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its weight becomes important. Aircraft bodies, rocket casing and liquid natural gas containers

are examples; they must be light, and at the same time they must be safe; that means that they

must not fail by yielding or by fast fracture. What are the best materials for their construction?

The following table summarizes the requirements

R

Function: pressure vessel

Constraints: must not fail by yielding; must not fail by fast fracture; diameter 2R and pressure

difference 4p specified

Objective: minimize mass m

Free variables: wall thickness t; choice of material

a. First write a performance equation for the mass m of the pressure vessel. Assume, for

simplicity, that it is spherical, of specified radius R, and that the wall thickness t (the

Pressure

difference

4p

free variable) is small compared with R. then the tensile stress in the wall is a =

ApR

2t

where 4p, the pressure difference across this wall, is fixed by the design. The first

constraint is that the vessel should not yield-that is, that the tensile stress in the wall

should not exceed øy. The second is that it should not fail by fast fracture; this requires

that the wall stress be less than K₁/√c, where Kic is the fracture toughness of the

material of which the pressure vessel is made and c is the length of the longest crack

that the wall might contain. Use each of these in tern to eliminate t in the equation for

m; use the result to derive the two material indices:

for each case.

questions

M₁ = and M₁ =7

as well as the coupling relation between them. It contains the crack length, c.

b. Plot the charts required and the coupling equation onto the charts for two values of c:

one of 3 mm, the other of 10 µm. Identify the lightest candidate materials for the vessel

K₁

Fig: 1