Induction heating has numerous technical advantages like high power densities enabling rapid temperature increases, direct heat generation within the target material, and the absence of contamination. Designing induction heating systems requires accurate modeling of magnetic and temperature fields, mechanical displacements, as well as their interaction effects. We focus on two critical aspects of induction heating: mechanical deformations caused by thermal expansion and the magnetic non-linearity of the heated material. Mechanical deformations are particularly significant for thin structures, as they are prone to thermal buckling. Large deformations alter the magnetic configuration, influencing both the temperature distribution and the mechanical behavior. Moreover, the magnetic material non-linearity prevents straightforward electromagnetic modeling in the frequency domain. To address these challenges, we propose a modeling strategy based on iteratively coupled finite element models for the different sub-problems. The stationary, non-linear eddy current problem is solved directly in the frequency domain using a harmonic balance approach. The heat equation with a convective term accounting for sheet motion is used to compute the temperature field. To compute the mechanical deformation of the thin sheet we use a computationally efficient shell description based on Kirchhoff-Love shell theory. Mechanical deformations are accounted for in the magnetic and heat problems by a mesh adaption scheme. Our results demonstrate the significance of accounting for non-linear material properties and large deformations in the system analysis. The parameter study conducted with the developed FEM algorithm illustrates the behavior of the model, highlighting the impact of the non-linear effects. The findings are compared against our analytical closed-form approximate solution.