Formulation of a nonlocal discrete model for anisotropic heat conduction problems

In this paper, we present the formulation of a nonlocal discrete model based on a lattice particle method for modeling anisotropic heat conduction problems. In the proposed model, material domain is decomposed into discrete material particles based on Bravais lattices. Nonlocal interaction between material particles is introduced via one-dimensional bond. The unique relationships between the bond thermal conductivities and the components of material thermal conductivity tensor are established by equating the heat transfer rate of a material particle between discrete description and its continuum counterpart under principal coordinate system. For anisotropic materials, rotation of the discretization lattice is developed to account for material orientation. The equivalency of the lattice rotation with the coordinate transformation used in continuum-based formulation is studied. Continuum-like measures such as temperature gradient and heat flux at a material particle location are also constructed based on discrete temperature solution. The proposed model has the following three distinct features: Firstly, the proposed model describes heat conduction problems using discretized integro-differential equation rather than partial differential equation. Secondly, the proposed model introduces nonlocal thermal interactions among discrete material particles via bond-like response function. Thirdly, material orientation is represented by rotating the discretization lattice instead of coordinate transformation to better capture the underlying material microstructural effects.