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

Published

### Author(s)

Vinod K. Tewary, Edward J. Garboczi

### Abstract

The Green's function (GF) for the steady state Laplace/Poisson equation is derived for an anisotropic finite two-dimensional (2D) composite material by solving a combined Boussinesq- Mindlin problem. The source term for the GF is a delta-function located somewhere in the bulk of the solid (Mindlin problem). The boundary condition for the GF is prescribed in terms of a delta function at a single point on the outer boundary (Boussinesq problem). The calculated GF is, therefore, quite general and can be used to find the full solution of the Laplace/Poisson equation for an arbitrary but integrable distribution of sources and boundary values in a composite material. The piecewise continuous composite, consisting of a host material containing one or more inclusions, is assumed to be infinite and periodic in one direction (X- direction) but of finite arbitrary size in the perpendicular direction in the plane (Z- direction). A semi-discrete semi-analytic model is developed for the calculation of the GF subject to the boundary conditions described above. The primary equation is solved by using a partial Fourier transform technique. Our model is semi-discrete in the sense that the solution space is discretized only in the periodic X-direction but not in the Z-direction. This allows the space to be filled up exactly in terms of the discrete elements even if the shape of the inclusions are geometrically irregular. Further, our model is semi-analytic in the sense that the Fourier integration in the Z direction is done exactly, so that a 2D problem has only a 1D discretization. An approximate analytical estimate shows that the numerical convergence of our semi-discrete model is at least an order of magnitude faster than conventional fully discretized models (e.g., finite-element or boundary-element models). As an application of the method, GF numerical results are presented at all points on the plane of 2D phosphorene containing a metallic inclusion.

Citation
Physical Review Materials

### Keywords

Boussinesq problem, Green's function, Laplace equation, Mindlin problem, phosphorene, Poisson equation, thermal electrostatic modeling, two-dimensional materials.

## Citation

Tewary, V. and Garboczi, E. (2019), Semi-discrete Green's function for solution of anisotropic thermal/electrostatic Boussinesq and Mindlin problems: Application to two-dimensional materials, Physical Review Materials, [online], https://doi.org/10.1016/j.enganabound.2019.10.004 (Accessed April 23, 2024)
Created October 3, 2019, Updated January 27, 2021