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

(Journal of the Society of Materials Science, Japan), Vol. 66, No. 11, pp. 853-860, Nov. 2017 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) 1980 11) 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 28 11 28 Received Nov. 28, 2016 c 2017 The Society of Materials Science, Japan 940-2188 Department of Mechanical Engineering, Nagaoka University of Technology, Kamitomioka, Nagaoka, 940-2188.

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. 4 4 1 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) 4 2 7 Step1: Step2: Step3: (14) Step4: (14) Step5: Step6: N Step7: Fig.7 y

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.10 x =0mmy =3mm Table1 4 λ 0.1 4.2 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 0.1 0.001 0.001 b 0.15 0.01 0.001 c 0.1 0.01 0.001 d 0.1 0.001 0.01 Fig. 8 Process of node generation. 1mm Material1 Singularty point 3mm Material2 3mm Fig. 9 Process of node relocation. Fig. 10 Bonded structure model.

857 Table 2 Numbers of nodes and elements for each meshes. Case node element a 936 1741 b 463 834 c 507 920 d 653 1199 (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. 6 2 1/4 Fig.12 Fig.12 Material1 Table3 Material2 E 2 70GPa ν 2 0.3 Bogy p λ Fig.13 Fig.13 (14) α(θ) 0.1 C 0.001Fig.8 δ 0.001 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 2 0.3 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. 12 Computational model and boundary conditions. Table 3 Material property (Material1) Case Young s modulus,gpa Poisson ratio a-1 20 a-2 30 0.3 a-2 40 a-3 50 b-1 20 b-2 30 0.4 b-1 40 b-3 50 c-1 20 c-1 30 c-2 40 0.5 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.

859 yy,mpa 15.0 14.0 13.0 12.0 11.0 10.0 9.0 1.000E-03 1.000E-02 1.000E-01 1.000E+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-1 0.5556 0.1944 0.0735 0.0645 0.00902 a-2 0.4000 0.1400 0.0384 0.0320 0.00644 a-3 0.2727 0.0956 0.0180 0.0154 0.00260 a-4 0.1667 0.0583 0.0068 0.0054 0.00141 b-1 0.5705 0.1604 0.1037 0.0898 0.01393 b-2 0.4182 0.1109 0.0616 0.0483 0.01326 b-3 0.2928 0.0702 0.0344 0.0253 0.00907 b-4 0.1878 0.0360 0.0171 0.0112 0.00586 c-1 0.5826 0.1248 0.1304 0.1094 0.02098 c-2 0.4331 0.0799 0.0833 0.0658 0.01749 c-3 0.3094 0.0428 0.0508 0.0352 0.01559 c-4 0.2053 0.0116 0.0284 0.0163 0.01206 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 Case1:0.416Case2:0.446Case3:0.438 0.5 Case2 Case3 Case2 Case3 σ yy 7 λ 15)

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