基于變分漸近法的雙周期梯形波紋板等效模型

鄧兵 鐘軼峰 席森彪 禹遼

摘要：雙周期梯形波紋板是沿兩個平面方向均呈周期性梯形波紋變化的新型輕質結構，現階段對其力學性能與結構參數之間關系的研究較少，限制了該結構的推廣應用。針對該問題，以凸起為中心的典型代表單元單胞為研究對象，利用變分漸近法建立單胞等效剛度的數值計算模型。再通過均勻化技術將雙周期梯形波紋板轉換為具有相同剛度特性的正交異性板進行分析。基于該方法分別計算出不同結構參數下的等效剛度、宏觀變形和屈曲模態，通過與三維有限元模擬結果對比驗證等效模型的精確性。計算結果表明：由于形貌變化，雙周期梯形波紋板的等效剛度較平板的拉伸剛度降低，彎曲剛度增加;隨著凸起間距的增大和板高度的減小，波紋板的拉伸剛度逐漸增大，彎曲剛度逐漸降低;而隨著板厚度和腰部長度的增加，彎曲剛度呈上升的趨勢。構建模型適用于波紋周期遠小于結構尺寸的情況，由于等效板模型的近似能量與原三維波紋板能量盡可能接近（通過對能量泛函變分主導項的漸近擴展分析加以保證），可用于計算不同材料波紋板等效彎矩和等效最大拉壓應力等，能有效減少計算量和節省計算資源。

關鍵詞：變分漸近法;梯形波紋板;等效剛度;等效模型;降維分析

中圖分類號：TU501;TB333 文獻標志碼：A 文章編號：20966717（2020）04017510

收稿日期：20191124

基金項目：國家自然科學基金（51778088）;重慶市自然科學基金 （cstc2017jcyjBX00036）

作者簡介：鄧兵（1993 ），男，主要從事輕質高強結構與材料研究，Email： 20171602025t@cqu.edu.cn。

鐘軼峰（通信作者），男，教授，博士生導師，Email： zhongjy58@sina.com。

Received：20191124

Foundation items：National Natural Science Foundation of China （No. 51778088）; Chongqing Natural Science Foundation （No. cstc2017jcyjBX00036）

Author brief：Deng Bing（1993 ）， main research interests： lightweight and high strength structure and materials， Email： ?20171602025t @cqu.edu.cn.

Zhong Yifeng（corresponding author）， professor， doctorial supervisor， Email： zhongjy58@sina.com.

Equivalent model of biperiodic trapezoidal corrugated plate based on variational asymptotic method

Deng Bing， Zhong Yifeng， Xi Senbiao， Yu Liao

（School of Civil Engineering; Key Laboratory of New Technology for Construction of Cities in Mountain Area， Ministry of Education， Chongqing University， Chongqing 400045， P.R. China）

Abstract： The biperiodic trapezoidal corrugated plate is a new type of lightweight structure with periodic trapezoidal corrugated changes along two plane directions. At present， there is little research on the relationship between its mechanical properties and structural parameters， limiting the popularization and application of this structure. In this article， a numerical model of the represent element （or called unit cell） is established by using the variational asymptotic method to calculate the equivalent stiffness. Then， the biperiodic trapezoidal corrugated plate is converted to orthotropic plate with the same stiffness characteristics by homogenization technology. Based on this method， the equivalent stiffness with different structural parameters， global deformation and buckling modes are calculated， and the accuracy of the equivalent model is verified by comparing with the threedimensional finite element simulation results. The results show that the equivalent stiffness of the biperiodic trapezoidal corrugated plate is lower than that of the plate due to the change of the shape， and the bending stiffness is increased. The tensile stiffness increase and the bending stiffness decrease with increasing interval length and decreasing plate height. While with the increase of plate thickness and web length， the bending stiffness shows an upward trend. The constructed model is suitable for the case that the period of corrugation is far less than the structure size. The approximate energy of the equivalent plate model is as close as that of the original threedimensional corrugation plate， which is guaranteed by the asymptotic expansion analysis of the leading terms of energy functional. The equivalent plate model can be used to calculate the equivalent bending moment and the equivalent maximum tensile/compressive stress of corrugated plates with different material properties， which can effectively reduce the calculation amount and save the calculation resources.

Keywords：variational asymptotic method; trapezoidal corrugated panel; equivalent stiffness; equivalent model; dimensional reduction analysis