双相介质垂直半空间界面裂纹及附近圆孔对SH波的散射
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摘要
本文在线弹性力学范畴内,采用SH波散射的“镜像叠加”思想,利用复变函数和多级坐标方法研究了直角平面区域出平面问题的Green函数解;利用Green函数方法与界面“契合”技术研究了双相介质垂直半空间界面附近圆孔对SH波的散射与动应力集中问题;利用界面“契合”与裂纹切割技术研究了SH波作用下双相介质垂直半空间界面裂纹对界面附近圆孔散射的影响。
研究直角平面区域出平面问题的Green函数解,问题的关键在于构造满足直角平面区域表面应力自由条件的散射波场。由于直角平面区域自由表面的存在,由圆形孔洞所激发的散射波将会在圆孔和直角平面区域自由表面上发生多次反射,致使能满足直角平面区域表面上应力自由边界条件波场的解析解很难给出。为了克服这一难点,可利用镜像叠加原理及移动坐标技术将该问题的解答转化为一个全空间问题进行求解,并通过具体的算例得出相应结论。
研究双相介质垂直半空间界面附近圆孔对SH波的散射与动应力集中问题,在该问题的求解过程中,一般将问题的模型视为“契合”问题:即可将所研究的问题沿其界面“剖分”为两个部分,其一为含有圆孔的直角平面区域,而另外一部分则可以是完整的,或含有圆孔的直角平面区域,并根据界面处位移连续性条件将解答归结为具有弱奇异性的第一类Fredholm积分方程组的求解,结合散射波的衰减特性,直接离散该方程组,积分方程组转化为线性代数方程组可得到该问题的数值结果,并给出了稳态载荷作用下圆形孔洞周围的动应力集中系数(DSCF)。
研究SH波作用下双相介质垂直半空间界面裂纹对界面附近圆形孔洞散射的影响。考虑到Green函数自身存在的“切割”功能,该问题按“契合”方式并采用裂纹切割技术即可构造出界面裂纹及其附近圆形孔洞的散射模型,在两个直角平面区域的水平表面上,分别加置待定的出平面荷载,并在欲出现裂纹位置加置出平面反力构造裂纹,然后利用Green函数写出界面上的位移连续条件,建立起确定待解外力系的第一类Fredholm积分方程组。最后给出具体算例分析了界面裂纹对圆孔散射的影响。
The scattering of SH-wave are studied in the field of linearly elastic dynamic mechanics in this paper. By using the methods of complex functions and multi-polar coordinates, the Green's function is researched in right-angle plane; the methods of Green's functions and interface conjunction are used to obtain the scattering of SH-wave by circular cavity near bi-material interface in vertical half-space and dynamic stress concentration; and the methods of interface conjunction and crack-division are used to get the influence of scattering of circular cavity near interface with bi-material interface crack impacted by SH-wave.
An analytical solution for the Green's function is studied based on the image method, which is the essential solution of displacement field for an elastic right-angle plane with a circular cavity impacted by anti-plane harmonic line source loading at horizontal surface. In this question, owing to the existence of the right-angle plane free surfaces, the elastic wave, which produced by the disturbance of out-plane line source force, is reflected and scattered many times. Thus the solution of displacement field satisfying the stress free boundary conditions is difficult to present. To overcome this difficulty, using image method and muti-polar coordinates method in here, the right-angle plane is extended half space. Then the problem of SH-wave scattering in the right-angle plane can be solved according to the half space with double circular cavity impacted by the out-plane harmonic line source force and the image line source force.
The scattering of SH-wave by circular cavity near bi-material interface in vertical half-space and dynamic stress concentration are studied based on the methods of Green's functions and interface conjunction in complex plane. The bi-material media is divided into two parts along the horizontal interface, one of which is elastic right-angle plane with a circular cavity and the other is either elastic complete right-angle plane or elastic right-angle plane with a circular cavity. Then horizontal surfaces of the two half space are loaded with undetermined anti-plane forces in order to satisfy continuity conditions at linking section. So a series of Fredholm integral equations of first kind for determining the unknown forces can be set up through continuity conditions that are expressed in terms of the Green's function. Finally, using the attenuation speciality of SH-wave, the integral equations can be dispersed to linear equations directly, so the numerical solution can be obtained.
The scattering of SH-wave by a bi-material interface crack and a circular cavity near interface is also discussed and the solution of dynamic stress concentration factors around circular cavity are obtained. The methods of combination and crack-division are used to make interface cracks. Then horizontal surfaces of the two half space are loaded with undetermined anti-plane forces in order to satisfy continuity conditions at linking section, with some forces to make cracks by means of crack-division technique. For the model of circular cavities near the interface, the added forces at linking interface can be employed to calculate the dynamic stress concentration factors around the edge of circular cavities. Then the problem is summarized into the solution of Fredholm integral equations of first kind by means of the Green's function. Finally, the dynamic stress concentration factors around the circular cavity are discussed to the cases of different parameters in numerical examples.
研究直角平面区域出平面问题的Green函数解,问题的关键在于构造满足直角平面区域表面应力自由条件的散射波场。由于直角平面区域自由表面的存在,由圆形孔洞所激发的散射波将会在圆孔和直角平面区域自由表面上发生多次反射,致使能满足直角平面区域表面上应力自由边界条件波场的解析解很难给出。为了克服这一难点,可利用镜像叠加原理及移动坐标技术将该问题的解答转化为一个全空间问题进行求解,并通过具体的算例得出相应结论。
研究双相介质垂直半空间界面附近圆孔对SH波的散射与动应力集中问题,在该问题的求解过程中,一般将问题的模型视为“契合”问题:即可将所研究的问题沿其界面“剖分”为两个部分,其一为含有圆孔的直角平面区域,而另外一部分则可以是完整的,或含有圆孔的直角平面区域,并根据界面处位移连续性条件将解答归结为具有弱奇异性的第一类Fredholm积分方程组的求解,结合散射波的衰减特性,直接离散该方程组,积分方程组转化为线性代数方程组可得到该问题的数值结果,并给出了稳态载荷作用下圆形孔洞周围的动应力集中系数(DSCF)。
研究SH波作用下双相介质垂直半空间界面裂纹对界面附近圆形孔洞散射的影响。考虑到Green函数自身存在的“切割”功能,该问题按“契合”方式并采用裂纹切割技术即可构造出界面裂纹及其附近圆形孔洞的散射模型,在两个直角平面区域的水平表面上,分别加置待定的出平面荷载,并在欲出现裂纹位置加置出平面反力构造裂纹,然后利用Green函数写出界面上的位移连续条件,建立起确定待解外力系的第一类Fredholm积分方程组。最后给出具体算例分析了界面裂纹对圆孔散射的影响。
The scattering of SH-wave are studied in the field of linearly elastic dynamic mechanics in this paper. By using the methods of complex functions and multi-polar coordinates, the Green's function is researched in right-angle plane; the methods of Green's functions and interface conjunction are used to obtain the scattering of SH-wave by circular cavity near bi-material interface in vertical half-space and dynamic stress concentration; and the methods of interface conjunction and crack-division are used to get the influence of scattering of circular cavity near interface with bi-material interface crack impacted by SH-wave.
An analytical solution for the Green's function is studied based on the image method, which is the essential solution of displacement field for an elastic right-angle plane with a circular cavity impacted by anti-plane harmonic line source loading at horizontal surface. In this question, owing to the existence of the right-angle plane free surfaces, the elastic wave, which produced by the disturbance of out-plane line source force, is reflected and scattered many times. Thus the solution of displacement field satisfying the stress free boundary conditions is difficult to present. To overcome this difficulty, using image method and muti-polar coordinates method in here, the right-angle plane is extended half space. Then the problem of SH-wave scattering in the right-angle plane can be solved according to the half space with double circular cavity impacted by the out-plane harmonic line source force and the image line source force.
The scattering of SH-wave by circular cavity near bi-material interface in vertical half-space and dynamic stress concentration are studied based on the methods of Green's functions and interface conjunction in complex plane. The bi-material media is divided into two parts along the horizontal interface, one of which is elastic right-angle plane with a circular cavity and the other is either elastic complete right-angle plane or elastic right-angle plane with a circular cavity. Then horizontal surfaces of the two half space are loaded with undetermined anti-plane forces in order to satisfy continuity conditions at linking section. So a series of Fredholm integral equations of first kind for determining the unknown forces can be set up through continuity conditions that are expressed in terms of the Green's function. Finally, using the attenuation speciality of SH-wave, the integral equations can be dispersed to linear equations directly, so the numerical solution can be obtained.
The scattering of SH-wave by a bi-material interface crack and a circular cavity near interface is also discussed and the solution of dynamic stress concentration factors around circular cavity are obtained. The methods of combination and crack-division are used to make interface cracks. Then horizontal surfaces of the two half space are loaded with undetermined anti-plane forces in order to satisfy continuity conditions at linking section, with some forces to make cracks by means of crack-division technique. For the model of circular cavities near the interface, the added forces at linking interface can be employed to calculate the dynamic stress concentration factors around the edge of circular cavities. Then the problem is summarized into the solution of Fredholm integral equations of first kind by means of the Green's function. Finally, the dynamic stress concentration factors around the circular cavity are discussed to the cases of different parameters in numerical examples.
引文
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