Asymptotic multiscale approach for the flexoelectric response of one-dimensional composite materials with imperfect interfaces

This work presents a multiscale asymptotic framework for the characterization of the electromechanical response of one-dimensional flexoelectric composites with imperfect interfaces. The approach is formulated within a hierarchical micro-meso-macro setting, in which a heterogeneous layered medium with thin adhesive layers is first described at the microscale. Through an asymptotic analysis of the interphase, the model is reduced to an equivalent mesoscale description featuring a non-classical imperfect interface with flexoelectric spring-type transmission conditions. At the macroscale, a two-scale asymptotic homogenization procedure is then employed to derive the effective constitutive coefficients of the composite. The proposed formulation captures the coupled influence of flexoelectricity, strain-gradient elasticity, and interfacial compliance on the overall material behavior. In particular, the presence of intrinsic length scales associated with higher-order electromechanical effects gives rise to size-dependent phenomena, including an apparent stiffening of the interface when strain-gradient contributions become dominant. The resulting homogenized model provides a consistent and analytically tractable tool to quantify how microstructural features and interface properties influence the macroscopic response of layered flexoelectric composites. Numerical examples are presented to validate the formulation and to illustrate the role of interface parameters and length-scale effects on the effective properties. The results confirm the convergence of the imperfect interface model to the perfect-contact limit and highlight the key role of the length-scale parameter in controlling the competition between local and nonlocal electromechanical mechanisms.