textile physics

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Woven Fabric GeometryWoven Fabric Geometry
Dr Jimmy LamDr Jimmy Lam
Institute of Textiles & ClothingInstitute of Textiles & Clothing

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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

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Introduction
The objectives of fabric geometry (math models forThe objectives of fabric geometry (math models for
fabric) is to:fabric) is to:
1.1. Prediction of the maximum sett (density) ofPrediction of the maximum sett (density) of
fabric and fabric dimensions;fabric and fabric dimensions;
2.2. Find out relationship between geometricalFind out relationship between geometrical
parameters (picks and ends);parameters (picks and ends);
3.3. Prediction of mechanical properties by combiningPrediction of mechanical properties by combining
fabric and yarn properties;fabric and yarn properties;
4.4. Understanding fabric performance (handle andUnderstanding fabric performance (handle and
surface effect).surface effect).

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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.

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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

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Mathematical Notation for
each model

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Pierce’s Model
(Classical Model)

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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.

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

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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.

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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.

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Pierce’s Elliptic Model

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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.

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

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Kemp Model
Results

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

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Hearle’s Model
Energy approach for Hearle’s model

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Hearle’s Model
Results

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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.

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

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