半空间垂直界面裂纹及圆夹杂对SH波的散射
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摘要
通常，在生产人工材料与结构时，将很难避免产生各种复杂的缺陷，例如孔洞、夹杂和裂纹。含有介质缺陷的材料在受到外力荷载作用时，由于材料的几何不连续性，将在缺陷附近产生动应力集中情况，进而决定材料的破坏程度。因此，为了满足理论与工程上的需要，研究缺陷附近的动应力集中分布情况是非常有意义的。由于SH波是弹性波散射理论中最简单的计算模型，因此具有相对比较成熟的理论。但是，仍然有许多其他边值问题的解析解答没有被解决。因此，基于线弹性理论，本文分别对双相介质半空间中含有单个圆形弹性夹杂、多个圆形弹性夹杂（孔洞）以及界面裂纹和圆形弹性夹杂组成的复合缺陷对SH波散射问题的解析解答进行了研究，同时给出介质缺陷附近的动应力集中情况以及半空间地表位移幅值的分布情况。本论文所涉及的主要工作可以概括为以下三个部分：
（1）第一部分主要研究了双相介质半空间垂直界面附近圆形弹性夹杂对SH波散射问题的解析解答，其主要应用了复变函数和Green函数方法。首先，构造适合此问题的Green函数，即：在一个含有圆形弹性夹杂的四分之一空间，在其垂直边界上作用一个任意的出平面线源荷载，此线源荷载在空间中产生的位移函数的解答。其可以被看成是由入射波和散射波的位移表达式组成的。前者可以采用“虚设点源”的方法构造，其满足四分之一空间水平边界应力自由条件，后者可以采用“镜像”的方法够造，其满足两个直角边界应力自由条件。然后，利用圆形弹性夹杂周边的位移和应力连续性边界条件，求解散射波表达式中未知系数。其次，采用“镜像”的方法分别构造SH波入射下满足边界条件的入射波、反射波、折射波以及散射波的位移表达式。然后，采用界面“契合”的思想将双相介质半空间沿着垂直界面划分为两部分，为了满足界面处的连续性条件，需要在剖分面两侧分别施加未知的出平面载荷，然后利用界面处位移和应力连续性条件建立可以确定未知外力系的第一类Fredholm积分方程组，考虑到散射波的衰减性质，采用离散方法求解未知力系。最后，通过具体算例给出圆形弹性夹杂周边动应力集中系数和半空间表面位移幅值的分布情况，分别讨论它们随无量纲参数变化的分布情况。
（2）第二部分主要研究了双相介质半空间垂直界面附近多个圆形弹性夹杂对SH波散射问题的解析解答，其主要应用了复变函数和多极坐标移动技术。首先，构造适合此问题的Green函数，即：在含有多个圆形弹性夹杂的四分之一空间，在其垂直边界上作用一个任意的出平面线源荷载，此线源荷载在空间中产生的位移函数解答。其次，采用界面“契合”的思想将双相介质半空间沿垂直界面处分开，在剖分面处施加外力荷载以满足界面处的连续性条件，同时利用Green函数表达式建立定解积分方程组。最后，通过具体的算例给出圆形弹性夹杂（孔洞）周边动应力集中情况，分别讨论它们随无量纲参数变化的分布情况。
（3）第三部分主要研究了双相介质半空间垂直界面裂纹及其附近圆形弹性夹杂对SH波散射问题的解析解答，其主要应用了复变函数和Green函数的方法。首先，构造适合此问题的Green函数，其与第一部分求解过程相同。其次，利用裂纹“切割”技术构造界面裂纹，即先将双相介质半空间沿垂直界面处剖开，在想要出现裂纹的地方施加与SH波作用下此处原有应力大小相等、方向相反的出平面应力。同时，需要在垂直界面其他位置施加外力荷载以满足界面处连续性条件，然后利用Green函数表达式建立定解积分方程组。最后，通过具体算例给出圆形弹性夹杂周边和III型裂纹尖端动应力集中系数和动应力力强度因子，分别讨论它们随无量纲参数变化的分布情况。
Generally, it is hard to avoid various complicated defects when working with the manualmaterials and structures, such as circular holes, inclusions and cracks. Under the externalforce, the material which contains defects will take place the phenomenon of dynamic stressconcentration near the defects because of the geometrical discontinuity. What’s more, itdecides the level of damage. So it is very significant to investigate dynamic stress state nearthe defects to satisfy the theoretical and the engineering needs. As the simplest one among thecalculative models of scattering problems of elastic waves, SH waves scattering problem hasrelative mature theories. However, there are still many boundary value problems unsolved. Inthis paper, the analysis solutions of the scattering of SH waves by the single circular elasticinclusion, the multiple circular elstic inclusions (holes) and the composite defects constitutedof the interfacial crack and circular elastic inclusion are considered respectively based on thelinear elastic theory. Meanwhile, some examples for dynamic stress concentration factor ofthe defects and the amplitude of ground surface displacement in half space are given. Themainwork in present paper can be summarized into three parts as follows:
（1） In part one, complex function and Green’s function methods are used toinvestigate the analysis solution of the scattering of SH-wave by the bi-material elastic halfspace which contains the vertical interface and a circualr elastic inclusion. Firstly, Green’sfunction is constructed to meet the needs of the problems, which is an essential solution ofdisplacement field for an elastic quarter plane containing a elastic cylindrical inclusion whilebearing out-of-plane harmonic line source load at any point of its vertical boundary. In thispaper, the method of fictitious line source force is used to constructe the expression ofincident waves which satisfies the stress free condition at the horizontal boundary in quarterplane, and the expression of scattering waves which satisfies the stress free conditions at thetwo vertical boundaries can be obtained with the aid of image method. Then, the unknowncoefficients can be determined by the continuous conditions of the stresses and displacementsaound the circular ealstic inclusion edge. Secondly, the expressions of the incident waves, thereflected waves and the refracted waves, which satisfy the boundary cinditions, can beconstructed using the image method. Then, the bi-material media is divided into two partsalong the vertical interface using the idea of interface “conjunction”, and the undetermined anti-plane forces are loaded at the linking sections respectively to satisfy continuity conditions.So a series of Fredholm integral equations of first kind for determining the unknown forcescan be set up through continuity conditions on interface. In the light of attenuationcharacteristic of the scattering waves, the unknown forces can be obtained by the method ofdirect discrete. Finally, some examples for dynamic stress concentration factor around thecircular elastic inclusion edge and the amplitude of ground surface displacement are given.Numerical results discuss the distribution of dynamic stress concentration factor andamplitude of ground surface with the changes of the nondimensional parameters.
（2）In part two, complex method and multi-polar coordinates technology are used toinvestigate the analysis solution for the multiple circular inclusions near the vertical interfacedisturbed by SH waves in bi-material half space. Firstly, the Green’s function should beconstructed in this problem, which is an essential solution to the displacement field for anelastic quarter plane with multiple circular inclusions disturbed by out-plane harmonic linesource loading at vertical surface. Secondly, the bi-material media is divided into two partsalong the bi-material interface based on the idea of interface “conjunction”, and the verticalsurfaces of the quarter space are loaded with undetermined anti-plane forces in order to satisfydisplacement continuity and stress continuity conditions at linking section. Then, the integralequations for determining the unknown forces can be set up through continuity conditions andthe Green’s function. Finally, some examples for dynamic stress concentration factor aroundthe circular elastic inclusion (hole) edge and the amplitude of ground surface displacement aregiven. Numerical results discuss the distribution of dynamic stress concentration factor withthe changes of the nondimensional parameters.
（3）In part three, complex method and Green’s function method are used toinvestigate the analysis solution for the circular inclusions and the vertical interfacial crackdisturbed by SH waves in bi-material half space. Firstly, the Green’s function should beconstructed in this problem, which is the same as the part one. Secondly, the interfacial crackis constructed with the aid of the crack-division technique. The detail is as follows: the bi-material media is divided into two parts along the vertical interface, and a pair of oppositeforces which are equal to the original stresses disturbed SH waves. Meanwhile, a series of theunknown forces must be loaded at the linking sections except the region of the crack to satisfycontinuity conditions. Then, the integral equations for determining the unknown forces can be set up through continuity conditions. Finally, some examples for dynamic stress concentrationfactor around the circular elastic inclusion edge and the dynamic stress intensity factors formode III at the crack tip are given. Numerical results discuss the distribution of dynamicstress concentration factor and the dynamic stress intensity factors with the changes of thenondimensional parameters.
（1）第一部分主要研究了双相介质半空间垂直界面附近圆形弹性夹杂对SH波散射问题的解析解答，其主要应用了复变函数和Green函数方法。首先，构造适合此问题的Green函数，即：在一个含有圆形弹性夹杂的四分之一空间，在其垂直边界上作用一个任意的出平面线源荷载，此线源荷载在空间中产生的位移函数的解答。其可以被看成是由入射波和散射波的位移表达式组成的。前者可以采用“虚设点源”的方法构造，其满足四分之一空间水平边界应力自由条件，后者可以采用“镜像”的方法够造，其满足两个直角边界应力自由条件。然后，利用圆形弹性夹杂周边的位移和应力连续性边界条件，求解散射波表达式中未知系数。其次，采用“镜像”的方法分别构造SH波入射下满足边界条件的入射波、反射波、折射波以及散射波的位移表达式。然后，采用界面“契合”的思想将双相介质半空间沿着垂直界面划分为两部分，为了满足界面处的连续性条件，需要在剖分面两侧分别施加未知的出平面载荷，然后利用界面处位移和应力连续性条件建立可以确定未知外力系的第一类Fredholm积分方程组，考虑到散射波的衰减性质，采用离散方法求解未知力系。最后，通过具体算例给出圆形弹性夹杂周边动应力集中系数和半空间表面位移幅值的分布情况，分别讨论它们随无量纲参数变化的分布情况。
（2）第二部分主要研究了双相介质半空间垂直界面附近多个圆形弹性夹杂对SH波散射问题的解析解答，其主要应用了复变函数和多极坐标移动技术。首先，构造适合此问题的Green函数，即：在含有多个圆形弹性夹杂的四分之一空间，在其垂直边界上作用一个任意的出平面线源荷载，此线源荷载在空间中产生的位移函数解答。其次，采用界面“契合”的思想将双相介质半空间沿垂直界面处分开，在剖分面处施加外力荷载以满足界面处的连续性条件，同时利用Green函数表达式建立定解积分方程组。最后，通过具体的算例给出圆形弹性夹杂（孔洞）周边动应力集中情况，分别讨论它们随无量纲参数变化的分布情况。
（3）第三部分主要研究了双相介质半空间垂直界面裂纹及其附近圆形弹性夹杂对SH波散射问题的解析解答，其主要应用了复变函数和Green函数的方法。首先，构造适合此问题的Green函数，其与第一部分求解过程相同。其次，利用裂纹“切割”技术构造界面裂纹，即先将双相介质半空间沿垂直界面处剖开，在想要出现裂纹的地方施加与SH波作用下此处原有应力大小相等、方向相反的出平面应力。同时，需要在垂直界面其他位置施加外力荷载以满足界面处连续性条件，然后利用Green函数表达式建立定解积分方程组。最后，通过具体算例给出圆形弹性夹杂周边和III型裂纹尖端动应力集中系数和动应力力强度因子，分别讨论它们随无量纲参数变化的分布情况。
Generally, it is hard to avoid various complicated defects when working with the manualmaterials and structures, such as circular holes, inclusions and cracks. Under the externalforce, the material which contains defects will take place the phenomenon of dynamic stressconcentration near the defects because of the geometrical discontinuity. What’s more, itdecides the level of damage. So it is very significant to investigate dynamic stress state nearthe defects to satisfy the theoretical and the engineering needs. As the simplest one among thecalculative models of scattering problems of elastic waves, SH waves scattering problem hasrelative mature theories. However, there are still many boundary value problems unsolved. Inthis paper, the analysis solutions of the scattering of SH waves by the single circular elasticinclusion, the multiple circular elstic inclusions (holes) and the composite defects constitutedof the interfacial crack and circular elastic inclusion are considered respectively based on thelinear elastic theory. Meanwhile, some examples for dynamic stress concentration factor ofthe defects and the amplitude of ground surface displacement in half space are given. Themainwork in present paper can be summarized into three parts as follows:
（1） In part one, complex function and Green’s function methods are used toinvestigate the analysis solution of the scattering of SH-wave by the bi-material elastic halfspace which contains the vertical interface and a circualr elastic inclusion. Firstly, Green’sfunction is constructed to meet the needs of the problems, which is an essential solution ofdisplacement field for an elastic quarter plane containing a elastic cylindrical inclusion whilebearing out-of-plane harmonic line source load at any point of its vertical boundary. In thispaper, the method of fictitious line source force is used to constructe the expression ofincident waves which satisfies the stress free condition at the horizontal boundary in quarterplane, and the expression of scattering waves which satisfies the stress free conditions at thetwo vertical boundaries can be obtained with the aid of image method. Then, the unknowncoefficients can be determined by the continuous conditions of the stresses and displacementsaound the circular ealstic inclusion edge. Secondly, the expressions of the incident waves, thereflected waves and the refracted waves, which satisfy the boundary cinditions, can beconstructed using the image method. Then, the bi-material media is divided into two partsalong the vertical interface using the idea of interface “conjunction”, and the undetermined anti-plane forces are loaded at the linking sections respectively to satisfy continuity conditions.So a series of Fredholm integral equations of first kind for determining the unknown forcescan be set up through continuity conditions on interface. In the light of attenuationcharacteristic of the scattering waves, the unknown forces can be obtained by the method ofdirect discrete. Finally, some examples for dynamic stress concentration factor around thecircular elastic inclusion edge and the amplitude of ground surface displacement are given.Numerical results discuss the distribution of dynamic stress concentration factor andamplitude of ground surface with the changes of the nondimensional parameters.
（2）In part two, complex method and multi-polar coordinates technology are used toinvestigate the analysis solution for the multiple circular inclusions near the vertical interfacedisturbed by SH waves in bi-material half space. Firstly, the Green’s function should beconstructed in this problem, which is an essential solution to the displacement field for anelastic quarter plane with multiple circular inclusions disturbed by out-plane harmonic linesource loading at vertical surface. Secondly, the bi-material media is divided into two partsalong the bi-material interface based on the idea of interface “conjunction”, and the verticalsurfaces of the quarter space are loaded with undetermined anti-plane forces in order to satisfydisplacement continuity and stress continuity conditions at linking section. Then, the integralequations for determining the unknown forces can be set up through continuity conditions andthe Green’s function. Finally, some examples for dynamic stress concentration factor aroundthe circular elastic inclusion (hole) edge and the amplitude of ground surface displacement aregiven. Numerical results discuss the distribution of dynamic stress concentration factor withthe changes of the nondimensional parameters.
（3）In part three, complex method and Green’s function method are used toinvestigate the analysis solution for the circular inclusions and the vertical interfacial crackdisturbed by SH waves in bi-material half space. Firstly, the Green’s function should beconstructed in this problem, which is the same as the part one. Secondly, the interfacial crackis constructed with the aid of the crack-division technique. The detail is as follows: the bi-material media is divided into two parts along the vertical interface, and a pair of oppositeforces which are equal to the original stresses disturbed SH waves. Meanwhile, a series of theunknown forces must be loaded at the linking sections except the region of the crack to satisfycontinuity conditions. Then, the integral equations for determining the unknown forces can be set up through continuity conditions. Finally, some examples for dynamic stress concentrationfactor around the circular elastic inclusion edge and the dynamic stress intensity factors formode III at the crack tip are given. Numerical results discuss the distribution of dynamicstress concentration factor and the dynamic stress intensity factors with the changes of thenondimensional parameters.
引文
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