2021/04/13 by Pieter D. Boom, Odysseas Kosmas, Boom, Pieter D. +5 · 3 citations
Engineering · Biochemistry, Genetics and Molecular Biology · #Advanced Materials and Mechanics #Cellular Mechanics and Interactions
paper · pdf · doi:10.48550/arxiv.2104.06000
A direct formulation of linear elasticity of cell complexes based on discrete\nexterior calculus is presented. The primary unknown are displacements,\nrepresented by primal vector-valued 0-cochain. Displacement differences and\ninternal forces are represented by primal vector-valued 1-cochain and dual\nvector-valued 2-cochain, respectively. The macroscopic constitutive relation is\nenforced at primal 0-cells with the help of musical isomorphisms mapping\ncochains to smooth fields and vice versa. The balance of linear momentum is\nestablished at primal 0-cells. The governing equations are solved as a Laplace\nequation with a non-local and non-diagonal material Hodge star. Numerical\nsimulations of several classical problems with analytic solutions are presented\nto validate the formulation. Good agreement with known solutions is obtained.\nThe formulation provides a method to calculate the relations between\ndisplacement differences and internal forces for any lattice structure, when\nthe structure is required to follow a prescribed macroscopic elastic behaviour.\nThis is also the first and critical step in developing formulations for\ndissipative processes in cell complexes.\n