  A new model of fracture mechanics considered previously by Sendova and Walton
\cite{SendovaWalton2010}, Zemlyanova \cite{Zemlyanova2013}, and Zemlyanova and
Walton \cite{Zemlyanova2012} is further developed on the example of a mixed
mode curvilinear interface fracture located on the boundary of a partially
debonded thin elastic inclusion embedded in an infinite thin elastic matrix.
The effect of the nano-structure of the material is incorporated into the model
in the form of a curvature-depended surface tension acting on the boundary of
the fracture. It is shown that the introduction of the surface tension allows
to eliminate the classical oscillating and power singularities of the order
$1/2$ present in the linear elastic fracture mechanics. The mathematical
methods used to solve the problem are based on the Muskhelishvili's complex
potentials and the Savruk's integral representations. The mechanical problem is
reduced to the system of singular integro-differential equations which is
further reduced to a system of weakly-singular integral equations. The
numerical computations and comparison with known results are presented.
