![]() ![]() The majority of these models utilize the bond-based version of PD. introduced a PD model for transient advection-diffusion problems. In these cases, the interaction between material points is prescribed by a thermal response function, and the spatial derivatives of temperature, involved in classical diffusion equations, are replaced by spatial integrals of the thermal response function. Bobaru and Duangpanya proposed a 1D PD heat conduction model in, and an extension of their work to 2D bodies with evolving discontinuities is presented in. for electromigration that accounts for heat conduction in a 1D body. The first PD diffusion model was developed by Gerstle et al. To address also such problems, PD is a promising approach. Thermal cracking, hydraulic fracturing, and pitting corrosion are just a few examples, where part of the solution is governed by diffusion, and, at the same time, they are subjected to the emergence of spontaneous discontinuities. Unfortunately, many of the nonlocal models are inapplicable for problems in which discontinuities (whether strong or weak) in the system emerge, interact, and evolve. Recently, miniaturization of devices has provided more cases where the application of nonlocal models rather than local ones is more appropriate. Therefore, nonlocal diffusion equations as well as related numerical methods have received a considerable attention in the literature. However, at this scale, nonlocal effects may play a crucial role for instance, in the case of heat conduction with steep temperature gradients, stochastic jump processes and anomalous diffusion are observed in heterogeneous environments. At the macroscale, most diffusion processes can be described well by local models based on Fourier’s law (heat conduction) as well as Fick’s law (mass transport). In this regard, numerous numerical methods (based on the classical local diffusion) have so far been employed, for example the finite element method (FEM), the finite difference method (FDM), the boundary element method (BEM), and meshfree methods (see, e.g., ). The mathematical description of many phenomena from areas such as fluid dynamics, chemistry, biology, information, environmental and materials sciences is governed by diffusion-type equations. In fact, the solution of diffusion problems is of great importance for engineering and physics applications concerning heat and mass transfer. The focus of the present study is on diffusion-type problems. As the horizon shrinks and asymptotically approaches zero, the interactions become local and the formulation reduces to that of the classical local theory when suitable regularity assumptions hold. The interactions are governed by a response function that includes all the material constitutive information. Each material point establishes direct interactions with other points within its neighborhood. PD models make use of a characteristic length scale called horizon that determines the region of the nonlocal interactions. ![]() In nonlocal continuum theories, the value of a physical quantity at a given material point is influenced by quantities within a finite neighborhood around that point we refer to such influences as nonlocal effects. To this respect, many PD models have so far been proposed and applied to a broad range of problems in solids including fracture and crack propagation to mention a few, see and for a comprehensive review one may refer to. Therefore, PD models are effective tools in modelling material discontinuities, and they consider fracture and damage as natural parts of the solution. In fact, the original formulation of PD introduces an equation of motion in solid mechanics based on integro-differential equations rather than on partial differential equations (PDEs) which are undefined at discontinuities. to handle material failure in solid structures, which is not an easy task for the classical continuum mechanics (CCM) theory. The theory was originally introduced by Silling and Silling et al. It has been widely exploited to solve various problems in mechanics and physics. Peridynamics (PD) is a recent nonlocal theory that has received a widespread attention in computational mechanics. ![]()
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