含裂纹和弱界面结构断裂分析中的辛方法

发布时间：2018-06-16 11:36

本文选题：哈密顿体系 + 粘弹性　；参考：《大连理工大学》2016年博士论文

【摘要】：随着我国先进材料科学和制造工艺的发展,先进复合材料的制备与应用对国家发展有重大意义。同时具备力、电、磁、热、声、光等两种或多种特性的功能复合材料,诸如聚合物基材料、压电材料、电磁材料、光伏材料等被广泛应用于航空航天、建筑工业、电子工业以及医疗器械等领域。层合板结构是工程中的基本结构,该类结构是由多层单层板通过聚合物胶黏剂粘合在一起组成整体的结构板。由于其界面的连接特点,脱粘是该类结构的主要破坏模式。因此,研究粘弹性断裂问题以及力电磁弹耦合界面断裂问题具有重要的实际意义。现有断裂问题的研究方法主要有解析法(复势函数法、积分变换法、权函数法等)和数值方法(有限单元法、边界单元法、无网格法等)。解析研究主要归结为高阶偏微分方程或积分方程的求解,在数学上受到求解体系(拉格朗日体系)的限制,它以提高控制方程的阶数为代价来减少变量个数,从而造成方程难以求解。在这种情况下,本博士论文将问题导入全新的哈密顿体系下进行求解,利用辛方法以增加基本变量为代价来降低微分方程阶数,利用高性能计算机来求解低阶微分方程。本博士论文以复合材料的层间界面断裂问题为研究对象,以粘弹性断裂问题为突破口,采用辛方法对电磁弹性材料弱界面断裂问题展开研究,主要研究工作如下：(1)提出一种求解粘弹性材料断裂问题的解析方法。首先,利用Laplace变换将粘弹性断裂问题转换为频域相关问题。然后,在频域内,通过拉格朗日函数和哈密顿函数,建立粘弹性材料断裂问题的哈密顿正则方程组和求解体系,将求解问题归结为辛本征值和辛本征解问题。最后,利用本征解之间的辛共轭正交关系以及展开定理,解析地表征出裂纹尖端的奇异性和域内的物理场,直接获得应力强度因子和J积分的解析表达式。数值结果验证了方法的收敛性和有效性,并且反映出粘弹性材料特有的应力松弛现象和蠕变现象,特别指出温度荷载作用对粘弹性断裂参数的影响。(2)将弱界面模型引入到平面弹性断裂问题中,并采用辛方法进行求解。将复合材料层间粘结问题简化为弱连接问题,弹簧模型作为弱连接条件的数学模型。首先推导出两种弹性材料的辛本征解形式,然后由弱连接条件以及裂纹面条件确定两种本征解共用的辛本征值,从而全域内的辛本征解由以上两种材料所在区域的本征解组成,并且在全域内满足辛共轭正交关系。最后利用辛本征解的展开定理、外边界边界条件和辛共轭正交关系,可直接得到裂纹尖端处的奇异性,域内应力等分布和广义应力强度因子等解析表达式。数值结果表明,该方法与已有经典问题结果相吻合,说明本文方法是有效的,并且该方法具有较高的精度。(3)将弱界面模型引入到电磁弹性材料反平面断裂问题中,并采用辛方法对多场耦合作用情况进行求解。首先,采用能量方法和勒让德变换,确定原变量(位移、电势和磁势)的对偶变量(应力、电位移、磁感应强度),从而将问题导入到哈密顿体系中。然后,以弹簧模型作为弱连接条件的数学模型,结合不同的力学、电学、磁学裂纹面条件,推导出弹簧模型连接的两区域内的辛本征解向量和共用的辛本征值,并验证全域内辛本征解之间存在辛共轭正交关系。最后,利用外边界条件和辛共轭正交关系,确定问题解的解析表达式以及断裂参数。数值结果表明,由于弱连接界面的存在,使得广义强度因子分为两类,且弱界面参数影响强度因子的大小,而裂纹面条件影响广义强度因子的存在与否。
[Abstract]:With the development of advanced materials science and manufacturing technology in China, the preparation and application of advanced composites have great significance for the development of the country. At the same time, two or more functional composite materials, such as polymer based materials, piezoelectric materials, electromagnetic materials and photovoltaic materials, are widely used in aerospace and aerospace. In the fields of construction industry, electronic industry and medical equipment. Laminate structure is the basic structure in engineering. This kind of structure is composed of multi layer monolayer bonded together by polymer adhesive. Because of its interface characteristics, debonding is the main failure mode of this kind of structure. Therefore, the study of viscoelastic fracture is studied. The problem and the fracture problem of the force electromagnetic elastic coupling interface are of great practical significance. The main research methods of the existing fracture problems are analytic method (complex potential function method, integral transformation method, weight function method etc.) and numerical methods (finite element method, boundary element method, meshless method, etc.). The analytical study is mainly attributed to the high order partial differential equation or integral square. The solution of the process is limited by the mathematical solution system (Lagrange system). It reduces the number of variables by increasing the order of the control equation. Thus, the equation is difficult to solve. In this case, the doctoral thesis introduces the problem into a new Hamilton system and uses the symplectic method to increase the basic variables. In order to reduce the order of differential equations and use high performance computer to solve the low order differential equation, this thesis takes the interlayer interface fracture problem of composite material as the research object, taking the problem of viscoelastic fracture as the breakthrough point, using symplectic method to study the problem of weak interface fracture of electromagnetic elastic material. The main research work is as follows: (1) proposed An analytical method for solving the fracture problem of viscoelastic materials is solved. First, the problem of viscoelastic fracture is converted into a frequency domain problem by Laplace transformation. Then, in the frequency domain, the Hamilton regular equation set and solution system for the fracture of viscoelastic materials are established by Lagrange's function and Hamiltonian function. The solution is reduced to symplectic problem. The eigenvalues and the symplectic eigensolutions are solved. Finally, the analytical expressions of the stress intensity factors and the J integral are obtained by using the symplectic conjugate orthogonal relations between the eigensolutions and the expansion theorem, and the analytical expressions of the stress intensity factors and the J integral are obtained. The numerical results verify the convergence and effectiveness of the square method and reflect the viscoelasticity. The special stress relaxation and creep phenomenon of the sexual material, especially the effect of the temperature load on the viscoelastic fracture parameters. (2) the weak interface model is introduced into the plane elastic fracture problem, and the symplectic method is used to solve the problem. The problem of interlayer bonding is simplified to the weak connection problem, and the spring model is used as the weak connection condition. The mathematical model. First, the symplectic eigensolution of two kinds of elastic materials is derived. Then the symplectic eigenvalues shared by the two eigensolutions are determined by the weak connection condition and the condition of the crack surface, thus the symplectic eigensolution in the whole domain is composed of the eigensolutions of the regions where the above two materials are located, and the symplectic conjugate orthogonal relationship is satisfied in the whole domain. The expansion theorem of eigensolution, the boundary boundary condition of the outer boundary and the symplectic conjugate orthogonality relation can directly obtain the analytic expressions of the singularity at the crack tip, the distribution of the stress in the domain and the generalized stress intensity factor. The numerical results show that the method is in agreement with the results of the existing classical problems, which shows that the method is effective and the method has a good effect. High precision. (3) the weak interface model is introduced to the anti plane fracture of electromagnetic elastic material and the symplectic method is used to solve the multi field coupling. First, the dual variable (stress, potential shift, magnetic induction intensity) of the original variable (displacement, potential and magnetic potential) is determined by the method of energy and Legendre transformation. In the Hamilton system, the symplectic eigenvalues and symplectic eigenvalues in the two regions connected by the spring model are derived by using the spring model as the mathematical model of the weak connection condition, combining the different mechanics, electrical and magnetic crack surface conditions, and the symplectic conjugate orthogonal relationship exists between the symplectic eigenvalues in the whole domain. Finally, the benefit of the symplectic conjugate is verified. By using the external boundary condition and the symplectic conjugate orthogonal relation, the analytic expression of the solution and the fracture parameters are determined. The numerical results show that the weak interface is divided into two types, and the weak interface parameters affect the intensity factor, while the cracked noodles affect the existence or not of the generalized intensity factor.
【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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