Modified Kirsch Problem Incorporating Surface Stresses under Plane Stress

  • Vakaeva, Aleksandra (St. Petersburg State University)
  • Grekov, Mikhail (St. Petersburg State University)

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Typical representatives of inhomogeneous materials are classes of fibrous composites with metal, polymer and ceramic matrices. In these materials, fi bers of different scale levels, ranging in diameter from hundreds of microns to several nanometers, are distributed in different ways. Strength and physicochemical properties of materials depend on the features of the stress strain state of near-surface and near-boundary layers in inhomogeneous systems. The development of plastic deformation and fracture processes in these areas determines the mechanical behavior of the material as a whole and is therefore of great interest. The traditional consideration of micro- and nanoscale heterogeneities in the framework of the classical theory of elasticity may lead to inaccuracies in determining the levels of real deformations and stresses. In this work, we present the modifi ed Kirsch problem a plane stress allowing for the surface elasticity and residual surface stress by the Gurtin-Murdoch model. The problem on a plane stress of a plate in presence of the surface stresses differs essentially from the corresponding problem on a plane strain of a body as the elastic parameters of the plate depend on the elastic parameters of the surface and plate thickness. The boundary conditions are derived according to the corresponding generalized Laplace-Young law. With the help of Goursat-Kolosov complex potentials and Muskhelishvili representations, the solution of the problem is reduced to the singular integro-differential equation. The algorithm of solving the integral equation is constructed in the form of a power series. Based on explicit forms of the analytical solution, we present numerical results for the stress eld near the boundary of the nanopore. The effect of plate thickness on the stress eld at the surface of the pore and the role of surface tension at this surface is shown.