Peridynamic theory of solids from the perspective of classical statistical mechanics

• This work is a classical statistical mechanical framework for peridynamic theory.
• Langevin thermostat is effective for peridynamics at smaller length scales.
• The formalism expands the applicability of peridynamics from macro to nano scales at any temperature.
• The ambiguity related to fluctuation–dissipation in peridynamic theory is unveiled.

In this paper the classical statistical mechanics has been explored in order to develop statistical mechanical framework for peridynamics. Peridynamic equation of motion is known as upscaled Newton’s equation. The peridynamic system consists of finite number of nonlocally interacting particles at nano and meso scales. This particle representation of peridynamics can be treated in terms of classical statistical mechanics. Hence, in this work the phase space is constructed based on the PD particle from their evolving momentum pi and positions xi. The statistical ensembles are derived by defining appropriate partition functions  . The algorithms for NVENVE and NPHNPH implemented in the classical molecular dynamics are revisited for equilibrium peridynamic models. The current work introduces Langevin dynamics to the peridynamic theory through fluctuation–dissipation principle. This introduces a heat bath to the peridynamic system which eliminates the ambiguity with the role of temperature in a peridynamic system. Finally, it was seen that the homogenization of a peridynamic model with finite number of particles approaches to a conventional continuum model. The upscaled non-equilibrium peridynamics has potential applications in modeling wide variety of multiscale–multiphysics problems from nano to macro scale or vice versa.