Page 117 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 97 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 97

FIGURE A2.2 Through-thickness crack in an infinite plate loaded in biaxial tension.

Comparing Equation (A2.34) and Equation (A2.35) gives

K I a = σπ (A2.36)
for the configuration in Figure A2.2. Note that π appears in Equation (A2.36) because K was
originally defined in terms of the energy release rate; an alternative definition of stress intensity
might be

K *
σ θ ( = ) = 0 I where K * a = σ
yy
2 x * I
for the plate in Figure A2.2.
Substituting Equation (A3.26) into Equation (A2.32) results in an expression of the Westergaard
stress function in terms of K :
I
K
Zz() = I (A2.37)
*
2π z *
where z * = z − a. It is possible to solve for the singular stresses at other angles by making the
following substitution in Equation (A2.37):

z * r = e iθ
where
 y 
r 2 x = − a( 2 + y) 2 and θ = tan −1
− 
 xa