Problem Sheet 5: Fracture and fatigue

The numbers in square brackets refer to the number of the problem in your textbook.

1. Assume a crack length of size lesser than clim to exist in the wall of a cylindrical thin-walled pressure vessel of radius R and thickness t and containing a pressure p. What is the condition for design such that the stresses are everywhere less than that required to make a crack of size clim propagate? What is the material index? Is this design fail-safe if a material is chosen according to the criterion? What is yield-before-break criterion? What is the material index for yield-before-break criterion? What is leak-before-break criterion? What is the material index for such a criterion? [Hint: Section 10.4 of your textbook]
2. Consider the engine of a Formula 1 racing car which has to tolerate speeds up to about 15000 rpm. Consider the connecting rods of such a high-performance engine. If you want a light, strong material that can undergo such high-cycle fatigue without failure, assuming uniform cross-sectional area and length, given a cyclic load of $\pm$ F, find the material index. [Hint: section 10.4 of your textbook]
3. [E10.1] Supersonic wind tunnels store air under high pressure in cylindrical pressure vessels — the pressure, when released, produces hypersonic rates of flow. The pressure vessels are routinely proof tested to ensure that they are safe. If such a cylinder, of diameter 400mm and wall thickness 20 mm, made of a steel with a fracture toughness of 42 MPa.m1/2, survives a proof test to 40 MPa, what is the length of the largest crack it might contain?
4. [E10.5] If you want to support a precision optical system (laser metrology equipment, for instance) on a stable platform, you put it on a granite slab supported on end plinths to bring it to working height. Granite can be ground to a flat surface and is thermally very stable and hard wearing. The granite chosen for one such table has a fracture toughness of 0.9 MPa m1/2 and is known, from NDT procedures, to contain internal cracks up to 5 mm in length. If the table is 2 m long and 1m deep, simply supported at its ends and must carry a uniformly distributed load of 2000N on its upper surface, what is the minimum thickness the slab must have? Include the self-weight of the slab in the analysis. Assume that at least one of the cracks will lie in the part of the beam that carries the highest tensile stress—that is, at the lower surface. The density of granite is 2700 kg / m3.
5. [E10.8] An adhesive has a toughness Gc = 100 J/m2 and a shear strength $\sigma_s$ = 0.1 MPa. What must the dimensions of the bonded area of a lap-joint be if it is to carry an in-plane tensile Ft of 100 N but allow peeling at a force Fp of 5N?
page revision: 3, last edited: 04 Feb 2009 09:47