拱形金属薄膜非线性力学行为数学建模及其计算

Mathematical modeling and simulation for nonlinear mechanical behavior of vaulted metallic membrane

对大变形金属薄膜结构塑性应力应变关系、几何关系和静力平衡关系进行整理和适当变换,将其转化成由3个微分方程和1个代数约束方程组成的初值问题的1阶微分代数方程。采用可变步长和变阶的Klopfenstein-Shampine数值微分方法和Newton-Raphson求解方法,可求得膜片任何位置在任意时刻的应力、应变和变形等力学参量,还可以估算出膜片的极限荷载。最后对一个实例作了数值分析,其计算结果与实验数据得到了较好的符合。

A method for plastic large deformation analysis of circular metallic membrane subjected to lateral pressure was presented. Stress-strain relationships of plasticity, geometrical equations, and static equilibrium conditions were acquired. Then these expressions were transformed to initial-value problems in differential algebraic equations of index 1, which consists of three ordinary differential equations and one algebraic constraint equation. Numerical solutions were carried out by using Newton-Raphson root-locating method and Klopfenstein-Shampine numerical differentiation formulas with varying step size and variable order. The stresses, strains, deformations of metallic membrane at specific times could be obtained and the ultimate pressure could also be estimated. Finally, numerical analysis for an example was completed. The comparison shows fairly close agreement between numerical and experimental results.