饱和土中的任意形状孔洞对弹性波的散射
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摘要
根据Biot波动理论建立了求解饱和土中任意形状孔洞对弹性波散射问题的复变函数方法.首先通过引入位移势函数把稳态条件下的Biot波动方程解耦为势函数所满足的Helmholtz方程.利用分离变量方法即得到Helmholtz方程完备的通解.根据所得位移势函数的通解,可得骨架位移、流体相对骨架的位移、应力和孔压的表达式.通过保角变换方法,把物理平面上的孔洞映射到像平面上单位圆.利用土骨架和流体的边界条件,即可确定波函数展开式中的未知系数.给出了一些数值结果.
In terms of Biot's dynamic theory, the complex variable method used to solve the problem of the scattering of elastic waves by holes of arbitrary shapes in saturated soil is estab-lished in the paper. The steady state Biot dynamic equations are decoupled by introducing the displacements potentials and reduced to Helmholtz equations that the potentials satisfy. Solving the Helmholtz equations by separating variable methods leads to the complete solutions of the potentials. By using the solutions of the potentials, the expressions of the displacements of solid matrix, the displacements of fluid relative to the solid matrix, the stresses of the bulk material and the pore pressure can be obtained. Through the conformal mapping method, the hole in physical plane can be mapped into unit circle in mapped plane. Utilizing the boundary conditions of the solid matrix and the fluid, the unknown coefficients in the potentials can be determined. Some numerical results are given in the paper.
In terms of Biot's dynamic theory, the complex variable method used to solve the problem of the scattering of elastic waves by holes of arbitrary shapes in saturated soil is estab-lished in the paper. The steady state Biot dynamic equations are decoupled by introducing the displacements potentials and reduced to Helmholtz equations that the potentials satisfy. Solving the Helmholtz equations by separating variable methods leads to the complete solutions of the potentials. By using the solutions of the potentials, the expressions of the displacements of solid matrix, the displacements of fluid relative to the solid matrix, the stresses of the bulk material and the pore pressure can be obtained. Through the conformal mapping method, the hole in physical plane can be mapped into unit circle in mapped plane. Utilizing the boundary conditions of the solid matrix and the fluid, the unknown coefficients in the potentials can be determined. Some numerical results are given in the paper.
引文
1Savin GN. Stress Concentration Around Holes. New York: Pergamon Press, 1961
2Pao YH. Diffraction of Elastic Wave and Dynamic Stress Concentrations. New York: Crane, Russak, 1973
3Manolis GD, Beskos DE. Boundary Elements Methods in Elastodynamics. London: Unwin Hyman, 1988
4Pao YH, Varathatajulu V. Principle condition and integral formulas for the scattering of elastic waves. J Acoust Soc Am, 1976, 59: 1361~1371
5Achenbach JD, Gautesen AK, McMaken H. Ray Methods for Waves in Elastic Solids. Boston, London: Pitman Advanced Publishing, 1982
6Liu DK, Gai BZ, Tao GY. Applications of the method of complex functions to dynamic stress concentrations. Wave Motion, 1982, 4: 293～304
7Zimmerman C. Scattering of plane compressional waves by spherical inclusions in a poroelastic medium.J Acoust Soc Am, 1993, 94(1): 527~536
8胡亚元,王立忠,陈云敏等 。饱和土中平面应变波在圆柱体上的散射和折射 ,地震学报,1998,ZO（3）:300～307（Hu Yayuan, Wang Lizhong, Chen Yunmin, et al. Scattering and diffracting of plane strain wave by circular cylinder in saturated soil. Acta Seismologica Sinica, 1998, 20(3): 300～307 (in Chinese))
9Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid, I, low frequency rang. J Acoust Soc Am, 1956, 28(2): 168～178
10Biot MA. Mechanics of deformation and acoustic propagation in porous media. J Appl Phys, 1962, 33(4): 1482～1498
11Biot MA. General theory of three dimensional consolidation. J Appl Phys, 1941, 12:155～164
2Pao YH. Diffraction of Elastic Wave and Dynamic Stress Concentrations. New York: Crane, Russak, 1973
3Manolis GD, Beskos DE. Boundary Elements Methods in Elastodynamics. London: Unwin Hyman, 1988
4Pao YH, Varathatajulu V. Principle condition and integral formulas for the scattering of elastic waves. J Acoust Soc Am, 1976, 59: 1361~1371
5Achenbach JD, Gautesen AK, McMaken H. Ray Methods for Waves in Elastic Solids. Boston, London: Pitman Advanced Publishing, 1982
6Liu DK, Gai BZ, Tao GY. Applications of the method of complex functions to dynamic stress concentrations. Wave Motion, 1982, 4: 293～304
7Zimmerman C. Scattering of plane compressional waves by spherical inclusions in a poroelastic medium.J Acoust Soc Am, 1993, 94(1): 527~536
8胡亚元,王立忠,陈云敏等 。饱和土中平面应变波在圆柱体上的散射和折射 ,地震学报,1998,ZO（3）:300～307（Hu Yayuan, Wang Lizhong, Chen Yunmin, et al. Scattering and diffracting of plane strain wave by circular cylinder in saturated soil. Acta Seismologica Sinica, 1998, 20(3): 300～307 (in Chinese))
9Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid, I, low frequency rang. J Acoust Soc Am, 1956, 28(2): 168～178
10Biot MA. Mechanics of deformation and acoustic propagation in porous media. J Appl Phys, 1962, 33(4): 1482～1498
11Biot MA. General theory of three dimensional consolidation. J Appl Phys, 1941, 12:155～164
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