Nonlinear Instability Modeling of Functionally Graded Sandwich Sanders’ Arches on Partial Hardening/Softening Elastic Foundations

This study investigates the instability and collapse behavior of sandwich arch structures with functionally graded material (FGM) face sheets and a metal core. The FGM domain is homogenized using the Mori–Tanaka approach. The third-order shear deformation theory is employed to approximate the in-plane displacement field, incorporating geometrically nonlinear kinematics and Sanders’ assumptions within the strain relations. Elastic constitutive laws are applied to the constituent materials, and a partial nonlinear elastic foundation is modeled beneath the arch. The Chebyshev Ritz method, known for its efficiency with discontinuous structures and rapid convergence, is utilized to derive the equilibrium equations. The nonlinear equations governing the arch’s response to a lateral uniform patch load are solved using the arc-length method. The snap-through instability characteristics are analyzed, demonstrating how the nonlinear elastic foundation’s coefficients influence the arch’s behavior, transitioning from hardening to softening. Finally, the collapse of the structure is examined, particularly when the softening response dominates. Understanding the stability and collapse of such advanced arch structures is crucial for applications in aerospace, civil engineering, and micro-electromechanical systems (MEMS). These structures are increasingly used in wing designs, bridge construction, and sensor technologies, where precise control of deformation and load-bearing capacity is paramount. The analysis of FGM sandwich arches with nonlinear foundations enables the optimization of structural designs for enhanced performance under extreme loading conditions, contributing to the development of more resilient and efficient engineering systems.