Integral-Type Stress Boundary Condition in the Complete Gurtin-Murdoch Surface Model with Accompanying Complex Variable Representation

In the large majority of papers utilizing the Gurtin-Murdoch (G-M) model of a material surface, the complete model is avoided in favor of various modified versions often because they lead to simpler representations of the corresponding stress boundary condition. We propose in this paper an integral-type stress boundary condition for the plane deformations of a bulk-interface composite system which allows for an equally simple implementation of the complete G-M model. Since the mechanical behavior of such composite systems is often analyzed using complex variable methods, we formulate our ideas accordingly, in this context. Remarkably, in contrast to what is often believed to be the case, we find that boundary value problems based on our formulation of the stress boundary condition offer no added difficulty when utilizing the complete G-M model versus its various simplified counterparts. This new representation of the stress boundary condition is concise in form and will prove to be extremely useful in, for example, the calculation of the elastic field in the vicinity of nano-inhomogeneities of irregular shape.