Page 230 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 210 Thursday, April 21, 2005 5:38 PM

210 Fracture Mechanics: Fundamentals and Applications

Beginning with the equation of motion, Equation (A4.1), taking an inner product of both sides
with displacement rate ˙ u i and rearranging gives

∂(σ u ˙ ) ∂( ˙ )
u
ji i
i
∂x j = ρ i + ˙˙ σu j i ∂x j
= ˙ + T ˙ w (A4.14)
where T and w are the kinetic energy and stress work densities, respectively, as defined in
Equation (4.27) to Equation (4.29). Equation (A4.14) is a general balance law that applies to all
material behavior. Integrating this relationship over an arbitrary volume, and applying the divergence
and transport theorems gives

d
∫ ∂ σ um dS = ji i j dt ∫ V ( + wT dV − ∫ ∂V V ( + wT V m dS (A4.15)
˙
)
)
j
j
where
ς = volume
m = outward normal to the surface ∂V
j
V = instantaneous velocity of ∂V
i
Consider now the special case of a crack in a two-dimensional body, where the crack is
propagating along the x axis and the origin is attached to the crack tip (Figure A4.2). Let us define
a contour C fixed in space, that contains the propagating crack and bounds the area A. The crack
o
tip is surrounded by a small contour Γ that is fixed in size and moves with the crack. The balance
law in Equation (A4.15) becomes

d
∫ C o σ ji i j dt ∫ A ( wT d − + ) A ∫ Γ [( wT V δ um dC = j + σ j 1 i u m dΓ (A4.16)
+
˙
)
˙
]
i
j
where V is the crack speed. The integral on the left side of Equation (A4.16) is the rate at which
energy is input into the body. The first term on the right side of this relationship is the rate of
increase in the internal energy in the body. Consequently, the second integral on the right side of

FIGURE A4.2 Conventions for the energy balance
for a propagating crack. The outer contour C o is fixed
in space, and the inner contour Γ and the x, y axes
are attached to the moving crack tip.