内部力源による半無限弾性体の変形について

抄録

Japanese seismologists succeeded in explaining the push-pull distribution of the initial motion of earthquakes by assuming two types stress distribution on the sphere which covers the hypocenter. Type A is the combination of hydrostatic pressure and pressure with distribution expressed in spherical harmonic P₂(cosθ). Type B is the distribution of pressure expressed in P₂¹(cosθ)cosφ. Generally the polar axis of these spherical harmonics does not coincide with vertical axis. On this point, Y. Sato obtained the formulae which express the transformation of the spherical harmonics by the rotation of coordinate system. According to his result,
P₂(cosθ₀)=P₂(cosθ)(1/4+3/4cos2χ)+P₂¹(cosθ)cos(φ-φ)(1/2sin2χ)+P₂²(cosθ)cos2(φ-φ)(1/8-1/8cos2χ)
P₂¹(cosθ₀)cosφ₀=[sinφA₂¹-cosφB₂¹]
where A₂¹=P₂¹(cosθ)sin(φ-φ)cosχ+P₂²(cosθ)sin2(φ-φ)(1/2sinχ)
B₂¹=P₂(cosθ)(3/2sin2χ)-P₂¹(cosθ)cos(φ-φ)cos2χ-P₂²(cosθ)cos2(φ-φ)(1/4sin2χ)
where (φ, φ, χ) is Euler angles which express the rotation of the coordinate.
In this paper, we calculated the strain produced in a semi-infinite elastic solid when hydrostatic pressure and pressure with distribution expressed in spherical harmonics P₂(cosθ), P₂¹(cosθ)cosφ, P₂²(cosθ)cos2φ were applied at the interior spherical cavity.
The deformations expressed in cylindrical coordinate (R, φ, z) at the surface of semiinfinite elastic solid are as follows:—
(1) The case in which hydrostatic pressure -P is applied
U_R=3a³P/4μR/(f²+R²)^3/2, Uφ=0, U_z=-3a³P/4μf/(f²+R²)^3/2
(2) The case of -PP₂(cosθ)
U_R=3a³P/46μ[-5P/(f+R^23/2)+18f²R/(f²+R²)^5/2], Uφ=0, U_z=-3a³P/46μ[-5f/(f²+R^23/2)+18f³/(f²+R²)^5/2]
(3) The case of -PP₂¹(cosθ)cosφ
U_R=54a³P/23μcosφ[-f((f²+R²)^3/2+f³(f²+R²)^5/2], Uφ45a⁵P/184μsinφf(R²+f²)^5/2, U_z=45a⁵P/184μsinφf(R²+f²)^5/2
(4) The case of -PP₂²(cosθ)cos2φ
U_R=9a³P/23μcos2φ[4f/R³-4f²/R³(R²+f²)^1/2+5R(f²+R²)^3/2-2f²/R(R²+f²)^3/2-6f²R