n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz

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1 1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 (1.4) a n1 x 1 + a n2 x a nn x n = b n (1.4) [A] {x} = {b} (1.5) (1.4) n a ij x j = b i (i = 1, 2, n) (1.6) j=1 1

2 n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz = 1( v 2 z + w ) y γ zx = 1( w 2 x + u ) z γ xy = 1( u 2 y + v ) x (1.8) ε ij = 1 2 (u i,j + u j,i ) (1.9) i, j x, y, z, i x i f x, f y, f z σ x x + τ xy y + τ xz z + f x = 0 τ yx (1.10) x + σ y y + τ yz z + f y = 0 (1.10) τ zx x + τ zy y + σ z z + f z = 0 σ ij,j + f i = 0 (1.11) λ, µ σ x = (2µ + λ)ε x + λε y + λε z σ y = λε x + (2µ + λ)ε y + λε z (1.12) σ z = λε x + λε y + (2µ + λ)ε z τ xy = 2µγ xy τ yz = 2µγ yz τ zx = 2µγ zx σ ij = E ijkl ε kl (1.13) E ijkl = λδ ij δ kl + µ(δ ik δ jl + δ il δ jk ) δ ij 2

3 2 u i ε ij ε ij = 1 2 (u i,j + u j,i ) (2.1) σ ij,j + f i = 0 (2.2) (2.2) u i (σ ij,j u i )d + (f i u i )d = 0 (2.3) (2.3) 1 Gauss (σ ij ε ij)d = (P i u i )d + (f i u i )d (2.4) ε ij = 1 2 (u i,j + u j,i) (2.5) P i = σ ij n j (2.6) n j (2.4) α i u αi u i Φ α u i (2.8) (2.5) (2.4) u αi (Φ α,j σ ij )d = u αi u i = Φ α u αi (2.7) u i = Φ α u αi (2.8) 2ε ij = Φ α,j u αi + Φ α,i u αj (2.9) (Φ α P i )d + u αi (Φ α f i )d (2.10) (2.10) (2.10) u αi (Φ α,j σ ij )d = Ω αi (2.11) Ω αi = (Φ α P i )d + 3 (Φ α f i )d (2.12)

4 (2.11) (2.11) Ω αi P i f i Φ α (2.2) (Φ α,j σ ij )d = Ω αi (2.13) Ω αi = (Φ α P i )d + (Φ α f i )d (2.14) 3 (2.11) (2.14) (2.11) (2.14) W σ ij = W ε ij (3.1) W W = 1 2 E ijklε ij ε kl (3.2) E ijkl 21 2 E ijkl = λδ ij δ kl + µ(δ ik δ jl + δ il δ jk ) (3.3) λ, µ E, ν λ = (3.2) (3.3) (3.5) (2.7) νe (1 2ν)(1 + ν) µ = E 2(1 + ν) (3.4) σ ij = E ijkl ε kl (3.5) σ ij = E ijkl 1 2 (Φ β,lu βk + Φ β,k u βl ) = E ijkl Φ β,l u βk (3.6) (2.11) Ω αi u βk K αiβk u βk = Ω αi (3.7) K αiβk = (Φ α,j E ijkl Φ βl )d (3.8) Ω αi = (Φ α P i )d + (Φ α f i )d 4

5 K αiβk (3.7) 4 σ ij ε ij ε ij ε ij. ε ij ε ij = ε ij + ε ij (4.1) f σ ij ε ij π f(σ ij, ε ij) = π (4.2) i ( ε ij = 0) f < π, π = 0 (4.3) ii (ε ij 0) ii ( ε ij = 0) f = π, π 0, f = π, π = 0, f σ ij σ ij > 0 (4.4) f σ ij σ ij 0 (4.5) (4.5) 3 (4.4) ε ij ε ij = Λ f σ ij (4.6) (3.5) (4.2) π ε ij f σ ij + f σ ij ε ij σ ij = E ijkl ε kl (4.7) ε ij = π ε ij ε ij (4.8) (4.1) (4.6) (4.7) (4.8) Λ (4.1) (4.6) (4.7) σ ij = C ijkl ε ij (4.9) C ijkl = E ijkl E ijpq ( f σ pq ) ( f σ rs ) Erskl ( f σ mn )( π ε ij 5 ) f f + E ε pqmn mn σ pq

6 (4.9) (4.4) i σ ij = D ijkl ε kl f < π, π = 0 D ijkl = E ijkl ii f = π, π 0, iii f = π, π = 0, f σ ij σ ij > 0 f σ ij σ ij 0 D ijkl = C ijkl D ijkl = C ijkl (4.10) (2.7) σ ij = D ijkl Φ β,l u βk (4.10) (2.13) K αiβk U βk = Ω αi (4.11) K αiβk = (Φ α,j D ijkl Φ β,l )d Ω αi = (Φ α P i )d + (Φ α f i )d D ijkl σ ij ε ij σ ij, ε ij 5 σ ij = t 0 G ijkl (t τ) ε kl τ G ijkl Maxwell dτ (5.1) G ijkl = G 0 δ ij δ kl + (δ ik δ jl + δ il δ jk ) N n=1 G n exp ( t τ τ n ) G n n τ n (2.7) ε ij τ = 1 ( U αi U ) αj Φα,j + Φ α,i 2 τ τ (5.2) (5.3) 6

7 (5.1) σ ij = t 0 G ijkl (t τ) U βl τ dτ Φ β,k (5.4) 0 t U i (τ) = t2 τ 2 t 2 U i (0) + τ 2 U i (t) τ = t2 2τ t 2 t 2 U i(t) + τ(t τ) t U i (0) + 2τ t U i(t) + t 2τ 2 t U i (0) (5.5) U i (0) (5.6) (5.4) σ ij = H ijkl Φ β,l U βk σ ij (5.7) H ijkl = σ ij = + t 0 t 0 t (5.9) (2.11) 0 2G ijkl (t τ) τ 2 t 2 dτ G ijkl (t τ) t2 2τ t 2 dτ Φ β,k U βl (0) G ijkl (t τ) t 2τ t dτ Φ β,k U βl (0) K αiβk U βk = Ω αi + Ω αi (5.8) K αiβk = (Φ α,j H ijkl Φ β,l )d Ω αi = (Φ α,j σij)d 0 t (5.1) Q α ij = A αβ ijkl qβ kl + Bαβ ijkl + ġβ kl (5.9) i, j, k, l = 1, 2, 3 α, β = 1, 2, L Q 1 ij = σ ij, q 1 kl = ε kl Q α ij = 0(α 1), q β kl = hβ kl (β 1) L h β kl (β 1) A αβ ijkl, Bαβ ijkl 7

8 t t ε ij q ij α = qα ij qij(0) α (5.10) t (5.10) (5.9) σ ij = K (1) ijkl ε kl + K (2)α ijkl h α kl + H (1) ijkl ε kl(0) + H (2)α ijkl h α kl(0) (5.11) K (2)α ijkl ε ij + K (3)αβ ijkl h β ij + H (2)α ijkl K (1) ijkl = A (1) ijkl + 1 t B(1) ijkl, ε ij (0) + H (3)αβ ijkl h β ij(0) = 0 (5.12) H ijkl = 1 t B(1) ijkl, K (2)α ijkl = A (2)α ijkl + 1 t B(2)α ijkl, Hα ijkl = 1 t B(2)α ijkl, K (3)αβ ijkl = A (3) ijkl + 1 t B(3)αβ ijkl, H αβ ijkl = 1 β t B(3)α ijkl, (5.12) (5.11) h β ij σ ij = D ijkl ε kl σ ij (5.13) h β ij = (1) β ε ij (2) β ε ij (0) (3) αβ hβ ij(0) (1) β = K 1(3)αβ ijkl K (2)αβ ijkl, (2) β = K 1(3)αβ ijkl H (2)α ijkl, (3) αβ = K 1(3)αγ ijkl H (3)γα ijkl, D ijkl = K (1) ijkl + K(2)γ ijkl K 1(3)γδ ijkl K (2)δ ijkl, σ ij = ( H (1) ijkl K(2)γ ) ijkl γ (3) εkl (0) ) h γ kl (0), + ( H (2)γ ijkl K (2)δ ijkl (3) δγ (5.13) D ijkl t σij t (5.12) ε ij h α ij (5.13) (2.11) K αi = Ω αi = K αiβk U βk = Ω αi + Ω αi (5.14) (Φ α,j D ijkl Φ β,k )d (Ω α,j σ ij)d (5.14) (5.8) 8

9 6 Praqer ij e ij ij = σ ij σ kk 3 δ ij (6.1) e ij = ε ij ε kk 3 δ ij (6.2) f(σ ij, e ij = π (6.3) e ij e ij e ij e ij ε σ e ij = e ij + e ij (6.4) σ = σ kk 3 ε = ε kk 3 (6.5) (6.6) σ = 3K ε (6.7) ij = 2G e ij (6.8) K G (6.8) Ṡ ij = 2G(ė ij ė ij) = 2G(ė ij ε ij) (6.9) ε ij ε ij (6.9) ε ij = ε ij + ε ij (6.10) ε ii = 0 (6.11) Praqer ε ij ε ij = 1 2η 1 2 (1 kj2 ) ij (6.12) 9

10 J 2 J 2 = 1 2 ij ij (6.13) 2 η k (6.7) (6.9) σ ij = {µ(δ ik δ jl + δ li δ kj ) + λδ ij δ kl } ε kl λ = 1 3 (3K 2G) = νe (1 + ν)(1 2ν) E µ = G = 2(1 + ν) µ(δ ik δ lj + δ li δ kj ) ε kl (6.14) (6.14), (6.12), (6.13) (6.3) (6.11) (6.13) σ ij = E ijkl ε kl σ ij (6.15) σij = 2µ ε ij ε ij = (1 kj2 ) ij, 2η J 2 = 1 2 ij ij E ijkl = µ(δ ik δ jl + δ il δ jk ) + λδ ij δ kl (2.7) (6.15) (2.11) K αiβk U βk = Ω αi + Ω αi (6.16) k αiβk = (Φ α,j E ijkl Φ β,l )d Ω αi = (Φ α,j σij)d (6.16) (6.16) K αiβk Ω αi 7 (2.1) (2.2) ζ i θ ζ i = θ,i (7.1) 10

11 ε ρ ε = σ ij ε ij + q i,i + ρh (7.2) ρ q i h η Clasius-Duhem T ρr η q i,i ρh + 1 T q it,i 0 (7.3) φ σ (7.4), (7.5) (7.2) φ = ε ηt (7.4) σ = σ ij ε ρ( φ + ηt ) (7.5) ρ φ = σ ij ε ij ρη T σ (7.6) ρt η = q i,i + ρh + σ (7.7) (7.6) (7.7) (7.7) θ 3 (θ ρt 0 η)d + (θ,iq i )d = (θ q i η i )d + (θ ρh)d (7.8) T T 0 (2.7) θ θ α Φ α θ = Φ α θ α (7.9) (7.9) (7.8) α θα (2.11) (Φ α ρt 0 η)d + (Φ α q i n i )d = Γ α (7.10) Γ α = (Φ α q i n i )d + (Φ α ρh)d (7.11) φ φ = 1 2 E ijklε ij ε kl B ij ε ij θ ρc 2T 0 (7.12) σ ij = φ ε ij = E ijkl ε kl B ij θ (7.13) ρη = φ θ = B ijε ij + ρc T 0 θ (7.14) 11

12 E ijkl = λδ ij δ kl + µ(δ ik δ jl + δ il δ jk ) B ij = b δ ij b = E 1 2ν α λ µ α C q i q i = κζ i (7.15) κ (7.13), (7.14), (7.15) (2.7), (7.9) σ ij = E ijkl Φ β,l U βk B ij Φ β θ β (7.16) ρη = B ij Φ β,i U βj + ρc Φ β θ β T 0 (7.17) q i = κφ β,i θ β (7.18) (7.16) (2.11) K αiβk = T αiβ = K αiβk U βk T αiβ θ = Ω αi (7.19) Ω αi = (Φ α,j E ijkl Φ β,l )d (Φ α,j B ij Φ β )d (Φ α ρ i )d + (Φ α f i )d (7.17) (7.18) (7.10) (7.20) M αβ θβ + N αiβ U αi + J αβ θ β = Γ α (7.20) M αβ = N αiβ = J αβ = Γ α = (Φ α ρcφ β )d (Φ α T B ij Φ β,j )d (Φ α,i κ Φ β,i )d (Φ α ρh)d + (Φ α q i n i )d (7.19) (7.20) (7.20) U αi J αβ θ = Γ α (7.21) 12

13 8 x i x i, z i U i U i = z i x i (8.1) F ij F ij z i,j = δ ij + U i,j (8.2) Green ε ij U i (8.3) ε ij = 1 2 (U i,j + U j,i + U m,j U m,j ) (8.3) Kirchhoff ij ρ f i (F ij jk ),k + ρf i = 0 (8.4) (8.4) U i { (Fij jk ),k U i } d 0 (ρ 0 f i U i )d = 0 (8.5) (8.5) 1 Green-Gauss 0 ( ij ε ij)d = Ω (8.6) ε ij = 1 2 (U i,j + Uj,i + Um,iU m,j + U m,i Um,j) (8.7) [ ] Ω = ij n j (δ ki + U k,i )Uk d + (ρ 0 f i Ui )d 0 0 n j A 0 ij = E ijkl ε kl (8.8) α i U αi U i Φ α (8.9) (8.3) (8.7) U i = Φ α U αi (8.9) 2 ε ij = Φ α,i U αj + Φ α,j U αi + Φ α,i Φ β,j U αm U βm (8.10) 2ε ij = Φ α,i U αj + Φ α,j U αi + Φ α,i Φ β,j U αmu βm + Φ α,i Φ β,j U αm U βm (8.11) (8.8) (8.6) 0 (E ijkl ε kl ε ij)d = Ω (8.12) 13

14 (8.10) (8.11) K αiβj U βj = Ω αi (8.13) [ K αiβj = Φα,k (δ li + Φ δ,l U δi )E klmn (δ mj Φ ] γ,mu γj )Φ β,n d0 Ω αi = (ρ i Φ α )d + (ρ 0 f i Φ α )d 0 0 P k = ij (δ ki + Φ β,i U βk )n j (8.13) Newton-Raphson 14

all.dvi

all.dvi 29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan