Theory of vibrations of carbon nanotubes like flexible micropolar mesh cylindrical shells taking into account shift

A theory of nonlinear dynamics of a flexible single-layer micropolar cylindrical shell of a network structure is constructed. The geometric nonlinearity is taken into account by the model of Theodor von Karman. We consider a nonclassical continuum shell model based on the Cosserat medium with constrained particle rotation (pseudocontinuum). It is assumed that the displacement and rotation fields are not independent. An additional independent material length parameter associated with the symmetric tensor of the rotation gradient is introduced into consideration. The equations of motion of the shell element, boundary and initial conditions are obtained from the variational principle of Ostrogradskii – Hamilton on the basis of kinematic hypotheses of the third approximation (Peleha – Sheremetyev – Reddy), allowing to take into account not only the rotation, but also the curvature of the normal after deformation. It is assumed that the cylindrical shell consists of n families of edges, each of which is characterized by an inclination angle with respect to the positive direction of the axis directed along the length of the shell and the distance between neighboring edges. The shell material is isotropic, elastic, and obeys Hooke's law. A dissipative mechanical system is considered. As a special case, the system of equations of motion for Kirchhoff – Love's micro-polar reticulated shell is presented. The theory constructed in this paper can be used, among other things, for studying the behavior of CNTs under the action of static and dynamic loads.