The two-dimensional problem of elasticity incorporating surface effect at a rugous interface of bimaterial is considered. The current study is directed towards developing the theoretical model for analysing the stress distribution along nanosized interface irregularities of isotropic bimaterial systems, as the fracture of heterogeneous materials is strongly influenced by the stress concentrations around topological defects in an interface region. Based on the simplified Gurtin-Murdoch model of surface elasticity and the boundary perturbation method combined with the complex variable technique, the original boundary value problem is reduced to the sequence of hypersingular integral equations for any-order approximation. For the first-order approximation and periodic shape of the interface, the solution is obtained in an explicit form of Fourier series. The closed form expression for the hoop stress at the interface is derived in the case of the harmonic shape of the interface. Some detailed numerical investigations of the influence of mechanical characteristics of the interface, the shape and the size of interface asperities, and the elastic properties of the bimaterial components on the hoop stress and stress concentration are presented. It is shown that the incorporation of an interfacial energy exhibits properties that cannot be obtained using classical approach. For instance, the size effect is revealed, namely, the stress concentration depends on the deviation amplitude of the interface undulation at the nanoscale when the ratio of this amplitude to the period is constant; allowing for the interface elasticity reduces the concentration of the hoop stresses if the interface tension is neglected; if external loading is absent or relatively small, the elastic field at the interphase region is determined by the value of the interface tension and depends on geometric and elastic properties of the interface.