Lagrangian formulism of elasticity with relevance to surface energy

TL;DR

This paper develops a Lagrangian framework incorporating surface energy into elasticity, deriving new boundary conditions and applying it to nanoparticle vibrations, revealing surface energy effects on natural frequencies.

Contribution

It introduces a novel Lagrangian formulation that explicitly includes surface energy effects in elastic materials, deriving new boundary conditions and a generalized Young-Laplace formula.

Findings

Surface energy causes a downward shift in nanoparticle vibration frequencies.

The model generalizes classical surface energy relations to elastic solids.

Numerical results show significant effects in soft matter nanoparticles.

Abstract

By introducing the divergence of a vector potential into the Lagrangian, a Lagrangian framework is developed to incorporate surface energy into elasticity. Besides the Euler-Lagrange equation and natural boundary condition, a new boundary constitutive equation is derived from the variation of the Lagrangian and configuration on which the Lagrangian is defined. On the boundary surface, explicit expression of the vector potential with respect to field variable and surface curvature is determined. Based on this framework, an elastic model with relevance to the surface energy is established. The Young-Laplace's formula is generalized into elastic solid in a new form. Making use of this model, we investigate the surface energy effect in the radial vibration of spherical nanoparticle. Numerical calculation shows that natural frequencies of nanoparticle will shift down due to the surface energy. This shift is especially apparent in the vibration of soft matter nanoparticles.