A polar-symmetric elastic diffusion problem is considered for an orthotropic solid multi-component cylinder under the action of an external uniformly distributed pressure over the surface. The applied loads initiate a mass transfer, which affects the cylinder’s stress-strain state. A coupled system of differential equations of elastic diffusion in a cylindrical coordinate system is used as a mathematical model. The model considers relaxation diffusion effects, implying finite velocities of diffusion fluxes propagation. We solved the problem by the equivalent boundary conditions method, which consists of the fact that first some auxiliary problem is considered, the solution of which is known. This problem differs from the original problem only in the boundary conditions. Then a relation connecting the right-hand sides of the boundary conditions of both problems is constructed. The specified ratio is an integral equation, the solution of which is sought using quadrature formulas. From this equation, the right-hand sides of the boundary conditions of the auxiliary problem are found. As a result, the solution to the original problem is found in convolutions of the Green’s functions of the auxiliary problem with the functions obtained by solving the above integral equation. The equivalent boundary conditions method was developed for initial-boundary value problems, the solution of which cannot be obtained by the method of separation of variables. For unsteady problems, it is semi-analytical; in stationary and static problems, it allows one to obtain a solution in an analytical form. Article considers calculus example based on a three-component material in which two components are independent. The study of the mechanical and diffusion fields interaction in a solid orthotropic cylinder is done. The limit transitions to static mechanodiffusion processes, as well as to classical models of elasticity, are investigated. The influence of relaxation effects on the kinetics of mass transfer is modeled in the continuum. The research results are presented in analytical and graphical forms.