Deformation-Induced Change in the Geometry of a General Material Surface and Its Relation to the Gurtin–Murdoch Model

Abstract

Small deformation theory plays an important role in analyzing the mechanical behavior of various elastic materials since it often leads to simple referential analytic results. For some specific mechanical problems however (for example, those dealing with small-scale materials/structures with significant surface energies or soft solids containing gas/liquid inclusions with high initial pressure), in order to obtain sufficiently accurate solutions, the classical boundary conditions associated with small deformation theory often require modification to incorporate the influence of deformation on the geometry of the boundary. In this note, we provide first-order approximate expressions characterizing the change in the geometry (normal vector, curvature tensor, etc.) of a general surface during deformation. In particular, using these expressions we recover without difficulty, the stress boundary condition in the original Gurtin–Murdoch surface model for an (initially) spherical interface with constant interface tension. We believe that the expressions established here will find widespread application in the mechanical analysis of problems requiring an extremely high level of accuracy in the description of the corresponding boundary conditions. In addition, higher-order approximate expressions representing the change in the geometry of a general surface during deformation could be also obtained using the same procedure.