Page 178 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C003.fm Page 158 Monday, May 23, 2005 5:42 PM

158 Fracture Mechanics: Fundamentals and Applications

On the crack face, T = dy = 0. Thus, J = J = 0 and J = −J . Therefore, any arbitrary
i
1
2
3
4
(counterclockwise) path around a crack will yield the same value of J; J is path-independent.
A3.3 J AS A NONLINEAR ELASTIC ENERGY RELEASE RATE
Consider a two-dimensional cracked body bounded by the curve Γ (Figure A3.4). Let A′ denote
the area of the body. The coordinate axis is attached to the crack tip. Under quasistatic conditions
and in the absence of body forces, the potential energy is given by

Π = ∫ ′ wdA − ∫ Γ′ A ′ Tu ds (A3.20)
ii

where Γ* is the portion of the contour on which the tractions are defined. Let us now consider the
change in potential energy resulting from a virtual extension of the crack:
dΠ = ∫ dw dA − ∫ T du i ds
da A ′ da Γ ′ i da (A3.21)

The line integration in Equation (A3.21) can be performed over the entire contour Γ because
du /da = 0 over the region where displacements are specified; also, dT /da = 0 over the region the
i
i
tractions are specified. When the crack grows, the coordinate axis moves. Thus a derivative with
respect to crack length can be written as
d = ∂ + x ∂ ∂ = ∂ − ∂
da a ∂ ∂ ax a ∂ x ∂ (A3.22)
∂
since ∂x/∂a = −1. Applying this result to Equation (A3.21) gives
dΠ =  ∂ w − ∂ w dA − ∫ T   u ∂ i − u ∂  ds
i
da A  ∫ ′  a ∂ x ∂   Γ′ i  a ∂ x ∂   (A3.23)

FIGURE A3.4 A two-dimensional cracked body bounded by the curve Γ′.