On the Influence of Residual Surface Stresses on the Properties of Structures at the Nanoscale
We discuss the influence of residual surface stresses on the effective (apparent) properties of materials at the nanoscale such as the stiffness of rods. The interest to the investigation of the surface effects is recently grown with respect to progress i
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Abstract We discuss the influence of residual surface stresses on the effective (apparent) properties of materials at the nanoscale such as the stiffness of rods. The interest to the investigation of the surface effects is recently grown with respect to progress in nanotechnologies. The surface and interface effects play an important role for nanofilms, nanocomposites, nanoporous materials, etc. Here we consider the Gurtin–Murdoch model of surface elasticity. With the help of the simple problem of uniaxial tension of a rod with residual surface stresses we analyze the behavior of the rod under tension and present the effective stiffness.
1 Introduction Recently, the interest to the model of surface elasticity by Gurtin and Murdoch [6] grows fast with respect to development of nanotechnologies, see [3, 16]. The model [6] predicts the size effect observed in the case of nanosized materials [15]. Unlike to macro- and microsized specimen where the size effect can be explained by various mechanisms, see the review [2], the size effect in nanomechanics can be related to surface phenomena only. An elastic body with surface stresses can be considered as a classical elastic body with glued elastic membrane. The stress resultant tensor acting H. Altenbach · V. A. Eremeyev (B) Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany e-mail: [email protected] V. A. Eremeyev South Scientific Center of RASci & South Federal University, Rostov on Don, Russia e-mail: [email protected]; [email protected] N. F. Morozov St. Petersburg State University, Bibliotechnaya sq. 2, 198904 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_2, © Springer-Verlag Berlin Heidelberg 2013
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in the membrane can be interpreted as surface stresses in the Gurtin–Murdoch model. The additional surface elasticity influences on the effective properties of materials [3, 16]. It was shown that within the linear theory of surface elasticity the presence of surface stresses lead to the stiffening of the material, see, for example, [1, 4, 17, 18]. The residual or initial surface stresses can play an important role with respect to the material behavior at the nanoscale. In particular, the residual surface stress change the free vibrations of materials [5, 10, 14]. The finite deformations of elastic solids within the framework of the model [6] including analysis of residual surface stresses are considered in [7–9]. Let us note that the using of nonlinear elasticity methods is necessary for the correct describing of the prestressed state of solids. The paper is organized as follows. In Sect. 2 we recall the basic equations of elasticity with surface stresses. In Sect. 3 we discuss the constitutive equations in more details. Here we introduce the natural configurations for the elastic body and resi
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