压电压磁固体断裂理论及数值方法研究
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摘要
本博士学位论文主要研究压电及电磁材料在力、电、磁载荷作用下的断裂模型及数值分析方法。在压电介质电塑性断裂的PS(Polarization Saturation)模型基础上,讨论了电磁材料条带电磁极化饱和(SEMPS-Strip Electric-Magnetic Polarization Saturation)的断裂行为,给出相应的解析解和数值方法或数值结果。然后,研究力、电、磁及其耦合载荷对断裂行为的影响。主要工作如下：
(1)利用位错理论和电位错概念,研究压电介质PS模型,得到由传统位错和电位错表示的局部K因子和局部J积分,为位错理论研究电磁材料的SEMPS'情况奠定基础。
(2)推广电位错,提出磁位错概念。将裂纹上下表面的磁势差看作是磁位错的连续分布,研究得到电磁材料断裂的广义位错理论。
(3)研究得到电磁材料SEMPS的基本方程和基本方法,其中裂纹尖端电塑性区中的电位移等于饱和电位移,磁塑性区中的磁感应强度等于饱和磁感应强度。根据电磁材料广义Stroh公式和广义位错模型,给出二维无限大电磁固体中电磁均不可穿透边界条件下Griffith裂纹的SEMPS的解析解。研究给出无限大电磁固体裂纹尖端电塑性区和磁塑性区的大小、广义强度因子和局部J积分；研究了力、电、磁等载荷对局部J积分的影响。结果表明：SEMPS模型中的电塑性区和磁塑性区的大小由材料常数、裂纹尺寸、外载荷、饱和电位移和饱和磁感应强度确定；广义强度因子和局部J积分与材料常数、施加的广义强度因子有关,而与饱和电位移和饱和磁感应强度无关。
(4)针对有限压电介质断裂问题,建立了压电介质的混合广义不连续位移-基本解(HEDD-FS——Hybrid Extended Displacement Discontinuity-Fundamental Solution)数值方法。在这一方法中,裂纹问题的解由无限体控制方程的基本解表出,其中的基本解包括源点作用在临近虚边界上的广义点力基本解和源点作用在裂纹面上及电塑性区上的广义Crouch基本解,其中未知系数由区域边界条件、裂纹面上的边界条件及塑性区上的边界条件来确定。该数值方法的关键是：在无限体中寻找一包含原区域的虚边界,在虚边界上配点,实现了用无限体基本解研究有限体断裂问题。运用有限弹性体和无限压电介质的断裂分析验证该数值方法的有效性和精度。结果表明：根据HEDD-FS方法研究得到的有限弹性体数值解与解析解吻合很好；当介质足够大或无限大,根据HEDD-FS方法研究得到的无限大压电介质PS模型的数值解与解析解同样很吻合；因此,本文所提出的数值方法是可靠的、可以接受的。
(5)采用所建立的HEDD-FS数值方法,给出了有限压电介质PS模型在不可穿透边界条件和半可穿透边界条件下的定量结果。
(6)采用广义不连续位移数值方法,研究电磁不可穿透边界条件下的无限大电磁材料SEMPS模型,为其他电磁边界条件的SEMPS模型的断裂研究提供数值方法。
In this doctoral dissertation, fracture model and quantitative analysis method of piezoelectric material and magnetoelectroelastic material are investigated. Based on the PS (Polarization Saturation) model in the piezoelectric media, the fracture behaviors of the magnetoelectroelastic media with the Strip Electric-Magnetic Polarization Saturation (SEMPS) are discussed, including their analytical solutions and numerical methods or results. After that, the effect of mechanical, electric and magnetic loadings on the fracture behaviors is displayed. The main works are as follows:
(1) By using the dislocation theory and the concept of electric dislocation, the local stress intensity factor and local J-intergal in the PS model are expressed by the conventional dislocation and electric dislocation, which is essential to extend the PS model to the SEMPS case in the magnetoelectroelastic media.
(2) The concept of magnetic dislocation is proposed by extending the electric dislocation. After that, the magnetic potential discontinuity penetrated the spatial region surrounded by the crack face is expressed by the continuous distribution of magnetic dislocation. The extended dislocation theory of fracture of the magnetoelectroelastic media is studied.
(3) Basic equations and approach to the study of the Strip Electric-Magnetic Polarization Saturation (SEMPS) are obtained for the fracture behavior of magnetoelectroelastic media, where the electric field in the strip electric yielding zone ahead of the crack tip is equal to the saturation electric displacement, while the magnetic induction in the strip magnetic yielding zone is equal to the saturation magnetic induction. By using the extended Stroh formalism and the extended dislocation modeling of a crack, the analytical solution of SEMPS for the Griffith crack problem under the electrically and magnetically impermeable boundary condition in a magnetoelectroelastic medium is obtained. The sizes of the electric yielding zone and the magnetic yielding zone, the extended intensity factors and the local J-integral are obtained. The effect of the combined mechanical-electric-magnetic loadings on the local J-integral is studied. The results show that the sizes of these two saturation zones are dependent on material constants, the size of crack, the applied mechanical-electric-magnetic loadings, as well as on the saturation electric displacement and the saturation magnetic induction. However, the local J-integral has nothing to do with the saturation electric displacement and the saturation magnetic induction.
(4) The Hybrid Extended Displacement Discontinuity-Fundamental Solution Method (HEDD-FSM) of the PS model of a crack in a piezoelectric medium is proposed. In this HEDD-FSM, the solution is expressed approximately by a combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack and on the electric yielding zone. The unknown constants in the solutions are determined by letting the approximated solution satisfy the prescribed conditions on the boundary of the domain, on the crack face and on the electric yielding zone. One key step of this method is choosing a suitable virtual boundary, which circles the real zone boundary. The efficiency and the accuracy of the HEDD-FS method are verified on fracture analysis of finite elastic media and infinite piezoelectric media. The obtained results exhibit that the numerical data on the basis of the HEDD-FS method for the case of finite elastic media agree well with the analytical solution. When the media becomes large enough or infinite, the numerical results of PS model on the basis of the HEDD-FS method for the case of infinite piezoelectric media also agree well with the analytical solution. In such case, the proposed numerical method is acceptable and reliable.
(5) By using the HEDD-FS method, the PS model of a crack is studied quantitatively in finite piezoelectric media under the electrically impermeable boundary condition and electrically semi-permeable boundary condition.
(6) The SEMPS model of a crack are studied in infinite magnetoelectroelastic media under the electrically and magnetrically impermeable boundary condition by using the extended displacement discontinuity (EDD) method, which could be used to study fracture of SEMPS model under other boundary condition.
(1)利用位错理论和电位错概念,研究压电介质PS模型,得到由传统位错和电位错表示的局部K因子和局部J积分,为位错理论研究电磁材料的SEMPS'情况奠定基础。
(2)推广电位错,提出磁位错概念。将裂纹上下表面的磁势差看作是磁位错的连续分布,研究得到电磁材料断裂的广义位错理论。
(3)研究得到电磁材料SEMPS的基本方程和基本方法,其中裂纹尖端电塑性区中的电位移等于饱和电位移,磁塑性区中的磁感应强度等于饱和磁感应强度。根据电磁材料广义Stroh公式和广义位错模型,给出二维无限大电磁固体中电磁均不可穿透边界条件下Griffith裂纹的SEMPS的解析解。研究给出无限大电磁固体裂纹尖端电塑性区和磁塑性区的大小、广义强度因子和局部J积分；研究了力、电、磁等载荷对局部J积分的影响。结果表明：SEMPS模型中的电塑性区和磁塑性区的大小由材料常数、裂纹尺寸、外载荷、饱和电位移和饱和磁感应强度确定；广义强度因子和局部J积分与材料常数、施加的广义强度因子有关,而与饱和电位移和饱和磁感应强度无关。
(4)针对有限压电介质断裂问题,建立了压电介质的混合广义不连续位移-基本解(HEDD-FS——Hybrid Extended Displacement Discontinuity-Fundamental Solution)数值方法。在这一方法中,裂纹问题的解由无限体控制方程的基本解表出,其中的基本解包括源点作用在临近虚边界上的广义点力基本解和源点作用在裂纹面上及电塑性区上的广义Crouch基本解,其中未知系数由区域边界条件、裂纹面上的边界条件及塑性区上的边界条件来确定。该数值方法的关键是：在无限体中寻找一包含原区域的虚边界,在虚边界上配点,实现了用无限体基本解研究有限体断裂问题。运用有限弹性体和无限压电介质的断裂分析验证该数值方法的有效性和精度。结果表明：根据HEDD-FS方法研究得到的有限弹性体数值解与解析解吻合很好；当介质足够大或无限大,根据HEDD-FS方法研究得到的无限大压电介质PS模型的数值解与解析解同样很吻合；因此,本文所提出的数值方法是可靠的、可以接受的。
(5)采用所建立的HEDD-FS数值方法,给出了有限压电介质PS模型在不可穿透边界条件和半可穿透边界条件下的定量结果。
(6)采用广义不连续位移数值方法,研究电磁不可穿透边界条件下的无限大电磁材料SEMPS模型,为其他电磁边界条件的SEMPS模型的断裂研究提供数值方法。
In this doctoral dissertation, fracture model and quantitative analysis method of piezoelectric material and magnetoelectroelastic material are investigated. Based on the PS (Polarization Saturation) model in the piezoelectric media, the fracture behaviors of the magnetoelectroelastic media with the Strip Electric-Magnetic Polarization Saturation (SEMPS) are discussed, including their analytical solutions and numerical methods or results. After that, the effect of mechanical, electric and magnetic loadings on the fracture behaviors is displayed. The main works are as follows:
(1) By using the dislocation theory and the concept of electric dislocation, the local stress intensity factor and local J-intergal in the PS model are expressed by the conventional dislocation and electric dislocation, which is essential to extend the PS model to the SEMPS case in the magnetoelectroelastic media.
(2) The concept of magnetic dislocation is proposed by extending the electric dislocation. After that, the magnetic potential discontinuity penetrated the spatial region surrounded by the crack face is expressed by the continuous distribution of magnetic dislocation. The extended dislocation theory of fracture of the magnetoelectroelastic media is studied.
(3) Basic equations and approach to the study of the Strip Electric-Magnetic Polarization Saturation (SEMPS) are obtained for the fracture behavior of magnetoelectroelastic media, where the electric field in the strip electric yielding zone ahead of the crack tip is equal to the saturation electric displacement, while the magnetic induction in the strip magnetic yielding zone is equal to the saturation magnetic induction. By using the extended Stroh formalism and the extended dislocation modeling of a crack, the analytical solution of SEMPS for the Griffith crack problem under the electrically and magnetically impermeable boundary condition in a magnetoelectroelastic medium is obtained. The sizes of the electric yielding zone and the magnetic yielding zone, the extended intensity factors and the local J-integral are obtained. The effect of the combined mechanical-electric-magnetic loadings on the local J-integral is studied. The results show that the sizes of these two saturation zones are dependent on material constants, the size of crack, the applied mechanical-electric-magnetic loadings, as well as on the saturation electric displacement and the saturation magnetic induction. However, the local J-integral has nothing to do with the saturation electric displacement and the saturation magnetic induction.
(4) The Hybrid Extended Displacement Discontinuity-Fundamental Solution Method (HEDD-FSM) of the PS model of a crack in a piezoelectric medium is proposed. In this HEDD-FSM, the solution is expressed approximately by a combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack and on the electric yielding zone. The unknown constants in the solutions are determined by letting the approximated solution satisfy the prescribed conditions on the boundary of the domain, on the crack face and on the electric yielding zone. One key step of this method is choosing a suitable virtual boundary, which circles the real zone boundary. The efficiency and the accuracy of the HEDD-FS method are verified on fracture analysis of finite elastic media and infinite piezoelectric media. The obtained results exhibit that the numerical data on the basis of the HEDD-FS method for the case of finite elastic media agree well with the analytical solution. When the media becomes large enough or infinite, the numerical results of PS model on the basis of the HEDD-FS method for the case of infinite piezoelectric media also agree well with the analytical solution. In such case, the proposed numerical method is acceptable and reliable.
(5) By using the HEDD-FS method, the PS model of a crack is studied quantitatively in finite piezoelectric media under the electrically impermeable boundary condition and electrically semi-permeable boundary condition.
(6) The SEMPS model of a crack are studied in infinite magnetoelectroelastic media under the electrically and magnetrically impermeable boundary condition by using the extended displacement discontinuity (EDD) method, which could be used to study fracture of SEMPS model under other boundary condition.
引文
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