Meccanica

, Volume 48, Issue 7, pp 1791–1804

# On the numerical solution of some two-dimensional boundary-contact delocalization problems

Article

## Abstract

The elastic equilibrium of a multi-layer confocal elliptic ring is studied. The ring consists of steel, rubber and celluloid layers which differ in thickness and in the order in which they are placed relative to one another. Using the solutions of the considered problems, the following delocalization problem is solved: for a three-layer elliptic body, the external elliptic boundary of which is loaded by normal point force and the internal boundary is stress-free and the layers of which are in rigid or sliding contact with one another, by an appropriate choice of layer thickness and arrangement of the layers relative to one another we can obtain a sufficiently uniform distribution of normal displacements on the internal elliptic boundary. Numerical solutions are obtained by the boundary element method and the related graphs are constructed. For the two-layer ellipse, exact and approximate solutions of the same problem are obtained respectively by the method of separation of variables and by the boundary element method. The results obtained by both methods are compared and the conclusion as to the reliability of the numerical boundary element method is made.

## Keywords

Boundary-contact problem Boundary element method Fictitious load Method of separation variables

## References

1. 1.
Love AEH (1927) A treatise on the mathematical theory of elasticity. Cambridge University Press, Cambridge
2. 2.
Koltunov MF, Vasil’ev YN, Chernykh VA (1970) Elasticity and solidity of cylindrical bodies. Izd “Vysshaya Shkola”, Moscow (in Russian) Google Scholar
3. 3.
Khomasuridze N (1998) Thermoelastic equilibrium of bodies in generalized cylindrical coordinates. Georgian Math J 5(6):521–544
4. 4.
Shestopalov Y, Kotik N (2006) Approximate decomposition for the solution of boundary value problems for elliptic systems arising in mathematical models of layered structures. In: Progress in electromagnetics research symposium, Cambridge, USA, March 26–29, pp 514–518 Google Scholar
5. 5.
Fabrikant VI (2011) Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space. Meccanica 46(6):1239–1263
6. 6.
Saint-Venant AJCB (1855) Memoire sur la torsion des prismes. Mem. divers savants, vol 14. pp 233–560 Google Scholar
7. 7.
von Mises R (1945) On Saint-Venant’s principle. Bull Am Math Soc 51:555–562
8. 8.
Ru CQ, Schiavone P, Miodochowski A (1999) Uniformity of stresses within a three-phase elliptic inclusion in anti-plane shear. J Elast 52:121–128
9. 9.
Li Y, Waas AM, Arruda EM (2011) A closed-form, hierarchical, multi-interphase model for composites? Derivation, verification and application to nanocomposites. J Mech Phys Solids 59:43–63
10. 10.
Lipinski P, El Barhdadi H, Cherkaoui M (2006) Micromechanical modelling of an arbitrary ellipsoidal multi-coated inclusion. Philos Mag 86(10):1305–1326
11. 11.
Giordano S, Palla PL, Colombo L (2008) Effective permittivity of materials containing graded ellipsoidal inclusions. Eur Phys J B 66:29–35
12. 12.
Li JY (2000) Thermoelastic behavior of composites with functionally graded interphase: a multi-inclusion model. Int J Solids Struct 37(39):5579–5597
13. 13.
Bermant AF (1958) Mapping linear coordinates. Transformation. Green’s formulas. Fizmatgiz, Moscow (in Russian) Google Scholar
14. 14.
Zirakashvili N (2009) The numerical solution of boundary-value problems for an elastic body with an elliptic hole and linear cracks. J Eng Math 65(2):111–123. doi:
15. 15.
Crouch SL, Starfield AM (1983) Boundary element methods in solid mechanics. Allen & Unwin, London
16. 16.
Sokolnikoff IS (1956) Mathematical theory of elasticity, 2nd edn. McGraw-Hill, New York
17. 17.
Muskhelishvili N (1953) Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen
18. 18.
Kantorovich LV, Krilov VI (1962) Approximate methods of higher analysis. Gos Izdat Phiz-Mat Lit, Moscow–Leningrad (708 p, in Russian) Google Scholar
19. 19.
Zirakashvili N (1992) Investigation and solution of infinite systems obtained by solution of some boundary value problems. Proc I Vekua Inst Appl Math 46:119–123 (in Russian)