Homogenization of Loaded Woven Membranes. The Case of Concentrated Point Forces

Abstract

This paper investigates new approaches to the averaging problem for loaded singular structures (geometric graphs) in the case of forces concentrated at the vertices. Conditions are derived for the closeness of physical parameters of two mechanical systems, one of which is a spatial network fixed on the boundary, while the other is a rubber-like continuum. It is shown that, under natural physical assumptions (equality of the concentrated forces to the integral of the distributed load over the corresponding cell and proportionality of the tension to the mesh size), the solution of the Dirichlet boundary value problem for the discrete system converges to the solution of the Dirichlet problem for the Poisson equation in a continuous membrane occupying the same domain. It is theoretically justified and numerically demonstrated that the finite-difference scheme arising from the force balance conditions in the network coincides with a stable conservative finite-difference approximation of the original continuous problem, ensuring uniform convergence of solutions as the mesh size tends to zero.