Proposal on Unstructured Triangular Mesh Generation Method for Singular Stress Field Analysis of Bonded Structures Based on Finite Element Method

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1 (Journal of the Society of Materials Science, Japan), Vol. 66, No. 11, pp , Nov 論文 Proposal on Unstructured Triangular Mesh Generation Method for Singular Stress Field Analysis of Bonded Structures Based on Finite Element Method by Takahiko KURAHASHI and Kengo YAMAGIWA In this paper, we present new unstructured triangular mesh generation method for singular stress field analysis of bonded structures based on finite element method. If tensile and bending loadings are applied to the bonded structure, stress concentration occurs around singular point. It is known that stress and strain distribution are proportional to r λ, i.e., r is distance from singular point and λ is order of singularity. In addition, in case of the stress analysis based on the FEM, it is known that the value of the stress component at singular point increases with decreasing mesh size around bonded structure. Therefore, fracture of the bonded structure evaluates by the intensity of stress singularity obtained by the stress distribution, and it is the most important that high accurate stress distribution is obtained. In this study, we introduce new mesh division procedure considering stress singularity near singular point, and some results for numerical experiments show in this paper. Key words: Automatic mesh generation, Unstructured triangular mesh, Finite element method, Stress analysis, Stress singularity, Bonded structures. 1 1),2) 3),4) 5) 6),7) 8),9) r λ σ ij r λ 3),4) λ 10) ) 12),13) 2 FEM (1)- (3) σ ij,j =0 (1) ϵ ij = 1 (ui,j + uj,i) (2) 2 σ ij = D ijkl ϵ kl (3) σ ijϵ kl u i D ijkl (1)-(3) (1)-(3) 3 2 λ Fig.1 2 σ ij Received Nov. 28, 2016 c 2017 The Society of Materials Science, Japan Department of Mechanical Engineering, Nagaoka University of Technology, Kamitomioka, Nagaoka,

2 854 σ r λ = r 1 p (4) p 0 <p<1 14) λ 0 < λ < 1 λ p Bogy 10) p Aβ 2 +2Bαβ + Cα 2 +2Dβ +2Eα + F =0 (5) αβ Dundurs { α = µ 2 (x 1 +1) µ 1 (x 2 +1) µ 2 (x 1 +1)+µ 1 (x 2 +1) β = µ 2(x 1 1) µ 1 (x 2 1) µ 2 (x 1 +1)+µ 1 (x 2 +1) (6) { 3 4µi (Plane strain) x i = 3 µ i 1+µ i (Plane stress) (7) E i µ i =, (i =1, 2) 2(1 + ν i) (8) E i µ i (5) AF A =4K(p, θ 1)K(p, θ 2) B =2p 2 sin 2 θ 1K(p, θ 2)+2p 2 sin 2 θ 2K(p, θ 1) C =4p 2 (p 2 1) sin 2 θ 1 sin 2 θ 2 + K(p, θ 1 + θ 2) D =2p 2 (sin 2 θ 1 sin 2 (pθ 2) sin 2 θ 2 sin 2 (pθ 1) E = D + K(p, θ 2) K(p, θ 1) F = K(p, θ 1 θ 2) (9) (9) K(p, x) K(p, x) =sin 2 (px) p 2 sin 2 x (10) Fig. 1 Bonded structure model r θ λ (11) σ ij(r, θ) =K ijf(θ)r λ +(Other terms) (11) K ij f(θ) (Other terms) (11) ϵ ij(r, θ) =A ijg(θ)r λ +(Other terms) (12) (12) ϵ rr r r r (12) (12) r ϵ rr r 3 λ 0 <λ<1 (13) (13) r 1 ϵ rrdr = A rrg(θ)r λ dr = α(θ)r λ dr α(θ) = λ +1 r λ+1 + C (13) (13) r (14) r = α(θ) λ +1 r λ+1 + C (14) C (14) Step1: Step2: Step3: (14) Step4: (14) Step5: Step6: N Step7: Fig.7 y

e 3 e 1 e 2 : Node : Interface (b) Step7-2 Fig. 4 Generation of new nodes in Step3,4.

3 855 Fig. 2 Input of node and data in Step1. Fig. 6 Node relocation in Step6. e 1 e 2 e 3 : Node : Interface (a) Step7-1 Fig. 3 Generation of rough meshes in Step2. e 3 e 1 e 2 : Node : Interface (b) Step7-2 Fig. 4 Generation of new nodes in Step3,4. (c) Comparison of node locations. Fig. 7 Node relocation on interface in Step7. Fig. 5 Generation of fine meshes in Step5. Step3,4 6 Step3,4 Fig.8 δ

856 r r r r A r B r C (15) r(x, y) =N A r A + N B r B + N C r C (15) N AN BN C Fig.8 P r P Step6 Fig.9 P (16) M M A ex e A ey e x P e=1 =, y M P e=1 = (16) M A e A e e=1 e=1 A e N 3 5 2 Fig.10 Fig.

4 856 r r r r A r B r C (15) r(x, y) =N A r A + N B r B + N C r C (15) N AN BN C Fig.8 P r P Step6 Fig.9 P (16) M M A ex e A ey e x P e=1 =, y M P e=1 = (16) M A e A e e=1 e=1 A e N Fig.10 Fig.10 x =0mmy =3mm Table1 4 λ Case Case Table2 Fig.11 Case α(θ) α(θ) C C δ δ Case-a Table 1 Mesh division parameter. Case α(θ) C δ a b c d Fig. 8 Process of node generation. 1mm Material1 Singularty point 3mm Material2 3mm Fig. 9 Process of node relocation. Fig. 10 Bonded structure model.

5 857 Table 2 Numbers of nodes and elements for each meshes. Case node element a b c d (d)generated mesh in Case-d. Fig. 11 Generated meshes around singularity point. (a)generated mesh in Case-a. (b)generated mesh in Case-b. (c)generated mesh in Case-c /4 Fig.12 Fig.12 Material1 Table3 Material2 E 2 70GPa ν Bogy p λ Fig.13 Fig.13 (14) α(θ) 0.1 C 0.001Fig.8 δ λ 4.2 Case Fig.12 r σ yy Fig.14 Fig.14 σ yy = K yyr λ λ K yy σ yy = K yyr λ Fig.14(a) Table3 Material1 Material2 E 2 70GPa ν Dundurs αβ λ f Bogy

858 λ λ λ f Table4 Table4 Dundurs αβ Fig.15 Fig.15 α β λ λ f α β 1mm yy=10mpa Material1 Singularty point 3mm (b)finite element mesh in Case-(a-2) (E 1 = 30GPa λ =0.0384) Material2 3mm Fig.

6 858 λ λ λ f Table4 Table4 Dundurs αβ Fig.15 Fig.15 α β λ λ f α β 1mm yy=10mpa Material1 Singularty point 3mm (b)finite element mesh in Case-(a-2) (E 1 = 30GPa λ =0.0384) Material2 3mm Fig. 12 Computational model and boundary conditions. Table 3 Material property (Material1) Case Young s modulus,gpa Poisson ratio a-1 20 a a-2 40 a-3 50 b-1 20 b b-1 40 b-3 50 c-1 20 c-1 30 c c-3 50 (c)finite element mesh in Case-(a-3) (E 1 = 40GPa λ =0.0180) (a)finite element mesh in Case-(a-1) (E 1 = 20GPa λ =0.0735) Fig. 13 (d)finite element mesh in Case-(a-4) (E 1 = 50GPa λ =0.0068) Finite element mesh generation considering order of singularity λ in interface of bonded structure.

7 859 yy,mpa E E E E+00 distance from singular point r,mm Case-(a-1) Fitting curve Case-(a-2) Fitting curve Case-(a-3) Fitting curve Case-(a-4) Fitting curve Fig. 14 Result of stress analysis for each case. Table 4 Relation between the absolute error λ λ f and Dundurs parameters,α and β. Case α β λ λ f λ λ f a a a a b b b b c c c c Fig. 15 Distribution of the absolute error λ λ f in field of Dundurs parameters α and β. Dundurs αβ λ f Bogy λ λ λ f α β λ λ f α β Dundurs α β 3 15) E 200GPa ν 0.3 Fig.16 () Case1 (1013,1936)Case2 (3671,7200)Case3 (870,1627) Case1:1.286mmCase2:0.189mm Case3:0.131mm Fig.17 y=0mm σ yy λ Case1:0.416Case2:0.446Case3: Case2 Case3 Case2 Case3 σ yy 7 λ 15)

8 860 Fig. 16 r Computational model. yy Fig. 17 Stress distribution from sigular point on line y=0mm. 5 PRIMERGY CX400 1) S. Ioka, K. Masuda and S. Kubo, Free-edge singular stress field of bonded dissimilar materials with an elastic-plastic interlayertransaction of Japan Society of Mechanical Engineers, Series A, Vol.73, No.729, pp (2007). 2) N. Noda, Z. Wang, K. Iida, Y. Sano and T. Miyazaki, Intensity of singular stress under tension for bonded pipe in comparison with bonded plate, Journal of the Society of Materials Science, Japan, Vol.65, No.6, pp (2016). 3) H. Koguchi, K. Hoshi and T. Kurahashi, Analysis for three-dimensional singular stress field at a vertex of bonded interface edge in single lap joint under tensile-shear load, Transaction of Japan Society of Mechanical Engineers, Series A, Vol.78, No.795, pp (2012). 4) T. Kurahashi, M. Nakajima, A. Ishikawa, K. Hoshi and H. Koguchi,Critical intensity of stress singularity at vertex in three dimensional dissimilar material joints (in case of rectangular bonded area), Transaction of Japan Society of Mechanical Engineers, Series A, Vol.78, No.794, pp (2012). 5) Y. Liu, S. Murakami and K. Hayakawa, Mesh dependence and stress singularity in local approach to creep-crack growth analysis Transaction of Japan Society of Mechanical Engineers, Series A, Vol.59, No.564, pp (1993). 6) S. Wang, M. Shiratori and Q. Yu, Evaluation of singular stress field at the end of the interface of dissimilar materials using akins singular element, Transaction of Japan Society of Mechanical Engineers, Series A, Vol.63, No.606, pp (1997). 7) T. Kurahashi, S. Oshima, K. Ibe, Y. Watanabe, T. Kondo and H. Koguchi,Remarks for relationship between iterfacial average stress in delamination and non-dimensional intensity of stress singularity for bonded strips (stress analysis based on FEM using singular element), Journal of the Society of Materials Science, Japan, Vol.64, No.12, pp (2015). 8) S.S. Pageau, and S.B. Biggers,JR, Enrichment of finite element with numerical solutions for singular stress fields, International Journal for Numerical Methods in Engineering, Vol.40, (1997). 9) C. Luangarpa. and H. Koguchi, Evaluation of intensity of singularity for three-materials joints with power-logarithmic singularities using an enriched finite element method, Journal of Computational Science and Technology, Vol.7, No.2, pp (2013). 10) D.B.Bogy, Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions Journal of Applied Mechanics,Vol.38,pp (1971). 11) M.A. Yerry and M.S. Shephard, Automatic three dimensional mesh generation by the modified octree technique, International Journal for Numerical Methods in Engineering, Vol.21, pp (1984). 12) S.H.Lo, A new mesh generation scheme for arbitorary planar domains, International Journal for Numerical Methods in Engineering, Vol.21, pp (1985). 13) K.Kashiyama and T.Okada, Automatic mesh generation method for shallow water flow analysis, International Journal for Numerical Methods in Fluids, Vol.15, pp (1992). 14) R.Yuuki, Mechanics of interface, P.25(1993),Baifukan. 15) T.Kurahashi, A.Sukigara, K.Yamagiwa, K.Maruoka and T.Iyama, Automatic mesh generation of triangular meshes in finite element analysis -mesh generation considering stress concentration near crack and notch tips-, Research Reports of National Institute of Technology, Nagaoka College, Vol.52, pp (2016).

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