断裂问题中的哈密顿体系方法及其应用
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摘要
随着科学技术的发展,多功能材料和智能材料越来越受到关注。电磁材料就是其中一种。利用这些材料的性质,许多智能结构及产品被用于工程结构中。基于该类材料特殊的力电磁能量转换特性,许多仪器和设备被设计,并在实际工程中得到广泛应用,如石油,化工,航空航天,军事,制造业,以及核工业等。这些仪器和设备在实际运行中,往往会受到力、电、磁、热耦合荷载作用。此外,由于受制造和运行环境的影响,会导致裂纹的出现,如疲劳裂纹等。裂纹往往会造成结构直接破坏和失效,因此,对其研究是完全必要的。特别是对精密仪器设备中的功能材料(如电磁材料等)和结构的断裂行为研究尤为重要。
研究和揭示材料和结构的断裂机理有利于提高设计和制造水平,由此可以有效的减少事故发生,并尽可能地延长设备的使用寿命。然而,研究该类问题需要系统的考察力电磁热相互作用效应和工况环境。虽然目前有很多相关的理论和方法,但仍需完善和改进,特别是对相关电磁材料的断裂行为研究方法等。从现有的方法看,其中大部分皆基于拉格朗日体系下的一类变量的控制方程。由此将面对高阶微分方程的求解和数值处理方法,这就给问题求解带来了相当的困难。可喜的是钟万勰院士首次将哈密顿体系引入到弹性力学和应用力学中,开创了一种全新的理念和方法,并建立了基于哈密顿体系的研究问题平台。在钟院士的带领下,他的科研团队对许多领域和研究方向系统和深入的展开探讨,并取得丰硕的研究成果。这些研究成果也为本论文的研究提供基础和依据。
本博士论文以带有边缘裂纹的弹性材料、压电材料和电磁弹性材料为研究对象,对裂纹尖端的奇异性和强度因子进行系统分析。并利用辛本征解展开方法和辛共轭正交关系,得到对偶变量和强度因子的解析表达式。该方法能克服传统半逆法的弱点,给出一种直接方法和系统方法。取得的研究结果为人们研究断裂问题提供了全新的认识。具体研究成果如下：
1.平面和空间弹性体的应力强度因子研究
在哈密顿体系下,位移和广义应力互为对偶变量。通过研究以混合变量描述的对偶正则方程,得到含断裂问题的辛本征解。在辛空间中构造出完备的辛本征解空间。哈密顿体系下的辛本征解可以分为两类：零本征值本征解和非零本征值本征解。零本征值本征解即是该问题对应的圣维南问题的解,代表了该问题对应的等效边界条件意义下的解。非零本征值本征解则包括圣维南原理所覆盖的解,即体现边缘效应和局部效应的解。研究工作以平面问题作为突破口,进而在空间问题展开。由于辛本征解之间存在辛共轭正交关系,问题的解可由辛本征解得展开得到,从而获得问题解得解析表达式。应力强度因子和T应力可由特殊的辛本征解和其系数直接表示。进一步利用边界条件和辛共轭正交关系,可确定所有展开级数的系数。这样Ⅰ型,Ⅱ型和Ⅲ型应力强度因子(KⅠ, KⅡ, KⅢ)同时被直接得到。此直接方法突显出更加方便和有效。利用边界积分等手段,将圆形外边界拓宽到非规则边界的裂纹问题,直接得到的半解析结果和数值结果。研究工作为进一步讨论动力问题提供了依据和基础。这些研究成果已经发表在Engineering Fracture Mechanics (2009,76(12):1866-1882), International Journal of Mechanical Sciences (2010,52(7):892-903)和Journal of Sound and Vibration (2011,330:1005-1017)。
2.含边缘裂纹压电材料的力/电强度因子和奇异性分析
将哈密顿体系求解方法应用于含边缘裂纹压电材料奇异性分析中。以—空间坐标模拟时间,采用弹性势能(应变能)和压电能表示拉格朗日函数和变分原理,得到广义位移(位移和电势)和广义应力(应力和电位移)的对偶关系。利用哈密顿原理构造出以广义位移和广义应力混合变量描述的对偶正则方程。利用哈密顿体系很好的性质和现代数学工具对含边缘裂纹压电材料问题展开研究和讨论。分析电可渗透和电不可渗透裂纹在尖端处的奇异性,并得到应力强度因子和电位移强度因子以及影响因素。结果表明,对于电可渗透裂纹,电场强度因子始终为零,即电场在裂纹尖端不存在奇异性；应变强度因子与材料常数无关,只与外边界荷载工况有关；应力强度因子和电位移强度因子可以用材料常数与广义位移强度因子的线性组合表示。相关成果已经发表在International Journal of Solids and Structures (2009,46(20):3577-3586)。
3.含边缘裂纹电磁弹性材料的耦合强度因子研究
构造含边缘裂纹电磁弹性材料问题的哈密顿体系结构,研究Ⅲ型裂纹问题的断裂行为。该类问题可归结为反平面问题。在哈密顿体系下,轴向位移与剪应力、电势与电位移、磁势与磁感应强度分别互为对偶变量。以这些变量和对偶变量组成的混合变量描述的基本问题对研究混合边界条件问题非常直接和特别有效。在得到辛本征解空间以后,将应力强度因子,电位移强度因子和磁感应强度因子等问题归结为线性代数方程组求解的问题。在此基础上,对电磁可渗透和电磁不可渗透裂纹问题分别进行分析和研究。得到电磁弹性材料反平面断裂问题的解析解和一些规律。研究结果表明,广义位移强度因子与材料常数无关,只与本征值为二分之一的本征解系数有关；广义应力变量在裂纹尖端处表现出传统-0.5阶次的奇异性,并且它们对应的强度因子可直接表示为材料常数和广义位移强度因子的函数；在电磁可渗透的裂纹问题中电场强度和磁场强度不出现奇异性,即对应的强度因子为零。研究成果已经发表在Engineering Fracture Mechanics (2010, 77(16):3157-3173)和Computers & Structures (2011,89:631-645)。
4.稳态和瞬态热弹性问题中的热应力强度因子
提出热传导方程和热弹性方程在空间坐标下可分离变量的哈密顿形式。研究工作分为两部分：首先在哈密顿体系下建立与热传导方程等价的正则方程,并求解温度场。温度场可由一系列辛本征解组合所表示,其中包括稳态和瞬态温度函数。然后利用所得的温度场构造热弹性问题的非齐次哈密顿对偶方程以及相应的初边条件。在此过程中,将时间变量只作为一个“空间坐标”,而将一空间坐标模拟为“时间坐标”。这样,提出
个全新地考虑问题思路。在这种观念下对问题求解,得到对应的辛本征解,即齐次正则方程的通解和非齐次方程特解。通过对解析解和数值结果的分析,得出结论：热应力问题的裂纹尖端奇异指数为-0.5；应力强度因子直接由第一阶非零本征解和温度函数表示和确定；最大热应力发生在裂纹尖端区域,并且成指数向外衰减。研究结果发现：在一定的温度环境下,热应力强度因子随裂纹长度增大而变小的现象。也就是裂纹会出现止裂的结果。这种现象对于工程设计和工程设备寿命评估是非常重要。根据这些研究工作,已经连续两篇文章发表在Journal of Thermal Stresses (2010,33(3):262-278; 2010,33(3)：279-301)。
With the development of science and technology, multifunctional materials and smart materials drew more and more attention of the designers. The magneto-electro-elastic material is one of these popular materials. Based on the properties of these materials, many intelligent structures and products are applied in the engineering structure. With the aid of magneto-electro-mechanical energy conversion, many instruments and facilities are designed and extensively used in petroleum industry, chemical Industry, aerospace industry, military affairs, manufacturing and nuclear industry. When subjected to coupled mechanical, magnetic, electric and thermal loads in service, these instruments and facilities are fail prematurely due to some defects arising during their manufacturing processes, i.e. fatigue crack. Therefore, it is of great importance to study the fracture behaviors of there composites and understand the cracking mechanism.
Studying and understanding the cracking mechanism of these materials and structures is useful to optimize the structural design and manufacturing standards. Moreover, it is pertinent to minimize the catastrophic failures for enhanced performance in fracture and wear resistance. However, the nature of the problem requires a systematic examination of the interplays among the electro-magneto-mechanical-thermal effects and working condition. Although some theoretical and experimental studies have been performed, many related challenges have not been fully resolved and need to be refine and improved, particularly in the fracture behaviors of the magneto-electro-elastic materials. In view of these literatures, it can be seen that all of the methods used the governing equation derived previously in Lagrangian sense involving only one kind of variables in terms of the energy. Since highter-order differential equations are not conducive to numerical solution methods, such elimination will cause problems in numerical analysis. Fortunately, Academician Zhong Wanxie developed an analytical symplectic approach for some basic problems in elasticity and in applied mechanics. It is a new concept and method which provids a research platform based on Hamiltonian system. Under the leadership of Academician Zhong, his associates have extendeded the method to many areas and directions. Some research results have been published and provide a basic technique for the dissertation.
In this dissertation, we study the singularities and intensity factors systematically for the edge-crack elastic, piezoelectric and magneto-electro-elastic materials. With the aid of symplectic expansion and symplectic adjoint orthogonality among the eigenfunctions, the analytic expressions for both of the intensity factors and dual variables are obtained. It overcomes the the defects of classical semi-inverse methods, and it is rational and systematic with a clearly defined, step-by-step derivation procedure. The present work provides a better understanding for fracture problem. The conclusions are listed below:
1. Analytic stress intensity factors for two-and three-dimensional problems
A Hamiltonian system is introduced by the energy method. The displacements and stresses are proved to be conjugating (dual) to each other. The eigensolutions for fracture problems are solved from the Hamilton equations based on the mixed variables. These eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions. In the symplectic space, the solution consists of two parts:zero eigenvalue solutions and non-zero eigenvalue solution. All the Saint Venant solutions have been identified as the zero-eigenvalue solutions and the Saint Venant solutions represent the average physical. The non-zero-eigenvalue solutions corresponding to effects which are coverd by the Saint-Venant principle, i.e. the local boundary layer effects. The plane problems are regarded as a breakthrough, the symplectic method is extended to the space problems. Based on symplectic adjoint orthogonality among the eigensolutions, the analytical solutions are obtained and can be expanded in terms of the symplectic eigensolutions. The stress intensity factors and T-stresses are identified to be the coefficients of certain eigenfunctions. The coefficients of the series are determined from the boundary conditions and the relationship of symplectic adjoint orthogonality. Thus, ModeⅠ,Ⅱ,Ⅲstress intensity factors are obtained simultaneously. It is a direct and effective method. In addition, a boundary integral technology is developed for the non-circular domains, semi-analytical or numerical results can be obtained. Moreover, the present work provides a way to solve the dynamic problems. These work has been published in the Engineering Fracture Mechanics (2009, 76(12):1866-1882), International Journal of Mechanical Sciences (2010,52(7):892-903) and Journal of Sound and Vibration (2011,330:1005-1017).
2. Stress/electric intensity factors and singularities analysis for the edge-crack piezoelectric materials
The Hamiltonian formalism is used to analyze singularities for the edge-crack piezoelectric materials. A space coordinate is defined as the longitudinal direction via an appropriate variable transformation to simulate the "time coordinate". With the aid of variational principle and the Lagrangian function which consists of elastic potential energy and piezoelectric energy, we generalized displacements (longitudinal direction displacement and electrical potential function) and stresses (shear stresses and electric displacements) as primary unknowns will result in a complete set of eigensolutions ensuring convergence and will give the generalized stresses directly as dual variables. Based on the properties of Hamilton system and modern mathematical tools, analysis of the edge-crack piezoelectric materials is preformed. The singularities and intensity factors for a permeable or impermeable crack in piezoelectric material are obtained. Furthermore, the influence factors are discussed. The results show that the electric field intensity factors for the electrically permeable crack is always of the zero value, or the electric field has no singularity at the crack tip. The strain intensity factors become independent of the material constants, it depends on the edge loading conditions only. The stress and the electric displacement intensity factors can be represented by a combination of material constants and the generalized displacement intensity factors. This work is published in International Journal of Solids and Structures (2009,46(20): 3577-3586).
3. The study of coupling intensity factors for the edge-crack in magneto-electro-elastic media
A Hamilton system is established for the edge-crack in magneto-electro-elastic media for studying the fracture behaviors. It can be reduced to the anti-plane problem. In symplectic space, it can be proved that the displacements and stresses, electric potentials and electric displacements, magnetic potentails and magnetic induction functions are conjugating (dual) variables respectively. It is convenient to solve the mixed boundary conditions with the aid of the mixed variables which consist of the original variables and dual variables. Using the exited eigensolutions, the solutions of stress, electric displacement and magnetic induction intensity factors are reduced to the solutions of a set of linear algebraic equations. Both of the permeable and impermeable electromagnetic boundary conditions at the crack surfaces are adopted and discussed. Some resultes and a closed form solution for-the anti-plane fracture problem of magneto-electro-elastic materials are obtained. The results show that the intensity factors can be obtained by the terms associated with the eigenvalue solutions having the coefficients of 1/2;The generalized stress variables at the crack tip show the traditional square root singularities and can be represented by a combination of material constants and the generalized displacement intensity factors; the field variables, which can permeate the crack surfaces, produce no singularities or their corresponding intensity factors always equal to zero. These work has been published in the Engineering Fracture Mechanics (77(16),2010, 3157-3173) and Computers & Structures (2011,89:631-645).
4. Analytic stress intensity factors for the steady and transient thermoelasticity
The equations of thermal thermal conduction and thermoelasticity are first rewritten in Hamiltonian form where the variables are separable in spatial coordinates. Our study will be considered in two parts:At first, a generalized Hamilton system will be introduced to the heat conduction problem and the temperature function will be represented by a symplectic series analytically. Using the existed solution, the temperature function for both of the steady and transient thermal conducitons will be obtained. Then, a set of inhomogeneous Hamiltion equations and corresponding boundary conditions are obtained by the temperature function which obtained from the first part. For the symplectic approach, the radial coordinate is defined as the longitudinal direction via an appropriate variable transformation to simulate the "time coordinate", so that it raises a new way to solve the problems. With the assumption, the eigensolutions are solved which inclued the general solution and particular solution of the inhomogeneous Hamiltion equations. The following conclusions can be drawn from the analysis of the problem:The singular order at the crack-tip is -0.5; the value of stress intensity factor can be represented by the combination of the first coefficient of the non-zero eigensolutions and the series of temperature functions; the distribution of radial stresses shows that the high stresses are always occurred nearby the center of the crack and have exponentially decaying distributions. It should be pointed out that the value of thermal stress intensity factor decreases as the length of crack increases. It is an important feature for engineering design and evaluations of fatigue life. Based on these results, two papers have been published in the Journal of Thermal Stresses (2010,33(3):262-278; 2010,33(3): 279-301).
研究和揭示材料和结构的断裂机理有利于提高设计和制造水平,由此可以有效的减少事故发生,并尽可能地延长设备的使用寿命。然而,研究该类问题需要系统的考察力电磁热相互作用效应和工况环境。虽然目前有很多相关的理论和方法,但仍需完善和改进,特别是对相关电磁材料的断裂行为研究方法等。从现有的方法看,其中大部分皆基于拉格朗日体系下的一类变量的控制方程。由此将面对高阶微分方程的求解和数值处理方法,这就给问题求解带来了相当的困难。可喜的是钟万勰院士首次将哈密顿体系引入到弹性力学和应用力学中,开创了一种全新的理念和方法,并建立了基于哈密顿体系的研究问题平台。在钟院士的带领下,他的科研团队对许多领域和研究方向系统和深入的展开探讨,并取得丰硕的研究成果。这些研究成果也为本论文的研究提供基础和依据。
本博士论文以带有边缘裂纹的弹性材料、压电材料和电磁弹性材料为研究对象,对裂纹尖端的奇异性和强度因子进行系统分析。并利用辛本征解展开方法和辛共轭正交关系,得到对偶变量和强度因子的解析表达式。该方法能克服传统半逆法的弱点,给出一种直接方法和系统方法。取得的研究结果为人们研究断裂问题提供了全新的认识。具体研究成果如下：
1.平面和空间弹性体的应力强度因子研究
在哈密顿体系下,位移和广义应力互为对偶变量。通过研究以混合变量描述的对偶正则方程,得到含断裂问题的辛本征解。在辛空间中构造出完备的辛本征解空间。哈密顿体系下的辛本征解可以分为两类：零本征值本征解和非零本征值本征解。零本征值本征解即是该问题对应的圣维南问题的解,代表了该问题对应的等效边界条件意义下的解。非零本征值本征解则包括圣维南原理所覆盖的解,即体现边缘效应和局部效应的解。研究工作以平面问题作为突破口,进而在空间问题展开。由于辛本征解之间存在辛共轭正交关系,问题的解可由辛本征解得展开得到,从而获得问题解得解析表达式。应力强度因子和T应力可由特殊的辛本征解和其系数直接表示。进一步利用边界条件和辛共轭正交关系,可确定所有展开级数的系数。这样Ⅰ型,Ⅱ型和Ⅲ型应力强度因子(KⅠ, KⅡ, KⅢ)同时被直接得到。此直接方法突显出更加方便和有效。利用边界积分等手段,将圆形外边界拓宽到非规则边界的裂纹问题,直接得到的半解析结果和数值结果。研究工作为进一步讨论动力问题提供了依据和基础。这些研究成果已经发表在Engineering Fracture Mechanics (2009,76(12):1866-1882), International Journal of Mechanical Sciences (2010,52(7):892-903)和Journal of Sound and Vibration (2011,330:1005-1017)。
2.含边缘裂纹压电材料的力/电强度因子和奇异性分析
将哈密顿体系求解方法应用于含边缘裂纹压电材料奇异性分析中。以—空间坐标模拟时间,采用弹性势能(应变能)和压电能表示拉格朗日函数和变分原理,得到广义位移(位移和电势)和广义应力(应力和电位移)的对偶关系。利用哈密顿原理构造出以广义位移和广义应力混合变量描述的对偶正则方程。利用哈密顿体系很好的性质和现代数学工具对含边缘裂纹压电材料问题展开研究和讨论。分析电可渗透和电不可渗透裂纹在尖端处的奇异性,并得到应力强度因子和电位移强度因子以及影响因素。结果表明,对于电可渗透裂纹,电场强度因子始终为零,即电场在裂纹尖端不存在奇异性；应变强度因子与材料常数无关,只与外边界荷载工况有关；应力强度因子和电位移强度因子可以用材料常数与广义位移强度因子的线性组合表示。相关成果已经发表在International Journal of Solids and Structures (2009,46(20):3577-3586)。
3.含边缘裂纹电磁弹性材料的耦合强度因子研究
构造含边缘裂纹电磁弹性材料问题的哈密顿体系结构,研究Ⅲ型裂纹问题的断裂行为。该类问题可归结为反平面问题。在哈密顿体系下,轴向位移与剪应力、电势与电位移、磁势与磁感应强度分别互为对偶变量。以这些变量和对偶变量组成的混合变量描述的基本问题对研究混合边界条件问题非常直接和特别有效。在得到辛本征解空间以后,将应力强度因子,电位移强度因子和磁感应强度因子等问题归结为线性代数方程组求解的问题。在此基础上,对电磁可渗透和电磁不可渗透裂纹问题分别进行分析和研究。得到电磁弹性材料反平面断裂问题的解析解和一些规律。研究结果表明,广义位移强度因子与材料常数无关,只与本征值为二分之一的本征解系数有关；广义应力变量在裂纹尖端处表现出传统-0.5阶次的奇异性,并且它们对应的强度因子可直接表示为材料常数和广义位移强度因子的函数；在电磁可渗透的裂纹问题中电场强度和磁场强度不出现奇异性,即对应的强度因子为零。研究成果已经发表在Engineering Fracture Mechanics (2010, 77(16):3157-3173)和Computers & Structures (2011,89:631-645)。
4.稳态和瞬态热弹性问题中的热应力强度因子
提出热传导方程和热弹性方程在空间坐标下可分离变量的哈密顿形式。研究工作分为两部分：首先在哈密顿体系下建立与热传导方程等价的正则方程,并求解温度场。温度场可由一系列辛本征解组合所表示,其中包括稳态和瞬态温度函数。然后利用所得的温度场构造热弹性问题的非齐次哈密顿对偶方程以及相应的初边条件。在此过程中,将时间变量只作为一个“空间坐标”,而将一空间坐标模拟为“时间坐标”。这样,提出
个全新地考虑问题思路。在这种观念下对问题求解,得到对应的辛本征解,即齐次正则方程的通解和非齐次方程特解。通过对解析解和数值结果的分析,得出结论：热应力问题的裂纹尖端奇异指数为-0.5；应力强度因子直接由第一阶非零本征解和温度函数表示和确定；最大热应力发生在裂纹尖端区域,并且成指数向外衰减。研究结果发现：在一定的温度环境下,热应力强度因子随裂纹长度增大而变小的现象。也就是裂纹会出现止裂的结果。这种现象对于工程设计和工程设备寿命评估是非常重要。根据这些研究工作,已经连续两篇文章发表在Journal of Thermal Stresses (2010,33(3):262-278; 2010,33(3)：279-301)。
With the development of science and technology, multifunctional materials and smart materials drew more and more attention of the designers. The magneto-electro-elastic material is one of these popular materials. Based on the properties of these materials, many intelligent structures and products are applied in the engineering structure. With the aid of magneto-electro-mechanical energy conversion, many instruments and facilities are designed and extensively used in petroleum industry, chemical Industry, aerospace industry, military affairs, manufacturing and nuclear industry. When subjected to coupled mechanical, magnetic, electric and thermal loads in service, these instruments and facilities are fail prematurely due to some defects arising during their manufacturing processes, i.e. fatigue crack. Therefore, it is of great importance to study the fracture behaviors of there composites and understand the cracking mechanism.
Studying and understanding the cracking mechanism of these materials and structures is useful to optimize the structural design and manufacturing standards. Moreover, it is pertinent to minimize the catastrophic failures for enhanced performance in fracture and wear resistance. However, the nature of the problem requires a systematic examination of the interplays among the electro-magneto-mechanical-thermal effects and working condition. Although some theoretical and experimental studies have been performed, many related challenges have not been fully resolved and need to be refine and improved, particularly in the fracture behaviors of the magneto-electro-elastic materials. In view of these literatures, it can be seen that all of the methods used the governing equation derived previously in Lagrangian sense involving only one kind of variables in terms of the energy. Since highter-order differential equations are not conducive to numerical solution methods, such elimination will cause problems in numerical analysis. Fortunately, Academician Zhong Wanxie developed an analytical symplectic approach for some basic problems in elasticity and in applied mechanics. It is a new concept and method which provids a research platform based on Hamiltonian system. Under the leadership of Academician Zhong, his associates have extendeded the method to many areas and directions. Some research results have been published and provide a basic technique for the dissertation.
In this dissertation, we study the singularities and intensity factors systematically for the edge-crack elastic, piezoelectric and magneto-electro-elastic materials. With the aid of symplectic expansion and symplectic adjoint orthogonality among the eigenfunctions, the analytic expressions for both of the intensity factors and dual variables are obtained. It overcomes the the defects of classical semi-inverse methods, and it is rational and systematic with a clearly defined, step-by-step derivation procedure. The present work provides a better understanding for fracture problem. The conclusions are listed below:
1. Analytic stress intensity factors for two-and three-dimensional problems
A Hamiltonian system is introduced by the energy method. The displacements and stresses are proved to be conjugating (dual) to each other. The eigensolutions for fracture problems are solved from the Hamilton equations based on the mixed variables. These eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions. In the symplectic space, the solution consists of two parts:zero eigenvalue solutions and non-zero eigenvalue solution. All the Saint Venant solutions have been identified as the zero-eigenvalue solutions and the Saint Venant solutions represent the average physical. The non-zero-eigenvalue solutions corresponding to effects which are coverd by the Saint-Venant principle, i.e. the local boundary layer effects. The plane problems are regarded as a breakthrough, the symplectic method is extended to the space problems. Based on symplectic adjoint orthogonality among the eigensolutions, the analytical solutions are obtained and can be expanded in terms of the symplectic eigensolutions. The stress intensity factors and T-stresses are identified to be the coefficients of certain eigenfunctions. The coefficients of the series are determined from the boundary conditions and the relationship of symplectic adjoint orthogonality. Thus, ModeⅠ,Ⅱ,Ⅲstress intensity factors are obtained simultaneously. It is a direct and effective method. In addition, a boundary integral technology is developed for the non-circular domains, semi-analytical or numerical results can be obtained. Moreover, the present work provides a way to solve the dynamic problems. These work has been published in the Engineering Fracture Mechanics (2009, 76(12):1866-1882), International Journal of Mechanical Sciences (2010,52(7):892-903) and Journal of Sound and Vibration (2011,330:1005-1017).
2. Stress/electric intensity factors and singularities analysis for the edge-crack piezoelectric materials
The Hamiltonian formalism is used to analyze singularities for the edge-crack piezoelectric materials. A space coordinate is defined as the longitudinal direction via an appropriate variable transformation to simulate the "time coordinate". With the aid of variational principle and the Lagrangian function which consists of elastic potential energy and piezoelectric energy, we generalized displacements (longitudinal direction displacement and electrical potential function) and stresses (shear stresses and electric displacements) as primary unknowns will result in a complete set of eigensolutions ensuring convergence and will give the generalized stresses directly as dual variables. Based on the properties of Hamilton system and modern mathematical tools, analysis of the edge-crack piezoelectric materials is preformed. The singularities and intensity factors for a permeable or impermeable crack in piezoelectric material are obtained. Furthermore, the influence factors are discussed. The results show that the electric field intensity factors for the electrically permeable crack is always of the zero value, or the electric field has no singularity at the crack tip. The strain intensity factors become independent of the material constants, it depends on the edge loading conditions only. The stress and the electric displacement intensity factors can be represented by a combination of material constants and the generalized displacement intensity factors. This work is published in International Journal of Solids and Structures (2009,46(20): 3577-3586).
3. The study of coupling intensity factors for the edge-crack in magneto-electro-elastic media
A Hamilton system is established for the edge-crack in magneto-electro-elastic media for studying the fracture behaviors. It can be reduced to the anti-plane problem. In symplectic space, it can be proved that the displacements and stresses, electric potentials and electric displacements, magnetic potentails and magnetic induction functions are conjugating (dual) variables respectively. It is convenient to solve the mixed boundary conditions with the aid of the mixed variables which consist of the original variables and dual variables. Using the exited eigensolutions, the solutions of stress, electric displacement and magnetic induction intensity factors are reduced to the solutions of a set of linear algebraic equations. Both of the permeable and impermeable electromagnetic boundary conditions at the crack surfaces are adopted and discussed. Some resultes and a closed form solution for-the anti-plane fracture problem of magneto-electro-elastic materials are obtained. The results show that the intensity factors can be obtained by the terms associated with the eigenvalue solutions having the coefficients of 1/2;The generalized stress variables at the crack tip show the traditional square root singularities and can be represented by a combination of material constants and the generalized displacement intensity factors; the field variables, which can permeate the crack surfaces, produce no singularities or their corresponding intensity factors always equal to zero. These work has been published in the Engineering Fracture Mechanics (77(16),2010, 3157-3173) and Computers & Structures (2011,89:631-645).
4. Analytic stress intensity factors for the steady and transient thermoelasticity
The equations of thermal thermal conduction and thermoelasticity are first rewritten in Hamiltonian form where the variables are separable in spatial coordinates. Our study will be considered in two parts:At first, a generalized Hamilton system will be introduced to the heat conduction problem and the temperature function will be represented by a symplectic series analytically. Using the existed solution, the temperature function for both of the steady and transient thermal conducitons will be obtained. Then, a set of inhomogeneous Hamiltion equations and corresponding boundary conditions are obtained by the temperature function which obtained from the first part. For the symplectic approach, the radial coordinate is defined as the longitudinal direction via an appropriate variable transformation to simulate the "time coordinate", so that it raises a new way to solve the problems. With the assumption, the eigensolutions are solved which inclued the general solution and particular solution of the inhomogeneous Hamiltion equations. The following conclusions can be drawn from the analysis of the problem:The singular order at the crack-tip is -0.5; the value of stress intensity factor can be represented by the combination of the first coefficient of the non-zero eigensolutions and the series of temperature functions; the distribution of radial stresses shows that the high stresses are always occurred nearby the center of the crack and have exponentially decaying distributions. It should be pointed out that the value of thermal stress intensity factor decreases as the length of crack increases. It is an important feature for engineering design and evaluations of fatigue life. Based on these results, two papers have been published in the Journal of Thermal Stresses (2010,33(3):262-278; 2010,33(3): 279-301).
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