April  2011, 4(2): 423-439. doi: 10.3934/dcdss.2011.4.423

Complete-damage evolution based on energies and stresses

1. 

Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin

Received  April 2009 Revised  August 2009 Published  November 2010

The rate-independent damage model recently developed in [1] allows for complete damage, such that the deformation is no longer well-defined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized $\Gamma$-convergence, we generalize the theory to convex, but non-quadratic elastic energies by providing $\Gamma$-convergence of energetic solutions from partial to complete damage under rather general conditions.
Citation: Alexander Mielke. Complete-damage evolution based on energies and stresses. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 423-439. doi: 10.3934/dcdss.2011.4.423
References:
[1]

G. Bouchitté, A. Mielke and T. Roubíček, A complete-damage problem at small strains,, Z. angew. Math. Phys. (ZAMP), 60 (2009), 205.   Google Scholar

[2]

G. Bouchitté and M. Valadier, Integral representation of convex functional on a space of measures,, J. Funct. Anal., 80 (1988), 398.  doi: 10.1016/0022-1236(88)90009-2.  Google Scholar

[3]

A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford University Press, (2002).  doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[4]

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.  doi: 10.1016/S0997-7538(01)01171-8.  Google Scholar

[5]

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

[6]

M. Frémond, K. Kuttler and M. Shillor, Existence and uniqueness of solutions for a dynamic one-dimensional damage model,, J. Math. Anal. Appl., 229 (1999), 271.  doi: 10.1006/jmaa.1998.6160.  Google Scholar

[7]

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

[8]

G. A. Francfort and J.-J. Marigo, Stable damage evolution in a brittle continuous medium,, European J. Mech. A Solids, 12 (1993), 149.   Google Scholar

[9]

G. Francfort 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

[10]

G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies,, J. reine angew. Math., 595 (2006), 55.  doi: 10.1515/CRELLE.2006.044.  Google Scholar

[11]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power,, Internat. J. Solids Structures, 33 (1996), 1083.  doi: 10.1016/0020-7683(95)00074-7.  Google Scholar

[12]

K. Hackl and H. Stumpf, Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-static brittle fracture,, Int. J. Solids Structures, 30 (2003), 1567.   Google Scholar

[13]

D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation,, Math. Models Meth. Appl. Sci. (M3AS), 18 (2008), 1529.   Google Scholar

[14]

D. Knees and A. Mielke, Energy release rate for cracks in finite-strain elasticity,, Math. Methods Appl. Sci., 31 (2008), 501.  doi: 10.1002/mma.922.  Google Scholar

[15]

D. Knees, C. Zanini and A. Mielke, Crack propagation in polyconvex materials,, Physica D, 239 (2010), 1470.  doi: 10.1016/j.physd.2009.02.008.  Google Scholar

[16]

P. M. Mariano and G. Augusti, Basic topics on damage pseudo-potentials,, Int. J. Solids Structures, 38 (2001), 1963.  doi: 10.1016/S0020-7683(00)00146-3.  Google Scholar

[17]

A. Mielke, Evolution in rate-independent systems (Ch. 6),, In, 2 (2005), 461.   Google Scholar

[18]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity,, Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177.   Google Scholar

[19]

A. Mielke and T. Roubíček, "Rate-Independent Systems: Theory and Application,", In preparation, (2011).   Google Scholar

[20]

A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis,, In H.-D. Alber, (1999), 117.   Google Scholar

[21]

A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151.   Google Scholar

[22]

U. Mosco, Approximation of the solutions of some variational inequalities,, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 373.   Google Scholar

[23]

A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. Part. Diff. Eqns., 31 (2008), 387.   Google Scholar

[24]

A. Mielke, T. Roubíček and J. Zeman, 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

[25]

F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$,, J. Math. Pures Appl. (9), 60 (1981), 309.   Google Scholar

[26]

M. Ortiz, A constitutive theory for the inelastic behavior of concrete,, Mech. Materials, 4 (1985), 67.  doi: 10.1016/0167-6636(85)90007-9.  Google Scholar

[27]

M. Sofonea, W. Han and M. Shillor, "Analysis and Approximation of Contact Problems with Adhesion or Damage," volume 276 of "Pure and Applied Mathematics (Boca Raton),", Chapman & Hall/CRC, (2006).   Google Scholar

[28]

M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain: existence and regularity results,, Z. angew. Math. Mech. (ZAMM), 90 (2010), 88.   Google Scholar

[29]

M. Thomas, "Damage Evolution for a Model with Regularization,", PhD thesis, (2009).   Google Scholar

[30]

A. Visintin, Strong convergence results related to strict convexity,, Comm. Partial Differential Equations, 9 (1984), 439.  doi: 10.1080/03605308408820337.  Google Scholar

[31]

C. Zălinescu, "Convex Analysis in General Vector Spaces,", World Scientific Publishing Co. Inc., (2002).  doi: 10.1142/9789812777096.  Google Scholar

show all references

References:
[1]

G. Bouchitté, A. Mielke and T. Roubíček, A complete-damage problem at small strains,, Z. angew. Math. Phys. (ZAMP), 60 (2009), 205.   Google Scholar

[2]

G. Bouchitté and M. Valadier, Integral representation of convex functional on a space of measures,, J. Funct. Anal., 80 (1988), 398.  doi: 10.1016/0022-1236(88)90009-2.  Google Scholar

[3]

A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford University Press, (2002).  doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[4]

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.  doi: 10.1016/S0997-7538(01)01171-8.  Google Scholar

[5]

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

[6]

M. Frémond, K. Kuttler and M. Shillor, Existence and uniqueness of solutions for a dynamic one-dimensional damage model,, J. Math. Anal. Appl., 229 (1999), 271.  doi: 10.1006/jmaa.1998.6160.  Google Scholar

[7]

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

[8]

G. A. Francfort and J.-J. Marigo, Stable damage evolution in a brittle continuous medium,, European J. Mech. A Solids, 12 (1993), 149.   Google Scholar

[9]

G. Francfort 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

[10]

G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies,, J. reine angew. Math., 595 (2006), 55.  doi: 10.1515/CRELLE.2006.044.  Google Scholar

[11]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power,, Internat. J. Solids Structures, 33 (1996), 1083.  doi: 10.1016/0020-7683(95)00074-7.  Google Scholar

[12]

K. Hackl and H. Stumpf, Micromechanical concept for the analysis of damage evolution in thermo-viscoelastic and quasi-static brittle fracture,, Int. J. Solids Structures, 30 (2003), 1567.   Google Scholar

[13]

D. Knees, A. Mielke and C. Zanini, On the inviscid limit of a model for crack propagation,, Math. Models Meth. Appl. Sci. (M3AS), 18 (2008), 1529.   Google Scholar

[14]

D. Knees and A. Mielke, Energy release rate for cracks in finite-strain elasticity,, Math. Methods Appl. Sci., 31 (2008), 501.  doi: 10.1002/mma.922.  Google Scholar

[15]

D. Knees, C. Zanini and A. Mielke, Crack propagation in polyconvex materials,, Physica D, 239 (2010), 1470.  doi: 10.1016/j.physd.2009.02.008.  Google Scholar

[16]

P. M. Mariano and G. Augusti, Basic topics on damage pseudo-potentials,, Int. J. Solids Structures, 38 (2001), 1963.  doi: 10.1016/S0020-7683(00)00146-3.  Google Scholar

[17]

A. Mielke, Evolution in rate-independent systems (Ch. 6),, In, 2 (2005), 461.   Google Scholar

[18]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity,, Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177.   Google Scholar

[19]

A. Mielke and T. Roubíček, "Rate-Independent Systems: Theory and Application,", In preparation, (2011).   Google Scholar

[20]

A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis,, In H.-D. Alber, (1999), 117.   Google Scholar

[21]

A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151.   Google Scholar

[22]

U. Mosco, Approximation of the solutions of some variational inequalities,, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 373.   Google Scholar

[23]

A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. Part. Diff. Eqns., 31 (2008), 387.   Google Scholar

[24]

A. Mielke, T. Roubíček and J. Zeman, 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

[25]

F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$,, J. Math. Pures Appl. (9), 60 (1981), 309.   Google Scholar

[26]

M. Ortiz, A constitutive theory for the inelastic behavior of concrete,, Mech. Materials, 4 (1985), 67.  doi: 10.1016/0167-6636(85)90007-9.  Google Scholar

[27]

M. Sofonea, W. Han and M. Shillor, "Analysis and Approximation of Contact Problems with Adhesion or Damage," volume 276 of "Pure and Applied Mathematics (Boca Raton),", Chapman & Hall/CRC, (2006).   Google Scholar

[28]

M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain: existence and regularity results,, Z. angew. Math. Mech. (ZAMM), 90 (2010), 88.   Google Scholar

[29]

M. Thomas, "Damage Evolution for a Model with Regularization,", PhD thesis, (2009).   Google Scholar

[30]

A. Visintin, Strong convergence results related to strict convexity,, Comm. Partial Differential Equations, 9 (1984), 439.  doi: 10.1080/03605308408820337.  Google Scholar

[31]

C. Zălinescu, "Convex Analysis in General Vector Spaces,", World Scientific Publishing Co. Inc., (2002).  doi: 10.1142/9789812777096.  Google Scholar

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