直角域中圆形夹杂与裂纹反平面动力的相互作用
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摘要
本文采用复变函数、多极坐标移动技术和Green函数方法研究了SH波作用下直角域中圆形孔洞(夹杂)及其附近任意方位直线裂纹的相互作用问题。首先,构造出了应用于求解本文问题的Green函数,该函数为时间谐和反平面线源荷载作用于含有圆形孔洞(夹杂)直角域时的位移函数解。利用构造出的Green函数,即可求解直角域中圆形孔洞(夹杂)及其附近裂纹对SH波的散射。其次,采用裂纹“切割”的方法构造裂纹,即在欲出现裂纹区域上加载与直角域中圆形孔洞(夹杂)对SH波散射产生的应力相对应大小相等、方向相反的连续反平面荷载,从而构造出裂纹,并因而得到直角域中圆形孔洞(夹杂)和裂纹同时存在条件下的位移与应力场。最后,讨论了直角域不同的介质参数、圆形孔洞(夹杂)埋深及裂纹方位和长度条件下圆形孔洞(夹杂)周边的动应力集中系数、直角域的表面位移与裂纹尖端的动应力强度因子变化情况。本文具体研究工作概括为以下几点:
1.研究了直角域中圆形孔洞对SH波的散射与地震动。求解问题的关键是要构造一个能够自动满足直角域表面应力自由边界的散射波,该散射波利用SH波散射自身的对称性质来构造,并由圆孔应力自由边界来确定。最终则可将散射波问题归结为一个无穷代数方程组的求解。最后给出了具体算例,讨论了直角域不同的介质参数和圆孔埋深对孔边动应力集中系数分布及直角域表面位移的影响。
2.求解了应用于论文研究工作的Green函数,即时间谐和反平面线源荷载作用于含有圆形孔洞(夹杂)弹性直角域时的位移函数解,并讨论了Green函数的连续性、奇异性等性状。
3.研究了SH波对直角域中圆形孔洞及其附近任意方位直线形裂纹的散射问题。利用构造出的Green函数,采用“人工切割”的方法构造裂纹,导出了圆形孔洞与裂纹相互作用的位移场、应力场,研究了圆形孔洞周边的动应力集中情况、以及裂纹尖端的动应力强度因子。针对具体的算例,讨论了直角域中不同入射波数、入射角度、圆孔埋深、裂纹方位与长度等因素对孔边动应力集中系数、裂纹尖端动应力强度因子的影响。
4.研究了SH波对直角域中圆形夹杂及其附近直线形裂纹的散射问题。利用构造出的Green函数,采用“人工切割”的方法构造裂纹,推导出了圆形夹杂与裂纹相互作用的位移、应力表达式。针对具体的算例,讨论了不同入射波数、入射角度、圆形夹杂介质的剪切模量、圆形夹杂埋深、裂纹几何方位与长度对上述问题的影响。
The problems of SH waves scattering, which is caused by circular cavity (cylindrical inclusion) and crack of arbitrary position and arbitrary length in right-angle plane, are studied using the methods of complex variables, muti-polar coordinates and Green’s Function. Firstly, a suitable Green’s function is constructed, which is an essential solution to the displacement field for elastic right-angle plane possessing circular cavity (cylindrical inclusion) while bearing in-of-plane harmonic line source load at arbitrary point. Then using the Green’s function, the scattering problem of SH waves is studied, which is caused by circular cavity (cylindrical inclusion) and crack of arbitrary position and arbitrary length in right-angle plane. Then using the method of crack division, the crack is established: reverse stresses are inflicted along the crack, that is, in-of-plane harmonic line source loads, which are equal in the quantity but opposite in the direction to the stresses produced for the reason of SH waves scattering by circular cavity (cylindrical inclusion), are loaded at the region where crack will appear, thus the crack can be made out. Thus expressions of displacement and stress are established while circular cavity (cylindrical inclusion) and crack are both in existent. Using the expressions, the dynamic stress concentration around the circular cavity (cylindrical inclusion), the ground motion of right-angle plane and the dynamic stress intensity factor at crack tip are discussed. The work in detail is as follows:
1. The problem of SH waves scattering caused by circular cavity in right-angle plane is investigated. The key point of the work is that: based on the symmetry of SH waves scattering and the method of multi-polar coordinate system, a scattering field which satisfies the stress-free conditions at the surfaces of right-angle plane caused by the circular cavity is constructed. Then the expression of scattering field can be determined by the stress-free boundary condition of circular cavity. Finally, the solution of this problem can be reduced to a series of algebraic equations, which can be solved numerically by truncating the infinite algebraic equations to the finite ones. Numerical examples are provided for cases, and some influencing factors to the problem are discussed, such as the wave number, the incident angle of SH waves andt he position of circular cavity.
2. The Green’s function is constructed compatibly, which is an essential solution to the displacement field for the elastic right-angle plane possessing a circular cavity (cylindrical inclusion) while bearing in-of-plane harmonic line source load at arbitrary point. The continuity, singularity and some other characteristics of the Green’s function are discussed as well.
3. The problem of scattering of SH waves by a circular cavity and a beeline crack of arbitrary position and arbitrary length in right-angle plane is investigated. Using the Green’s function which is suitable to the present problem, the expressions of displacement and stress are deduced with crack-division technique while the circular cavity and the crack are both in existent. The dynamic stress concentration around the circular cavity and the dynamic stress intensity factor at the crack tip are discussed. Furthermore, some examples and results are given. Finally, some influencing factors to the problem are discussed, such as the wave number, the incident angle of SH-wave, the position of circular cavity , and the position, angle and length of crack.
4. The problem of scattering of SH waves by a cylindrical inclusion and a beeline crack of arbitrary position and arbitrary length in right-angle plane is investigated. Using the Green’s function which is suitable to the present problem, the expressions of displacement and stress are deduced with crack-division technique while the interaction of the cylindrical inclusion and crack are both in existent. The dynamic stress concentration around the circular cavity and the dynamic stress intensity factor at the crack tip are discussed. Furthermore, some examples and results are given. Finally, some influencing factors to the problem are discussed, s such as the wave number, the incident angle of SH-wave, the shearing modulus of cylindrical inclusion medium, the position of cylindrical inclusion , and the position, angle and length of crack.
1.研究了直角域中圆形孔洞对SH波的散射与地震动。求解问题的关键是要构造一个能够自动满足直角域表面应力自由边界的散射波,该散射波利用SH波散射自身的对称性质来构造,并由圆孔应力自由边界来确定。最终则可将散射波问题归结为一个无穷代数方程组的求解。最后给出了具体算例,讨论了直角域不同的介质参数和圆孔埋深对孔边动应力集中系数分布及直角域表面位移的影响。
2.求解了应用于论文研究工作的Green函数,即时间谐和反平面线源荷载作用于含有圆形孔洞(夹杂)弹性直角域时的位移函数解,并讨论了Green函数的连续性、奇异性等性状。
3.研究了SH波对直角域中圆形孔洞及其附近任意方位直线形裂纹的散射问题。利用构造出的Green函数,采用“人工切割”的方法构造裂纹,导出了圆形孔洞与裂纹相互作用的位移场、应力场,研究了圆形孔洞周边的动应力集中情况、以及裂纹尖端的动应力强度因子。针对具体的算例,讨论了直角域中不同入射波数、入射角度、圆孔埋深、裂纹方位与长度等因素对孔边动应力集中系数、裂纹尖端动应力强度因子的影响。
4.研究了SH波对直角域中圆形夹杂及其附近直线形裂纹的散射问题。利用构造出的Green函数,采用“人工切割”的方法构造裂纹,推导出了圆形夹杂与裂纹相互作用的位移、应力表达式。针对具体的算例,讨论了不同入射波数、入射角度、圆形夹杂介质的剪切模量、圆形夹杂埋深、裂纹几何方位与长度对上述问题的影响。
The problems of SH waves scattering, which is caused by circular cavity (cylindrical inclusion) and crack of arbitrary position and arbitrary length in right-angle plane, are studied using the methods of complex variables, muti-polar coordinates and Green’s Function. Firstly, a suitable Green’s function is constructed, which is an essential solution to the displacement field for elastic right-angle plane possessing circular cavity (cylindrical inclusion) while bearing in-of-plane harmonic line source load at arbitrary point. Then using the Green’s function, the scattering problem of SH waves is studied, which is caused by circular cavity (cylindrical inclusion) and crack of arbitrary position and arbitrary length in right-angle plane. Then using the method of crack division, the crack is established: reverse stresses are inflicted along the crack, that is, in-of-plane harmonic line source loads, which are equal in the quantity but opposite in the direction to the stresses produced for the reason of SH waves scattering by circular cavity (cylindrical inclusion), are loaded at the region where crack will appear, thus the crack can be made out. Thus expressions of displacement and stress are established while circular cavity (cylindrical inclusion) and crack are both in existent. Using the expressions, the dynamic stress concentration around the circular cavity (cylindrical inclusion), the ground motion of right-angle plane and the dynamic stress intensity factor at crack tip are discussed. The work in detail is as follows:
1. The problem of SH waves scattering caused by circular cavity in right-angle plane is investigated. The key point of the work is that: based on the symmetry of SH waves scattering and the method of multi-polar coordinate system, a scattering field which satisfies the stress-free conditions at the surfaces of right-angle plane caused by the circular cavity is constructed. Then the expression of scattering field can be determined by the stress-free boundary condition of circular cavity. Finally, the solution of this problem can be reduced to a series of algebraic equations, which can be solved numerically by truncating the infinite algebraic equations to the finite ones. Numerical examples are provided for cases, and some influencing factors to the problem are discussed, such as the wave number, the incident angle of SH waves andt he position of circular cavity.
2. The Green’s function is constructed compatibly, which is an essential solution to the displacement field for the elastic right-angle plane possessing a circular cavity (cylindrical inclusion) while bearing in-of-plane harmonic line source load at arbitrary point. The continuity, singularity and some other characteristics of the Green’s function are discussed as well.
3. The problem of scattering of SH waves by a circular cavity and a beeline crack of arbitrary position and arbitrary length in right-angle plane is investigated. Using the Green’s function which is suitable to the present problem, the expressions of displacement and stress are deduced with crack-division technique while the circular cavity and the crack are both in existent. The dynamic stress concentration around the circular cavity and the dynamic stress intensity factor at the crack tip are discussed. Furthermore, some examples and results are given. Finally, some influencing factors to the problem are discussed, such as the wave number, the incident angle of SH-wave, the position of circular cavity , and the position, angle and length of crack.
4. The problem of scattering of SH waves by a cylindrical inclusion and a beeline crack of arbitrary position and arbitrary length in right-angle plane is investigated. Using the Green’s function which is suitable to the present problem, the expressions of displacement and stress are deduced with crack-division technique while the interaction of the cylindrical inclusion and crack are both in existent. The dynamic stress concentration around the circular cavity and the dynamic stress intensity factor at the crack tip are discussed. Furthermore, some examples and results are given. Finally, some influencing factors to the problem are discussed, s such as the wave number, the incident angle of SH-wave, the shearing modulus of cylindrical inclusion medium, the position of cylindrical inclusion , and the position, angle and length of crack.
引文
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