A Variational Perturbation Method for kinematically constrained manifolds in multiscale structural dynamics

초록

This work develops a unified Variational Perturbation Method (VPM) for the dimensional reduction of heterogeneous continua to equivalent beam and plate theories. The central idea is to identify macroscopic state variables directly from the fluctuation spectrum of the underlying continuum. The fluctuation field is decomposed into a quasi-static corrector, which refines the constitutive description without enlarging the macroscopic state space, and homogeneous microeigenmodes, some of which may be promoted to independent macroscopic degrees of freedom. This corrector-promotion decomposition provides the organizing principle of the framework. The formulation adopts the same multiscale kinematic decomposition as classical two-scale approaches but treats the fluctuation space variationally: unresolved components are relaxed through a micro-equilibrium problem, while retained micro-modes define the macroscopic kinematic manifold. As a representative example, the Timoshenko rotation emerges as the promoted amplitude of the lowest admissible shear micro-eigenmode, and the classical shear correction factor kappa = 5/6 follows automatically from variational stationarity. The framework generates a hierarchy of structural models, ranging from Euler-Bernoulli and Timoshenko beam models to higher-order shear-deformable and micromorphic theories, within a single variational setting. The formulation is validated through laminated and sandwich composite plates under cylindrical bending (generalized plane strain) as well as periodic metamaterials. For laminated plates in this configuration, frequency prediction errors are reduced to below 8% (VPM-based higher-order models) in the moderately thick regime, compared with errors exceeding 60% for classical lamination theory. For high-contrast sandwich plates, errors exceeding 100% in classical theories are reduced to accurate predictions across the practical slenderness range. For periodic metamaterials, micro-mode promotion extends the wavelength-valid domain by approximately 25%, delaying artificial branch crossings and improving dispersion predictions beyond classical homogenization limits. Because the energy functional is not truncated order-by-order, positive definiteness holds within the admissible variational manifold under the coercivity assumptions of the underlying elasticity tensor. The proposed framework provides a systematic variational pathway from microscale fluctuations to macroscopic structural theories at finite slenderness.

키워드