收稿日期: 2020-01-22
网络出版日期: 2020-05-10
基金资助
国家重点研发计划资助项目(2018YFC0808402)
Application of a meshless method based on the S-R decomposition theorem in functionally graded plates
Received date: 2020-01-22
Online published: 2020-05-10
为了研究功能梯度板的非线性变形问题, 以 S-R 和分解定理为基础, 从虚功率原理出发, 结合更新拖带坐标系法、无网格 Galerkin 法, 推导出用于求解三维几何非线性问题的离散方程. 利用 MATLAB 编写无网格法程序, 对功能梯度板的非线性弯曲问题进行求解, 并研究板的体积分数指数和宽厚比对板弯曲的影响. 将计算结果与已有成果进行了比较, 验证了三维 S-R 无网格法求解功能梯度板大变形问题的合理性.
关键词： S-R 和分解定理; 几何非线性问题; 无网格 Galerkin 法; 功能梯度板
宋彦琦, 石博康, 李向上 . 基于S-R和分解定理的无网格法在功能梯度板中的应用[J]. 上海大学学报(自然科学版), 2022 , 28(4) : 702 -714 . DOI: 10.12066/j.issn.1007-2861.2266
To study the nonlinear deformation problem of functionally graded plates, this study uses the S-R decomposition theorem combined with the updated comoving coordinate system method and meshless Galerkin method to derive a discrete equation for solving three-dimensional geometric nonlinear problems. The meshless method is programmed by MATLAB. The nonlinear bending problem of the functionally graded plate is first solved, and the effects of the volume fraction index and width-thickness ration on the bending of plates are studied. Results are compared with existing results, and the rationality of solving the large deformation problem of functionally graded plates using the three-dimensional S-R meshless method is verified.
| [1] | 贾金正. 功能梯度材料结构的非线性弯曲和过屈曲[D]. 兰州: 兰州理工大学, 2019: 1-2. |
| [2] | 贾金正, 马连生. 纵横荷载作用下功能梯度梁的弯曲和过屈曲[J]. 应用力学学报, 2019, 37(1): 231-238. |
| [3] | 王雪, 赵伟东. 功能梯度梁在热-机械荷载作用下的几何非线性分析[J]. 应用数学和力学, 2019, 40(5): 508-517. |
| [4] | Khabbaz R S, Manshadi B D, Abedian A. Nonlinear analysis of FGM plates under pressure loads using the higher-order shear deformation theories[J]. Composite Structures, 2009, 89(3): 333-344. |
| [5] | 陈至达. 有限变形力学基础[M]. 北京: 中国矿业大学出版社, 2001: 28-30. |
| [6] | Hassan Z, Mehran K. Nonlinear material and geometric analysis of thick functionally graded plates with nonlinear strain hardening using nonlinear finite element method[J]. Aerospace Science and Technology, 2019, 92: 930-944. |
| [7] | Nam V N, Hoang X N, Lee S, et al. Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates[J]. Advances in Engineering Software, 2018, 126: 110-126. |
| [8] | 张雄, 刘岩, 马上. 无网格法的理论及应用[J]. 力学进展, 2009, 39(1): 1-36. |
| [9] | 聂旭涛, 范大鹏, 陈峰军. 无网格法在几何非线性力学中的应用[J]. 机械强度, 2007, 29(1): 437-441. |
| [10] | Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229-256. |
| [11] | 叶志强, 刘高联, 陈波. 二维翼型绕流的无网格差分解法[J]. 上海大学学报(自然科学版), 2009, 15(1): 26-31. |
| [12] | 刘世宇, 陈波, 刘高联, 等. 二维可压翼型绕流的新型无网格解法[J]. 上海大学学报(自然科学版), 2009, 15(5): 464-469. |
| [13] | Li G, Belytschko T. Element-free Galerkin method for contact problems in metal forming analysis[J]. Engineering Computations, 2001, 18: 62-78. |
| [14] | 张弛, 校金友, 张硕. 用无网格法分析功能梯度材料圆板的自由振动[J]. 科学技术与工程, 2014, 14(13): 145-150. |
| [15] | 陈海昌, 程昌昀. 求解功能梯度材料板弯曲问题的无网格方法[J]. 上海大学学报(自然科学版), 2007, 13(3): 299-303. |
| [16] | 陈芳祖, 罗丹. 基于S-R和分解定理的几何非线性问题的数值计算分析[J]. 湖南大学学报(自然科学版), 2012, 39(1): 42-46. |
| [17] | 罗丹. 基于S-R和分解定理的几何非线性问题的无网格 Galerkin 法分析[D]. 长沙: 湖南大学, 2011. |
| [18] | 宋彦琦, 郝亮钧, 李向上. 基于S-R和分解定理的几何非线性问题的数值计算分析[J]. 应用数学和力学, 2017, 38(9): 1029-1040. |
| [19] | 宋彦琦, 周涛. 基于S-R和分解定理的三维几何非线性无网格法[J]. 力学学报, 2018, 50(4): 853-862. |
| [20] | 李平. 非线性大变形有限元分析的更新拖带坐标法及其应用[D]. 北京: 中国矿业大学(北京), 1991. |
| [21] | Zhu P, Zhang L W, Liew K M. Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov-Galerkin approach with moving Kriging interpolation[J]. Composite Structures, 2014, 107: 298-314. |
| [22] | Zhao X, Liew K M. Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198: 2796-2811. |
| [23] | Do V N V, Lee C H. Nonlinear analyses of FGM plates in bending by using a modified radial point interpolation mesh-free method[J]. Applied Mathematical Modelling, 2018, 57: 1-20. |
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