In all cases where an expression for the stress-intensity
factor cannot be obtained from existing solutions, finite-element analysis can
be used to determine K [Chan, et al., 1970; Byskov, 1970; Tracey, 1971;
Walsh, 1971]. Certain aircraft
structural configurations have to be analyzed by finite-element techniques
because of the influence of complex geometrical boundary conditions or complex
load transfer situations. In the case
of load transfer, the magnitude and distribution of loadings may be
unknown. With the application of
finite-element methods, the required boundary conditions and applied loadings
must be imposed on the model.
Complex structural configurations and multicomponent structures
present special problems for finite-element modeling. These problems are associated with the structural
complexity. When they can be solved,
the stress-intensity factor is determined in the same way as in the case of
simpler geometry. This subsection deals
with the principles and procedures that permit the determination of the
stress-intensity factor from a finite-element solution.
Usually quadrilateral, triangular, or rectangular
constantain elements are used, depending on the particular finite-element
structural analysis computer program being used. For problems involving holes or other stress concentrations, a
fine-grid network is required to accurately model the hole boundary and
properly define the stress and strain gradients around the hole or stress
Within the finite-element grid system of the structural
problem, the crack surface and length must be simulated. Usually, the location and direction of crack
propagation is perpendicular to the maximum principal stress direction. If the maximum principal stress direction is
unknown, then an uncracked stress analysis of the finite-element model should
be conducted to establish the location of the crack and the direction of
The crack surfaces and lengths are often simulated by
double-node coupling of elements along the crack line. Progressive crack extension is then
simulated by progressively “unzipping” the coupled nodes along the crack
line. Because standard finite-element
formulations do not treat singular stress behavior in the vicinity of the ends
of cracks, special procedures must be utilized to determine the
stress-intensity factor. Three basic
approaches to obtain stress-intensity factors from finite-element solutions
have been rather extensively studied.
These approaches are as follows:
Method. The numerical results of
stress, displacement, or crack-opening displacement are fitted to analytical
forms of crack-tipess-displacement fields to obtain stress-intensity
Method. The stress-intensity follows
from its relation to other quantities such as compliance, elastic energy, or
work energy for crack closure.
Element. A hybrid-cracked element
allowing a stress singularity is incorporated in the finite-element grid system
and stress-intensity factors are determined from nodal point displacements
along the periphery of the cracked element.
These approaches can be applied to determine both Mode 1 and
Mode 2 stress-intensity factors.
Application of methods has been limited to two-dimensional planar
problems. The state-of-the-art for
treating three-dimensional structural crack problems is still a research area.