k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x

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1 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 0 12

2 (e) e (f) (e) (f) non-linear deformation S. Karato, Deformation of Earth Materials, Cambridge University Press (c) (2.1) (c) 3 3 ij = C ijkl e kl = C ijkl e kl (i, j = 1, 2, 3 or x, y, z) (2.2) k=1 l=1 k C ijkl (1.14) (1.23) (2.2) i j k l 3x3x3x3= ij kl (C ijkl = C klij ) U d Q p V (2.3) dq d Q - T (2.3) pdv p (1.9) (1.25) (1.9) ij = pδ ij (1.25) dv/v = de ii pdv ij de ij 13

3 traction t i Aki and Richards, Quantitative Seismology, 2.2 (2.3) (2.4) U(S, V ) (2.4) ( ) U T =, (2.5) S e ij S (a) e= ΔL /L L Δt v = L /Δt L O (b) T (2.4) (2.6) T ( ) F S =, (2.7) T e ij [- ] (2.5) (2.7) C p C V K S K T K S /K T = C p /C V 1+γαT α T 1 γ Mie-Grunneisen Karato(2008) W U F ( ) W ij = (2.8) e ij 14 S or T

4 W strain energy (2.2) C ijkl = ij e kl = 2 W e kl e ij = 2 W e ij e kl = kl e ij = C klij ij kl = C ijkl = C klij W e ij = 0 W = 0 (2.8) ij (2.9) (2.10) e ij e ij (2.11) x k kx 2 /2 (2.10) k > (23.2) k ij x i x j /2 k ij = k ji 2.2 Elastic constants for isotropic media C ijkl k C ijkl anisotropy x xx (a) x 15

5 xx y x e y y e xx <0 >0 e =0 xy xx e, e y y, e =0 xx xy (a) (b) e xx y z e yy e zz e xy (a) Karato (2008) xy z z z monoclinic yz z x x orthorhombic z hexagonal z xy (a) transverse isotropy 16

6 C ijkl (a) Lamé constants λ µ (2.10) (2.12) Einstein l, i, k C ijkl C ijkl = λδ ij δ kl + µ (δ ik δ jl + δ il δ jk ) (2.13) (2.14) f i = kx i 2.4 e ik = 0 (2.12) (2.14) µ [- ] u i (b) Young modulus and Poisson s ratio (b) (a) x xx x e xx > 0 (2.15) E x (y z ) E x y z ν (2.16) 17

7 [- ] (2.14)-(2.16) E ν (c) bulk modulus and shear modulus bulk modulus incompressibility K (2.17) K 2.5 (2.7) S T 1.9 (1.25) (2.14) p = 1 3 ( xx + yy + zz ) = 3λ + 2µ 3 (e xx + e yy + e zz ) = (2.17) 3λ + 2µ V 3 V (2.18) µ (a) µ K µ free oscillation (λ, µ) (E, ν) (K, µ) (c) p p S z ρ z g p(z) z p(z) z + z p(z + z) ρgdv = ρgs z (2.19) 1.9 (c) V/V 18

8 2.3 Elasto-dynamic equation md 2 x i /dt 2 = f i = kx i m x, y, z ρ V = x y z u = (u x, u y, u z ) u(x, y, z, t) z Δx Δz Δy xx xy xz x y y x xx xy xz x Δx y x ρ V 2 u x t 2 = ρ 2 u x t 2 x y z f = (f x, f y, f z ) f x V = f x x y z traction A B x x ij i x j x xx A x (x + x, y, z) y z xx (x + x, y, z, t) y z B x (x, y, z) xx (x, y, z, t) y z xx (x + x, y, z, t) y z xx (x, y, z, t) y z ) ( xx (x, y, z, t) + xx x x y z xx (x, y, z, t) y z = xx x x y z 19

9 y x yx (x, y + y, z, t) yx (x, y, z, t) x z yx (x, y + y, z, t) x z yx (x, y, z, t) x z yx y x y z z zx zx (x, y, z + z, t) x y zx (x, y, z, t) x y zx z x y z = ( ) xx x y z + f x x y z ρ 2 u x t 2 x y z = x + yx y + zx z y z (2.20) (2.21) (2.22) (2.23) elasto-dynamic equation homogeneous2.14 θ (1.25) 2 Laplacian [- ] 2.24 θ e xx + e yy + e zz = u x x + u y y + u z z = u j x j = u j,j 2 2 x y z 2 (2.24). Seismic wave velocities P S P (2.24) 20

10 Aki-Richards [- ] (a) period T (b) frequency f (c) velocity v (d) wavelength λ (e) angular frequency ω (f) wavenumber k k 1/( ) f ω k ω [- ] t x f(x, t) x v f(x, t) =f(t x/v) x f(x, t) P S decoupled x v f(t x/v) ω monochromatic ω ω cos(ωt) x v cos(ω(t x/v)) cos sin e iθ = cos θ + i sin θ (2.25) { ( )} (2.26) P x wavefront amplitudep x (a) x u =(u x,u y,u z ) (u, v, w) u y x 21

11 P A α (2.26) u(x, t) = v(x, t) = w(x, t) = (2.27) (2.24) f i f i (2.27) (2.24) u i =1 θ = u x + v y + w z = u ( x = iω ) { ( A exp iω t x )} α α ( θ x = A iω ) 2 { ( exp iω t x )} α α ( 2 u = 2 u x u y u z 2 = 2 u x 2 = A 2 u t 2 =(iω)2 A exp iω α ) 2 exp { ( iω t x )} α { ( iω t x )} α { ( (iω) 2 ρa exp iω t x )} ( = (λ + µ)a iω ) 2 { ( exp iω t x )} ( + µa iω ) 2 { ( exp iω t x )} α α α α α A exp( ) ρω 2 = λ + µ α 2 ω 2 µ α 2 ω2 P α (2.28) S (b) y S β P, B u(x, t) = v(x, t) = (2.29) w(x, t) = (2.24) S (2.30) λ µ α >β P P λ µ α 3β 1.7β [- ] S θ 22

note1.dvi

note1.dvi (1) 1996 11 7 1 (1) 1. 1 dx dy d x τ xx x x, stress x + dx x τ xx x+dx dyd x x τ xx x dyd y τ xx x τ xx x+dx d dx y x dy 1. dx dy d x τ xy x τ x ρdxdyd x dx dy d ρdxdyd u x t = τ xx x+dx dyd τ xx x dyd