Variational formulation and numerical algorithm for the structural non-uniform surface wear-out by rigid punch problem solution

Аuthors

Bobylov A. A.^*, Belashova I. S.^**

Moscow Automobile and Road Construction State Technical University (MADI), MADI, 64, Leningradsky Prospect, Moscow, 125319, Russia

*e-mail: abobylov@gmail.com
**e-mail: irina455@inbox.ru.

Abstract

The paper considers a flat contact problem on an elastic half-plane with structurally non-uniform surface wear-out by the rigid wear-free punch. While stating the problem we suppose that elastic characteristics of the material are structurally tolerant and identical in all points of the body, and parameters characterizing wear- resistant properties of the material depend on the value of linear wear. We also suppose predetermined the main vector and the main moment of force applied to the rigid punch.

To solve the contact wear-out problem under consideration we implemented variational approach. As a result, we obtained the problem formulation in tensions in terms of the system of quasivariational inequality of evolutionary type and first-order differential equation. For the problem time sampling, Euler explicit difference scheme was used. Thus, to determine contact pressure an elliptic inequity or extremum problem equal to it should be solved on each time step. We performed the problems quantization over spatial coordinates using the space of integrated fundamental solutions of the Flamant on the effect of normal concentrated force over the surface of elastic half-plane. We used boundary element approach to plot the surface of integrated fundamental solutions. Elements with a uniform distribution of contact pressure were used. As a result of sampling we formulated the problem of quadratic programming with restrictions in the form of equalities. The linear transformation of variables, allowing simplification of the restrictions was suggested. For numerical solution of the problem we used a variant of conjugate gradients method taking into account specificity of restrictions.

The calculations carried out showed that structural heterogeneity over the depth of the wear-out surface significantly influences the nature of the interface wear-in process.

The developed computational algorithm can be implemented for study of the efficiency of various technologies for superficial hardening by the method of computing experiment.

Keywords:

structurally non-uniform surface, wear-out contact problem, variational inequality, integrated fundamental solutions, boundary element method

References

1. Belashova I.S., Shashkov D.P. Poverkhnostnoe uprochnenie instrumental’nykh stalei s primeneniem lazernogo nagreva (Surface Hardening of Tool Steels Using Laser Heating), Moscow, Tehpoligrafcentr, 2004, 147 p.

2. Shashkov D.P., Belashova I.S. Poverkhnostnoe uprochnenie instrumental’nykh stalei (Surface Hardening of Tool Steels), Moscow, Tehpoligrafcentr, 2004, 376 p.

3. Galin L.A. Kontaktnye zadachi teorii uprugosti i vyazkouprugosti (Contact Problems of Theory of Elasticity and Viscoelasticity), Moscow, Nauka, 1980, 303 p.

4. Goryacheva I.G., Dobychin M.N. Kontaktnye zadachi v tribologii (Contact problems in tribology), Moscow, Mashinostroenie, 1988, 256 p.

5. Goryacheva I.G. Mekhanika friktsionnogo vzaimodeistviya (Mechanics of Friction Interaction), Moscow, Nauka, 2001, 478 p.

6. Soldatenkov I.A. Iznosokontaktnaya zadacha s prilozheniyami k inzhenernomu raschetu iznosa (Wear-out Contact Problem with Application for Engineering Calculation of Wear-out), Moscow, Fizmatkniga, 2010, 160 p.

7. Goryacheva I.G., Soldatenkov I.A. Mekhanika kontaktnykh vzaimodeistvii. Sbornik statei, Moscow, FIZMATLIT, 2001, pp. 438-458.

8. Duvaut G., Lions J.-L. Les Inequations en Mecanique et en Physique, Paris, Dunod, 1972, 405 p.

9. Kalker J.J. Variational principles of contact elastostatics. Journal of the Institute of Mathematics and its Applications, 1977, vol. 20 (2), pp. 199-219.

10. Kravchuk A.S. Variatsionnye i kvazivariatsionnye neravenstva v mekhanike (Variational and Quasivariational Inequalities in Mechanics), Moscow, MGAPI, 1997, 340 p.

11. Panagiotopoulos P. Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Boston-Basel-Stuttgart, Birkhauser, 1985, 412 p.

12. Bobylov A.A. Sovremennye problemy mekhaniki kontaktnykh vzaimodeistvii. Sbornik statei, Dnepropetrovsk, DGU, 1990, pp. 49-52.

13. Temam R. Mathematical Problems in Plasticity, Paris, Gauthier-Villars, 1985, 354 p.

14. Aleksidze M.A. Fundamental’nye funktsii v priblizhennykh resheniyakh granichnykh zadach (Fundamental Functions in Approximate Solution of Boundary Problems), Moscow, Nauka, Fizmatlit, 1991, 352 p.

15. Ugodchikov A.G., Khutoryanskii N.M. Metod granichnykh elementov v mekhanike deformiruemogo tverdogo tela (Boundary Element Method in Solid Mechanics), Kazan’, Izdatel’stvo Kazanskogo universiteta, 1986, 295 p.

16. Ciarlet P. The Finite Element Method for Elliptic Problems, Amsterdam-New York-Oxford, North-Holland, 1978, 530 p.

17. Bobylov A.A. Reshenie prikladnykh zadach matematicheskoi fiziki i diskretnoi matematiki. Sbornik statei, Dnepropetrovsk, DGU, 1987, pp. 23-29.