Page 112 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C02.fm Page 92 Thursday, April 14, 2005 6:28 PM

92 Fracture Mechanics: Fundamentals and Applications

where ∆ is the local load line displacement and P is the applied load. Differentiating
Equation (A2.14) gives

 ∂∆  ∂∆
+
d∆=   ∆ a  P da +   ∂ P  a dP C dP = M 0 (A2.15)
T
assuming ∆ depends only on load and crack length. We can make this same assumption about the
energy release rate:

 ∂ G   ∂ G 
dG =   da +   dP (A2.16)
 a ∂  P  ∂ P  a

Dividing both sides of Equation (A2.16) by da and fixing ∆ yields
T
 dG  =   ∂ G   ∂ G   dP 

 da  ∆ T  a ∂   +   ∂ P   da  ∆ T (A2.17)
P
a
which, upon substitution of Equation (A2.15), leads to

 dG  =   ∂ G   ∂ G   ∂∆  C +   ∂∆  −1
 
 da  ∆ T  a ∂   −   ∂ P   a ∂     M  ∂ P   (A2.18)
a
P
a
P
A virtually identical expression for the J integral (Equation 3.52) can be derived by assuming J
depends only on P and a, and expanding dJ into its partial derivatives.
Under dead-loading conditions, C =∞ , and all but the first term in Equation (A2.18) vanish.
M
Conversely,C = 0 corresponds to an infinitely stiff system, and Equation (A2.18) reduces to the
M
pure displacement control case.
A2.3 CRACK-TIP STRESS ANALYSIS

A variety of techniques are available for analyzing stresses in cracked bodies. This section focuses
on two early approaches developed by Williams [11, 37] and Westergaard [8]. These two analyses
are complementary; the Williams approach considers the local crack-tip fields under generalized in-
plane loading, while Westergaard provided a means for connecting the local fields to global boundary
conditions in certain configurations.
Space limitations preclude listing every minute step in each derivation. Moreover, stress, strain,
and displacement distributions are not derived for all modes of loading. The derivations that follow
serve as illustrative examples. The reader who is interested in further details should consult the original
references.

A2.3.1 Generalized In-Plane Loading

Williams was the first to demonstrate the universal nature of the 1 r singularity for elastic crack
problems, although Inglis [1], Westergaard, and Sneddon [10] had earlier obtained this result in
specific configurations. Williams actually began by considering stresses at the corner of a plate
with various boundary conditions and included angles; a crack is a special case where the included
angle of the plate corner is 2π and the surfaces are traction free (Figure A2.1).