电磁弹耦合材料的断裂问题研究
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摘要
电磁弹耦合材料作为一种新型的智能材料,其应用范围涉及航天航空、精密制造、电子封装、核电设备等各个领域,电磁弹耦合材料的压电性能、压磁性能以及电磁耦合性能也日益受到广大学者的重视。本文从经典断裂理论和非局部理论两个方面对电磁弹耦合材料的断裂问题进行了分析研究,得到了一些具有重要实用价值的结论。
第一章分析评述了电磁弹耦合材料断裂问题研究现状和存在的问题,根据相关研究现状和存在问题,确立了本文要研究的内容。
第二章求解了功能梯度电磁弹性材料的多个非对称裂纹反平面断裂静态问题以及共线界面裂纹的反平面动态问题。利用裂纹上下面的位移间断函数和Fourier变换将断裂问题转化为成对的对偶积分方程的求解,再通过Jacobi多项式将位移间断函数展开,进而采用Schmidt方法求解,求得了裂纹附近应力场、电位移场和磁感应强度场的表达式以及裂纹尖端的应力强度因子、电位移强度因子和磁感应强度因子的数学表达式,给出了裂纹个数、裂纹长短、裂纹分布情况、材料属性变化情况、载荷类型等因素对于材料断裂特性的影响规律,结果表明裂纹的分布情况,材料的梯度参数等对材料的断裂特性的影响是明显的。
第三章研究了具有限制性电场和磁场导通性裂纹边界条件下多裂纹相互作用的平面断裂问题。首先利用广义Almansi理论给出本问题的基本解,进而将问题的求解转化为对偶积分方程的求解,随后利用Schmidt方法对此对偶积分方程进行求解,求得了有限导通电磁边界条件下多裂纹间相互作用的问题,进而得到了裂纹尖端的应力、电位移和磁感应强度因子的数学表达式,首次给出了裂纹内部介质的介电常数和磁导率对电磁弹耦合材料中裂纹断裂特性的影响规律,结果表明裂纹内部介质的介电常数和磁导率等对裂纹尖端附近电场、磁场的影响是明显的,而对应力场影响却很小。
第四章在考虑了裂纹内部介质介电常数和磁导率的情况下,利用广义Almansi理论求得了电磁弹性材料中矩形裂纹的基本解,给出了裂纹内部介质的介电常数、磁导率、裂纹间距离以及裂纹的几何形状对电磁弹耦合材料中矩形裂纹断裂特性的影响规律,给出了矩形裂纹的薄弱位置。
第五章首次把非局部理论推广应用到电磁弹耦合材料和功能梯度材料的断裂性能分析当中,克服了以往求解此类问题时遇到的数学问题,经过复杂的数学推导,最终将断裂问题转化为对偶积分方程的求解,与经典断裂理论所得的解相比,本文所得的非局部理论解中的应力场、电位移场和磁感应强度场在裂纹尖端处没有奇异性,而是一个具有明确物理意义的有限值,从而可以采用这一有限值来预测裂纹是否达到破坏的条件,同时也给出了裂纹内部介质的介电常数和磁导率对电磁弹性材料中裂纹尖端附近非奇异应力、电位移和磁感应强度场的影响规律。
本文关于电磁弹耦合材料断裂性能的研究,对其设计、生产和安全使用具有重要参考价值。
The megnetoelectroelastic composite material, as a kind of intelligent materials, which has been used in aeronautics/astronautics, precision manufacturing, electric packaging, nuclear equipment, and so on, attracts extensive attention of many researchers due to its properties of piezoelectric effect, piezomagnetic effect and coupled effect of magneto-electric. The fracture problem of megnetoelectroelastic materials was investigated in this article using the classical theory or the non-local theory. Some useful conclusions were achieved in analyzing the fracture mechanism of megnetoelectroelastic materials.
In chapter one, the research and problem about the fracture of megnetoelectroelastic composite materials are reviewed, the main research contents of this article were confirmed according to these comments.
In chapter two, the anti-plane static fracture problems of non-symmetrical multi-cracks and the anti-plane dynamic fracture problems of collinear cracks were solved in functionally graded megnetoelectroelastic composite materials. The problems were formulated through Fourier transform into dual integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the Schmidt method was used. Then the expressions of the full field stress, electric displacement, magnetic induction, the stress intensity factor, the electric displacement intensity factor and the magnetic induction intensity factor were obtained. The numerical results reveal the effects of the number, length and distributing of cracks, the changing of material properties and the loading of kind on the fracture characteristics of magnetoelectroelastic material. And it can be concluded that the material gradient parameter has obvious effects on the fracture characteristics of magnetoelectroelastic material.
In chapter three, the multiple mode-I cracks fracture problems were studied with the electromagnetic limited-permeable boundary conditions. The basic solution of multiple mode-I crack problem was formulated by using general Almansi theory. Then this problem was translated into the dual integral equations which could be solved using the Schmidt method. Finally, the stress intensity factors, electric displacement intensity factors and magnetic induction intensity factors at the crack tips were achieved. The effects of the dielectric permittivities and magnetic permeabilities of the medium inside cracks on fracture characteristics of cracks were drawn. It can be found that the dielectric permittivities and magnetic permeabilities of the medium inside cracks have clear impact on electric and magnetic fields near the crack tips.
In chapter four, considering the dielectric permittivities and magnetic permeabilities of the medium inside cracks, the basic solutions of the rectangular cracks in megnetoelectroelastic composite material were derived in three-dimension using general Almansi theory. As a result, the effects of the dielectric permittivities and magnetic permeabilities of the medium inside cracks, the distance between two rectangular cracks and the rectangular geometry of cracks were summarized on fracture characteristics of rectangular cracks in magnetoelectroelastic composite material. And the weak parts of the rectangular crack were given.
In chapter five, the non-local theory was firstly employed to study the fracture problem of megnetoelectroelastic material or functionally graded material. The mathematical difficult encountered in similar problems in previous literature were conquered by transforming the fracture problems into the dual integral equations. Different from the classical solutions, that the present non-local theory solution exhibits no stress, electric displacement and magnetic flux singularities at the crack tips in a magnetoelectroelastic media. The finite value of the stress fields at the crack tips exhibits the clear physical meaning, which make it possible using maximum stress criterion to predict the destroy condition of crack. At the same time, the rules of the effects of the dielectric permittivities and magnetic permeabilities of the medium inside cracks on fracture characteristics of cracks were given.
In this article, the investigation on fracture characteristics of megnetoelectroelastic composite material is very important to designing, manufacture and application of megnetoelectroelastic composite material.
第一章分析评述了电磁弹耦合材料断裂问题研究现状和存在的问题,根据相关研究现状和存在问题,确立了本文要研究的内容。
第二章求解了功能梯度电磁弹性材料的多个非对称裂纹反平面断裂静态问题以及共线界面裂纹的反平面动态问题。利用裂纹上下面的位移间断函数和Fourier变换将断裂问题转化为成对的对偶积分方程的求解,再通过Jacobi多项式将位移间断函数展开,进而采用Schmidt方法求解,求得了裂纹附近应力场、电位移场和磁感应强度场的表达式以及裂纹尖端的应力强度因子、电位移强度因子和磁感应强度因子的数学表达式,给出了裂纹个数、裂纹长短、裂纹分布情况、材料属性变化情况、载荷类型等因素对于材料断裂特性的影响规律,结果表明裂纹的分布情况,材料的梯度参数等对材料的断裂特性的影响是明显的。
第三章研究了具有限制性电场和磁场导通性裂纹边界条件下多裂纹相互作用的平面断裂问题。首先利用广义Almansi理论给出本问题的基本解,进而将问题的求解转化为对偶积分方程的求解,随后利用Schmidt方法对此对偶积分方程进行求解,求得了有限导通电磁边界条件下多裂纹间相互作用的问题,进而得到了裂纹尖端的应力、电位移和磁感应强度因子的数学表达式,首次给出了裂纹内部介质的介电常数和磁导率对电磁弹耦合材料中裂纹断裂特性的影响规律,结果表明裂纹内部介质的介电常数和磁导率等对裂纹尖端附近电场、磁场的影响是明显的,而对应力场影响却很小。
第四章在考虑了裂纹内部介质介电常数和磁导率的情况下,利用广义Almansi理论求得了电磁弹性材料中矩形裂纹的基本解,给出了裂纹内部介质的介电常数、磁导率、裂纹间距离以及裂纹的几何形状对电磁弹耦合材料中矩形裂纹断裂特性的影响规律,给出了矩形裂纹的薄弱位置。
第五章首次把非局部理论推广应用到电磁弹耦合材料和功能梯度材料的断裂性能分析当中,克服了以往求解此类问题时遇到的数学问题,经过复杂的数学推导,最终将断裂问题转化为对偶积分方程的求解,与经典断裂理论所得的解相比,本文所得的非局部理论解中的应力场、电位移场和磁感应强度场在裂纹尖端处没有奇异性,而是一个具有明确物理意义的有限值,从而可以采用这一有限值来预测裂纹是否达到破坏的条件,同时也给出了裂纹内部介质的介电常数和磁导率对电磁弹性材料中裂纹尖端附近非奇异应力、电位移和磁感应强度场的影响规律。
本文关于电磁弹耦合材料断裂性能的研究,对其设计、生产和安全使用具有重要参考价值。
The megnetoelectroelastic composite material, as a kind of intelligent materials, which has been used in aeronautics/astronautics, precision manufacturing, electric packaging, nuclear equipment, and so on, attracts extensive attention of many researchers due to its properties of piezoelectric effect, piezomagnetic effect and coupled effect of magneto-electric. The fracture problem of megnetoelectroelastic materials was investigated in this article using the classical theory or the non-local theory. Some useful conclusions were achieved in analyzing the fracture mechanism of megnetoelectroelastic materials.
In chapter one, the research and problem about the fracture of megnetoelectroelastic composite materials are reviewed, the main research contents of this article were confirmed according to these comments.
In chapter two, the anti-plane static fracture problems of non-symmetrical multi-cracks and the anti-plane dynamic fracture problems of collinear cracks were solved in functionally graded megnetoelectroelastic composite materials. The problems were formulated through Fourier transform into dual integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the Schmidt method was used. Then the expressions of the full field stress, electric displacement, magnetic induction, the stress intensity factor, the electric displacement intensity factor and the magnetic induction intensity factor were obtained. The numerical results reveal the effects of the number, length and distributing of cracks, the changing of material properties and the loading of kind on the fracture characteristics of magnetoelectroelastic material. And it can be concluded that the material gradient parameter has obvious effects on the fracture characteristics of magnetoelectroelastic material.
In chapter three, the multiple mode-I cracks fracture problems were studied with the electromagnetic limited-permeable boundary conditions. The basic solution of multiple mode-I crack problem was formulated by using general Almansi theory. Then this problem was translated into the dual integral equations which could be solved using the Schmidt method. Finally, the stress intensity factors, electric displacement intensity factors and magnetic induction intensity factors at the crack tips were achieved. The effects of the dielectric permittivities and magnetic permeabilities of the medium inside cracks on fracture characteristics of cracks were drawn. It can be found that the dielectric permittivities and magnetic permeabilities of the medium inside cracks have clear impact on electric and magnetic fields near the crack tips.
In chapter four, considering the dielectric permittivities and magnetic permeabilities of the medium inside cracks, the basic solutions of the rectangular cracks in megnetoelectroelastic composite material were derived in three-dimension using general Almansi theory. As a result, the effects of the dielectric permittivities and magnetic permeabilities of the medium inside cracks, the distance between two rectangular cracks and the rectangular geometry of cracks were summarized on fracture characteristics of rectangular cracks in magnetoelectroelastic composite material. And the weak parts of the rectangular crack were given.
In chapter five, the non-local theory was firstly employed to study the fracture problem of megnetoelectroelastic material or functionally graded material. The mathematical difficult encountered in similar problems in previous literature were conquered by transforming the fracture problems into the dual integral equations. Different from the classical solutions, that the present non-local theory solution exhibits no stress, electric displacement and magnetic flux singularities at the crack tips in a magnetoelectroelastic media. The finite value of the stress fields at the crack tips exhibits the clear physical meaning, which make it possible using maximum stress criterion to predict the destroy condition of crack. At the same time, the rules of the effects of the dielectric permittivities and magnetic permeabilities of the medium inside cracks on fracture characteristics of cracks were given.
In this article, the investigation on fracture characteristics of megnetoelectroelastic composite material is very important to designing, manufacture and application of megnetoelectroelastic composite material.
引文
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