November  2021, 14(11): 3865-3924. doi: 10.3934/dcdss.2021067

Discrete approximation of dynamic phase-field fracture in visco-elastic materials

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

* Corresponding author: Sven Tornquist

Received  August 2020 Published  November 2021 Early access  June 2021

Fund Project: The authors gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 "Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis" within the project "Reliability of Efficient Approximation Schemes for Material Discontinuities Described by Functions of Bounded Variation" – Project Number 255461777 (TH 1935/1-2). MT also gratefully acknowledges the partial financial support by the DFG within the Collaborative Research Center 1114 "Scaling Cascades in Complex Systems", Project C09 "Dynamics of rock dehydration on multiple scales"

This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively $ 1 $-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

Citation: Marita Thomas, Sven Tornquist. Discrete approximation of dynamic phase-field fracture in visco-elastic materials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (11) : 3865-3924. doi: 10.3934/dcdss.2021067
References:
[1]

S. AlmiS. Belz and M. Negri, Convergence of discrete and continuous unilateral flows for Ambrosio-Tortorelli energies and application to mechanics, ESAIM M2AN, 53 (2018), 659-699.  doi: 10.1051/m2an/2018057.  Google Scholar

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M. AmbatiT. Gerasimov and L. De Lorenzis, Phase-field modeling of ductile fracture, Computational Mechanics, 55 (2015), 1017-1040.  doi: 10.1007/s00466-015-1151-4.  Google Scholar

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E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials, Ann. Inst. H. Poincré Anal. Non Linéaire, 25 (2008), 1187–1208. doi: 10.1016/j. anihpc. 2007.05.009.  Google Scholar

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B. BourdinG. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids, 48 (2000), 797-826.  doi: 10.1016/S0022-5096(99)00028-9.  Google Scholar

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S. Bartels, M. Milicevic, M. Thomas, S. Tornquist and N. Weber, Approximation schemes for materials with discontinuities, WIAS Preprint 2799, 2020. Google Scholar

[6]

S. Bartels, M. Milicevic, M. Thomas and N. Weber, Fully discrete approximation of rate-independent damage models with gradient regularization, WIAS Preprint 2707, 2020. Google Scholar

[7]

H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert., North Holland, 1973.  Google Scholar

[8]

M. J. BordenC. V. VerhooselM. A. ScottT. J. R. Hughes and C. M. Landis, A phase-field description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering, 217 (2012), 77-95.  doi: 10.1016/j.cma.2012.01.008.  Google Scholar

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B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Springer Berlin Heidelberg, 2012.  Google Scholar

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M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^{p}(0, T;B)$, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 3072-3077.  doi: 10.1016/j.na.2011.12.004.  Google Scholar

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G. Dal MasoG. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4.  Google Scholar

[12]

G. Dal MasoC. J. Larsen and R. Toader, Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition, Journal of the Mechanics and Physics of Solids, 95 (2016), 697-707.  doi: 10.1016/j.jmps.2016.04.033.  Google Scholar

[13]

G. Dal Maso, C. J. Larsen and R. Toader, Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition, Journal de Mathématiques Pures et Appliquées, 127 (2019), 160–191. doi: 10.1016/j. matpur. 2018.08.006.  Google Scholar

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G. Dal Maso, C. J. Larsen and R. Toader, Elastodynamic Griffith fracture on prescribed crack paths with kinks, Nonlinear Differential Equations and Applications NoDEA, 27 (2020). doi: 10.1007/s00030-019-0607-1.  Google Scholar

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M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, Journal of Convex Analysis, 13 (2006), 151.  Google Scholar

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G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids, 46 (1998), 1319-1342.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[17]

A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures, Calculus of Variations and Partial Differential Equations, 22 (2005), 129-172.  doi: 10.1007/s00526-004-0269-6.  Google Scholar

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C. Heinemann and C. Kraus, Existence of weak solutions for Cahn–Hilliard systems coupled with elasticity and damage, Adv. Math. Sci. Appl., (2011), 321–359.  Google Scholar

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C. HeinemannC. KrausE. Rocca and R. Rossi, A temperature-dependent phase-field model for phase separation and damage, Arch. Rational Mech. Anal., 225 (2017), 177-247.  doi: 10.1007/s00205-017-1102-7.  Google Scholar

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R. Henstock, The Calculus and Gauge Integrals, arXiv: Classical Analysis and ODEs, 2016. Google Scholar

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R. HerzogC. Meyer and G. Wachsmuth, Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions, Journal of Mathematical Analysis and Applications, 382 (2011), 802-813.  doi: 10.1016/j.jmaa.2011.04.074.  Google Scholar

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B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, J. Mécanique, 14 (1975), 39-63.   Google Scholar

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C. Hesch and K. Weinberg, Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture, International Journal for Numerical Methods in Engineering, 99 (2014), 906-924.  doi: 10.1002/nme.4709.  Google Scholar

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C. Kuhn and R. Müller, A continuum phase field model for fracture, Engineering Fracture Mechanics, 77 (2010), 3625-3634.  doi: 10.1016/j.engfracmech.2010.08.009.  Google Scholar

[26]

D. KneesR. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), 565-616.  doi: 10.1142/S021820251250056X.  Google Scholar

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D. KneesR. Rossi and C. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Nonlin. Anal. Ser. B: Real World Appl., 24 (2015), 126-162.  doi: 10.1016/j.nonrwa.2015.02.001.  Google Scholar

[28]

D. KneesR. Rossi and C. Zanini, Balanced viscosity solutions to a rate-independent system for damage, European Journal of Applied Mathematics, 30 (2019), 117-175.  doi: 10.1017/S0956792517000407.  Google Scholar

[29]

D. Knees and A. Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, Mathematical Methods in the Applied Sciences, 35 (2012), 1859-1884.  doi: 10.1002/mma.2598.  Google Scholar

[30]

D. Knees and C. Zanini, Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads, Discrete & Continuous Dynamical Systems-S, (2018). doi: 10.3934/dcdss. 2020332.  Google Scholar

[31]

G. Lazzaroni, R. Rossi, M. Thomas, and R. Toader, Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics, Journal of Physics: Conference Series, 727 (2016), 012009. doi: 10.1088/1742-6596/727/1/012009.  Google Scholar

[32]

G. Lazzaroni, R. Rossi, M. Thomas and R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, J. Dyn. Diff. Equat., (2018). doi: 10.1007/s10884-018-9666-y.  Google Scholar

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C. MieheM. Hofacker and F. Welschinger, A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 2765-2778.  doi: 10.1016/j.cma.2010.04.011.  Google Scholar

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M. Marcus and V. J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Functional Analysis, 33 (1979), 217-229.  doi: 10.1016/0022-1236(79)90113-7.  Google Scholar

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A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 177-209.  doi: 10.1142/S021820250600111X.  Google Scholar

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A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences, Springer New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

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A. Mielke and T. Roubíček, Rate-independent Systems: Theory and Application, Applied Mathematical Sciences, 193, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

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A. MielkeR. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete & Continuous Dynamical Systems, 25 (2009), 585-615.  doi: 10.3934/dcds.2009.25.585.  Google Scholar

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A. MielkeR. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 36-80.  doi: 10.1051/cocv/2010054.  Google Scholar

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A. Mielke and F. Theil, On rate–independent hysteresis models, NODEA, 11 (2004), 151-189.  doi: 10.1007/s00030-003-1052-7.  Google Scholar

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H. Royden and P. Fitzpatrick, Real Analysis (Classic Version), Pearson Modern Classics for Advanced Mathematics Series, Pearson, 2017. Google Scholar

[43]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics, Birkhäuser Basel, 2006. Google Scholar

[44]

T. Roubíček, Models of dynamic damage and phase-field fracture, and their various time discretisations, in Topics in Applied Analysis and Optimisation, Springer, 2019, 363–396. Google Scholar

[45]

E. Rocca and R. Rossi, "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM J. Math. Anal., 74 (2015), 2519-2586.  doi: 10.1137/140960803.  Google Scholar

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R. Rossi and M. Thomas., Coupling rate-independent and rate-dependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017), 1419-1494.  doi: 10.1137/15M1051567.  Google Scholar

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T. RoubíčekM. Thomas and C. G. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Anal. Real World Appl., 22 (2015), 645-663.  doi: 10.1016/j.nonrwa.2014.09.011.  Google Scholar

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M. ThomasC. Bilgen and K. Weinberg, Phase-field fracture at finite strains based on modified invariants: A note on its analysis and simulations, GAMM-Mitteilungen, 40 (2018), 207-237.  doi: 10.1002/gamm.201730004.  Google Scholar

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show all references

References:
[1]

S. AlmiS. Belz and M. Negri, Convergence of discrete and continuous unilateral flows for Ambrosio-Tortorelli energies and application to mechanics, ESAIM M2AN, 53 (2018), 659-699.  doi: 10.1051/m2an/2018057.  Google Scholar

[2]

M. AmbatiT. Gerasimov and L. De Lorenzis, Phase-field modeling of ductile fracture, Computational Mechanics, 55 (2015), 1017-1040.  doi: 10.1007/s00466-015-1151-4.  Google Scholar

[3]

E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials, Ann. Inst. H. Poincré Anal. Non Linéaire, 25 (2008), 1187–1208. doi: 10.1016/j. anihpc. 2007.05.009.  Google Scholar

[4]

B. BourdinG. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids, 48 (2000), 797-826.  doi: 10.1016/S0022-5096(99)00028-9.  Google Scholar

[5]

S. Bartels, M. Milicevic, M. Thomas, S. Tornquist and N. Weber, Approximation schemes for materials with discontinuities, WIAS Preprint 2799, 2020. Google Scholar

[6]

S. Bartels, M. Milicevic, M. Thomas and N. Weber, Fully discrete approximation of rate-independent damage models with gradient regularization, WIAS Preprint 2707, 2020. Google Scholar

[7]

H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert., North Holland, 1973.  Google Scholar

[8]

M. J. BordenC. V. VerhooselM. A. ScottT. J. R. Hughes and C. M. Landis, A phase-field description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering, 217 (2012), 77-95.  doi: 10.1016/j.cma.2012.01.008.  Google Scholar

[9]

B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Springer Berlin Heidelberg, 2012.  Google Scholar

[10]

M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^{p}(0, T;B)$, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 3072-3077.  doi: 10.1016/j.na.2011.12.004.  Google Scholar

[11]

G. Dal MasoG. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Archive for Rational Mechanics and Analysis, 176 (2005), 165-225.  doi: 10.1007/s00205-004-0351-4.  Google Scholar

[12]

G. Dal MasoC. J. Larsen and R. Toader, Existence for constrained dynamic Griffith fracture with a weak maximal dissipation condition, Journal of the Mechanics and Physics of Solids, 95 (2016), 697-707.  doi: 10.1016/j.jmps.2016.04.033.  Google Scholar

[13]

G. Dal Maso, C. J. Larsen and R. Toader, Existence for elastodynamic Griffith fracture with a weak maximal dissipation condition, Journal de Mathématiques Pures et Appliquées, 127 (2019), 160–191. doi: 10.1016/j. matpur. 2018.08.006.  Google Scholar

[14]

G. Dal Maso, C. J. Larsen and R. Toader, Elastodynamic Griffith fracture on prescribed crack paths with kinks, Nonlinear Differential Equations and Applications NoDEA, 27 (2020). doi: 10.1007/s00030-019-0607-1.  Google Scholar

[15]

M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, Journal of Convex Analysis, 13 (2006), 151.  Google Scholar

[16]

G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids, 46 (1998), 1319-1342.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[17]

A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures, Calculus of Variations and Partial Differential Equations, 22 (2005), 129-172.  doi: 10.1007/s00526-004-0269-6.  Google Scholar

[18]

A. A. Griffith, VI. The phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society of London, Series A, Containing Papers of a Mathematical or Physical Character, 221 (1921), 163-198.   Google Scholar

[19]

C. Heinemann and C. Kraus, Existence of weak solutions for Cahn–Hilliard systems coupled with elasticity and damage, Adv. Math. Sci. Appl., (2011), 321–359.  Google Scholar

[20]

C. HeinemannC. KrausE. Rocca and R. Rossi, A temperature-dependent phase-field model for phase separation and damage, Arch. Rational Mech. Anal., 225 (2017), 177-247.  doi: 10.1007/s00205-017-1102-7.  Google Scholar

[21]

R. Henstock, The Calculus and Gauge Integrals, arXiv: Classical Analysis and ODEs, 2016. Google Scholar

[22]

R. HerzogC. Meyer and G. Wachsmuth, Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions, Journal of Mathematical Analysis and Applications, 382 (2011), 802-813.  doi: 10.1016/j.jmaa.2011.04.074.  Google Scholar

[23]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, J. Mécanique, 14 (1975), 39-63.   Google Scholar

[24]

C. Hesch and K. Weinberg, Thermodynamically consistent algorithms for a finite-deformation phase-field approach to fracture, International Journal for Numerical Methods in Engineering, 99 (2014), 906-924.  doi: 10.1002/nme.4709.  Google Scholar

[25]

C. Kuhn and R. Müller, A continuum phase field model for fracture, Engineering Fracture Mechanics, 77 (2010), 3625-3634.  doi: 10.1016/j.engfracmech.2010.08.009.  Google Scholar

[26]

D. KneesR. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), 565-616.  doi: 10.1142/S021820251250056X.  Google Scholar

[27]

D. KneesR. Rossi and C. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Nonlin. Anal. Ser. B: Real World Appl., 24 (2015), 126-162.  doi: 10.1016/j.nonrwa.2015.02.001.  Google Scholar

[28]

D. KneesR. Rossi and C. Zanini, Balanced viscosity solutions to a rate-independent system for damage, European Journal of Applied Mathematics, 30 (2019), 117-175.  doi: 10.1017/S0956792517000407.  Google Scholar

[29]

D. Knees and A. Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, Mathematical Methods in the Applied Sciences, 35 (2012), 1859-1884.  doi: 10.1002/mma.2598.  Google Scholar

[30]

D. Knees and C. Zanini, Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads, Discrete & Continuous Dynamical Systems-S, (2018). doi: 10.3934/dcdss. 2020332.  Google Scholar

[31]

G. Lazzaroni, R. Rossi, M. Thomas, and R. Toader, Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics, Journal of Physics: Conference Series, 727 (2016), 012009. doi: 10.1088/1742-6596/727/1/012009.  Google Scholar

[32]

G. Lazzaroni, R. Rossi, M. Thomas and R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, J. Dyn. Diff. Equat., (2018). doi: 10.1007/s10884-018-9666-y.  Google Scholar

[33]

J. Mawhin, Analyse. Fondements, Techniques, Évolution. (Analysis. Foundations, Techniques, Evolution), Accès Sciences. De Boeck Université, Brussels, 1997. Available from: https://www.researchgate.net/publication/266367922.  Google Scholar

[34]

C. MieheM. Hofacker and F. Welschinger, A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 2765-2778.  doi: 10.1016/j.cma.2010.04.011.  Google Scholar

[35]

M. Marcus and V. J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Functional Analysis, 33 (1979), 217-229.  doi: 10.1016/0022-1236(79)90113-7.  Google Scholar

[36]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 177-209.  doi: 10.1142/S021820250600111X.  Google Scholar

[37]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences, Springer New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[38]

A. Mielke and T. Roubíček, Rate-independent Systems: Theory and Application, Applied Mathematical Sciences, 193, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[39]

A. MielkeR. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete & Continuous Dynamical Systems, 25 (2009), 585-615.  doi: 10.3934/dcds.2009.25.585.  Google Scholar

[40]

A. MielkeR. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 36-80.  doi: 10.1051/cocv/2010054.  Google Scholar

[41]

A. Mielke and F. Theil, On rate–independent hysteresis models, NODEA, 11 (2004), 151-189.  doi: 10.1007/s00030-003-1052-7.  Google Scholar

[42]

H. Royden and P. Fitzpatrick, Real Analysis (Classic Version), Pearson Modern Classics for Advanced Mathematics Series, Pearson, 2017. Google Scholar

[43]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics, Birkhäuser Basel, 2006. Google Scholar

[44]

T. Roubíček, Models of dynamic damage and phase-field fracture, and their various time discretisations, in Topics in Applied Analysis and Optimisation, Springer, 2019, 363–396. Google Scholar

[45]

E. Rocca and R. Rossi, "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM J. Math. Anal., 74 (2015), 2519-2586.  doi: 10.1137/140960803.  Google Scholar

[46]

R. Rossi and M. Thomas., Coupling rate-independent and rate-dependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017), 1419-1494.  doi: 10.1137/15M1051567.  Google Scholar

[47]

T. RoubíčekM. Thomas and C. G. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Anal. Real World Appl., 22 (2015), 645-663.  doi: 10.1016/j.nonrwa.2014.09.011.  Google Scholar

[48]

E. Schechter, An introduction to the gauge integral, Webpage at Vanderbilt University, 2009. Available from: https://math.vanderbilt.edu/schectex/ccc/gauge/. Google Scholar

[49]

A. SchlüterC. KuhnR. MüllerM. TomutC. TrautmannH. Weick and C. Plate, Phase field modelling of dynamic thermal fracture in the context of irradiation damage, Continuum Mechanics and Thermodynamics, 29 (2017), 977-988.  doi: 10.1007/s00161-015-0456-z.  Google Scholar

[50]

G. Scilla and F. Solombrino, A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension, Journal of Differential Equations, 267 (2019), 6216-6264.  doi: 10.1016/j.jde.2019.06.018.  Google Scholar

[51]

A. SchlüterA. WillenbücherC. Kuhn and R. Müller, Phase field approximation of dynamic brittle fracture, Computational Mechanics, 54 (2014), 1141-1161.  doi: 10.1007/s00466-014-1045-x.  Google Scholar

[52]

M. ThomasC. Bilgen and K. Weinberg, Phase-field fracture at finite strains based on modified invariants: A note on its analysis and simulations, GAMM-Mitteilungen, 40 (2018), 207-237.  doi: 10.1002/gamm.201730004.  Google Scholar

[53]

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Figure 1.  Qualitative shape of $ w_{ \mathbb{C}}: \mathbb{R}\to[w_{0},w^{*}] $: The function is constant on the intervals $ (-\infty,0]\cup[z^{*},\infty) $, monotonously increasing on $ \mathbb{R} $, and convex on the interval $ (-\infty,z_{*}) $ with $ z_{*}>1 $ but non-convex on $ [z_{*},z^{*}) $. The points $ z_{\ominus}\ll0 $ and $ z_{\oplus}\gg z^{*} $ will play a role later in the proof of Theorem 4.1, Formula (41), when showing that solutions $ z_{\tau}^{k} $ of the space-continuous problem (2b) are bounded in $ [0,1] $
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