臧勇等：基于渐进变分法的蛋盒型结构等效刚度分析及优化 ·1395· 合公式，通过修改目标函数及限制条件可以快速实现 orthines,2010.15(1):59 蛋盒型结构参数的优化设计. [9]Nowpada S,Chirwa E C.Myler P,et al.Aluminum "egg-box" panel as an energy absorber for pedestrian protection.Adu Eng 3结论 Mater,2010,12(7):591 [10]Sashikumar S,Chirwa EC.Myler P,et al.Numerical investiga- (1)实现了蛋盒型结构等效刚度特性的数值计 tion into the collapse behaviour of an aluminium egg-box under 算方法，并对计算结果进行了验证，计算结果表明拉 quasi-static loading.Int Crashuorthines,2012,17(6):582 伸刚度降低，弯曲刚度升高，同时表现出了负泊松比 [11]Javid F,Smith-Roberge E,Innes M C,et al.Dimpled elastic 的特性，蛋盒型结构比普通薄板呈现出了更高的承 sheets:a new class of non-porous negative Poisson's ratio materi- 载能力. als.Sci Rep-UK,2015,5:18373 (2)计算不同厚度H和间距p的蛋盒型结构的等 [12]Xia YY,Friswell M I.Equivalent models of corrugated lami- 效刚度特性，得到结构参数变化对等效刚度特性的影 nates for morphing skins//Proceedings of the Active and Passive Smart Structures and Integrated Systems V.San Diego,2011: 响规律：随后对等效刚度特性进行拟合，拟合公式可以 79771-1 很好地预测蛋盒型结构参数与等效刚度特性的关系. [13]Xia Y,Friswell M I,Flores E I S.Equivalent models of corruga- (3)根据拟合公式，可在不同的约束条件和优化 ted panels.Int Solids Struct,2012,49(13):1453 目标的情况下快速实现蛋盒型结构的主动优化设计. [14]Briassoulis D.Equivalent orthotropic properties of corrugated 未来结合成型过程厚度的变化及蛋盒型结构的吸能能 sheets.Comput Struct,1986,23(2):129 力、稳定性等特性进行分析，可以使其在汽车、航空航 [15]Peng L X,Liew K M,Kitipornchai S.Analysis of stiffened cor- rugated plates based on the FSDT via the mesh-free method.IntJ 天、船舶重工、建筑等领域具有更广泛的应用前景. Mech Sci,2007,49(3):364 [16] Ye Z,Berdichevsky VL,Yu W B.An equivalent classical plate 参考文献 model of corrugated structures.Int J Solids Struct,2014,51(11- [1]Ashmead M.Energy-Absorbing Structures:US Patent,6547280. 12):2073 2003-4-15 [17]Berdichevsky V.Variational Principles of Continuum Mechanies: [2]Deshpande V S,Fleck N A.Energy absorption of an egg-box ma- Il.Applications.Berlin:Springer Science Business Media, terial.J Mech Phys Solids,2003,51(1):187 2009 [3]Zupan M,Chen C.Fleck N A.The plastic collapse and energy [18]Ye Z.Enhance Variational Asymptotic Method for Unit Cell Hom- absorption capacity of egg-box panels.Int J Mech Sci.2003,45 ogenization VAMUCH)for Real Engineering Structures and Ma- (5):851 terials Dissertation].Logan Utah:Utah State University,2013 [4]Akisanya A R,Fleck N A.Plastic collapse of thin-walled frusta [19]Yu W B,Tang T.Variational asymptotic method for unit cell and egg-box material under shear and normal loading.Int Mech homogenization of periodically heterogeneous materials.IntSol- Sci,2006,48(7):799 ids Struct,2007,44(11-12):3738 [5]Chung J G,Chang S H,Sutcliffe M P F.Deformation and energy [20]Wang X C.Finite Element Method.Beijing:Tsinghua University absorption of composite egg-box panels.Compos Sci Technol, Press,2003 2007,67(11-12):2342 (王勖成.有限单元法.北京：清华大学出版社，2003) [6]Yoo S H,Chang S H.An experimental study on energy absorbing [21]Huang A X.A method for treating periodical boundary condi- structures made of fabric composites.Compos Struct,2008,86(1- tions.J Num Method Comp Appl,1983,4(1):37 3):211 (黄艾香.周期性边界条件的一种处理方法.数值计算与计 [7]Yoo S H,Chang S H,Sutcliffe M P F.Compressive characteris- 算机应用，1983,4(1)：37) tics of foam-filled composite egg-box sandwich panels as energy ab- [22]Shen G L.Hu G K.Mechanies of Composite Materials.Beijing: sorbing structures.Composites Part A,2010,41(3):427 Tsinghua University Press,2006 [8]Nowpada S,Chirwa E C,Myler P,et al."Egg-box"panel for (沈观林，胡更开.复合材料力学.北京：清华大学出版社， commercial vehicle front-compressive loading tests.Int Crash- 2006)臧 勇等: 基于渐进变分法的蛋盒型结构等效刚度分析及优化 合公式,通过修改目标函数及限制条件可以快速实现 蛋盒型结构参数的优化设计. 3 结论 (1) 实现了蛋盒型结构等效刚度特性的数值计 算方法,并对计算结果进行了验证,计算结果表明拉 伸刚度降低,弯曲刚度升高,同时表现出了负泊松比 的特性,蛋盒型结构比普通薄板呈现出了更高的承 载能力. (2) 计算不同厚度 H 和间距 p 的蛋盒型结构的等 效刚度特性,得到结构参数变化对等效刚度特性的影 响规律;随后对等效刚度特性进行拟合,拟合公式可以 很好地预测蛋盒型结构参数与等效刚度特性的关系. (3) 根据拟合公式,可在不同的约束条件和优化 目标的情况下快速实现蛋盒型结构的主动优化设计. 未来结合成型过程厚度的变化及蛋盒型结构的吸能能 力、稳定性等特性进行分析,可以使其在汽车、航空航 天、船舶重工、建筑等领域具有更广泛的应用前景. 参 考 文 献 [1] Ashmead M. Energy鄄Absorbing Structures: US Patent, 6547280. 2003鄄鄄4鄄鄄15 [2] Deshpande V S, Fleck N A. Energy absorption of an egg鄄box ma鄄 terial. J Mech Phys Solids, 2003, 51(1): 187 [3] Zupan M, Chen C, Fleck N A. The plastic collapse and energy absorption capacity of egg鄄box panels. Int J Mech Sci, 2003, 45 (5): 851 [4] Akisanya A R, Fleck N A. Plastic collapse of thin鄄walled frusta and egg鄄box material under shear and normal loading. Int J Mech Sci, 2006, 48(7): 799 [5] Chung J G, Chang S H, Sutcliffe M P F. Deformation and energy absorption of composite egg鄄box panels. Compos Sci Technol, 2007, 67(11鄄12): 2342 [6] Yoo S H, Chang S H. An experimental study on energy absorbing structures made of fabric composites. Compos Struct, 2008, 86(1鄄 3): 211 [7] Yoo S H, Chang S H, Sutcliffe M P F. Compressive characteris鄄 tics of foam鄄filled composite egg鄄box sandwich panels as energy ab鄄 sorbing structures. Composites Part A, 2010, 41(3): 427 [8] Nowpada S, Chirwa E C, Myler P, et al. “Egg鄄box冶 panel for commercial vehicle front鄄compressive loading tests. Int J Crash鄄 worthines, 2010, 15(1): 59 [9] Nowpada S, Chirwa E C, Myler P, et al. Aluminum “ egg鄄鄄 box冶 panel as an energy absorber for pedestrian protection. Adv Eng Mater, 2010, 12(7): 591 [10] Sashikumar S, Chirwa E C, Myler P, et al. Numerical investiga鄄 tion into the collapse behaviour of an aluminium egg鄄box under quasi鄄static loading. Int J Crashworthines, 2012, 17(6): 582 [11] Javid F, Smith鄄Roberge E, Innes M C, et al. Dimpled elastic sheets: a new class of non鄄porous negative Poisson蒺s ratio materi鄄 als. Sci Rep鄄UK, 2015, 5: 18373 [12] Xia Y Y, Friswell M I. Equivalent models of corrugated lami鄄 nates for morphing skins / / Proceedings of the Active and Passive Smart Structures and Integrated Systems V. San Diego, 2011: 79771I鄄1 [13] Xia Y, Friswell M I, Flores E I S. Equivalent models of corruga鄄 ted panels. Int J Solids Struct, 2012, 49(13): 1453 [14] Briassoulis D. Equivalent orthotropic properties of corrugated sheets. Comput Struct, 1986, 23(2): 129 [15] Peng L X, Liew K M, Kitipornchai S. Analysis of stiffened cor鄄 rugated plates based on the FSDT via the mesh鄄free method. Int J Mech Sci, 2007, 49(3): 364 [16] Ye Z, Berdichevsky V L, Yu W B. An equivalent classical plate model of corrugated structures. Int J Solids Struct, 2014, 51(11鄄 12): 2073 [17] Berdichevsky V. Variational Principles of Continuum Mechanics: II. Applications. Berlin: Springer Science & Business Media, 2009 [18] Ye Z. Enhance Variational Asymptotic Method for Unit Cell Hom鄄 ogenization (VAMUCH) for Real Engineering Structures and Ma鄄 terials [Dissertation]. Logan Utah: Utah State University, 2013 [19] Yu W B, Tang T. Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials. Int J Sol鄄 ids Struct, 2007, 44(11鄄12): 3738 [20] Wang X C. Finite Element Method. Beijing: Tsinghua University Press, 2003 (王勖成. 有限单元法. 北京: 清华大学出版社, 2003) [21] Huang A X. A method for treating periodical boundary condi鄄 tions. J Num Method Comp Appl, 1983, 4(1): 37 (黄艾香. 周期性边界条件的一种处理方法. 数值计算与计 算机应用, 1983, 4(1): 37) [22] Shen G L, Hu G K. Mechanics of Composite Materials. Beijing: Tsinghua University Press, 2006 (沈观林, 胡更开. 复合材料力学. 北京: 清华大学出版社, 2006) ·1395·