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 引用本文: 李煜冬, 傅卓佳, 汤卓超. 功能梯度碳纳米管增强复合材料板弯曲和模态的广义有限差分法. 力学学报, 2022, 54(2): 414-424.
Li Yudong, Fu Zhuojia, Tang Zhuochao. Generalized finite difference method for bending and modal analysis of functionally graded carbon nanotube-reinforced composite plates. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 414-424.
 Citation: Li Yudong, Fu Zhuojia, Tang Zhuochao. Generalized finite difference method for bending and modal analysis of functionally graded carbon nanotube-reinforced composite plates. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 414-424.

## GENERALIZED FINITE DIFFERENCE METHOD FOR BENDING AND MODAL ANALYSIS OF FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED COMPOSITE PLATES

• 摘要: 由复合材料构成的板结构一直以来受到很大关注, 其中功能梯度碳纳米管增强复合材料(functionally graded carbon nanotube-reinforced composite, FG-CNTRC)具有异常优越的力学性能, 使得诸多学者展开了对功能梯度碳纳米管增强复合材料板结构力学行为的研究. 本文以FG-CNTRC板为研究对象, 将一种新型的区域型无网格方法——广义有限差分法应用于求解基于一阶剪切变形的FG-CNTRC板结构的静态线性弯曲和自振模态问题. 广义有限差分法(generalized finite difference method, GFDM)基于函数的泰勒展开式和移动最小二乘法将计算区域中任意一子区域中心点处函数值的各阶偏导数表示成该支撑域节点上函数值的线性叠加. 该方法不仅无需网格划分和数值积分而且避免了全域无网格配点法通常遇到的病态稠密矩阵问题, 使得这类方法具有形式简单、易于应用和实现等优点, 目前广泛应用于各种科学和工程计算问题. 本文首先介绍了基于一阶剪切变形理论的功能梯度碳纳米管增强复合材料板的广义有限差分法离散模型. 随后通过基准算例, 检验了广义有限差分法的计算精度与收敛性. 最后数值分析和讨论了碳纳米管中不同分布型、体积分数、碳纳米管旋转角度、宽厚比、板倾斜角度和长宽比等对FG-CNTRC板结构弯曲和模态的影响.

Abstract: Composite plates have always received much attention. In view of excellent mechanical properties of functionally graded carbon nanotube-reinforced composite (FG-CNTRC), it is particularly important to study the mechanical behavior of FG-CNTRC plates by scholars. Based on the first order shear deformation theory, a novel meshless collocation method, generalized finite difference method (GFDM), is applied to the bending and modal analysis of FG-CNTRC plates. Based on the multivariate Taylor series expansion and the moving least-squares theory, the partial derivatives of the underdetermined displacements at a certain node can be represented by a linear combination of the displacements of its neighboring nodes in the GFDM implementation. The proposed GFDM not only has the advantages of avoiding meshing generation and numerical integration, but also provides the sparse system, which overcomes the highly ill-conditioned assembled matrix issue existed in most of meshless collocation methods. Hence the method has advantages of simple-form, easy-to-use and -implement, which is generally used in a variety of scientific and engineering problems. The numerical model for the bending and modal analysis of FG-CNTRC plates in the GFDM implementation is firstly proposed. Then the computational validity and convergence of the GFDM are analysed by some benchmark cases. Finally, the influences of different distributional types, volume fraction, rotational angle of CNTs, inclination angle of plate, thickness to span ratio, length-width ratio and boundary conditions on the structural bending and modal are investigated in details.

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