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Mathematical Study of Boundary-Value Problems of Linear Elasticity with Surface Stresses

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Part of the Advanced Structured Materials book series (STRUCTMAT,volume 30)

Abstract

Following [1, 2] a mathematical investigation of initial-boundary and boundary-value problems of statics, dynamics and natural oscillations for elastic bodies including surface stresses is presented. The weak setup of the problems based on mechanical variational principles is given with introducing of corresponding energy spaces. Theorems of uniqueness and existence of the weak solution in energy spaces of static and dynamic problems are formulated and proved. The studies are performed applying the functional analysis techniques. Solutions of the problems under consideration are more smooth on the boundary surface than solutions of corresponding problems of the classical linear elasticity. The weak setup of the eigen-value problems is based on the Rayleigh variational principle. Certain spectral properties are established for the problems under consideration. In particular, bounds for the eigenfrequencies of an elastic body with surface stresses are presented. These bounds demonstrate increases in both the rigidity of the body and of the eigenfrequencies over those of the body with surface stresses neglected. The considered weak statements of the initial and boundary problems constitute the mathematical foundation for some numerical methods, in particular, for the finite element method.

Keywords

  • Surface Stress
  • Weak Setup
  • Rayleigh Variational Principle
  • Classical Linear Elasticity
  • Eigenfrequencies

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgments

The second author was supported by the DFG grant No. AL 341/33-1 and by the RFBR with the grant No. 12-01-00038.

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Correspondence to Holm Altenbach .

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Altenbach, H., Eremeyev, V.A., Lebedev, L.P. (2013). Mathematical Study of Boundary-Value Problems of Linear Elasticity with Surface Stresses. In: Altenbach, H., Morozov, N. (eds) Surface Effects in Solid Mechanics. Advanced Structured Materials, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35783-1_1

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