Transformational model of deformation of a rod-strip with a section of double-sided fastening in a support element with specified displacements

A refined transformational mathematical model is proposed to describe the deformation process of a rod-strip having fixed and non-fixed sections along its length. It is assumed that the rod in the fixed section is connected to a support element, which has displacement components prescribed (known) at the points of connection with the rod, which makes it possible, in particular, to simulate the process of kinematic loading of the rod during tensile and compression tests. To describe the process of deformation of the non-fixed section of the rod, tangential displacements are approximated by a third-degree polynomial along the transverse coordinate, and deflection by a second-degree polynomial. In the fixed section, the approximations of the displacements that were assumed for the non-fixed section are transformed into other approximation functions along the transverse coordinate due to their compliance with the kinematic conditions of the two-sided connection with a support element with prescribed displacements. The conditions for the kinematic coupling of the fixed and non-fixed parts of the rod are formulated, taking them into account using the D’Alembert–Lagrange variational principle, the equations of equilibrium and motion of the marked parts, their corresponding boundary conditions, as well as the force conditions for coupling the fixed and non-fixed sections of the rod are obtained.