Page 149 - Fiber Fracture

134 P.K. Gupta

bond failure and a perfect crystalline lattice vary by as much as 100% (France et al.,
1985). Several models relate the theoretical strength to a product of Young’s modulus
E and surface tension y. However, it is not clear whether the intrinsic strength is
simply related to E. This is because E is determined by the harmonic part of the pair
interaction. The strain to break a bond, on the other hand, is determined by the inflection
point (Le., the anharmonic part) of the interaction. Similarly, the surface tension depends
on the depth of the potential well but contains little information about the inflection
point. Therefore, a product of E and y will not have any information about the inflection
point of the interaction potential and therefore may not be directly related to the intrinsic
strength.
Molecular dynamics (MD) simulations have shed some light on the atomistics of the
fracture behavior of silica glass. For example, Soules (1985) has shown that the random
structure weakens silica glass, relative to the cristobalite crystalline structure, by about a
factor of 3. Simmons (Simmons et al., 1991; Simmons, 1998) has shown that structural
relaxation and surface reconstruction play important roles at the crack tip. However,
MD simulations do not provide an accurate estimate of intrinsic strength largely because
of uncertainty in the anharmonic part of the interaction potentials. It is clear that
accurate calculations of theoretical strength will require (1) a detailed knowledge of the
long-range behavior of the interatomic interaction potentials (or the nonlinear aspects
of the interatomic forces); (2) a knowledge of the covalent bonds which are necessarily
three- or multi-body interactions; (3) a knowledge of the intermediate range structure
of the glass network; (4) a knowledge of the effect of topological disorder on the stress
concentration; and (5) use of finite temperature to allow for structural changes.

Diameter and Length Dependence of Inert, Extrinsic Strength, & (d, L)

Because the probability of finding a flaw of a given severity increases with increase
in the volume or surface of a fiber sample, the average strength tends to decrease with
increase in length or diameter. Using the weakest link model, the cumulative probability
of failure of a fiber can be shown to be (Hunt and McCartney, 1979):
P(So, Ld) = 1 - exp [-(L/LR)(d/dR>k(SO/~R,)m] (6)
where k = 1 for surface flaws and k = 2 for volume (or bulk) flaws, and LR and dR are
reference fiber length and fiber diameter, respectively.
From Eq. 6, it can be shown that the average strength < SO > decreases with increase
in L or in d according to the following equation:

It can also be shown that, according to Eq. 6, the coefficient of variation in strength is
independent of the fiber length or fiber diameter.

Fatigue Strength, S(et,X, T)

Because of fatigue, strength measured under non-inert conditions increases with
increase in strain rate, st. Strength measured at a constant (but moderate) strain rate at