Mathematical Study of Boundary-Value Problems of Linear Elasticity with Surface Stresses
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.
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.
- 3.Ciarlet, P.G.: Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity. North-Holland, Amsterdam (1988)Google Scholar
- 4.Ciarlet, P.G.: Mathematical Elasticity. Vol. III: Theory of Shells. North-Holland, Amsterdam (2000)Google Scholar
- 5.Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM, Philadelphia (2002)Google Scholar
- 6.Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Wiley, Singapore (1989)Google Scholar
- 7.Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2008)Google Scholar
- 8.Fichera, G.: Existence theorems in elasticity. In: S. Flügge (ed.) Handbuch der Physik, vol. VIa/2, pp. 347–389. Springer, Berlin (1972)Google Scholar
- 9.Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)Google Scholar
- 10.Javili, A., McBride, A., Steinmann, P.: Numerical modelling of thermomechanical solids with mechanically energetic (generalised) Kapitza interfaces. Comput. Mater. Sci. (2012). doi: 10.1016/j.commatsci.2012.06.006
- 11.Javili, A., McBride, A., Steinmann, P., Reddy, B.D.: Relationships between the admissible range of surface material parameters and stability of linearly elastic bodies. Phil. Mag. (2012). doi: 10.1080/14786435.2012.682175
- 14.Lebedev, L.P., Vorovich, I.I.: Functional Analysis in Mechanics. Springer, New York (2003)Google Scholar
- 15.Podstrigach, Y.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids (in Russian). Naukova Dumka, Kiev (1985)Google Scholar
- 20.Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., Wang, T.: Surface stress effect in mechanics of anostructured materials. Acta Mechanica Solida Sinica 24, 52–82 (2011)Google Scholar