The mechanics of anisotropic active plates with applications to cell alignment on curved substrates

We develop a continuum mechanics framework for active anisotropic plates within the Föppl–von Kármán limit, incorporating a preferential direction and inelastic active contractions in geometrically nonlinear plate theory. Through asymptotic expansion, we derive coupled equilibrium equations for plates with transversely isotropic and possibly inhomogeneous reinforcement undergoing spatially varying active contractions through their thickness. The framework highlights the coupling between material anisotropy and active deformations, with target curvatures that compete with geometric constraints. To demonstrate its capabilities, we apply the model to curvature-induced cell alignment, where substrate geometry, cytoskeletal anisotropy, and contractility interact to determine orientation. For cylindrical substrates, the model predicts a supercritical bifurcation in preferred orientation, from perpendicular to parallel through an oblique orientation, governed by the ratio of active contractility to substrate curvature. For ellipsoidal geometries, we capture stable parallel, perpendicular, and oblique configurations set by principal curvatures, whereas spherical substrates show no preferred alignment. These predictions qualitatively reproduce experimental observations across cell types, providing a mechanistic interpretation of the distinct behaviors of contractile epithelial cells and stiffer fibroblasts. As a further illustration, we analyze the buckling of active anisotropic rings, showing how reinforcement and contractility jointly modulate the instability threshold. More broadly, the model applies to thin fiber-reinforced active structures arising in soft robotics, morphogenesis, and tissue engineering