Page 113 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 93 Thursday, April 14, 2005 6:28 PM

Linear Elastic Fracture Mechanics 93

FIGURE A2.1 Plate corner configuration analyzed by Williams. A crack is formed when ψ = 2π: (a) plate
corner with included angle ψ and (b) special case of a sharp crack. Taken from Williams, J.G. and Ewing,
P.D., International Journal of Fracture Mechanics, Vol. 8, 1972.

For the configuration shown in Figure A2.1(b), Williams postulated the following stress function:

Φ= r λ +1 [ c 1 λsin( + θ 1) * + c 2 + θ 1) * + λ cos( 3 − c * +λsin( 4 λcos( −c θ 1) θ 1) * ]
= r λ +1 Φ(, (A2.19)
*
θλ)
*
where c , c , c , and c are constants, and θ is defined in Figure A2.1(b). Invoking Equation (A2.13)
1
2
4
3
gives the following expressions for the stresses:
F+
σ rr λ−1 [ F ′′( θ = * λ r 1) ( )] (A2.20a)
*
θ + ) (
σ θθ λ−1 λ λ= + r 1) ( )] (A2.20b)
F
*
θ [(
θr
τ θ r = λ−1 [ − λ F ′( )] (A2.20c)
*
*
where the primes denote derivatives with respect to θ . Williams also showed that Equation (A2.19)
λ
implies that the displacements vary with r . In order for displacements to be finite in all regions of the
body, λ must be > 0. If the crack faces are traction free, σ θθ σ () 0 θθ π= ) = τ (2 θ r τ () 0 = θ r = (2 π ) = , 0
which implies the following boundary conditions:
F F() 0 = ) = F (2π ′ F() 0 = ′ (2 π ) = 0 (A2.21)
Assuming the constants in Equation (A2.19) are nonzero in the most general case, the boundary
conditions can be satisfied only when sin(2πλ = 0 . Thus,
)
n
λ = , where n = 1, 2, 3,…
2
There are an infinite number of λ values that satisfy the boundary conditions; the most general
solution to a crack problem, therefore, is a polynomial of the form
N  n  n 
Φ=  ∑ r 2 +1 F   θ ,  (A2.22)
*
n =1   2 