Page 229 - Handbook of Properties of Textile and Technical Fibres

204 Handbook of Properties of Textile and Technical Fibres

By an approximation method (Abramowitz and Stegun, 1964), Eqns. (6.17)e(6.19)
can be reduced to forms suitable for evaluation of the relaxation spectrum S(s) from
stress relaxation and creep data for the time interval (s 1 , s 2 )

s 2
1 Cg C ln
s 1
GðtÞ¼ þ oðCÞ (6.21)
s 2
1 þ C ln
s 1

t t
1 C E 1 E 2
C þ s 2 C þ s 1
JðtÞ¼ þ oðCÞ (6.22)
C þ s 2
1 C ln
C þ s 1
where g is Euler’s constant (g ¼ 0.5772) and o(C) is a small residue much smaller than
C. The residue varies slowly with time. Neglecting the residue o(C) we have the
following relationships:

s 2
1 Cg C ln
s 1
GðtÞ¼ (6.23)
s 2
1 þ C ln
s 1
t t

1 C E 1 E 2
C þ s 2 C þ s 1
JðtÞ¼ (6.24)
C þ s 2
1 Cln
C þ s 1
dGðtÞ C
¼ (6.25)
dlnðtÞ 1 C[

dJðtÞ C
¼ (6.26)
dlnðtÞ C þ s 2
1 Cln
C þ s 1
With stress relaxation and/or creep data, we can calculate C, s 1 , and s 2 in different
ways.
From experimental data expressed in the form of a reduced relaxation function
defined by Eq. (6.1), we obtain the slope S R of the middle portion of the relaxation
function plotted against the logarithm of time, a typical point [G(t 0 ), t 0 ] in the middle
portion of the reduced relaxation function, and the value G(N), when relaxation