衬砌与裂纹对SH波的散射
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摘要
本文在弹性动力学范畴内,采用Green函数、复变函数和多极坐标方法研究了半空间中圆形衬砌与单个裂纹,圆形衬砌与多个裂纹,双相介质界面附近圆形衬砌及其附近任意方位有限长度裂纹对SH波的散射问题。
首先,给出了研究本文三个问题需要用到的五个Green函数：一,出平面线源荷载作用在含有圆形衬砌的弹性半空间表面的Green函数；二,出平面线源荷载作用在含有圆形衬砌的弹性半空间内的Green函数；三,出平面线源荷载作用在含有圆形衬砌和一个裂纹的弹性半空间表面的Green函数；四,出平面线源荷载作用在含有圆形衬砌和一个裂纹的弹性半空间内的Green函数；五,出平面荷载作用在完整的弹性半空间表面的Green函数。
其次,利用前面给出的Green函数,分别求解了半空间中圆形衬砌与单个裂纹,圆形衬砌与多个裂纹,双相向介质界面附近圆形衬砌及其附近任意方位有限长度裂纹对SH波的散射问题。具体工作如下：
1.研究了半无限空间中圆形衬砌与裂纹对SH波的散射问题。求解该问题的关键是构造能自动满足含圆形衬砌的弹性半空间自由表面上应力为零边界条件的散射波和衬砌内的驻波。这个散射波可以利用SH波散射具有的对称性质和多极坐标方法来构造,应用衬砌的边界条件来确定。再利用适合该问题的Green函数,采用裂纹“切割”方法构造裂纹：沿裂纹位置施加反向应力,即在欲出现裂纹区域加置与圆形衬砌对SH波散射产生应力相对应的的大小相等,方向相反的出平面荷载,从而构造出裂纹,并得到圆形衬砌和裂纹同时存在条件下的位移场与应力场。
2.研究了半无限空间中圆形衬砌与多个裂纹对SH波的散射问题。本问题的求解关键是构造适合本问题的Green函数,即含有圆形衬砌和多个任意长度任意位置直线型裂纹的弹性半空间内任意一点承受时间谐和的出平面线源荷载的位移函数的基本解。首先含有圆形衬砌和一个裂纹的的弹性半空间内任意一点承受时间谐和的出平面线源荷载的位移函数的基本解已求出,利用其可求出含有圆形衬砌和两个裂纹的的弹性半空间内任意一点承受时间谐和的出平面线源荷载的位移函数的基本解,依此类推,即可求出适合本问题的Green函数。其次,用“切割”方法构造裂纹,导出圆形衬砌与多个裂纹同时存在条件下的位移场和应力场。
3.研究SH波对双相介质界面附近圆形衬砌和裂纹的散射问题,求解过程中将问题的模型视为“契合”问题：即可将所研究的问题沿其界面“剖分”为两个部分,其一为含有圆形衬砌和裂纹的弹性半空间,而另外一部分则是完整的弹性半空间。若在两个半无限空间的自由表面上,分别加置待定的出平面荷载,利用求出的Green函数写出界面上的连续条件,建立起确定待解外力系的第一类Fredholm积分方程组,采用直接离散的方法将定解积分方程组转化为线性代数方程组计算求解,从而可以得到双相介质界面附近圆形衬砌和裂纹对SH散射的位移场、应力场的解析表达式。
最后,针对上述三个问题,结合具体算例,分析了不同的介质参数、入射波数、入射角度、衬砌到界面的距离与衬砌内半径的比、裂纹与衬砌的距离与衬砌内半径的比、裂纹的角度、裂纹长度等参数对地表位移、衬砌周边动应力集中系数、裂纹尖端动应力强度因子的影响规律。
The scattering of SH-wave by half-space circular lining structure and crack, circular lining structure and multiple cracks, circular lining structure near the bimaterial interface and crack of arbitrary length and arbitrary position, are studied in this paper beyond the field of elastodynamics respectively, and the methods of Green's Function, complex variables function and multi-polar coordinates are used here.
First, five Green's functions are given for solving three problems in this paper:the first one is an essential solution of the displacement field for the elastic half-space possessing circular lining structure while bearing out-of-plane harmonic line source loads at half space surface arbitrary point; the second Green's function is an essential solution of the displacement field for the elastic half-space possessing circular lining structure while bearing out-of-plane harmonic line source loads in half space arbitrary point; the third, an essential solution of the displacement field for the elastic half-space possessing circular lining structure and a crack while bearing out-of-plane harmonic line source loads at half space surface arbitrary point; the fourth, an essential solution of the displacement field for the elastic half-space possessing circular lining structure and a crack while bearing out-of-plane harmonic line source loads in half space arbitrary point; the fifth one is an essential solution of the displacement field for the elastic half-space bearing out-of-plane harmonic line source loads at half space surface arbitrary point.
Second, by using these Green's functions, the problems of SH-wave scattering, which is caused by half-space circular lining structure and crack, circular lining structure and multiple cracks, circular lining structure and crack near the bimaterial interface are investigated respectively. The work in details is as follows:
1. The problem of SH-wave scattering caused by circular lining structure and crack in infinite half space is investigated. The key point of the work is that:a scattering wave which can satisfy the stress-free condition at the horizontal surface in half space caused by the circular lining structure is constructed based on the symmetry of SH-wave scattering and the method of multi-polar coordinate system. The expression of scattering wave can be determined by virtue of the boundary condition of circular lining structure. Then by using the Green's function which is suitable to the present problem and "crack-division", the crack is established:reverse stresses are inflicted along the crack, that is, out-of-plane harmonic line source loading, which are equal in the quantity but opposite in the direction to the stresses produced for the reason of SH-wave scattering by circular lining structure, are loaded at the region where crack will appear, the crack can be made out. Thus expressions of displacement and stress are established while crack and inclusion are both in existent.
2. The scattering of SH-wave caused by circular lining structure and multiple cracks in infinite half space is investigated. The key point of the work is the construction of a suitable Green's function, which is an essential solution of the displacement field for the elastic half-space possessing circular lining structure and multiple cracks while bearing out-of-plane harmonic line source loads in half space arbitrary point. First of all, the Green's function which is an essential solution of the displacement field for the elastic half-space possessing circular lining structure and a crack while bearing out-of-plane harmonic line source loads in half space arbitrary point is solved, by using this, the Green's function which is an essential solution of the displacement field for the elastic half-space possessing circular lining structure and two cracks while bearing out-of-plane harmonic line source loads in half space arbitrary point can be solved, then the rest may be deduced by analogy, the Green's function which is suitable to the present problem can be solved. Secondly, the crack is established by using "crack-division", thus expressions of displacement and stress are established while cracks and inclusion are both in existent.
3. The scattering of SH-wave by crack of arbitrary position and a circular lining structure near the bimaterial interface is investigated, the problem can be regarded as harmony model:the bimaterial media is divided into two parts along the horizontal interface, one is an elastic half space with a circular lining structure and a crack, and the other is a complete elastic half space. The horizontal surfaces of the two parts are loaded with undetermined anti-plane forces in order to satisfy at the linking section. A series of Fredholm integral equations can be set up through continuity conditions. Then, the solution of the problem can be reduced to a series of algebraic equations and solved numerically by truncating the finite terms of the infinite integral equations, the expressions of displacement field and stress field of scattering of SH-wave by crack and circular lining structure near the bimaterial interface are given.
Finally, to solve the problems mentioned above, numerical examples are provided to show the influences of wave numbers, incident angle, the ratio of distance between the center of circular lining structure and horizontal surfaces and inner radius of circular lining structure, the ratio of distance between the center of circular lining structure and crack and inner radius of circular lining structure, the angle of crack, the length of crack, and parameter combinations of different media upon the horizontal surface displacement, dynamic stress concentration factor (DSCF) around the circular lining structure and dynamic stress intensity factor (DSIF) at crack tip.
首先,给出了研究本文三个问题需要用到的五个Green函数：一,出平面线源荷载作用在含有圆形衬砌的弹性半空间表面的Green函数；二,出平面线源荷载作用在含有圆形衬砌的弹性半空间内的Green函数；三,出平面线源荷载作用在含有圆形衬砌和一个裂纹的弹性半空间表面的Green函数；四,出平面线源荷载作用在含有圆形衬砌和一个裂纹的弹性半空间内的Green函数；五,出平面荷载作用在完整的弹性半空间表面的Green函数。
其次,利用前面给出的Green函数,分别求解了半空间中圆形衬砌与单个裂纹,圆形衬砌与多个裂纹,双相向介质界面附近圆形衬砌及其附近任意方位有限长度裂纹对SH波的散射问题。具体工作如下：
1.研究了半无限空间中圆形衬砌与裂纹对SH波的散射问题。求解该问题的关键是构造能自动满足含圆形衬砌的弹性半空间自由表面上应力为零边界条件的散射波和衬砌内的驻波。这个散射波可以利用SH波散射具有的对称性质和多极坐标方法来构造,应用衬砌的边界条件来确定。再利用适合该问题的Green函数,采用裂纹“切割”方法构造裂纹：沿裂纹位置施加反向应力,即在欲出现裂纹区域加置与圆形衬砌对SH波散射产生应力相对应的的大小相等,方向相反的出平面荷载,从而构造出裂纹,并得到圆形衬砌和裂纹同时存在条件下的位移场与应力场。
2.研究了半无限空间中圆形衬砌与多个裂纹对SH波的散射问题。本问题的求解关键是构造适合本问题的Green函数,即含有圆形衬砌和多个任意长度任意位置直线型裂纹的弹性半空间内任意一点承受时间谐和的出平面线源荷载的位移函数的基本解。首先含有圆形衬砌和一个裂纹的的弹性半空间内任意一点承受时间谐和的出平面线源荷载的位移函数的基本解已求出,利用其可求出含有圆形衬砌和两个裂纹的的弹性半空间内任意一点承受时间谐和的出平面线源荷载的位移函数的基本解,依此类推,即可求出适合本问题的Green函数。其次,用“切割”方法构造裂纹,导出圆形衬砌与多个裂纹同时存在条件下的位移场和应力场。
3.研究SH波对双相介质界面附近圆形衬砌和裂纹的散射问题,求解过程中将问题的模型视为“契合”问题：即可将所研究的问题沿其界面“剖分”为两个部分,其一为含有圆形衬砌和裂纹的弹性半空间,而另外一部分则是完整的弹性半空间。若在两个半无限空间的自由表面上,分别加置待定的出平面荷载,利用求出的Green函数写出界面上的连续条件,建立起确定待解外力系的第一类Fredholm积分方程组,采用直接离散的方法将定解积分方程组转化为线性代数方程组计算求解,从而可以得到双相介质界面附近圆形衬砌和裂纹对SH散射的位移场、应力场的解析表达式。
最后,针对上述三个问题,结合具体算例,分析了不同的介质参数、入射波数、入射角度、衬砌到界面的距离与衬砌内半径的比、裂纹与衬砌的距离与衬砌内半径的比、裂纹的角度、裂纹长度等参数对地表位移、衬砌周边动应力集中系数、裂纹尖端动应力强度因子的影响规律。
The scattering of SH-wave by half-space circular lining structure and crack, circular lining structure and multiple cracks, circular lining structure near the bimaterial interface and crack of arbitrary length and arbitrary position, are studied in this paper beyond the field of elastodynamics respectively, and the methods of Green's Function, complex variables function and multi-polar coordinates are used here.
First, five Green's functions are given for solving three problems in this paper:the first one is an essential solution of the displacement field for the elastic half-space possessing circular lining structure while bearing out-of-plane harmonic line source loads at half space surface arbitrary point; the second Green's function is an essential solution of the displacement field for the elastic half-space possessing circular lining structure while bearing out-of-plane harmonic line source loads in half space arbitrary point; the third, an essential solution of the displacement field for the elastic half-space possessing circular lining structure and a crack while bearing out-of-plane harmonic line source loads at half space surface arbitrary point; the fourth, an essential solution of the displacement field for the elastic half-space possessing circular lining structure and a crack while bearing out-of-plane harmonic line source loads in half space arbitrary point; the fifth one is an essential solution of the displacement field for the elastic half-space bearing out-of-plane harmonic line source loads at half space surface arbitrary point.
Second, by using these Green's functions, the problems of SH-wave scattering, which is caused by half-space circular lining structure and crack, circular lining structure and multiple cracks, circular lining structure and crack near the bimaterial interface are investigated respectively. The work in details is as follows:
1. The problem of SH-wave scattering caused by circular lining structure and crack in infinite half space is investigated. The key point of the work is that:a scattering wave which can satisfy the stress-free condition at the horizontal surface in half space caused by the circular lining structure is constructed based on the symmetry of SH-wave scattering and the method of multi-polar coordinate system. The expression of scattering wave can be determined by virtue of the boundary condition of circular lining structure. Then by using the Green's function which is suitable to the present problem and "crack-division", the crack is established:reverse stresses are inflicted along the crack, that is, out-of-plane harmonic line source loading, which are equal in the quantity but opposite in the direction to the stresses produced for the reason of SH-wave scattering by circular lining structure, are loaded at the region where crack will appear, the crack can be made out. Thus expressions of displacement and stress are established while crack and inclusion are both in existent.
2. The scattering of SH-wave caused by circular lining structure and multiple cracks in infinite half space is investigated. The key point of the work is the construction of a suitable Green's function, which is an essential solution of the displacement field for the elastic half-space possessing circular lining structure and multiple cracks while bearing out-of-plane harmonic line source loads in half space arbitrary point. First of all, the Green's function which is an essential solution of the displacement field for the elastic half-space possessing circular lining structure and a crack while bearing out-of-plane harmonic line source loads in half space arbitrary point is solved, by using this, the Green's function which is an essential solution of the displacement field for the elastic half-space possessing circular lining structure and two cracks while bearing out-of-plane harmonic line source loads in half space arbitrary point can be solved, then the rest may be deduced by analogy, the Green's function which is suitable to the present problem can be solved. Secondly, the crack is established by using "crack-division", thus expressions of displacement and stress are established while cracks and inclusion are both in existent.
3. The scattering of SH-wave by crack of arbitrary position and a circular lining structure near the bimaterial interface is investigated, the problem can be regarded as harmony model:the bimaterial media is divided into two parts along the horizontal interface, one is an elastic half space with a circular lining structure and a crack, and the other is a complete elastic half space. The horizontal surfaces of the two parts are loaded with undetermined anti-plane forces in order to satisfy at the linking section. A series of Fredholm integral equations can be set up through continuity conditions. Then, the solution of the problem can be reduced to a series of algebraic equations and solved numerically by truncating the finite terms of the infinite integral equations, the expressions of displacement field and stress field of scattering of SH-wave by crack and circular lining structure near the bimaterial interface are given.
Finally, to solve the problems mentioned above, numerical examples are provided to show the influences of wave numbers, incident angle, the ratio of distance between the center of circular lining structure and horizontal surfaces and inner radius of circular lining structure, the ratio of distance between the center of circular lining structure and crack and inner radius of circular lining structure, the angle of crack, the length of crack, and parameter combinations of different media upon the horizontal surface displacement, dynamic stress concentration factor (DSCF) around the circular lining structure and dynamic stress intensity factor (DSIF) at crack tip.
引文
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