This question illustrates the dichotomy in fracture mechanics between high strength and high toughness.  Generally speaking, as strength level increases in an alloy class, the toughness decreases.  This has led to some problems for designers who have been schooled in yield orientated design.   Certain alloys, such as the maraging steels, combine high strength and high toughness through careful compositional and microstructural design.  Maraging steels are used in applications like undercarriage legs for aircraft, missile cases, cannon recoil springs and fan shafts in jet engines.  They can achieve strength/toughness combinations ranging from about 1500 MPa/120 MPam^½ to 2000 MPa/60 MPam^½.  Such steels typically have very high nickel, cobalt and molybdenum contents and very low carbon.  Their high strength and toughness derives from age hardening (precipitation of intermetallic compounds) of a low carbon, iron-nickel lath martensite matrix.  Further information on these steels can be found in:

ASM Handbook, Vol. 1 10^th edition (1990), Properties and Selection: Irons, Steels and High Performance Alloys, American Society for Materials, Materials Park, Ohio, pp.793-800.

This problem is the first one that uses hoop stress, calculated using stress analysis for a thin walled pressure vessel.  Pressurised vessels are commonly found in engineering practice and hence turn up regularly in fracture mechanics examples.  Thin walled theory is approximate, and can be used for ratios of diameter/wall thickness of > 10.  For ratios lower than this, the thick walled theory should be used.  The two relevant stress equations are given in example 4.4.1, which deals with fatigue crack growth.  In pressurised vessels with internal surface-breaking cracks, one has to combine the stress intensity factors arising from the hoop stress and the internal pressure (see the Theory card in example 2.16 for information on superposition of K values).

Thin and thick walled pressure vessel theory is given in the following website:

http://www.engin.brown.edu/courses/En175/Elasticity2/Elasticity2.htm

The problem should take around 15 minutes.

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You are involved in the design and manufacture of 6.6m diameter (D) rocket motor cases with a wall thickness (t) of 18.5 mm, and an operational pressure (p) of 6.6 MPa.  These components are presently manufactured from a Grade 200 maraging steel with a yield strength of 1515 MPa and K_1C = 136.5 MPam^½.  In order to save weight, a design engineer has proposed changing to a Grade 250 maraging steel with a yield strength of 1650 MPa and a plane strain fracture toughness value of 72.5 MPam^½, and has requested an fracture analysis of allowable defect size against failure stress.

Failure of the motor case can be assumed to occur from embedded elliptical defects orientated perpendicular to the hoop stress.   Typical elliptical defects resulting from the welding process can be detected by NDT, and are known to occur with sizes up to 5.5 mm by 35.5 mm.  Determine design data for these alloys of fracture stress against allowable defect size, over a range of major axis lengths from 20 mm to 50 mm.

Based on this data, make recommendations to the designer as to the appropriate material to use when welding defects are likely to be present.  Also determine whether the use of LEFM, based on K_1C, is likely to be conservative or not for the given dimensions of defects and motor case, i.e. whether plane strain conditions exist.

Hoop stress is given by pD/2t, and the stress intensity solution for an embedded crack, at the semi-minor axis position (crack depth a), is given below:

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Answer:  The Grade 200 steel is the best choice, and use of K_1C is conservative. 

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