What is Pierce Geometrical Model?

The properties of the fabric depend on the fabric structure. The formal structure of a woven fabric is defined by weave, thread density, crimp and yarn count. The interrelation between fabric parameters can be obtained by considering a geometrical model of the fabric. The geometrical model is mainly concerned with the shape taken up by the yarn in the warp or weft cross-section of the fabric.

The basic relationship between geometrical parameters:
Pierce Geometrical Model

Figure represents a unit cell interlacement in which the yarns are considered inextensible and flexible. The yarns have circular cross-section and consist of straight and curved segments. The main advantages in considering this simple geometry are: 
1. Guide to the maximum density of yarn packing possible in the cloth. 
2. Helps to establish relationship between various geometrical parameters. 
3. Provide information on the relative resistance of the cloth to the passage of air, water or light. 
4. Able to calculate the resistance of the cloth to mechanical deformation such as initial extension, bending and shear in terms of the resistance to deformation of individual fibres. 

From the two-dimensional unit cell of a plain woven fabric, geometrical parameters such as thread-spacing, weave angle, crimp and fabric thickness are related by deriving a set of equations. The symbols used to denote these parameters are listed below: 
d - Diameter of thread, 
p - Thread spacing, 
h - Maximum displacement of thread axis normal to the plane of cloth (crimp height), 
θ - Angle of thread axis to the plane of cloth (weave angle in radians) 
l - Length of thread axis between the planes through the axes of consecutive cross- threads (modular length) 
c - Crimp (fractional) 
D = d1 + d2 [Suffix 1 and 2 to the above parameters represent warp and weft threads respectively.] 

In the above figure projection of yarn axis parallel and normal to the cloth plane gives the following equations:
Pierce Geometrical Model
Three similar equations are obtained for the weft direction by interchanging suffix from 1 to 2 or vice-versa as below:
Pierce Geometrical Model
Also,
d1+d2=h1+h2=D

In all, there are seven equations connecting eleven variables. If any four variables are known then the equations can be solved and the remaining variables can be determined. 

The model is not merely an exercise in mathematics. It is not only useful in determining the entire structure of a fabric from a few values given in technological terms but it also establishes a base for calculating various changes in fabric geometry when the fabric is subjected to known extensions in a given direction or known compressions or complete swelling in aqueous medium.

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