User login

Navigation

You are here

One, no one, and one hundred thousand crack propagation laws: A generalized Barenblatt and Botvina dimensional analysis approach

Mike Ciavarella's picture

One,
no one, and one hundred thousand crack propagation laws: A generalized
Barenblatt and Botvina dimensional analysis approach to fatigue crack
growth

Journal of the Mechanics and Physics of Solids, Volume 56, Issue 12, December 2008, Pages 3416-3432
Michele Ciavarella, Marco Paggi, Alberto Carpinteri

 

Abstract | Figures/TablesFigures/Tables | ReferencesReferences

Abstract

Barenblatt
and Botvina with elegant dimensional analysis arguments have elucidated
that Paris’ power-law is a weak form of scaling, so that the Paris’
parameters C and m
should not be taken as material constants. On the contrary, they are
expected to depend on all the dimensionless parameters of the problem,
and are really “constants” only within some specific ranges of all
these. In the present paper, the dimensional analysis approach by
Barenblatt and Botvina is generalized to explore the functional
dependencies of m and C
on more dimensionless parameters than the original Barenblatt and
Botvina, and experimental results are interpreted for a wider range of
materials including both metals and concrete. In particular, we find
that the size-scale dependencies of m and C and the resulting correlation between C and m
are quite different for metals and for quasi-brittle materials, as it
is already suggested from the fact the fatigue crack propagation
processes lead to m=2–5 in metals and m=10–50
in quasi-brittle materials. Therefore, according to the concepts of
complete and incomplete self-similarities, the experimentally observed
breakdowns of the classical Paris’ law are discussed and interpreted
within a unified theoretical framework. Finally, we show that most
attempts to address the deviations from the Paris’ law or the empirical
correlations between the constants can be explained with this approach.
We also suggest that “incomplete similarity” corresponds to the
difficulties encountered so far by the “damage tolerant” approach
which, after nearly 50 years since the

introduction of Paris’ law, is still not a reliable calculation of damage, as Paris himself admits in a recent review.

Article Outline

1. Introduction
2. BB's approach
3. BB's generalized
4. Analysis of the functional dependencies of the Paris’ law parameters
5. Correlations between the Paris’ law parameters
6. Other complete and incomplete similarity laws
6.1. Representation based on Young's modulus
6.2. Representation based on the stress ratio or on the maximum stress-intensity factor
7. Discussion and conclusions
Acknowledgements
References

Figures

View image in article

Fig. 1. Π1-dependence of the Paris’ law parameter m. (a) A typical da/dN curve for steel (R=0, View the MathML source, View the MathML source). (b) Effective Paris’ slope m vs. Π1=ΔK/KIC computed from (a).

View image in article

Fig. 2. Dependence of the crack growth rate on View the MathML source.

View image in article

Fig. 3. Z-dependence of the Paris’ law parameter m.
(a) Aluminium alloys ([Yarema and Ostash, 1975] and [Ostash et al.,
1977]). (b) 4340 steel (Heiser and Mortimer, 1972). (c) ASTM steels
(Clark and Wessel, 1970). (d) Low carbon steel ([Ritchie and Knott,
1974] and [Ritchie et al., 1975]). (e) High strength concrete (data
from Bažant and Shell, 1993 reinterpreted by Spagnoli, 2005). (f)
Normal strength concrete (data from Bažant and Xu, 1991 reinterpreted
by Spagnoli, 2005).

View image in article

Fig. 4. Slope of the m vs. Z relationship as a function of Π4=(E/σy)avg.

View image in article

Fig. 5. Assessment of incomplete self-similarity in Π5 in metals (C evaluated using ΔK in View the MathML source and da/dN in m/cycle). (a) and (b) 4340 steel (Heiser and Mortimer, 1972). (c) and (d) ASTM steels (Clark and Wessel, 1970).

View image in article

Fig. 6. Assessment of incomplete self-similarity in Π5 in concrete (C evaluated using ΔK in View the MathML source and da/dN
in m/cycle). (a) and (b) High strength concrete (data from Bažant and
Shell, 1993 reinterpreted by Spagnoli, 2005). (c) and (d) Normal
strength concrete (data from Bažant and Xu, 1991 reinterpreted by
Spagnoli, 2005).

View image in article

Fig. 7. The effect of R on the mZ and logCZ relationships for tempered steels (experimental data from Evans et al., 1971, C evaluated using ΔK in View the MathML source and da/dN in m/cycle).

View image in article

Fig. 8. Paris’ law parameter m vs. ΔKth/KIC (reprinted from Fleck et al., 1994 with permission).

View image in article

Fig. 9. Correlations between C and m (C evaluated using ΔK in View the MathML source and da/dN
in m/cycle). Dashed lines refer to the correlation by Carpinteri and
Paggi (2007), whereas dashed-dotted lines refer to the correlation by
Tanaka (1979). (a) 4340 steel (Heiser and Mortimer, 1972). (b) ASTM
steels (Clark and Wessel, 1970). (c) High strength concrete (data from
Bažant and Shell, 1993 reinterpreted by Spagnoli, 2005). (d) Normal
strength concrete (data from Bažant and Xu, 1991 reinterpreted by
Spagnoli, 2005).

 

 

 

Subscribe to Comments for "One, no one, and one hundred thousand crack propagation laws: A generalized Barenblatt and Botvina dimensional analysis approach"

Recent comments

More comments

Syndicate

Subscribe to Syndicate