板塑性屈曲的变分原理

Variational principle of plastic buckling of plates

在金属板料塑性成形加工中 ,起皱是常见的 ,它影响成形件质量 ,预测与防止起皱已成为人们长期以来关注的热点问题。起皱实质上是金属板料的一种压缩不稳定现象。本文把板料塑性屈曲前后的材质状态视为刚塑性 ,板材塑性皱曲时 ,所满足的全部方程 :平衡 ,机动 ,屈服及流动定律 ,归结为一个泛函的求极值问题 ,在这个泛函中 ,内力场与速度场是彼此独立的自变函数。用直接法可以方便地求得金属板料塑性屈曲问题的近似解析解。文中以环缺在边界纵向拉压力作用下的皱曲问题为例 。

Wrinkling is one of common defects in sheet metal forming processes, which has bad effect on formed parts. Prediction and prevent of wrinkling has been considered as a hot problem for long time. Wrinkling essentially is a compressive instability phenomenon of sheet metals. In this paper, the sheet metal before and after wrinkling is assumed to be the rigid-plastic material model. During plastic wrinkling, the sheet metal satisfies all equations, including differential equations of equilibrium, strain compatibility equations, yield criteria and flow rules. Thus this problem comes as to solution of the maximum value of a variation function. In the variation function, the internal force field and the velocity field are independent variable functions, which are dependent each other. Therefore, an approximately analytical solution to the plastic wrinkling in sheet metal processes can be conveniently obtained by the direct method. As an example, the wrinkling load, the internal force and velocity field are given for annular plate with the longitudinal tensile and compressive force acting on its boundary.