高精度、高效率及显式刚度矩阵的八节点固体壳单元与四节点膜单元

发布时间：2018-08-06 09:05

【摘要】：固体壳单元是一种对于具有板壳类拓扑特性的工程结构进行高效有限元分析的新型三维有限元模型,被广泛地应用于非线性板壳、复合材料层合结构以及金属薄板冲压成型等领域。但是目前固体壳单元的研究还远不够完善,容易出现各种自锁现象,因此近年来固体壳单元的开发也成为国际计算力学界的研究热点。四节点膜单元是在考虑面内弹性变形问题以及四节点板壳单元的开发中应用最广泛的一种二维有限元模型。由于最早提出的基于位移法的双线性Q4单元在承受面内弯曲的情况时容易出现剪切自锁现象,几十年来专家学者们一直致力于精确、高效、可靠的平面四节点膜单元的开发,这对平面单元列式的理论基础创新和更加高效地解决工程问题都具有重要的现实意义。鉴于上述两种单元的研究现状与应用前景,本文的研究工作主要包括以下内容:本文采用拟协调元方法推导了一个具有显式单元刚度矩阵的八节点固体壳单元。该单元每个节点仅具有3个位移自由度,共计24个节点位移参数。根据固体壳单元中各应力分量的特点,拟协调固体壳单元在单元内假设了合理的应变场,从而可有效地避免固体壳单元中容易出现的各种自锁现象。拟协调固体壳单元的另一个显著优点是可以得到显式单元刚度矩阵,这极大地提高了所得单元的计算效率。此外,本文还采用了基于弹性力学平面问题解析解的位移试探函数近似单元围面上的位移场,从而提高所得固体壳单元的计算精度。算例表明,本文所给出的八节点拟协调固体壳单元不仅有效地克服了剪切自锁,而且拥有很高的计算效率和良好的计算精度。在笛卡尔直角坐标系内,本文利用拟协调元方法构造一个四节点四边形平面单元,相应的每个节点具有两个位移自由度(属于Q4类型膜单元)。该精确、高效的四节点拟协调膜单元的假设应变场仅有五个独立的应变参数,并且考虑了泊松效应的影响;此单元假设应变场还与由平面弹性问题控制方程给出的位移解析解相一致协调。此外,本文还给出在1991年提出的另一个基于假设应变场的四节点膜单元的性能考核。以上两个四节点膜单元均没有任何单元内部参数,并且在确定应变参数时不涉及任何数值积分,它们的单元刚度矩阵均可在笛卡尔直角坐标系内显式地计算出来。因此,这两个四节点拟协调膜单元的列式极其简单,具有非常高的计算效率;同时它们均能够通过分片试验,无剪切自锁。与其他四边形膜单元的数值结果对比表明这两个四节点拟协调四边形膜单元不仅可靠稳定,而且给出的位移和应力结果都非常精确。
[Abstract]:Solid shell element is a new kind of three-dimensional finite element model which is widely used in nonlinear plate and shell, which is used for efficient finite element analysis of engineering structures with the topological characteristics of plates and shells. Composite laminated structure and metal sheet stamping and other fields. However, the research of solid shell element is far from perfect, and it is easy to appear various self-locking phenomena. Therefore, in recent years, the development of solid shell element has become a research hotspot in the field of international computational mechanics. Four-node membrane element is the most widely used two-dimensional finite element model in the consideration of in-plane elastic deformation and the development of four-node plate-shell element. Since the first bilinear Q4 element based on displacement method is prone to shear self-locking when it is subjected to in-plane bending, experts and scholars have been devoting themselves to the development of accurate, efficient and reliable planar four-node film element for decades. This is of great practical significance to the theoretical foundation innovation of plane element formulation and to solving engineering problems more efficiently. In view of the research status and application prospect of the two kinds of elements mentioned above, the research work in this paper mainly includes the following contents: in this paper, an eight-node solid shell element with explicit element stiffness matrix is derived by using the quasi-conforming element method. Each node of the unit has only 3 degrees of freedom and a total of 24 node displacement parameters. According to the characteristics of each stress component in the solid shell element, the reasonable strain field is assumed by the quasi-conforming solid shell element in the element, which can effectively avoid all kinds of self-locking phenomena which are easy to occur in the solid shell element. Another significant advantage of the quasi-conforming solid shell element is that the explicit element stiffness matrix can be obtained, which greatly improves the computational efficiency of the resulting element. In addition, the displacement-heuristic function based on the analytical solution of the plane problem of elasticity is used to approximate the displacement field on the circumplane of the element, so as to improve the calculation accuracy of the obtained solid shell element. The numerical examples show that the proposed eight-node quasi-conforming solid shell element not only overcomes the shear self-locking effectively, but also has high calculation efficiency and good calculation accuracy. In the Cartesian Cartesian Cartesian coordinate system, a quadrilateral plane element with four nodes is constructed by using the quasi-conforming element method. Each node has two degrees of displacement (belonging to the Q4 type membrane element). The assumed strain field of the four-node quasi-conforming membrane element has only five independent strain parameters, and the Poisson effect is taken into account. This element assumes that the strain field is also consistent with the analytical solution of displacement derived from the governing equation of the plane elastic problem. In addition, the performance evaluation of another four-node membrane element based on the hypothetical strain field proposed in 1991 is also presented in this paper. Neither of the above two four-node membrane elements has any internal parameters and no numerical integration is involved in the determination of strain parameters. Their element stiffness matrices can be calculated explicitly in Cartesian Cartesian coordinate system. Therefore, the formulation of these two quasi-conforming membrane elements is extremely simple and highly efficient, and both of them can pass the shearing self-locking experiment. Compared with other quadrilateral membrane elements, the numerical results show that the two quadrilateral quasi-conforming quadrilateral membrane elements are not only reliable and stable, but also the results of displacement and stress are very accurate.
【学位授予单位】：天津大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TB115

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