A lightweight design of turbofan disc for an aero-engine has failed during a run-up test. The disc can be taken to be of uniform thickness 125mm and has a central hole of diameter 300mm and an outer rim diameter of 700mm. The outer rim was loaded with 40 blades, each of mass 5kg evenly pitched around the periphery at an effective radius of 400mm. The disc had previously survived a run-up test with no blades attached. The titanium alloy used in the new design was expected to have a uniaxial tensile yield stress of 1250MPa, Poisson's Ratio 0.32, density 4500kgm^-3, plane strain fracture toughness 100MPam and Young's Modulus 116GPa. These properties are ideal values, but it is known that, if the forging and heat-treatment conditions are not correct, local areas may have a much-reduced fracture toughness, as low as 15 MPam, but with little effect on the other properties.

3.1 a) Assuming the Tresca Criterion and an elastic-perfectly plastic constitutive model, find the rotational speeds at which yielding first occurs:

• for the disc without blades
• for the fully bladed disc

3.2 b) Plot the elastic hoop and radial stress distributions in the disc at its design rotational speed of 500sec^-1.

note: Sketch should include relevant values of the stresses in MPa.

3.3 One possible failure mode for the disc is the propagation of radial cracks emanating from the internal diameter of the disc. Assuming the crack to behave like a single-edged through-thickness notch (Y=1.12), calculate the critical defect size at the design speed for the disc with and without the blades. You should first verify that the crack tip is in plane strain.

3.4 Identify the part of the disc that is most susceptible to propagation of circumferential cracks and determine the relevant opening stress at the design speed for the disc with and without blades. Taking Y = 1 for a through-thickness circumferential crack (of length 2a), again, calculate the critical defect size at the design speed for each of the two cases