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第2期 马文锁等：二维编织复合材料几何结构的平面群分析 .231. [5]Quek S C.Waas A.Shahwan K W,et al.Compressive response and failure of braided textile composites:Part 2-Computations. Int J Non Linear Mech.2004.39(4):649 [6]庞宝君，杜善义，韩杰才.三维四向编织复合材料细观组织及 分析模型.复合材料学报，1999,16(3)：135 [7]Sun W.Lin F,Hu X.Computer aided design and modeling of p3m1.p31m p6mm composite unit cells.Compos Sci Technol.2001.61(2):289 [8]Adanur S.Liao T.3D modeling of textile composite performs 图7由平面群p3ml、p31m和p6mm推导平面编织几何结构 Compos Part B,1998,29(6):787 时出现的不合理图案 [9]Brown D.Morgan M.Mellhagger R.A system for the automatic Fig.7 Yarn segments combination patterns with the symmetrical generation of solid models of woven structures-Compos Part A. characteristic of plane groups p3ml,p31m and p6 mm 2003,34(6):511 [10]Lomov S V.Textile geometry preprocessor for meso mechanical 6 结论 models of woven composites.Compos Sci Technol,2000.60 (11):2083 由平面点群到平面群可以推导出4个编织系， [11]Naik N K,Kuchibhotla R.Analytical study of strength and fail- 对应的17个平面群不仅可以充分概括二维平面编 ure behaviour of plain weave fabric composites made of twisted yarns.Compos Part A.2002.33(5):697 织的现有几何结构，而且可以以点符号的组合、交叉 [12]Huang Z M.The mechanical properties of composites reinforced 原则，通过建立无对称单元、基本对称单元，遵从不 with woven and braided fabrics.Compos Sci Technol,2000.60 同二维空间点阵描述的对称性，最终推导和预测不 (4):479 同几何结构的编织方法，在编织方案的确定，优化 [13]Subhash G.Influence of strain rate on the uniaxial compressive 材料性能方面无疑是有益的， behavior of 2-D braided textile composites.Compos Part A. 2001,32(11):1583 在正方编织系和六角编织系对应的平面群中， [14]Bystrom J.Jekabsons N,Varna J.An evaluation of different 对应p4mm、p4gm和p3m1、p31m、p6mm五种 models for prediction of elastic properties of woven composites. 空间群，由于存在3、4、6次轴和镜面对称的组合对 Compos Part B.2000.31:7 称操作，无法构成可行的平面编织方式，笔者倾向 [15]Ivanov I.Tabiei A.Three-dimensional computational micro me- 于其无对应平面编织结构，这种结论还有待于进一 chanical model for woven fabric composites.Compos Struct. 2001,54(4):489 步加以证实 [16]Lee S K,Byun J H.Hong S H.Effect of fiber geometry on the 参考文献 elastic constants of the plain woven fabric reinforced aluminum matrix composites.Mater Sci Eng.2003.A347(1/2):346 [1]Mouritz A P,Bannister M K.Fakzon P J.et al.Review of appli- [17]kabsons NJ.Bystrom J.On the effect of stacked fabric layers on cations for advanced three-dimensional fibre textile composites. the stiffness of a woven composite.Compos Part B.2002.33 Comp0 s Part A.1999,30(12):1445 (8):619 [2]Wang Y Q,Wang A S D.Spatial distribution of yarns and me- [18]Alif N,Carlsson L A.Boogh L.The effect of weave pattern and chanical properties in 3D braided tubular composites.Appl Com- crack propagation direction on mode I delamination resistance of pos Mater,1997,4(2):121 woven glass and carbon composites.Compos Part B.1998,29 [3]JanZ,Michal S.Homogenization of balanced plain weave com- (5):603 posites with imperfect microstructure:PartTheoretical formu [19]Ning Q G.Chou T W.A general analytical model for predicting lation.Int J Solids Struct.2004.41(22/23):6549 the transverse effective thermal conductivities of woven fabric [4]Chiu C H.Tsai K H.Huang W J.Crush-failure modes of 2D tri- composites.Compos Part A.1998,29(33):315 axially braided hybrid composite tubes.Compos Sci Technol. [20]刘木兰，冯克勤.群论.北京：国防工业出版社，1992：145 1999,59,1713 [21]王仁卉，郭可信。晶体学中的对称群，北京：科学出版社， 1990:1 (下转第246页)图7 由平面群 p3m1、p31m 和 p6mm 推导平面编织几何结构 时出现的不合理图案 Fig．7 Yarn segments combination patterns with the symmetrical characteristic of plane groups p3m1p31m and p6mm 6 结论 由平面点群到平面群可以推导出4个编织系 对应的17个平面群不仅可以充分概括二维平面编 织的现有几何结构而且可以以点符号的组合、交叉 原则通过建立无对称单元、基本对称单元遵从不 同二维空间点阵描述的对称性最终推导和预测不 同几何结构的编织方法．在编织方案的确定优化 材料性能方面无疑是有益的． 在正方编织系和六角编织系对应的平面群中 对应 p4mm、p4gm 和 p3m1、p31m、p6mm 五种 空间群由于存在3、4、6次轴和镜面对称的组合对 称操作无法构成可行的平面编织方式．笔者倾向 于其无对应平面编织结构这种结论还有待于进一 步加以证实． 参 考 文 献 ［1］ Mouritz A PBannister M KFalzon P Jet al．Review of applications for advanced three-dimensional fibre textile composites． Compos Part A199930（12）：1445 ［2］ Wang Y QWang A S D．Spatial distribution of yarns and mechanical properties in3D braided tubular composites．Appl Compos Mater19974（2）：121 ［3］ Jan ZMichal S．Homogenization of balanced plain weave composites with imperfect microstructure：PartⅠ—Theoretical formulation．Int J Solids Struct200441（22／23）：6549 ［4］ Chiu C HTsai K HHuang W J．Crush-failure modes of2D triaxially braided hybrid composite tubes．Compos Sci Technol 199959：1713 ［5］ Quek S CWaas AShahwan K Wet al．Compressive response and failure of braided textile composites：Part 2—Computations． Int J Non Linear Mech200439（4）：649 ［6］ 庞宝君杜善义韩杰才．三维四向编织复合材料细观组织及 分析模型．复合材料学报199916（3）：135 ［7］ Sun WLin FHu X．Computer-aided design and modeling of composite unit cells．Compos Sci Technol200161（2）：289 ［8］ Adanur SLiao T．3D modeling of textile composite performs． Compos Part B199829（6）：787 ［9］ Brown DMorgan MMcIlhagger R．A system for the automatic generation of solid models of woven structures．Compos Part A 200334（6）：511 ［10］ Lomov S V．Textile geometry preprocessor for meso-mechanical models of woven composites．Compos Sci Technol200060 （11）：2083 ［11］ Naik N KKuchibhotla R．Analytical study of strength and failure behaviour of plain weave fabric composites made of twisted yarns．Compos Part A200233（5）：697 ［12］ Huang Z M．The mechanical properties of composites reinforced with woven and braided fabrics．Compos Sci Technol200060 （4）：479 ［13］ Subhash G．Influence of strain-rate on the uniaxial compressive behavior of 2—D braided textile composites．Compos Part A 200132（11）：1583 ［14］ Bystrom JJekabsons NVarna J．An evaluation of different models for prediction of elastic properties of woven composites． Compos Part B200031：7 ［15］ Ivanov ITabiei A．Three-dimensional computational micro-mechanical model for woven fabric composites．Compos Struct 200154（4）：489 ［16］ Lee S KByun J HHong S H．Effect of fiber geometry on the elastic constants of the plain woven fabric reinforced aluminum matrix composites．Mater Sci Eng2003A347（1／2）：346 ［17］ kabsons N JBystrom J．On the effect of stacked fabric layers on the stiffness of a woven composite．Compos Part B200233 （8）：619 ［18］ Alif NCarlsson L ABoogh L．The effect of weave pattern and crack propagation direction on mode I delamination resistance of woven glass and carbon composites．Compos Part B199829 （5）：603 ［19］ Ning Q GChou T W．A general analytical model for predicting the transverse effective thermal conductivities of woven fabric composites．Compos Part A199829（33）：315 ［20］ 刘木兰冯克勤．群论．北京：国防工业出版社1992：145 ［21］ 王仁卉郭可信．晶体学中的对称群北京：科学出版社 1990：1 （下转第246页） 第2期 马文锁等： 二维编织复合材料几何结构的平面群分析 ·231·