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# Section 11.2.2.1. Green's Function Technique

The Green's function technique takes advantage of the additive property of the stress-intensity factor and is based on generalized point load solutions of crack problems.  For example, the point load solution for the central crack problem described in Figure 11.2.7 is given by:

 (11.2.11)

This solution can be used to obtain the stress-intensity factor for stresses distributed over the crack faces by noting that the point load per unit thickness (P/B) in Equation 11.2.11 can be replaced by the product of the pressure stress (s (x)) and the distance over which it acts (dx).  Thus, the stress-intensity factor for the distributed stresses applied to the crack becomes:

 (11.2.12)

The stress-intensity factor for the case of uniform opening stresses applied to the crack, where s is a constant, is determined to be K = sÖpa , as was expected from the discussion of the method of superposition described previously.

Figure 11.2.7.  Point Load (P) Applied to the Crack Faces for a Central Crack Located in an Infinite Plate

Figure 11.2.8.  Distributed Loading Applied to Crack Faces of the Central Crack

Given a point force solution for a geometry of concern, it is then possible to define the summation process that would integrate the effects of stress loading over the crack faces.  Integral equations such as that defined by Equation 11.2.12 utilize the stress solutions from the uncracked body problem.  A number of point force stress-intensity factor solutions are presented in the tables given in Section 11.3 and an extensive review of the availability and application of Green's functions can be found in Cartwright & Rooke [1979].  Other reviews can be found in Cartwright & Rooke [1978] and Cartwright [1979].

One of the cases reviewed by Cartwright and Rooke [Cartwright & Rooke, 1979] is of particular interest to structural engineers.  They presented the work by Hsu and Rudd [1978] on the development of a Green's function for a diametrically cracked hole.  The Hsu and Rudd Green's function was based on a series of finite-element determined stress-intensity factor solutions for a symmetrical set of point forces of the type shown in Figure 11.2.9.  The finite-element point force solutions were developed as a function of position for X (=x/a) < 0.9 and a limiting expression was given for X > 0.9.  The Hsu and Rudd Green's function is shown in Figure 11.2.10 for several values of a/R; also shown are Green's functions for an edge crack and for a central crack.  Note that all the Green's functions tend to infinity as X approaches 1.  It should also be noted that the Green's functions presented are based on the following format

 (11.2.13)

which has been widely used.  Hsu and Rudd based their presentation of the Green's function on an approach taken by Hsu, et al. [1978], wherein the Green's function G(x,a) in Equation 11.2.13 is obtained by multiplying the Hsu, et al. value GH by p, i.e.

 (11.2.14)

The complete table of GH(x,a) derived by Hsu, et al. can be found in Table 11.2.1.  Other work by Hsu and co-workers on lug-type problems can be found in Section 11.3.

Figure 11.2.9.  Diametrically Cracked Hole With Symmetrically Located Point Focus

Figure 11.2.10.  Green's Function for Geometry and Loading Described in Figure 11.2.9 [Cartwright & Rooke, 1979; Hsu & Rudd, 1978; Hsu, et al., 1978]

Table 11.2.1.  Green's Function For A Double Crack Emanating From An Open Hole In An Infinite Plate [Hsu, et al., 1978]

 x/a a/r .20 .30 .40 .50 .60 70 80 .90 1.00 1.40 1.60 2.00 2.40 3.00 .00 .664 .629 .603 .595 .568 .575 .572 .554 .548 .571 .582 .594 .600 .611 .10 .676 .639 .615 .604 .582 .583 .586 .569 .563 .587 .596 .603 .603 .615 .20 .688 .645 .628 .617 .599 .596 .600 .589 .578 .604 .612 .613 .609 .624 .30 .699 .658 .646 .633 .621 .613 .623 .610 .598 .627 .630 .625 .619 .639 .40 .718 .679 .671 .656 .651 .639 .655 .639 .624 .656 .653 .642 .635 .664 .45 .740 .691 .689 .671 .673 .657 .674 .658 .643 .674 .665 .654 .647 .680 .50 .760 .708 .712 .689 .698 .681 .701 .682 .668 .692 .678 .670 .662 .699 .55 .781 .732 .739 .712 .730 .711 .733 .708 .699 .709 .695 .692 .679 .723 .60 .802 .764 .762 .746 .770 .752 .766 .739 .737 .730 .725 .721 .702 .753 .70 .889 .868 .837 .838 .865 .867 .850 .827 .847 .819 .801 .811 .760 .842 .75 .960 .946 .907 .911 .913 .960 .912 .911 .929 .888 .859 .884 .817 .905 .80 1.071 1.089 1.044 1.030 .989 1.056 1.018 .995 1.021 .985 .955 .979 .904 .977 .85 1.234 1.254 1.245 1.211 1.141 1.252 1.177 1.187 1.192 1.130 1.130 1.120 1.042 1.101 .90 1.429 1.432 1.434 1.436 1.437 1.438 1.440 1.441 1.442 1.445 1.446 1.448 1.449 1.451 *For x/a >  0.9,

There are two cautionary remarks that must be made about the use of Green's function techniques for solving crack problems.  First, if all the loading across the crack tip is not tensile, and if the stress-intensity factor is positive at the crack tip of interest, the crack faces at some distance away from the crack tip may have (mathematically) merged in a nonphysical overlapping manner and the estimated stress-intensity factor might be unconservatively low.  Accordingly, one should check to determine if the crack displacements all along the crack are positive and thus non-overlapping to ensure validity of the solution.  Second, it is important in displacement boundary value problems to derive a Green's function that accounts for the requirement that there be zero displacement on those boundaries where displacement conditions are applied when estimating the stress-intensity factor from the uncracked geometry solution.  Typically, neglecting this requirement for displacement boundary value problems produces a stress-intensity factor that is conservatively high.  These two cautions apply equally well to the weight function technique.