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Linear Elastic Fracture Mechanics 41
it is possible to obtain a significant amount of stable crack growth. If an instability occurs during
the test, the R curve cannot be defined beyond the point of ultimate failure.
EXAMPLE 2.3
Evaluate the relative stability of a DCB specimen (Figure 2.9) in load control and displacement control.
Solution: From the result derived in Example 2.2, the slope of the driving force curve in load control
is given by
2
dG = 2 Pa = 2 G
da P BEI a
In order to evaluate displacement control, it is necessary to express G in terms of ∆ and a. From beam
theory, load is related to displacement as follows:
P = 3 ∆ EI
2 a 3
Substituting the above equation into expression for energy release rate gives
2
G = 9 ∆ EI
4Ba 4
Thus
dG =− 9 ∆ 2 EI =− 4 G
da ∆ Ba 5 a
Therefore, the driving force increases with crack growth in load control and decreases in displacement
control. For a flat R curve, crack growth in load control is always unstable, while displacement control
is always stable.
2.5.3 STRUCTURES WITH FINITE COMPLIANCE
Most real structures are subject to conditions between load control and pure displacement
control. This intermediate situation can be schematically represented by a spring in series with
the flawed structure (Figure 2.12). The structure is fixed at a constant remote displacement ∆ ;
T
the spring represents the system compliance C . Pure displacement control corresponds to an
m
infinitely stiff spring, where C . = 0. Load control (dead loading) implies an infinitely soft
m
spring, i.e., C . = ∞.
m
When the system compliance is finite, the point of fracture instability obviously lies somewhere
between the extremes of pure load control and pure displacement control. However, determining
the precise point of instability requires a rather complex analysis.
At the moment of instability, the following conditions are satisfied:
G = R (2.34a)
