Email this article The problem of collective indentation of an elastic half-plane by a system of rigid punches elastically connected to a common platform
- Authors: Bobylev A.A.1
-
Affiliations:
- Lomonosov Moscow State University
- Issue: Vol 89, No 2 (2025)
- Pages: 280-294
- Section: Articles
- URL: https://bakhtiniada.ru/0032-8235/article/view/306633
- DOI: https://doi.org/10.31857/S0032823525020078
- EDN: https://elibrary.ru/ilomzx
- ID: 306633
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Abstract
The problem is considered for the indentation of an elastic half-plane by a system of rigid punches elastically connected to a common rigid platform. A variational formulation of the problem are obtained in the form of a boundary variational inequality using the Poincare–Steklov operator for an elastic half-plane. A minimization problem equivalent to the variational inequality is given, for approximation of which the boundary-element approach is used. As a result, a quadratic programming problem with equality and inequality restrictions is obtained, for the numerical solution of which an algorithm based on the conjugate gradient method was used. Patterns of collective indentation of elastic half-plane by a system of rigid punches elastically connected to a common platform have been investigated by computational experiment.
About the authors
A. A. Bobylev
Lomonosov Moscow State University
Author for correspondence.
Email: abobylov@gmail.com
Moscow, Russia
References
- Walley S.M. Historical origins of indentation hardness testing // Mater. Sci.&Technol., 2012, vol. 28, no. 9–10, pp. 1028–1044.
- Golovin Yu.I. Nanoindentation and Its Capabilities. Moscow: Mashinostroenie, 2009. 312 p. (in Russian)
- Sadovnichy V.A., Goryacheva I.G., Akaev A.A. et al. Application of Contact Mechanics Methods in the Diagnosis of Pathological Conditions of Soft Biological Tissues. Moscow: MSU, 2009. 306 p. (in Russian)
- Gorinevsky DM, Formalsky AM, Schneider AJ. Manipulation Systems Management Based on Effort Information. Moscow: Nauka, 1994. 368 p. (in Russian)
- Argatov I.I., Jin X., Keer L.M. Collective indentation as a novel strategy for mechanical palpation tomography // J. of the Mech.&Phys. of Solids, 2020, vol. 143, art. no. 104063.
- Muskhelishvili N.I. Some Basic Problems of the Mathematical Theory of Elasticity. Dordrecht: Springer, 1977. 732 p.
- Timoshenko S., Goodier J.N. Theory of Elasticity. N.Y.: McGraw-Hill, 1970. 591 p.
- Bobylev A.A. On the positive definiteness of the Poincaré–Steklov operator for elastic half-plane // Moscow Univ. Mech. Bull., 2021, vol. 76, no. 6, pp. 156–162.
- Kravchuk A.S., Neittaanmäki P.J. Variational and Quasi-Variational Inequalities in Mechanics. Dordrecht: Springer, 2007. 329 p.
- Eck C., Jarušek J., Krbec M. Unilateral Contact Problems: Variational Methods and Existence Theorems. N.Y.: CRC Press, 2005. 398 p.
- Sofonea M., Matei A. Mathematical Models in Contact Mechanics. Cambridge: Univ. Press, 2012. 280 p.
- Capatina A. Variational Inequalities and Frictional Contact Problems. Cham: Springer, 2014. 235 p.
- Steinbach O. Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. N.Y.: Springer, 2008. 386 p.
- Bobylev A.A. Application of the conjugate gradient method to solving discrete contact problems for an elastic half-plane // Mech. of Solids, 2022, vol. 57, no. 2, pp. 317–332.
- Zabreiko P.P., Koshelev A.I., Krasnoselsky M.A. et al. Integral Equations. Moscow: Nauka, 1968. 448 p. (in Russian)
- Khludnev A.M. Elasticity Problems in Non-Smooth Domains. Moscow: Fizmatlit, 2010. 252 p. (in Russian)
- Gwinner J., Stephan E.P. Advanced Boundary Element Methods. Treatment of Boundary Value, Transmission and Contact Problems. Cham: Springer, 2018. 652 p.
- Rjasanow S., Steinbach O. The Fast Solution of Boundary Integral Equations. N.Y.: Springer, 2007. 284 p.
- Sauter S.A., Schwab C. Boundary Element Methods. Berlin;Heidelberg: Springer, 2011. 652 p.
- Davis P.J. Circulant Matrices. N.Y.: Wiley, 1979. 250 p.
- Wang Q.J., Sun L., Zhang X. et al. FFT-based methods for computational contact mechanics // Front. Mech. Eng., 2020, vol. 6, no. 61, pp. 92–113.
- Bobylev A.A. Algorithm for solving discrete contact problems for an elastic strip // Mech. of Solids, 2022, vol. 57, no. 7, pp. 1766–1780.
- Bobylev A.A. The unilateral discrete contact problem for a functionally graded elastic strip // Moscow Univ. Mech. Bull., 2024, vol. 79, no. 2, pp. 56–68.
- Bobylev A.A. Algorithm for solving unilateral discrete contact problems for a multilayer elastic strip // J. Appl. Mech. Tech. Phys., 2024, vol. 65, no. 2, pp. 382–392.
- Bobylev A.A. Algorithm for solving discrete contact problems for an elastic layer // Mech. of Solids, 2023, vol. 58, no. 2, pp. 439–454.
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