Page 75 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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Linear Elastic Fracture Mechanics 55

FIGURE 2.24 Determination of K I for a semielliptical surface crack under internal pressure p by means of
the principle of superposition.

EXAMPLE 2.5

Determine the stress intensity factor for a semielliptical surface crack subjected to an internal pressure
p (Figure 2.24(a)).

Solution: The principle of superposition enables us to construct the solution from known cases. One
relevant case is the semielliptical surface flaw under uniform remote tension p (Figure 2.24(b)). If we
impose a uniform compressive stress −p on the crack surface (Figure 2.24(c)), K I = 0 because the crack
faces close, and the plate behaves as if the crack were not present. The loading configuration of interest
is obtained by subtracting the stresses in Figure 2.24(c) from those of Figure 2.24(b):
K I a () = K I b ( ) − K I c ( )

πa πa
= s Q f φ − λ p = λ () 0 s p Q f φ ()

Example 2.5 is a simple illustration of a more general concept, namely, stresses acting on the
boundary (i.e., tractions) can be replaced with tractions that act on the crack face, such that the two
loading configurations (boundary tractions vs. crack-face tractions) result in the same stress intensity
factor. Consider an uncracked body subject to a boundary traction P(x), as illustrated in Figure 2.25.
This boundary traction results in a normal stress distribution p(x) on Plane A-B. In order to confine
the problem to Mode I, let us assume that no shear stresses act on Plane A-B. (This assumption is
made only for the sake of simplicity; the basic principle can be applied to all three modes of
loading.) Now assume that a crack that forms on Plane A-B and the boundary traction P(x) remains
fixed, as Figure 2.26(a) illustrates. If we remove the boundary traction and apply a traction p(x)
on the crack face (Figure 2.26(b)), the principle of superposition indicates that the applied K will
I
be unchanged. That is

K I a () = K I b () − K I c ( ) = K I b () ( since K I c () = ) 0