Page 274 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C005.fm Page 254 Monday, May 23, 2005 5:47 PM

254 Fracture Mechanics: Fundamentals and Applications

Equation (5.24) implies that the arrest toughness is single valued; a microcrack always prop-
agates above K , but always arrests at or below K . Experimental data, however, indicate that arrest
o
o
can occur over a range of K values. The data in Figure 5.27 exhibit a sigmoidal shape, while the
truncated Weibull is nearly linear near the threshold.
A computer simulation of cleavage propagation in a polycrystalline material [42, 43] resulted
in a prediction of P as a function of the applied K; these results fit an offset power law expression
pr
(Equation (5.25)). The absolute values obtained from the simulation are questionable, but the
predicted trend is reasonable. Inserting Equation (5.25) into Equation (A5.13) gives

 K K 3 
F   − ∫ K o α ( K =− exp − K 1 o ) β 4 Θ 4 K dK   (A5.15)

The integral in Equation (A5.15) has a closed-form solution, but it is rather lengthy. The above
distribution exhibits a sigmoidal shape, much like the experimental data in Figure 5.27. Unfortu-
nately, it is very difficult to fit experimental data to Equation (A5.22). Note that there are four
fitting parameters in this distribution: α, β, K , and Θ . Even with fewer unknown parameters, the
K
o
form of Equation (A5.15) is not conducive to curve-fitting because it cannot be linearized.
Equation (A5.15) can be approximated with a conventional three-parameter Weibull distribution
with the slope fixed at 4 (Equation (5.26)). The latter expression also gives a reasonably good fit
of experimental data (Figure 5.27). The three-parameter Weibull distribution is sufficiently flexible
to model a wide range of behavior. The advantage of Equation (5.26) is that there are only two
parameters to fit (the Weibull-shape parameter is fixed at 4.0) and it can be linearized. Wallin [46]
has shown that Equation (5.26) is rigorously correct if P is given by Equation (5.32).
pr

REFERENCES

1. Knott, J.F., ‘‘Micromechanisms of Fracture and the Fracture Toughness of Engineering Alloys.’’
Fracture 1977, Proceedings of the F ourth International Conference on Fracture (ICF4), Waterloo,
Canada, Vol. 1, 1977, pp. 61–91.
2. Knott, J.F., ‘‘Effects of Microstructure and Stress-State on Ductile Fracture in Metallic Alloys.” In:
K. Salama, et al. Advances in Fracture Research, Proceedings of the Seventh International Conference
on Fracture (ICF7). Pergamon Press, Oxford, 1989, pp. 125–138.
3. Wilsforf, H.G.F., ‘‘The Ductile Fracture of Metals: A Microstructural Viewpoint.” Materials Science
and Engineering, Vol. 59, 1983, pp. 1–19.
4. Garrison, W.M., Jr. and Moody, N.R., ‘‘Ductile Fracture.” Journal of the Physics and Chemistry of
Solids, Vol. 48, 1987, pp. 1035–1074.
5. Knott, J.F., ‘‘Micromechanisms of Fibrous Crack Extension in Engineering Alloys.” Metal Science,
Vol. 14, 1980, pp. 327–336.
6. Argon, A.S., Im, J., and Safoglu, R., ‘‘Cavity Formation from Inclusions in Ductile Fracture.”
Metallurgical Transactions, Vol. 6A, 1975, pp. 825–837.
7. Beremin, F.M., ‘‘Cavity Formation from Inclusions in Ductile Fracture of A 508 Steel.” Metallurgical
Transactions, Vol. 12A, 1981, pp. 723–731.
8. Brown, L.M. and Stobbs, W.M., ‘‘The Work-Hardening of Copper-Silica vs. Equilibrium Plastic
Relaxation by Secondary Dislocations.” Philosophical Magazine, 1976, Vol. 34, pp. 351–372.
9. Goods, S.H. and Brown, L.M., ‘‘The Nucleation of Cavities by Plastic Deformation.” Acta Metallur-
gica, Vol. 27, 1979, pp. 1–15.
10. Van Stone, R.H., Cox, T.B., Low, J.R., Jr., and Psioda, P.A., ‘‘Microstructural Aspects of Fracture by
Dimpled Rupture.” International Metallurgical Reviews, Vol. 30, 1985, pp. 157–179.
11. Thomason, P.F., Ductile Fracture of Metals. Pergamon Press, Oxford, 1990.
12. Rice, J.R. and Tracey, D.M., ‘‘On the Ductile Enlargement of Voids in Triaxial Stress Fields.” Journal
of the Mechanics and Physics of Solids, Vol. 17, 1969, pp. 201–217.