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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

Alice Fiaschi 1,

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.
Keywords: Partial damage, Young measures., rate-independent systems, brittle damage.
Mathematics Subject Classification: Primary: 74B20, 74R20, 28A33; Secondary: 74R05, 49J4.
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.

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