Abstract Few researches have focused on elastoplastic mechanical performances of functionally graded plates and shells. In this paper, elastoplastic and plastic buckling behaviors of externally pressured cylindrical shells made from functionally graded materials are investigated by using Donnell shell theory. Either $J_2$ flow theory or $J_2$ deformation theory helps to found the constitutive relation of functionally graded materials. The material properties vary smoothly through the thickness, and a multi-linear hardening elastoplasticity is used in the analysis. The buckling governing equations are solved by Galerkin method, and the expression of the elastoplastic critical external pressure is given analytically. Numerical results from the present theory are derived by an iterative arithmetic developed in MATHEMATICA code. The theoretical critical loads of the present $J_2$ deformation theory are well verified by experimental and numerical results in literature. The elastic, elastoplastic, and plastic buckling regions of functionally graded cylindrical shells can be effectively distinguished by the present method, and various effects of the dimensional parameters, the power law exponent, and the elastoplastic material parameters are investigated.

Highlights • Investigation on the elastoplastic buckling of functionally graded materials (FGM) plates and shells is not reported in literature at least in our awareness. • J 2 flow theory and J 2 deformation theory in the case of buckling of elastoplastic FGM cylindrical shells under external pressure are compared. Results show enormous deviations as in the case of axially compressed isotopic cylindrical shells. • The theoretical results of the present J 2 deformation theory are well verified by experimental and numerical results in literature. By using J 2 deformation theory, the elastic, elastoplastic, and plastic buckling districts of FGM cylindrical shells can be effectively distinguished. • The plastic critical external pressure of FGM cylindrical shells can be much lower than its elastic counterpart due to the plastic deformation, and the critical external pressure decreases with the increase of the radius-to-thickness ratio and the length-to-radius ratio of the shells, as well as the power law exponent or the volume fraction of the metallic constituent. Meanwhile, it increases with the tangent modulus and the yield limit of the metallic constituent and the stress transfer parameter. Among these factors, the dimensional parameters, the power law exponent, and the tangent modulus of the metallic constituent effect more apparently on the critical load, and the buckling mode is affected primarily by the dimensional parameters, but seems insensitive to change of the power law exponent and the elastoplastic material parameters of FGMs.