Fig. 1: Buckling of a magnetic shell under combined pressure and magnetic loading.

a Photograph of the experimental apparatus: (1) a magnetic shell is positioned in between (2) a set of Helmholtz coils and depressurized using a pneumatic-loading system, comprising (3) a syringe pump and (4) a pressure sensor. The Helmholtz coils are driven by the current I in the direction shown in b, output from (5) a DC power supply . b Computed field of the vector (arrow) and magnitude (contour line) of the magnetic flux density \({\hat{{\bf{B}}}}^{{\rm{a}}}\) generated by the coils, under current ||I|| = 1 A (Supplementary Note 2). The field is uniaxial and uniform in the central region with a flux density of B^a = B^ae₃. Given axisymmetry, the field is represented in the x–z plane (y = 0). Coordinates x and z are normalized by the coil radius R_c = 46.5 mm. c Photograph of the representative imperfect shell specimen (top left). Image of its cross-section (top right) scanned through x-ray μCT and the reconstructed 3D view (bottom, one quarter was artificially hidden to aid with visualization). Scale bars are 5 mm. d Loading curves of pressure, \(\overline{p}=p/{p}_{{\rm{c}}}\), versus volume change, \({{\Delta }}\overline{V}={{\Delta }}V/{{\Delta }}{V}_{{\rm{c}}}\), for the shell shown in c. The peak value represented by the open circle on each curve is the critical buckling pressure. The buckling test is performed at different levels of external flux density B^a. Inset: schematic of a shell subjected to a magnetic field B^a, which is opposite (left) or parallel (right) to the shell magnetization B^r.