Page 228 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)
P. 228
1656_C004.fm Page 208 Thursday, April 21, 2005 5:38 PM
208 Fracture Mechanics: Fundamentals and Applications
Let us now introduce a moving coordinate system (x, y) attached to the crack tip, where
x = X − a(t) and y = Y. The rate of change of each wave potential can be written as
d ψ i = ∂ ψ i − V ∂ ψ i
dt t ∂ x ∂ (i = 1, 2) (A4.6)
where V(= −dx/dt) is the crack speed. Differentiating Equation (A4.6) with respect to time gives
∂ 2 ψ 2 ψ 2 ψ ∂ ∂ ψ ∂
˙˙ ψ = V 2 i − 2 V i + i − V ˙ i (A4.7)
i x ∂ 2 ∂∂ t ∂ 2 x ∂
xt
According to Equation (A4.5) the first term on the right-hand side of Equation (A4.7) is proportional
to the stress tensor. This term should dominate close to the crack tip, assuming there is a stress
singularity. Substituting the first term of Equation (A4.7) into Equation (A4.4) leads to
∂ 2 ψ ∂ 2 ψ
β 2 1 + 1 = 0 (A4.8a)
1 ∂x 2 ∂y 2
and
∂ 2 ψ ∂ 2 ψ
β 2 2 + 2 = 0 (A4.8b)
2 ∂x 2 ∂y 2
where
2 V 2
V
β =− and β =−
2
2
1
1
c
1 2 c 2
1
Note that the governing equations depend only on instantaneous crack speed; the term that contains
crack acceleration in Equation (A4.7) is negligible near the crack tip.
If we scale y by defining new coordinates, y = β y and y = β y, Equation (A4.8) becomes
1
1
2
2
the Laplace equation. Freund and Clifton [32] applied a complex variable method to solve Equation
(A4.8). The general solutions to the wave potentials are as follows:
Fz
ψ = Re[ ( )] (A4.9)
1 1
and
Gz
ψ = Im[ ( )]
2 2
where F and G are as yet unspecified complex functions, z = x + iy and z = x + iy .
1
1
2
2
The boundary conditions are the same as for a stationary crack: σ = τ = 0 on the crack surfaces.
xy
yy
Freund and Clifton showed that these boundary conditions can be expressed in terms of second
derivatives for F and G at y = 0 and x < 0:
1+ ( β 2 ) 2 ′′( ) + + ′′( )]Fx − + 2 [G ′′( )x + + ′′( )]G x − = 0 (A4.10a)
β [Fx
2
G
Fx
2 [ ′′ ( ) − ′′ ( )]+ 1 β Fx +( 1 ) 2 [ ′′ x G ( )] = 0 (A4.10b)
( ) − ′′ x
β
1 + − + −
