王之洋^1,3,4,,
李幼铭^2,3,4,,,
陈朝蒲^1,3,4,
白文磊^1,3,4
1. 北京化工大学信息科学与技术学院, 北京 100029
2. 中国科学院地质与地球物理研究所, 中国科学院油气资源研究重点实验室, 北京 100029
3. 高铁地震学联合研究组, 北京 100029
4. 非对称性弹性波动方程联合研究组, 北京 100029

基金项目: 国家自然科学基金项目(41630319)，国家自然科学基金联合基金项目(U20B2014)，国家重点研发计划项目(2018YFF01013503)共同资助


详细信息

作者简介: 王之洋, 男, 1987年生, 2015年博士毕业于中国科学院地质与地球物理研究所, 主要从事地震波数值模拟和逆时偏移成像方法研究.E-mail: wangzy@mail.buct.edu.cn

通讯作者: 李幼铭, 男, 1939年生, 中国科学院地质与地球物理研究所研究员.E-mail: ymli@mail.iggcas.ac.cn

中图分类号: P631

收稿日期:2020-09-25
修回日期:2021-02-09
上线日期:2021-07-10

Numerical modelling for elastic wave equations based on the second-order strain gradient theory

WANG ZhiYang^1,3,4,,
LI YouMing^2,3,4,,,
CHEN ChaoPu^1,3,4,
BAI WenLei^1,3,4
1. College of Information Science and Technology, Beijing University of Chemical Technology, Beijing 100029, China
2. Key Laboratory of Petroleum Resource Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
3. The Joint Research Group of High-Speed Rail Seismology, Beijing 100029, China
4. The Joint Research Group of Asymmetric Elastic Wave Equations, Beijing 100029, China


More Information

Corresponding author: LI YouMing,E-mail:ymli@mail.iggcas.ac.cn

MSC: P631

--> Received Date: 25 September 2020
Revised Date: 09 February 2021
Available Online: 10 July 2021


摘要
摘要:介质微结构相互作用会使介质存在不均匀性，而这种不均匀性，则会引发新的响应.当位移场/旋转场存在强烈空间/时间变化时，这种由介质微结构相互作用所导致的不均匀性会愈加明显.应变梯度通过在应变能密度函数中引入应变的一阶或者高阶导数，以描述这种由介质微结构相互作用导致的不均匀性，由于引入高阶导数，应变梯度理论可以描述更小尺度的微结构相互作用，但是其存在计算量大以及物理解释困难等问题.单参数二阶应变梯度理论作为应变梯度理论的一种特例或者简化版本，将二阶应变梯度视为对应变能密度函数的附加影响.本文从非局部理论出发，推导单参数二阶应变梯度理论的本构方程，进而结合几何方程和运动微分方程，给出非对称弹性波动方程的数学表达式.并应用该非对称弹性波动方程在各向同性均匀介质模型和Marmousi模型上进行数值模拟，合成地震记录.将该地震记录与传统弹性波动方程所生成的合成地震记录进行对比，研究分析应用二阶应变梯度描述介质微结构相互作用对地震记录的影响规律，给出以下结论：(1)基于单参数二阶应变梯度理论的非对称弹性波动方程所描述的位移扰动对纵波和横波的传播都产生了影响，且对横波的影响较大；(2)介质更小尺度的微结构相互作用可以在地震记录中被反映出来，我们需要考虑其对地震波传播的影响.
关键词: 非对称地震学/
应变梯度理论/
非局部理论/
弹性波动方程/
数值模拟

Abstract:The existence of microstructure interactions results in the heterogeneity of the media, which leads to the new responses in seismic wave propagation. The heterogeneity caused by microstructure interactions can be generated and amplified when the temporal or spatial variations of the displacement field or rotating field are strong. By introducing the higher derivative of strain into the strain energy density function, the strain gradient theory can describe the heterogeneity of the media caused by the microstructure interactions. Due to the introduction of higher derivative, the strain gradient theory can describe the smaller-scale microstructural interactions, but a mass of computation and difficult interpretation in physical need to be faced. The single-parameter second-order strain gradient theory is regarded as a simplified version of the strain gradient theory, which consider the second-order strain gradient as an additional effect of the strain energy density functions. In this paper, the constitutive equations of the single-parameter second-order strain gradient theory has been investigated starting from the nonlocal theory, we can derive the mathematical expression of the asymmetric elastic wave equations by incorporating the geometric equations and the differential equations of motion. Then, we obtain the synthetic seismograms for the isotropic homogeneous model and Marmousi model using the asymmetric elastic wave equations. Comparing the synthetic seismograms generated by the conventional elastic wave equations with these generated by the asymmetric elastic wave equations, we study and analyze the law of influence on the seismograms when using the second-order strain gradient to describe the microstructure interactions and draw the following conclusions: (1) The displacement perturbation described by the asymmetric elastic wave equation based on the one-parameter second-order strain gradient theory has an influence on the propagation of P- and S-waves, and has a greater influence on the S-wave. (2) The smaller scale microstructure interactions of the medium can be reflected in the seismograms, and we need to consider the influence of the smaller scale microstructure interactions of the medium on the propagation of seismic waves.
Key words:Asymmetric seismology/
Strain gradient theory/
Nonlocal theory/
Elastic wave equation/
Numerical modelling



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