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Balanced viscosity solutions to a rate-independent system for damage

Published online by Cambridge University Press:  24 January 2018

DOROTHEE KNEES ,
RICCARDA ROSSI  and
CHIARA ZANINI
DOROTHEE KNEES
Affiliation:
Institute of Mathematics, University of Kassel, Heinrich-Plett Str. 40, 34132 Kassel, Germany email: [email protected]
RICCARDA ROSSI
Affiliation:
Department DIMI, University of Brescia, Via Branze 38, 25133 Brescia, Italy email: [email protected]
CHIARA ZANINI
Affiliation:
Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy email: [email protected]
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Abstract

This article is the third one in a series of papers by the authors on vanishing-viscosity solutions to rate-independent damage systems. While in the first two papers (Knees, D. et al. 2013 Math. Models Methods Appl. Sci.23(4), 565–616; Knees, D. et al. 2015 Nonlinear Anal. Real World Appl.24, 126–162) the assumptions on the spatial domain Ω were kept as general as possible (i.e., non-smooth domain with mixed boundary conditions), we assume here that ∂Ω is smooth and that the type of boundary conditions does not change. This smoother setting allows us to derive enhanced regularity spatial properties both for the displacement and damage fields. Thus, we are in a position to work with a stronger solution notion at the level of the viscous approximating system. The vanishing-viscosity analysis then leads us to obtain the existence of a stronger solution concept for the rate-independent limit system. Furthermore, in comparison to [18, 19], in our vanishing-viscosity analysis we do not switch to an artificial arc-length parameterization of the trajectories but we stay with the true physical time. The resulting concept of Balanced Viscosity solution to the rate-independent damage system thus encodes a more explicit characterization of the system behaviour at time discontinuities of the solution.

Keywords

Rate-independent damage systemvanishing-viscosity approximationBalanced Viscosity solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 1 , February 2019 , pp. 117 - 175
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

D. Knees acknowledges the partial financial support through the DFG-Priority Program SPP 1962 Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization.

References

[1] Agmon, S., Douglis, A. & Nirenberg, Louis (1964) Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17, 35–92.Google Scholar
[2] Ambrosio, L., Gigli, N. & Savaré, G. (2008) Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd ed., Birkhäuser Verlag, Basel.Google Scholar
[3] Ambrosio, L. & Tortorelli, V. M. (1990) Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commun. Pure Appl. Math. 43 (8), 999–1036.Google Scholar
[4] Bonfanti, G., Frémond, M. & Luterotti, F. (2000) Global solution to a nonlinear system for irreversible phase changes. Adv. Math. Sci. Appl. 10 (1), 1–24.Google Scholar
[5] Babadjian, J.-F. Francfort, G. & Mora, M. G. (2012) Quasistatic evolution in non-associative plasticity – The cap model. SIAM J. Math. Anal. 44, 245–292.Google Scholar
[6] Babadjian, J.-F. & Millot, V. (2014) Unilateral gradient flow of the Ambrosio–Tortorelli functional by minimizing movements. Ann. Inst. Henri Poincaré 31 (4), 779–822.Google Scholar
[7] Brézis, H. (1973) Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing, New York.Google Scholar
[8] Crismale, V. & Lazzaroni, G. (2016) Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model. Calc. Var. Partial Differ. Equ. 55 (1), Art. 17, 54.Google Scholar
[9] Crismale, V. & Lazzaroni, G. (2017) Quasistatic crack growth based on viscous approximation: A model with branching and kinking. NoDEA Nonlinear Differ. Equ. Appl. 24 (1), Art. 7, 33.Google Scholar
[10] Dal Maso, G., DeSimone, A. & Solombrino, F. (2011) Quasistatic evolution for cam-clay plasticity: A weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differ. Equ. 40, 125–181.Google Scholar
[11] Efendiev, M. & Mielke, A. (2006) On the rate–independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13, 151–167.Google Scholar
[12] Giusti, E. (2003) Direct Methods in the Calculus of Variations. World Scientific Publishing.Google Scholar
[13] Gilardi, G. & Rocca, E. (2007) Well-posedness and long-time behaviour for a singular phase field system of conserved type. IMA J. Appl. Math. 72 (4), 498–530.Google Scholar
[14] Jerome, J. W. (1983) Approximation of Nonlinear Evolution Systems, Mathematics in Science and Engineering, Vol. 164, Academic Press, Orlando, FL.Google Scholar
[15] Kato, T. (1984) Perturbation Theory for Linear Operators.Google Scholar
[16] Knees, D., Mielke, A. & Zanini, C. (2008) On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18 (9), 1529–1569.Google Scholar
[17] Knees, D. & Negri, M. (2017) Convergence of alternate minimization schemes for phase field fracture and damage. Math. Models Methods Appl. Sci. 27 (9), 1743–1794.Google Scholar
[18] Knees, D., Rossi, R. & Zanini, C. (2013) A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23 (4), 565–616.Google Scholar
[19] Knees, D., Rossi, R. & Zanini, C. (2015) A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains. Nonlinear Anal. Real World Appl. 24, 126–162.Google Scholar
[20] Mielke, A. (2011) Differential, energetic and metric formulations for rate-independent processes (Ch. 3). In: Ambrosio, L. & Savaré, G. (editors), Nonlinear PDEs and Applications. C.I.M.E. Summer School, Cetraro, Italy 2008, Springer, Heidelberg, pp. 87–170.Google Scholar
[21] Maz'ya, V. G. & Rossmann, J. (2003) Weighted Lp estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains. Z. Angew. Math. Mech. 83 (7), 435–467.Google Scholar
[22] Mielke, A., Rossi, R. & Savaré, G. (2012) BV solutions and viscosity approximation of rate-independent systems. ESAIM Control Optim. Calc. Var. 18, 36–80.Google Scholar
[23] Mielke, A., Rossi, R. & Savaré, G. (2012) BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim. Calc. Var. 18 (1), 36–80.Google Scholar
[24] Mielke, A., Rossi, R. & Savaré, G. (2013) Nonsmooth analysis of doubly nonlinear evolution equations. Calc. Var. Partial Differ. Equ. 46, 253–310.Google Scholar
[25] Mielke, A., Rossi, R. & Savaré, G. (2016) Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. 18 (9), 2107–2165.Google Scholar
[26] Mielke, A. & Theil, F. (2004) On rate–independent hysteresis models. Nonlinear Diff. Eqns. Appl. (NoDEA) 11, 151–189.Google Scholar
[27] Negri, M. (2017) A unilateral L 2- gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement. Adv. Calc. Var. Published Online: 2017-06-04 DOI: https://doi.org/10.1515/acv-2016-0028Google Scholar
[28] Renardy, M. & Rogers, R. C. (2004) An Introduction to Partial Differential Equations, 2nd ed., Springer, New York, NY.Google Scholar
[29] Savaré, G. (1998) Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1), 176–201.Google Scholar
[30] Simon, J. (1987) Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. (4) 146, 65–96.Google Scholar
[31] Valent, T. (1978) Teoremi di esistenza e unicità in elastostatica finita. Rend. Semin. Mat. Univ. Padova 60, 165–181.Google Scholar