磁电弹性复合材料断裂分析及其辛数值方法

发布时间：2018-07-11 15:12

本文选题：哈密顿体系 + 压电/电磁弹性材料　；参考：《大连理工大学》2016年博士论文

【摘要】：随着材料科学和电子技术的发展,具有良好力、电、磁耦合效应的压电／电磁等功能复合材料,被广泛用于制造传感器、探测器、超声成像器等智能器件。然而,由于材料性质间的不匹配及自身的脆性,该类器件在制作和使用过程中,不可避免会产生裂纹或孔洞,从而导致缺陷附近的物理场奇异(力、电和磁场集中现象),影响结构的完整性,并可能引发结构功能失效。因此,研究压电／电磁复合结构的断裂问题,具有重要的理论和实际意义,是智能结构设计和评估的重要基础和前提。本博士论文通过哈密顿体系辛方法,建立了统一形式的有限尺寸压电／电磁弹性智能复合材料界面断裂问题的对偶控制方程,获得了以本征解展开形式表示的裂纹尖端物理场的解析表达式,以及表征力、电、磁场奇异程度的物理场强度因子。此外,基于本征解函数和传统有限元方法构造出一种能够克服网格敏感和路径敏感的辛离散有限元方法。本文主要研究工作如下：(1)建立了有限尺寸压电／电磁弹性智能复合材料界面断裂问题的哈密顿求解体系在哈密顿理论体系下,压电／电磁智能材料的位移和应力、电势和电位移、磁势和磁感应强度互为对偶变量。将上述变量构成的全状态向量作为基本未知量,构造出具有统一形式的力、电、磁智能复合材料的哈密顿正则方程。利用分离变量法,原问题归结为辛空间下的本征值和本征解问题。各物理场的解通过辛本征解的线性组合表示,其中本征解的待定系数可以利用边界条件和辛共轭正交关系求解。考虑四种理想电磁裂纹面条件,利用断裂力学公式获得裂纹强度因子和能量释放率的解析表达式。该方法突破了传统半逆法的局限,是一种理性的直接求解方法。研究结果表明,裂纹尖端物理场均具有-1/2奇异性,应力、电位移、磁感应强度因子和能量释放率与材料常数相关并可由广义位移强度因子线性组合表示；而应变、电场和磁场强度因子与材料常数无关,只与本征值为1/2的本征解系数相关。该方法适用于不同类型边界条件,包括复杂的混合边界条件。数值计算表征出裂纹尖端的机械场、电场和磁场特性,揭示了材料参数、几何尺寸和外加荷载对断裂参数的影响。(2)提出一种针对压电／电磁弹性复合材料断裂问题的辛离散有限元方法利用裂纹尖端附近的解析辛本征解函数结合传统有限元方法构造出适用于压电／电磁弹性复合材料断裂分析的辛离散有限元方法。首先,将裂纹结构整体划分为含裂纹尖端的近场奇异区域和以及不含裂纹尖端的远场非奇异区域,并对整体结构进行有限元网格划分。其次,在近场中以辛本征解展开作为全局函数,将近场内大量广义节点位移未知量转化为辛本征解展开系数,而远场中的节点未知量保持不变。最后,通过求解出的本征解待定系数直接获得近场内物理场的显式表达式以及裂纹的断裂参数。与其它数值方法相比,辛离散有限元方法具有三点优势：(i)计算过程中,无需引入特殊的奇异单元和网格加密,消除传统有限元对网格敏感问题；(ii)近场内大量的节点位移转化为少量的本征解待定系数,极大程度缩小了刚度矩阵的维度,从而大幅提高了计算效率和精度。(iii)无需额外的后处理程序,断裂参数可以直接通过求解出来的辛本征解系数表示,消除路径敏感问题。数值结果验证了辛离散有限元方法的精确性。给出的数值算例,包括含有多裂纹,分叉裂纹和椭圆孔边缘开裂裂纹等问题,计算结果为压电、电磁智能复合材料的研发、设计、制造、可靠性分析及寿命评估提供直接的理论指导和技术支持。
[Abstract]:With the development of material science and electronic technology, the piezoelectric / electromagnetic composite materials with good force, electricity and magnetic coupling effect are widely used in the manufacture of sensors, detectors, ultrasonic imagers and other intelligent devices. However, because of the mismatch between material properties and their own brittleness, this kind of device can not be avoided in the process of production and use. The crack or hole can be avoided, which leads to the singularity of the physical field near the defect (force, electricity and magnetic field), which affects the integrity of the structure and may cause the failure of the structural function. Therefore, it is of great theoretical and practical significance to study the fracture of the piezoelectric / electromagnetic composite structure, which is an important basis for the design and evaluation of the intelligent structure. By means of the Hamilton system symplectic method, this thesis establishes a unified form of the dual control equation for the interfacial fracture problem of a finite size piezoelectric / electromagnetic elastic composite material, and obtains an analytic expression of the crack tip physical field expressed in the form of eigensolution, as well as the properties of the singularity of the force, electricity and magnetic field. In addition, based on the eigensolution function and the traditional finite element method, a symplectic discrete finite element method can be constructed to overcome the sensitivity of the grid and the path sensitivity. The main research work of this paper is as follows: (1) the Hamilton solution system of the interface fracture of the finite size piezoelectric / electromagnetic elastic intelligent composite material is established in Hazakhstan. The displacement and stress of piezoelectric / electromagnetic intelligent materials, potential and potential shift, magnetomotive force and magnetic induction intensity are dual variables under the system of the mill theory, and the whole state vector made up of the above variables is used as the basic unknown quantity to construct a Hamiltonian regular equation with a unified form of force, electric and magnetic compound material. The original problem is attributed to the eigenvalue and the eigensolution under the symplectic space. The solutions of the physical fields are represented by the linear combination of the symplectic eigensolutions. The undetermined coefficients of the eigensolutions can be solved by the boundary conditions and the symplectic conjugate orthogonal relations. The crack strength factors and energy are obtained by using the fracture mechanics formula to consider the four ideal surface conditions for the electromagnetic crack. This method breaks through the limitations of the traditional semi inverse method and is a rational direct solution. The results show that the physical field of the crack tip has -1/2 singularity, stress, potential shift, magnetic induction intensity factor and energy release rate related to the material constant and can be linear combination of generalized displacement intensity factors. The strain, the electric field and the magnetic field intensity factor are independent of the material constants, which are only related to the eigenvalues of the eigenvalues of 1/2. This method is suitable for different types of boundary conditions, including complex mixed boundary conditions. The numerical calculation shows the mechanical field, the electric field and magnetic field characteristics at the crack tip, and reveals the material parameters, geometry size and addition. The effect of load on the fracture parameters. (2) a symplectic discrete finite element method for the fracture problem of piezoelectric / electromagnetic elastic composite materials is proposed. The symplectic discrete finite element method is constructed by using the analytic symplectic intrinsic solution function near the crack tip and the traditional finite element method. First, a symplectic discrete finite element method is constructed for the fracture analysis of piezoelectric / electromagnetic elastic composite material. The crack structure is divided into the near field singular region with the crack tip and the nonsingular region of the far field without the crack tip, and the finite element mesh of the whole structure is divided. Secondly, in the near field, the symplectic intrinsic solution is used as a global function, and the unknown displacement of a large number of wide sense nodes in the field is transformed into the symplectic expansion system. The unknown quantity of the nodes in the far field remains unchanged. Finally, the explicit expressions of the near field physical fields and the fracture parameters are obtained by the undetermined coefficients of the eigensolutions. In comparison with other numerical methods, the symplectic discrete finite element method has three advantages: (I) no special singular elements need to be introduced in the (I) calculation process. The grid is encrypted to eliminate the sensitive problem of the traditional finite element to the grid. (II) the displacement of a large number of nodes in the near field is converted into a small number of eigenvalues, which greatly reduces the dimension of the stiffness matrix, thus greatly improves the computational efficiency and accuracy. (III) no additional post-processing program is needed, and the fracture parameters can be solved directly through the solution. The Xin Benzheng solution coefficient is expressed and the path sensitivity is eliminated. The numerical results verify the accuracy of the symplectic discrete finite element method. The numerical examples are given, including the problems of multiple cracks, bifurcation cracks and the edge cracks on the edge of the elliptical hole. The calculation results are the research, design, manufacture, reliability analysis and life of the piezoelectric, electromagnetic intelligent composite materials. The assessment provides direct theoretical guidance and technical support.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O346.1

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