Page 179 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 159 Monday, May 23, 2005 5:42 PM

Elastic-Plastic Fracture Mechanics 159

By applying the same assumptions as in Equation (A3.15) and Equation (A3.16), we obtain:

∂w = ∂w ∂ε ij = σ ∂  u i
∂ 
∂a ∂ε ij ∂a ij ∂ x j   a  (A3.24)
∂ 

Invoking the principle of virtual work gives

∫ A′ σ ij ∂    u ∂   dA = ∫ Γ ′ T i u ∂ a ∂ i ds (A3.25)
i
x ∂
a ∂ 
j
which cancels with one of the terms in the line integral in Equation (A3.23), resulting in the
following:

dΠ = ∫ T u ∂ i ds − ∫ ∂ w dA
da Γ′ i x ∂ A′ x ∂ (A3.26)

Applying the divergence theorem and multiplying both sides by –1 leads to

u
dΠ  ∂ 
− =  ∫ −  wn T i  ds
x
da ′ Γ x i ∂ 
(A3.26a)
∂u
=   ∫  wdy − T i i ds  
′ Γ ∂x 
since n ds = dy. Therefore, the J contour integral is equal to the energy release rate for a linear or
x
nonlinear elastic material under quasistatic conditions.

A3.4 THE HRR SINGULARITY

Hutchinson [7] and Rice and Rosengren [8] independently evaluated the character of crack-tip
stress fields in the case of power-law-hardening materials. Hutchinson evaluated both plane stress
and plane strain, while Rice and Rosengren considered only plane-strain conditions. Both articles,
which were published in the same issue of the Journal of the Mec hanics and Physics of Solids ,
argued that stress times strain varies as 1/r near the crack tip, although only Hutchinson was able
to provide a mathematical proof of this relationship.
The Hutchinson analysis is outlined below. Some of the details are omitted for brevity. We
focus instead on his overall approach and the ramifications of this analysis.
Hutchinson began by defining a stress function Φ for the problem. The governing differential
equation for deformation plasticity theory for a plane problem in a Ramberg-Osgood material is
more complicated than the linear elastic case:

α
∆Φ + 4 Φ σ (, , ,, ) = γ 0 (A3.27)
rn
e
where the function g differs for plane stress and plane strain. For the Mode I crack problem,
Hutchinson chose to represent Φ in terms of an asymptotic expansion in the following form:

Φ= C r () θ s + C r () θ t +  (A3.28)
1 2