压电弹性体孔边裂纹问题研究
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摘要
在过去的一百多年中，压电材料的各种性能（弹性、压电性、介电性、热释电性、铁电性、光电性）相继被人们发现。而伴随着压电陶瓷制备技术的日渐完善，压电材料的应用也日趋广泛。如今，压电材料及其结构在日常生活和高科技的各个领域中随处可见，如：雷达通讯、水声超声、医学成像、红外探测、航空航天、动物仿生和电子测量等。在工程实际中，开孔结构是一种较为常见的结构。在复杂的加载环境下，含孔洞的结构不可避免的会出现应力集中，在孔边诱发裂纹，从而导致材料的断裂破坏。因此，研究压电弹性体孔边裂纹问题具有重要的理论意义和工程价值。
本文利用复变函数和保角变换的方法，结合柯西积分，针对无限大压电体内孔边裂纹问题进行理论和数值的研究，导出了某些经典问题的解析解；针对有限大压电体内的孔边裂纹问题，采用半解析半数值的边界元方法，分析了裂尖场的变化规律。主要工作如下：
(1)利用复变函数和保角变换的方法，提出了研究压电弹性体中任意形状孔边裂纹的反平面问题的解法。对于经典的圆孔/椭圆孔孔边裂纹问题，在前人的基础上，给出了改进后的保角变换函数，从而得到了这些经典问题的解析解；对于任意形状的孔边裂纹问题，提出了用数值保角变换来求得近似的保角变换函数的方法，从而得到该问题的近似解。通过对孔边裂纹问题的分析，探寻裂尖场强度因子和能量释放率随裂纹尺寸、圆孔直径和外载的变化关系，从而简单判断裂纹的扩展。
(2)发展了双材料中含椭圆孔孔边界面裂纹的无限大压电弹性体反平面问题的解法。利用椭圆孔的保角变换函数将椭圆孔及其裂纹保角变换到新平面内的直线裂纹，从而将椭圆孔孔边裂纹双材料问题转化为新平面内的直线界面裂纹问题。利用Stroh公式，分别得到了在电不可穿透裂纹假设和电可穿透裂纹假设下的本问题的复势函数和场强度因子的解析表达式。分析了在两种电边界条件下裂尖的奇异性，以及场强度因子随椭圆大小、裂纹尺寸的变换关系，从理论上找到降低裂尖强度因子的方法。
(3)考虑热应力对结构的影响，探讨了复杂形状孔边裂纹的广义二维问题的一般方法。首先由椭圆孔的保角变换函数导出由椭圆孔演化而来的复杂形状孔边非对称裂纹的保角变换函数；其次，基于绝热边界条件，导出了热复势函数表达式；再次，根据热复势函数，假设出电弹场的复势函数表达式，利用无穷远处的场有界性条件、位移单值条件、边界自由和电不可穿透条件，得到该复势函数的表达式；最后，分析了热流、孔周环向应力和环向电位移随裂纹尺寸变化趋势，讨论了裂纹尺寸、椭圆几何形状及外载的变化对场强度因子的影响，从理论上寻找降低场强度因子的方法。
(4)针对任意形状孔边裂纹的广义二维问题，将椭圆孔的存在考虑入Green函数基本解中，从而建立了间接边界元的半解析半数值解法。研究任意形状孔边裂纹裂尖应力场及电位移场的分布规律，分析了各种几何尺寸和外载对裂尖场强度因子的影响。由于在基本解中已经考虑了椭圆孔的存在，因此在边界的离散过程中，该椭圆孔不再作为边界离散。当椭圆孔退化为裂纹时，避免了裂纹尖端边界离散的困难，极大地提高了裂尖场的计算精度。
During the past century, the properties of piezoelectric materials, such as those of elasticity,piezoelectricity, dielectricity, pyroelectricity, ferroelectricity and photoelectricity, have been found.With the piezoelectric ceramic manufacturing technique beening improved gradually, the application ofpiezoelectric materials has become more and more wide, e.g. they hace benn used in radarcommunications, ultrasound, medical imaging, infrared detection, aerospace, animal bionic, electronicmeasurement and so on.
Structures with holes are very common in most practical engineering. Under complicated loadingenvironment, this kind of structures will produce the so-called “stress concentration” phenomenon,which will leads to the damage of the materials. So, it is very important to study the cracked-holeproblems in the peizoelectric materials.
In the present paper, the cracked-hole problems in an infinite piezoelectric body are studied byusing the complex variable method and the conformal mapping function, combined with the Cauchyintegral, and the analytical solutions are derived for some classic problems. On the other hand, theboundary element method is used to deal with the abritrary shape cracked-hole problems in a finitepiezoelectric body. The main works can be summarized as follows:
(1) By using the complex variable method and the conformal mapping function, the cracked-holeanti-plane problems in the transversely isotropic piezoelectric materials are solved. For the classicproblems, the improved mapping function is obtained, and the explicit and exact expressions forthe complex potentials, field intensity factors and energy release rates are presented respectivelyon the assumption that the surface of the cracks and hole is electrically impermeable. For thearbitrary shape cracked-hole problems, the method of numerical conformal mapping is used toobtain the mapping problem which maps the outside of arbitrary shape with a crack into theoutside of a circular hole. Based on the mapping function, the approximate expressions for thecomplex potentials, field intensity factors and energy release rates are presented, respectively.Numerical analysis is then conducted to discuss the influences of crack length and appliedmechanical/electric loads on the field intensity factors and energy release rates for one and twoedge cracks, respectively.
(2) The solution of anti-plane problems for the bimaterials which contain an elliptic hole with twoedge cracks is obtained. The mapping function which maps the ellipse to a line crack is obtained. Based on the assumption of permeable or impermeable crack, the expressions for the complexpotentials and field intensity factors are obtained by using the Stroh formulation, respectively. Thesingularity of the crack tip, the field intensity factors are studied.
(3) The cracked-hole problems in the generalized two-dimensional electrical materials is studied,where the heat flow is considered. Firstly, the mapping function based on the elliptic hole has beenobtained, which maps the outer of the cracked-hole into the outer of the circle hole. Secondly, theheat complex function is solved based on the adiabatic condition. Thirdly, the expressions for thecomplex potentials and field intensity factors are presented under the condition that the stress,strain and electric displacement are bounded at infinity, single-valued displacement, theequilibrium of mechanical, and electrical electrically impermeable boundary condition. Andfinally, some numerical analyses are also made to discuss the influences of crack length and heatflux on the electroelastic fields and fields intensity factors, in order to find a way to decrease thefield intensity factors.
(4) By using the boundary element method (BEM), the cracked-hole problems in the finitepiezoelectric body is studied. The stress field and the electrical displacement field around thecrack tip are analyzed. For the arbitrary shape of hole, the influence of the geometric dimensionsand the external loads on the field intensity factors are discussed. Since the elliptic hole or thecrack has been considered in the fundamental solution, there is no need to discrete the ellipticalhole boundary or the crack boundary in the discretization process, which avoids the problem ofdiscretization caused by the singularity at the crack tip, and greatly improves the accuracy of thecalculation at the crack tip.
本文利用复变函数和保角变换的方法，结合柯西积分，针对无限大压电体内孔边裂纹问题进行理论和数值的研究，导出了某些经典问题的解析解；针对有限大压电体内的孔边裂纹问题，采用半解析半数值的边界元方法，分析了裂尖场的变化规律。主要工作如下：
(1)利用复变函数和保角变换的方法，提出了研究压电弹性体中任意形状孔边裂纹的反平面问题的解法。对于经典的圆孔/椭圆孔孔边裂纹问题，在前人的基础上，给出了改进后的保角变换函数，从而得到了这些经典问题的解析解；对于任意形状的孔边裂纹问题，提出了用数值保角变换来求得近似的保角变换函数的方法，从而得到该问题的近似解。通过对孔边裂纹问题的分析，探寻裂尖场强度因子和能量释放率随裂纹尺寸、圆孔直径和外载的变化关系，从而简单判断裂纹的扩展。
(2)发展了双材料中含椭圆孔孔边界面裂纹的无限大压电弹性体反平面问题的解法。利用椭圆孔的保角变换函数将椭圆孔及其裂纹保角变换到新平面内的直线裂纹，从而将椭圆孔孔边裂纹双材料问题转化为新平面内的直线界面裂纹问题。利用Stroh公式，分别得到了在电不可穿透裂纹假设和电可穿透裂纹假设下的本问题的复势函数和场强度因子的解析表达式。分析了在两种电边界条件下裂尖的奇异性，以及场强度因子随椭圆大小、裂纹尺寸的变换关系，从理论上找到降低裂尖强度因子的方法。
(3)考虑热应力对结构的影响，探讨了复杂形状孔边裂纹的广义二维问题的一般方法。首先由椭圆孔的保角变换函数导出由椭圆孔演化而来的复杂形状孔边非对称裂纹的保角变换函数；其次，基于绝热边界条件，导出了热复势函数表达式；再次，根据热复势函数，假设出电弹场的复势函数表达式，利用无穷远处的场有界性条件、位移单值条件、边界自由和电不可穿透条件，得到该复势函数的表达式；最后，分析了热流、孔周环向应力和环向电位移随裂纹尺寸变化趋势，讨论了裂纹尺寸、椭圆几何形状及外载的变化对场强度因子的影响，从理论上寻找降低场强度因子的方法。
(4)针对任意形状孔边裂纹的广义二维问题，将椭圆孔的存在考虑入Green函数基本解中，从而建立了间接边界元的半解析半数值解法。研究任意形状孔边裂纹裂尖应力场及电位移场的分布规律，分析了各种几何尺寸和外载对裂尖场强度因子的影响。由于在基本解中已经考虑了椭圆孔的存在，因此在边界的离散过程中，该椭圆孔不再作为边界离散。当椭圆孔退化为裂纹时，避免了裂纹尖端边界离散的困难，极大地提高了裂尖场的计算精度。
During the past century, the properties of piezoelectric materials, such as those of elasticity,piezoelectricity, dielectricity, pyroelectricity, ferroelectricity and photoelectricity, have been found.With the piezoelectric ceramic manufacturing technique beening improved gradually, the application ofpiezoelectric materials has become more and more wide, e.g. they hace benn used in radarcommunications, ultrasound, medical imaging, infrared detection, aerospace, animal bionic, electronicmeasurement and so on.
Structures with holes are very common in most practical engineering. Under complicated loadingenvironment, this kind of structures will produce the so-called “stress concentration” phenomenon,which will leads to the damage of the materials. So, it is very important to study the cracked-holeproblems in the peizoelectric materials.
In the present paper, the cracked-hole problems in an infinite piezoelectric body are studied byusing the complex variable method and the conformal mapping function, combined with the Cauchyintegral, and the analytical solutions are derived for some classic problems. On the other hand, theboundary element method is used to deal with the abritrary shape cracked-hole problems in a finitepiezoelectric body. The main works can be summarized as follows:
(1) By using the complex variable method and the conformal mapping function, the cracked-holeanti-plane problems in the transversely isotropic piezoelectric materials are solved. For the classicproblems, the improved mapping function is obtained, and the explicit and exact expressions forthe complex potentials, field intensity factors and energy release rates are presented respectivelyon the assumption that the surface of the cracks and hole is electrically impermeable. For thearbitrary shape cracked-hole problems, the method of numerical conformal mapping is used toobtain the mapping problem which maps the outside of arbitrary shape with a crack into theoutside of a circular hole. Based on the mapping function, the approximate expressions for thecomplex potentials, field intensity factors and energy release rates are presented, respectively.Numerical analysis is then conducted to discuss the influences of crack length and appliedmechanical/electric loads on the field intensity factors and energy release rates for one and twoedge cracks, respectively.
(2) The solution of anti-plane problems for the bimaterials which contain an elliptic hole with twoedge cracks is obtained. The mapping function which maps the ellipse to a line crack is obtained. Based on the assumption of permeable or impermeable crack, the expressions for the complexpotentials and field intensity factors are obtained by using the Stroh formulation, respectively. Thesingularity of the crack tip, the field intensity factors are studied.
(3) The cracked-hole problems in the generalized two-dimensional electrical materials is studied,where the heat flow is considered. Firstly, the mapping function based on the elliptic hole has beenobtained, which maps the outer of the cracked-hole into the outer of the circle hole. Secondly, theheat complex function is solved based on the adiabatic condition. Thirdly, the expressions for thecomplex potentials and field intensity factors are presented under the condition that the stress,strain and electric displacement are bounded at infinity, single-valued displacement, theequilibrium of mechanical, and electrical electrically impermeable boundary condition. Andfinally, some numerical analyses are also made to discuss the influences of crack length and heatflux on the electroelastic fields and fields intensity factors, in order to find a way to decrease thefield intensity factors.
(4) By using the boundary element method (BEM), the cracked-hole problems in the finitepiezoelectric body is studied. The stress field and the electrical displacement field around thecrack tip are analyzed. For the arbitrary shape of hole, the influence of the geometric dimensionsand the external loads on the field intensity factors are discussed. Since the elliptic hole or thecrack has been considered in the fundamental solution, there is no need to discrete the ellipticalhole boundary or the crack boundary in the discretization process, which avoids the problem ofdiscretization caused by the singularity at the crack tip, and greatly improves the accuracy of thecalculation at the crack tip.
引文
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