( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =

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1 ; C. H. Scholz, The Mechanics of Earthquakes and Faulting : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ = λ/ σ t sin πr a dr a = λσ t λ π σ t E/π γ Ea/4π 5-1 GPa defect 4 crack brittle dislocation plastic ; : schizosphere plastosphere 1

2 1.1. Griffith : σ B. 3σ b, c c/b σ σ t σ σ 1 + c/b 5 c b σ σ 1 + c/ρ σ c/ρ 6 U U = W + U e + U s 7 W : ; U e : ; U s : du dc = 8 : U e = 1 σ E yσ 1 y = 9 E E : G I = K I E, K I = πc σ... G I = πσ c E c Π = G I dc = πσ c E U e = σ y + πc E E = ye y + πc 1 σ W = σy E σ = πσ c E E 13 U s = 4cγ 14

3 11, 13, U = πσ c E σ t = 4Eγ πc + 4cγ Griffith? 3, 4 λ σ t = Eγ a ρ a σ f σ t = σ f c a 18 17, 18 σ f = Eγ 4c Obreimoff Griffith Obreimoff W = C. U e = Ed3 h 8c 3 U s = cγ du/dc = : 3Ed 3 h 1/4 c = 1 16γ I :, 3

4 II : in-plane III : antiplane near field D. σ ij = K n / πrf ij θ Kn r u i = E π f iθ 3 r θ 1.6 K n : stress intensity factor; K I, K II, K III 3 Griffith energy release rate G 7 G d W + U e 4 dc E. G : G I = K I E E = G II = K II E 5 G III = 1 + ν E K III { E E/1 ν 6 7, 8 du dc = d W + U e + du s dc dc = = G + γ = 7 I G Ic = K Ic E = γ 8 G c K c critical F. K I = σ yy πc K II = σ xy πc 9 K III = σ zy πc 4

5 5 G I = πc E σ yy G II = πc E σ xy 3 G III = 1 + ν E πcσ zy 5, 9 G c = Γ 31 8 Γ lumped parameter stress free / : σ 1 > σ > σ 3 tensile σ 1 = fσ, σ 3 3 σ 3 = T 33 T Coulomb τ = τ + µσ n 34 µ ; tan φ φ 1.8 Mohr Mohr σ σ 1 + σ 3 + τ = σ1 σ

6 θ = π 4 φ 36 σ 1 µ = tan φ = BO AB BO AB σ 1, σ 3, τ, µ { } { } σ 1 µ + 1 µ σ 3 µ µ = τ σ 1 -σ 3 σ 1 { } C = τ µ µ : { } { } [ ] σ 1 µ + 1 µ σ 3 µ µ = τ σ1 C 1 C T /4τ [ ] σ 3 = T σ1 < C 1 C T /4τ 4 Griffith? Coulomb-based ; : Griffith tensile crack : : I. σ 1 σ 3 8T σ 1 + σ 3 = σ 1 3σ 3 σ 3 = T σ 1 < 3σ 3 41 Mohr τ = 4T σ n + T 4 cos θ = 1 σ 1 σ 3 /σ 1 + σ 3 43 Griffith Griffith 41 Griffith McClintock and Walsh 196 J. 6

7 : normal stress σ y σ c ; normal stress σ n = σ y - σ c ; stress τ f = µσ n : { 1 µ µ} σ 1 σ 3 = 4T 1 + σc /T + µσ 3 σ c 44 Mohr : τ = T 1 + σc /T + µσ n σ c 45 Coulomb σ 1 -σ 3 τ σ n σ c { 1 µ µ} σ 1 σ 3 = 4T + µσ 3 46 τ = T + µσ n 47 τ = T Coulomb µ A. H. Cottrell, The Mechanical Properties of Matter, Krieger, 1981.,,, 1999.,,, J. C. Jaeger, N. G. W. Cook, Fundamentals of Rock Mechanics, 3rd ed., Chapman and Hall, A. Airy Airy stress function: U χ σ 11 = U x Compatibility equation plane stress, σ = U x, σ 1 = U 48 1 x 1 x U ε 1 x 1 x = ε 11 x + ε x 1 x x U = 5 B. σ rr σ rθ r + 1 r θ σ rθ r + 1 r σ θθ θ + σ rr σ θθ r + σ rθ r = 51 = 5 7

8 Airy 51, 5 σ rr = 1 r U r + 1 U r θ σ rθ = 1 U r θ 1 U r r θ σ θθ = U r 53 Compatibility r + 1 r r + 1 r θ U = 54 θ σ rθ, θ : d 4 U dr 4 + r dσ rr dr d 3 U dr 3 + σ rr σ θθ r 1 r d U dr U = A ln r + Br ln r + Cr + D = 55 + r 3 du dr = 56 σ rr = A + B1 + ln r + C 57 r σ θθ = A + B3 + ln r + C r σ rθ = A = B = r = B dislocation B = σ rr = A r + C σ θθ = A + C 58 r σ rr = σ cos θ = 1 σ1 + cos θ σ rθ = σ sin θ cos θ = 1 σ sin θ 59 σ θθ = σ sin θ = 1 σ1 cos θ θ : σ rr = 1 σ, σ rθ =, σ θθ = 1 σ σ rr = 1 σ cos θ, σ rθ = 1 σ sin θ, σ θθ = 1 σ cos θ 6 8

9 r : σ rr = σ rθ = σ rr = σ rθ = r = r 61 σ θ 58 σ rr = σ σ θθ = σ 1 r r 1 + r r 6 σ U = fr cos θ σ rr = 1 r cos θ { r dfr dr } 4fr } σ rθ = { r sin θ r dfr fr dr σ θθ = cos θ d fr dr 59 θ 54 d dr + 1 r d dr 4 r fr = 64 fr = Er + F r 4 + G r + H 65 E = σ 4, F =, G = r4 4 σ, H = r σ 66 σ σ : σ rr = σ σ rθ = σ σ θθ = σ r = r { 1 r r + 1 3r4 { 1 + r r θ = ±π/ σ θθ = 3σ 1 + 3r4 r 4 + r r 1 + 3r4 r 4 cos θ } r 4 4r r sin θ 67 } cos θ σ θθ = σ1 cos θ 68 C. 1.4 M Mc = F c 69 9

10 ρ + y ρ ε = y ρ = 1 + εdx dx 7 71 y σ = Eε = E y ρ 7 M M = σyda 73 = A y da = EI ρ ρ 74 I I A y da I = = A d/ 1 d/ d/ d/ y dx dy 75 y dy 76 = d 3 / ρ = M EI σ = My I 1 ρ = d y dx 8 78 EI d y = Mx 81 dx Mx = F x y = F 6EI x3 3c x + c 3 8 y y max = F c3 3EI = h 83 F = 3EIh c u = 1 σε = 1 E My 85 I 1

11 U e = V udv = = c c = F EI A M M EI y dadx EI dx c x dx = F c 3 6EI = 9E I h c 6 c 3 6EI = 3EIh c 3 = Eh d 3 8c 3 86 D., Goursat z Ux, y = x Re φz + y Im φz + Re ψzdz 87 σ y + σ x / = φ z + φ z = Re φ z σ y σ x / + iτ xy = zφ z + ψ z 88 Gu + iv = κφz zφ z ψz 89 κ = { 3 4ν 3 ν/1 + ν 9 Gw = Re ζz 91 Westergaard Z I z = φ Iz Z II z = iφ IIz 9 d ZI z dz = Z I z 93 Ux, y = Re ZI z + y Im Z I z y Re Z II z 94 σ x σ y = ReZ I ReZ I y Im Z I +y Im Z I + Im Z II +y Re Z II y ReZ II 95 τ xy y Re Z I Re Z II y Im Z II 11

12 G u v = κ 1 Re Z I y Im Z I κ+1 Im Z I y Re Z I + κ+1 Im Z II +y ReZ II κ 1 Re Z II y ImZ II 96 Gω = κ + 1 Im Z I Re Z II 97 Z III z = iζ z 98 τ xz iτ yz = ζ z 99 τ xz τ yz = { Im Z III z Re Z III z } 1 z = x + iy σ x = σx, σ y = σy, τ xy = τxy Goursat 88 z A, B, α, β φz = Az + α, ψz = Bz + β φz = A z, ψz = B + ib z, 88 σ y + σ x = A, A, B, B φz = σ y + σ x 4 Westergaard σ y σ x + iτ xy = B + ib σ y σx z, ψz = + iτ xy z 11 Z I z = σ y, Z II z = τ xy + iσ y σ x / 1 III τ xz = τxz τ yz = τyz ζz = τxz iτyz z, ZIII z = τyz + iτxz 13 I II λ n n =, ±1, ±,... A n = A In + ia IIn, B n = B In + ib IIn φz = n ψz = n A n z λ n, 14 B n z λ n 15 1

13 88 φ z = n φ z = n φ z = n ψ z = n A n λ n r λn 1 e iλn 1θ A n λ n r λn 1 e iλn 1θ A n λ n λ n 1r λn e iλ n θ B n λ n r λn 1 e iλ n 1θ 88 1 σ x σ y + iτ xy = n λ n r λ n 1 [A n + B n + λ n 1A n e iθ e iλ n 1θ + A n e iλ n 1θ ] θ = π, π r σ y = τ xy = A n, B n e iπλ n e iπλ n = λ n A n + B n e iπλ n + A n e iπλ n = 16 λ n A n + B n e iπλn + A n e iπλn = 17 e iπλ n e iπλ n sin πλ n = λ n = n/ n =, ±1, ±,... n 17 λ n = n/ n = 1,, B n = n A n 1 n A n = 88 σ x σ y τ xy = n A In n=1 n=1 r n 1 n A IIn r n 1 { } + 1 n + n cos n 1 θ n 1 cos n { 3 } 1 n n cos n 1 θ + n 1 cos n 3 { } 1 n + n sin n 1 θ + n 1 sin n 3 { } 1 n + n sin n 1 θ n 1 sin n 3 { } + 1 n n sin n 1 θ + n 1 cos n 3 { } 1 n n cos n 1 θ n 1 cos n u = v n=1 AIn n=1 [ r n G AIIn G r n κ cos n θ n cos n θ + { n + 1n} cos nθ κ sin n θ + n sin n θ { n + 1n} sin nθ [ κ sin n θ n sin n θ + { n 1n} sin nθ κ cos n θ n cos n θ { n 1n} cos nθ ] ] 11 A I1 = K I / π, A II1 = K II / π 111 n = 1 13

14 I σ x σ y = K I cos θ πr 1 sin θ sin 3θ 1 + sin θ sin 3θ 11 τ xy u v = K { I r G π sin θ cos 3θ cos θ κ 1 + sin θ sin θ κ + 1 cos θ } 113 II σ x σ y τ xy u v = K II πr = K II r G π sin θ + cos θ cos 3θ sin θ cos θ cos 3θ cos θ 1 sin θ sin 3θ { } sin θ κ cos θ cos θ κ 1 sin θ III ζz = n C n z λn 116 { τ xz n = C nr n/ 1 sin n/ 1θ cos n/ 1θ τ yz n=1,3,5,... } + n=,4,6,... { n C nr n/ 1 cos n/ 1θ sin n/ 1θ } 117 C1 = /πk III near field n = 1 τ xz τ yz = K III πr sin θ cos θ 118 w 91 w = K III r G π sin θ 119 E. 4.4a δa δa a c GδA : 1. δa δa.. y = θ = K I A 11 σ y x = K IA πx 1 14

15 δb x δa δa = δbδa K I = K I A + δa 113 θ = ±π, r = δa x vx = ± κ + 1 δa x G K IA + δa π c x σ y x vx x = δa sin θ δa δa σ y v GδBδa = δb δbdx = δb π K IAK I A + δa κ + 1 G δa x dx = x δa δa x dx x [ ] xx δa x δa + δa arcsin x + δa = π δa 1 GδBδa = κ + 1 8G δbδak IAK I A + δa 13 δa G = 1 E K I, G = κ + 1 8G K I 14 { E E = E/1 ν G = lim δa δa 3 δa σ y v + τ xy u + τ yz w dx 16 G = 1 K E I + KII 1 + G K III 17 G = G I + G II + G III, 18 G I = K I E, G II = K II E, G III = 1 + ν E K III F. Z I z Z II z Z III z = σ y τ xy τ yz z 19 z a 15

16 Westergaard Z I z Z I z Z I z = σ y z a 1/ 13 Z Iz = σ y a z a 3/ 131 z = re iθ, z a = r 1 e iθ1, z + a = r e iθ Z I z σ y = re iθ r 1 e iθ1 r e iθ 1/ = = r r1 r e i{θ θ 1+θ /} r [cos θ θ 1 + θ + i sin θ θ ] 1 + θ r1 r 13 Z I z σ y = a ei{3θ1+θ/} r 1 r 3/ = a r 1 r 3/ [ cos { 3 θ 1 + θ } { }] 3 i sin θ 1 + θ 133 y = r sin θ 95 cos θ θ 1 + θ a sin θ sin 3 σ x r σ y = σ y r 1 r θ 1 + θ cos θ θ 1 + θ + a sin θ sin 3 r1 r r 1 r θ 1 + θ τ xy a sin θ cos 3 r 1 r θ 1 + θ 134 Z I z = r 1 r e iθ 1+θ / = ] θ1 + θ θ1 + θ r 1 r [cos + i sin σ y : G u v = σ y κ 1 r1 r κ + 1 cos θ 1 + θ sin θ 1 + θ r sin θ sin θ θ 1 + θ r 1 r r sin θ cos θ θ 1 + θ r 1 r 136? r a, r 1 = r = r, θ 1 = θ = θ σ y = σ x = σ y, τ xy = θ =, ±π, θ 1 + θ = ±π = 16

17 19 Z I z σ y = σ x = σy 19 Z II z, Z III z τ xy = τxy, τ yz = τyz 11, 114, 118 σ z = ν σ x + σ y 137 x θ = σ x 1 σ y 1 σ z τ xy τ yz τ xz θ= = K I πr ν + K II πr 1 + K III πr 1 σ z σ z = K I σ y, σ x K II = lim πr τ xy r K III τ yz θ= 138 K I K II K III = lim πx a x a + σ y τ xy τ yz 139 σ x = σx, τ xz = τxz, σ z = σz σ y, τ xy, τ yz σx, σy, τxy, τxz, τyz σz 1, 13 Z I z Z II z Z III z = z z a σ y τ xy τ yz + i σ x σ y / 95, 1 x σ x σ y τ xy τ xz τ yz x = z a σ y σ y τ xy τ yz + σ x τ xz σ y 139 τ xz K I = σ y πa, KII = τxy πa, KIII = τyz πa 14 σx, σz, τxz G. Curvilinear coordinates 17

18 : z = ωζ 143 z = x + iy, ζ = ξ + iη ζ P ξ, η z P x, y η = η const. PA dx + idy = dz = ω ζdζ = ω ζ[dξ + idη] 144 = Me iδ [dξ + idη] dη = P A Pξ e iδ = ω ζ/ω ζ 146 dy = tan δ 147 dx ξ = ξ const. z Pη P 145 dξ = dy = cot δ 148 dx ξ = ξ, η = η z σ x = σ x cos θ + τ xy sin θ cos θ + σ y sin θ 149 σ y = σ x sin θ τ xy sin θ cos θ + σ y cos θ 15 τ x y = 1 σ y σ x sin θ + τ xy cos θ , 15, 151 u = u cos θ + v sin θ v = v cos θ u sin θ 15 σ y σ x + iτ x y = σ y σ x + iτ xy e iθ 153 trace σ x + σ y = σ x + σ y u + iv = u + ive iθ , 89 [ ] σ ξ + σ η = σ x + σ y = φ z + φ z 156 σ η σ ξ + iτ ξη = σ y σ x + iτ xy e iδ [ ] = [ zφ z + ψ z] ω ζ/ω ζ

19 Gu ξ + iu η = Gu + ive iδ [ ] = κφz zφ z ψz ω ζ/ω ζ 158 φ z = dφ dζ dζ dz = 1 dφ ω ζ dζ 159 elliptic coordinates 143 z = x + iy = c cosh ζ = c coshξ + iη 16 x = c cosh ξ cos η y = c sinh ξ sin η 161 ξ = ξ const. x-y semi-axes ξ = x = c x = c slit η = η const. x y c cosh + ξ c sinh = 1 16 ξ a = c cosh ξ, b = c sinh ξ 163 x y c cos η c sin = η H. 161, 16, 163 ξ = ξ a Ox β p : z ζ φz = 1 4 p ce ξ cos β cosh ζ p c1 e ξ+iβ sinh ζ 165 ψz = 1 4 p c [ cosh ξ cos β + e ξ sinh ζ ξ iβ ] cosechζ 166 z = ωζ = c cosh ζ 167 dz dζ = ω ζ = c sinh ζ , 166 ξ = ξ 168 φ z = dφ dζ dζ dz = 1 4 p e ξ cos β p 1 e ξ+iβ coth ζ

20 σ t σ ξ =, σ t = σ η, 169, 156 σ t = {φ ξ + iη + φ ξ iη} = p e ξ cos β + 1 p 1 e ξ +iβ cothξ + iη + 1 p 1 e ξ iβ cothξ iη = p e ξ sinh ξ cos β + p cosh ξ cos η p e ξ cos β sinh ξ + sin β sin η cosh ξ cos η sinh ξ + cos β e ξ cos β η = p cosh ξ cos η ab + a b cos β a + b cos β η = p a + b a b cos η η x = r cos θ, y = r sin θ 161, 163 β = : tan θ = y x = tanh ξ tan η = b tan η 17 A θ = η = σ t = B θ = η = π/ σ t = 1 + b a p β = π/ A θ = η = σ t B θ = η = π/ σ t = p 1 + b a p flat elliptic crack ξ = α = cosh ξ cos η 1 156, 169 σ ξ + σ η = p cos β + αp {1 cos β sinh ξ sin β sin η} 173 σ ξ σ η = αp cosh ξ cos η β + α p {1 cos βcos η 1 sinh ξ cosh ξ cos β + cos η β cosh ξ sin β sin η} 174 τ ξη = 1 p α sinh ξ sin β η + 1 p α {sinh ξ sin βcos η cos βcosh ξ 1 sin η} u ξ u η 89 Gu + iv = κφz zφ z ψz 176 x-y ξ ξ flat crack crack 165, φz = 1 4 p c [ e ξ cosh ζ 1 + e ξ sinh ζ ] 177 ψz = 1 4 p c [ 1 + cosh ξ e ξ sinh ζ ξ ] cosechζ 178 φ z = 1 4 p [ e ξ 1 + e ξ coth ζ ] 179 8Gu + iv cp = κe ξ cosh ζ + κ1 + e ξ sinh ζ + [ e ξ 1 + e ξ coth ζ ] cosh ζ + [ 1 + cosh ξ e ξ sinh ζ ξ ] cosech ζ 18 p

21 flat crack ξ = ξ =, ζ = iη, cos η = x/c 18 v = [κ + 1p /4G] c x 181 O-y η = π/, ζ = ξ + π i, sinh ζ = i cosh ξ, cosh ζ = i sinh ξ, 18 u = ξ 8Gu + iv cp = κe ξ sinh ξ + κ1 + e ξ cosh ξ + [ e ξ 1 + e ξ tanh ξ ] sinh ξ + [ 1 + cosh ξ + e ξ sinh ξ ξ ] sech ξ 18 8Gu + iv cp = κ cosh ξ sinh ξ + 3 sinh ξ cosh ξ η = π/ sech ξ + ξ [κcosh ξ sinh ξ + 3 sinh ξ 3 cosh ξ + sech ξ] 183 sinh ξ = y/c, cosh ξ = 1 + y /c, 184 8Gu + iv cp = 3 κ y c + κ y /c + + ξ {3 κ y c + κ y /c y /c } 1 + y /c 185 κ = 3 ν/1 + ν v = cp νy + 1 ν 1 + y E c c ν + c ξ p ν 1 + y E c 1 + y c y c ν y c p y = b cξ v = cξ 185 y/c = ξ ξ p = 4Gξ κ β σ π + β σ 1 σ 1 cos β + σ sin β = 4Gξ κ I. Griffith : H. a = c cosh ξ, b = c sinh ξ σ 1, σ 1.13 σ x = σ 1 sin β + σ cos β σ y = σ 1 cos β + σ sin β 189 τ xy = 1 σ 1 σ sin β 19 1

22 17 σ 1, σ σ t = σ 1 + σ sinh ξ + σ 1 σ [ e ξ cos β η cos β ] 191 cosh ξ cos β 189, 19 σ t = σ y sinh ξ + τ xy [ 1 + sinh ξ cot β e ξ cos β ηcosec β ] cosh ξ cos β 19 ξ flat crack η A σ t = ξ σ y ητ xy ξ + η 193 η σ t 193 dσ t /dη = η τ xy ηξ σ y ξτ xy = 194 σ y ± σy + τxy or η = ξ τ xy σ t ξ σ t = σ y σy + τxy 196 σ t tensile tensile stress σ e η ξ σ e = σ y σy + τxy 197 = σ 1 cos β + σ sin β σ1 cos β + σ sin β 198 η ξ = σ y + σy + τxy 199 τ xy = σ 1 cos β + σ sin β + σ1 cos β + σ sin β σ σ 1 sin β 198 σ e σ 1, σ β 198 dσ e ξ dβ = σ1 σ σ σ 1 + sin β cos β 1 σ1 cos β + σ sin β β =, π/, cos β = 1 σ 1 σ /σ 1 + σ cos β < σ 1 + 3σ >. 3 σ e = σ 1 σ 4σ 1 + σ ξ 4

23 σ 3 198, 191 β = π/ σ e = σ ξ 5 tensile stress T 5, σ = T σ e = T /ξ 4 σ 1 σ 8T σ 1 + σ = σ 1 3σ σ = T σ 1 < 3σ 6 σ = σ 1 = C C = 8T 7 σ m = σ 1 + σ /, τ m = σ 1 σ / 6 τ m = 4T σ m if σ m > τ m τ m = σ m + T if σ m < τ m 8 BC Mohr σ m, τ m 8 1 σ σ m + τ = τ m σ σ m + τ = 4T σ m 9 fσ m = σ σ m + τ 4T σ m fσ m =, f σ m = σ σ m + T = 1 9, 1 τ = 4T σ + T 11 envelope J. Griffith I. 188 σ y = σ 1 cos β + σ sin β = 4Gξ κ

24 σ c σ n = σ y σ c τ f = µσ n 1.14a τ xy B, B τ f b,c σ n shear τ f σ y σ n = σ c shear stress τ xy + τ f = τ xy + µσ y σ c d b,c a d 193 σ t 189, 19 σ t = ξ σ c η {τ xy + µσ y σ c } ξ + η 13 = ξ σ c + ησ ξ + η 14 σ = σ 1 σ sin β µ cos β µσ 1 + σ σ c η σ t η = σ c ± 4σ c + σ ξ σ σ e : σ c ± 4σ c + σ ξ β σ e dσ /dβ = 15 tan β = 1 µ. 18 sin β = 1 µ + 1, cos β = µ µ σ e = T /ξ T 17 4σc + σ σ c = T [ ] σ 1 µ + 1 µ σ c or σ = 4T 1 + σc /T [ ] σ µ µ = 4T 1 + σc /T µσ c [ ] σ 1 µ + 1 µ reduce Coulomb 4 [ ] σ µ µ = 4T 1 4

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II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11