# American Institute of Mathematical Sciences

February  2013, 6(1): 17-42. doi: 10.3934/dcdss.2013.6.17

## Young-measure quasi-static damage evolution: The nonconvex and the brittle cases

 1 IMATI-CNR, v. Ferrata 1, I-27100, Pavia, Italy

Received  May 2011 Revised  September 2011 Published  October 2012

A rate-independent model for incomplete damage is considered, with nonconvex energy density, mixed boundary condition, and nonzero external load, both for non-brittle and brittle materials. An existence result for a Young measure quasi-static evolution is proved.
Citation: Alice Fiaschi. Young-measure quasi-static damage evolution: The nonconvex and the brittle cases. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 17-42. doi: 10.3934/dcdss.2013.6.17
##### References:
 [1] H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization,'' MPS/SIAM Series on Optimization, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006. [2] J. F. Babadjian, A quasi-static evolution model for the interaction between fracture and damage, Arch. Rational Mech. Anal., 200 (2011), 945-1002. doi: 10.1007/s00205-010-0379-6. [3] J. M. Ball, A version of the fundamental theoremfor Young measures, in "PDE's and Continuum Models of Phasetransitions (Nice, 1988)'', Lecture Notes in Physics, Springer-Verlag, Berlin, (1989), 207-215. [4] G. Bouchitté, A. Mielke, and T. Roubíček, A complete-damage problem at small strains, ZAMP Z. Angew. Math. Phys., 60 (2009), 205-236. doi: 10.1007/s00033-007-7064-0. [5] G. Dal Maso, A. De Simone, M. G. Mora and M. Morini, Time-dependent systems of generalized Young measures, Netw. Heterog. Media, 2 (2007), 1-36. [6] G. Dal Maso, A. De Simone, M. G. Mora and M. Morini, Globally stable quasi-static evolution in plasticitywith softening, Netw. Heterog. Media, 3 (2008), 567-614. [7] G. Dal Maso, G. Francfort and R. Toader, Quasi-static crack growth in finite elasticity, Preprint SISSA, Trieste, 2004 (http://www.sissa.it/fa/). [8] A. DeSimone, J. J. Marigo and L. Teresi, A damage mechanics approach to stress softening and its application to rubber, Eur. J. Mech. A, Solids, 20 (2001), 873-892. doi: 10.1016/S0997-7538(01)01171-8. [9] E. De Souza Neto, D. Peric and D. Owen, A phenomenological three-dimensional rate-independent continuum damage model for highly filled polymers: Formulation and computational aspects, J. Mech. Phys. Solids, 42 (1994), 1533-1550. doi: 10.1016/0022-5096(94)90086-8. [10] I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'' North Holland, Amsterdam, 1976. Translation ofAnalyse convexe et problèmes variationnels. Dunod, Paris, 1972. [11] A. Fiaschi, A Young measure approach toquasi-static evolution for a class of material models with nonconvexelastic energies, ESAIM Control Optim. Calc. Var., 15 (2009), 245-278. doi: 10.1051/cocv:2008030. [12] A. Fiaschi, Rate-independent phase transitions in elastic materials: a Young-measure approach, Netw. Heterog. Media, 5 (2010), 257-298. [13] A. Fiaschi, D. Knees and U. Stefanelli, Young-measure qusi-static damage evolution, Arch. Rational Mech. Anal., 203 (2012), 415-453. doi: 10.1007/s00205-011-0474-3. [14] A. Fiaschi, D. Knees and S. Reichelt, "Global Higher Integrability of Minimizers of Variational Problems with Mixed Boundary Conditions," WIAS Preprint 1664. [15] I. Fonseca, D. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition, Calc. Var. PDEs, 2 (1994), 283-313. [16] G. Francfort and A. Garroni, A variational view of partial brittle damage evolution, Arch. Rational Mech. Anal., 182 (2006), 125-152. doi: 10.1007/s00205-006-0426-5. [17] G. Francfor and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech Phys. Solids, 46 (1998), 1319-1342. doi: 10.1016/S0022-5096(98)00034-9. [18] A. Garroni and C. Larsen, Threshold-based quasi-static brittle damage evolution, Arch. Rational Mech. Anal., 194 (2009), 585-609. doi: 10.1007/s00205-008-0174-9. [19] M. Giaquinta and E. Giusti, Quasi-minima, Ann. Inst. H. Poincaré, Analyse non lineaire, 1 (1984), 79-107. [20] E. Giusti, "Direct Methods in the Calculus of Variations,'' World Scientific Publishing Co., Inc., River Edge, NJ, 2003. [21] K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), 679-687. doi: 10.1007/BF01442860. [22] Y. Hamiel, O. Katz, V. Lyakhovsky, Z. Reches and Y. Fialko, Stable and unstable damage evolution in rocks with implications to fracturing of granite, Geophys. J. Int., 167 (2006), 1005-1016. doi: 10.1111/j.1365-246X.2006.03126.x. [23] Y. Hamiel, V. Lyakhovsky, S. Stanchits, G. Dresen and Y. Ben-Zion, Brittle deformation and damage-induced seismic wave anisotropy in rocks, Geophys. J. Int., 178 (2009), 901-909. doi: 10.1111/j.1365-246X.2009.04200.x. [24] D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal., 4 (1994), 59-90. [25] V. Lyakhovsky, Z. Reches, R. Weiberger and T. E. Scott, Nonlinear elastic behaviour of damaged rocks, Geophys. J. Int., 130 (1997), 157-166. doi: 10.1111/j.1365-246X.1997.tb00995.x. [26] V. Lyakhovsky and Y. Ben-Zion, Scaling relations of earthquakes and aseismic deformation in a damage rheology model, Geophys. J. Int., 172 (2008), 651-662. doi: 10.1111/j.1365-246X.2007.03652.x. [27] 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. [28] A. Mielke, T. Roubíček and J. Zemam, Complete damage in elastic and viscoelastic media and its energetics, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1242-1253. doi: 10.1016/j.cma.2009.09.020. [29] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7. [30] P. Pedregal, "Parametrized Measures and Variational Principles,'' Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser Verlag, Basel, 1997. [31] M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain - Existence and regularity results, ZAMM Z. Angew. Math. Mech., 90 (2010), 88-112. doi: 10.1002/zamm.200900243. [32] M. Valadier, Young measures, in "Methods of nonconvex analysis (Varenna,1989)'', Lecture Notes in Math., Springer-Verlag,Berlin, (1990), 152-188.

show all references

##### References:
 [1] H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization,'' MPS/SIAM Series on Optimization, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006. [2] J. F. Babadjian, A quasi-static evolution model for the interaction between fracture and damage, Arch. Rational Mech. Anal., 200 (2011), 945-1002. doi: 10.1007/s00205-010-0379-6. [3] J. M. Ball, A version of the fundamental theoremfor Young measures, in "PDE's and Continuum Models of Phasetransitions (Nice, 1988)'', Lecture Notes in Physics, Springer-Verlag, Berlin, (1989), 207-215. [4] G. Bouchitté, A. Mielke, and T. Roubíček, A complete-damage problem at small strains, ZAMP Z. Angew. Math. Phys., 60 (2009), 205-236. doi: 10.1007/s00033-007-7064-0. [5] G. Dal Maso, A. De Simone, M. G. Mora and M. Morini, Time-dependent systems of generalized Young measures, Netw. Heterog. Media, 2 (2007), 1-36. [6] G. Dal Maso, A. De Simone, M. G. Mora and M. Morini, Globally stable quasi-static evolution in plasticitywith softening, Netw. Heterog. Media, 3 (2008), 567-614. [7] G. Dal Maso, G. Francfort and R. Toader, Quasi-static crack growth in finite elasticity, Preprint SISSA, Trieste, 2004 (http://www.sissa.it/fa/). [8] A. DeSimone, J. J. Marigo and L. Teresi, A damage mechanics approach to stress softening and its application to rubber, Eur. J. Mech. A, Solids, 20 (2001), 873-892. doi: 10.1016/S0997-7538(01)01171-8. [9] E. De Souza Neto, D. Peric and D. Owen, A phenomenological three-dimensional rate-independent continuum damage model for highly filled polymers: Formulation and computational aspects, J. Mech. Phys. Solids, 42 (1994), 1533-1550. doi: 10.1016/0022-5096(94)90086-8. [10] I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'' North Holland, Amsterdam, 1976. Translation ofAnalyse convexe et problèmes variationnels. Dunod, Paris, 1972. [11] A. Fiaschi, A Young measure approach toquasi-static evolution for a class of material models with nonconvexelastic energies, ESAIM Control Optim. Calc. Var., 15 (2009), 245-278. doi: 10.1051/cocv:2008030. [12] A. Fiaschi, Rate-independent phase transitions in elastic materials: a Young-measure approach, Netw. Heterog. Media, 5 (2010), 257-298. [13] A. Fiaschi, D. Knees and U. Stefanelli, Young-measure qusi-static damage evolution, Arch. Rational Mech. Anal., 203 (2012), 415-453. doi: 10.1007/s00205-011-0474-3. [14] A. Fiaschi, D. Knees and S. Reichelt, "Global Higher Integrability of Minimizers of Variational Problems with Mixed Boundary Conditions," WIAS Preprint 1664. [15] I. Fonseca, D. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition, Calc. Var. PDEs, 2 (1994), 283-313. [16] G. Francfort and A. Garroni, A variational view of partial brittle damage evolution, Arch. Rational Mech. Anal., 182 (2006), 125-152. doi: 10.1007/s00205-006-0426-5. [17] G. Francfor and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech Phys. Solids, 46 (1998), 1319-1342. doi: 10.1016/S0022-5096(98)00034-9. [18] A. Garroni and C. Larsen, Threshold-based quasi-static brittle damage evolution, Arch. Rational Mech. Anal., 194 (2009), 585-609. doi: 10.1007/s00205-008-0174-9. [19] M. Giaquinta and E. Giusti, Quasi-minima, Ann. Inst. H. Poincaré, Analyse non lineaire, 1 (1984), 79-107. [20] E. Giusti, "Direct Methods in the Calculus of Variations,'' World Scientific Publishing Co., Inc., River Edge, NJ, 2003. [21] K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), 679-687. doi: 10.1007/BF01442860. [22] Y. Hamiel, O. Katz, V. Lyakhovsky, Z. Reches and Y. Fialko, Stable and unstable damage evolution in rocks with implications to fracturing of granite, Geophys. J. Int., 167 (2006), 1005-1016. doi: 10.1111/j.1365-246X.2006.03126.x. [23] Y. Hamiel, V. Lyakhovsky, S. Stanchits, G. Dresen and Y. Ben-Zion, Brittle deformation and damage-induced seismic wave anisotropy in rocks, Geophys. J. Int., 178 (2009), 901-909. doi: 10.1111/j.1365-246X.2009.04200.x. [24] D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal., 4 (1994), 59-90. [25] V. Lyakhovsky, Z. Reches, R. Weiberger and T. E. Scott, Nonlinear elastic behaviour of damaged rocks, Geophys. J. Int., 130 (1997), 157-166. doi: 10.1111/j.1365-246X.1997.tb00995.x. [26] V. Lyakhovsky and Y. Ben-Zion, Scaling relations of earthquakes and aseismic deformation in a damage rheology model, Geophys. J. Int., 172 (2008), 651-662. doi: 10.1111/j.1365-246X.2007.03652.x. [27] 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. [28] A. Mielke, T. Roubíček and J. Zemam, Complete damage in elastic and viscoelastic media and its energetics, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1242-1253. doi: 10.1016/j.cma.2009.09.020. [29] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7. [30] P. Pedregal, "Parametrized Measures and Variational Principles,'' Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser Verlag, Basel, 1997. [31] M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain - Existence and regularity results, ZAMM Z. Angew. Math. Mech., 90 (2010), 88-112. doi: 10.1002/zamm.200900243. [32] M. Valadier, Young measures, in "Methods of nonconvex analysis (Varenna,1989)'', Lecture Notes in Math., Springer-Verlag,Berlin, (1990), 152-188.
 [1] Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks and Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257 [2] Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585 [3] Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 [4] Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i [5] T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215 [6] Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257 [7] Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332 [8] Riccarda Rossi, Giuseppe Savaré. A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 167-191. doi: 10.3934/dcdss.2013.6.167 [9] Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649 [10] Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 [11] Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070 [12] Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks and Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145 [13] Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i [14] G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks and Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1 [15] Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256 [16] Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591 [17] Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations and Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411 [18] Gilles Pijaudier-Cabot, David Grégoire. A review of non local continuum damage: Modelling of failure?. Networks and Heterogeneous Media, 2014, 9 (4) : 575-597. doi: 10.3934/nhm.2014.9.575 [19] Gianni Dal Maso, Flaviana Iurlano. Fracture models as $\Gamma$-limits of damage models. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1657-1686. doi: 10.3934/cpaa.2013.12.1657 [20] Alexander Mielke. Complete-damage evolution based on energies and stresses. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 423-439. doi: 10.3934/dcdss.2011.4.423

2021 Impact Factor: 1.865