Page 180 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 160 Monday, May 23, 2005 5:42 PM

160 Fracture Mechanics: Fundamentals and Applications

where C and C are constants that depend on q, the angle from the crack plane. Equation (A3.28) is
1
2
analogous to the Williams expansion for the linear elastic case (Appendix 2.3). If s < t, and t is less
than all subsequent exponents on r, then the first term dominates as r → 0. If the analysis is
restricted to the region near the crack tip, then the stress function can be expressed as follows:

Φ =κσ o r
Φ s ˜ θ () (A3.29)
Φ
where k is the amplitude of the stress function and ˜ is a dimensionless function of q. Although
Equation (A3.27) is different from the linear elastic case, the stresses can still be derived from
Φ through Equation (A2.6) or Equation (A2.13). Thus the stresses, in polar coordinates, are
given by

σ rr κ = σ o s− σ r rr θ ˜ () κ = σ o r s−2 2 ( ΦΦ ′′)
˜
s
˜
+
˜
σ θθ κ σ = o − σ r θθ θ ˜ () κ σ = o r s 2 s ( s − 1)Φ (A3.30)
− s 2
σ θ κ σ = o s− σ r r θ r θ ˜ () κ σ = o r s−2 2 1 ( − s )Φ ′
˜
σ )
σ κ = σ s− σr θ  κ = σ s−2 2 σ r  () 2 σ +  2 σ − ( σ   + 3  2 1 2
e o e o rr θθ rr θθ r θ
θ
The boundary conditions for the crack problem are as follows:
Φ ˜ ( ± ) Φ =π ˜ ( ′ ± ) =π 0

In the region close to the crack tip where Equation (A3.29) applies, elastic strains are negligible
compared to plastic strains; only the second term in Equation (A3.27) is relevant in this case.
Hutchinson substituted the boundary conditions and Equation (A3.29) into Equation (A3.27) and
obtained a nonlinear eigenvalue equation for s. He then solved this equation numerically for a range
of n values. The numerical analysis indicated that s could be described quite accurately (for both
plane stress and plane strain) by a simple formula:

s = 2 n +1 (A3.31)
n +1

which implies that the strain energy density varies as 1/r near the crack tip. This numerical analysis
also yielded relative values for the angular functions ˜ σ ij . The amplitude, however, cannot be
obtained without connecting the near-tip analysis with the remote boundary conditions. The J
contour integral provides a simple means for making this connection in the case of small-scale
yielding. Moreover, by invoking the path-independent property of J, Hutchinson was able to obtain
a direct proof of the validity of Equation (A3.31).
Consider two circular contours of radius r and r around the tip of a crack in small-scale yielding,
2
1
as illustrated in Figure A3.5. Assume that r is in the region described by the elastic singularity,
1
while r is well inside of the plastic zone, where the stresses are described by Equation (A3.30).
2
When the stresses and displacements in Table 2.1 and Table 2.2 are inserted into Equation (A3.26),
and the J integral is evaluated along r , one finds that J K = I 2 E / ′ as expected from the previous
1
section. Since the connection between K and the global boundary conditions is well established for
I
a wide range of configurations, and J is path-independent, the near-tip problem for small-scale
yielding can be solved by evaluating J at r and relating J to the amplitude (k).
2