Approximation of dynamic and quasi-static evolution problems in elasto-plasticity by cap models

Babadjian, Jean-François; Mora, Maria

doi:10.1090/S0033-569X-2015-01376-8

Authors: Jean-François Babadjian and Maria Giovanna Mora
Journal: Quart. Appl. Math. 73 (2015), 265-316
MSC (2010): Primary 74C05, 74C10; Secondary 74G65, 49J45, 35Q72
DOI: https://doi.org/10.1090/S0033-569X-2015-01376-8
Published electronically: February 3, 2015
MathSciNet review: 3357495
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Abstract: This work is devoted to the analysis of elasto-plasticity models arising in soil mechanics. Contrary to the typical models mainly used for metals, it is required here to take into account plastic dilatancy due to the sensitivity of granular materials to hydrostatic pressure. The yield criterion thus depends on the mean stress, and the elasticity domain is unbounded and not invariant in the direction of hydrostatic matrices. In the mechanical literature, so-called cap models have been introduced, where the elasticity domain is cut in the direction of hydrostatic stresses by means of a strain-hardening yield surface, called a cap. The purpose of this article is to study the well-posedness of plasticity models with unbounded elasticity sets in dynamical and quasi-static regimes. An asymptotic analysis as the cap is moved to infinity is also performed, which enables one to recover solutions to the uncapped model of perfect elasto-plasticity.

References

Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
Gabriele Anzellotti, On the existence of the rates of stress and displacement for Prandtl-Reuss plasticity, Quart. Appl. Math. 41 (1983/84), no. 2, 181–208. MR 719504, DOI https://doi.org/10.1090/S0033-569X-1983-0719504-7
G. Anzellotti and S. Luckhaus, Dynamical evolution of elasto-perfectly plastic bodies, Appl. Math. Optim. 15 (1987), no. 2, 121–140. MR 868903, DOI https://doi.org/10.1007/BF01442650
A. Bensoussan and J. Frehse, Asymptotic behaviour of the time dependent Norton-Hoff law in plasticity theory and $H^1$ regularity, Comment. Math. Univ. Carolin. 37 (1996), no. 2, 285–304. MR 1399003
Sören Bartels, Alexander Mielke, and Tomáš Roubíček, Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM J. Numer. Anal. 50 (2012), no. 2, 951–976. MR 2914293, DOI https://doi.org/10.1137/100819205
H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
Krzysztof Chełmiński, Perfect plasticity as a zero relaxation limit of plasticity with isotropic hardening, Math. Methods Appl. Sci. 24 (2001), no. 2, 117–136. MR 1808687, DOI https://doi.org/10.1002/1099-1476%2820010125%2924%3A2%3C117%3A%3AAID-MMA201%3E3.0.CO%3B2-%23
Krzysztof Chełmiński, Coercive approximation of viscoplasticity and plasticity, Asymptot. Anal. 26 (2001), no. 2, 105–133. MR 1832581
Gianni Dal Maso, Gilles A. Francfort, and Rodica Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005), no. 2, 165–225. MR 2186036, DOI https://doi.org/10.1007/s00205-004-0351-4
Gianni Dal Maso, Antonio DeSimone, and Maria Giovanna Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal. 180 (2006), no. 2, 237–291. MR 2210910, DOI https://doi.org/10.1007/s00205-005-0407-0
Gianni Dal Maso, Antonio DeSimone, and Francesco Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling, Calc. Var. Partial Differential Equations 40 (2011), no. 1-2, 125–181. MR 2745199, DOI https://doi.org/10.1007/s00526-010-0336-0
F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J. 33 (1984), no. 5, 673–709. MR 756154, DOI https://doi.org/10.1512/iumj.1984.33.33036
F. Demengel and R. Temam, Convex function of a measure: the unbounded case, FERMAT days 85: mathematics for optimization (Toulouse, 1985) North-Holland Math. Stud., vol. 129, North-Holland, Amsterdam, 1986, pp. 103–134. MR 874363, DOI https://doi.org/10.1016/S0304-0208%2808%2972396-X
A. Demyanov, Regularity of stresses in Prandtl-Reuss perfect plasticity, Calc. Var. Partial Differential Equations 34 (2009), no. 1, 23–72. MR 2448309, DOI https://doi.org/10.1007/s00526-008-0174-5

F. L. DiMaggio and I. S. Sandler, Material model for granular soils, J. Engrg. Mech. 97 (1971) 935–950.

D. C. Drucker and W. Prager, Soil mechanics and plastic analysis or limit design, Quart. Appl. Math. 10 (1952), 157–165. MR 48291, DOI https://doi.org/10.1090/S0033-569X-1952-48291-2

G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 (French). Travaux et Recherches Mathématiques, No. 21. MR 0464857

Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943

Gilles A. Francfort and Alessandro Giacomini, Small-strain heterogeneous elastoplasticity revisited, Comm. Pure Appl. Math. 65 (2012), no. 9, 1185–1241. MR 2954614, DOI https://doi.org/10.1002/cpa.21397

Casper Goffman and James Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159–178. MR 162902

R. Hill, The Mathematical Theory of Plasticity, Oxford, at the Clarendon Press, 1950. MR 0037721

Claes Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl. (9) 55 (1976), no. 4, 431–444. MR 438867

Robert Kohn and Roger Temam, Dual spaces of stresses and strains, with applications to Hencky plasticity, Appl. Math. Optim. 10 (1983), no. 1, 1–35. MR 701898, DOI https://doi.org/10.1007/BF01448377

J. Lubliner, Plasticity Theory, Macmillan Publishing Company, New York (1990).

S. Luckhaus, Elastisch-plastische Materialien mit Viskosität, Preprint no. 65 of Heidelberg University, SFB 123.

Andreas Mainik and Alexander Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations 22 (2005), no. 1, 73–99. MR 2105969, DOI https://doi.org/10.1007/s00526-004-0267-8

H. Matthies, G. Strang, and E. Christiansen, The saddle point of a differential program, Energy methods in finite element analysis, Wiley, Chichester, 1979, pp. 309–318. MR 537013

Alexander Mielke, Tomáš Roubíček, and Ulisse Stefanelli, $\Gamma $-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations 31 (2008), no. 3, 387–416. MR 2366131, DOI https://doi.org/10.1007/s00526-007-0119-4

J.-J. Moreau, Application of convex analysis to the treatment of elasto-plastic systems, Springer Lecture Notes in Math., no. 503, Ed. P. Germain and B. Nayroles (1975).

S. Repin and G. Serëgin, Existence of a weak solution of the minimax problem arising in Coulomb-Mohr plasticity, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, RI, 1995, pp. 189–220. MR 1334144, DOI https://doi.org/10.1090/trans2/164/09

L. Resende and J. B. Martin, Formulation of Drucker-Prager Cap Model, J. Engrg. Mech. 111 (1985) 855–881.

L. Resende and J. B. Martin, Closure of “Formulation of Drucker-Prager Cap Model”, J. Engrg. Mech. 113 (1987) 1257–1259.

Tomáš Roubíček, Rate-independent processes in viscous solids at small strains, Math. Methods Appl. Sci. 32 (2009), no. 7, 825–862. MR 2507935, DOI https://doi.org/10.1002/mma.1069

J. Salençon, Application of the Theory of Plasticity to Soil Mechanics, Wiley Ser. Geotechnical Engrg., John Wiley, New York (1977).

G. A. Serëgin, Differential properties of the stress tensor in the Coulomb-Mohr theory of plasticity, Algebra i Analiz 4 (1992), no. 6, 234–252 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 6, 1257–1272. MR 1199642

Pierre-M. Suquet, Sur les équations de la plasticité: existence et régularité des solutions, J. Mécanique 20 (1981), no. 1, 3–39 (French, with English summary). MR 618942

Pierre-M. Suquet, Evolution problems for a class of dissipative materials, Quart. Appl. Math. 38 (1980/81), no. 4, 391–414. MR 614549, DOI https://doi.org/10.1090/S0033-569X-1981-0614549-X

Pierre-M. Suquet, Un espace fonctionnel pour les équations de la plasticité, Ann. Fac. Sci. Toulouse Math. (5) 1 (1979), no. 1, 77–87 (French, with English summary). MR 533600

Roger Temam, Problèmes mathématiques en plasticité, Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science], vol. 12, Gauthier-Villars, Montrouge, 1983 (French). MR 711964

Roger Temam, A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity, Arch. Rational Mech. Anal. 95 (1986), no. 2, 137–183. MR 850094, DOI https://doi.org/10.1007/BF00281085

Roger Temam and Gilbert Strang, Existence de solutions relaxées pour les équations de la plasticité: étude d’un espace fonctionnel, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A515–A518 (French, with English summary). MR 512094

Roger Temam and Gilbert Strang, Functions of bounded deformation, Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 7–21. MR 592100, DOI https://doi.org/10.1007/BF00284617

Roger Temam and Gilbert Strang, Duality and relaxation in the variational problems of plasticity, J. Mécanique 19 (1980), no. 3, 493–527 (English, with French summary). MR 595981