Semi-discrete Green's function for solution of anisotropic thermal/electrostatic Boussinesq and Mindlin problems: Application to two-dimensional material systems

Abstract

Green's function (GF) for steady state Laplace/Poisson equation is derived for an anisotropic, finite two-dimensional (2D) composite material by solving a combined Boussinesq–Mindlin problem. A semi-discrete model of the material is developed in which only one Cartesian coordinate axis is discretized, while the other is treated as a continuous variable. The Fourier integral for the continuous coordinate is obtained analytically. Thus, a 2D problem needs only a 1D discretization. An approximate analytical estimate shows that the numerical convergence of our model is at least an order of magnitude better than fully discretized models. Numerical results are reported for the GF for a phosphorene composite containing an array of metallic inclusions. The GF is useful as a starting solution in boundary element calculations. It can be used for deriving the full solution of the Laplace/Poisson equation for an arbitrary distribution of sources and boundary values, used for modeling heat flow and electrostatic potential distribution in a 2D composite. These material systems are of strong topical interest because of their potential application in revolutionary new solid-state devices for energy conversion and quantum computing. This paper is another step towards developing GF based characterization techniques for modern 2D materials.