### Abstract

Macroscopic behavior of solids is widely studied from microscopic viewpoints such as molecular dynamics and lattice dynamics. However, there are some difficulties with the averaging methods in the microscopic expressions of stress, higher-order stress and heat flux. In this study, we discuss the microscopic expressions of macroscopic variables and macroscopic balance equations of a solid that is modeled as an assembly of atoms, on the basis of generalized Cosserat continuum theories. The concept of mesodomain is introduced to relate microscopic quantities of atoms to macroscopic quantities of continua. Microscopic expressions of stresses and heat flux are described as area averages of microscopic quantities such as velocities of atoms, interatomic potential forces. Balance equations for stress and higher-order stresses are derived from the equations of atomic motion. The energy equation, represented by the averages for the values in the mesodomain, is obtained by dividing the velocity of atoms into macroscopic motion and thermal motion. Since the generalized polar effects are taken into account in this model, derived macroscopic balance equations have the same form as the equations of generalized Cosserat continua. Moreover, microscopic descriptions obtained here show that the higher-order mechanical power of the generalized Cosserat continua is equivalent to the thermodynamic quantities of simple materials. The values of velocities of atoms calculated by molecular dynamics simulation are substituted into the newly obtained microscopic expressions of stress and higher-order stresses. The obtained values of stresses and the theoretical values derived from Eringen's formula correspond well, which demonstrates the usefulness of the present microscopic expressions for the practical application in computer simulations.

Original language | English |
---|---|

Pages (from-to) | 533-549 |

Number of pages | 17 |

Journal | Archive of Applied Mechanics |

Volume | 70 |

Issue number | 8-9 |

Publication status | Published - 2000 |

### Fingerprint

### Keywords

- Atom
- Balance equation
- Mesodomain
- Micromorphic material
- Stress

### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials

### Cite this

*Archive of Applied Mechanics*,

*70*(8-9), 533-549.

**Microscopic discussions of macroscopic balance equations for solids based on atomic configurations.** / Nakane, M.; Shizawa, Kazuyuki; Takahashi, K.

Research output: Contribution to journal › Article

*Archive of Applied Mechanics*, vol. 70, no. 8-9, pp. 533-549.

}

TY - JOUR

T1 - Microscopic discussions of macroscopic balance equations for solids based on atomic configurations

AU - Nakane, M.

AU - Shizawa, Kazuyuki

AU - Takahashi, K.

PY - 2000

Y1 - 2000

N2 - Macroscopic behavior of solids is widely studied from microscopic viewpoints such as molecular dynamics and lattice dynamics. However, there are some difficulties with the averaging methods in the microscopic expressions of stress, higher-order stress and heat flux. In this study, we discuss the microscopic expressions of macroscopic variables and macroscopic balance equations of a solid that is modeled as an assembly of atoms, on the basis of generalized Cosserat continuum theories. The concept of mesodomain is introduced to relate microscopic quantities of atoms to macroscopic quantities of continua. Microscopic expressions of stresses and heat flux are described as area averages of microscopic quantities such as velocities of atoms, interatomic potential forces. Balance equations for stress and higher-order stresses are derived from the equations of atomic motion. The energy equation, represented by the averages for the values in the mesodomain, is obtained by dividing the velocity of atoms into macroscopic motion and thermal motion. Since the generalized polar effects are taken into account in this model, derived macroscopic balance equations have the same form as the equations of generalized Cosserat continua. Moreover, microscopic descriptions obtained here show that the higher-order mechanical power of the generalized Cosserat continua is equivalent to the thermodynamic quantities of simple materials. The values of velocities of atoms calculated by molecular dynamics simulation are substituted into the newly obtained microscopic expressions of stress and higher-order stresses. The obtained values of stresses and the theoretical values derived from Eringen's formula correspond well, which demonstrates the usefulness of the present microscopic expressions for the practical application in computer simulations.

AB - Macroscopic behavior of solids is widely studied from microscopic viewpoints such as molecular dynamics and lattice dynamics. However, there are some difficulties with the averaging methods in the microscopic expressions of stress, higher-order stress and heat flux. In this study, we discuss the microscopic expressions of macroscopic variables and macroscopic balance equations of a solid that is modeled as an assembly of atoms, on the basis of generalized Cosserat continuum theories. The concept of mesodomain is introduced to relate microscopic quantities of atoms to macroscopic quantities of continua. Microscopic expressions of stresses and heat flux are described as area averages of microscopic quantities such as velocities of atoms, interatomic potential forces. Balance equations for stress and higher-order stresses are derived from the equations of atomic motion. The energy equation, represented by the averages for the values in the mesodomain, is obtained by dividing the velocity of atoms into macroscopic motion and thermal motion. Since the generalized polar effects are taken into account in this model, derived macroscopic balance equations have the same form as the equations of generalized Cosserat continua. Moreover, microscopic descriptions obtained here show that the higher-order mechanical power of the generalized Cosserat continua is equivalent to the thermodynamic quantities of simple materials. The values of velocities of atoms calculated by molecular dynamics simulation are substituted into the newly obtained microscopic expressions of stress and higher-order stresses. The obtained values of stresses and the theoretical values derived from Eringen's formula correspond well, which demonstrates the usefulness of the present microscopic expressions for the practical application in computer simulations.

KW - Atom

KW - Balance equation

KW - Mesodomain

KW - Micromorphic material

KW - Stress

UR - http://www.scopus.com/inward/record.url?scp=0007669093&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0007669093&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0007669093

VL - 70

SP - 533

EP - 549

JO - Archive of Applied Mechanics

JF - Archive of Applied Mechanics

SN - 0939-1533

IS - 8-9

ER -