Page 147 - Fiber Fracture

132 P.K. Gupta

S. When measured strength values fall on a straight line (with slope m), the data imply
a (unimodal) Weibull distribution of strengths (Epstein, 1948; Freudenthal, 1968; Hunt
and McCartney, 1979; Katz, 1998):
P(S) = 1 - exp [-(s/s#] (1)
Here SR is a scaling parameter which is related to the average strength, (S), as follows:

(S) = SRr(l+ l/m) (2)
where r(x) is the Gamma function of x. The coefficient of variation, COV, of strength
is related to the Weibull modulus ‘m’ according to the following relation:

cov = {[r(i +2/m)/r2(1 + 1/m)1- I I”* % 1.28/m (3)
According to Eq. 3, the higher the Weibull modulus the lower is the value of COV.
For example, a 3% COV corresponds to an m of about 40 and a 12% COV corresponds
to an m of about 10.
When the measured strengths do not fall on a straight line in a Weibull plot, one
can fit the data to a combination of straight line segments. In this case, the Weibull
distribution is referred to as bimodal (if two lines are sufficient to describe the data) or
multi-modal (if more than two lines are needed).

Intrinsic Strength, S*

When there are no flaws present, the measured strengths are called intrinsic. The
strength of pristine fibers therefore provides the intrinsic strength of a glass composition.
The intrinsic strength is denoted by S*. Intrinsic strengths are measured when the
following three conditions are satisfied:
(1) Measured strength is constant with respect to the fiber diameter and length.
(2) COV (strength) % 2 COV (diameter). Diameter variations are always present
in fibers. The magnitude of these variations depends on the method of making fibers
(Kurkjian and Paek, 1983). However, the primary source of diameter uncertainty in
a strength measurement lies in the fact that a high-strength glass fiber upon fracture
disintegrates into a large number of small pieces and the fracture surfaces are not
available. Therefore, the diameter cannot be measured at the point of fracture and is
typically measured at some distance away from the point of fracture. A second source of
uncertainty, especially in the case of thin fibers, lies in the measurement precision when
using optical microscopy. For example, a 0.1 pm measurement uncertainty in a 10 p,m
diameter fiber gives rise to a 2% uncertainty in the fiber strength.
(3) Measured strengths are amongst the highest ones measured (typically >E/20, E
being Young’s modulus).
Because the probability of the presence of an extrinsic flaw increases with increase in
volume or in total surface of test samples, intrinsic strength measurements require fiber
samples of as small a diameter and as small a length as possible. The two-point bend
technique (Matthewson et al., 1986) currently provides the simplest way of carrying
out such experiments provided the fiber is not too thin (diameter > 50 pm). This
technique is routinely used for testing silica fibers (typical diameter of 125 pm). For