Deformation of elastic plates under configuration-dependent loading

Modeling the deformation of low-dimensional elastic structures under complex loading remains a long-standing challenge in engineering science, largely because conventional analyses treat external loads as configuration-independent, which can conflict with practical observations. A landmark study by Chen and Yuan (2025) introduced game theory and the configuration-dependent loading (CDL) mechanism to predict membrane deformation under liquid self-weight, demonstrating that load redistribution induced by deformation can fundamentally alter equilibrium shapes. However, extending this insight to bending-dominated structures is nontrivial: the membrane framework neglects bending stiffness and typically requires prescribed initial stresses. Here we generalize the CDL concept to elastic plates in pure bending by incorporating bending stiffness and enforcing liquid volume conservation, which yields a linear integro-differential formulation capturing the two-way coupling between plate deformation and liquid redistribution. Closed-form solutions are derived for simply supported and clamped plates under three equilibrium configurations. The equilibrium morphology is governed by a single dimensionless coupling parameter quantifying the competition between hydrostatic loading and bending stiffness, and a critical threshold emerges that separates plate-dominated and liquid-dominated regimes. Notably, the threshold equilibrium profile coincides for the fourth-order plate model and the second-order membrane model of Chen and Yuan (2025) revealing a universal CDL balance mechanism across structural orders. Experiments validate the theoretical predictions and show pronounced deviations from classical, configuration-independent load models in the strong-coupling regime, establishing CDL as a general structure–liquid interaction mechanism for engineering applications.