Page 201 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 181 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 181

dimension. Assuming the plate is elastic, the displacements must also be proportional to the nominal
applied strain; thus

σ σ
u x x a = α E and u y y a = α E (4.8)

where α and α are dimensionless constants. (Note that quantitative estimates for α and α near
x
y
y
x
the crack tip in the quasistatic case can be obtained by applying the relationships in Table 2.2.)
The kinetic energy is equal to half the mass times the velocity squared. Therefore, E for the cracked
k
plate (assuming a unit thickness) is given by
 
E  ∫∫
E 1 ρ a = 2 V 2 σ 2 x ( 2 + ) α α 2 dxdy (4.9)
k
2  y
where ρ is the mass density of the material and V( = ˙ a ) is the crack speed. Assuming the integrand
1
depends only on position, E can be written in the following form:
k
 
E 1 k = a 2 V ρ 2 σ 2 (4.10)
k
2  E 
where k is a constant. Applying the modified Griffith energy balance (Equation (4.7)) gives

a
 
G() = t 1 d πσ 22 − k ρ aV 2 σ 2   = 2 w (4.11)
2

2 da  E 2  E   f
where w is the work of fracture, defined in Chapter 2; in the limit of an ideally brittle material,
f
w = γ , the surface energy. Note that Equation (4.11) assumes a flat R curve (constant w ). At
f
s
f
initiation, the kinetic energy term is not present, and the initial crack length a can be inferred
o
from Equation (2.22):
2 Ew
a = πσ 2 f (4.12)
o
Substituting Equation (4.12) into Equation (4.11) and solving for V leads to

2π  a 
V = c 1 − o (4.13)
k o  a 

where c = E/ρ , the speed of sound for one-dimensional wave propagation. Mott [22] actually
o
obtained a somewhat different relationship from Equation (4.13), because he solved Equation (4.11)
by making the erroneous assumption that dV/da = 0. Dulaney and Brace [27] and Berry [28] later
corrected the Mott analysis and derived Equation (4.13).
Roberts and Wells [29] obtained an estimate for k by applying the Westergaard stress
function (Appendix 2.3) for this configuration. After making a few assumptions, they showed that
2π/k ≈ 0.38.

1 In a rigorous dynamic analysis, α x and α y and thus k depend on the crack speed.