Elastic Theory of Nanomaterials Based on Surface Energy Density

Abstract

Recent investigations into surfaceenergy density of nanomaterials lead to a ripe chance to propose, within the framework of continuum mechanics, a new theory for nanomaterials based on surfaceenergy density. In contrast to the previous theories, the linearly elastic constitutive relationship that is usually adopted to describe the surface layer of nanomaterials is not invoked and the surface elastic constants are no longer needed in the new theory. Instead, a surfaceinduced traction to characterize the surface effect in nanomaterials is derived, which depends only on the Eulerian surfaceenergy density. By considering samplesize effects, residual surface strain, and external loading, an explicit expression for the Lagrangian surfaceenergy density is achieved and the relationship between the Eulerian surfaceenergy density and the Lagrangian surfaceenergy density yields a conclusion that only two material constants—the bulk surfaceenergy density and the surfacerelaxation parameter—are needed in the new elastic theory. The new theory is further used to characterize the elastic properties of several fcc metallic nanofilms under biaxial tension, and the theoretical results agree very well with existing numerical results. Due to the nonlinear surface effect, nanomaterials may exhibit a nonlinearly elastic property though the inside of nanomaterials or the corresponding bulk one is linearly elastic. Moreover, it is found that externally applied loading should be responsible for the softening of the elastic modulus of a nanofilm. In contrast to the surface elastic constants required by existing theories, the bulk surfaceenergy density and the surfacerelaxation parameter are much easy to obtain, which makes the new theory more convenient for practical applications.