The Research On Membrane Element With Drilling Degree Of Freedom Based On Thin Shell Finite Element Method

Most injection-molded parts are complicated thin-shell like structure, so the thin shell finite element method is usually used to analyze the warpage and deformation of the parts. Generally, we have several types of thin shell elements to be used: flat-shell element, degenerated shell element and multi-variable shell element, of which the flat-shell element is adopted most widely due to its simplicity and is also selected by most famous finite element softwares, such as ABAQUS, ADINA, ANSYS, MARC, NASTRAN and SAP. However, the flat-shell has its own disadvantages: The precision of membrane element is not perfect and it is difficult to connect one-dimensional element and two-dimensional element.To resolve the problems, some scholars thought that the drilling degree of freedom should be added to membrane element. In all of the related researches, Allman's work is doubtlessly a milestone. Since he successfully introduced the drilling degree of freedom for planar triangle element in 1984, some representative elements with the drilling degree of freedom have been developed quickly. It has been proved that the membrane element with the drilling degree of freedom could effectively improve element precision. On the other hand, those elements have also been found some shortcomings: some of them have the so-called spurious zero energy mode and the drilling degree of freedom in most elements has no exact physical meaning. At the same time the correctness of the drilling degree of freedom has never been proved, so it is doubtful whether the drilling degree of freedom could correctly connect one-dimensional and two-dimensional elements and pass the twist moment.The above problems seem to exist in most membrane elements with drilling degree of freedom. To provide a robust element for warpage and shrinkage analysis in CAE of the polymer processing, the paper is mainly concentrated on the study of membrane element with drilling degree of freedom, and at the same time develops a flat-shell element. The main work is as follows:1. Based on the 6-node linear triangular element, the middle nodes of edges were removed, and an element with drilling degree of freedom was derived by means of linear interpolation in tangential and quadratic interpolation in normal. At the same time a parameter was introduced. By letting the drilling degree offreedom at the elemental nodes equal to the rotation defined in elasticity, the value of the parameter was determined, which also led to an additional equation that eliminated the spurious zero energy mode. The concrete examples show that the new element OAT3 has same precision as the original Allman's element did, but no spurious zero energy mode. Most importantly, it is proved that the drilling degree of freedom in the present element can correctly pass the twist moment.2. Based on the 10-node linear triangular element, an element with drilling degree of freedom (AT4) was derived. The nodes on the element sides were removed by means of linear interpolation in tangential and cubic interpolation in normal. The examples show that the new element AT4 has better precision than the element without the drilling degree of freedom, and no spurious zero energy mode. Its disadvantage is that stiffness matrix is too rigid to use in some cases.3. According to the discrete Kirchhoff theory, a bending plate element was derived, and the one-dimensional beam element was also derived in 3-D space. The concrete examples show that the flat-shell element whose membrane element adopted the element OAT3 has better precision than the other flat-shell element. The problems of mentioned above are also resolved.