Inward buckling of spiral layered structures subject to volumetric expansion and external constraint

Abstract

Spiral layered structures subjected to thermal or diffusion-induced expansion often develop inward collapse, yet predictive models that connect geometry, confinement, and contact remain limited. Here we formulate a planar beam-based theory for an Archimedean spiral confined by a rigid circular ring, in which eigenstrain is converted into compressive circumferential resultants through kinematic constraint and unilateral contact. Using a variational framework, we derive spiral-path Föppl–von Kármán–type governing equations including a one-sided Winkler penalty representation of wall contact. A Rayleigh–Ritz energy estimate is then used to obtain closed-form scaling relations for the onset of instability. As an analytical benchmark, we reduce the spiral to the zero-pitch limit and recover a circular ring model, yielding mode-dependent critical compressive resultants for detached and fully contacting pre-buckling states. Complementary Dynamic/Explicit finite-element simulations under prescribed thermal expansion validate the theory and provide robust onset identification via internal-energy release, kinetic-energy bursts, and Fourier-mode growth. Finally, we study physically realistic multi-turn spirals, where turn-to-turn self-contact produces intrinsic kinematic locking without artificial end constraints, and we nondimensionalize the parameter space using the inner radius as the reference length to characterize the response in terms of thickness and turn number. The results establish a unified eigenstrain-to-compression mechanism for inward buckling, quantify the roles of confinement and contact state, and provide dimensionless design trends for layered spiral structures.