Page 80 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 60 Thursday, April 14, 2005 6:28 PM

60 Fracture Mechanics: Fundamentals and Applications

Assume that the plate has a unit thickness. Let us now apply a compressive stress field to the crack
faces between x = 0 and x = ∆a of sufficient magnitude to close the crack in this region. The work
required to close the crack at the tip is related to the energy release rate:

 U
∆ 
G = lim (2.57)
∆ 
a
∆→0  a fixedload
where ∆U is the work of crack closure, which is equal to the sum of contributions to work from
x = 0 to x = ∆a:

=∆
∆ =U ∫ x x =0 a d U () (2.58)
x

and the incremental work at x is equal to the area under the force-displacement curve:

x u x dx
dU x() =× 1 F x u x() = σ () () (2.59)
2
()
2 y y yy y
The factor of 2 on the work is required because both crack faces are displaced an absolute distance
u (x). The crack-opening displacement u for Mode I is obtained from Table 2.2 by setting θ = π.
y
y
ax
(κ + Ka + ) 1 ( ∆ a) ∆−
u = I (2.60)
y
2 µ 2 π
where K (a + ∆a) denotes the stress intensity factor at the original crack tip. The normal stress
I
required to close the crack is related to K for the shortened crack:
I
Ka()
σ = I (2.61)
yy
2 πx

Combining Equation (2.57) to Equation (2.61) gives
(κ + ( )Ka + )Ka ∆ 1 ) a ∆a ∆−
ax
(
G = lim I I ∫ dx
a
∆→0 4πµ ∆a 0 x
(κ + ) 1 K 2 K 2
= I = I (2.62)
8 µ ′ E
Thus, Equation (2.56) is a general relationship for Mode I. The above analysis can be repeated for
other modes of loading; the relevant closure stress and displacement for Mode II are, respectively,
τ and u , and the corresponding quantities for Mode III are τ and u . When all three modes of
z
yz
x
yx
loading are present, the energy release rate is given by
K 2 K 2 K 2
G = I + II + III (2.63)
′ E ′ E 2µ

Contributions to G from the three modes are additive because energy release rate, like energy, is a
scalar quantity. Equation (2.63), however, assumes a self-similar crack growth, i.e., a planar crack