Closed-form solutions to the static transverse deformation of a spider orb-web

This paper deals with the study of the Dirichlet and Poisson problem for the infinitesimal transverse deformation of a spider orb-web in statical regime. The mechanical model is a pre-stressed membrane and attention is focused on axially-symmetric webs formed by radial threads connected with circumferential threads belonging to concentric circles. The governing equation is an elliptic scalar equation with variable coefficients having a singularity at the center of the web. Under suitable assumption on the tensile pre-stress acting in the referential configuration, closed-form series representation of the solution to the Dirichlet and Poisson problem are derived. To the authors's knowledge, these are the first closed-form solutions to the statical deformation of a continuous membrane model of spider orb-web. Possible applications of the obtained results will concern both the analysis of the most relevant mechanical parameters that rule the response of the web and the estimate of the initial pre-stress via approaches of inverse analysis.