Abstract
The peridinamics theory, proposed by Silling in 2000, is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives and may cope with discontinuous displacement fields commonly occurring in fracture mechanics. Instead of spatial differential operators, integration over differences of the displacement field is used to describe the existing, possibly nonlinear, forces between particles of the solid body. Beside an overview of the peridynamics modelling and its application, results concerning the mathematical solution of the governing equation, which is a partial integro-differential equation with second-order time derivative, are presented. In this paper we consider well-posed Cauchy problem for the singular periodic integro-differential equation of peridinamics. In case of two-dimensional space the existence and uniqueness of solution are proved in Sobolev spaces. In multidimensional space the unique solvability of the problem in logarithmic scale of Hilbert spaces is proved.
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