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 & 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, (2006).   Google Scholar

[2]

J. F. Babadjian, A quasi-static evolution model for the interaction between fracture and damage,, Arch. Rational Mech. Anal., 200 (2011), 945.  doi: 10.1007/s00205-010-0379-6.  Google Scholar

[3]

J. M. Ball, A version of the fundamental theoremfor Young measures,, in, (1989), 207.   Google Scholar

[4]

G. Bouchitté, A. Mielke, and T. Roubíček, A complete-damage problem at small strains,, ZAMP Z. Angew. Math. Phys., 60 (2009), 205.  doi: 10.1007/s00033-007-7064-0.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

G. Dal Maso, G. Francfort and R. Toader, Quasi-static crack growth in finite elasticity,, Preprint SISSA, (2004).   Google Scholar

[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, 20 (2001), 873.  doi: 10.1016/S0997-7538(01)01171-8.  Google Scholar

[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.  doi: 10.1016/0022-5096(94)90086-8.  Google Scholar

[10]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'', North Holland, (1976).   Google Scholar

[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.  doi: 10.1051/cocv:2008030.  Google Scholar

[12]

A. Fiaschi, Rate-independent phase transitions in elastic materials: a Young-measure approach,, Netw. Heterog. Media, 5 (2010), 257.   Google Scholar

[13]

A. Fiaschi, D. Knees and U. Stefanelli, Young-measure qusi-static damage evolution,, Arch. Rational Mech. Anal., 203 (2012), 415.  doi: 10.1007/s00205-011-0474-3.  Google Scholar

[14]

A. Fiaschi, D. Knees and S. Reichelt, "Global Higher Integrability of Minimizers of Variational Problems with Mixed Boundary Conditions,", WIAS Preprint 1664., (1664).   Google Scholar

[15]

I. Fonseca, D. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition,, Calc. Var. PDEs, 2 (1994), 283.   Google Scholar

[16]

G. Francfort and A. Garroni, A variational view of partial brittle damage evolution,, Arch. Rational Mech. Anal., 182 (2006), 125.  doi: 10.1007/s00205-006-0426-5.  Google Scholar

[17]

G. Francfor and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem,, J. Mech Phys. Solids, 46 (1998), 1319.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[18]

A. Garroni and C. Larsen, Threshold-based quasi-static brittle damage evolution,, Arch. Rational Mech. Anal., 194 (2009), 585.  doi: 10.1007/s00205-008-0174-9.  Google Scholar

[19]

M. Giaquinta and E. Giusti, Quasi-minima,, Ann. Inst. H. Poincaré, 1 (1984), 79.   Google Scholar

[20]

E. Giusti, "Direct Methods in the Calculus of Variations,'', World Scientific Publishing Co., (2003).   Google Scholar

[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.  doi: 10.1007/BF01442860.  Google Scholar

[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.  doi: 10.1111/j.1365-246X.2006.03126.x.  Google Scholar

[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.  doi: 10.1111/j.1365-246X.2009.04200.x.  Google Scholar

[24]

D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces,, J. Geom. Anal., 4 (1994), 59.   Google Scholar

[25]

V. Lyakhovsky, Z. Reches, R. Weiberger and T. E. Scott, Nonlinear elastic behaviour of damaged rocks,, Geophys. J. Int., 130 (1997), 157.  doi: 10.1111/j.1365-246X.1997.tb00995.x.  Google Scholar

[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.  doi: 10.1111/j.1365-246X.2007.03652.x.  Google Scholar

[27]

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

[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.  doi: 10.1016/j.cma.2009.09.020.  Google Scholar

[29]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151.  doi: 10.1007/s00030-003-1052-7.  Google Scholar

[30]

P. Pedregal, "Parametrized Measures and Variational Principles,'', Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser Verlag, (1997).   Google Scholar

[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.  doi: 10.1002/zamm.200900243.  Google Scholar

[32]

M. Valadier, Young measures,, in, (1990), 152.   Google Scholar

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, (2006).   Google Scholar

[2]

J. F. Babadjian, A quasi-static evolution model for the interaction between fracture and damage,, Arch. Rational Mech. Anal., 200 (2011), 945.  doi: 10.1007/s00205-010-0379-6.  Google Scholar

[3]

J. M. Ball, A version of the fundamental theoremfor Young measures,, in, (1989), 207.   Google Scholar

[4]

G. Bouchitté, A. Mielke, and T. Roubíček, A complete-damage problem at small strains,, ZAMP Z. Angew. Math. Phys., 60 (2009), 205.  doi: 10.1007/s00033-007-7064-0.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

G. Dal Maso, G. Francfort and R. Toader, Quasi-static crack growth in finite elasticity,, Preprint SISSA, (2004).   Google Scholar

[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, 20 (2001), 873.  doi: 10.1016/S0997-7538(01)01171-8.  Google Scholar

[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.  doi: 10.1016/0022-5096(94)90086-8.  Google Scholar

[10]

I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'', North Holland, (1976).   Google Scholar

[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.  doi: 10.1051/cocv:2008030.  Google Scholar

[12]

A. Fiaschi, Rate-independent phase transitions in elastic materials: a Young-measure approach,, Netw. Heterog. Media, 5 (2010), 257.   Google Scholar

[13]

A. Fiaschi, D. Knees and U. Stefanelli, Young-measure qusi-static damage evolution,, Arch. Rational Mech. Anal., 203 (2012), 415.  doi: 10.1007/s00205-011-0474-3.  Google Scholar

[14]

A. Fiaschi, D. Knees and S. Reichelt, "Global Higher Integrability of Minimizers of Variational Problems with Mixed Boundary Conditions,", WIAS Preprint 1664., (1664).   Google Scholar

[15]

I. Fonseca, D. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition,, Calc. Var. PDEs, 2 (1994), 283.   Google Scholar

[16]

G. Francfort and A. Garroni, A variational view of partial brittle damage evolution,, Arch. Rational Mech. Anal., 182 (2006), 125.  doi: 10.1007/s00205-006-0426-5.  Google Scholar

[17]

G. Francfor and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem,, J. Mech Phys. Solids, 46 (1998), 1319.  doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[18]

A. Garroni and C. Larsen, Threshold-based quasi-static brittle damage evolution,, Arch. Rational Mech. Anal., 194 (2009), 585.  doi: 10.1007/s00205-008-0174-9.  Google Scholar

[19]

M. Giaquinta and E. Giusti, Quasi-minima,, Ann. Inst. H. Poincaré, 1 (1984), 79.   Google Scholar

[20]

E. Giusti, "Direct Methods in the Calculus of Variations,'', World Scientific Publishing Co., (2003).   Google Scholar

[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.  doi: 10.1007/BF01442860.  Google Scholar

[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.  doi: 10.1111/j.1365-246X.2006.03126.x.  Google Scholar

[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.  doi: 10.1111/j.1365-246X.2009.04200.x.  Google Scholar

[24]

D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces,, J. Geom. Anal., 4 (1994), 59.   Google Scholar

[25]

V. Lyakhovsky, Z. Reches, R. Weiberger and T. E. Scott, Nonlinear elastic behaviour of damaged rocks,, Geophys. J. Int., 130 (1997), 157.  doi: 10.1111/j.1365-246X.1997.tb00995.x.  Google Scholar

[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.  doi: 10.1111/j.1365-246X.2007.03652.x.  Google Scholar

[27]

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

[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.  doi: 10.1016/j.cma.2009.09.020.  Google Scholar

[29]

A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151.  doi: 10.1007/s00030-003-1052-7.  Google Scholar

[30]

P. Pedregal, "Parametrized Measures and Variational Principles,'', Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser Verlag, (1997).   Google Scholar

[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.  doi: 10.1002/zamm.200900243.  Google Scholar

[32]

M. Valadier, Young measures,, in, (1990), 152.   Google Scholar

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