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