(a) For the thin-walled spherical tank discussed in Design Example 8.1, on the basis of th...

(a) For the thin-walled spherical tank discussed in Design Example 8.1, on the basis of the criticalcrack size-criterion [as addressed in part (a)], rank the following polymers from longest to shortest critical crack length: nylon 6,6 (50% relative humidity), polycarbonate, poly(ethylene terephthalate), and poly(methyl methacrylate). Comment on the magnitude range of the computed values used in the ranking relative to those tabulated for metal alloys as provided in Table 8.3. For these computations, use data contained in Tables B.4 and B.5 in Appendix B.

(b) Now rank these same four polymers relative to maximum allowable pressure according to the leak-before-break criterion, as described in part (b) of Design Example 8.1. As before, comment on these values in relation to those for the metal alloys tabulated in Table 8.4.

Table 8.3 Ranking of Several Metal Alloys Relative to Critical Crack Length (Yielding Criterion) for a Thin-Walled Spherical Pressure Vessel

Table 8.4 Ranking of Several Metal Alloys Relative to Maximum Allowable Pressure (Leak-before-Break Criterion) for a Thin-Walled Spherical Pressure Vessel

Table B.4 Typical Room-Temperature Yield Strength, Tensile Strength, and Ductility (Percent Elongation) Values for Various Engineering Materials

aThe strength of graphite, ceramics, and semiconducting materials is taken as flexural strength.

bThe strength of concrete is measured in compression.

cFlexural strength value at 50% fracture probability.

Sources: ASM Handbooks, Volumes 1 and 2, Engineered Materials Handbooks, Volumes 1 and 4, Metals Handbook: Properties and Selection: Nonferrous Alloys and Pure Metals, Vol. 2, 9th edition, Advanced Materials&Processes, Vol. 146, No. 4, and Materials&Processing Databook (1985), ASM International, Materials Park, OH; Modern Plastics Encyclopedia ’96, The McGraw-Hill Companies, New York, NY; R. F. Floral and S. T. Peters, “Composite Structures and Technologies,” tutorial notes, 1989; and manufacturers’ technical data sheets.

Table B.5 Room-Temperature Plane Strain Fracture Toughness and Strength Values for Various Engineering Materials

aFor metal alloys and polymers, strength is taken as yield strength; for ceramic materials, flexural strength is used.

Sources: ASM Handbooks, Volumes 1 and 19, Engineered Materials Handbooks, Volumes 2 and 4, and Advanced Materials&Processes, Vol. 137, No. 6, ASM International, Materials Park, OH.

DESIGN EXAMPLE 8.1

Material Specification for a Pressurized Spherical Tank

Consider a thin-walled spherical tank of radius r and thickness t (Figure 8.11) that may be used as a pressure vessel.

(a) One design of such a tank calls for yielding of the wall material prior to failure as a result of the formation of a crack of critical size and its subsequent rapid propagation. Thus, plastic distortion of the wall may be observed and the pressure within the tank released before the occurrence of catastrophic failure. Consequently, materials having large critical crack lengths are desired. On the basis of this criterion, rank the metal alloys listed in Table B.5, Appendix B, as to critical crack size, from longest to shortest.

(b) An alternative design that is also often utilized with pressure vessels is termed leak-beforebreak. On the basis of principles of fracture mechanics, allowance is made for the growth of a crack through the thickness of the vessel wall prior to the occurrence of rapid crack propagation (Figure 8.11). Thus, the crack will completely penetrate the wall without catastrophic failure, allowing for its detection by the leaking of pressurized fluid. With this criterion the critical crack length ac (i.e., one-half the total internal crack length) is taken to be equal to the pressure vessel thickness t. Allowance for ac = t instead of ac = t/2 ensures that fluid leakage will occur prior to the buildup of dangerously high pressures. Using this criterion, rank the metal alloys in Table B.5, Appendix B, as to the maximum allowable pressure.

For this spherical pressure vessel, the circumferential wall stress s is a function of the pressure p in the vessel and the radius r and wall thickness t according to

For both parts (a) and (b), assume a condition of plane strain

Figure 8.11 Schematic diagram showing the cross section of a spherical tank that is subjected to an internal pressure p and that has a radial crack of length 2a in its wall.