We investigate H-convergence for a fourth-order system of one-dimensional equations modeling the bending of anisotropic elastic rods with axially varying coefficients. The model is obtained via dimension reduction of three-dimensional linearized elasticity with variable cross-sections and leads to a coupled system for transversal displacements governed by a matrix-valued bending tensor.
We introduce a notion of H-convergence adapted to this fourth-order rod system and prove a compactness result. Owing to the one-dimensional structure of the domain, the analysis simplifies compared with higher-dimensional problems: strong convergence of the stress sequence replaces the compensated compactness arguments typically required in multidimensional settings.
As a consequence, we show that the H-limit of any bounded sequence of uniformly elliptic bending tensors can be identified explicitly and is given by the harmonic mean, even without periodicity assumptions. In addition, the key structural properties of H-convergence, such as locality, preservation of ellipticity bounds, and independence of boundary conditions, remain valid in this framework. These results provide a homogenization theory for fourth-order rod operators and yield an explicit characterization of the effective bending tensor.
Possible applications are in optimal design problems, where homogenization is a well-established tool for relaxation and was the original motivation of Tartar and Murat for introducing H-convergence.