Asymptotic limits and wrinkling patterns in a pressurised shallow spherical cap

• Asymmetric bifurcation problems for pressurised shallow spherical caps are re-visited for the case when the applied pressure acts either internally or externally.
• Asymptotic formulae for both the critical pressure and the number of wrinkles are derived in both cases studied.
• Comparisons with direct numerical simulations confirm the accuracy of the analytical results.

The asymmetric bifurcation problem for a shallow spherical cap is examined. The applied pressure can act either external or internal to the cap and both cases are treated here. Assuming a non-linear axisymmetric basic state, the linearised bifurcation equations for the pressurised shell are investigated in the limit when the thickness of the cap is much less than the maximum rise of the shell mid-surface. Within this regime the wrinkling patterns in both cases are confined to a narrow zone near the edge of the shell, making it possible to solve asymptotically the corresponding equations and derive analytical predictions for both the critical pressure and the corresponding number of wrinkles. Some comparisons with direct numerical simulations are included as well.