Basic Equations of Solid Mechanics Based on Lattice Dynamics and Molecular Dynamics

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In conventional continuum mechanics, macroscopic state variables such as stress, strain, and temperature are defined at each macroscopic point, and only the macroscale is referenced. On the other hand, since these values are determined from the motions of atoms or molecules, a microscopic scale must be introduced to observe such internal motions from the microscopic standpoint.The domain in which this mesoscale is used to measure distance is called the mesodomain, i.e., it is sufficiently smaller than the macroscopic domain, yet sufficiently larger than the microscopic volume.This study proposes a method for expressing stresses from motions of atoms, and for deriving the conservation laws for solids microscopically. Constitutive equations and elastic constants not only for stresses but also for higher-order stresses are also derived microscopically. For the derivation of equations, a concept of a hierarchical Reynolds decomposition is introduced. The hierarchical deviation terms are ex … More pressed by characteristic tensors which can be called P-tensors. The P-tensors are the indexes of atomic configurations and are used effectively for the expression of the elastic constants.The microscopic expressions described above are useful for obtaining the values of the macroscopic stress, higher-order stresses or constitutive equations from the results of molecular dynamics simulations. A three-dimensional simulation model with 10 x 10 x 10 Al atoms is introduced for calculations. Loads are applied to the atoms on the horizontal surface, changing proportionally with the gradient of 7.06 μN/m. The values of velocities of atoms calculated by molecular dynamics are substituted into the present microscopic expressions of stresses, higher-order stresses and constitutive equations. The numerical values and the theoretical values correspond well.Based on these results, a notion of one dimensional thermal polar materials are introduced. These materials can be said as thermal beams. A model of the thermal beam is proposed, and thermal cross-sectional constants for the model are obtained by FEM calculations. Less