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Visco-energetic solutions to some rate-independent systems in damage, delamination, and plasticity

Riccarda Rossi

Dipartimento di Ingegneria Meccanica e Industriale, Università di Brescia, via Branze 38, I–25133 Brescia, Italy

https://doi.org/10.1142/S0218202519500179Cited by:0 (Source: Crossref)

Abstract

This paper revolves around a newly introduced weak solvability concept for rate-independent systems, alternative to the notions of Energetic ( $E$ $E$ ) and Balanced Viscosity ( $BV$ $BV$ ) solutions. Visco-Energetic ( $VE$ $VE$ ) solutions have been recently obtained by passing to the time-continuous limit in a time-incremental scheme, akin to that for $E$ $E$ solutions, but perturbed by a “viscous” correction term, as in the case of $BV$ $BV$ solutions. However, for VE solutions this viscous correction is tuned by a fixed parameter. The resulting solution notion turns out to describe a kind of evolution in between Energetic and BV evolution. In this paper we aim to investigate the application of $VE$ $VE$ solutions to nonsmooth rate-independent processes in solid mechanics such as damage and plasticity at finite strains. We also address the limit passage, in the $VE$ $VE$ formulation, from an adhesive contact to a brittle delamination system. The analysis of these applications reveals the wide applicability of this solution concept, in particular to processes for which $BV$ $BV$ solutions are not available, and confirms its intermediate character between the $E$ $E$ and $BV$ $BV$ notions.

Communicated by T. Roubicek

Keywords:

AMSC: 35A15, 34A60

References

1. L. Ambrosio, N. Gigli and G. Savaré , Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn., Lectures in Mathematics (Birkhäuser, 2008). Crossref, Google Scholar
2. J.-F. Babadjian, G. Francfort and M. Mora , Quasistatic evolution in non-associative plasticity — the cap model, SIAM J. Math. Anal. 44 (2012) 245–292. Crossref, Web of Science, Google Scholar
3. A. Braides , Local Minimization, Variational Evolution and Γ-Convergence, Lecture Notes in Mathematics, Vol. 2094 (Springer, 2014). Crossref, Google Scholar
4. A. Braides, B. Cassano, A. Garroni and D. Sarrocco , Quasi-static damage evolution and homogenization: A case study of non-commutability, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 309–328. Crossref, Web of Science, Google Scholar
5. V. Crismale and G. Lazzaroni , Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model, Calc. Var. Partial Differential Equations 55 (2016) 54. Crossref, Web of Science, Google Scholar
6. G. Dal Maso, A. DeSimone and M. Mora , Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal. 180 (2006) 237–291. Crossref, Web of Science, Google Scholar
7. G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini , A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Ration. Mech. Anal. 189 (2008) 469–544. Crossref, Web of Science, Google Scholar
8. 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) 125–181. Crossref, Web of Science, Google Scholar
9. 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) 101–135. Crossref, Web of Science, Google Scholar
10. M. Efendiev and A. Mielke , On the rate–independent limit of systems with dry friction and small viscosity, J. Convex Anal. 13 (2006) 151–167. Web of Science, Google Scholar
11. G. Francfort and A. Garroni , A variational view of partial brittle damage evolution, Arch. Ration. Mech. Anal. 182 (2006) 125–152. Crossref, Web of Science, Google Scholar
12. G. Francfort and A. Mielke , Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math. 595 (2006) 55–91. Crossref, Web of Science, Google Scholar
13. G. A. Francfort and J.-J. Marigo , Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998) 1319–1342. Crossref, Web of Science, Google Scholar
14. M. Frémond , Non-Smooth Thermomechanics (Springer, 2002). Crossref, Google Scholar
15. K. Hackl, A. Mielke and D. Mittenhuber , Dissipation distances in multiplicative elastoplasticity, in Analysis and Simulation of Multifield Problems, eds. W. Wendland and M. Efendiev (Springer, 2003), pp. 87–100. Crossref, Google Scholar
16. B. Halphen and Q. Nguyen , Sur les matériaux standards généralisés, J. Mécanique 14 (1975) 39–63. Google Scholar
17. D. Knees, A. Mielke and C. Zanini , On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci. 18 (2008) 1529–1569. Link, Web of Science, Google Scholar
18. D. Knees, R. Rossi and C. Zanini , A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci. 23 (2013) 565–616. Link, Web of Science, Google Scholar
19. D. Knees, R. Rossi and C. Zanini , Balanced viscosity solutions to a rate-independent system for damage, Europ. J. Appl. Math. 30 (2019) 117–175. Crossref, Web of Science, Google Scholar
20. G. Lazzaroni and R. Toader , A model for crack propagation based on viscous approximation, Math. Models Methods Appl. Sci. 21 (2011) 2019–2047. Link, Web of Science, Google Scholar
21. A. Mainik and A. Mielke , Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations 22 (2005) 73–99. Crossref, Web of Science, Google Scholar
22. A. Mainik and A. Mielke , Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci. 19 (2009) 221–248. Crossref, Web of Science, Google Scholar
23. A. Mielke , Finite elastoplasticity Lie groups and geodesics on $SL (d)$ $SL (d)$ , in Geometry, Mechanics, and Dynamics (Springer, 2002), pp. 61–90. Crossref, Google Scholar
24. A. Mielke , Differential, energetic, and metric formulations for rate-independent processes, in Nonlinear PDE’s and Applications, Lecture Notes in Mathematics, Vol. 2028 (Springer, 2011), pp. 87–170. Crossref, Google Scholar
25. A. Mielke , On evolutionary $Γ$ $Γ$ -convergence for gradient systems, in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Lecture Notes in Applied Mathematics and Mechanics, Vol. 3 (Springer, 2016), pp. 187–249. Crossref, Google Scholar
26. A. Mielke, R. Rossi and G. Savaré , Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc. 18 (2016) 2107–2165. Crossref, Web of Science, Google Scholar
27. A. Mielke, R. Rossi and G. Savaré , Global existence results for viscoplasticity at finite strain, Arch. Ration. Mech. Anal. 227 (2018) 423–475. Crossref, Web of Science, Google Scholar
28. A. Mielke and T. Roubíček , Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci. 16 (2006) 177–209. Link, Web of Science, Google Scholar
29. A. Mielke and T. Roubíček , Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, Vol. 193 (Springer, 2015). Crossref, Google Scholar
30. A. Mielke, T. Roubíček and U. Stefanelli , $Γ$ $Γ$ -limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations 31 (2008) 387–416. Crossref, Web of Science, Google Scholar
31. A. Mielke and F. Theil , A mathematical model for rate-independent phase transformations with hysteresis, in Proc. Workshop on “Models of Continuum Mechanics in Analysis and Engineering”, eds. H.-D. Alber, R. Balean and R. Farwig (Shaker-Verlag, 1999), pp. 117–129. Google Scholar
32. A. Mielke and F. Theil , On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl. 11 (2004) 151–189. Crossref, Web of Science, Google Scholar
33. A. Mielke, F. Theil and V. I. Levitas , A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal. 162 (2002) 137–177. Crossref, Web of Science, Google Scholar
34. L. Minotti , Visco-energetic solutions to one-dimensional rate-independent problems, Discrete Contin. Dyn. Syst. 37 (2017) 5883–5912. Crossref, Web of Science, Google Scholar
35. L. Minotti and G. Savaré , Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems, Arch. Ration. Mech. Anal. 227 (2018) 477–543. Crossref, Web of Science, Google Scholar
36. M. Negri , A comparative analysis on variational models for quasi-static brittle crack propagation, Adv. Calc. Var. 3 (2010) 149–212. Crossref, Web of Science, Google Scholar
37. M. Negri , A unilateral $L^{2}$ $L^{2}$ -gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement, Adv. Calc. Var. 12 (2019) 1–29. Crossref, Web of Science, Google Scholar
38. 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) 167–191. Web of Science, Google Scholar
39. R. Rossi and G. Savaré , From Visco-Energetic to Energetic and Balanced Viscosity solutions of rate-independent systems, in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, P. colli, A. Favini, E. Rocca, G. Schimperna and J. Sprekels , Springer INdAM Series, Vol. 22 (Springer, 2017), pp. 489–531. Crossref, Google Scholar
40. T. Roubíček, L. Scardia and C. Zanini , Quasistatic delamination problem, Continuum Mech. Thermodynam. 21 (2009) 223–235. Crossref, Web of Science, Google Scholar
41. E. Sandier and S. Serfaty , Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math. 57 (2004) 1627–1672. Crossref, Web of Science, Google Scholar
42. S. Serfaty , Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. 31 (2011) 1427–1451. Crossref, Web of Science, Google Scholar
43. U. Stefanelli , The Brézis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim. 47 (2008) 1615–1642. Crossref, Web of Science, Google Scholar
44. R. Temam and G. Strang , Duality and relaxation in the variational problems of plasticity, J. Mécanique 19 (1980) 493–527. Google Scholar
45. M. Thomas , Quasistatic damage evolution with spatial BV-regularization, Discrete Contin. Dyn. Syst. Ser. S 6 (2013) 235–255. Crossref, Web of Science, Google Scholar
46. M. Thomas and A. Mielke , Damage of nonlinearly elastic materials at small strain: Existence and regularity results, Zeit. Angew. Math. Mech. 90 (2010) 88–112. Crossref, Web of Science, Google Scholar
47. R. Toader and C. Zanini , An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital. (9) 2 (2009) 1–35. Google Scholar
48. A. Visintin , Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differential Equations 47 (2013) 273–317. Crossref, Web of Science, Google Scholar