The Elastic-Plastic Finite Element Alternating Method ... - Springer

5.1Example 1 ± Elastic-plastic stable crack growthThis example is taken from the ®rst veri®cation study(Brust, Nishioka, Nakagaki, and Atluri (1985) and Brust,McGowan, and Atluri (1986)), and illustrates several importantfeatures of T . Consider an A533B steel which ismade into a number of compact tension specimens andtested. These specimens were analyzed using elastic plastic®nite element analysis from the beginning of the test,through crack initiation and growth, through failure. Sincethese were displacement control tests, the analysis inputconsists of the load point displacement versus crackgrowth record and the predictions consist of the loads, theT -integral, and other fracture parameters. Figure 3 illustratesthe behavior of T throughout one of the tests. Thisanalysis was performed using four different mesh re®nementswith near crack tip mesh size (h from Fig. 2; hereh ˆ Da, the crack growth step size) de®nitions. Fig. 3,which is the T resistance curve for this material fore ˆ 0:6 mm (see Fig. 2), shows that T is mesh size independent.Note that the solution using the Da ˆ 0:6 mmmesh re®nement is very coarse, and violates the rule ofthumb discussed in the previous section requiring a bufferzone of two elements for the path de®nition, but still doesFig. 3. T resistance curves for different near-®eld meshre®nementnot deviate much from the converged solutions representedby the Da ˆ 0:1; 0:15 mm solutions in Fig. 3. Alsoobserve from Fig. 3 that T rises up and reaches a saturationvalue at steady state. This is a typical feature of theT resistance curve, and also is observed under creep crackgrowth conditions.Finally, an example illustrating the importance of consideringhistory dependence is presented (Brust,McGowan, and Atluri (1986). An ASTM A508 steel compacttension specimen was loaded beyond crack initiation,then completely unloaded and reloaded. The response ofthis specimen was predicted using three different resistancecurves based on J, CTOA, and T . The resistancecurves were developed from a generation phase analysis ofanother A508 compact specimen which was not unloaded.The J-resistance curve rises continuously, the CTOA resistancecurve starts with a high value and lowers to aconstant value, while the T resistance curve is of the formof Fig. 3, except numerically different since the material isdifferent. The unload/reload specimen was analyzed threedifferent times in an application phase prediction to observewhich resistance curve can capture the history dependentdamage caused by the unload/reload cycle. Thepredictions are compared with experimental results in Fig.4. Figure 4 shows that the crack growth reinitiation afterthe unload/reload cycle was accurately predicted using theT methodology since the history of crack growth andunloading is accounted for with T . The analyses based onJ or CTOA yielded nonconservative predictions. While thisis a simple example, it was the ®rst which clearly illustratedthe advantages of T compared with the othercurrently accepted fracture parameters. It should bementioned that J and CTOA are still widely used in manyindustries to predict elastic-plastic crack growth. It is timethat T methodology begins to work its way into the applications'arena.5.2Example 2 ± Aircraft fuselage multi-site damage predictionsThe example summarized above represented the ®rst developmentof a practical application procedure and veri-375Fig. 4. Experimental and analytical loadversus crack-growth curves for A508 steel