Page 428 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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408 Fracture Mechanics: Fundamentals and Applications

Thus, there is a unique relationship between T and J , and the T -J curve can be defined
R
R
R
R
unambiguously. Suppose, for example, that the J-R curve is fit to a power law:
J R C = 1 ( a a − o ) C 2 (9.51)
The material tearing modulus is given by
E CJ E
T = σ 2 o aa ) = σ ( 2 o CC 1/ C 2 J ( R C 2 C −1)/ 2 (9.52)
R
2
R
−
21
o
The applied tearing modulus curve is less clearly defined, however. There are a number of
approaches for defining the T -J curve, depending on the application. Figure 9.15 illustrates two
app
possible approaches, which are discussed next.
Suppose that the initial crack size a is known, and one wishes to determine the loading
o
conditions (P, ∆, and ∆ ) at failure. In this case, the T should be computed at various points on
T
app
the R curve. Since J = J during stable crack growth, the applied J at a given crack size can be
R
inferred from the J-R curve (Equation (9.50)). The remote displacement ∆ increases as the loading
T
progresses up the J-R curve (Figure 9.13); instability occurs at ∆ T(4) . The final load, local displace-
ment, crack size, and stable crack extension can be readily computed, once the critical point on
the J-R curve has been identified.
The T -J curve can also be constructed by fixing one of the loading conditions (P, ∆, or ∆ ),
T
app
and determining the critical crack size at failure, as well as a . For example, if we fix ∆ at ∆ T(4)
T
o
in the structure, we would predict the same failure point as the previous analysis but the T -J
app
curve would follow a different path (Figure 9.15). If, however, we fix the remote displacement at
a different value, we would predict failure at another point on the T -J curve; the critical crack
R
R
size, stable crack extension, and a would be different from the previous example.
o
9.3.4 SOME PRACTICAL CONSIDERATIONS
If the material is sufficiently tough or if crack-like flaws in the structure are small, the structure
will not fail unless it is loaded into the fully plastic regime. When performing fracture analyses in
this regime, there are a number of important considerations that many practitioners overlook.
n+1
In the fully plastic regime, the J integral varies with P for a power-law material; a slight
increase in load leads to a large increase in the applied J. The J vs. the crack-length driving force
curves are also very steep in this regime. Consequently, the failure stress and critical crack size are
insensitive to toughness in the fully plastic regime; rather, failure is governed primarily by the flow
properties of the material. The problem is reduced to a limit load situation, where the main effect
of the crack is to reduce the net cross section of the structure.
Predicting the failure stress or critical crack size under fully plastic conditions need not be
complicated. A detailed tearing instability analysis and a simple limit load analysis should lead to
similar estimates of failure conditions.
Problems arise, however, when one tries to compute the applied J at a given load and crack
size. Since J is very sensitive to load in the fully plastic regime, a slight error in P produces a
significant error in the estimated J. For example, a 10% overestimate in the yield strength s will
o
produce a corresponding error in P , which will lead to an underestimate of J by a factor of 3.2
o
for n = 10. Since flow properties typically vary by several percent in different regions of a plate,
and heat-to-heat variations can be much larger, accurate estimates of the applied J at a fixed load
are extremely difficult.
If estimates of the applied J are required in the fully plastic regime, the displacement, not the
load, should characterize conditions in the structure. While the plastic J is proportional to P , J pl
n+1
scales with ∆ n + )/ n ( according to Equation (9.29) and Equation (9.31). Thus a J-∆ plot is nearly linear
1
p
in the fully plastic regime, and displacement is a much more sensitive indicator of the applied