Variational formulation, asymptotic analysis, and finite element simulation of wrinkling phenomena in modified plate equations modeling biofilms growing on agar substrates

The expansion of a bacterial biofilm on an agar substrate is modeled as a system of Föppl-von Kármán equations modified to include growth and coupling to a viscoelastic substrate. Analysis shows that wrinkles appear on the biofilm as a result of growth incompatibility and their frequency increases due to interaction with the agar layer. Simple cases of homogeneous radial and azimuthal growth are approximated by cone and corona solutions of the Monge-Ampère equation that are corrected by corner and boundary layers. A weak formulation of the problem allows us to express in-plane elastic strains and Airy potential solely in terms of the vertical displacement. We have developed a numerical method based on finite elements and simulated biofilm deformation for wide spectra of different growths. For heterogeneous growth, we find wrinkled patterns that are combinations of cones and coronas.