基于非协调F?ppl-von Kármán方程组的冷轧带材后屈曲分析

Post buckling analysis of cold rolled strip based on incompatible F?ppl-von Kármán equations

为了揭示冷轧带材前屈曲面内残余应力与后屈曲挠度、后屈曲残余应力的关系,引入非协调F?ppl-von Kármán方程组,建立了两边自由无限长带条后屈曲的非线性偏微分方程组边值问题模型。根据冷轧带材后屈曲挠度具有轧制方向单波长周期性变化的特点,将非线性偏微分方程组边值问题分离变量而形成非线性常微分方程组边值问题。将边值问题中涉及的各物理量无量纲化,并分析这些物理量的数量级,进而确定出带有待定系数的无量纲挠度函数的形式。然后将总势能写成只与无量纲挠度函数有关的形式,并利用Ritz法确定各待定系数。最后采用其他文献中的计算结果与本文提出方法的计算结果进行对比,发现较为吻合,并解释了产生误差的原因。同时针对某冷轧厂产品计算出后屈曲释放后的残余应力,并计算了使带钢保持平直的最小张应力,为板形仪的合理应用提供了参考。

In order to reveal the relationship between pre-buckling residual stress and post buckling deflection, post buckling residual stress of cold rolled strip, the incompatible F6ppl-von Kgtrmfin equations is introduced to establish the model of boundary value problem of nonlinear partial differential equations of infinite strip with two free boundaries. The boundary value problem of nonlinear partial differential equations is simplified to a boundary value problem of non- linear ordinary differential equations with consideration of harmonic pattern of deflection. After non-dimensionalization of the involved quantities, the non-dimensional deflection with undetermined coefficients are determined by analyzing the magnitude of quantities. The total potential energy can be written only by using non-dimensional deflection and the undetermined coefficients are calculated by using Ritz method. The calculated deflection agrees well with results men- tioned in other literature, and the deviation is explained. Aiming at a typical product of a cold rolling mill, the post buck- ling released residual stress is calculated and the minimum tension is determined, which provides a reference to applica- tion of flatness measurement.