Surface Stress in an Elastic Plane with a Nearly Circular Hole
A boundary value problem on a nanometer hole in an elastic plane under arbitrary remote loading is solved. It is assumed that complementary surface stress is acting at the boundary of the hole. The outer surface of the hole is supposed to be conformally m
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Abstract A boundary value problem on a nanometer hole in an elastic plane under arbitrary remote loading is solved. It is assumed that complementary surface stress is acting at the boundary of the hole. The outer surface of the hole is supposed to be conformally mapped on the outer surface of the circle by means of a power function. The Gurtin–Murdoch surface elasticity model is applied to take into account the surface stress effect. Based on the Goursat–Kolosov complex potentials and Muskhelishvili’s technique, the solution of the problem is reduced to a singular integro-differential equation in an unknown surface stress. For a nearly circular hole, the boundary perturbation method is used that leads to successive solutions of hypersingular integral equations. Numerical results based on the first-order approximate solution are specified for an elliptical nearly circular hole.
1 Introduction In traditional continuum mechanics, the effect of surface energy is ignored as it is small compared to the bulk energy. At the same time, the surface effects become significant for nanoscale materials and structures due to the high surface/volume ratio [6, 19, 22]. In particular, the surface stresses are directly related to the size effect, that means the material properties of a specimen depend on its size [4–6, 19]. Besides the results of theoretical analysis and numerical calculations, the size effect was observed in a number of experimental measurements of nanowires and nanotubes [22]. M. A. Grekov (B) Saint-Petersburg State University, Universitetski pr. 35 ,198504 St.-Petersburg, Russia e-mail: [email protected] A. A. Yazovskaya Saint-Petersburg State University, Universitetski pr. 28 ,198504 St.-Petersburg, Russia e-mail: [email protected] H. Altenbach and N. F. Morozov (eds.), Surface Effects in Solid Mechanics, Advanced Structured Materials 30, DOI: 10.1007/978-3-642-35783-1_7, © Springer-Verlag Berlin Heidelberg 2013
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M. A. Grekov and A. A. Yazovskaya
Solid surface stresses are known like the prestress in a prestressed membrane that is perfectly fitted to the bounding surface of a bulk material. The inclusion of such a surface stress in an otherwise traction-free surface of the bulk material leads to additional loads applied to this surface [7]. The general form of a boundary condition with surface stress, namely, the generalized Young–Laplace equation, was derived in [11, 17]. Surface effects have been considered in numerous theoretical investigations based on the generalized Young-Laplace equation and the theory of surface elasticity which had been developed in [11, 15]. In particular, the theory of elasticity with surface stresses is applied to the modifications of the two-dimensional theories of nanosized plates and shells in [1–3]. Various application of the Gurtin–Murdoch model [11, 15] in nanomechanics is presented in the literature which is reviewed by Wang et al. in [22]. Using this model, Tian and Rajapakse [19, 20] gave the solutions of two-dimensional size-dependent elastic fields of a matrix
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