基于非协调F?ppl-von Kármán方程组的冷轧带材后屈曲分析

李秾; 李洪波; 张杰

doi:10.13228/j.boyuan.issn0449-749x.20160554

钢铁 ›› 2017, Vol. 52 ›› Issue (5) : 47-54. DOI: 10.13228/j.boyuan.issn0449-749x.20160554

压力加工

基于非协调F?ppl-von Kármán方程组的冷轧带材后屈曲分析

李秾¹，李洪波¹，张杰¹，贾生晖²，褚玉刚²，刘海军²

作者信息 +

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

李秾¹，李洪波¹，张杰¹，贾生晖²，褚玉刚²，刘海军²

Author information +

文章历史 +

摘要

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。

Abstract

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 F?ppl-von Kármán 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 nonlinear 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 mentioned in other literature，and the deviation is explained. Aiming at a typical product of a cold rolling mill，the post buckling released residual stress is calculated and the minimum tension is determined，which provides a reference to application of flatness measurement.

关键词

冷轧薄带材 / 后屈曲 / 非协调F?ppl-von Kármán方程组 / Rayleigh-Ritz法

Key words

thin cold rolled strip / post buckling / incompatible F?ppl-von Kármán equations / Rayleigh-Ritz method

图表

引用本文

EndNote

Ris (Procite)

Bibtex

导出引用

李秾, 李洪波, 张杰. 基于非协调F?ppl-von Kármán方程组的冷轧带材后屈曲分析[J]. 钢铁, 2017, 52(5): 47-54 https://doi.org/10.13228/j.boyuan.issn0449-749x.20160554

LI Nong, LI Hong-Bei, ZHANG Jie. Post buckling analysis of cold rolled strip based on incompatible F?ppl-von Kármán equations[J]. Iron and Steel, 2017, 52(5): 47-54 https://doi.org/10.13228/j.boyuan.issn0449-749x.20160554

参考文献

[1] Lewicka M, Mahadevan L, Pakzad R. The F?ppl-von Kármán equations for plates with incompatible strains [J]. Proceedings of Royal Society A. 2011, 467(2126): 402-426.
[2] 李秾, 李洪波, 张杰, 等. 基于平面弹性复变方法的冷轧带钢分布位错-残余应力模型[J]. 工程科学学报, 2016, 38(3): 410-416. (LI Nong, LI Hong-bo, ZHANG Jie, et al. Distributed dislocation-residual stress model of cold rolled strips based on the complex variable function method of plane elasticity [J]. Chinese Journal of Engineering, 2016, 38(3): 410-416.)
[3] 宋小芳, 何安瑞, 邱增帅. 基于初始温差的高强钢板层冷过程前屈曲分析[J]. 钢铁, 2016, 51(5): 52-56. (SONG Xiao-fang, HE An-rui, QIU Zeng-shuai. Influence of initial temperature difference on high strength strip buckling in laminar cooling [J]. Iron and Steel, 2016, 51(5): 52-56.)
[4] Sharon E, Roman B, Swinney H L. Geometrically driven wrinkling observed in free plastic sheets and leaves [J]. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2007, 25(11): 77-85.
[5] 戴杰涛, 张清东. 冷轧薄板中浪板形缺陷的屈曲及后屈曲理论与轧制试验研究[J]. 机械工程学报, 2011, 47(2): 44-50. (DAI Jie-tao, ZHANG Qing-dong. Analysis and experiment on central buckling and post buckling of thin cold-rolled sheet [J]. Journal of Mechanical Engineering, 2011, 47(2): 44-50.)
[6] Nakhoul R, Montmitonnet P, Potier-Ferry M. Multi-scale method for modeling thin sheet buckling under residual stresses in the context of strip rolling [J]. International Journal of Solids and Structures, 2015, 66(4): 62-76.
[7] 王澜, 曹建国, 贾生晖, 等. 冷轧带钢板形屈曲变形失稳限的有限元分析[J]. 中南大学学报(自然科学版). 2007, 38(6): 1157-1161. (WANG Lan, CAO Jian-guo, JIA Sheng-hui, et al. Finite element analysis of shape buckling load for cold rolled strips. Journal of Central South University (Science and Technology), 2007, 38(6): 1157-1161.)
[8] Abdelkhalek S, Zahrouni H, Legrand N, Potier-Ferry M. Post-buckling modeling for strips under tension and residual stresses using asymptotic numerical method [J]. International Journal of Mechanical Sciences. 2015, 104: 126-137.
[9] QIN Jian, ZHANG Qing-dong, HUANG Ke-fu. Oblique and herringbone buckling analysis of steel strip by spline FEM [J]. Journal of Iron and Steel Research International. 2011. 18(9), 21-26.
[10] 杨荃. 冷轧带钢屈曲理论与板形控制目标的研究[D]. 北京: 北京科技大学, 1992. ( YANG Quan. Study on the Cold Rolled Strip Buckling and the Target Shape in the Automatic Flatness Control [D]. Beijing: University of Science and Technology Beijing, 1992)
[11] Fischer F D, Rammerstorfer F G, Friedl N. Residual stress-induced center wave buckling of the rolled strip [J]. ASME Journal of Applied Mechanics. 2003, 70(1): 84-90.
[12] 张清东, 卢兴福, 张晓峰. 具有初始翘曲缺陷冷轧薄带钢板形瓢曲变形为研究[J]. 工程力学, 2014, 31(8): 243-249. (ZHANG Qing-dong, LU Xing-fu, ZHANG Xiao-feng. Analysis of buckling deformation for thin cold-rolled strip with initial warping defect. Engineering Mechanics, 2014, 31(8): 243-249.)
[13] 边宇虹, 刘宏民. 求解带材轧后大挠度屈曲变形的一个通用方法[J]. 机械工程学报, 1994, 30(增刊): 21-27. (BIAN Yu-hong, LIU Hong-min. Universal method analyzing the large deflection buckling deformation of rolled strip [J]. Chinese Journal of Mechanical Engineering, 1994, 30(Suppl.)：21-27.)
[14] 常铁柱. 薄宽带钢板形斜向和横向屈曲变形行为研究[D]. 北京: 北京科技大学, 2009. (CHANG Tie-zhu. Deformation of Herringbone Buckling and Transverse Buckling for Thin and Wide Strip [D]. Beijing：University of Science and Technology Beijing, 2009.)
[15] 潘立宇, 王蜀. 均布载荷下矩形板大挠度问题的摄动变分解[J]. 应用数学和力学, 1986, 7(8): 675-688. (PAN Li-yu, WANG Shu. A perturbation-variational solution of the large deflection of rectangular plates under uniform load [J]. Applied Mathematics and Mechanics, 1986, 7(8): 675―688.)