When aerogel materials are subjected to extreme compressive strains reaching 50% or 80%, their macroscopic stress-strain behavior deviates from linear Hookean elasticity, exhibiting nonlinear regimes: an initial elastic domain, an extended stress plateau spanning large strains, and a final densification region driven by pore collapse. During cyclic compression and unloading, the stress-strain curves form large hysteresis loops, demonstrating mechanical energy dissipation per cycle. Formulating constitutive models for the nonlinear elastic buckling and energy dissipation of aerogel nanonetworks represents an area in solid mechanics.

The origin of this nonlinear elasticity lies in the elastic buckling of nanoscale struts and walls within the three-dimensional network under mechanical loads. Unlike dense metals or ceramics that undergo plastic yielding or dislocation slip, the nanoparticle chains or thin-wall cells of aerogels undergo elastic instability and periodic folding when subjected to axial loads. This microscopic wall buckling relieves localized mechanical stresses without inducing brittle chemical bond rupture.

To quantify this multi-scale nonlinear mechanical response, mechanicians formulate advanced constitutive models based on statistical micromechanics and continuum mechanics. These constitutive frameworks treat aerogels as statistical assemblies of interconnected micro-struts or thin-walled cells with distributed geometrical properties. Under macroscopic strain, total strain energy density is partitioned into elastic deformation energy, strut buckling strain energy, and frictional dissipation energy generated by contact between adjacent pore walls.

By introducing nonlinear constitutive differential equations incorporating power-law scaling relationships with relative density and frictional dissipation parameters, researchers simulate the cyclic compressive response, energy absorption efficiency, and elastic recovery of aerogels. These constitutive models advance the understanding of nanomaterial mechanics, supporting the design of aerogels for impact absorption, aerospace protective shielding, and damping systems.