Page 231 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 211 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 211

Equation (A4.16) corresponds to the rate at which energy is lost from the body due to flux through
Γ. By defining n = −m on Γ, we obtain the following expression for the energy flux into Γ:
j
j
F ∫ Γ w( ) Γ T= V [( + j σ j 1 i u ) δ ˙ ] n j d + Γ (A4.17)
i

In the limit of a vanishingly small contour, the flux is independent of the shape of Γ. Thus, the
energy flux to the crack tip is given by

F → ∫ w = + T lim V [( σ u) δ ˙ ] n d + Γ (A4.18)
Γ 0 Γ j j 1 i i j

In an increment of time dt, the crack extends by da = Vdt and the energy expended is Φ dt. Thus,
the energy release rate is given by

F
J = (A4.19)
V

Substituting Equation (A4.18) into Equation (A4.19) will yield a generalized expression for the J
integral. First, however, we must express the displacement rate in terms of crack speed. By analogy
with Equation (A4.6), the displacement rate can be written as

u ∂ u ∂
˙ u V =− i + i (A4.20)
i x ∂ t ∂

Under steady-state conditions, the second term in Equation (A4.20) vanishes; the displacement at
a fixed distance from the propagating crack tip remains constant. Close to the crack tip, the
displacement changes rapidly with position (at a fixed time) and the first term in Equation (A4.20)
dominates in all cases. Thus, the J integral is given by

i
J → ∫   w = + T lim ) δ −( σ u ∂   ndΓ
Γ 0 Γ  j j 1 i x ∂  j
(A4.21)
 ∂u 
− (w T
=  ∫ + lim )dy σ n i d Γ 
Γ →0 Γ  ji j ∂x 
Equation (A4.21) applies to all types of material response (e.g., elastic, plastic, viscoplastic, and
3
viscoelastic behavior), because it was derived from a generalized energy balance. In the special
case of an elastic material (linear or nonlinear), w is the strain energy density, which displays the
properties of an elastic potential:

∂ w
σ = ε ∂ ij (A4.22)
ij

Recall from Appendix 3 that Equation (A4.22) is necessary to demonstrate the path independence
of J in the quasistatic case. In general, Equation (A4.21) is not path independent except in a local

3 Since the divergence and transport theorems were invoked, there is an inherent assumption that the material behaves as a
continuum with smoothly varying displacement fields.