## 1 Introduction  
Three-dimensional graphene foams (3D-GrFs) [1; 2; 3], unlike conventional polymeric open-cell foam materials, are a very new kind of nanofoam materials with three-dimensionally interconnected constituent graphene flakes. These 3D-GrFs have been synthesized by some approaches such as hard templating [4], sol-gel reaction [5], solution processing [6], powder metallurgy [7], 3D printing [8], freeze drying [9], hydrothermal reduction [10], and chemical vapor deposition [11]. These 3D-GrFs possess excellent properties of good electrical conductivity, high energy dissipation, super low density, superelasticity, and electrochemical stability [12; 13; 14; 15]. These combined properties enable 3D-GrFs to be applied in electronics and energy storage/conversion systems [16; 17], gas detection [15], sorbent materials [18], stretchable electronics [11], and so on.  
In contrast to many experimental studies on the mechanical property of 3D-GrFs, few computational and theoretical investigations have been conducted to evaluate the relationship between the macro-mechanical characteristics and the intrinsic micro/nanostructures. [PERSON] et al. [19] investigated and discussed the effects of graphene layer number and 3D-GrF defects on the electrical and mechanical properties of 3D-GrFs. Based on the coarse-grained molecular dynamics simulation, the constitutive relation between 3D-GrFs and microscopic deformation mechanisms were investigated by [PERSON] et al. [20]. The mechanical properties in both tension and compression of 3D-GrFs at macro and nanoscales were evaluated by [PERSON] et al. [21]. [PERSON] et al. [3] stated that 3D-GrFs have an exceptionally high tensile strength and they are 10 times as strong as mild steel. [PERSON] et al. [22] studied the dynamic mechanical properties of 3D-GrFs by nanoindentation technique.  
As material size changes from macroscale to nano/microscale, size effect on mechanical properties of materials should be taken into account. It is reported that size effects in micro/nanostructures have been experimentally observed [23; 24; 25]. Due to the exclusion of size effect in the classical theory (CT), several non-classical theories were developed for analysis of micro/nanostructures. One of the non-classical theories incorporating size effect is the modified couple stress theory [26; 27]. This theory has been employed in many aspects to interpret the size effect in micro/nanostructures [28; 29; 30; 31; 32; 33; 34].  
Micro/nanoshells are an important and widely used form of micro/nanostructures in various engineering fields. [PERSON] et al. [35] analyzed the dynamic stability and axial buckling of functionally graded cylindrical microshells. Based on the strain gradient elasticity theory, [PERSON] et al. [36] developed a shear deformable functionally graded microshell model. [PERSON] and [PERSON] [37] studied the nonlinear dynamical characteristics of doubly curved shallow microshells. [PERSON] et al. [38] investigated the influences of various temperature distributions on the vibration of functionally graded rotating cylindrical microshells. Afterward, the buckling behavior of functionally graded sandwich microshells under axial loads was studied by [PERSON] and [PERSON] [39]. [PERSON] et al. [40] studied nonlinear vibrations of cylindrical nanoshells conveying fluid in the framework of the surface stress elasticity theory.  
To date, no work has been published on the mechanical properties of 3D-GrF microshells. In this paper, we aim to conduct size-dependent buckling and free vibration analysis of 3D-GrF cylindrical microshells. The couple stress theory and [PERSON]'s thin shell theory are used to derive the governing equations. After that, natural frequencies and critical buckling loads are solved by adopting the Navier method and Galerkin method. Finally, the results are illustrated for different parameters.