Technical Paper by J.H. Greenwood - IGS - International ...

GREENWOOD D Designing to Residual Strength Instead of Stress-Rupture
where: a = crack length; A = a constant; n = index; and, K = stress concentration factor
which is equal to σYa ½ for simple geometries, where σ = applied stress and Y =geometrical
factor (Evans and Wiederhorn 1974). For any value of crack length, a, thereisa
corresponding residual strength σ t with an initial value of σ 0 , corresponding to a critical
value of K=K IC . By expressing σ t as a function of time and integrating, it can be shown
that σ t is given by the following expression:
σ n−2
t
= σ n−2
0
− σ n t B
where B =2/((n - 2) AY 2 K n−2
IC
), omitting any effects of crack growth during tensile testing.
The stress-rupture curve is given by setting σ = σ t which requires solving the following
equation:
(2)
σ n−2
t
= σ n−2
0
− σ n t
t B
Since n is usually large and σ 0 > σ t , Equation 3 can be approximated using the following
expression:
σ n−2
0
= σ n t
t B
which plots as a double logarithmic stress-rupture diagram, and has a gradient of -(1/n).
As mentioned above, for polyester the stress-rupture gradient is of the order of -1/40
and thus n is equal to 40. Thus, if the values σ 0 =1,B =10 - 6 and n = 40 are substituted
into Equation 2, then for an applied load of σ = 0.6 (i.e. 60% of tensile strength), the
time to failure is 748 hours. After 720 hours, the residual strength is 91.7%, after 730
hours it is 90.6%, and after 740 hours it is 88.7%, before reducing to 60% at failure.
This example illustrates how a stress-rupture curve with a low gradient ( -1/n), as observed
in geotextiles, may be associated with a high residual strength over most of the
design life. This index implies that the majority of crack growth takes place just before
failure, and thus the residual strength remains close to the original tensile strength. A
complete model would have to incorporate the processes of primary and secondary
creep, with decreasing and constant strain rates, respectively, as well as the chemical
aspects of the fracture of polymer chains (Horrocks 1996).
(3)
(4)
5 REVISED DESIGN PROCEDURE
5.1 Residual Strength Due to Sustained Load Alone
It is proposed that instead of using the stress-rupture curve directly, the design
strength could be based on the residual strength curve, T R , corresponding to the anticipated
design strength, T D .AvalueofT D must be found such that at the end of the design
life, a factor of safety, f m , will remain between the design strength and the residual or
available strength, T RD , such that T D =T RD /f m (Figure 5). In the absence of an analytical
6 GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NO. 1