The paper deals with an optimization problem related to axisymmetric membrane shells made of homogeneous, linear and isotropic materials and subject to a cyclic internal pressure. The optimization problem considers the possible crack growth in the shell as a structural constraint, which is accounted for by the well-known Paris law, and hence obeys the Linear Elastic Fracture Mechanics (LEFM) principles. The design variables are the meridian shape and the thickness distribution along the axis of the shell. After recalling the general background on the mechanics of thin-walled shells, the optimization problem is stated, formulated and solved in closed form by means of Pontryagin's Principle, showing two LEFM-based design criteria capable to help the designer in selecting the desired elastic performance through which the shell undergoes. It is shown that the derived optimal solutions are ellipsoids with variable thicknesses, whose distributions depend on the mass constraint and the applied pressure. Nevertheless, the optimal shapes guarantee the uniformity of either the meridian or the hoop stress, and such uniformity has been verified within a numerical example by an accurate finite element analysis.
