Boundary equation from a lattice model and modification of the Peierls equation

ABSTRACT

The boundary equation establishes the relationship between the displacement and the external force on the boundary of a solid. In the short-range interaction approximation, the boundary equation of a half-infinite lattice can be formally derived in lattice statics. Based on a model of cubic lattice, the boundary matrix as the kernal of the boundary equation is studied and calculated. In particular, the boundary matrix is completely presented for quasi one-dimensional problem and the leading order correction to the results in the elastic continuum theory is explicitly determined. The boundary equation can be used to derive the dislocation equation as a generalisation of classical Peierls equation. As an application, the modification of the Peierls equation is obtained and relevant coefficients are successfully related to bulk properties of solids.

KEYWORDS:

• Boundary equation
• lattice model
• Peierls equation

Disclosure statement

No potential conflict of interest was reported by the author.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Additional information

Funding

The work is supported by the National Natural Science Foundation of China (Grant No. 11874093).