A variational formulation governed by two bipotentials for a frictionless contact model
Abstract
We consider a frictionless contact model whose constitutive law and contact condition are described by means of subdifferential inclusions. For this model, we deliver a variational formulation based on two bipotentials. Our formulation envisages the computation of a three-field unknown consisting of the displacement vector, the stress tensor and the normal stress on the contact zone, the contact being described by a generalized Winkler condition. Subsequently, we obtain existence and uniqueness results. Some properties of the solution are also discussed, focusing on the data dependence.
Keyword : contact condition, subdifferential inclusion, bipotentials, three-field weak solution, data dependence
How to Cite
Matei, A., & Osiceanu, M. (2024). A variational formulation governed by two bipotentials for a frictionless contact model. Mathematical Modelling and Analysis, 29(1), 109–124. https://doi.org/10.3846/mma.2024.17944
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Matei and M. Osiceanu. Weak solvability via bipotentials and approximation results for a class of bilateral frictional contact problems. Commun. Nonlinear Sci. Numer. Simul., 119:107135, 2023. https://doi.org/10.1016/j.cnsns.2023.107135
A. Matei and M. Osiceanu. Weak solvability via bipotentials for contact problems with power-law friction. J. Math. Anal. Appl., 524(1):127064, 2023. https://doi.org/10.1016/j.jmaa.2023.127064
U. Mosco. Convergence of convex sets and of solutions of variational inequalities. Adv. in Math., 3(4):510–585, 1969. https://doi.org/10.1016/0001-8708(69)90009-7
C.P. Niculescu and L.-E. Persson. Convex Functions and their Applications. A Contemporary Approach. Springer-Verlag, 2006. https://doi.org/10.1007/0-387-31077-0_2
P.D. Panagiatopoulos. Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkha¨user, Basel, 1985.
M. Sofonea. Problèmes Non-Linéaires dans la Théorie de l’Élasticité, Cours de Magister de Mathématiques Appliquées. Université de Setif, Algérie, 1993.
M. Sofonea and A. Matei. Mathematical Models in Contact Mechanics. Cambridge University Press, 2012. https://doi.org/10.1017/CBO9781139104166
L. Tao, Y. Li, Z.Q. Feng, Y.J. Cheng and H.J. Chen. Bi-potential method applied for dynamics problems of rigid bodies involving friction and multiple impacts. Nonlinear Dyn., 106:1823–1842, 2021. https://doi.org/10.1007/s11071-021-06916-z
S. Zeng, S. Migórski and A. Khan. Nonlinear quasi-hemivariational inequalities: existence and optimal control. SIAM J. Control Optim., 59:1246–1274, 2021. https://doi.org/10.1137/19M1282210
S. Zeng, S. Migórski and Y. Liu. Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities. SIAM J. Optim., 31:2829–2862, 2021. https://doi.org/10.1137/20M1351436
V. Barbu and T. Precupanu. Convexity and Optimization in Banach Spaces. Springer, New York, 2012. https://doi.org/10.1007/978-94-007-2247-7
H. Bauschke and Y. Lucet. What is... a Fenchel conjugate. Notices of the American Mathematical Society, 59(1):44–46, 2012. https://doi.org/10.1090/noti788
G. Bodovillé and G. de Saxcé. Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach. Eur. J. Mech. A Solids, 20(1):99–112, 2001. https://doi.org/10.1016/S0997-7538(00)01109-8
M. Buliga, G. de Saxcé and C. Vallée. Existence and construction of bipotentials for graphs of multivalued laws. J. Convex Anal., 15(1):87–104, 2008. https://doi.org/10.48550/arXiv.math/0608424
M. Buliga, G. de Saxcé and C. Vallée. Bipotentials for non monotone multivalued operators: fundamental results and applications. Acta Appl. Math., 110(2):955– 972, 2010. https://doi.org/10.1007/s10440-009-9488-3
M. Buliga, G. de Saxcé and C. Vallée. A variational formulation for constitutive laws described by bipotentials. Math. Mech. Solids, 18(1):78–90, 2013. https://doi.org/10.1177/1081286511436136
H.J. Chen, Z.Q. Feng, Y.H. Du, Q.W. Chen and H.C. Miao. Spectral finite element method for efficient simulation of nonlinear interactions between lamb waves and breathing cracks within the bi-potential framework. Int. J. Mech. Sci., 215:106954, 2022. https://doi.org/10.1016/j.ijmecsci.2021.106954
N. Costea, M. Csirik and C. Varga. Weak solvability via bipotential method for contact models with nonmonotone boundary conditions. Z. Angew. Math. Phys., 66(5):2787–2806, 2015. https://doi.org/10.1007/s00033-015-0513-2
G. de Saxcé and Z.Q. Feng. New inequality and functional for contact with friction: the implicit standard material approach. Mech. Struct. Mach., 19(3):301–325, 1991. https://doi.org/10.1080/08905459108905146
W. Fenchel. On conjugate convex functions. Can. J. Math., 1(1):73–77, 1949. https://doi.org/10.4153/CJM-1949-007-x
J. Fortin, O. Millet and G. de Saxcé. Numerical simulation of granular materials by an improved discrete element method. Int. J. Numer. Methods Eng., 62(5):639–663, 2005. https://doi.org/10.1002/nme.1209
M. Hjiaj, G. Bodovillé and G. de Saxcé. Matériaux viscoplastiques et loi de normalité implicites. C. R. Acad. Sci., 328(7):519–524, 2000. https://doi.org/10.1016/S1620-7742(00)00007-6
L.B. Hu, Y. Cong, P. Joli and Z.Q. Feng. A bi-potential contact formulation for recoverable adhesion between soft bodies based on the rcc interface model. Comput. Methods Appl. Mech. Eng., 390:114478, 2022. https://doi.org/10.1016/j.cma.2021.114478
Y. Liu, S. Migórski, V. Nguyen and S. Zeng. Existence and convergence results for elastic frictional contact prob- lem with nonmonotone subdifferential boundary condtions. Acta Math. Sci., 41:1–18, 2021. https://doi.org/10.1007/s10473-021-0409-5
A. Matei. A variational approach via bipotentials for unilateral contact problems. J. Math. Anal. Appl., 397(1):371–380, 2013. https://doi.org/10.1016/j.jmaa.2012.07.065
A. Matei. A variational approach via bipotentials for a class of frictional contact problems. Acta Appl. Math., 134(1):45–59, 2014. https://doi.org/10.1007/s10440-014-9868-1
A. Matei and C. Niculescu. Weak solutions via bipotentials in mechanics of deformable solids. J. Math. Anal. Appl., 379(1):15–25, 2011. https://doi.org/10.1016/j.jmaa.2010.12.016
A. Matei and M. Osiceanu. Two-field variational formulations for a class of nonlinear mechanical models. Math. Mech. Solids, 27(11):2532–2547, 2022. https://doi.org/10.1177/10812865211066123
A. Matei and M. Osiceanu. Two-field weak solutions for a class of contact models. Mathematics, 10(3):369, 2022. https://doi.org/10.3390/math10030369
A. Matei and M. Osiceanu. Weak solvability via bipotentials and approximation results for a class of bilateral frictional contact problems. Commun. Nonlinear Sci. Numer. Simul., 119:107135, 2023. https://doi.org/10.1016/j.cnsns.2023.107135
A. Matei and M. Osiceanu. Weak solvability via bipotentials for contact problems with power-law friction. J. Math. Anal. Appl., 524(1):127064, 2023. https://doi.org/10.1016/j.jmaa.2023.127064
U. Mosco. Convergence of convex sets and of solutions of variational inequalities. Adv. in Math., 3(4):510–585, 1969. https://doi.org/10.1016/0001-8708(69)90009-7
C.P. Niculescu and L.-E. Persson. Convex Functions and their Applications. A Contemporary Approach. Springer-Verlag, 2006. https://doi.org/10.1007/0-387-31077-0_2
P.D. Panagiatopoulos. Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkha¨user, Basel, 1985.
M. Sofonea. Problèmes Non-Linéaires dans la Théorie de l’Élasticité, Cours de Magister de Mathématiques Appliquées. Université de Setif, Algérie, 1993.
M. Sofonea and A. Matei. Mathematical Models in Contact Mechanics. Cambridge University Press, 2012. https://doi.org/10.1017/CBO9781139104166
L. Tao, Y. Li, Z.Q. Feng, Y.J. Cheng and H.J. Chen. Bi-potential method applied for dynamics problems of rigid bodies involving friction and multiple impacts. Nonlinear Dyn., 106:1823–1842, 2021. https://doi.org/10.1007/s11071-021-06916-z
S. Zeng, S. Migórski and A. Khan. Nonlinear quasi-hemivariational inequalities: existence and optimal control. SIAM J. Control Optim., 59:1246–1274, 2021. https://doi.org/10.1137/19M1282210
S. Zeng, S. Migórski and Y. Liu. Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities. SIAM J. Optim., 31:2829–2862, 2021. https://doi.org/10.1137/20M1351436