the bonded jont behavour read as follows: 2 p d xx ( x) p 2 xx ( x) dx 2 p a d xx ( x) ( x) e xz p 2 dx (7) wth: G a 1 1 e a e p Ex es E Ga xx eae p Es s (8) σ p xx (x), e p and E x are, respectvely, the longdtudnal stress component, the thckness and the Young s modulus of the bonded composte. σ a xz (x), e a and G a are, respectvely, the shear stress component, the thckness and the shear modulus of the adhesve. e s and E s are, - 7 -

8 respectvely, the thckness and the Young s modulus of the substrate. The substrate s subjected to a tensle stress σ n the x-drecton. Ths dfferental equaton may be solved xx analytcally [14] when the thckness s constant. However, as explaned above, the adhesve thckness varatons are modelled leadng to a non-unform adhesve thckness dstrbuton. We have, therefore, used a fnte dfference model to solve ths dfferental equaton. 4.2 Stochastc model A stochastc model s developed here to take nto account the uncertan spatal varatons n the adhesve thckness. For ths purpose, the adhesve thckness s modelled by a splne curve wth nne nterpolaton ponts. The abscssae of these ponts are dstrbuted accordng to a geometrc dstrbuton (see Fgure 5). The mean value of the adhesve thckness s determned by the decomposton of the stochastc feld (see Secton 2.2), and t corresponds to t mean (x). These nne thcknesses X represent the varables of the problem that consttute the vector X wth the stochastc characterstcs dentfed n secton 3. These varables follow a lognormal a dstrbuton. The value of shear stress peak n the adhesve xz (0) enables us to evaluate the bonded jont falure through a lmt state whch s defned n the next secton. 4.3 Relablty model The stochastc parameters defned n equaton (6) are ntegrated n a relablty model. The lmt state G(X)=0 characterzes the bonded jont falure. It s wrtten as follows: a a G( X ) S (0) (9) xz The shear strength S a of the adhesve here s equal to 40 MPa. The shear stress n the adhesve s calculated wth (7). There s falure of the bonded jont when G ( X ) 0. The problem s then to calculate the probablty to have G ( X ) 0. The Frst Order Relablty Method (FORM) here s used [14-15]. It conssts n an soprobablstc transformaton, whch transforms the random vector X n the physcal space to a random vector U n the standard Gaussan space where the mage of lmt state s denoted as H(U)=0. The mnmal dstance between the orgn and the lmt state H(U)=0 represents the relablty ndex. The closest pont on the lmt state represents the desgn pont U*. A falure probablty P f s then approxmated from the ndex : P f Φ( β) (10) represents the standard normal dstrbuton. Determnaton of the ndex poses the constraned optmzaton problem: 2 2 mn U, subject to H ( U ) 0 (11) - 8 -

9 The optmzaton gves the relablty ndex and the falure probablty P f through Equaton (10). For ths purpose, the relablty software FERUM v4.0 toolbox s used [18]. Note that the Second Order Relablty Method (SORM) has been used too and results are very close to results calculated wth the FORM. 5 Applcaton 5.1 Determnaton of the falure probablty Smulatons are done wth the stochastc parameters calculated n Secton 3. We consder that the substrate s subjected to a tensle test wth a stress equal to 110 MPa. We consder two types of models n order to hghlght the nfluence of uncertan varatons n adhesve thckness: the determnstc model and the stochastc model. The determnstc model takes nto account the varablty of the adhesve thckness wthout uncertan varatons. In ths case, the adhesve thckness s equal to t mean (x). The stochastc model takes nto account uncertan varatons. In ths case, the adhesve thckness s equal to t a ( x, ω). Usng the determnstc model, the shear stress peak can be calculated and s equal to 31.3 MPa, a whch corresponds to 78% of the strength S of the adhesve. Usng the stochastc model, the falure probablty s equal to 83%. We have a sgnfcant falure probablty although a a determnstc calculaton equal only to 78% of the adhesve strength S. Ths mportant result clearly shows that the falure probablty of the bonded jont s very hgh (83%) when we consder the uncertan varatons n the adhesve thckness despte good results n the determnstc case (78% of the strength S a ). Fgure 6 represents the mportance factors defned as follows: I T JU, x D (12) T J D) U, x J, represents the Jacoban of the soprobablstc transformaton. D represents the U x dagonal matrx of T J U x JU, x,. The vector s defned as: H ( U ) * U * U H ( U ) (13) For the calculaton of the mportance factors, the relablty software FERUM s used [18]. The mportance factors do not have a symmetrc dstrbuton. Indeed, they are very mportant near the rght free edge. Ths s due to the fact that the adhesve works essentally near the free edge of the composte patch. So the adhesve thckness has no nfluence far away from the free edge. Moreover, the mean adhesve thckness s lower near the rght free edge of the jont. So the uncertan varatons n the adhesve thckness have more nfluence on ths part of the jont

10 5.2 Determnaton of safety coeffcent The am of ths secton s to determne a safety coeffcent n order to have a falure probablty lower than 0.01%. Safety coeffcent s expressed as a multplcaton factor of the adhesve thckness. The dea s to change the ntal settng of the adhesve thckness durng the jont fabrcaton process to account for uncertan varatons n the adhesve thckness. Calculatons are done wth safety coeffcent Cs ntegrated n the mechancal model. For ths purpose, smulatons are done wth a safe adhesve thckness e C a S (x) equal to: C ea S ( x) Csea ( x) (14) Fgure 7 represents the falure probablty accordng to Cs. Ths fgure enables us to analyze the evoluton of the falure probablty as a functon of Cs. value. The falure probablty s very hgh when Cs s lower than 1 and decreases sgnfcantly to reach 1.2. Afterwards, ths coeffcent s determned so as to have a falure probablty less than 0.01%. In ths case, Cs s equal to Ths sgnfes that the ntal adhesve thckness settng must be multpled by 1.54.e., an adhesve characterstc thckness equal to 0.23 mm nstead of 0.15 mm. 6. Concluson Ths work hghlghts the nfluence of adhesve thckness varablty on the falure of bonded jont. Measurements are done usng a three-dmensonal measurng machne and expermentally show ths varablty. An dentfcaton procedure enables us to have access to the stochastc parameters of the expermental adhesve thckness feld. These parameters are dentfed and the stablty of the dentfcaton process s checked. These dentfed parameters are ntegrated nto a relablty model from whch a falure probablty of the bonded jont s calculated. Fnally, the relablty model s used to determne a safety coeffcent to decrease the falure probablty to 0.01%. The future of ths study les n mprovng the mechancal models. In partcular, the thckness of the composte adherend can be progressvely reduced near the end to reduce the shear stress ampltude. Furthermore, the geometry of the adhesve layer at the free edge ( square end edges, spew fllets) can be taken nto account too. Ths means that the soluton n ths more realstc case can probably only be carred out wth a numercal model such as a fnte element model, despte the problems whch are generally encountered when modelng bonded jonts wth such a tool.. Moreover, surface approxmaton method may be used n order to ncrease the accuracy of the falure probablty calculaton, especally f fnte element models are used. Acknowledgement The Ateler Industrel de l Aéronautque de Clermont-Ferrand s gratefully acknowledged for ts support durng ths study. REFERENCES

11 [1] L. Hollaway and M. Leernng, Strengthenng of renforced concrete structures usng externally-bonded FRP compostes n structural and cvl engneerng, Woodhead PublshngLtd, Cambrdge, (1999). [2] A.A. Baker, Composte Structures, 2, , (1984). [3] M.J Davs, n: Proceedng of the 3rd. Int. Workshop on Repar of Metallc Structures usng Compostes, 4, 1-15, (1995). [4] O. Volkersen, Luftfahrtforschung, 15, 41 47, (1938). [5] M. Tsa and J. Morton, Composte Structures, 32, , (1995). [6] D. Oplnger, Intl. J. Solds Structures, 31, , (1994). [7] L. Hart-Smth, Adhesve-bonded sngle-lap jonts, Tech. Rep. CR , NASA, (1973). [8] L. Hart-Smth, Adhesve-bonded double-lap jonts, Tech. Rep. CR , NASA, (1973). [9] D. Bgwood and A. Crocombe, Intl. J. Adheson Adhesves, 10, 31 41, (1990) [10] P. Majda and J. Skrodzewcz, Intl. J. Adheson Adhesves; 29, , (2009) [11] G. J, Z. Ouyang, G. L, S. Ibekwe and S.-S. Pang, Intl. J. Solds Structures, 47, , (2010) [12] R. Kahraman, M. Sunar and B. Ylbas, Mater. Process. Technol., 205, , (2008) [13] A. A. Tab, R. Boukhl, S. Achou, S. Gordon and H. Boukehl, Intl. J. Adheson Adhesves, 26, , (2006) [14] M. Lemare, Structural Relablty, John Wley & Sons, (2009) [15] O. Dtlevsen, and H.O. Madsen, Structural Relablty Methods, John Wley and Sons, (1996) [16] C.G. Bucher and U. Bourgund, Structural Safety, 7, 57-66, (1990). [17] L. Bng, Z. Meln and X. Ka, Relablty Eng..System Safety, 67, , (2000)

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13 Table 1 - Mechancal propertes of the materals 1. E x (GPa) E y (GPa) xy (-) G xy (GPa) S (shear strength) (MPa) Composte Alumnum Adhesve E x (respectvely E y ) corresponds to the Young s modulus n the x-drecton (respectvely n the y-drecton). xy represents the Posson s rato n the plane (x,y). G xy represents the shear modulus n the plane (x,y)

14 Fg. 1 Schematc vew of the specmen and the path of the feeler

15 Mean determnstc feld t mean (x) of replca thckness (mm) x (mm) replca Stochastc feld Y ( x, ω) of replca Expermental adhesve thckness t adh ( x, y) thckness (mm) x (mm) Fg. 2 Scheme of the decomposton of the expermental thckness t adh (x,y).the expermental thckness t adh (x,y) presents two types of varatons: the frst s global and s represented by the mean determnstc feld t mean (x) for each measure n the x-drecton; the second s modelled by a stochastc feld Y ( x, ω) for each measure n the x-drecton

16 a Total adhesve thckness t adh (x,y) b Smoothng example of a replca ( x, ω) t a c Mean felds t mean (x) d Stochastc felds Y (x,) Fg. 3 Results of the adhesve thckness feld decomposton t adh (x,y). Each measure t a ( x, ω) n the x-drecton s smoothed n order to obtan the mean felds t mean (x). Then, the stochastc felds Y (x,) are obtaned by subtractng the mean felds t mean (x) from the adhesve feld t adh (x,y)

17 Fg. 4 Coeffcent of varaton accordng to L and L u

18 Lognormal varatons thckness (mm) t mean (x) Local varatons thckness (mm) f X X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 Varables =110 MPa Composte patch Fg. 5 Stochastc model: varables X are consdered to model varatons n the adhesve thckness. These varables follow a lognormal dstrbuton and ther abscssae are dstrbuted accordng to a geometrc dstrbuton

19 Fg. 6 Importance factors of the X - varables. The varables located near the free edge (X 7, X 8, X 9 ) have sgnfcant mportance factors. Therefore, ther nfluences on the falure probablty of the bonded jont are hgher than the nfluences of the other varables

20 Fg. 7 Falure probablty accordng to the safety coeffcent C s