Page 41 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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History and Overview 21

When the plate width is finite (Figure 1.13(b)), an additional dimension is required to describe the
problem:

σ ij E  r W 
a a
σ ∞ = F 2  σ ∞ ,, , νθ ,  (1.15)

Thus, one might expect Equation (1.14) to give erroneous results when the crack extends across a
significant fraction of the plate width. Consider a large plate and a small plate made of the same
material (same E and ν), with the same a/W ratio, loaded to the same remote stress. The local
stress at an angle θ from the crack plane in each plate would depend only on the r/a ratio, as long
as both plates remained elastic.
When a plastic zone forms ahead of the crack tip (Figure 1.13(c)), the problem is complicated
further. If we assume that the material does not strain harden, the yield strength is sufficient to
define the flow properties. The stress field is given by

σ ij  E σ YS r W r y 
νθ 
σ = F   σ , σ , , , a , ,  (1.16)
a a
3
The first two functions, F and F , correspond to LEFM, while F is an elastic-plastic relationship.
2
1
3
Thus, dimensional analysis tells us that LEFM is valid only when r << a and σ ∞ <<σ . In Chapter 2,
YS
y
the same conclusion is reached through a somewhat more complicated argument.
REFERENCES
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