##
**Balanced viscosity solutions to a rate-independent coupled elasto-plastic damage system.**
*(English)*
Zbl 1483.35072

In nonlinear elasticity, rate-independent systems are idealized models where internal oscillations and viscous dissipations are neglected, since the (slower) scale of external loadings is dominant. On the other hand, in the latter time scale the system presents time discontinuities, corresponding to fast transitions between equilibria. In such transitions, a major role is played by the viscous dissipations.

A well-known method to study e.g. damage models is to consider a system where the flow rule for the damage variable is viscously regularized; next, one passes to the limit as the viscosity parameter tends to zero. The time discontinuities of the resulting evolution can be interpolated by means of transitions governed again by viscosity.

In this paper the authors study a model for damage coupled with plasticity, affected by viscosity both in the damage evolution and in the elastoplastic evolution. Moreover, a further dissipation source may come from a hardening process. Viscosity and hardening provide regularizing terms in the PDE system.

In the rate-independent idealization, one would neglect both viscosity and hardening. To rigorously see this, the authors consider a singularly perturbed PDE system where the terms related to viscosity and hardening are modulated by small parameters tending to zero. By tuning the speed of the convergence of such coefficients, one may model a system where the elastic and the plastic strain converge to rate-independent evolution with the same rate, or with a faster rate, than the damage variable.

Specifically, the system analyzed by the authors features three coefficients: a hardening parameter \(\mu\), a viscosity parameter \(\varepsilon\) related to damage, and a viscosity coefficient \(\varepsilon\nu\) related to plasticity. In fact, \(\nu\) is a rate parameter that modulates the rate of convergence of the damage variable with respect to the plastic strain. The authors study the convergence of the system as \(\varepsilon\to0\) while \(\nu,\mu\) are fixed, or as \(\varepsilon,\nu\to0\), or as all parameters \(\varepsilon,\nu,\mu\) converge to zero. In the limit, they obtain different notions of solutions, showing in the time discontinuities a single-rate or a multi-rate character. Studying various notions of rate-independent solutions is important in order to understand which of them captures the behavior of the system for small viscosities.

A well-known method to study e.g. damage models is to consider a system where the flow rule for the damage variable is viscously regularized; next, one passes to the limit as the viscosity parameter tends to zero. The time discontinuities of the resulting evolution can be interpolated by means of transitions governed again by viscosity.

In this paper the authors study a model for damage coupled with plasticity, affected by viscosity both in the damage evolution and in the elastoplastic evolution. Moreover, a further dissipation source may come from a hardening process. Viscosity and hardening provide regularizing terms in the PDE system.

In the rate-independent idealization, one would neglect both viscosity and hardening. To rigorously see this, the authors consider a singularly perturbed PDE system where the terms related to viscosity and hardening are modulated by small parameters tending to zero. By tuning the speed of the convergence of such coefficients, one may model a system where the elastic and the plastic strain converge to rate-independent evolution with the same rate, or with a faster rate, than the damage variable.

Specifically, the system analyzed by the authors features three coefficients: a hardening parameter \(\mu\), a viscosity parameter \(\varepsilon\) related to damage, and a viscosity coefficient \(\varepsilon\nu\) related to plasticity. In fact, \(\nu\) is a rate parameter that modulates the rate of convergence of the damage variable with respect to the plastic strain. The authors study the convergence of the system as \(\varepsilon\to0\) while \(\nu,\mu\) are fixed, or as \(\varepsilon,\nu\to0\), or as all parameters \(\varepsilon,\nu,\mu\) converge to zero. In the limit, they obtain different notions of solutions, showing in the time discontinuities a single-rate or a multi-rate character. Studying various notions of rate-independent solutions is important in order to understand which of them captures the behavior of the system for small viscosities.

Reviewer: Giuliano Lazzaroni (Firenze)

### MSC:

35D40 | Viscosity solutions to PDEs |

35B25 | Singular perturbations in context of PDEs |

35A15 | Variational methods applied to PDEs |

34A60 | Ordinary differential inclusions |

35Q74 | PDEs in connection with mechanics of deformable solids |

74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |

### Keywords:

rate-independent systems; variational models; vanishing viscosity; BV solutions; damage; elasto-plasticity
PDF
BibTeX
XML
Cite

\textit{V. Crismale} and \textit{R. Rossi}, SIAM J. Math. Anal. 53, No. 3, 3420--3492 (2021; Zbl 1483.35072)

### References:

[1] | R. Alessi, V. Crismale, and G. Orlando, Fatigue effects in elastic materials with variational damage models: A vanishing viscosity approach, J. Nonlinear Sci., 29 (2019), pp. 1041-1094. · Zbl 1417.74005 |

[2] | J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure Appl. Math. (New York), John Wiley & Sons, New York, 1984. · Zbl 0641.47066 |

[3] | L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2005. · Zbl 0957.49001 |

[4] | L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. · Zbl 1145.35001 |

[5] | S. Almi, L. Lazzaroni, and I. Lucardesi, Crack growth by vanishing viscosity in planar elasticity, Math. Eng., 2 (2020), pp. 141-173. |

[6] | S. Almi, Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening, ESAIM Control Optim. Calc. Var., 23 (2017), pp. 791-826. · Zbl 1373.49011 |

[7] | R. Alessi, J.J. Marigo, and S. Vidoli, Gradient damage models coupled with plasticity and nucleation of cohesive cracks, Arch. Ration. Mech. Anal., 214 (2014), pp. 575-615. · Zbl 1321.74006 |

[8] | R. Alessi, J.J. Marigo, and S. Vidoli, Gradient damage models coupled with plasticity: variational formulation and main properties, Mech. Mater. B, 80 (2015), pp. 351-367. |

[9] | J.-F. Babadjian, G. Francfort, and M.G. Mora, Quasistatic evolution in non-associative plasticity-the cap model, SIAM J. Math. Anal., 44 (2012), pp. 245-292. · Zbl 1379.74006 |

[10] | S. Bartels, A. Mielke, and T. Roubíček, Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation, SIAM J. Numer. Anal., 50 (2012), pp. 951-976. · Zbl 1248.35105 |

[11] | E. Bonetti, E. Rocca, R. Rossi, and M. Thomas, A rate-independent gradient system in damage coupled with plasticity via structured strains, in Gradient Flows: From Theory to Application, ESAIM Proc. Surveys 54, EDP, Les Ulis, 2016, pp. 54-69. · Zbl 1366.74009 |

[12] | V. Crismale and G. Lazzaroni, Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model, Calc. Var. Partial Differential Equations, 55 (2016), 17. · Zbl 1333.74021 |

[13] | V. Crismale and G. Lazzaroni, Quasistatic crack growth based on viscous approximation: a model with branching and kinking, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 7. · Zbl 1372.35376 |

[14] | V. Crismale and G. Orlando, A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in \(W^{1,n}\), NoDEA Nonlinear Differential Equations Appl., 25 (2018), 16. · Zbl 1388.49011 |

[15] | V. Crismale and G. Orlando, A lower semicontinuity result for linearised elasto-plasticity coupled with damage in \(W^{1,\gamma}, \gamma>1\), Math. Eng., 2 (2020), pp. 101-118. |

[16] | V. Crismale, Globally stable quasistatic evolution for a coupled elastoplastic-damage model, ESAIM Control Optim. Calc. Var., 22 (2016), pp. 883-912. · Zbl 1342.74026 |

[17] | V. Crismale, Globally stable quasistatic evolution for strain gradient plasticity coupled with damage, Ann. Mat. Pura Appl. (4), 196 (2017), pp. 641-685. · Zbl 1365.74033 |

[18] | G. Dal Maso, A. DeSimone, and F. Solombrino, Quasistatic evolution for cam-clay plasticity: a weak formulation via viscoplastic regularization and time rescaling, Calc. Var. Partial Differential Equations, 40 (2011), pp. 125-181. · Zbl 1311.74024 |

[19] | G. Dal Maso, A. DeSimone, and M.G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), pp. 237-291. · Zbl 1093.74007 |

[20] | G. Dal Maso, G. Orlando, and R. Toader, Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case, Calc. Var. Partial Differential Equations, 55 (2016), Art. 45, 39. · Zbl 1358.49015 |

[21] | G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures: Existence and approximation results, Arch. Ration. Mech. Anal., 162 (2002), pp. 101-135. · Zbl 1042.74002 |

[22] | E. Davoli, T. Roubíček, and U. Stefanelli, Dynamic perfect plasticity and damage in viscoelastic solids, ZAMM Z. Angew. Math. Mech., 99 (2019), e201800161. |

[23] | M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Anal., 13 (2006), pp. 151-167. · Zbl 1109.74040 |

[24] | G.A. Francfort and A. Giacomini, Small-strain heterogeneous elastoplasticity revisited, Comm. Pure Appl. Math., 65 (2012), pp. 1185-1241. · Zbl 1396.74036 |

[25] | G. Francfort and U. Stefanelli, Quasistatic evolution for the Armstrong-Frederick hardening-plasticity model, Appl. Maths. Res. Express, 2 (2013), pp. 297-344. · Zbl 1273.35265 |

[26] | C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J., 31 (1964), pp. 159-178. · Zbl 0123.09804 |

[27] | A.D. Ioffe and V.M. Tihomirov, Theory of Extremal Problems, Stud. Math. Appl. 6, North-Holland, Amsterdam, 1979. · Zbl 0407.90051 |

[28] | D. Knees, A. Mielke, and C. Zanini, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci., 18 (2008), pp. 1529-1569. · Zbl 1151.49014 |

[29] | D. Knees, R. Rossi, and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), pp. 565-616. · Zbl 1262.74030 |

[30] | D. Knees, R. Rossi, and C. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Nonlinear Anal. Real World Appl., 24 (2015), pp. 126-162. · Zbl 1330.35438 |

[31] | R. Kohn and R. Temam, Dual spaces of stresses and strains, with applications to Hencky plasticity, Appl. Math. Optim., 10 (1983), pp. 1-35. · Zbl 0532.73039 |

[32] | G. Lazzaroni and R. Toader, A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci., 21 (2011), pp. 2019-2047. · Zbl 1277.74066 |

[33] | B.S. Mordukhovich, Variational Analysis and Generalized Differentiation. II, Grundlehren Math. Wiss. 331, Springer-Verlag, Berlin, 2006. |

[34] | A. Mielke and R. Rossi, Balanced viscosity solutions to infinite-dimensional multi-rate systems, in preparation. · Zbl 1357.35007 |

[35] | A. Mielke, R. Rossi, and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), pp. 585-615. · Zbl 1170.49036 |

[36] | A. Mielke, R. Rossi, and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), pp. 36-80. · Zbl 1250.49041 |

[37] | A. Mielke, R. Rossi, and G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan J. Math., 80 (2012), pp. 381-410. · Zbl 1255.49078 |

[38] | A. Mielke, R. Rossi, and G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Partial Differential Equations, 46 (2013), pp. 253-310. · Zbl 1270.35289 |

[39] | A. Mielke, R. Rossi, and G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc. (JEMS), 18 (2016), pp. 2107-2165. · Zbl 1357.35007 |

[40] | A. Mielke, R. Rossi, and G. Savaré, Balanced-viscosity solutions for multi-rate systems, J. Phys. Conf. Ser., (2016), 010210. · Zbl 1465.70060 |

[41] | D. Melching, R. Scala, and J. Zeman, Damage model for plastic materials at finite strains, ZAMM Z. Angew. Math. Mech., 99 (2019), e201800032. |

[42] | A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, H.-D. Alber, R.M. Balean, and R. Farwig, eds., Shaker-Verlag, Aachen, 1999, pp. 117-129. |

[43] | A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), pp. 151-189. · Zbl 1061.35182 |

[44] | M. Negri, Quasi-static rate-independent evolutions: Characterization, existence, approximation and application to fracture mechanics, ESAIM Control Optim. Calc. Var., 20 (2014), pp. 983-1008. · Zbl 1301.49017 |

[45] | M. Negri, A unilateral \(L^2\)-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement, Adv. Calc. Var., 12 (2019), pp. 1-29. · Zbl 07003374 |

[46] | P. Perzyna, Thermodynamic theory of Viscoplasticity, Adv. Appl. Mech., 11 (1971), pp. 313-354. |

[47] | R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 564-614. · Zbl 1116.34048 |

[48] | R. Rossi and G. Savaré, A characterization of energetic and BV solutions to one-dimensional rate-independent systems, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), pp. 167-191. · Zbl 1270.34027 |

[49] | R. Rossi and M. Thomas, Coupling rate-independent and rate-dependent processes: existence results, SIAM J. Math. Anal., 49 (2017), pp. 1419-1494. · Zbl 1368.35009 |

[50] | T. Roubíček and J. Valdman, Perfect plasticity with damage and healing at small strains, its modeling, analysis, and computer implementation, SIAM J. Appl. Math., 76 (2016), pp. 314-340. · Zbl 1383.74016 |

[51] | T. Roubíček and J. Valdman, Stress-driven solution to rate-independent elasto-plasticity with damage at small strains and its computer implementation, Math. Mech. Solids, 22 (2017), pp. 1267-1287. · Zbl 1371.74055 |

[52] | J. Simon, Compact sets in the space \(L^p(0,T;B)\), Ann. Mat. Pura Appl., 146 (1987), pp. 65-96. · Zbl 0629.46031 |

[53] | F. Solombrino, Quasistatic evolution problems for nonhomogeneous elastic plastic materials, J. Convex Anal., 16 (2009), pp. 89-119. · Zbl 1166.74006 |

[54] | F. Solombrino, Quasistatic evolution in perfect plasticity for general heterogeneous materials, Arch. Ration. Mech. Anal., 212 (2014), pp. 283-330. · Zbl 1293.35327 |

[55] | R. Temam, Problèmes mathématiques en plasticité, Méthodes Mathématiques de l’Informatique 12, Gauthier-Villars, Montrouge, 1983. · Zbl 0547.73026 |

[56] | R. Temam and G. Strang, Duality and relaxation in the variational problems of plasticity, J. Mécanique, 19 (1980), pp. 493-527. · Zbl 0465.73033 |

[57] | M. Valadier, Young measures, in Methods of Nonconvex Analysis (Varenna, 1989), Springer, Berlin, 1990, pp. 152-188. |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.