新型的三维板方法分析层合板弯曲问题

A New 3-D Plate Theory for the Analysis of Laminate Bending Problem

在层合板弯曲问题的数值模拟过程中,经常会采用不同假设理论来近似和简化处理,包括经典层合板理论、一阶剪切理论、三阶剪切理论等高阶剪切理论,这些理论都有各自的优缺点.基于方向的无网格和有限元耦合的三维板方法,即在板厚度方向的各子层内使用满足C0连续性的有限元近似,而在板的面内方向使用基于径向基插值的无网格近似,无需事先假设横向剪切理论,是一种新型的三维板方法,它采用的位移近似理论能够呈现整体满足连续性的位移和各面内应变分量,及满足"层间不连续性"的各剪切应变分量,为得到"层间连续性"的各剪切应力分量提供了可能性.然后,通过面内的二维节点离散及"映射技术"较快捷地实现三维位移场的近似,并利用三维的几何关系和物理关系分别构造应变场、应力场近似.同时具有插值性的耦合位移近似使得能够方便地施加本质边界条件.基于最小势能原理,利用无单元Galerkin法建立系统离散代数方程组.通过分析数值结果,探讨该方法在实现过程中可能出现的各种自锁现象,并给出解决自锁的方案.最后,通过数值算例验证了它的有效性和高精度.

Different hypothesis of the plate are often used to approximate and simplify the treating of the bending problems of laminated plates, including classic laminated plate theory （CLPT）, first order shear deformation plate theory （FSDT）, the third order shear deformation plate theory （TSDT） and the other higher order shear deformation plate theory （HOST）. These theories have their own advantages and disadvantages. 3-D plate method based on the couple of the meshless and finite element, namely, the finite element approximation of the Co continuity is used in the sub-layers of the plate thickness direction, in-plane direction is based on the meshless approximation of the radial basis interpolation, and the transverse shear theory is not required. It is a new 3-D plate method. The displacement approximation theory shows an overall satisfy the continuity of displacement and the in-plane strain components and meets“discontinuities between the layer” of the shear strain components. It brings the possibility to the “interlayer continuity” of the sheer force components. Then, the 3-D displacement field approximation is quickly realized by the 2-D node discretization and “mapping technology”, and the strain field, stress field are respectively approximated by the 3-D geometric relationship and physical relations. At the same time, the coupling displacement approximation with interpolation can impose essential boundary conditions easily. Based on the principle of minimum potential energy, the discrete algebraic equations of the system were established by using the element free Galerkin method. By the analysis of numerical results, the possibility of self-locking in the process were discussed, and the scheme of solving the difficulty was given. Finally, the validity and accuracy of the method wer verified by numerical examples.