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1656_C009.fm Page 394 Monday, May 23, 2005 3:58 PM
394 Fracture Mechanics: Fundamentals and Applications
The corresponding influence coefficients for the free surface (f = 0°) are as follows:
G = Q 4 2 N + 4 N + 1 N + (9.20a)
2 π 5 3 1 7 2 2 3
G = Q 4 1 N + 4 N + 2 N + (9.20b)
3 π 7 2 1 9 2 5 3
G = Q 4 2 N + 4 N + 1 N + (9.20c)
4 π 9 5 1 11 2 3 3
Therefore, if one wishes to apply the influence coefficient approach, it is not necessary to
compute the higher-order G solutions with finite element analysis. If solutions for uniform and
n
linear crack-face loading are available, the weight function coefficients can be computed from
Equation (9.17) and Equation (9.18). For more complex load cases, K can be inferred by inte-
I
grating Equation (9.12). Alternatively, if the stress distribution can be represented by a polynomial
expression, the higher-order influence coefficients can be computed from Equation (9.19) and
Equation (9.20) and K can be inferred from Equation (9.7). The advantage of the latter approach
I
is that numerical integration is not required. Consequently, the influence coefficient approach is
more conducive to spreadsheet calculations than the weight function method.
Equations (9.13)–(9.20) apply only to two locations on the crack front: f = 0° and f = 90°.
Wang and Lambert [15] have developed a weight function expression that applies to the full
range 0 ≤≤ 90 for semielliptical surface cracks. Anderson et al. [11] have integrated this expres-
φ
sion to solve for the influence coefficients (G ) at arbitrary crack front angles. The resulting equations
i
are rather lengthy, and consequently are omitted for the sake of brevity.
9.1.4 PRIMARY, SECONDARY, AND RESIDUAL STRESSES
Section 2.4 introduced the concept of load control and displacement control, where a structure or
specimen is subject to applied forces or imposed displacements, respectively. The applied energy
release rate, and therefore the stress intensity factor, is the same, irrespective of whether a given
stress distribution is the result of applied loads or imposed displacements. Section 2.5 and
Section 3.4.1 discuss the relative stability of cracked components in load control and displacement
control. Crack growth tends to be unstable in load control but can be stable in displacement control.
There are very few practical situations in which a cracked body is subject to pure displacement
control. Figure 2.12, which is a simple representation of the more typical case, shows a cracked
plate subject to a fixed remote displacement ∆ . The spring in series represents the system compli-
t
ance C . For example, a long cracked plate in which the ends are fixed would have a large system
m
compliance. If C is large, there is essentially no difference between (remote) displacement control
m
and load control as far as the crack is concerned. See Section 9.3.3 for further discussion of the
effect of system compliance on crack stability.
Some design codes for structures such as pressure vessels and piping refer to load-controlled
stresses as primary and displacement-controlled stresses as secondary. Hoop stress due to internal
pressure in a pipe or pressure vessel is an example of a primary stress because the pressure
constitutes applied forces on the boundary of the structure. Thermal expansion (or contraction)
leads to imposed displacements, so thermal stresses are usually considered secondary. As long as
the total stress is well below the yield strength, the classification of stresses as primary and secondary
is not important. When plastic deformation occurs, however, secondary stresses redistribute and
may relax from their initial values. For this reason, design codes that classify stresses as primary
and secondary usually permit higher allowable values of the latter.
A special case of a displacement-controlled stress is residual stress due to weld shrinkage. Weld
residual stresses are usually not taken into account in design because they do not affect the failure
