Research Article
Year 2023,
Volume: 44 Issue: 2, 336 - 344, 30.06.2023
Abstract
The elasto-plastic contact problem with an unknown contact domain (UCD) has attracted mathematicians, mechanics and engineers for decades. So, the problem of determining the stresses in the UCD is very important nowadays in terms of engineering and applied mathematics. To improve the finite element model, the remeshing algorithm is used for the considered indentation problem. The algorithm allows the determination of the UCD at each step of the indentation with high accuracy. This paper presents the analysis and numerical solution of the boundary value problem for the Lame system, and the modeling of the contact problem for rigid materials. By using barycentric coordinates, the finite difference approximation of the mathematical model of the deformation problem with undetermined bounded is obtained and the relations between the finite elements and finite differences are investigated.
References
- [1] Hlavacek I., Haslinger J., Necas J., Lovisek J., Solution of Variational Inequalities in Mechanics. Berlin: Springer, (1988) 109-262.
- [2] Duvaut G., Lions J.L., Inequalities in Mechanics and Physics. Berlin: Springer-Verlag, (1989) 102-227.
- [3] Kinderlehrer D., Stampacchia G., An Introduction to Variational Inequalities and Their Applications. New York: Academic Press, (1980) 1-274.
- [4] Signorini A., Sopra Alcune Questioni di Elastostatica, Atti della Societa Italiana per il Progresso delle Scienze, 2 (2) (1933) 231-251.
- [5] Fichera G., Existence Theorems in Elasticity. In: Truesdell C., (Eds). Linear Theories of Elasticity and Thermoelasticity. Springer-Verlag Berlin Heidelberg, (1973) 347-389.
- [6] Lions J.L., Some Methods of Solving Non-Linear Boundary Value Problems. Paris: Dunod-Gauthier-Villars, (1969) 280-309.
- [7] Glovinski R., Lions J.L., Trémolierès R., Analyse Numérique des Inéquations Variationnelles. Paris: Dunod, (1976) 1-268.
- [8] Lee A., Komvopoulos K., Dynamic Spherical Indentation of Elastic-Plastic Solids, International Journal of Solids and Structures, 146 (2018) 180-191.
- [9] Wagih A., Maim P., Blanco N., Trias D., Predictive Model for the Spherical Indentation of Composite Laminates with Finite Thickness, Composite Structures, 153 (2016) 468-477.
- [10] Gasanov A.I., Numerical Method for Solving a Contact Problem of Elasticity Theory in the Absence of Friction Forces, Differ. Uravn., 18 (7) (1982) 1156–1161.
- [11] Hasanov A., Seyidmamedov Z, Qualitative Behaviour of Solutions of Signorini Problem with Perturbing the Unknown Boundary II. The Multigrid Method, Applied Mathematics and Computation, 109 (2000) 261-271.
- [12] Weng P., Yin X., Hu W., Yuan H., Chen C., Ding H., Yu B., Xie W., Jiang L., Wang H., Piecewise linear deformation characteristics and a contact model for elastic-plastic indentation considering indenter elasticity, Tribology International, 162 (2021) 107-114.
- [13] Ilyushin A.A., Plasticity. Moscow: Gostekhizdat, (1948) 144-268, in Russian.
- [14] Norrie D.H., De Vries G., An Introduction to Finite Element Analysis. New York: Academic Press, (1978) 1-301.
- [15] Ciarlet P.G., The Finite Element Method for Elliptic Problems. 2nd ed. SIAM, (2002) 287-380.
Year 2023,
Volume: 44 Issue: 2, 336 - 344, 30.06.2023
Abstract
References
- [1] Hlavacek I., Haslinger J., Necas J., Lovisek J., Solution of Variational Inequalities in Mechanics. Berlin: Springer, (1988) 109-262.
- [2] Duvaut G., Lions J.L., Inequalities in Mechanics and Physics. Berlin: Springer-Verlag, (1989) 102-227.
- [3] Kinderlehrer D., Stampacchia G., An Introduction to Variational Inequalities and Their Applications. New York: Academic Press, (1980) 1-274.
- [4] Signorini A., Sopra Alcune Questioni di Elastostatica, Atti della Societa Italiana per il Progresso delle Scienze, 2 (2) (1933) 231-251.
- [5] Fichera G., Existence Theorems in Elasticity. In: Truesdell C., (Eds). Linear Theories of Elasticity and Thermoelasticity. Springer-Verlag Berlin Heidelberg, (1973) 347-389.
- [6] Lions J.L., Some Methods of Solving Non-Linear Boundary Value Problems. Paris: Dunod-Gauthier-Villars, (1969) 280-309.
- [7] Glovinski R., Lions J.L., Trémolierès R., Analyse Numérique des Inéquations Variationnelles. Paris: Dunod, (1976) 1-268.
- [8] Lee A., Komvopoulos K., Dynamic Spherical Indentation of Elastic-Plastic Solids, International Journal of Solids and Structures, 146 (2018) 180-191.
- [9] Wagih A., Maim P., Blanco N., Trias D., Predictive Model for the Spherical Indentation of Composite Laminates with Finite Thickness, Composite Structures, 153 (2016) 468-477.
- [10] Gasanov A.I., Numerical Method for Solving a Contact Problem of Elasticity Theory in the Absence of Friction Forces, Differ. Uravn., 18 (7) (1982) 1156–1161.
- [11] Hasanov A., Seyidmamedov Z, Qualitative Behaviour of Solutions of Signorini Problem with Perturbing the Unknown Boundary II. The Multigrid Method, Applied Mathematics and Computation, 109 (2000) 261-271.
- [12] Weng P., Yin X., Hu W., Yuan H., Chen C., Ding H., Yu B., Xie W., Jiang L., Wang H., Piecewise linear deformation characteristics and a contact model for elastic-plastic indentation considering indenter elasticity, Tribology International, 162 (2021) 107-114.
- [13] Ilyushin A.A., Plasticity. Moscow: Gostekhizdat, (1948) 144-268, in Russian.
- [14] Norrie D.H., De Vries G., An Introduction to Finite Element Analysis. New York: Academic Press, (1978) 1-301.
- [15] Ciarlet P.G., The Finite Element Method for Elliptic Problems. 2nd ed. SIAM, (2002) 287-380.
There are 15 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | June 30, 2023 |
| Submission Date | January 17, 2022 |
| Acceptance Date | April 30, 2023 |
| Published in Issue | Year 2023Volume: 44 Issue: 2 |