基于分层法的功能梯度三明治壳线性弯曲无网格分析

Linear Bending Analysis of Functionally Graded Sandwich Shells With the Meshless Method Based on the Layer⁃Wise Theory

基于3D连续壳理论和一阶剪切变形理论,采用分层法,提出了一种求解功能梯度三明治壳线性弯曲问题的移动最小二乘无网格法.通过映射技术,将随动坐标系上的二维无网格节点信息映射到三维壳中,并在随动坐标系上形成移动最小二乘近似的形函数.因基于3D连续壳理论的壳数值解答无法像特定壳一样给出其厚度方向的显式表达式,该文将功能梯度三明治材料壳结构中材料参数变化的部分划分成若干层,得到每层的材料参数为常数.利用最小势能原理,推导出了功能梯度三明治壳线性弯曲的无网格控制方程.通过引入一个厚度方向的线性变换,使得每层厚度方向的Gauss积分均在-1至1区间内,不违背一阶剪切变形理论.采用完全转化法施加本质边界条件.以功能梯度三明治板、柱壳、双曲扁壳经典几何形状壳为例,讨论了不同梯度系数、径厚比和曲率半径等对数值结果的影响,并将计算结果与文献解对比.研究表明,该方法在求解不同形状的功能梯度三明治壳线性弯曲问题时,具有收敛性好、计算精度高的特点.

Based on the 3D continuous shell theory and the 1st-order shear deformation theory,a moving least squares meshless method was proposed for solving the linear bending problem of functionally graded sandwich shells with the layered method.With the mapping technology,the 2D meshless node information on the convec-ted coordinate system was mapped on the 3D shell,and the shape function of moving least squares(MLS)ap-proximation was formed on the convected coordinate system.Due to the inability of shell numerical solutions based on the 3D continuous shell theory to provide an explicit expression for the thickness direction of a specif-ic shell,the portion of material parameter changes in the functionally graded sandwich material shell structure was divided into several layers,and the material parameters of each layer were obtained as constants.The gov-erning meshless equation for linear bending of functionally graded sandwich shells was derived under the princi-ple of minimum potential energy.Through introduction of a linear transformation in the thickness direction,the Gaussian integral in the thickness direction of each layer was bounded within the range of-1 to 1,without vio-lation of the 1st-order shear deformation theory(FSDT).The essential boundary conditions were employed with the complete transformation method.Finally,the effects of different gradient coefficients,diameter to thickness ratios,and curvature radii on numerical results were discussed through examples of functionally gra-ded sandwich plates,cylindrical shells,and hyperbolic shallow shells with classical geometric shapes.The cal-culated results were compared with the literature solutions.The results show that,the proposed method has the characteristics of good convergence and high computation accuracy in solving linear bending problems of func-tional graded sandwich shells with different shapes.