Analytical solution of peridynamics for heat conduction with an external source

Abstract

This study provides a novel analytical approach for solving one- and two-dimensional heat conduction problems with an external heat source within the peridynamic framework. By employing Duhamel’s principle, the inhomogeneous governing equation with an external source is transformed into a homogeneous equation, enabling the derivation of peridynamic analytical solutions through separation of variables. An innovation involves introducing a nonlocal factor by ensuring compatibility between the spatial functions in the peridynamic solution and their classical continuum mechanics counterparts. Numerical examples demonstrate the methodology’s effectiveness for both one- and two-dimensional heat conduction problems subjected to sinusoidal and constant external sources, and the nonlocal effect is measured based on the nonlocal factor. Unlike peridynamic systems without external sources, these examples reveal that the nonlocal factor appears not only in the exponential term but also in other components of the solution, indicating persistent nonlocal effects even as time approaches infinity. Furthermore, the analysis shows that the kernel function with n = 0 exhibits the strongest nonlocal influence, while n = 2 results in the weakest nonlocal behavior. The findings demonstrate that peridynamic analytical solutions for heat conduction can be systematically derived from classical continuum mechanics solutions through appropriate incorporation of the nonlocal factor. This work significantly expands the scope of peridynamic analytical solutions and offers new insights into nonlocal heat conduction phenomena, providing a valuable foundation for future studies in this field.