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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 the residual stress in the buckling surface and the post-buckling residual stress and the post-buckling residual stress of the cold-rolled strip, a non-linear Föppl-von Kármán equation was introduced to establish the nonlinear partial post-buckling nonlinearity Differential Equation System Boundary Value Problem Model. According to the characteristic that the buckling deflection of cold-rolled strip has periodic change of single wavelength in the rolling direction, boundary value problems of nonlinear partial differential equations are separated by variables to form boundary value problems of nonlinear ordinary differential equations. The dimensionless quantities of the physical quantities involved in the boundary value problem are analyzed and the orders of magnitude of these physical quantities are analyzed to determine the form of dimensionless deflection function with undetermined coefficients. Then the total potential energy is written as a form related only to the dimensionless deflection function and the Ritz method is used to determine the undetermined coefficients. In the end, we compare the calculated results in other literatures with the calculated ones proposed in this paper and find that they are in good agreement with each other and explain the reasons for the errors. At the same time, the residual stress after post-buckling release was calculated for a cold-rolling mill product, and the minimum tensile stress for keeping the strip straight was calculated, which provides a reference for the reasonable application of the shape meter.