Page 183 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 163 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 163

Small-Scale Yielding

Rice et al. analyzed the local stresses and displacements at a growing crack by modifying the
classical Prandtl slip-line field to account for elastic unloading behind the crack tip. They assumed
small-scale yielding conditions and a nonhardening material; the details of the derivation are omitted
for brevity. The RDS crack growth analysis resulted in the following expression:

R
δ ˙ α = J ˙ + β σ o a ˙ ln   for r → 0 (A3.38)
σ o E  
r

where
δ ˙ = rate of crack-opening displacement at a distance R behind the crack tip
J ˙ = rate of change in the J integral
˙ a = crack-growth rate
a, b, and R = constants 9

The asymptotic analysis indicated that b = 5.083 for n = 0.3 and b = 4.385 for n = 0.5. The
other constants, a and R, could not be inferred from the asymptotic analysis. Rice et al. [15]
performed an elastic-plastic finite element analysis of a growing crack and found that R, which has
units of length, scales approximately with the plastic zone size, and can be estimated by

λ EJ
R = where l ≈ 0.2 (A3.39)
σ 2 o

˙ a
The dimensionless constant α can be estimated by considering a stationary crack ( = 0):

J
δ α = (A3.40)
σ o

Referring to Equation (3.48), a obviously equals d when d is defined by the 90° intercept method.
n
The finite element analysis performed by Rice et al. indicated that a for a growing crack is nearly
equal to the stationary crack case.
Rice et al. performed an asymptotic integration of (Equation (A3.38)) for the case where the
crack length increases continuously with J, which led to

αrdJ σ  eR 
δ = + βr o ln (A3.41)
σ da E  r 
o
where d, in this case, is the crack-opening displacement at a distance r from the crack tip, and
e (= 2.718) is the natural logarithm base. Equation (A3.41) can be rearranged to solve for the nondi-
mensional tearing modulus:

EdJ Eδ β  eR 
T ≡ = − ln (A3.42)
σ o 2 da ασ o r α  r 

9 The constant a in the RDS analysis should not be confused with the dimensionless constant in the Ramberg-Osgood
relationship (Equation (3.22)), for which the same symbol is used.