1 Woven Fabric Geometry Dr Jimmy Lam Institute of Textiles & Clothing

2 Learning Objectives 1.Introduction 2.Fabric Geometry Models 1.Pierce’s Model 2.Modified Pierce’s Model 3.Kemp’s racetrack Model 4.Hearle’s Lenticular Model 3.Mathematical descriptions of each model 4.Limitations on fabric geometry

3 Introduction The objectives of fabric geometry (math models for fabric) is to: 1.Prediction of the maximum sett (density) of fabric and fabric dimensions; 2.Find out relationship between geometrical parameters (picks and ends); 3.Prediction of mechanical properties by combining fabric and yarn properties; 4.Understanding fabric performance (handle and surface effect).

4 Geometry Theories Approach 1.In conventional approaches, the general character of fabrics was idealized into simple geometrical forms (circle, ellipse, rectangle) 2.They treated the micro-mechanics of fabrics on the basis of the unit-cell approach, ie fabrics are considered as a repeating network of identical unit cells in the form of crimp weaves and constant yarn cross-section in the woven structure. 3.By combining this kind of geometry with or without physical parameters (material), mathematical deductions could be obtained.

5 Four Fabric Models (geometry models) By using circle, ellipse, rack-track approaches, four fabric geometrical models are formed 1.Pierce model 2.Modified model (ellipse) 3.Kemp’s race track model (rectangle & circle) 4.Hearle’s lenticular model

6 Mathematical Notation for each model

7 Pierce’s Model (Classical Model)

8 Pierce’s Model (1) In this model, a two-dimensional unit cell of fabric was built by superimposing linear and circular yarn segments to produce the desired shaped. The yarns were assumed to be circular in cross- section and highly incompressible, but perfectly flexible so that each set of yarns had a uniform curvature imposed by the circular cross-sectional shape of interlacing yarns. Geometrical parameters such as thread spacing (p), weave crimp, weave angle and fabric thickness (h) can be found.

9 Pierce’s Model (2) Results Pick spacing (p1) and end spacing (p2), warp thickness (h1), weft thickness (h2) can be found from this model

10 Pierce’s Model Limitations This model is convenient for calculation and is valid for open structure (loose density) However, the assumptions of circular cross- section, uniform structure along the longitudinal direction, perfect flexibility and incompressibility are all unrealistic.

11 Pierce’s Elliptic Model In more tightly woven fabrics, however, the inter-thread pressures setup during weaving cause considerable thread flattening normal to the plane of cloth. Pierce recongized this and proposed an elliptic section theory as shown in Fig 3.2 Because such model would be too complex and laborious in operation, he adopted an approximate treatment, which involved merely replacing the circular thread diameter in his circular-thread geometry with minor diameter as shown in Fig 3.2 This modified model is good for reasonable open fabric but cannot be applied for very closed jammed fabric.

12 Pierce’s Elliptic Model

13 Kemp Model (Race-track section) To overcome the jammed structure, Kemp proposed a racetrack section to modified cross-section shape. The model consisted of a rectangle enclosed by two semi-circular ends and had the advantage that it allowed the relatively simple relations of circular-thread geometry, already worked out by Pierce, to be applied to a flatted threads.

14 Kemp Model (Race-track section) A rectangle and semi-circular cross section of Kemp Model

15 Kemp Model Results

16 Hearle’s Model Using energy method for calculations in fabric mechanics, a lenticular geometry was proposed by Hearle as shown in Fig 3.5

17 Hearle’s Model Energy approach for Hearle’s model

18 Hearle’s Model Results

19 Limitations Fabric Geometry Models 1.Firstly, fabrics are complicated materials that do not conform even approximately to any of the ideal features suggested by these four fabric models. 2.Secondly, the measurement of geometrical parameters is not easy in practice. 3.Thirdly, the relationship between fabric mechanic (tensile, elongation, bending) to fabric geometry is not fully explored.

20 Conclusion What is fabric geometry? Why are objectives to study fabric geometry? Suggest Pierce’s model and its limitations

21 References Structure and mechanics of woven fabrics by Jinlan HU Chapter 3 Structural properties of fabric pp61-89