Crack Shape Parameter
Flaw types that often go undetected in metallic pressure vessels are the «urface and embedded flaws. The flaw size required to cause fracture at a given applied stress level is called the critical size. If the vessel contains an initial flaw which exceeds the critical size at the proof-stress level, catastrophic failure can be expected during proof testing. Failure during service operation will occur when the initial flaw is less than the critical size at the proof-stress level, but grows with service usage until it reaches the critical size at the operating stress level. Pressure vessel leakage occurs when an initial flaw grows through the thickness of the vessel wall prior to reaching critical size.
In clastic stress fields, the critical sizes for surface and internal flaws depend on the planc-sfrain critical stress-intensity or fracture toughness values (Kjc) of the vessel materials, and the applied stress levels. If the critical flaw sizes are small with respect to the wall thickness of the pressure vessel, the vessel is termed "thick walled." If the critical sizes approach or exceed the wall thickness, the vessel is termed "thin walled."
The critical flaw sizes for surface flaws in uniformly stressed thick-walled vessels can be calculated using the following expression:
For small interna! flaws the same expression can be used except the 1.21 coefficient is decreased to unity.
Figure 1 shows the relationship between the flaw-shape parameter, Q, and the flaw depth-to-length ratio; figure 2 is a graphical representation of equation (1).
To predict critical flaw sizes (as well as failure modes and operational life) of thin-walled pressure vessels, it is necessary to know the stress intensity for flaws that become very deep with respect to the wall thickness. The stress-intensity solution shown in equation (1) for the semiclliptical surface flaw was derived by Irwin (ref. 4) and was found to be reasonably accurate for flaw depths up to about 50 percent of the material thickness. At greater depths, the applied stress intensity is magnified by the effect of the free surface near the flaw tip. This means that in thin-walled vessels, the flaw-tip stress intensity can attain the critical value (i.e., the Kjc value) at a flaw size significantly smaller than that which would be predicted using equation (1).
Flaw-shape parameter, Q
Flaw-shape parameter, Q
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Proof stress
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uJira-omla/Qli,/i a
F'gure 2. - Applied stress vs critics! flaw size.
Kobayashi and Smith developed approximate solutions for deep surface flaws that are very long with respect to their depth (i.e., small a/2c values) and for semicircular surface flaws (i.e., ,a/2c - 0.5), respectively (refs. 5 and 6). Results of their solutions are shown in terms of a stress-intensity magnification factor, Mj^, versus.a/t in figure 3. Reference 7 shows an estimate made by NASA/MSG of how varies as a function of a/2c between values of a/2c of 0 and 0.5. The Mj^. factor is applied to the original Irwin equation to obtain the stress intensity for deep surface flaws. The magnification reaches a maximum value of less than 10 percent for semicircular flaws, whereas there is an increase of about 60 percent for flaws having smaller values of a/2c.
Experimental data obtained on several materials with varying flaw sizes and flaw shapes appear to provide a fair degree of substantiation of the available approximate solutions (ref. 8). An exact numerical solution for deep, semielliptical, surface flaws with varying values of a/2c is under development, and additional experimental investigations are being performed.
To illustrate the effect of the deep-flaw stress-intensity magnification on predicted critical flaw sizes, it is convenient to assume that the vessel contains flaws which are long with respect to their depth. When the flaw-shape parameter, Q, is approximately equal to unity (i.e., for long flaws), the flaw size can be described in terms of the flaw depth, a. A predicted critical flaw-size curve (obtained using Kobayashi's M^ curve) for
Flaw-depth-to-wall thickness ratio, a/t
Figure 3. - Stress-intensity magnification factors for dssp surface flaws.
Flaw-depth-to-wall thickness ratio, a/t
Figure 3. - Stress-intensity magnification factors for dssp surface flaws.
a typical tank material and wall thickness is shown in figure 4. Also shown for comparison is the critical flaw-size curve for the same material in a thick-walled vessel. The curve for the thin-walled vessel is characterized by a significant reduction in failing stress at a given flaw size as compared to that for the thick-walled vessel. The life and potential failure modes of these thin-walled vessels are schematically illustrated in figure 5. The failure mode for thin-walled vessels can be complete fracture if the critical flaw depth is less than the wall thickness at the operating stress level (figure 5A). Figure 5B illustrates the case where the critical flaw depth is greater than the wall thickness at the operating stress level and the resulting failure mode is leakage.
From equation (1) it is apparent that to predict the critical sizes for surface and internal flaws it is necessary to know the pla o-strain fracture toughness (Kjc) values for the vessel materials (i.e., parent metal, welds, etc.). In heavy-gage, high-strength materials or in thin-gage materials that are relatively brittle, it is generally a straightfoiward task to obtain Kic values from laboratory tests. Several types of test specimens are used to measure Kjcvalues. These include fatigue-cracked bend specimens, surface-flawed specimens, crack-line loaded specimens, center-cracked and edge-cracked sheet specimens, and fatigue-cracked round notched-bar specimens. Testing requirements, limitations, advantages, and disadvantages of these various types of test specimens are discussed in considerable detail in references 2 and 3.
Figure 4. - Critical flaw-size curves at L02 tempsrature for 221S-T87 aluminum
Figure 4. - Critical flaw-size curves at L02 tempsrature for 221S-T87 aluminum
- Applied stress, O
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Minimum vessel strength
Minimum vessel strength
For predicting critical flaw sizes in aerospace pressure vessels, the surface-flawed specimen has probably been the most widely used. However, the fatigue-cracked bend specimen has the distinct advantage of being the only test specimen for which a detailed proposed recommended practice has been published by the American Society for Testing Materials (ref. 9).
In thin-gaged materials with moderate-to-high toughness, as'weii as all oilier situations where the fracture stress levels exceed the yield strength, it is necessary to obtain critical flaw-size data empirically. This was generally accomplished by testing a series of surface-flawed specimens with thickness equal to the pressure-vessel wall thickness and having various initial flaw sizes. Examples of such specimen tests are included in references 10 and 11. Also, an example of such test data is shown in figure 6. These data were obtained from reference 12.
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