Discrete and Continuum Thermomechanics

TL;DR

This paper presents a method to derive continuum thermomechanical equations from discrete atomic models for various crystal systems, including heat transfer and stress relations.

Contribution

It introduces a systematic approach for transitioning from discrete particle models to continuum descriptions in thermomechanics of solids, including derivation of equations of state and heat transfer.

Findings

Derived macroscopic balance equations from particle motion.

Established equations of state relating thermal pressure and energy.

Analyzed unsteady ballistic heat transfer in harmonic crystals.

Abstract

In the present chapter, we discuss an approach for transition from discrete to continuum description of thermomechanical behavior of solids. The transition is carried out for several anharmonic systems: one-dimensional crystal, quasi-one-dimensional crystal (a chain possessing longitudinal and transversal motions), two- and tree-dimensional crystals with simple lattice. Macroscopic balance equations are derived from equations of motion for particles. Macroscopic parameters, such as stress, heat flux, deformation, thermal energy, etc., are represented via parameters of the discrete system. Closed form equations of state relating thermal pressure, thermal energy and specific volume are derived. Description of the heat transfer in harmonic approximation is discussed. Unsteady ballistic heat transfer in a harmonic one-dimensional crystal is considered. The heat transfer equation for this system is rigorously derived.