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Article

Kovtunenko, Victor A. ; Krejčí, Pavel ; Bauer, Erich ; Siváková, Lenka ; Zubkova, Anna V.
On Lyapunov stability in hypoplasticity. (English). In: Mikula, Karol (ed.): Proceedings of Equadiff 14. Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017. Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, Bratislava, 2017. pp. 107-116
MSC: 34D20, 37B25, 74C15
Keywords:
Nonlinear ODE, rate-independent problem, asymptotic behavior, attractor, Lyapunov function, proportional loading, hypoplasticity, granular media
Summary:
We investigate the Lyapunov stability implying asymptotic behavior of a nonlinear ODE system describing stress paths for a particular hypoplastic constitutive model of the Kolymbas type under proportional, arbitrarily large monotonic coaxial deformations. The attractive stress path is found analytically, and the asymptotic convergence to the attractor depending on the direction of proportional strain paths and material parameters of the model is proved rigorously with the help of a Lyapunov function.
References:
[1] Annin, B. D., Kovtunenko, V. A., Sadovskii, V. M.: Variational and hemivariational inequalities in mechanics of elastoplastic, granular media, and quasibrittle cracks. in Analysis, Modelling, Optimization, and Numerical Techniques, G. O. Tost, O. Vasilieva, eds.,Springer Proc. Math. Stat., 121 (2015), 49–56. MR 3354167
[2] Bauer, E.: Modelling limit states within the framework of hypoplasticity. AIP Conf. Proc., 1227, J. Goddard, P. Giovine and J. T. Jenkin, eds., AIP, 2010, pp. 290–305.
[3] Bauer, E., Wu, W.: A hypoplastic model for granular soils under cyclic loading. Proc. Int. Workshop Modern Approaches to Plasticity, D. Kolymbas, ed., Elsevier, 2010, pp. 247–258.
[4] Brokate, M., Krejčí, P.: Wellposedness of kinematic hardening models in elastoplasticity. RAIRO Modél. Math. Anal. Numér., (1998), 177–209. DOI 10.1051/m2an/1998320201771 | MR 1622607
[5] Gudehus, G.: Physical Soil Mechanics. Springer, Berlin, Heidelberg, 2011.
[6] Huang, W., Bauer, E.: Numerical investigations of shear localization in a micro-polar hypoplastic material. Int. J. Numer. Anal. Meth. Geomech., 27 (2003), pp. 325–352. DOI 10.1002/nag.275
[7] Khludnev, A.M., Kovtunenko, V.A.: Analysis of Cracks in Solids. WIT-Press, Southampton, Boston, 2000.
[8] Kolymbas, D.: An outline of hypoplasticity. Arch. Appl. Mech., 61 (1991), pp. 143–151.
[9] Kovtunenko, V.A., Zubkova, A.V.: Mathematical modeling of a discontinuous solution of the generalized Poisson–Nernst–Planck problem in a two-phase medium. Kinet. Relat. Mod., 11 (2018), pp.119–135. DOI 10.3934/krm.2018007 | MR 3708185
[10] Niemunis, A.: Extended Hypoplastic Models for Soils. Habilitation thesis, Ruhr University, Bochum, 2002.
[11] Niemunis, A., Herle, I.: Hypoplastic model for cohesionless soils with elastic strain range. Mech. Cohes.-Frict. Mat., 2 (1997), pp. 279–299. DOI 10.1002/(SICI)1099-1484(199710)2:4<279::AID-CFM29>3.0.CO;2-8
[12] Svendsen, B., Hutter, K., Laloui, L.: Constitutive models for granular materials including quasi-static frictional behaviour: toward a thermodynamic theory of plasticity. Continuum Mech. Therm., 4 (1999), pp. 263–275. DOI 10.1007/s001610050115 | MR 1710675
[13] Wu, W., Bauer, E., Kolymbas, D.: Hypoplastic constitutive model with critical state for granular materials. Mech. Mater., 23 (1996), pp. 45–69. DOI 10.1016/0167-6636(96)00006-3
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