Report98.dvi
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- しょうじ いなくら
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1 n Hooke updated Lagrange
2 /
3 4 1, t t, t. Hooke, [1] [7] A, B,,.,,,. e e e e p. e e e + e p 1.1
4 5 σ E. σ Ee e p 1., Hooke. σ C e :e e p 1., σ, e, e p Cauchy,,, C e 4 Hooke, :. 1.,. σ C e :ė ė p ,, A B e e e p e 1.1:
5 6. σ C ep : ė 1.5, flow rule,,, ,, 9., vonmises Tresca,.,,, 1.1 A.,, B. von Mises., 1.1.1, 1.1., 1.1., 1.1.4, 1.1.5, n, 1.1.7,
6 ,. Mises σ. σ σ ij σ ij 1 1.6, σ ij. σ ij σ ij 1 σ kkδ ij 1.7 Mises σ ij J,, σ 11 0 σ σ 11. J tr σ σ ii 1.8 σ ii 1 σ kkδ ii J 1 { } tr σ tr σ σ ijσ ij 1.11, σ ij σ ij σ ij 1 σ kkδ ij σ ij 1 σ kkδ ij 1.1 σ ij σ ij σ kkδ ij σ ij σ kk δ ij δ ij 1.1 σ ij σ ij σ kkσ ii σ kk 1.14 σ ij σ ij 1 σ kk 1.15, σ 11 0,. σ ij σ ij σ ijσ ij 1 σ kk σ 11 σ 11 1 σ 11 σ
7 8,,,.,.,. 1 F σ σ y 1.17 F σ σ y 1.18 σ y. F σ y σ ij σ ij 1.:,,, λ Ψ. ė p ij λ Ψ 1.19
8 9 associated flow rule., ė p ij λ 1.0,,. normality rule. 1.0 / t 1.. ė p ij t λ t 1.1 ė p ij σ ij λ F 1., F 0, F 0, 1. ė p ij σ ij λ F 0 1., 0,.,,.. von Mises σ 1 σ ij σ ij F σ σ y 1.5 F σ σ y 1.6 ė p ij λ 1.7,.
9 10. σ ij. e p ij 1.: F 0 F 0, σ y. 1.4, / F σ ij σ ij C ijkl ė kl ė p kl 1.9 C ijkl ė kl λ 1.0 σ kl σ ij C ijkl ė kl C ijkl λ σ kl λ. λ C ijkl ė kl 1. C ijkl σ kl
10 σ ij C ijkl ė kl C ijkl C ijcd C abcd ė cd C abcd σ cd σ cd C abkl C abcd σ cd σ kl 1. ė kl 1.4 / 1,, 1 F σ σ y F σ σ y 1 σ σ 1 σ kl σ kl σ kl σ kl 1.5 { σ kl 1 } σ σ mmδ kl σ kl 1.6 ij δ ik δ jl 1 δ ijδ kl σ kl 1.7 σ σ ij 1.8 { σ rs 1 } σ σ mmδ rs σ rs 1.9 ij δ ir δ js 1 δ ijδ rs σ rs 1.40 σ ij /, 1.,, 1.4, 1 F σ σ y λ C ijkl ė kl C ijkl σ ij σ Cijkl ė kl σ σ ij Cijkl σ σ kl σ kl
11 1 σ σ ij C ijklė kl σ ij C ijklσ kl 1.44 F σ σ y σ ij C ijkl C ijcd C ijcd C ijkl C abkl σ cd C abcd σ cd σ σ cd σ σ ab C ijkl C ijcdσ cd σ ab C abcd C abkl σ ab C abcd σ cd σ σ ab ė kl 1.45 σ σ cd C abkl ė kl 1.46 ė kl 1.47 λ C ijkl ė kl C ijkl σ σ ij σ kl 1.48 ij Cijkl ė kl C ijkl σ kl σ ij C ijklė kl σ ij C ijklσ kl 1.50 σ ij C ijkl C ijcd C abkl σ cd C abcd σ cd C ijcd σ cd C ijkl σ ab C abcd σ ab C ijkl C ijcdσ cd σ ab C abkl σ ab C abcdσ cd σ cd ė kl 1.51 C abkl ė kl 1.5 ė kl 1.5,,. Hooke C ijkl λ, μ Lamé C ijkl λδ ij δ kl +μδ ik δ jl 1.54
12 1. μ G. C ijcd σ cd λδ ij δ cd +μδ ic δ jd σ cd 1.55 λδ ij σ cc +μσ ij 1.56 μσ ij 1.57 Gσ ij 1.58 σ abc abkl Gσ kl 1.59 σ abc abcd σ cd Gσ abσ ab G σ 1.61,. 1 F σ σ y F σ σ y λ σ σ jic ijkl ė kl σ ij C 1.6 ijklσ kl σ Gσ ij ėkl 1.6 4G σ σ klėkl 1.64 σ σ ij C ijkl C ijcdσ cd σ ab C abkl σ ab C ė abcdσ kl 1.65 cd Gσ ij Gσ kl C ijkl ė 4G kl 1.66 σ C ijkl Gσ ij σ kl ė σ kl 1.67 λ 1 σ ijc ijkl ė kl σ ij C ijklσ kl 1.68
13 14 1 Gσ ij ėkl 4G σ 1.69 σ klėkl σ 1.70 σ ij C ijkl C ijcdσ cd σ ab C abkl σ ab C ė abcdσ kl 1.71 cd Gσ ij Gσ kl C ijkl ė 4G kl 1.7 σ C ijkl Gσ ij σ kl ė σ kl von Mises σ σ ij σ ij F σ σ y 1.75 F σ σ y 1.76 ė p ij λ 1.77, σ y, S y,.,,., S y ē p.,. 1 F σ S y ē p 1.78 F σ S y ē p 1.79
14 15 σ y σ ij σ ij Sy 1.4:, ē p. 1 ē p e p ije p ij 1.80 ē p ē p dt 1.81, σ He p S y Hē p 1.8. H., S y ē p H 1.8 von Mises σ σ ij σ ij
15 16 1 F σ S y ē p 1.85 F σ S y ē p 1.86 ė p ij λ ē p e p ije p ij F 0 F 0. F σ ij + S y S y ē ē p p 1 F σ S y ē p F σ S y ē p F σ ij H ē p F σ ij S y H ē p σ ij σh ē p 0 1.9, , σ S y. / 1 F σ S y ē p F σ S y ē p σ σ ij 1.9 σ ij 1.94, 1.87,. 1 F σ S y ē p ė p ij λ σ σ ij 1.95
16 17 F σ S y ē p ė p ij λσ ij , 1.96, 1.88, 1 F σ S y ē p ē p λ σ λ 1 σ ij σ σ ij 1.97 λ 1.98 F σ S y ē p ē p 1 λσ λσ ij ij 1.99 σ λ , F σ S y ē p F σ S y ē p 1.4, 1.87, F σ ij H λ F σ ij 4 σ H λ σ ij C ijkl ė kl ė p kl 1.10 C ijkl ė kl / λ σ kl σ ij C ijkl ė kl C ijkl λ σ kl 1.105
17 , 1.10 λ 1 F σ S y ē p H λ C ijkl ė kl C ijkl λ σ kl F σ S y ē p λ C ijkl ė kl C ijkl σ kl + H 4 σ H λ C ijkl ė kl C ijkl λ σ kl λ C ijkl ė kl C ijkl σ kl +4 σ H 1.107, F σ S y ē p F σ S y ē p σ ij C ijkl ė kl C ijkl C ijcd σ ij C ijkl ė kl C ijkl C abcd ė cd σ cd + H C abcd σ cd C abkl C abcd σ cd + H C abcd ė cd C abcd C ijcd σ cd σ kl σ cd +4 σ H C abkl C abcd σ cd +4 σ H ė kl σ kl 1.11 ė kl 1.11, /, ,, ,
18 19 1 F σ S y ē p λ C ijkl ė kl C ijkl σ ij σ Cijkl ė kl σ σ ij σ kl + H C ijkl σ σ kl + H σ σ ij C ijklė kl σ ij C ijklσ kl σ 9 H F σ S y ē p σ ij C ijkl C ijcd σ cd C abkl ė C abcd σ cd + H kl C ijcd σ C σ cd σ σ ab C abkl ijkl ė kl σ σ ab C abcd σ σ cd + H C ijkl C ijcdσ cd σ ab C abkl σ ab C abcd σ cd + 4 σ ė kl H λ C ijkl ė kl C ijkl 1.10 σ kl +4 σ H σ ij Cijkl ė kl 1.11 σ ij C ijkl σ kl +4 σ H σ ij C ijkl C ijkl C ijkl 1 σ ijc ijkl ė kl σ ij C ijklσ kl σ 9 H C ijcd σ cd C abcd C ijcd σ cd σ ab C abkl σ cd +4 σ H σ ab C abcd σ cd C ijcdσ cd σ ab C abkl σ ab C abcdσ cd + 4 σ 9 H C abkl ė kl 1.1 ė kl σ H ė kl 1.15
19 0,,., Hooke, ,,. 1 F σ S y ē p λ σ σ σ ij C ijklė kl σ ij C ijklσ kl σ 9 H Gσ kl ėkl G σ + 4 σ 9 H σ H G σ klėkl σ ij C ijkl C ijcdσ cd σ ab C abkl σ ab C abcdσ cd + ė kl σ 9 H Gσ ij Gσ kl C ijkl ė 4G kl 1.10 σ + 4 σ 9 H C ijkl Gσ ij σ kl σ ė kl H G F σ S y ē p ē p λ σ klėkl σ H G λ 1 σ ijc ijkl ė kl σ ij C ijklσ kl σ 9 H 1 Gσ kl ėkl G σ + 4 σ 9 H σ ij C ijkl σ klėkl σ H G C ijcdσ cd σ ab C abkl σ ab C abcdσ cd + 4 σ 9 H ė kl 1.16
20 1 Gσ ij Gσ kl C ijkl ė 4G kl 1.17 σ + 4 σ 9 H C ijkl Gσ ij σ kl σ ė kl H G σ klėkl ē p σ λ σ σ σ klėkl 1+ H σ H G G H 0, 1.1., S y,, S y.,,, von Mises.,,. 1 F σ S y W p F σ S y W p 1.141, W p. Ẇ p σ ij ė p ij 1.14 W p σ ij e p ijdt 1.14, σ He p S y Hē p 1.144
21 , , H. S y ē p H Ẇ p σ ij ė p ij σ ē p 1.146, vonmises,.,., F σ S y ē p F σ S y ē p ė p ij ė p ij λ λ σ σ ij λ λσ ij 1.148,. 1 F σ S y ē p F σ S y ē p ē p ē p λ σ λ 1 σ ij σ σ ij λ λσ λσ ij ij σ λ 1.150,. 1 F σ S y ē p Ẇ p σ ij ė p ij σ ij λ σ σ ij λ σ σ ē p 1.151
22 F σ S y ē p Ẇ p σ ij ė p ij σ ij λσ ij λ σ σ ē p ,,,. 1 F σ S y W p F σ S y W p Ẇ p σ ē p σ ij ė p ij σ ij λ σ σ ij λ σ 1.15 ē p λ Ẇ p σ ē p σ ij ė p ij σ ij λσ ij λ σ , 1.148,, 1 F σ S y W p F σ S y W p ē p λ σ ė p ijėp ij λ σ σ ij λ σ σ ij λ ē p ė p ijėp ij λσ ij λσ ij λ σ ē p ē p ėp ijėp ij 1/ von Mises σ 1 σ ijσ ij F σ S y W p 1.161
23 4 F σ S y W p 1.16 ė p ij λ 1.16 Ẇ p σ ij ė p ij σ ē p F 0 F 0. 1 F σ S y W p F σ S y W p F σ ij + S y S y W Ẇ p p σ ij + S y ē p S y ē p W σ p ij λ F σ ij H 1 σ σ σ λ ij ij σ σ ij σ ij H λ F σ ij σh 1 σ σ σ λσ ij ij ij σ ij 4H λ σ 1.170, σ S y , , 1.10,,. 1.4, 1.16 σ ij C ijkl ė kl ė p kl C ijkl ė kl λ 1.17 σ kl
24 / σ ij C ijkl ė kl C ijkl λ 1.17 σ kl , λ 1 F σ S y W p H λ C ijkl ė kl C ijkl λ σ kl F σ S y W p λ C ijkl ė kl C ijkl σ kl + H 4 σ H λ C ijkl ė kl C ijkl λ σ kl λ C ijkl ė kl C ijkl σ kl +4 σ H 1.175, F σ S y W p F σ S y W p σ ij C ijkl ė kl C ijkl C ijcd σ ij C ijkl ė kl C ijkl C abcd ė cd σ cd + H C abcd σ cd C abkl C abcd σ cd + H C abcd ė cd C abcd C ijcd σ cd σ kl σ cd +4 σ H C abkl C abcd σ cd +4 σ H ė kl σ kl ė kl 1.181
25 6, /σ ij, , , 1 F σ S y W p λ C ijkl ė kl C ijkl σ ij σ Cijkl ė kl σ σ ij σ kl + H 1.18 C ijkl σ σ kl + H 1.18 σ σ ij C ijklė kl σ ij C ijklσ kl σ 9 H σ ij C ijkl C ijcd σ cd C abkl ė C abcd σ cd + H kl C ijcd σ C σ cd σ σ ab C abkl ijkl ė kl σ σ ab C abcd σ σ cd + H C ijkl C ijcdσ cd σ ab C abkl σ ab C abcd σ cd + 4 σ ė kl H F σ S y W p λ C ijkl ė kl C ijkl σ kl +4 σ H σ ij Cijkl ė kl σ ij C ijkl σ kl +4 σ H 1 σ ij C ijklė kl σ ij C ijklσ kl σ 9 H σ ij C ijkl C ijcd σ cd C abkl ė C abcd σ cd +4 σ H kl C ijcd σ cd σ ab C abkl C ijkl ė kl 1.19 σ ab C abcd σ cd +4 σ H
26 C ijkl C ijcdσ cd σ ab C abkl σ ab C abcdσ cd + 4 σ 9 H 7 ė kl 1.19,,., Hooke, ,,. 1 F σ S y W p λ σ σ ij C ijklė kl σ ij C ijklσ kl σ 9 H σ Gσ kl ėkl G σ + 4 σ 9 H σ klėkl σ H G σ ij C ijkl C ijcdσ cd σ ab C abkl σ ab C abcdσ cd + ė kl σ 9 H Gσ ij Gσ kl C ijkl ė 4G kl σ + 4 σ 9 H C ijkl Gσ ij σ kl σ ė kl H G F σ S y W p ē p λ σ klėkl σ H G λ 1 σ ij C ijklė kl σ ij C ijklσ kl σ 9 H 1 Gσ kl ėkl 1.0 4G σ + 4 σ 9 H σ klėkl σ H G
27 8 σ ij C ijkl C ijcdσ cd σ ab C abkl σ ab C abcdσ cd + ė kl σ 9 H Gσ ij Gσ kl C ijkl ė 4G kl 1.05 σ + 4 σ 9 H C ijkl Gσ ij σ kl σ ė kl H G ē p σ λ σ σ klėkl σ σ klėkl 1+ H σ H G G H 0, , ,,,, Bauschinger effect.,. back stress,. Prager.. von Mises σ 1 σ ij σ ij F σ S y ē p 1.09 F σ S y ē p 1.10 ė p ij λ 1.11, S y,,,.
28 9 1.5:,, Cauchy α ij. 1.1 σ ij σ ij α ij 1.1 1/ σ σ ij σ ij F σ σ y 1.14 F σ σ y 1.15., σ y.,.. von Mises σ ij σ ij α ij 1.16
29 0 σ y σ ij α ij σ ij σ ~ ij σ ij α ij 1.6: σ 1 σ ij σ ij F σ σ y 1.18 F σ σ y 1.19 ė p ij λ 1.0,. ē p ėp ijėp ij Prager. α ij K p ė p ij 1. K p. α ij, α ij,. ė p ij 0 α ij 0
30 1. K p. Prager α ij K p ė p ij, α ij ė p ij. K p ė p ij 0 α ij 0.,. α ij C p ė p ij C p 1. F 0 F 0. F σ ij σ ij / σ ij 1 F σ σ y σ ij σ σ ij 1.5 F σ σ y σ ij 1.6 σ ij, 1.0. ė p ij λ σ σ ij 1.7 F σ σ y ė p ij λ σ ij ,1.8, 1.1,. 1 F σ σ y { ē p λ σ σ ij λ } 1 σ σ ij 1.9 λ 1.0
31 F σ σ y ē p { λ σ ij λ } 1 σ ij 1.1 σ λ , 1.0 σ ij C ijkl ė kl ė p kl 1. C ijkl ė kl λ σ kl α ij, /., σ ij α ij } {C ijkl ė kl λ α ij σ kl 1.5 σ kl 1.6 σ kl σ ij, 1.4, } {C ijkl ė kl λ α ij 1.7 σ kl 1.0, 1., λ. 1 F σ σ y α ij C p ė p ij 1.8 C p λ 1.9 C p λ σ σ ij 1.40 λ Cp σ σ ij 1.41
32 F σ σ y α ij C p ė p ij 1.4 C p λ 1.4 C p λ σ ij 1.44 λ C p σ ij 1.45, α ij λm ij } {C ijkl ė kl λ σ λm ij 1.47 kl,. λ C ijkl ė kl 1.48 C ijkl σ kl + M ij σ ij C ijkl ė kl C ijkl C abcd ė cd C abcd σ cd + M ab C ijcd σ cd C abkl C abcd σ cd + M ab σ kl 1.49 ė kl 1.50, / , F σ σ y λ C ijkl ė kl C ijkl σ kl + M ij 1.51
33 4 σ σ ij σ ij σ Cijkl ė kl C ijkl + σ σ kl σ σ ij 1.5 Cp σ σ ij σ ij C ijkl C ijkl C ijkl F σ σ y σ ij C ijklė kl σ σ ij C ijkl σ kl + C 1.5 p σ ij σ ij C ijcd σ cd C abcd σ cd σ σ ab C ijcd C abcd C abkl + M ab σ σ cd σ σ cd C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + C p σ ab σ ab + ė kl 1.54 σ σ ab Cabkl σ σ ab Cp σ σ ab ė kl 1.55 ė kl 1.56 λ C ijkl ė kl C ijkl σ kl + σ ij C ijkl σ kl 1.57 M ij σ ij Cijkl ė kl σ ij C p σ ij σ ijc ijkl ė kl σ ij C ijkl σ kl + C p σ ij σ ij 1.59, Hooke, ,. 1 F σ σ y λ σ σ σ ij C ijklė kl ij C ijkl σ kl + C p σ ij σ ij G σ kl ėkl 1.60 { } 1.61 σ G σ kl σ kl + Cp σ G σ klėkl { } 1.6 G σ σ + Cp σ
34 5 σ σ σ klėkl 1+ Cp G σ klėkl H G C ijcd σ σ ij C ijkl cd σ ab C abkl σ ab C abcd σ cd + C ė p σ kl 1.65 ab σ ab C ijkl G σ ijg σ kl G σ cd σ cd + ė kl 1.66 Cp σ C ijkl C ijkl C ijkl G σ ij σ kl σ 1+ Cp G G σ ij σ kl σ 1+ Cp G G σ ij σ kl σ 1+ H G ė kl 1.67 ė kl 1.68 ė kl 1.69 ē p λ σ σ klėkl H G, Cp H. F σ σy λ σ ij C ijklė kl σ ij C ijkl σ kl + C 1.71 p σ ij σ ij G σ kl ėkl { } 1.7 G σ kl σ kl + Cp σ G σ klėkl 1.7 G σ + Cp σ σ klėkl σ 1+ Cp G 1.74
35 6 σ klėkl σ 1+ H G 1.75 C ijcd σ σ ij C ijkl cd σ ab C abkl σ ab C abcd σ cd + C ė p σ kl 1.76 ab σ ab C ijkl G σ ij G σ kl G σ cd σ cd + ė kl 1.77 Cp σ C ijkl G σ ij σ kl σ 1+ Cp G ė kl 1.78 C ijkl C ijkl G σ ij σ kl σ + 1+ Cp G ė kl 1.79 G σ ij σ kl ė kl 1.80 σ + 1+ H G ē p σ λ σ σ klėkl H G, Cp H. 1.64,1.69,1.70, 1.75,1.80,1.81,, C p H ,, , 1.1.,., 1.1.4,.
36 7,.. von Mises σ ij σ ij α ij 1.8 σ σ ij σ ij F σ S y ē p 1.84 F σ S y ē p 1.85 ė p ij λ 1.86 Prager α ij C p λ C p ė p ij ē p 1.88 ėp ijėp ij F 0 F 0. F σ ij + S y σ ij S y ē ē p p 1 F σ S y ē p F σ S y ē p F σ ij S y σ ij ē ē p p F S y σ ij S y σ ij ē ē p p σ ij σ ij σ S y ē p ē p 0 1.9, σ S y. / σ ij
37 8 1 F σ S y ē p F σ S y ē p, σ ij σ σ ij 1.9 σ ij σ ij 1.94 σ kl σ kl σ ij 1.95,. 1 F σ ij S y ē p ė p ij λ σ σ ij 1.96 ē p λ 1.97 F σ S y ē p ė p ij λ σ ij , 1.9, 1.97, F σ ij S y ē p F σ S y ē p 1.4, 1.86 ē p σ λ 1.99 F σ ij S y σ ij ē λ p F σ ij 4 σ S y σ ij ē λ p σ ij C ijkl ė kl ė p ij 1.0 C ijkl ė kl λ σ kl 1.0
38 9 1.0 α ij, / σ ij α ij } {C ijkl ė kl λ α ij σ kl / σ ij σ ij. Prager, 1 F σ ij S y ē p S y ē λ p } {C ijkl ė kl λ α ij σ kl {C ijkl ė kl λ σ kl } λc p F σ S y ē p λ 4 σ S y ē λ p λ C ijkl σ kl C ijkl ė kl + Sy ē p C p } {C ijkl ė kl λ α ij σ kl {C ijkl ė kl λ σ kl C ijkl σ kl C ijkl ė kl +4 σ Sy ē p } λc p C p 1.07, F σ ij S y ē p σ ij C ijkl ė kl C ijkl F σ S y ē p σ ij C ijkl ė kl C abcd σ cd C abcd σ cd C abcd ė cd + Sy ē p C ijcd σ cd C abkl C abcd σ cd + Sy ē p + C p + C p C abcd ė cd +4 σ Sy ē p + C p σ kl σ kl σ kl 1.11 ė kl
39 C ijkl C abcd σ cd C ijcd σ cd C abkl +4 σ Sy ē p + C p σ kl 40 ė kl 1.14, 1.9, 1.94, , 1.10, 1.1, F σ S y ē p λ C ijkl σ kl σ σ ij C ijkl ė kl C ijkl + Sy ē p σ σ kl + C p σ ij σ Cijkl ė kl + Sy + C ē p p 1.15 σ σ ij σ σ ij 1.16 σ σ σ ij C ijklė kl σ ij C ijkl σ kl + 4 σ 9 Sy ē p Cp σ ij C ijklė kl σ ij C ijkl σ kl σ 9 H σ ij C ijkl C ijkl C ijkl C ijkl C abcd σ cd σ σ ab F σ S y ē p λ C ijcd σ cd C abkl + Sy ē p C ijcd C abcd σ σ cd C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + 4 σ 9 + C p σ σ cd σ ab σ Cabkl + Sy + C ē p p σ σ ab Sy ē p C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + 4 σ 9 H C ijkl σ kl C ijkl ė kl +4 σ Sy ē p + Cp σ ij Cijkl ė kl σ ij C ijkl σ kl +4 σ Sy + C ē p p σ ij ė kl 1.19 σ σ ab ė kl 1.0 ė kl 1.1 ė kl C p 1.4 σ ij
40 41 σ ij C ijkl C ijkl C ijkl C ijkl 1 σ ij C ijklė kl σ ij C ijkl σ kl + 4 σ 9 Sy ē p Cp 1 σ ij C ijklė kl σ ij C ijkl σ kl σ 9 H C abcd σ cd σ ab C ijcd σ cd C abkl +4 σ Sy ē p C ijcd σ C abcd σ cd C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + 4 σ 9 + C p σ cd ab Cabkl +4 σ Sy Sy ē p C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + 4 σ 9 H, H Sy ē p + C ē p p σ ab + Cp C + Cp. ė kl 1.7 σ ab ė kl 1.8 ė kl 1.9 ė kl 1.0, Hooke, ,. 1 F σ S y ē p λ σ σ σ ijc ijkl ė kl σ ij C ijkl σ kl σ 9 H G σ kl ėkl 1. 4G σ + 4 σ 9 H σ klėkl σ H G σ ij C ijkl C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + ė kl σ 9 H G σ ij G σ kl C ijkl ė 4G kl 1.5 σ + 4 σ 9 H
41 4 C ijkl G σ ij σ kl σ ė kl H G F σ S y ē p ē p λ σ klėkl σ H G λ 1 σ ij C ijklė kl σ ij C ijkl σ kl σ 9 H 1 G σ kl ėkl 1.9 4G σ + 4 σ 9 H σ klėkl σ H G σ ij C ijkl C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + ė kl σ 9 H G σ ij G σ kl C ijkl ė 4G kl 1.4 σ + 4 σ 9 H C ijkl G σ ij σ kl σ ė kl H G σ klėkl ē p σ λ σ σ σ klėkl 1+ H σ H G G,. H 0, C p 0 α ij 0,. 1., 1.0 C p 0 α ij 0, S y / σ p n,,,,, 10%
42 4 n [8]., n,. Hē p c 1 ē p + c n 1.45 c 1, c, n 0. 0.,. n, H ē p c 1 nē p + c n F σ S y ē p 1.47 F σ S y ē p 1.48 S y S y Hē p 1.49, n., n., α ij σ ij σ ij α ij 1.50 σ 1/ σ ij σ ij 1.51, 1 F σ σ y 1.5 F σ σ y 1.5
43 44 σ y. Prager α ij C p λ C p ė p ij F σ σ y ė p ij λ σ σ ij 1.55 F σ σ y ė p ij λ σ ij 1.56, trė p ij 0,, Prager, α ij C p ė p ij 1.57, tr α ij 0. α ij α ij, ė p ij ėp ij 1.58 α ij C pė p ij 1.59 α ij α ij C p ėp ij ė p ij 1.60 ᾱ C p ē p 1.61, Mises ᾱ α ij α ij 1.6 H Prager C p H C p 1.6
44 45. n, H ē p c 1 nē p + c n C p c 1nē p + c n , 1 F σ S y ē p 1.66 F σ S y ē p 1.67 S y S y 1 rσ y + rhē p 0 r S y ē p r H ē p rh 1.69 C p 1 rh 1.70, r 0, r ,. Ziegler.. von Mises σ ij σ ij α ij 1.71 σ σ ij σ ij 1 1.7
45 46 1 F σ σ y 1.7 F σ σ y 1.74 ė p ij λ 1.75,. ē p ėp ijėp ij Prager, Ziegler Ziegler. Prager α ij K p ė p ij 1.77 Ziegler α ij K z σ ij 1.78 Ziegler α ij K 1 σ ij K α ij 1.79 K p,k z,k 1,K. α ij, α ij,. ė p ij 0 α ij 0. K p,k z,k 1,K. i Prager α ij K p ė p ij α ij ė p ij. K p ė p ij 0 α ij 0. ii Ziegler α ij K z σ ij ė p ij 0 σ ij 0 ė p ij 0 K z 0.. K z C z ē p C z 1.80,. α ij C z ē p σ ij 1.81
46 47 ii Ziegler α ij K 1 σ ij K α ij ė p ij 0 σ ij 0,α ij 0 ė p ij 0 K 1 0,K 0.. K 1 C 1 ē p C K C ē p C 1.8,. Prager α ij C p ė p ij C p 1.84 Ziegler α ij C z ē p σ ij C z 1.85 Ziegler α ij C 1 ē p σ ij C ē p α ij C 1,C 1.86 F 0 F 0. F σ ij σ ij 1 F σ σ y σ ij σ σ ij 1.88 F σ σ y σ ij 1.89 σ ij, ė p ij λ σ σ ij 1.90 F σ σ y ė p ij λ σ ij 1.91
47 ,1.91, 1.76,. 1 F σ σ y { ē p λ σ σ ij λ } 1 σ σ ij 1.9 λ 1.9 F σ σ y ē p { λ σ ij λ } 1 σ ij 1.94 σ λ , 1.75, σ ij C ijkl ė kl ė p kl 1.96 C ijkl ė kl λ σ kl α ij, /., σ ij α ij } {C ijkl ė kl λ α ij σ kl 1.98 σ kl 1.99 σ kl σ ij, 1.87, } {C ijkl ė kl λ α ij σ kl 1.9, 1.95, λ.
48 49 1 F σ σ y Prager α ij C p ė p ij C p λ 1.40 C p λ σ σ ij 1.40 λ Cp σ σ ij Ziegler α ij C z ē p σ ij C z λ σij λ C z σ ij Ziegler α ij C 1 ē p σ ij C ē p α ij C 1 λ σij C λαij λ C 1 σ ij C α ij F σ σ y Prager α ij C p ė p ij C p λ 1.41 C p λ σ ij 1.41 λ C p σ ij Ziegler α ij C z ē p σ ij C z σ λ σ ij λ C z σ σ ij Ziegler α ij C 1 ē p σ ij C ē p α ij C 1 σ λ σ ij C σ λα ij λ C 1 σ σ ij C σα ij 1.40
49 50, α ij λm ij } {C ijkl ė kl λ σ λm ij 1.4 kl,. λ C ijkl ė kl 1.4 C ijkl σ kl + M ij σ ij C ijkl ė kl C ijkl C abcd ė cd C abcd σ cd + M ab C ijcd σ cd C abkl C abcd σ cd + M ab σ kl 1.44 ė kl 1.45, / , F σ σ y λ C ijkl ė kl C ijkl σ kl σ σ ij + M ij 1.46 σ ij σ Cijkl ė kl C ijkl + σ σ kl σ σ ij M ij 1.47 σ ij C ijklė kl σ σ ij C ijkl σ kl + σ kl M ij 1.48 σ ij C ijkl C ijcd σ cd C abcd σ cd C abkl + M ab ė kl 1.49
50 C ijkl F σ σ y C ijkl σ σ ab C ijcd C abcd σ σ cd + σ σ cd C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + σ σ ab M ab σ σ ab Cabkl σ σ ab M ab 51 ė kl 1.40 ė kl 1.41 λ C ijkl ė kl C ijkl σ kl + σ M ij 1.4 ij Cijkl ė kl σ ij C ijkl σ kl σ ij M ij σ ij C ijklė kl σ ij C ijkl σ kl + σ ij M ij 1.44 σ ij C ijkl C ijkl C ijkl C ijcd σ cd C abcd σ cd σ ab C abkl C ijcd σ cd C abcd σ cd C ijcd σ cd σ ab C abkl + M ab σ σ ab C abcd σ cd + 1 σ ab M ab ab Cabkl + σ ab ė kl 1.45 M ab ė kl 1.46 ė kl 1.47, Hooke, ,. 1 F σ σ y λ σ ij C ijklė kl σ σ ij C ijkl σ kl + σ ij M ij G σ kl ėkl σ G σ kl σ kl + σ ij M ij G σ klėkl σ G σ + σ ij M ij
51 5 σ klėkl σ + 1 G σ ij M ij σ ij C ijkl C ijkl C ijkl C ijkl C σ ijcd σ cd σ ab C abkl σ σ ab C abcd σ cd + σ ab M ė kl 1.44 ab σ G σ ijg σ kl σ G σ cd σ cd + σ ab M ė kl 1.44 ab σ G σ ijg σ kl G σ σ + σ ab M ė kl ab G σ ij σ kl σ + σ G σ ab M ė kl ab ē p λ σ klėkl σ + 1 G σ ij M ij F σ σ y λ σ ijc ijkl ė kl σ ij C ijkl σ kl + σ ij M ij G σ kl ėkl G σ kl σ kl + σ ij M ij G σ klėkl 6G σ + σ ij M ij σ klėkl σ + 1 G σ ij M ij σ ij C ijkl C ijcd σ cd σ ab C abkl σ ab C abcd σ cd + σ ab M ė kl ab C ijkl G σ ij G σ kl G σ cd σ cd + σ ab M ė kl 1.45 ab C ijkl G σ ijg σ kl 4G σ + σ ab M ė kl 1.45 ab
52 C ijkl ē p σ λ σ G σ ij σ kl σ + 1 4G σ ab M ab σ klėkl σ + 1 G σ ij M ij 5 ė kl σ klėkl σ + 1 4G σ σ ij M ij Prager, M ij Cp σ σ ij F σ σ y, M ij C p σ ij F σ σy,. 1 F σ σ y λ σ klėkl σ + 1 G σ ij M ij σ klėkl σ + 1 G σ ij σ σ σ klėkl 1+ Cp G σ klėkl Cp σ σ ij H G σ ij C ijkl G σ ij σ kl σ + σ G σ ab M ab ė kl C ijkl G σ ij σ kl σ + σ G σ ab Cp σ σ ij ė kl C ijkl σ ij σ kl σ 1+ Cp G ė kl 1.46 σ ij σ kl C ijkl σ ė 1+ H kl 1.46 G ē p λ σ klėkl σ 1+ H G 1.464
53 54, Cp H. F σ σy λ σ klėkl σ + 1 G σ ij M ij σ klėkl σ + 1 G σ ij Cp σ ij σ klėkl σ 1+ Cp σ ij C ijkl σ klėkl σ G 1+ H G G σ ij σ kl σ + 1 4G σ ab M ab ė kl G σ ij C ijkl σ kl ė σ + 1 4G σ ab Cp σ kl ab C ijkl G σ ij σ kl σ 1+ Cp G ė kl C ijkl G σ ij σ kl σ ė 1+ H kl 1.47 G ē p σ λ σ σ klėkl H G, Cp H ,1.445,1.446, 1.450,1.454,1.455,, M ij
54 Hooke, Hooke. Hooke. F Cauchy T elastic material. T t ff t f. ff fq F Q ff Q T F, F O,O, O O Q., P ff ff P V. T fv fv fq V Q T Q fv Q T V, V O,O, O O Q. fv isotropic tensor function T, V, T fv φ 0 I + φ 1 V + φ V 1.479
55 56., φ i i 0, 1, V. representation theorem V B 1/. T gb gb gq B Q T Q gb Q T 1.481, gb,., B. T ψ 0 I + ψ 1 B + ψ B 1.48 ξ 0 I + ξ 1 B + ξ 1 B Hooke., E L V I + 1 {u x + x u} E L 1 {u x + x u} 1.485, E L T φ 0 + φ 1 + φ I +φ 1 +φ E L η 0 I + η 1 E L 1.487, η 0, η 1 E L. T E L, Hooke. T λtr E L I +μe L 1.488
56 57, λ, μ Lamé., Hooke , 1.480, B Almansi A A 1 I B 1.489,.. T ha ha hq A Q T Q ha Q T A, A O,O, O O Q. ha,. T ha ζ 0 I + ζ 1 A + ζ A 1.49 Hooke T λ tr AI +μa A E L 1.494, λ, μ Lamé W, T, A T, Å. T Ṫ W T + T W Å Ȧ W A + A W 1.496
57 58. T λtr ÅI +μå 1.497,,,,,.,, ÅC D, F t τ R t τ, U t τ I T J T O T C T G Å J ÅO ÅC ÅG T J Ṫ W T + T W T J T O + D T + T D 1.50 T J T C D T T D 1.50 T G Ṫ Ω T + T Ω W Ω T λ tr DI +μd T Kirchhoff ˆT t τ J t τt τ ˆT t t λ tr DI +μd , ˆT t t J T J + T tr D ˆT t t O T O + T tr D ˆT t t C T C + T tr D Hooke
58 v v e v p v v e + v p 1.511, L D D D e + D p σ ij T ij, e p ij D p ij., C ep ijkl T ij C ep ijkld kl 1.51.,,, 1.51, Cauchy Kirchhoff ˆT ij C ep ijkld kl , Kirchhoff ˆα t τ J t τατ Kirchhoff, Jaumann.., T ij ˆT ij ˆα ij 1.516,,. G T T C ep ij kl ijkl C ijkl σ 1 + H G 1.517
59 60 T ij, T ij T ij 1 T kk δ ij., 1F σ S y ē p, F σ S y ēp. pe. 1F σ S y ē p λ T kl D kl σ H G T kl D kl σ H G F σ Sy ēp T kl λ D kl σ H G Ziegler, Ziegler, M ij 1F σ σ y Ziegler M ij C z Tij 1.51 Ziegler M ij C 1 Tij C α ij 1.5 F σ σ y Ziegler M ij C z σ T ij 1.5 Ziegler M ij C 1 σ T ij C σα ij 1.54,,,. 1F σ σ y C ep ijkl C ijkl G T ij T kl σ 1+ 1 T 1.55 G σ ab M ab ē p T kl D kl σ 1+ 1 T 1.56 Gσ ij M ij
60 61 λ T kl D kl σ 1+ 1 T 1.57 Gσ ij M ij F σ σ y C ep ijkl C ijkl G T ij T kl σ G σ T ab M ab 1.58 ē p λ T kl D kl σ Gσ ij M ij 1.59 T kl D kl σ Gσ ij M ij Pager ˆαij K p D p ij 1.51 Ziegler ˆαij K z Tij 1.5 Ziegler ˆα ij K 1 Tij K α ij ,,., t e p ij t 0 τė p ij dτ 1.54 t σ ij t 0 t τ σ ij dτ 1.55 τ C ep ijkl τ ė kl dτ t t σ ij t σ ij τ C ep ijkl τ ė kl dτ 1.57 t
61 6. t C ep ijkl, 1.56, 1.57,. forward-euler t t backward-euler t. forward-euler explicit,,,. backward-euler implicit,.,,, forward-euler. i t 1. t 1 t, 1.1.1, t 1 t t 1, t t 1 C e ijkl. t σ ij t 1 σ ij + t 1 C e ijklδe ij 1.58 Δe ij t 1 t. t σ ij, t t.. ii t 1. t 1 t
62 6.. t 1 t t 1, t 1 t t 1 C ep ijkl. Δσ ij t 1 C ep ijklδe ij 1.59 Δē p, F σ S y,,. 1 Δē p t 1 σ kl Δe kl t 1 σ 1+ t H μ Δλ Δα ij Δλ C p t1 σ ij σ Ziegler Δē p t 1 σ kl Δe kl Δλ 1.54 t 1 σ 1+ 1 G t 1 σ t 1 σ t 1Mij ij Δα ij Δλ t 1 M ij 1.54 t 1 M ij C 1 t1 σ ij C t 1 α ij Δλ Δē p, t t. Δλ Δē p,.,. Δσ ij t 1 C e ijkl 1.545, t t., forward-euler.
63 64,. Kirchhoff.,, t C ep ijkl, t C e ijkl. t T ij t T ij + t T ij + t T ij + t T ij + t T ij + t t t t t t t t t t τ T ij dτ {τ ˆTτ ij tr τ D τ T ij } dτ {τ ˆTτ ij + τ W ik τ ˆTτ kj τ ˆTτ ik τ W kj tr τ D τ T ij } {τ ˆTτ ij + τ W ik τ T kj τ T ik τ W kj tr τ D τ T ij } dτ dτ { τ C ep ijkl τ D kl + τ W ik τ T kj τ T ik τ W kj tr τ D τ T ij } dτ t T ij + {t C ep ijkl t D kl + t W ik t T kj t T ik t W kj tr t D t T ij } Δt 1.551, Cauchy, t t Kirchhoff.,. t α ij t α ij + t α ij + t α ij + t α ij + t t t t t t t t t τ α ij dτ 1.55 {τ ˆατ ij tr τ D τ α ij } dτ 1.55 {τ ˆατ ij + τ W ikτ ˆα τ kj τ ˆα τ ik τ W kj tr τ D τ α ij } {τ ˆατ ij + τ W ik τ α kj τ α ik τ W kj tr τ D τ α ij } dτ dτ } t α ij + {C t λt Tij + τ W τ ik α kj τ α τ ik W kj tr τ D τ α ij dτ t } t α ij + {C t λt Tij + t W t ik α kj t α t ik W kj tr t D t α ij Δt 1.557
64 V,v, S, s. s t, u, v g.. T Cauchy. x T + ρg T T n t u u D ij 1 ui + u j x j x i ˆT ij C ep ijkl D kl, T ij t 0 T ij dt ,1.559 T T, u U. Ť T, ǔ U,. x Ť + ρg ǔ dv v 1.56,., s t,s u t, u s, s s t + s u. Ť :ǔ x dv t ǔ ds + n Ť u ds + v s t s u v ρg ǔdv 1.564, s u w 0 w W. ǔ U, w W ǔ + w U., Cauchy T, T :ǔ x dv t ǔ ds + n T u ds + ρg ǔdv v s t s u v
65 v T : {ǔ + w x } dv t ǔ + wds + n T u ds + s t s u v 66 ρg ǔ + wdv T :w x dv t w ds + ρg wdv v s t v. T : δa L dv v δv t w ds + v ρg w dv δa L, w W Almange. δa Lij 1 wi + w j x j x i ,,.,.,. find u h W h such that w h Qu h F 0 w h W h 1.570, W h W, W h W. Q, F, , Qu n F 1.571
66 67, F {F k }., 0F 0 < F 1 < F < < F n F 1.57 Qu k F k 1.57 u k u k 1. F 0 0 u 0 0. a. forward-euler u k 1, F k F k 1 u k. K k u k u k 1 F k F k K k Q u uu k F k F k 1 K k Q u k u k 1 1.7: u k
67 68 K., forward-euler, step n, F ΔF 1 n F, Δuk. K k Δu k ΔF u k. u k u k 1 + Δu k k Δu i i1 1.8.,step,., 1step step, forward-euler.
68 69 K ΔF ΔF K Q K 1 ΔF u 1 u u Δu 1 Δu Δu u 1.8: forward-euler b. Newton-Raphson Newton-Raphson. u k 1, u k 1.9.,
69 70 F k F F k Q k 0 K k 1 F k Q k 1 F k Q k Q k 1 K k Q k Q k F k 1 Δu k 1 Δu k u k 0 u k 1 u k 1 u k u k u 1.9: Newton-Raphson. K., u k 0 u k Q k 0 Qu k K k 1 Q u uu k 0 K k 1 Δuk 1 F k Q k u k 1 uk 0 +Δuk u k 1, uk 0 uk 1 uk., u k i 1 Q k i 1 Quk i
70 K k i Q u 71 uu k i K k i Δuk i F k Q k i u k i uk i 1 +Δuk i u k i. u k i 1 uk i uk, Q F k, F k Q k 0, u k i u k, F k Q k i 0., F k Q k i 0,., F k, step n, F k k n F ,,,, step.,. u k 1, u k. K k Δu k F k Qu k u k u k 1 + Δu k forward-euler. forward-euler. forward-euler,, forward-euler.
71 7 ΔF ΔF F F F 1 K K K Qu K F Qu 1 Qu F Qu ΔF K 1 Qu 1 u 1 u 1 u u u u Δu 1 Δu Δu Δu 1 Δu Δu 1.10: forward-euler c.??, forward-euler newton-raphson.,. Prager, n. Young 1959 kgf/mm, Poisson 0., 5 kgf/mm,n0.,c , forward-euler Newton-Raphson,, step,.
72 7 Newton-Raphson load-stroke, forward-euler load-stroke forward-euler Newton-Raphson without iteration 8 load ton stroke mm 1.11: load-stroke
73 forward-euler Newton-Raphson without iteration 10 8 load ton stroke mm 1.1: load-stroke
74 load ton forward-euler Newton-Raphson without iterarion stroke mm 1.1: load-stroke forward-euler, Newton-Raphson, step step. forward-euler 1.14, Newton-Raphson step forward-euler, Newton-Raphson. 1.15, Newton-Raphson, step 1000 step step. 1.14, forward-euler step. 1.16, forward-euler, step.
75 76,, Newton-Raphson step 1000 step 8 load ton stroke mm 1.14: load-stroke forward-euler
76 step 1000 step load ton stroke mm 1.15: load-stroke Newton-Raphson
77 forward-euler Newton-Raphson without iteration load ton stroke mm 1.16: load-stroke 1000step updated Lagrange Newton-Raphson, updated Lagrange.. δa ij T ij dv δr v, δr, A ij, Almange
78 79. δa ij 1 δui + δu j 1.59 x j x i T ij δa ij T ij., δa ij T ij. δa ij T ij δa 11 T 11 + δa 1 T 1 + δa 1 T 1 + δa 1 T 1 + δa T + δa T + δa 1 T 1 + δa T + δa T 1.59 δa 11 T 11 + δa T + δa T +δa 1 T 1 +δa T +δa 1 T 1, δa ij,t ij i, j., {δa} T {δa 11 δa δa δa 1 δa δa 1 } T {T } {T 11 T T T 1 T T 1 } 1.596, δa ij T ij dv {δa} T {T } dv v, δa ij. δa ij. v δa 11 δu 1 x δa δu x δa δu x δa 1 δu 1 + δu x x δa δu + δu x x 1.60 δa 1 δu x 1 + δu 1 x 1.60
79 80, u i N k u k i u i x j Nk u k i x j, δu i δu i N k δu k i δu i x j Nk δu k i x j., {δu} { } T δu 1 1 δu 1 δu 1 δu 1 δu δu δu n 1 δu n δu n 1.608,. {δa} [B][δu] 1.609, [B] [B k ],. [ ] B [[ B 1][ B ] [B n]] ] [B k N k x 1 N k x N k x N k x N k x 1 N k x N k x N k x N k x ,. δa ij T ij dv {δu} T [B] T {T } dv 1.61 v v
80 Q, Q [B] T [T ]dv 1.61 v., V, δa ij. δa ij 1. δui X j + δu n X i , Newton-Raphson, Q u K. K Q u, 1.61 u, Q Q t Q u u t Q u u,, 1.591, ,. F, J. F x i X j e i e j J detf 1.618, Piola-Kirchhoff S. T 1 J F S F T 1.619
81 8 S Green-Lagrange E δe. E 1 ui + u j + u k u k e i e j 1.60 X j X i X i X j δe 1 δui + δu j + δu k u k + u k δu k 1.61 X j X i X i X j X i X j δa δe. δe F T δa F 1.6, δe : S dv F T δa F :JF 1 T F T 1 dv 1.6 v v J δa : T dv 1.64 v Total-Lagrange. δe : Ṡ + 1 δf T Ḟ + Ḟ T δf : S dv 1.65 v. Piola-Kirchhoff S t t S t t Ṡ. Ṡ JF 1 S t t F T , δe : Ṡ F T δa F :JF 1 S t t F T 1.67 JδA : Ṡtt 1.68, δe : Ṡ δa : Ṡttdv 1.69 v v
82 8, Ḟ δf,., L, Ḟ L F 1.60 δf δf t t F 1.61 L u i e i e j 1.6 x j., 1.65, 1 δf T Ḟ + F T δf : S 1 F T δf t t T L F + F T L T δf t t F : JF 1 T F T 1.6 J 1 δf t t T L + L T δf t t : T 1.64 V 1 δf T Ḟ + F T 1 δf : S dv δf t t T L + L T δf t t : T dv v 1.65 updated Lagrange. δa : Ṡtt+ 1 δf t t T L + L T δf t t : T dv δṙ 1.66 v. { } 1 δa ij Ṡ t t ij dv + δf tt ki L kj + L ki δf t t kj T ij dv δṙ 1.67 v v, 1., Ṡ t t ṠF tt F. δa ij T ij. { } δa ij Ṡ ij {δa} T Ṡ } Ṡ {Ṡ11 Ṡ Ṡ Ṡ 1 Ṡ Ṡ
83 84, Ṡ ij D ij. D ij 1 ui + u j x j x i,. Ṡ ij C ijkl D kl Ṡ ij C ij11 D 11 + C ij1 D 1 + C ij1 D 1 + C ij1 D 1 + C ij D + C ij D + C ij1 D 1 + C ij D + C ij D 1.64 C ij11 D 11 + C ij D + C ij D + 1 C ij1 + C ij1 D C ij + C ij D + 1 C ij1 + C ij1 D , } [ }, {Ṡ C]{ D. C ijkl 1 C ijkl + C ijlk Ṡ 11 Ṡ Ṡ Ṡ 1 Ṡ Ṡ 1 C 1111 C11 C11 C111 C11 C111 C 11 C C C1 C C1 C 11 C C C1 C C1 C 111 C1 C1 C11 C1 C11 C 11 C C C1 C C1 C 111 C1 C1 C11 C1 C11 D 11 D D D 1 D D δa ij, {D} [B] { u} 1.646
84 { u} 85 { } T u 1 1 u 1 u 1 u 1 u u u n 1 u n u n 1.647, 1. [ ] δa ij Ṡ t t ij dv {δu} T [B] T D [B] { u} dv v v. T ij T ji,. 1 δf kil kj + L ki δf kj T ij 1 T ijl ki δf kj + 1 T ijδf ki L kj δf ki T ij L kj , δf ki T ij L kj {δf} T [Σ] {L} {δf} {δf 11 δf 1 δf 1 δf 1 δf δf δf 1 δf δf } 1.65 {L} {L 11 L 1 L 1 L 1 L L L 1 L L } 1.65 T 11 T 1 T 1 [T ] T 1 T T T 1 T T [T ] [0] [0] [Σ] [0] [T ] [0] [0] [0] [T ], δf ij,l ij. δf ij δu i x j L ij u i x j 1.657
85 86. { } δu {δf} [Z] {δu} x { } u {L} [Z] { u} x, { } { } δu δu1 δu 1 δu 1 δu δu δu δu δu δu x x 1 x x x 1 x x x 1 x x { } { } u u1 u 1 u 1 u u u u u u x x 1 x x x 1 x x x 1 x x [ ] Z N 1 x 1 N x 1 N 1 x N x N 1 x N x N 1 x 1 N x 1 N 1 x N x N 1 N x x N 1 N x 1 x 1 N 1 N x x N 1 N x x N n x 1 N n x N n x, δf ki T ij L kj. N n x 1 N n x N n x N n x 1 N n x N n x 1.66 δf ki T ij L kj {δf} T [Σ] {L} 1.66 [Z] T [Σ] [Z]. ] { } T N i N [G i N i 11 T 1 T 1 ij T x 1 x x 1 T T T 1 T T {δu} T [Z] T [Σ] [Z] { u} N i x 1 N i x N i x
86 87 [G 11 ] [G 1n ] [Z] T [Σ] [Z].. [G] [G n1 ] [G nn ] v δf ki T ij L kj dv v {δu} [G] { u} dv [ ] δa ij S t t ij + δf ki T ij L kj dv {δu} T [B] T D [B]+[G] dv { u} v 1.67 V, δa ij 1.614, Ṡ t t ij Caushy , 1.668,.,. 1 1 dv, v 1 [ ] J v det J dr 1 dr dr x 1 x 1 x 1 r r x x x r r x x x r r [J]. x i r j Nk x k i Nk X k i r j r j, [B], Nk x j [J]. N k x 1 N k x N k x r r x 1 x 1 x 1 r r x x x r r x x x + u k i N k N k r N k r 1.67
87 88 [ ] 1 J r r x 1 x 1 x 1 r r x x x r r x x x, [J] [ ] J X 1 X 1 X X X 1 r r X r r X X X r r ,. X i r j Nk X k i r j ,. Ṡ ij C ijkl D kl D kl D lk, C ijkl 1 C ijkl + C ijlk D, D ij [ C]. Ṡ 11 Ṡ Ṡ Ṡ 1 Ṡ Ṡ 1 C 1111 C11 C11 C111 C11 C111 C 11 C C C1 C C1 C 11 C C C1 C C1 C 111 C1 C1 C11 C1 C11 C 11 C C C1 C C1 C 111 C1 C1 C11 C1 C11 D 11 D D D 1 D D
88 89 [ C], Cijkl C klij,. Hooke Cijkl e λδ ij δ kl +μδ ik δ jl C ijkl e λδ ij δ kl + μ δ ik δ jl + δ il δ jk C klij e λδ kl δ ij + μ δ ki δ lj + δ kj δ li λδ ij δ kl + μ δ ki δ lj + δ il δ jk δ mn δ nm 1.68 C ijkl e 1.68, Hooke C 1111 e λδ 11δ 11 + μ δ 11 δ 11 + δ 11 δ 11 λ +μ C 11 e λδ 11δ + μ δ 11 δ 11 + δ 11 δ 11 λ C 11 e λδ 11δ + μ δ 1 δ 1 + δ 1 δ 1 λ C 111 e λδ 11δ 1 + μ δ 11 δ 1 + δ 1 δ C 11 e λδ 11δ + μ δ 1 δ 1 + δ 1 δ C 111 e λδ 11δ 1 + μ δ 1 δ 11 + δ 11 δ C e λδ δ + μ δ δ + δ δ λ +μ C e λδ δ + μ δ δ + δ δ λ C 1 e λδ δ 1 + μ δ 1 δ + δ δ C e λδ δ + μ δ δ + δ δ C 1 e λδ δ 1 + μ δ δ 1 + δ 1 δ C e λδ δ + μ δ δ + δ δ λ +μ C 1 e λδ δ 1 + μ δ 1 δ + δ δ C e λδ δ + μ δ δ + δ δ
89 90 C 1 e λδ δ 1 + μ δ δ 1 + δ 1 δ C 11 e λδ 1 δ 1 + μ δ 11 δ + δ 1 δ 1 μ C 1 e λδ 1 δ + μ δ 1 δ + δ 1 δ C 11 e λδ 1 δ 1 + μ δ 1 δ 1 + δ 11 δ C e λδ δ + μ δ δ + δ δ C 1 e λδ δ 1 + μ δ δ 1 + δ 1 δ C 11 e λδ 1δ 1 + μ δ δ 11 + δ 1 δ 1 μ Hooke.. λ +μ λ λ λ λ+μ λ [ C λ λ λ+μ e ] μ μ μ Lamé λ, μ E, ν. νe λ 1 + ν1 ν E μ 1+ν
90 91, 1.705,. Eν1 ν Eν1 ν ν1 ν 1+ν1 ν [ C e ] E1 ν 1+ν1 ν Eν1 ν 1+ν1 ν Eν1 ν 1+ν1 ν E1 ν 1+ν1 ν Eν1 ν 1+ν1 ν Eν1 ν 1+ν1 ν E1 ν 1+ν1 ν E ν E ν E ν Ziegler C p ijkl C p ijkl G σ ij σ kl σ 1+ H G G σ σ ij σ kl σ 1+ 1 σ G σ ab M ab,,.,,., G 1 A σ 1+ H G G Ziegler A σ 1+ 1 σ G σ ab M 1.71 ab C p ijkl 1 C p ijkl A σ ij σ kl 1.71 C p ijkl + Cp ijlk C p ijkl 1.714, C p ijkl C p klij 1.715
91 9, C p ijkl 6 6,. C p 1111 A σ 11 σ C p 11 A σ 11 σ C p 11 A σ 11 σ C p 111 A σ 11 σ C p 11 A σ 11 σ 1.70 C p 111 A σ 11 σ C p A σ σ 1.7 C p A σ σ 1.7 C p 1 A σ σ C p A σ σ 1.75 C p 1 A σ σ C p A σ σ 1.77 C p 1 A σ σ C p A σ σ 1.79 C p 1 A σ σ C p 11 A σ 1 σ C p 1 A σ 1 σ 1.7 C p 11 A σ 1 σ C p A σ σ 1.74 C p 1 A σ σ C p 11 A σ 1 σ
92 9. A σ 11 σ 11 A σ 11 σ A σ 11 σ A σ 11 σ 1 A σ 11 σ A σ 11 σ 1 A σ σ 11 A σ σ A σ σ A σ σ 1 A σ σ A σ σ 1 [ C A σ p ] σ 11 A σ σ A σ σ A σ σ 1 A σ σ A σ σ 1 A σ 1 σ 11 A σ 1 σ A σ 1 σ A σ 1 σ 1 A σ 1 σ A σ 1 σ 1 A σ σ 11 A σ σ A σ σ A σ σ 1 A σ σ A σ σ 1 A σ 1 σ 11 A σ 1 σ A σ 1 σ A σ 1 σ 1 A σ 1 σ A σ 1 σ {S} { } σ 11 σ σ σ 1 σ σ ,. [ Cp ] A {S}{S} T 1.79, Kirchhoff Jaumann, D. t t ˆT J C ep : D C t tṡ C : D. t tṡ t t ˆT J D T T D C ep : D D T T D 1.74 t tṡij C ep ijkl D kl D ik T kj T ik D kj 1.74 C ep ijkl D kl δ il T kj D kl T ik δ jl D kl { C ep ijkl 1 δ ijt kj + δ ik T lj 1 } T ikδ ij + T il δ jk D kl 1.745
93 94. C ep ijkl Cep ijkl 1 δ ilt kj + δ ik T lj 1 T ikδ jl + T lj δ jk t tṡij. t tṡij C ij11 D 11 + C ij1 D 1 + C ij1 D 1 + C ij1 D 1 + C ij D + C ij D + C ij1 D 1 + C ij D + C ij D C ij11 D 11 + C ij D + C ij D + 1 Cij1 + C ij1 D1 + 1 Cij + C ij D + 1 Cij1 + C ij1 D ,. 1 C 1111 C11 C11 C111 + C 111 C1 + C 1 t tṡ11 t tṡ t tṡ t tṡ1 t tṡ t tṡ1 C 11 C C 1 C 11 C C 1 C 111 C1 C1 1 C 11 C C 1 C 111 C1 C1 1 C1 + C 1 C11 + C 11 C1 + C 1 C11 + C 11,. 1 C11 + C 11 C + C C + C C1 + C 1 C + C C1 + C C111 + C 111 C1 + C 1 C1 + C 1 C11 + C 11 C1 + C 1 C11 + C D 11 D D D 1 D D 1 C 1111 C δ 11T 11 + δ 11 T 11 1 T 11δ 11 + T 11 δ 11 C 1111 T C 11 C 11 1 δ 1T 1 + δ 1 T 1 1 T 1δ 1 + T 1 δ 1 C C 11 C 11 1 δ 1T 1 + δ 1 T 1 1 T 1δ 1 + T 1 δ 1 C C 111 C δ 1T 11 + δ 11 T 1 1 T 11δ 1 + T 1 δ 11 C T 1 + T 1 C 111 T
94 95 C 111 C δ 11T 1 + δ 1 T 11 1 T 1δ 11 + T 11 δ 1 C T 1 + T 1 C 111 T C 11 C 11 1 δ 1T 1 + δ 1 T 1 1 T 1δ 1 + T 1 δ 1 C C 11 C 11 1 δ 1T 1 + δ 1 T 1 1 T 1δ 1 + T 1 δ 1 C C 111 C δ 11T 1 + δ 1 T 11 1 T 1δ 11 + T 11 δ 1 C T 1 + T 1 C 111 T C 111 C δ 1T 11 + δ 11 T 1 1 T 11δ 1 + T 1 δ 11 C T 1 + T 1 C 111 T C 11 C 11 1 δ 1T 1 + δ 1 T 1 1 T 1δ 1 + T 1 δ 1 C C C 1 δ T + δ T 1 T δ + T δ C T C C 1 δ T + δ T 1 T δ + T δ C C 1 C 1 1 δ T 1 + δ 1 T 1 T 1δ + T δ 1 C 1 1 T 1 + T 1 C 1 T C 1 C 1 1 δ 1T + δ T 1 1 T δ 1 + T 1 δ C 1 1 T 1 + T 1 C 1 T C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C T C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C T C 1 C 1 1 δ 1T + δ T 1 1 T δ 1 + T 1 δ C C 1 C 1 1 δ T 1 + δ 1 T 1 T 1δ + T δ 1 C C 11 C 11 1 δ 1T 1 + δ 1 T 1 1 T 1δ 1 + T 1 δ 1 C
95 96 C C 1 δ T + δ T 1 T δ + T δ C C C 1 δ T + δ T 1 T δ + T δ C T C 1 C 1 1 δ T 1 + δ 1 T 1 T 1δ + T δ 1 C C 1 C 1 1 δ 1T + δ T 1 1 T δ 1 + T 1 δ C C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C T 1.77 C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C T C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C T C 1 C 1 1 δ 1T + δ T 1 1 T δ 1 + T 1 δ C 1 1 T 1 + T 1 C 1 T C 1 C 1 1 δ T 1 + δ 1 T 1 T 1δ + T δ 1 C 1 1 T 1 + T 1 C 1 T C 111 C δ 11T 1 + δ 11 T 1 1 T 11δ 1 + T 11 δ 1 C T 1 + T 1 C 111 T C 1 C 1 1 δ 1T + δ 1 T 1 T 1δ + T 1 δ C 1 1 T 1 + T 1 C 1 T C 1 C 1 1 δ 1T + δ 1 T 1 T 1δ + T 1 δ C C 11 C 11 1 δ 1T 1 + δ 11 T 1 T 11δ + T 1 δ 1 C 11 1 T 11 + T C 11 C 11 1 δ 11T + δ 1 T 1 1 T 1δ 1 + T 11 δ C 11 1 T 11 + T 1.78
96 97 C 1 C 1 1 δ 1T + δ 1 T 1 T 1δ + T 1 δ C 1 1 T C 1 C 1 1 δ 1T + δ 1 T 1 T 1δ + T 1 δ C 1 1 T C 11 C 11 1 δ 11T + δ 1 T 1 1 T 1δ 1 + T 11 δ C 11 1 T C 11 C 11 1 δ 1T 1 + δ 11 T 1 T 11δ + T 1 δ 1 C 11 1 T C 11 C 11 1 δ 1T 1 + δ 1 T 1 1 T 1δ 1 + T 1 δ 1 C C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C T C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C T C 1 C 1 1 δ T 1 + δ 1 T 1 T 1δ + T δ 1 C 1 1 T C 1 C 1 1 δ 1T + δ T 1 1 T δ 1 + T 1 δ C 1 1 T C 1 C 1 1 δ 1T + δ T 1 1 T δ 1 + T 1 δ C 1 1 T C C 1 δ T + δ T 1 T δ + T δ C 1 T + T 1.79 C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C C 1 δ T + δ T 1 T δ + T δ C 1 T + T C 1 C 1 1 δ 1T + δ T 1 1 T δ 1 + T 1 δ C 1 1 T C 1 C 1 1 δ T 1 + δ 1 T 1 T 1δ + T δ 1 C 1 1 T C 111 C δ 1T 11 + δ 1 T 11 1 T 1δ 11 + T 1 δ 11 C 111 T C 111 C δ 1T 11 + δ 1 T 11 1 T 1δ 11 + T 1 δ 11 C 111 T
97 98 C 1 C 1 1 δ T 1 + δ T 1 1 T δ 1 + T δ 1 C C 1 C 1 1 δ T 1 + δ T 1 1 T δ 1 + T δ 1 C 1 T C 11 C 11 1 δ T 11 + δ 1 T 1 1 T 1δ 1 + T δ 11 C 11 1 T 1.80 C 11 C 11 1 δ 1T 1 + δ T 11 1 T δ 11 + T 1 δ 1 C 11 1 T 1.80 C 1 C 1 1 δ T 1 + δ T 1 1 T δ 1 + T δ 1 C 1 1 T C 1 C 1 1 δ T 1 + δ T 1 1 T δ 1 + T δ 1 C 1 1 T C 11 C 11 1 δ 1T 1 + δ T 11 1 T δ 11 + T 1 δ 1 C 11 1 T 11 + T C 11 C 11 1 δ T 11 + δ 1 T 1 1 T 1δ 1 + T δ 11 C 11 1 T 11 + T T T 1 0 T 1 0 T 0 T 1 T T T 1 T T 1 T 1 T T T 1 1 T 0 T T 1 T 1 1 T + T 1 T 1 T 1 0 T 1 1T 1T 1 1 T 11 + T ,. following force follower force, load stiffness matrix.
98 99. load stiffness matrix, following force. load stiffness matrix,.,, δ W ext t δ uds Sṫ δ W ext Sṫ t u X X : : : : :X. :X., current configuration tds reference configuration, ds.,. t t, t t ds ds t ṫ ds ds + t ds ds , u, C t u D t u
99 100. t C t + D t u 1.81, self-adjoint. D t u D t u ds u, u Sṫ u u u on S u D t u, D t u., D t D t u, D t D t u 1.815, u u Δ u, D t D t Δ D t 1.816, 1.81, Sṫ D t Δ u Δ D t u ds Δ Δ u δ,. D t δ u δ D t u ds Sṫ D t u, δ Δ , D t δ uds δ D t uds Sṫ Sṫ δ W ext D t, D t δ uds 1 D t δ uds + Sṫ Sṫ Sṫ δ D t uds 1.80
100 1 δ Sṫ 101 D t uds 1.81, D t,. W ext Sṫ C t + 1 D t uds 1.8,, D t u δ Δ.,, D t u p. t ds. t pn ds ds pjf T N 1.8, Nanson nds detf F T NdS 1.84 n N F x i X j e i e j J detf : : : :
101 , t ṗjf T N pjf T N pjf T N 1.85 F T, İ F T F T,. F T F T + F T F T F T F T F T F T 1.87, J, L L Ḟ F J JtrL 1.89, J Jtr Ḟ F , 1.87, 1.80 t ṗjf T N pjtr Ḟ F 1 F T N + pjf T Ḟ T F T N , u. x x i e i t t ds t f. Δt t / t f t t t ds f t Δt ds / t f t ṫ t ds f t Δt ds.
102 10 trl u 1 + u + u x u u x x 1 x x 1.8 L u i e i e j u x x j 1.84 L T u i e j e i x u x j 1.85 C t ṗjf T N 1.86 D t pjtr Ḟ F 1 F T N + pjf T F T F T N 1.87, J 1, F I, N n, 1.86, C 0t ṗn 1.88 D 0t ptr F t t n + p F T t t n 1.89 p trl n + pl T n Ḟ u L 1.88, 1.87, D t, D 0t u D t δ u δ D t u ds Sṫ sṫ D t0 δ u δ D t0 u ds , , D 0t δ u δ D 0t u ds 1.84 Sṫ p [ δ u { trli + L } T u {δtrli + δl }] T nds sṫ sṫ 1.84 pn { x u δ u + δ u x u x δ u u u x δ u} ds
103 pn rot u δ u ds rot sṫ p u δ u dr C Stokes rotv nds v dr S C dr S C , load stiffness matrix sṫ C δ u 0. u δ u 0 u δ u. u δ u dr 4. C 1.17.,.
104 105 P P1, 1: P 1 C C 4: P C u z a u b u u 1/8, : a x dr y : b y z 1.17: 1..,, N, n., 8. N n N r 11 r 1 r N r 11 r 1 r N r 11+r 1 r 1.850
105 106 N r 11+r 1 r N r 11 r 1+r 1.85 N r 11 r 1+r 1.85 N r 11+r 1+r N r 11+r 1+r r r 6 4 r : X u. X r 1,r,r u r 1,r,r 8 N n X n X n n1 8 N n u n n1 r 1,r,r 1 1, ,, r 1
106 N n 0 n N r 11 r N r 11 r N r 11+r N r 11+r ,.,, 4. dr 1 dr, dr 1,dr dx 1 X dr 1 dx 1 x dr 1 dx X r dr 1.86 dx x r dr X x. load stiffness matrix NdS, nds. NdS dx 1 dx X X dr 1 dr r X i X j e ijk dr 1 dr e k r x nds dx 1 dx x dr 1 dr r
107 108 e ijk e ijk x i x j r dr 1 dr e k Hibbit [9]... δ W p Sṫ t δ uds 1.869, updated Lagrange C δ W p t 0 δ uds t 0 + D t 0 δ uds sṫ , { δ W p ṗn δ uds + p trl n + plt n } δ uds sṫ sṫ e ijk ṗ x j x k δ u i dr 1 dr 1.87 r D t 0.. { D t 0 kds p u1 x 1 + u x + u x k 1 [ D t 0 1ds p u1 x 1 + u x + u x sṫ x i x j e ijk + p u } c x a x b e abc dr 1 dr r x k r 1.87 x x x x r r
108 { u1 + p x 1 + u x 1 x x x x r r x1 x r x x 1 r + u x }] 1 dr 1 dr x x 1 x 1 x r r p u 1 x 1 x x r u x x x r u x x x r + u 1 x 1 x x r + u x x x r + u x x x r + u 1 x 1 x x r + u x 1 x x 1 r + u x 1 x 1 x r u 1 x 1 x x r u x 1 x 1 x r u x 1 x x 1 r + u x x u x x x r x r + u x x x r u x x x r dr 1 dr ,. { t 0 1ds p x1 u + x u + x u x x 1 x x r x1 u + + x u + x u x x 1 x x r x x1 u + x u + x u r x 1 r x r x + x x1 u + x u + x } u dr 1 dr r x 1 r x r x u i x 1 u i x 1 + x u i x + x u i x
109 110 u i r x 1 r u i x 1 + x r u i x + x r u i x t 0 1ds p u x + u x x u + x u dr 1 dr r r r r e ijk,. D 0t uj x k 1 ds pe 1jk + x j u k dr 1 dr r r D t 0, D t , { δ W p e ijk ṗ x j x k xj u k + p + u } j x k δ u i dr 1 dr r r r 1.880,., ṗ , δ W pii. δ W pii e ijk xj u k p + u j x k r r δ u i dr 1 dr ,. [K] [ δu n i T ] [K] [ ] u n i load stiffness matrix p e ijk xj u k r + u j x k r 1.88 δ u i 1.88
110 δ u 1 x u r x u r +δ u x u 1 r x 1 u r. 0 ] [δ x x x u 1 δ u δ u 0 x 1 x x 1 0. δ u 1 u x r u x r [ ] u [δ u] [A] r u x 1 +δ u u 1 x r r. 0 ] [δ u 1 δ u δ u x r 0 x r x r x 1 r x r x 1 r 0 x1 u +δ u x u 1 r r u 1 r u r u r u1 x +δ u u x 1 r r u 1 u u [ ] u [δ u] [B]
111 11 N n [ u]. u 1 N 1 N N 4 u N 1 N... N 4 u N 1 N N 4. [ u] 1 [ [N] 1 ] u n i 1 1 [δ u] [N]. [ ] u u 1 u u N 1 N N 4 N 1 N... N 4. [ ] [ ] u N [ N 1 N N ] u n i 1 1 u 1 1 u 1 u 1 u 1. u 4 1 u 4 u u 1 1 u 1 u 1 u 1. u 4 1 u 4 u
112 11, [ δ W pii [ δ u n i δ u n i [ [ δ u n i δ u n i [ ] [ ] u N [ r r 1 1 T ] [ N T ] [A] [ N r T ] [ N T ] [B] [ N ] { T [N T p ] [A] ] T [ ] [ K 1 L n n ] u n i [ ] N r ] u n i 1 1 ] [ ] u n i ] [ ] u n i, loadstifness matrix : [ K 1 L]. [ K 1 L] + [ N ] [ ]} N [ T [B] dr 1 dr ] u n i {α ij } [ ] [ ] N T N [A] r {α 11 } {α 1 }... {α 14 }. {α 1 }.. {α4 }. {α ij }. {α 41 } {α 44 } 0 N i x N j r N i x N j r N i x N j r 0 N i x 1 N i x N j r N i x 1 N j r 0 [ ] [ ] N T N [B] N j r {β 11 } {β 1 }... {β 14 }. {β 1 }.. {β4 }. {β ij }. {β 41 } {β 44 }
113 114 {β ij } 0 N i x r N j N i x N i x N j r 0 N i x 1 N i x r N j N i x 1 r N j 0 N j r N j r 1.90 x i, x i r. i 1,, N n x i N1 x 1 i + N x i Nn x n i 1.90 [ K 1 L].
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72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G
第5章 偏微分方程式の境界値問題
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β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t,
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k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m A f i x i B e e e e 0 e* e e (2.1) e (b) A e = 0 B = 0 (c) (2.1) (d) e
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第10章 アイソパラメトリック要素
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構造と連続体の力学基礎
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e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6
W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
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II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w
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5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0
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