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Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity

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Abstract

Mathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin–Murdoch model to the Steigmann–Ogden model of surface elasticity using techniques from the theory of Sobolev’s spaces and methods of functional analysis. The Steigmann–Ogden model accounts for the bending stiffness of the surface film; it is a generalization of the Gurtin–Murdoch model. Weak setups of the problems, based on variational principles formulated, are employed. Some uniqueness-existence theorems for weak solutions of static and dynamic problems are proved in energy spaces via functional analytic methods. On the boundary surface, solutions to the problems under consideration are smoother than those for the corresponding problems of classical linear elasticity and those described by the Gurtin–Murdoch model. The weak setups of eigenvalue problems for elastic bodies with surface stresses are based on the Rayleigh and Courant variational principles. For the problems based on the Steigmann–Ogden model, certain spectral properties are established. In particular, bounds are placed on the eigenfrequencies of an elastic body with surface stresses; these demonstrate the increase in the body rigidity and the eigenfrequencies compared with the situation where the surface stresses are neglected.

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Authors and Affiliations

  1. Institute of Mechanics, Faculty of Mechanical Engineering, Otto von Guericke-University, Universitätplatz 2, 39106, Magdeburg, Germany

    Victor A. Eremeyev

  2. South Scientific Center of RASci & South Federal University, Milchakova St. 8a, 344090, Rostov on Don, Russia

    Victor A. Eremeyev

  3. Departamento de Matemáticas, Universidad Nacional de Colombia, Carrera 45 N 26-85, Bogotá, Colombia

    Leonid P. Lebedev

Authors
  1. Victor A. Eremeyev
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  2. Leonid P. Lebedev
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Correspondence to Victor A. Eremeyev.

Additional information

Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.

In honor of Prof. David Steigmann

The first author acknowledges the supports by the RFBR with the Grant No. 15-01-01492.

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Eremeyev, V.A., Lebedev, L.P. Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity. Continuum Mech. Thermodyn. 28, 407–422 (2016). https://doi.org/10.1007/s00161-015-0439-0

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  • DOI: https://doi.org/10.1007/s00161-015-0439-0

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