Microbuckling of a doublewalled carbon nanotube (DWCNT) in an elastic (polymer) matrix is studied. The investigations are made within the scope of the piecewise homogeneous body model by utilizing the three-dimensional linearized theory of stability of deformable bodies. Flexural and axisymmetric microbuckling modes are considered. The DWCNT is modeled as concentrically-nested two circular hollow cylinders between which there is free space. It is assumed that on the inner surface of the outer tube (cylinder) and on the outer surface of the inner tube (cylinder) of the DWCNT full slipping conditions occur. At the same time, it is assumed that the difference between the radial displacements of the adjacent surfaces of the tubes resists with the van der Waals forces. On the interface between the matrix and DWCNT complete contact conditions are satisfied. Numerical results on the influence of the problem parameters on the critical deformation are presented and discussed. Also, numerical results related to the cases where the interlayer space is ignored and where full contact between the tubes is assumed are presented and compared with the mentioned results. In particular, it is established that full slipping between the tubes causes the values of the critical deformation to decrease significantly with respect to those obtained in the case where complete contact conditions occur between the tubes. Moreover, it is established that an increase in the values of the van der Waals forces also causes a decrease in the values of the critical compressing strain and the magnitude of this decrease depends on the thicknesses of the tubes of the DWCNT. (C) 2013 Elsevier Ltd. All rights reserved.