Axisymmetric static and dynamic buckling of hollow microspheres

• Buckling of microballoons used in syntactic foams is theoretically studied.
• Predictions of Donnell, Sanders–Koiter, and Teng–Hong theories are compared.
• Full non-linear buckling analyses are compared to simplified treatments.
• Analytical predictions on glass particles are validated with finite element results.
• Sanders–Koiter theory yields accurate predictions at a limited computational cost.

Syntactic foams are particulate composites that are obtained by dispersing thin hollow inclusions in a matrix material. The wide spectrum of applications of these composites in naval and aerospace structures has fostered a multitude of theoretical, numerical, and experimental studies on the mechanical behavior of syntactic foams and their constituents. In this work, we study static and dynamic axisymmetric buckling of single hollow spherical particles modeled as non-linear thin shells. Specifically, we compare theoretical predictions obtained by using Donnell, Sanders–Koiter, and Teng–Hong non-linear shell theories. The equations of motion of the particle are obtained from Hamilton׳s principle, and the Galerkin method is used to formulate a tractable non-linear system of coupled ordinary differential equations. An iterative solution procedure based on the modified Newton–Raphson method is developed to estimate the critical static load of the microballoon, and alternative methodologies of reduced complexity are further discussed. For dynamic buckling analysis, a Newmark-type integration scheme is integrated with the modified Newton–Raphson method to evaluate the transient response of the shell. Results are specialized to glass particles, and a parametric study is conducted to investigate the effect of microballoon wall thickness on the predictions of the selected non-linear shell theories. Comparison with finite element predictions demonstrates that Sanders–Koiter theory provides accurate estimates of the static critical load for a wide set of particle wall thicknesses. On the other hand, Donnell and Teng–Hong theories should be considered valid only for very thin particles, with the latter theory generally providing better agreement with finite element findings due to its more complete kinematics. In this context, we also demonstrate that a full non-linear analysis is required when considering thicker shells, while simplified treatment can be utilized for thin particles. For dynamic buckling, we confirm the accuracy of Sanders–Koiter theory for all the considered particle thicknesses and of Teng–Hong and Donnell theories for very thin particles.