Persoz’s gephyroidal model described by a maximal monotone differential inclusion. (English) Zbl 1161.74303
Summary: Persoz’s gephyroidal model, which consists of elementary rheological models (dry friction element and linear spring), can be covered by the existence and uniqueness theory for maximal monotone operators. Moreover, classical results of numerical analysis allow one to use a numerical implicit Euler scheme, with convergence order of the scheme equal to one. Some numerical simulations are presented.
MSC:
| 74A20 | Theory of constitutive functions in solid mechanics |
| 74H20 | Existence of solutions of dynamical problems in solid mechanics |
| 74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |
Keywords:
gephyroidal model; Persoz’s model; elastoplasticity; maximal monotone operator; differential inclusionReferences:
| [1] | Bastien, J.: Étude théorique et numérique d’inclusions différentielles maximales monotones. Applications à des mod��les élastoplastiques. Ph.D. thesis, Université Lyon I, numéro d’ordre: 96 (2000) |
| [2] | Bastien J. and Schatzman M. (2000). Schéma numérique pour des inclusions différentielles avec terme maximal monotone. CR Acad. Sci. Paris Sér. I Math. 330(7): 611–615 · Zbl 0951.65059 |
| [3] | Bastien J. and Schatzman M. (2002). Numerical precision for differential inclusions with uniqueness. M2AN Math. Model. Numer. Anal. 36(3): 427–460 · Zbl 1036.34012 · doi:10.1051/m2an:2002020 |
| [4] | Bastien J., Schatzman M. and Lamarque C.H. (2000). Study of some rheological models with a finite number of degrees of freedom. Eur. J. Mech. A Solids 19(2): 277–307 · Zbl 0954.74011 · doi:10.1016/S0997-7538(00)00163-7 |
| [5] | Bastien J., Schatzman M. and Lamarque C.H. (2002). Study of an elastoplastic model with an infinite number of internal degrees of freedom. Eur. J. Mech. A Solids 21(2): 199–222 · Zbl 1023.74009 · doi:10.1016/S0997-7538(01)01205-0 |
| [6] | Bastien J., Michon G., Manin L. and Dufour R. (2007). An analysis of the modified Dahl and Masing models: Application to a belt tensioner. J. Sound Vib. 302(4–5): 841–864 · Zbl 1242.74025 · doi:10.1016/j.jsv.2006.12.013 |
| [7] | Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50) (1973) |
| [8] | Coleman T.F. and Li Y. (1996). A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables. SIAM J. Optim. 6(4): 1040–1058 · Zbl 0861.65053 · doi:10.1137/S1052623494240456 |
| [9] | Gill P.E., Murray W. and Wright M.H. (1981). Practical optimization. Academic/Harcourt Brace Jovanovich, London · Zbl 0503.90062 |
| [10] | Persoz, B. (ed.): La Rhéologie: recueil de travaux des sessions de perfectionnement, Institut national des sciences appliquTes, Lyon. Monographies du Centre d’actualisation scientifique et technique, 3, Masson, Paris (1969) · Zbl 0179.55702 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
