© 2021, The Brazilian Society of Mechanical Sciences and Engineering.Nanotubes are the most important structural elements of nanomaterials used in nanotechnological applications and are also used in the production of nanocomposite materials. In the reinforcement (in the fibers) of the structure of composite materials, the initial curvature occurring due to structural reasons or technological processes causes the appearance of self-balancing stresses. The magnitudes of these stresses can lead to exceeding of the safety limits of the material. Therefore, theoretical investigation of the mechanical behaviors of the material, under tension and compression in the direction of the strengthening (fibers), is important for engineering. In this study, composite materials containing the double-walled nanotube are investigated in the scope of the piecewise homogeneous body model with the use of the geometric nonlinear exact equations of the three-dimensional theory of elasticity. The investigation involves stress analysis on the outer surface of the outer layer of the carbon nanotube (CNT) and on the matrix intersection, and non-ideal contact conditions between CNT and the matrix are used. Van der Waals forces which exist between the CNT walls are considered. The results include the impact of the geometric linearity and geometric nonlinearity on the parameters of the problem at the interface of the study. Numerical results on the influence of the problem parameters on the stress distribution are presented and discussed. The effect of van der Waals force on the stresses is shown by comparison with the case where the interlayer gap is disregarded and full contact between the CNT is assumed. It is seen that the maximum absolute values of the stress increase significantly when there is no van der Waals force. In addition, it has been observed that the values of the stress change monotonically with the increase of the thickness of the nanotube. Moreover, it is established that the increase of elasticity constants significantly affects the stresses.