参数非均质球体的力学问题及其在地球物理学中的应用

The mechanical problems of sphere with inhomogeneous parameters and their applications in geophysics

周宇澄

11749216

硕士

航天工程

黄克服

2019-05-28

2019-07-20

哈尔滨工业大学

深圳

本文分别给出了非均质性球对称问题和空间轴对称问题的解析通解，构建了地球内部各物理和力学参数由简洁的函数关系所表达的弹性地球分层结构模型，并给出了模型的弹性力学解。 对于非均质性球对称问题的研究，推导了非均质性问题的位移控制方程。首先考虑了杨氏模量随半径指数分布的情形，给出了问题的解析解。其次考虑了杨氏模量随半径为一般线性规律变化的情形，采用常微分方程的幂级数解法给出了问题的级数解。文中对球对称万有引力的分布进行了计算，其中假设密度函数沿半径方向呈指数分布，并给出了有体力非齐次方程的解析位移解。对于非均质空间轴对称问题的研究，考虑杨氏模量为最一般分布的情形，采用弹性力学应力解法求解柱坐标系中的轴对称应力平衡方程，给出了无体力问题的解析通解，并探讨了轴对称球体问题在柱坐标系中的表示方法以及求解程序。 作为非均质性球体问题的应用，文中对地球内部结构模型进行了探讨。首先通过对现有的地球物理学测量数据进行拟合得到了地球内部地震波波速的函数分布关系，进而得到了地球内部弹性常数沿深度分布的函数关系，构建出弹性地球的力学模型，并采用Runge-Kutta 数值方法计算了在给定边界条件时杨氏模量二次多项式分布情形的分层球对称问题，分别得到了考虑万有引力体力和不考虑体力时的弹性力学数值解。 同时文中对构建的弹性地球内部结构模型从几个不同的方面进行了验证，包括地球的质量、转动惯量和各分层的地震波波速以及密度分布等。探究了考虑轴对称自转向心力时构建轴对称地球模型的程序和方法，同时也讨论了地球结构模型的几个实际应用以及还可以改进的地方。

In this thesis, analytical general solutions of the inhomogeneous spherically symmetric problem and the spatial axisymmetric problem are given respectively. The elastic earth hierarchical structure model is constructed by the simple functional relationship between the physical and mechanical parameters of the earth, and elastic mechanics solutions of the earth model is given.For the study of inhomogeneous spherically symmetric problems, the displacement control equations for them are derived in the beginning. Firstly, considering the case that Young's modulus is distributed exponentially with the radius, and the analytical solutions of the problem are given. Secondly, considering Young’s modulus changes with the radius of the general linear rule, the series solutions of the problem are given by the power series method of the ordinary differential equation. In this thesis, the distribution of spherically symmetric universal gravitation is calculated, assuming that the density function is exponentially distributed along the radial direction, and analytical displacement solutions of the non-homogeneous equation with body force are given. For the heterogeneous spatial axisymmetric problem, considering the most general distribution of Young’s modulus, the elastic stress solution method is used to solve the axisymmetric stress equilibrium equations in cylindrical coordinate system, analytical general solutions without body force are given, and the expression method and solving procedure of axisymmetric sphere problems are discussed in the cylindrical coordinates.The internal structure model of the earth is discussed as an application of the inhomogeneous sphere problems. The functional relationships of seismic wave velocity within the earth are obtained by fitting the existing geophysical measurement data at first, then the functions of elastic constants distribution along the depth are obtained in the earth's interior, and the mechanical model of the elastic earth is constructed. The hierarchical spherical symmetry problem for given boundary conditions is calculated by Runge-Kutta numerical method, in which the Young's modulus is quadratic polynomial distribution along the depth, and the elastic numerical solutions are obtained when considering the universal gravitation and not considering body force respectively.At the same time, the internal structure model of elastic earth is verified from several different aspects, including the earth’s mass, moment of inertia and the of seismic wave velocities, density distribution in each layer. The procedure and method of constructing the axisymmetric earth model by considering the axisymmetric self-steering force are discussed. At the same time, several practical applications of the earth’s internal structure model and some improving aspects are also discussed in the paper.

非均质性 球对称 轴对称 球体 地球模型

inhomogeneous spherical symmetry axial symmetry sphere earth model

中文

联合培养

南方科技大学

周宇澄. 参数非均质球体的力学问题及其在地球物理学中的应用[D]. 深圳. 哈尔滨工业大学,2019.