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

在传统的连续介质力学中，宏观状态变量如应力、应变和温度在每个宏观点上都是定义的，并且只参考宏观尺度。另一方面，由于这些值是由原子或分子的运动决定的，因此必须引入微观尺度来从微观的角度观察这种内部运动。这个中尺度用来测量距离的区域被称为中尺度，即它比宏观区域足够小，但又比微观体积足够大。本研究提出了一种从原子运动中表达应力的方法，并在微观上推导了固体的守恒定律。并从微观上推导了高阶应力和应力的本构方程和弹性常数。对于方程的推导，引入了层次Reynolds分解的概念。层次偏差项更多地由特征张量（p张量）所压制。p张量是原子构型的指标，可有效地用于弹性常数的表示。上述微观表达式可用于从分子动力学模拟结果中获得宏观应力、高阶应力或本构方程的数值。采用10 × 10 × 10 Al原子的三维模拟模型进行计算。在水平面上对原子施加载荷，以7.06 μN/m的梯度成比例变化。将分子动力学计算的原子速度值代入目前的微观应力表达式、高阶应力表达式和本构方程。数值与理论值吻合较好。在此基础上，引入了一维热极性材料的概念。这些材料可以说是热光束。提出了一种热光束模型，并通过有限元计算得到了该模型的热截面常数。少

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