Mechanics of Biological Membranes from Lattice Homogenization

Abstract

The goal of this chapter is to set up a novel methodology for the calculation of the effective mechanical properties of biological membranes viewed as repetitive networks of elastic filaments, basing on the discrete asymptotic homogenization method. We will show that for some lattice configurations, internal structure mechanisms at the unit cell scale lead to additional flexional effects at the continuum scale, accounted for by an internal length associated to a micropolar behavior. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes for a given network topology, as closed form expressions of the geometrical and mechanical micro-parameters. A new approach, based on general beam equations, is proposed to tackle the non-linear constitutive behavior of the network, accounting for large strains and large rotations. Thereby, a perturbed equilibrium problem is set up at the unit cell level, solved by the Newton-Raphson method. This localization problem interacts with the homogenization procedure allowing the construction of the Cauchy and couple stress tensors, both steps leading to an update the network geometry and constitutive behaviour. A classification of lattices with respect to the choice of the equivalent continuum model is proposed: the Cauchy continuum and a micropolar continuum are adopted as two possible effective medium, for a given beam model. The relative ratio of the characteristic length of the micropolar continuum to the unit cell size determines the relevant choice of the equivalent medium. Calculation of the equivalent mechanical properties of the peptodoglycan membrane illustrates the proposed methodology.