基于连续位错模型的热释电材料断裂理论
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摘要
热释电陶瓷作为功能材料，由于固有的压电、介电和热释电性质，被广泛的应用于传感器、转换器和制动器等智能系统中。然而，热释电材料本身呈脆性，在制备和使用过程中极易产生裂纹、孔洞和夹杂等缺陷。因此，针对含裂纹等缺陷热释电材料的热、电、应力场的分析是非常有必要的。弄清楚热释电材料断裂的物理和力学机制，从而可以更好的为这些智能系统的结构强度、稳定性、使用寿命和设计等方面提供理论参考。基于扩展的Stroh型解、Green函数方法和奇异积分方程方法，本文针对热释电材料直裂纹扩展、裂纹尖端分叉、界面裂纹和裂纹面热-电边界条件等问题作了系统的研究。具体内容如下：
首先，针对热-电-机械多场耦合载荷作用下含裂纹的热释电材料，建立了求解应力场和电位移场的裂纹部分接触模型。裂纹模型分别假设为电绝缘的和电流可导通的。基于Green函数方法，问题简化为具有封闭解的一系列奇异积分方程的求解。在裂纹面应力和电荷自由的假定下，除非有一足够大的远场拉应力作用，否则将有裂纹一端的张开位移为负值。因此，上述热释电介质的热-电-力耦合问题转化为裂纹尖端存在部分接触的断裂模型重新求解。作为外部机械载荷、电位移载荷和热流载荷的函数，分别给出了接触区长度、裂纹尖端应力强度因子和电位移强度因子等重要断裂参量的封闭解。结果显示，对于电绝缘裂纹模型来说，电位移不仅在裂纹尖端具有奇异性，而且在接触区前沿也具有奇异性。对于电流可导通裂纹模型，只有裂纹尖端电位移才具有平方根的奇异性。
其次，研究了热释电材料在承受热-电-机械多场耦合载荷作用下裂纹尖端分叉的问题，其中裂纹模型假设为电绝缘的。给出了裂纹与热释电位错（即，位于同一点的热学位错、力学位错和电偶极子）相互作用的Green函数。问题转化为求解在分叉裂纹上具有未知热释电位错密度函数的一组奇异积分方程。并获得了重要的断裂力学参量，如：分叉裂纹尖端的应力和电位移强度因子、能量释放率等。通过数值结果分析了热流和电位移载荷对裂纹扩展路径的影响。
众所周知，在理论分析方面，双材料界面裂纹尖端存在着振荡奇异性。为了去除这种不合理的界面断裂振荡奇异性，提出了修正界面位错模型。在数学上，界面位错基本解中的Dirac函数由正态分布函数来表示；在物理上，Dirac函数所表征的单位点力解释为区域分布力，成功的消除了裂纹尖端振荡奇异性。基于上述模型，界面裂纹问题从而转化为求解一组第一类奇异积分方程问题。获得了特定材料配比的应力强度因子、断裂混合度和能量释放率等断裂参量。并成功的将修正界面位错模型应用到解释热释电材料界面裂纹问题。
最后，基于有限厚度裂纹模型，研究了裂纹面热电边界条件对热释电材料断裂力学问题的影响。假设裂纹内部充满空气（或真空）。利用Green函数和奇异积分方程方法，获得了有限厚度裂纹尖端的应力和电位移强度因子、以及裂纹内部热流和电位移的封闭解，并与理想裂纹面热电边界条件的数值结果进行对比。结果发现对于裂纹面电边界条件来说，电绝缘裂纹模型是更为合理的。而对于裂纹面热边界条件来说，完全导热和热绝缘裂纹模型都不能准确的反映相关的断裂物理参量，裂纹内部空气类介质的导热率是不能忽略的。
Thermopiezoelectric ceramics, as a typical smart material, are widely used assensors, transducers and actuators in intelligent systems due to their pronouncedpiezoelectric, dielectric and pyroelectric properties. However, these materials are brittleand susceptible to cracking. It is of vital importance to investigate thethermoelectroelastic fields as a result of the presence of defects, such as crack andinclusions in thermal environments. The requirements of structural strength, reliabilityand lifetime of these intelligent systems call for a better understanding of the fracturemechanics of thermopiezoelectric materials. The problems of crack, crack branching,interface crack and thermoelectric boundary conditions of crack surfaces areinvestigated based on the extended Stroh formalism, Green’s function method andsingular integral equation technique in this paper. The main research contents andconclusions are as follows:
A partial contact zone model is developed for the stress and electric displacementfields due to the obstruction of a uniform heat flux in thermopiezoelectric materials inthe charpters two and three. The crack is assumed electrically impermeable andelectrically permeable, respectively. Green’s function method is used to reduce theproblem to a set of singular integral equations which are solved in closed-form. Whenthe crack is assumed to be traction free, the crack opening displacement is found to benegative over one half of the crack unless a sufficiently large far field tensile stress issuperposed. The problem is reformulated assuming a contact zone at one crack tip. Theextent of this zone, the stress and electric displacement intensity factors at each crack tipare obtained as functions of the applied mechanical stress and heat flux. The resultshows that the electric displacement intensity is not only singular at both crack tips, butalso at contact zone tip for the electrically impermeable crack. For the electricallypermeable, the singularity only exists at the crack tips.
Solutions are presented for an electrically impermeable crack branching out of thecrack plane in a thermopiezoelectric medium under thermo-electro-mechanical loadsbased on Stroh formalism in charpter four. Explicit Green’s functions for the interactionof a crack and a thermopiezoelectric dislocation (i.e., a thermal dislocation, amechanical dislocation and an electric dipole located at the same point) are developed.The problem then can be expressed in terms of coupled singular integral equations forthe thermopiezoelectric dislocation density functions associated with a branched crack. Some essential fracture mechanics parameters, such as stress and electric displacementintensity factors, and energy release rate at the branched crack tip are obtained.Numerical results are presented for the effect of applied thermal flux loads and electricfield on the crack propagation path.
It is well known that there is oscillation singularity near the interface crack tip,which is physically unreasonable. A modified interface dislocation model is developedto remove such oscillation singularity for interface fracture in charpter five. The Diracdelta function in the fundamental solution of interface dislocation is explained bylocally-distributed continuous function. The oscillation singularity at the crack tip ofrinterface fracture is eliminated. Then the problem is reduced to the solution of a systemof singular integral equations. The critical interfacial fracture mechanics parameters,such as the stress intensity factor, the mode mixity and the energy release rates forparticular materials group of the isotropic solids are obtained. Finally, the modifiedinterface dislocation model is extended to the thermopiezoelectric bimaterials interfacecrack problem.
The applicability and effect of the crack surfaces thermoelectric boundaryconditions in thermopiezoelectric fracture mechanics problem are discussed by usingthe finite thickness notch approach in charpter six. The notch in these materials is full ofair (or vacuum). The stress and electric displacement intensity factors at the notch tips,and thermal flux and electric displacement inside the notch are derived in closed-form.The numerical results are compared with the ideal crack solutions. It is found that theelectrically impermeable crack boundary condition assumption is a reasonable one, andthe thermal conductivity of air or vacuum inside the crack must be considered.
首先，针对热-电-机械多场耦合载荷作用下含裂纹的热释电材料，建立了求解应力场和电位移场的裂纹部分接触模型。裂纹模型分别假设为电绝缘的和电流可导通的。基于Green函数方法，问题简化为具有封闭解的一系列奇异积分方程的求解。在裂纹面应力和电荷自由的假定下，除非有一足够大的远场拉应力作用，否则将有裂纹一端的张开位移为负值。因此，上述热释电介质的热-电-力耦合问题转化为裂纹尖端存在部分接触的断裂模型重新求解。作为外部机械载荷、电位移载荷和热流载荷的函数，分别给出了接触区长度、裂纹尖端应力强度因子和电位移强度因子等重要断裂参量的封闭解。结果显示，对于电绝缘裂纹模型来说，电位移不仅在裂纹尖端具有奇异性，而且在接触区前沿也具有奇异性。对于电流可导通裂纹模型，只有裂纹尖端电位移才具有平方根的奇异性。
其次，研究了热释电材料在承受热-电-机械多场耦合载荷作用下裂纹尖端分叉的问题，其中裂纹模型假设为电绝缘的。给出了裂纹与热释电位错（即，位于同一点的热学位错、力学位错和电偶极子）相互作用的Green函数。问题转化为求解在分叉裂纹上具有未知热释电位错密度函数的一组奇异积分方程。并获得了重要的断裂力学参量，如：分叉裂纹尖端的应力和电位移强度因子、能量释放率等。通过数值结果分析了热流和电位移载荷对裂纹扩展路径的影响。
众所周知，在理论分析方面，双材料界面裂纹尖端存在着振荡奇异性。为了去除这种不合理的界面断裂振荡奇异性，提出了修正界面位错模型。在数学上，界面位错基本解中的Dirac函数由正态分布函数来表示；在物理上，Dirac函数所表征的单位点力解释为区域分布力，成功的消除了裂纹尖端振荡奇异性。基于上述模型，界面裂纹问题从而转化为求解一组第一类奇异积分方程问题。获得了特定材料配比的应力强度因子、断裂混合度和能量释放率等断裂参量。并成功的将修正界面位错模型应用到解释热释电材料界面裂纹问题。
最后，基于有限厚度裂纹模型，研究了裂纹面热电边界条件对热释电材料断裂力学问题的影响。假设裂纹内部充满空气（或真空）。利用Green函数和奇异积分方程方法，获得了有限厚度裂纹尖端的应力和电位移强度因子、以及裂纹内部热流和电位移的封闭解，并与理想裂纹面热电边界条件的数值结果进行对比。结果发现对于裂纹面电边界条件来说，电绝缘裂纹模型是更为合理的。而对于裂纹面热边界条件来说，完全导热和热绝缘裂纹模型都不能准确的反映相关的断裂物理参量，裂纹内部空气类介质的导热率是不能忽略的。
Thermopiezoelectric ceramics, as a typical smart material, are widely used assensors, transducers and actuators in intelligent systems due to their pronouncedpiezoelectric, dielectric and pyroelectric properties. However, these materials are brittleand susceptible to cracking. It is of vital importance to investigate thethermoelectroelastic fields as a result of the presence of defects, such as crack andinclusions in thermal environments. The requirements of structural strength, reliabilityand lifetime of these intelligent systems call for a better understanding of the fracturemechanics of thermopiezoelectric materials. The problems of crack, crack branching,interface crack and thermoelectric boundary conditions of crack surfaces areinvestigated based on the extended Stroh formalism, Green’s function method andsingular integral equation technique in this paper. The main research contents andconclusions are as follows:
A partial contact zone model is developed for the stress and electric displacementfields due to the obstruction of a uniform heat flux in thermopiezoelectric materials inthe charpters two and three. The crack is assumed electrically impermeable andelectrically permeable, respectively. Green’s function method is used to reduce theproblem to a set of singular integral equations which are solved in closed-form. Whenthe crack is assumed to be traction free, the crack opening displacement is found to benegative over one half of the crack unless a sufficiently large far field tensile stress issuperposed. The problem is reformulated assuming a contact zone at one crack tip. Theextent of this zone, the stress and electric displacement intensity factors at each crack tipare obtained as functions of the applied mechanical stress and heat flux. The resultshows that the electric displacement intensity is not only singular at both crack tips, butalso at contact zone tip for the electrically impermeable crack. For the electricallypermeable, the singularity only exists at the crack tips.
Solutions are presented for an electrically impermeable crack branching out of thecrack plane in a thermopiezoelectric medium under thermo-electro-mechanical loadsbased on Stroh formalism in charpter four. Explicit Green’s functions for the interactionof a crack and a thermopiezoelectric dislocation (i.e., a thermal dislocation, amechanical dislocation and an electric dipole located at the same point) are developed.The problem then can be expressed in terms of coupled singular integral equations forthe thermopiezoelectric dislocation density functions associated with a branched crack. Some essential fracture mechanics parameters, such as stress and electric displacementintensity factors, and energy release rate at the branched crack tip are obtained.Numerical results are presented for the effect of applied thermal flux loads and electricfield on the crack propagation path.
It is well known that there is oscillation singularity near the interface crack tip,which is physically unreasonable. A modified interface dislocation model is developedto remove such oscillation singularity for interface fracture in charpter five. The Diracdelta function in the fundamental solution of interface dislocation is explained bylocally-distributed continuous function. The oscillation singularity at the crack tip ofrinterface fracture is eliminated. Then the problem is reduced to the solution of a systemof singular integral equations. The critical interfacial fracture mechanics parameters,such as the stress intensity factor, the mode mixity and the energy release rates forparticular materials group of the isotropic solids are obtained. Finally, the modifiedinterface dislocation model is extended to the thermopiezoelectric bimaterials interfacecrack problem.
The applicability and effect of the crack surfaces thermoelectric boundaryconditions in thermopiezoelectric fracture mechanics problem are discussed by usingthe finite thickness notch approach in charpter six. The notch in these materials is full ofair (or vacuum). The stress and electric displacement intensity factors at the notch tips,and thermal flux and electric displacement inside the notch are derived in closed-form.The numerical results are compared with the ideal crack solutions. It is found that theelectrically impermeable crack boundary condition assumption is a reasonable one, andthe thermal conductivity of air or vacuum inside the crack must be considered.
引文
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