Lagrangian formulism of elasticity with relevance to surface energy
  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. 
Zaixing Huang
cond-mat.mtrl-sci cond-mat.soft physics.chem-ph