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 constant-strain 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 concentration.

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 propagation.

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:

a)      Direct Method.  The numerical results of stress, displacement, or crack-opening displacement are fitted to analytical forms of crack-tip-stress-displacement fields to obtain stress-intensity factors.

b)      Indirect Method.  The stress-intensity follows from its relation to other quantities such as compliance, elastic energy, or work energy for crack closure.

c)      Cracked 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.