25. Continuum Theory for Deformable Interfaces/Surfaces with Multi-field Coupling

Continuum mechanics is a well-demonstrated powerful tool for dealing with comprehensive physical problems of macroscopic bodies involving deformation and flow processes. It has also been successfully and extensively applied to deformable bodies at micro- and nano-scales without or with appropriate modifications. One particular and interesting phenomenon of small-sized subjects is that they usually exhibit size-dependent characteristics, which cannot be well explained within the conventional framework of continuum mechanics. A modified version that accounts for the so-called surface elasticity has been developed. The rigorous derivation of surface elasticity is due to Gurtin and Murdoch as early in 1975 by creating a two-dimensional set of elasticity. This chapter aims to introduce a relatively traditional method to derive the interface/surface elasticity, which further incorporates the coupling among multiple physical fields (e.g., elastic, electric, and magnetic). The method is illustrated here by only considering the electroelastic coupling, but there is completely no difficulty to make a further step toward the situation where more fields are involved. The method first assumes a finite thickness of the interface/surface so that an interphase layer model is actually adopted, which is governed by the traditional three-dimensional theory of piezoelectricity. Then, the state-space formalism is derived, based on which a transfer relation between the state vectors at the upper and lower surfaces of the interphase layer can be easily obtained. By series expansion and truncation, various theories of interface/surface piezoelectricity of different orders can then be established. Comparison is made with those reported in the literature, which shows good agreement and hence validates the present approach.