Page 77 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 57 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 57

where Γ and A are the perimeter and area of the body, respectively, and u are the displacements
i
(2)
in the x and y directions. Since loading systems (1) and (2) are arbitrary, it follows that K cannot
I
(1)
(1)
depend on K and u . Therefore, the function
i
I
E u ∂ ()
1
hx() = i (2.50)
i
1
2 K () a ∂
I
where x represents the x and y coordinates, must be independent of the nature of loading system
i
(1). Bueckner [16] derived a similar result to Equation (2.50) two years before Rice, and referred
to h as a weight function.
Weight functions are first-order tensors that depend only on the geometry of the cracked body.
Given the weight function for a particular configuration, it is possible to compute K from Equation
I
(2.49) for any boundary condition. Moreover, the previous section invoked the principle of super-
position to show that any loading configuration can be represented by appropriate tractions applied
directly to the crack face. Thus K for a two-dimensional cracked body can be inferred from the
I
following expression:
K I ∫ p = x h x () () d x (2.51)
c Γ
where p(x) is the crack face traction (equal to the normal stress acting on the crack plane when the
body is uncracked) and Γ is the perimeter of the crack. The weight function h(x) can be interpreted
c
as the stress intensity resulting from a unit force applied to the crack face at x, and the above integral
represents the superposition of the K values from discrete opening forces along the crack face.
I

EXAMPLE 2.6

Derive an expression for K I for an arbitrary traction on the face of a through crack in an infinite plate.

Solution: We already know K I for this configuration when a uniform tensile stress is applied:
K I a = σπ

where a is the half-crack length. From Equation (A2.43), the opening displacement of the crack faces
in this case is given by
2σ
(
u =± E′ xa − x)2
y
where the x-y coordinate axis is defined in Figure 2.27(a). Since the crack length is 2a, we must
differentiate u y with respect to 2a rather than a:

∂u y 2σ x
∂( 2 ) a =± E 2 ax
−

Thus, the weight function for this crack geometry is given by

hx() =± 1 x
π a 2 ax
−