The paper addresses the problem of the compression of elastic cylindrical specimens under axial end loads. Approximate deformation models of various orders are developed for slender specimens by postulating that the displacement field admits an expansion in radial polynomials of different orders whose coefficients are unknown functions of the axial coordinate. The approximate models are derived from Lagrange’s variational principle. A reduced energy functional is obtained by integrating the three-dimensional elastic energy over the radial coordinate. Applying the Kantorovich variational method, the problem is then reduced to systems of ordinary differential equations with corresponding boundary conditions; for homogeneous bodies these systems exhibit constant coefficients that depend on Poisson’s ratio. Closed-form solutions are constructed for the simplified models, and both rod-like (axial) and boundary-layer solution modes are identified. The proposed models are verified against finite-element simulations for both constant and spatially heterogeneous Lame parameters, and a series of computational experiments demonstrates that the models may be used for slender specimens to estimate their accuracy and to support the solution of diverse applied problems.