Stability and Size-Dependency of Temperature-Related Cauchy-Born Hypothesis In Three-Dimensional Applications

Continuum models are expected to cover small deformation in large nano scale simulation, as an alternative for Molecular Dynamics (MD) since they are feasible for expensive simulation. The constitutive model which is used in Finite Elements (FEM) and multiscale methods is usually based on the Cauchy-Born (CB) hypothesis. It seeks the intrinsic characteristics of the material via the atomistic information and it is valid in small deformation. The main purpose of this paper is to investigate the temperature effect on stability and size dependency of CB. Three-dimensional temperature-related Cauchy-Born equations for crystalline structure are developed and the stability and size dependency of temperature-related Cauchy-Born hypothesis is investigated by means of direct comparison between atomistic and continuous mediums information. In modelling of crystalline solids, the deformation is assumed to be homogeneous and atoms have the same local vibration mode while couple vibration of different atom is negligible. The temperature control in MD simulation is performed at constant temperature rather than constant energy. In order to control the temperature, the Nose-Hoover thermostat is used. Since the Helmholtz free energy is temperature dependent; the first Piola-Kirchhoff stress can be explicitly computed as the first derivative of the Helmholtz free energy density to the deformation gradient. It is numerically shown that the temperature expands the validity surface in small specimens and the validity surface shrinks with temperature in large specimens. It is also observed that the material stability decreases with increasing in ambient temperature.

کلیدواژه ها:

Temperature-related Cauchy-Born (TCB) ، Size effect ، Molecular Dynamics ، Nose-Hoover thermostat ، Temperature