A priori analysis of a higher-order nonlinear elasticity model for an atomistic chain with periodic boundary condition. (English) Zbl 1511.65130
Summary: Nonlinear elastic models are widely used to describe the elastic response of crystalline solids, for example, the well-known Cauchy-Born model. While the Cauchy-Born model only depends on the strain, effects of higher-order strain gradients are significant and higher-order continuum models are preferred in various applications such as defect dynamics and modeling of carbon nanotubes. In this paper we rigorously derive a higher-order nonlinear elasticity model for crystals from its atomistic description in one dimension. We show that, compared to the second-order accuracy of the Cauchy-Born model, the higher-order continuum model in this paper is of fourth-order accuracy with respect to the interatomic spacing in the thermal dynamic limit. In addition we discuss the key issues for the derivation of higher-order continuum models in more general cases. The theoretical convergence results are demonstrated by numerical experiments.
MSC:
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
65D32 | Numerical quadrature and cubature formulas |
65D05 | Numerical interpolation |
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
65N15 | Error bounds for boundary value problems involving PDEs |
35J30 | Higher-order elliptic equations |
74B20 | Nonlinear elasticity |
74E15 | Crystalline structure |
82D20 | Statistical mechanics of solids |
82D80 | Statistical mechanics of nanostructures and nanoparticles |
81V45 | Atomic physics |