International Journal of Mechanical Engineering and Applications

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1 Interntionl Journl of Mechnicl Engineering nd Applictions 203; (2) : Published online My 30, 203 ( doi: 0.648/j.ijme Evlution of intensity of stress singulrity for 3d dissimilr mteril joints bsed on mesh free method (reltionship between interfce width nd intensity of stress singulrity). Kurhshi, A. Ishikw 2, H. Koguchi Deprtment of Mechnicl Engineering, Ngok University of echnology, Ngok, Niigt , Jpn 2 Grdute School of Ngok University of echnology, Ngok, Niigt , Jpn Emil ddress: kurhshi@mech.ngokut.c.jp (. Kurhshi) o cite this rticle:. Kurhshi, A. Ishikw, H. Koguchi. Evlution of Intensity of Stress Singulrity for 3D Dissimilr Mteril Joints Bsed on Mesh Free Method (Reltionship between Interfce Width nd Intensity of Stress Singulrity), Interntionl Journl of Mechnicl Engineering nd Applictions. Vol., No. 2, 203, pp doi: 0.648/j.ijme Abstrct: We present the intensity of stress singulrity for 3D dissimilr mteril joints bsed on mesh free method. When lod is pplied to surfce of the bonded structure, stress t vertex on interfce drsticlly increses nd it ppers tht this stress singulrity occur delmintion of the bonded structure. Intensity of stress singulrity cn be expressed by stress distribution nd the intensity of stress singulrity. herefore, it is necessry to obtin the stress distribution precisely. In this study, we focus on the mesh free method for the computtion of the stress distribution. When this method is pplied to compute stress distribution, incomptible cell cn be employed nd geometricl dt for trget structure cn be simply prepred. o confirm the vlidity of the results of mesh free method, comprison of the intensity of stress singulrity between the mesh free nd the boundry element methods is crried out. Keywords: Stress Anlysis by Mesh Free Method, 3D Dissimilr Mteril Joints, Intensity of Stress Singulrity, Order of Singulrity. Introduction In this study, evlution of intensity of singulrity for three-dimensionl dissimilr mteril joints bsed on the mesh free method(mfm) [] is crried out. If the stress nlysis is crried out for 3D dissimilr mteril joints, the highest vlue is obtined t vertex on the interfce of the dissimilr mteril joints. his stress vlue depends on the element size, the vlue is pproched to infinity in cse tht the element size is grdully smll. herefore, the stress vlue cn t be pplied to the design stndrd for 3D dissimilr mteril joints. o solve this problem, we focus on the evlution by the intensity of singulrity. It is well known tht if distnce from crck front is r, nd stress distribution ner crck front is expressed by 0.5 σ / r. Similrly, stress distribution ner edge of ij interfce for dissimilr mteril joints is lso expressed by the reltion eqution with respect to distnce from edge of interfce. If the distnce from edge of interfce is expressed by r nd order of singulrity λ is introduced, the reltion eqution between stress σ ij nd distnce r is expressed by σ /. he order of singulrity λ is determined by ij r λ combintion of mterils nd configurtion of edge or vertex. Here, in cse tht the order of singulrity λ is replced by -p, the stress distribution is written s λ p p σij / r = / r = r. In ddition, becuse the stress is expressed s grdient of the displcements u i, the reltionship the displcements nd distnce r is given p p u r by the integrtion of σ ij r. he prmeter p i is refer to s the chrcteristic root, the investigtions for the chrcteristic root of 2D dissimilr mteril joints is crried out by Bogy [2]. his methodology is nlyticlly pproch, it is sid tht it is difficult this methodology is directly pplied to 3D dissimilr mteril joints. On the other hnd, there is methodology tht the order of singulrity is numericlly obtined. Ymd et. l. [3]

2 Interntionl Journl of Mechnicl Engineering nd Applictions 203 ; (2) : developed the numericl evlution method for the singulrity bsed on the finite element procedure. he p interpoltion function is expressed s the function of r, nd the comprison of the stress intensity fctor between the computtionl nd the theoriticl solutions ws crried out for cylindricl br model with circumferentil crck. In ddition, Pgeu et. l. [4] pplied this formultion to 3D dissimilr mteril joints, nd the chrcteristic eqution ws derived expressed by the chrcteristic root p. he chrcteristic root p indictes the eigen vlue of the chrcteristic eqution, nd this vlue is obtined by eigen vlue nlysis. In the numericl experiments, the chrcteristic root p t crck front clculted by chnging the number of Guss points nd mesh division, nd it is reported tht there is tendency the chrcteristic root converges to n unique vlue in cse tht lot of finite elements re generted, even if the number of Guss points re chnged. Moreover, n ppliction exmple for models of n nisotoropic three-mteril junction with free edge is introduced, the numericl results for the chrcteristic root is shown. In ddition, Koguchi et. l. evlute the intensity of singulrity for 3D dissimilr mteril joints bsed on stress nlysis results nd the order of singulrity λ [5], [6], [7]. he stress nlysis is crried out by the boundry element method, nd the order of singulrity λ is obtined by the chrcteristic root p tht is clculted bsed on the methodology by Pgeu et. l.. Especilly, in reference [7], it ws clrified tht there is possibility for reltionship between delmintion force nd the intensity of singulrity. If numericl nlysis for delmintion of mteril is crried out, the remeshing technique is usully introduced. In cse tht the MFM is pplied to the numericl nlysis, the remeshing process cn be ignored. hough lot of studies for crck propgtion problems using the MFM hve been crried out [8], [9], it is difficult to sy tht reserches for evlution of intensity of singulrity using the MFM bsed on methodology such s the previous studies [5], [6], [7] re crried out enough. herefore, the formultion for evlution of intensity of singulrity using the MFM is crried out, nd some numericl results nd remrks re shown in this pper. 2. Discretiztion of Elstic Equtions by Mesh Free Method he equilibrium eqution, the strin-displcement reltion nd the stress-strin reltion re written s Eqution (). σij, j = 0, εij = ( ui, j + uj, i ), σij = Dijklεkl () 2 where σ ij, ε ij, u i nd D ijkl indicte stress nd strin nd displcement elstic coefficient mtrix. Here, the Equtions.() re represented s Eqution (2). { } = { }, { } = [ ]{ }, { } = [ ]{ } c 0 e B u s D e (2) where { c }, { e }, [ B ], { u }, { s } nd [ ] Eqution (3). { c } = { σxj, j σyj, j σzj, j} ( j; summtion convention) { e } = { εx εy εz γxy γyz γzx} [ B ] x y z = y x z z y x { u } = { u v w} { s } = { σx σy σz τxy τyz τzx} [ ] D indicte λ + 2µ λ λ λ λ 2µ λ λ λ λ + 2µ D = (3) µ µ µ In Eqution (3), λ nd µ indicte the Lme s constnts, nd the constnts re written s λ = νe, µ = E ( 2ν )( + ν ) 2( + ν ) * Multiplying weighting function u ( x ) for both sides of equilibrium eqution nd integrting domin influence (See Fig..), Eqution (5) is obtined. * { ( )} { ( )} d = 0 (4) u x c x (5) Applying the Green theorem to Eqution (5), Eqution (6) is obtined. { * ( )} [ ] { ( )} d = * ( ) { } { ( )} u x B s x u x t x d (6) where t indictes trction force, nd is written s { } = { tx ty tz} = { σxjnj σyjnj σzjnj } t (7) ( j; summtion convention) Substituting the stress-strin reltion nd the Γ

3 30. Kurhshi et l: Evlution of intensity of stress singulrity for 3d dissimilr mteril joints bsed on mesh free method (reltionship between interfce width nd intensity of stress singulrity) displcement-strin reltion in Equtions (2) to Eqution (6), the Eqution (6) is represented s Eqution (8). { * ( )} [ ] [ ][ ]{ ( )} d= * ( ) { } { ( )} u x B D B u x u x t x d (8) Γ defined by sphericl configurtion whose rdius r is r 0, nd the sphericl coordinte system is introduced (See Fig. 2.). As the finl form of the derived eqution, the eqution on the sphericl surfce is obtined. herefore, the surfce domin is divided into finite elements, nd the computtion for the chrcteristic eqution is crried out. In the reference of Pgeu et. l.[4], the qudrtic isoprmetric element is employed s the finite elements. 7 η = η 6 5 α z 8 4 ξ = ξ = 2 3 η = x ξ α = α = φ O θ r r O Q y Figure. Domin influence If the weighting function u * nd displcement u t n rbitrry point x is interpolted by ech vlues t referred nodes in domin of influence bsed on the Glerkin procedure, the interpoltion function for ech vlues re written s Equtions (9) nd (0). u ( x) = q ( x) u + q2 ( x) u2 + q3 ( x) u3 + qn ( x) un = { q( x) } * * * * * (9) ( x) = ( x) + 2 ( x) ( x) 3 + n ( x) n = { ( x) } u q u q u q u q u q (0) where q indictes shpe function, nd n indictes number of referred nodes in the domin influence. he shpe functions re determined by the moving lest squre method, nd liner bsis nd qudrtic spline functions re employed s the bsis function nd weighting function. Applying the interpoltion functions (Equtions (9) nd (0)) to the Eqution (8), Eqution () is finlly obtined. [ ] d { } { } Κ u = f dγ () where the left hnd side coefficient mtrix nd the right hnd side vector indicte the stiffness mtrix nd externl force vector, nd Γ indictes the boundry on domin.he Legendre-Guss formul is employed s the numericl integrtion for Eqution (). In ddition, the penlty function method is employed s tretment of essentil boundry condition. 3. Computtion of Order of Stress Singulrity In this section, the derivtion of the chrcteristic eqution shown in the reference of Pgeu et. l.[4] is simply introduced. In this formultion, the computtionl region is Γ Figure 2. Computtionl model nd qudrtic isoprmetric element. If displcements for ech element re expressed by interpoltion function shown in Eqution (2) nd the interpoltion function is substituted to eqution of the principle of virtul work, chrcteristic eqution is finlly derived s shown in Eqution (3). r (, θ, ϕ ) u r r = p 8 hu 0 i ri r (2) i= ( p 2 [ ] + p[ ] + [ ]){ } = { } A B C x 0 (3) where u i is expressed by ui uo, nd u i nd u o represent sphericl displcements t n rbitrry point in the sphericl surfce. In ddition, h i indictes the shpe function. In the Eqution (3), p indictes chrcteristic root nd vector { x } denotes superposed displcement vector in entire domin, nd mtrices [ A ], [ B ] nd [ C ] represent the coefficient mtrices derived by finite element procedure. Detil of this formultion is shown in reference [4]. he chrcteristic root p is obtined by solving the Eqution (3) bsed on eigen nlysis. Reltionship between the chrcteristic root nd order of singulrity λ is λ = Re p. If the prmeter λ is expressed by ( ) < λ < 0, it is denoted tht stress fields hs stress singulrity. On the other hnd, if the prmeter λ is 0 < λ, it is denoted tht the stress singulrity disppers. 4. Numericl Exmples A bonded structure consists of iron nd luminum is employed s the computtionl model. Scle of computtionl model is shown in Fig. 3. In this study, the

4 Interntionl Journl of Mechnicl Engineering nd Applictions 203 ; (2) : width b of the bonded structure is set 0.25, 0.5,.0, 2.0, 4.0, 6.0, 8.0 nd 50.0mm, nd vrition of intensity of stress singulrity is investigted. Mteril properties for ech mteril nd totl number of evlution points nd cells re shown in bs. nd 2. /8model x Figure 3. Computtionl model nd nodl distribution t vertex on interfce Mteril σ z z o Fe Al b[mm] ble. Mteril properties Young s modulus E (GP).0[mm] 6.0[mm] Singulr poi h[mm] Poisson rtio ν Fe Al y found tht if the lest squre pproximtion for Fig.5 by K = αb β, i.e., α nd β re fitting coefficients, is θθ crried out, α nd β is obtined 7.72 nd 0.3, nd the vlue of β is close to the order of stress singulrity λ t vertex. Nextly, the results obtined by the MFM re compred with the results obtined by the boundry element method. he boundry element mesh in cse of b=.0mm is shown in Fig.6. nd totl number of nodes nd elements for ech cse re shown in b.3. he tensile stress sme s the cse of the MFM is given on the top surfce of the bonded structure. Computtionl results re shown in Figs. 7 nd 8. Fig. 7 shows the comprison of stress distribution of σ θθ in cse of the MFM nd the BEM. he stress indictes the vlue on the interfce for the rdius direction from vertex t ϕ =45deg nd θ =90deg. In ddition, Fig.8 shows the reltionship between the intensity of the stress singulrity nd the width of the bonded structure obtined by the MFM nd the BEM. In Fig.7, it is seen tht the stress distribution obtined by the BEM is linerly obtined in semi-logrithmic grph compring to the results obtined by the MFM. In ddition, in Fig.8, it is found tht the intensity of the stress singulrity obtined by the MFM is close to tht obtined by the BEM. However, it is seen tht though the grdient of the intensity of the stress singulrity with respect to the width of the bonded structure obtined by the MFM is not constnt compring with the result obtined by the BEM. herefore it ppers tht some improvements re needed to obtin results with high relibility Al-Fe interfce (θ=90deg, φ=45deg) [mm] b=0.25 b=0.5 b=.0 b=2.0 b=4.0 b=6.0 b=8.0 b=50.0 Curve fitting ble 2. otl number of evlution points nd cells for ech cse Width b (mm) Number of evlution points Number of cells ,663 4, ,034 3,68.0 6,40 4, ,89 4, ,546 5, ,506 7,00 8.0,466 8,636 Stress σ θθ,mp ,255 5,976 Computtionl results by the MFM re shown below. Figs. 4 nd 5 show the stress distribution of σθθ on the interfce for the rdius direction t ϕ =45deg nd θ =90deg nd the vrition of the stress singulrity with respect to model width b. Here, the order of singulrity λ t vertex is 0.2. In Fig.4, it is seen tht stress distribution ner vertex decreses with decresing the width b. In ddition, it is Distnce from origin r, mm σ Figure 4. Stress distribution of θθ for rdius direction from vertex (φ=45deg θ=90deg)

5 32. Kurhshi et l: Evlution of intensity of stress singulrity for 3d dissimilr mteril joints bsed on mesh free method (reltionship between interfce width nd intensity of stress singulrity) Intensity of stress singulrity K θθ,mpmm Al-Fe interfce (θ=90deg, φ=45deg) Curve fiting Width b, mm 2 4 Stress σ θθ,mp Al-Fe interfce (θ=90deg) [mm] MFM_b=2.0 BEM_b=2.0 MFM_b=.0 BEM_b=.0 MFM_b=0.5 BEM_b=0.5 Curve fitting Distnce from origin r, mm Figure 7. Comprison of stress distribution of θθ for rdius direction from vertex in cse of MFM nd BEM (φ=45deg θ=90deg) σ 2 3 Figure 5. Vrition of intensity of stress singulrity with respect to model width b Fe Fe Al Intensity of stress singulrity K θθ,mpmm Al-Fe interfce (θ=90deg) MFM BEM Al Minimum nodl distnce [µm] Width b, mm Figure 8. Comprison of intensity of stress singulrity with respect to model width b in cse of MFM nd BEM 2 Figure 6. Boundry element mesh nd mgnified figure round vertex (b=.0mm) ble 3. otl number of nodes nd elements for boundry element model Width b(mm) Number of evlution point Number of evlution cell 0.5 8,930 2, ,576 2, ,554 6,84 5. Conclusions In this pper, we present the intensity of stress singulrity bsed on the stress distribution by the MFM. As the computtionl model, the bonded structure consists of iron nd luminum is employed. he reltionship between width of the structure nd the intensity of the stress singulrity is investigted by numericl experiments. he width of the bonded structure is vried 0.25, 0.5,.0, 2.0, 4.0, 6.0, 8.0, 50.0mm, nd liner reltionship is consequently obtined between the intensity of the stress singulrity nd the width of the bonded structure in double-logrithmic grph nd if the lest squre pproximtion is crried out for the intensity of the stress singulrity nd the width of interfce by K b β θθ = α, the vlue of β is close to the order of stress singulrity λ t vertex. In ddition, it is found tht the results obtined by the MFM re close to the results obtined by the BEM. However, in cse tht detiled comprison is crried out for the results between the MFM nd the BEM, it

6 Interntionl Journl of Mechnicl Engineering nd Applictions 203 ; (2) : is seen tht though the grdient of intensity of the stress singulrity with respect to the width obtined by the BEM is pproximtely constnt, tht obtined by the MFM is not constnt. herefore, it ppers tht it is necessry to improve the MFM to increse the relibility of the intensity of the stress singulrity. Acknowledgements his work ws supported by Grnt-in-Aid for Scientific Reserch (B) (No ). References []. Belytschko, Y. Y.Lu nd L. Gu, Element-free Glerkin methods, Int. J. Numer. Meth. Engrg. 994; 37(5), pp [2] D. B. Bogy, wo Edge-Bonded Elstic Wedges on Different Mterils nd Wedge Angles Under Surfce rctions, J. Appl. Mech. 97; 38, pp [3] Y.Ymd, Y.Ezw nd I.Nishiguchi, Reconsidertions on singulrity or crck tip elements, Int.J.Numer.in Engng. 979;4: pp [4] S.S.Pgeu nd S.B.Bigger,JR, Finite element evlution of free-edge singulr stress field in nisotropic mterils, Int.J.Numer.in Engng. 995;38: pp [5] H.Koguchi nd.niguchi, Evlution of interfce strength t the 3D-Corner in Si-resin joint considering residul therml stresses, he ASME 2007 InterPACK Conference 2007, [6] P.Monchi nd H.Koguchi, Boundry element nlysis of the stress field t the singulrity lines in three-dimensionl bonded joints under therml loding, J. Mechnics of Mterils nd Structures, 2 (), pp [7] H.Koguchi nd M.Nkjim, Evlution of the bonding strength t the three-dimensionl vertex in silicon-resin joints he ASME 2009 InterPACK Conference 2009, [8] R.D.Borst, M.A.Guierrez, G.N.Wells, J.J.C.Remmers nd H.Askes, Cohesive-zone models, higher-order continuum theories nd relibility methods for computtionl filure nlysis, Int.J.Numer.in Engng. 2004;60: pp [9] B.C.Simonsen nd S.Li, "Mesh-free simultion of ductile frcture, Int.J.Numer.in Engng. 2004;60 : pp