In this paper, a simple example to illustrate what is basically known from the Gauss’ times interplay between geometry and mechanics in thin shells is presented. Specifically, the eigen-mode spectrum in spontaneously curved (i.e., up-down asymmetric) extensible polymerized or elastic membranes is studied. It is found that in the spontaneously curved crystalline membrane, the flexural mode is coupled to the acoustic longitudinal mode, even in the harmonic approximation. If the coupling (proportional to the membrane spontaneous curvature) is strong enough, the coupled modes dispersions acquire the imaginary part, i.e., effective damping. The damping is not related to the entropy production (dissipation); it comes from the redistribution of the energy between the modes. The curvature-induced mode coupling makes the flexural mode more rigid, and the acoustic mode becomes softer. As it concerns the transverse acoustical mode, it remains uncoupled in the harmonic approximation, keeping its standard dispersion law. We anticipate that the basic ideas inspiring this study can be applied to a large variety of interesting systems, ranging from still fashionable graphene films, both in the freely suspended and on a substrate states, to the not yet fully understood lipid membranes in the so-called gel and rippled phases.
