  Nanomechanical resonances of two-dimensional (2D) materials are sensitive
probes for condensed-matter physics, offering new insights into magnetic and
electronic phase transitions. Despite extensive research, the influence of the
spin dynamics near a second-order phase transition on the nonlinear dynamics of
2D membranes has remained largely unexplored. Here, we investigate nonlinear
magneto-mechanical coupling to antiferromagnetic order in suspended
FePS$_3$-based heterostructure membranes. By monitoring the motion of these
membranes as a function of temperature, we observe characteristic features in
both nonlinear stiffness and damping close to the N\'{e}el temperature
$T_{\rm{N}}$. We account for these experimental observations with an analytical
magnetostriction model in which these nonlinearities emerge from a coupling
between mechanical and magnetic oscillations, demonstrating that
magneto-elasticity can lead to nonlinear damping. Our findings thus provide
insights into the thermodynamics and magneto-mechanical energy dissipation
mechanisms in nanomechanical resonators due to the material's phase change and
magnetic order relaxation.
