×
Knees, Dorothee; Rossi, Riccarda; Zanini, Chiara

A vanishing viscosity approach to a rate-independent damage model. (English) Zbl 1262.74030

Math. Models Methods Appl. Sci. 23, No. 4, 565-616 (2013).
Summary: We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps.{ }Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparametrization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

Cited in 1 Review
Cited in 82 Documents

MSC:

74R05 Brittle damage
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
35D40 Viscosity solutions to PDEs
35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators
49J40 Variational inequalities

Keywords:

rate-independent damage evolution; vanishing viscosity method; arclength reparametrization; time discretization
Cite Review PDF
Full Text: DOI

References:

[1] Ambrosio, L., Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 19, 191, 1995 · Zbl 0957.49029
[2] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems, 2000, The Clarendon Press · Zbl 0957.49001
[3] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2008, Birkhäuser-Verlag · Zbl 1145.35001
[4] Aubin, J.-P.; Ekeland, I., Applied Nonlinear Analysis, 1984, John Wiley & Sons · Zbl 0641.47066
[5] Babadjian, J.-F.; Francfort, G.; Mora, M., SIAM J. Math. Anal., 44, 245, 2012 · Zbl 1379.74006
[6] Bonetti, E.; Bonfanti, G., Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 1187, 2008 · Zbl 1152.35505
[7] Bonetti, E.; Schimperna, G., Cont. Mech. Thermodyn., 16, 319, 2004 · Zbl 1066.74048
[8] Bonetti, E.; Schimperna, G.; Segatti, A., J. Differential Equations, 218, 91, 2005 · Zbl 1078.74048
[9] Bouchitté, G.; Mielke, A.; Roubíček, T., ZAMP Z. Angew. Math. Phys., 60, 205, 2009 · Zbl 1238.74005
[10] Brezis, H., Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, 1973, North-Holland · Zbl 0252.47055
[11] Brezis, H.; Mironescu, P., J. Evolutionary Equations, 1, 387, 2001 · Zbl 1023.46031
[12] Cagnetti, F., Math. Models Methods Appl. Sci., 18, 1027, 2008 · Zbl 1154.49005
[13] Dal Maso, G., Arch. Rational Mech. Anal., 189, 469, 2008
[14] Dal Maso, G.; DeSimone, A.; Solombrino, F., Calc. Var. Partial Differential Equations, 40, 125, 2011 · Zbl 1311.74024
[15] Dal Maso, G.; DeSimone, A.; Solombrino, F., Calc. Var. Partial Differential Equations, 44, 495, 2012 · Zbl 1311.74025
[16] Efendiev, M.; Mielke, A., J. Convex Anal., 13, 151, 2006 · Zbl 1109.74040
[17] Fiaschi, A.; Knees, D.; Stefanelli, U., Arch. Rational Mech. Anal., 203, 415, 2012 · Zbl 1281.74005
[18] Francfort, G.; Garroni, A., Arch. Rational Mech. Anal., 182, 125, 2006 · Zbl 1098.74006
[19] Frémond, M.; Kenmochi, N., Adv. Math. Sci. Appl., 16, 697, 2006 · Zbl 1158.74310
[20] Frémond, M.; Nedjar, B., Int. J. Solids Struct., 33, 1083, 1996 · Zbl 0910.73051
[21] Garroni, A.; Larsen, C., Arch. Rational Mech. Anal., 194, 585, 2009 · Zbl 1222.74014
[22] Giacomini, A., Calc. Var. Partial Differential Equations, 22, 129, 2005 · Zbl 1068.35189
[23] Giusti, E., Direct Methods in the Calculus of Variations, 2003, World Scientific · Zbl 1028.49001
[24] Gröger, K., Math. Ann., 283, 679, 1989 · Zbl 0646.35024
[25] Heinemann, C.; Kraus, C., Adv. Math. Sci. Appl., 21, 321, 2011 · Zbl 1253.35179
[26] Herzog, R.; Meyer, C.; Wachsmuth, G., J. Math. Anal. Appl., 382, 802, 2011 · Zbl 1419.74125
[27] Ioffe, A. D., SIAM J. Control Optim., 15, 521, 1977 · Zbl 0361.46037
[28] Ioffe, A. D.; Tihomirov, V. M., Theory of Extremal Problems, 6, 1979, North-Holland · Zbl 0407.90051
[29] Kachanov, L., Introduction to Continuum Damage Mechanics, 10, 1986, Martinus Nijhoff · Zbl 0596.73091
[30] Knees, D.; Mielke, A.; Zanini, C., Math. Models Methods Appl. Sci., 18, 1529, 2008 · Zbl 1151.49014
[31] D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, WIAS-preprint 1633, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2011.
[32] Knees, D.Schröder, A., Disc. Cont. Dynam. Syst. Ser. S6(1), (2013).
[33] Knees, D.; Zanini, C.; Mielke, A., Physica D, 239, 1470, 2010 · Zbl 1201.49013
[34] Lazzaroni, G.; Toader, R., Math. Models Methods Appl. Sci., 21, 1, 2011 · Zbl 1277.74066
[35] Mielke, A., Cont. Mech. Thermodyn., 15, 351, 2003 · Zbl 1068.74522
[36] Mielke, A., Handbook of Differential Equations, Evolutionary Equations2, eds. Dafermos, C.Feireisl, E. (Elsevier, 2005) pp. 461-559. · Zbl 1120.47062
[37] Mielke, A., Nonlinear PDEs and Applications. C.I.M.E. Summer School, Cetraro, Italy 2008, eds. Ambrosio, L.Savaré, G. (Springer, 2011) pp. 461-559.
[38] Mielke, A.; Rossi, R.; Savaré, G., Disc. Cont. Dynam. Syst., 25, 585, 2009 · Zbl 1170.49036
[39] Mielke, A.; Rossi, R.; Savaré, G., ESAIM Control Optim. Calc. Var., 18, 36, 2012 · Zbl 1250.49041
[40] A. Mielke, R. Rossi and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Partial Differential Equations, doi:10.1007/s00526-011-0482-z. · Zbl 1270.35289
[41] A. Mielke, R. Rossi and G. Savaré, BV solutions to infinite-dimensional rate-independent systems, in preparation.
[42] Mielke, A.; Roubíček, T., Math. Models Methods Appl. Sci., 16, 177, 2006 · Zbl 1094.35068
[43] Mielke, A.Theil, F., A mathematical model for rate-independent phase transformations with hysteresis, Proc. of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, eds. Alber, H.-D.Balean, R.Farwig, R. (Shaker-Verlag, 1999) pp. 117-129.
[44] Mielke, A.; Theil, F., Nonl. Diff. Eqns. Appl. (NoDEA), 11, 151, 2004 · Zbl 1061.35182
[45] A. Mielke and S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Technical Report to appear. · Zbl 1295.35036
[46] Nochetto, R. H.; Savaré, G.; Verdi, C., Commun. Pure Appl. Math., 53, 525, 2000 · Zbl 1021.65047
[47] Rossi, R.; Mielke, A.; Savaré, G., Ann. Sc. Norm. Super. Pisa Cl. Sci., VII, 97, 2008 · Zbl 1183.35164
[48] Rossi, R.; Savaré, G., ESAIM Control Optim. Calc. Var., 12, 564, 2006 · Zbl 1116.34048
[49] Thomas, M.; Mielke, A., ZAMM Z. Angew. Math. Mech., 90, 88, 2010 · Zbl 1191.35159
[50] Toader, R.; Zanini, C., Boll. Unione Mat. Ital., 2, 1, 2009
[51] Valadier, M., Methods of Nonconvex Analysis, 1446 (Springer, 1990) pp. 152-188. · Zbl 0738.28004
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.