含裂纹结构加固问题中的哈密顿体系方法
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摘要
工程中的设备和结构等的安全问题一直受到人们的关注。特别是含裂纹结构的加固问题是关系到结构安全,延长结构寿命以及充分节约资源等各个方面。然而,加固后的结构常常出现界面的层裂等问题。因此,对其机理的研究,并提供一种具有高科技含量的加固技术等是十分重要的。目前的研究方法多采用有限元方法。该方法在计算应力强度因子时对单元网格密度和极限路径有一定的局限。本文尝试提出一种有效的方法并对含裂纹结构的问题进行分析,揭示其机理,研究含裂纹结构加固过程中的关键科学问题。本博士学位论文以含裂纹结构加固问题作为研究背景,以结构问题的哈密顿体系方法和奇异元数值方法作为研究对象开展比较系统的研究和分析,得到一些重要结果和结论。具体研究成果如下：
将哈密顿体系推广到空间各向异性弹性基本问题中。以一空间坐标模拟时间,利用弹性势能得到对偶变量,并运用哈密顿原理构造出哈密顿体系下的对偶正则方程组。在哈密顿体系下,基本问题可归结为辛几何空间中的辛本征值和辛本征解的问题。通过再次引入子辛体系,因而建立起一种本征值和本征解直接求解方法。结果表明所有的零本征值本征解即是圣维南问题的解,而非零本征值本征解恰是由圣维南原理所覆盖的解。在完备的本征解空间提出一种新的辛共轭双正交关系,从而完成和完善了求解体系。之后,研究含裂纹或缺陷的材料和结构中的哈密顿体系方法。以含裂纹双材料问题为突破口,提出一种哈密顿控制方程和辛本征解的分区域表示方法,实现本征解和一般解皆能统一表述出来。注意到非零本征值本征解具有局部性特点,特别是本征值为1／2的本征解揭示了应力的奇异性特征。在引入分区域积分的泛函情况下,提出一种独特的本征解之间辛共轭正交关系。借助于此关键技术,得到了问题的解,并且可以直接给出应力强度因子的表达式。基于这些成果,可构造一种含裂纹的单元模型。将辛本征解函数与单元形函数密切结合,得到以及本征解级数系数与单元节点位移等的对应关系,直接给出单元刚度阵等。应力强度因子有明确的表达式。把该辛奇异单元与有限元软件结合形成一种独特的数值方法。该方法可对含裂纹结构分析,并可提高应力强度因子的数值精度。这种方法可应用于含裂纹结构加固问题中。经过对含裂纹结构加固问题的分析,得到了系列研究结果和结论。结果表明,根据结构裂纹的长度,结构材料和加固材料常数,几何尺寸以及粘结强度的特点可合理粘连,可以实现结构裂纹和粘结强度均处于安全范围的合理设计。降低两材料连接处的裂纹应力强度因子以及原结构裂纹应力强度因子,就可达到最佳的加固方案。研究成果为含裂纹结构在加固技术上提供了可靠依据。
The security of equipments and structures has been drawn more attention in engineering. Specially, the reinforcement problem of cracked structures has relation to security of structures, extending of service life of structures and resource saving. However, structures may appear sometimes spallations on the interfaces after they are reinforced. Thus the mechanism of layer crack problem need to be researched and the reinforcement craft is demanded to be improved with technology content. So far, the finite element method is used almost for analyzing the problem. Nevertheless, for computing stress intensity factors, the mesh density of the elements and the path of limit affect directly precision of the factors. For the problem, in this degree thesis, an impactful method is presented to analyze cracked structures, to reveal the cracking mechanism and to study the key scientific problem for the reinforcement of cracked structures. In this dissertation, the reinforcement problem of cracked structures is discussed as a background and application, and the method of Hamiltonian system for structures and the model of symplectic singular element are researched and analyzed systemically. Some main results are obtained and listed below:
The Hamiltonian system is generalized to spatial fundamental problem of anisotropic elasticity. In the system, one spatial coordinate is simulated to the time and the dual variable is obtained from the elastic potential energy. The dual equations, in Hamiltonian system, are shown with the aid of the Hamiltonian principle. Thus, the fundamental problem is reduced to symplectic eigenvalues and eigensolutions in symplectic space. Since sub-symplectic system is introduced again, a direct method is presented for solving symplectic eigenvalues and eigensolutions. Results show that Zero eigenvalue solutions belong to Saint Venant solutions and non-zero-eigenvalue solutions corresponding to the solutions that are coverd by the Saint Venant principle. In the complete space of eigensolutions, a new symplectic adjoint relationship of biorthogonality is presented among the eigensolutions. Thus, complete and perfect system for solving solutions is founded. In further research, analyzing singularity for cracked bi-materials is discussed as a breach. A method, which shows Hamiltonian governing equations and eigensolutions divided regions, is presented. Therefore, eigensolutions and general solutions can be expressed uniformly. It is taken notice of that non-zero-eigenvalue solutions have local character. Especially,1/2eigenvalue solutions reveals the singularity of stresses. Namely, the stress intensity factors can be identified directly to be the coefficients of the series of certain eigenvalue solutions. After the functional of subregional integration is introduced, a new and idiographic symplectic relationship of adjoint orthogonality is presented. Based on the key technique, the relationship, the coefficients of the series, which is expanded by symplectic eigenvalue solutions, are determined from the boundary conditions. Thus, the stress intensity factors can be given directly an expression. Based on the results, a model of symplectic singular element is presented. Corresponding relationships between the coefficients of the series symplectic eigenvalue solutions and node displacements are obtained aid the close combination of symplectic functions of eigensolutions and shape functions of elements. Farther, the element stiffness matrix is given directly. The stress intensity factors have clear expressions in the system. The method of symplectic singular element combining with finite element software gives a numerical method for analyzing cracked structures and improves the accuracy of stress intensity factors. As an application of symplectic singular element, it combines with finite element software to analyze reinforcement problem of cracked structures. By analyzing and discussing, a series of results and conclusions is obtained. Results show that crack of the structure and bond strength can be in safety range under reasonable bond and design based on the length of crack in structure, material of cracked structure, constants of reinforced material, geometry size and strength of bond. The optimum reinforcement scheme is to reduce stress intensity factors at the tip of the crack on the interface of two-materials, as well as the cracked structure. The research provides reliable basis on reinforcement technology for the cracked structure.
将哈密顿体系推广到空间各向异性弹性基本问题中。以一空间坐标模拟时间,利用弹性势能得到对偶变量,并运用哈密顿原理构造出哈密顿体系下的对偶正则方程组。在哈密顿体系下,基本问题可归结为辛几何空间中的辛本征值和辛本征解的问题。通过再次引入子辛体系,因而建立起一种本征值和本征解直接求解方法。结果表明所有的零本征值本征解即是圣维南问题的解,而非零本征值本征解恰是由圣维南原理所覆盖的解。在完备的本征解空间提出一种新的辛共轭双正交关系,从而完成和完善了求解体系。之后,研究含裂纹或缺陷的材料和结构中的哈密顿体系方法。以含裂纹双材料问题为突破口,提出一种哈密顿控制方程和辛本征解的分区域表示方法,实现本征解和一般解皆能统一表述出来。注意到非零本征值本征解具有局部性特点,特别是本征值为1／2的本征解揭示了应力的奇异性特征。在引入分区域积分的泛函情况下,提出一种独特的本征解之间辛共轭正交关系。借助于此关键技术,得到了问题的解,并且可以直接给出应力强度因子的表达式。基于这些成果,可构造一种含裂纹的单元模型。将辛本征解函数与单元形函数密切结合,得到以及本征解级数系数与单元节点位移等的对应关系,直接给出单元刚度阵等。应力强度因子有明确的表达式。把该辛奇异单元与有限元软件结合形成一种独特的数值方法。该方法可对含裂纹结构分析,并可提高应力强度因子的数值精度。这种方法可应用于含裂纹结构加固问题中。经过对含裂纹结构加固问题的分析,得到了系列研究结果和结论。结果表明,根据结构裂纹的长度,结构材料和加固材料常数,几何尺寸以及粘结强度的特点可合理粘连,可以实现结构裂纹和粘结强度均处于安全范围的合理设计。降低两材料连接处的裂纹应力强度因子以及原结构裂纹应力强度因子,就可达到最佳的加固方案。研究成果为含裂纹结构在加固技术上提供了可靠依据。
The security of equipments and structures has been drawn more attention in engineering. Specially, the reinforcement problem of cracked structures has relation to security of structures, extending of service life of structures and resource saving. However, structures may appear sometimes spallations on the interfaces after they are reinforced. Thus the mechanism of layer crack problem need to be researched and the reinforcement craft is demanded to be improved with technology content. So far, the finite element method is used almost for analyzing the problem. Nevertheless, for computing stress intensity factors, the mesh density of the elements and the path of limit affect directly precision of the factors. For the problem, in this degree thesis, an impactful method is presented to analyze cracked structures, to reveal the cracking mechanism and to study the key scientific problem for the reinforcement of cracked structures. In this dissertation, the reinforcement problem of cracked structures is discussed as a background and application, and the method of Hamiltonian system for structures and the model of symplectic singular element are researched and analyzed systemically. Some main results are obtained and listed below:
The Hamiltonian system is generalized to spatial fundamental problem of anisotropic elasticity. In the system, one spatial coordinate is simulated to the time and the dual variable is obtained from the elastic potential energy. The dual equations, in Hamiltonian system, are shown with the aid of the Hamiltonian principle. Thus, the fundamental problem is reduced to symplectic eigenvalues and eigensolutions in symplectic space. Since sub-symplectic system is introduced again, a direct method is presented for solving symplectic eigenvalues and eigensolutions. Results show that Zero eigenvalue solutions belong to Saint Venant solutions and non-zero-eigenvalue solutions corresponding to the solutions that are coverd by the Saint Venant principle. In the complete space of eigensolutions, a new symplectic adjoint relationship of biorthogonality is presented among the eigensolutions. Thus, complete and perfect system for solving solutions is founded. In further research, analyzing singularity for cracked bi-materials is discussed as a breach. A method, which shows Hamiltonian governing equations and eigensolutions divided regions, is presented. Therefore, eigensolutions and general solutions can be expressed uniformly. It is taken notice of that non-zero-eigenvalue solutions have local character. Especially,1/2eigenvalue solutions reveals the singularity of stresses. Namely, the stress intensity factors can be identified directly to be the coefficients of the series of certain eigenvalue solutions. After the functional of subregional integration is introduced, a new and idiographic symplectic relationship of adjoint orthogonality is presented. Based on the key technique, the relationship, the coefficients of the series, which is expanded by symplectic eigenvalue solutions, are determined from the boundary conditions. Thus, the stress intensity factors can be given directly an expression. Based on the results, a model of symplectic singular element is presented. Corresponding relationships between the coefficients of the series symplectic eigenvalue solutions and node displacements are obtained aid the close combination of symplectic functions of eigensolutions and shape functions of elements. Farther, the element stiffness matrix is given directly. The stress intensity factors have clear expressions in the system. The method of symplectic singular element combining with finite element software gives a numerical method for analyzing cracked structures and improves the accuracy of stress intensity factors. As an application of symplectic singular element, it combines with finite element software to analyze reinforcement problem of cracked structures. By analyzing and discussing, a series of results and conclusions is obtained. Results show that crack of the structure and bond strength can be in safety range under reasonable bond and design based on the length of crack in structure, material of cracked structure, constants of reinforced material, geometry size and strength of bond. The optimum reinforcement scheme is to reduce stress intensity factors at the tip of the crack on the interface of two-materials, as well as the cracked structure. The research provides reliable basis on reinforcement technology for the cracked structure.
引文
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