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 材料工程  2017, Vol. 45 Issue (4): 102-107 PDF
http://dx.doi.org/10.11868/j.issn.1001-4381.2015.000479
0

#### 文章信息

ZHOU Yu-bo, LI Yan-xia, LI Min, GU Yi-zhuo, ZHANG Zuo-guang, SONG Yong-zhong, YU Li-qiong, CHENG Jia

Theory and Examples of Mathematical Modeling for Fine Weave Pierced Fabric

Journal of Materials Engineering, 2017, 45(4): 102-107.
http://dx.doi.org/10.11868/j.issn.1001-4381.2015.000479

### 文章历史

1. 北京航空航天大学 材料科学与工程学院 空天材料与服役教育部重点实验室, 北京 100191;
2. 航天材料及工艺研究所, 北京 100076

Theory and Examples of Mathematical Modeling for Fine Weave Pierced Fabric
ZHOU Yu-bo1, LI Yan-xia1 , LI Min1, GU Yi-zhuo1, ZHANG Zuo-guang1, SONG Yong-zhong2, YU Li-qiong2, CHENG Jia2
1. Key Laboratory of Aerospace Materials and Performance (Ministry of Education), School of Materials Science and Engineering, Beihang University, Beijing 100191, China;
2. Aerospace Research Institute of Materials & Processing Technology, Beijing 100076, China
Abstract: A mathematical abstraction and three-dimensional modeling method of three-dimensional woven fabric structure was developed for the fine weave pierced fabric, taking parametric continuity splines as the track function of tow. Based on the significant parameters of fine weave pierced fabric measured by MicroCT, eight kinds of the three-dimensional digital models of the fabric structure were established with two kinds of tow sections and four kinds of tow trajectory characteristic functions. There is a good agreement between the three-dimensional digital models and real fabric by comparing their structures and porosities. This mathematical abstraction and three-dimensional modeling method can be applied in micro models for sub unit cell and macro models for macroscopic scale fabrics, with high adaptability.
Key words: fine weave pierced fabric    preform    mathematical abstraction    computer modeling

1 数学建模用织物样例

 图 1 细编穿刺织物的MicroCT照片 Fig. 1 MicroCT photo of fine weave pierced fabric
2 细编穿刺织物的数学抽象

2.1 丝束轨迹特性

 (1)

 (2)
2.1.1 贝塞尔样条线的参数定义

 (3)

 (4)

(1) 若丝束不具有周期性，有：

 (5)

(2) 若丝束是周期性的，满足：

 (6)

2.1.2 自然立方样条线的参数定义

 (7)

 (8)

(1) 若丝束不具有周期性，则在样条线端头处 (t=at=b)，满足：

 (9)

(2) 若针对织物的单胞建模，丝束具有周期性，则在样条线端头处 (t=at=b)，满足：

 (10)

 图 2 四种样条线的对比 Fig. 2 Comparison of four kinds of spline
2.2 丝束截面特性

2.2.1 幂次椭圆截面

 (11)
 图 3 幂次椭圆 (a) 与透镜形 (b) 示意图 Fig. 3 Power ellipse (a) and lenticular (b)
2.2.2 透镜形截面

 (12)

 (13)

2.3 丝束实体与表面特性

 (14)

 (15)

 图 4 建模坐标系 Fig. 4 Coordinate system of modeling

 (16)

 图 5 截面插值示意图 (a) 线性插值；(b) 平滑修正后的插值 Fig. 5 Sketch of section interpolation (a) linear interpolation; (b) interpolation after smooth

 (17)

 (18)

3 细编穿刺织物的三维仿真模型建立

 图 6 采用椭圆形截面、周期性贝塞尔样条线的织物模型的透视图 Fig. 6 Scenograph of models with ellipse section and periodic Bezier spline

 图 7 周期性贝塞尔样条线 (a) 与自然立方样条线 (b) 的区别 Fig. 7 Difference of periodic Bezier spline (a) with periodic natural cubic spline (b)

 图 8 采用椭圆形截面、周期性贝塞尔样条线的织物模型的断面图 Fig. 8 Sectional view of models with ellipse section and periodic Bezier spline

 图 9 不同模型的孔隙率及其与实际值的比较 Fig. 9 Comparison of porosities from different models and with reality

4 结论

(1) 与传统的跑道型轨迹模型相比，本工作中采用的贝塞尔样条线与自然立方样条线轨迹模型的出发点是丝束的参数连续性。相对于只具有零阶参数连续性的跑道型轨迹模型，贝塞尔样条线与自然立方样条线轨迹模型分别具有一阶和二阶参数连续性，因而其更接近于自然状态下的织物，建立的模型更符合真实的丝束状态。

(2) 以细编穿刺三维立体织物为样例，本工作分别建立了基于两种典型截面、四种轨迹样条线的八种模型，并将它们的孔隙率与真实织物的实测孔隙率进行比较，发现模型与真实织物的孔隙率数据吻合较好，说明所建立的数学模型抽象与建模方法能够较好地描述织物的真实状况，为穿刺织物渗透率及力学性能等参数预测提供了重要基础。

(3) 该织物微观结构建模方法也可用于其他类型的2D，2.5D及3D织物等预成型体的抽象建模，并可用于预成型体的渗透率、导电性、导热性等物理性能以及力学性能的预测与分析，可以提高针对预成型体相关性能研究的效率，降低研究的成本。

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