Fiber bending | Fiber flexural rigidity properties

BENDING OF FIBERS:

The bending of fibers influences the drape and handle of fabrics, and recovery from bending is a factor in creasing. Twisting and bending both plays a part in the arrangement of fibers in a yarn and transverse compressive forces are involved when tension is applied to twisted yarn. Bending strength and shear strength is also important in wear.

FLEXURAL RIGIDITY:

The flexural rigidity (or stiffness) of a fiber is defined as the couple required to bend the fiber to un radius of curvature. By this definition, the direct effect of the length has not been considered.

The flexural rigidity may be calculated in terms of other fiber properties: the problem is similar to that of the bending of beams.Suppose we have a specimen of length l, bent through an angle Ө to a radius of curvature r, . Its outer layers will be extended and its inner layers compressed, but a plane in the centre, known as the neutral plane, will be unchanged in length. As a result of the extension and compression, stresses will be set up

giving an internal couple to balance the applied couple.

Consider an element of area of cross-section    δA, at a distance x from the neutral plane:

                          Elongation of element = xӨ  = xl/r……………………….(1)

and, therefore,

                          Tension element = [(xl/r)/l]  YδA  = (x/r) YδA…………..(2)

 where Y = Young’s modulus.

Its moment about an axis in the neutral plane = (x/r) YδA . x

                                                                                       2

                                                                         = (Y/r) x   δA………………(3)

Therefore                                              

Total internal couple = Y/r ∑         

                                 = YAk²/r ………………………………………………… (4)

where A = area of cross-section, and                     

         Ak² =  S  ……………………………………………………………(5)

k² is thus analogous to a radius of gyration, taken about the neutral plane. It may be related to a shape factor  η , which is 1 for a circular fiber, by the expression:                                     

k²=(1/4π)ηA……………………………………..(6)                                                                                                               

Since  A = (T/ r) x 10^-5……….………………………………………...(7)                                            

where   r= density in gm/cm³ , and

            T= count of filament in tex,

and,

                    Y  =  rE  x  10-⁵.……………………………………………………(8)

where E = modulus in g-wt/tex,

We get:                                                                                                     

             total couple = [ (1/4 π ) (  η ET² / rr )  x] 10^-5    g-wt cm……………….(9)

Therefore:      Flexural Rigidity = [ (1/4 π ) (η ET² / r)  x  ] 10^-5  g-wt cm² ……(10)

It follows from this relation that the flexibility of a fiber depends on its shape, its tensile modulus, its density, and, most of all, its thickness.

The density of the ordinary textile fiber varies between 1.1 to 1.6, so  here it can be neglected.