Theory of continuous filament yarns extension

Yarns are extended or elongated under loading. Stress analysis based on assumptions of small deformations and fixed thread diameter or theory of continuous filament yarns extension have been explained as below.

h: length of one twist yarn. 

δh: the amount of extension. 

r: radius of the cylinder. 

R: radius of the yarn package. 

θ: twist angle at the helical cylinder. 

l: length of fibre in one turn of twist on the helical cylinder. 

L: length of fibre in one turn of twist at the surface.

If the elongation rate of the yarn is indicated by ε_y then, 

ε_y = δh/h ……………………………….(1)

When the length of the twisting fibre in the yarn length, h is indicated by l then, 

l² = h² + 4π²r² ………………………….(2)

If the differential of equation (2) is taken,

2ldl = 2hdh

or, dl/dh = h/l …………………………………….(3) 

If the elongation rate of the fibre is indicated by ε_f then, 

ε_f =  δl/l = 1/l (dl/dh)  δh ………………………………(4)

Now from equation (3) and (4), 

ε_f = h²/l² (δh/h) ……………………………………………………(5)

If youngs modulus of fiber is E_f, then according to Hooke’s law: 

= ε_y cos²𝜃 …………………………………………………………………..(6) [since h/l = cos𝜃 and from equation (1)]

Fibre Elongation, 

= E_f ε_f …………………………………………………………………………………(7)

= E_f ε_y cos²𝜃 ……………………………………………………………………………….(8)