Page 74 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 54 Thursday, April 14, 2005 6:28 PM

54 Fracture Mechanics: Fundamentals and Applications

The above equation can be expressed in the form of Equation (2.47):

P f  a  = P f  a  W a π Y = σ a π
BW  w  BW  w  π a

where

Y f =  a  W
 W  π a

In the limit of a small flaw, the geometry correction factor in Table 2.4 becomes

lim f  a  = π a ( .752 0 . )
+
37
0
aW→0  W  W
Thus,

.
lim fY () = 112
aW→0

2.6.4 PRINCIPLE OF SUPERPOSITION

For linear elastic materials, individual components of stress, strain, and displacement are additive.
For example, two normal stresses in the x direction imposed by different external forces can be
added to obtain the total σ , but a normal stress cannot be summed with a shear stress. Similarly,
xx
stress intensity factors are additive as long as the mode of loading is consistent. That is
K (total ) K I ( A) K = I ( B) + K + ( I C)
I

but

K K K≠ + K +
(total ) I II III

In many instances, the principle of superposition allows stress intensity solutions for complex
configurations to be built from simple cases for which the solutions are well established. Consider,
for example, an edge-cracked panel (Table 2.4) subject to combined membrane (axial) loading P ,
m
and three-point bending P . Since both types of loading impose pure Mode I conditions, the K I
b
values can be added:
K (total ) = K I (membrane ) + K I (bending )
I
= 1  Pf  a  + Pf  a    (2.48)
BW  mm  W  bb  W  


where f and f are the geometry correction factors for membrane and bending loading, respectively,
m
b
listed in Table 2.4 and plotted in Figure 2.23.