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

基于晶格动力学和分子动力学的固体力学基本方程

基本信息

批准号：
12650094
负责人：
TAKAHASHI Kunihiro
金额：
$ 2.05万
依托单位：
Keio University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

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

在传统的连续介质力学中，应力、应变和温度等宏观状态变量是在每个宏观点上定义的，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，所以必须引入微观尺度才能从微观角度观察这种内部运动。用这种介观尺度来测量距离的区域称为介观区域，即它足够小于宏观区域，但又足够大于微观体积。本研究提出了一种从原子运动中表示应力的方法，并从微观上推导了固体的守恒定律。从微观上推导了应力的本构方程和弹性常数，以及高阶应力的本构方程和弹性常数。对于方程的推导，引入了分层雷诺分解的概念。分级偏差项为ex…更多地受到特征张量的挤压，这可以被称为P张量。P张量是原子构型的指标，可以有效地用来表示弹性常数。上述微观表达式对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值是有用的。引入了一个含10×10×10个Al原子的三维模拟模型进行计算。对水平表面上的原子施加载荷，按7.06μN/m的梯度成比例变化，用分子动力学计算的原子速度值代入到目前的应力、高阶应力和本构方程的微观表达式中。在此基础上，提出了一维热极性材料的概念。这些材料可以说是热束。建立了热梁模型，并通过有限元计算得到了模型的热截面常数。较少