Page 115 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 95 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 95

It is convenient in most cases to treat the symmetric and antisymmetric stresses separately. The
constants s and t can be replaced by the Mode I and Mode II stress intensity factors, respectively:
1
1
K
s =− I (A2.27a)
1 2π
K
t = II (A2.27b)
1 2π

The crack-tip stress fields for symmetric (Mode I) loading (assuming the higher-order terms are
negligible) are given by

σ = I 5 cos  θ K − 1 cos  3 θ     (A2.28a)

rr
2
2 πr  4   4  2  
σ = I 3 cos  θ K + 1 cos  3 θ     (A2.28b)

r
θθ
2 π  4   4  2  
2
1
τ = I 1 sin  θ K + sin  3 θ     (A2.28c)

θ r
2
2 πr  4   4  2  
The singular stress fields for Mode II are given by
K  θ  5 3  3 θ   
σ = II  − sin + sin  (A2.29a)
rr
2 πr  4   4  2  
2
K  θ  3 3  3 θ   
σ = II  − sin − sin  (A2.29b)
r
θθ
2
2 π  4   4  2  
τ = II 1 cos  θ K + 3 cos  3 θ     (A2.29c)

θ r
2 πr  4   4  2  
2
The relationships in Table 2.1 can be obtained by converting Equation (A2.28) and Equation (A2.29)
to Cartesian coordinates.
The stress intensity factor defines the amplitude of the crack-tip singularity; all the stress and
strain components at points near the crack tip increase in proportion to K, provided the crack is
stationary. The precise definition of the stress intensity factor is arbitrary, however; the constants s 1
and t would serve equally well for characterizing the singularity. The accepted definition of stress
1
intensity stems from the early work of Irwin [9], who quantified the amplitude of the Mode I singularity
with GE , where G is the energy release rate. It turns out that the π in the denominators of
Equation (A2.28) and Equation (A2.29) is superfluous (see Equation (A2.34)– (A2.36), but convention
established over the last 35 years precludes redefining K in a more straightforward form.
Williams also derived relationships for radial and tangential displacements near the crack tip.
We will postpone the evaluation of displacements until the next section, however, because the
Westergaard approach for deriving displacements is somewhat more compact.

A2.3.2 The Westergaard Stress Function

Westergaard showed that a limited class of problems could be solved by introducing a complex
stress function Z(z), where z = x + iy and i =−1 . The Westergaard stress function is related to