# American Institute of Mathematical Sciences

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), 205-236.  Google Scholar [2] G. Bouchitté and M. Valadier, Integral representation of convex functional on a space of measures, J. Funct. Anal., 80 (1988), 398-420. 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), 873-892. 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), 1533-1550. 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), 271-294. 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), 125-152. 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), 149-189.  Google Scholar [9] G. Francfort 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.  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), 55-91. 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), 1083-1103. 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), 1567-1584. 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), 1529-1569.  Google Scholar [14] D. Knees and A. Mielke, Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci., 31 (2008), 501-528. doi: 10.1002/mma.922.  Google Scholar [15] D. Knees, C. Zanini and A. Mielke, Crack propagation in polyconvex materials, Physica D, 239 (2010), 1470-1484. 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), 1963-1974. doi: 10.1016/S0020-7683(00)00146-3.  Google Scholar [17] A. Mielke, Evolution in rate-independent systems (Ch. 6), In "C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Evolutionary Equations," vol. 2, pages 461-559. Elsevier B.V., Amsterdam, 2005.  Google Scholar [18] A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177-209.  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, R. Balean and R. Farwig, editors, Proceedings of the Workshop on "Models of Continuum Mechanics in Analysis and Engineering," pages 117-129, Aachen, 1999. Shaker-Verlag. Google Scholar [21] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151-189.  Google Scholar [22] U. Mosco, Approximation of the solutions of some variational inequalities, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 373-394. 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), 387-416.  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), 1242-1253. 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), 309-322.  Google Scholar [26] M. Ortiz, A constitutive theory for the inelastic behavior of concrete, Mech. Materials, 4 (1985), 67-93. 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, Boca Raton, FL, 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), 88-112. Google Scholar [29] M. Thomas, "Damage Evolution for a Model with Regularization," PhD thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, 2009. Google Scholar [30] A. Visintin, Strong convergence results related to strict convexity, Comm. Partial Differential Equations, 9 (1984), 439-466. doi: 10.1080/03605308408820337.  Google Scholar [31] C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific Publishing Co. Inc., River Edge, NJ, 2002. doi: 10.1142/9789812777096.  Google Scholar

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##### References:
 [1] G. Bouchitté, A. Mielke and T. Roubíček, A complete-damage problem at small strains, Z. angew. Math. Phys. (ZAMP), 60 (2009), 205-236.  Google Scholar [2] G. Bouchitté and M. Valadier, Integral representation of convex functional on a space of measures, J. Funct. Anal., 80 (1988), 398-420. 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), 873-892. 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), 1533-1550. 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), 271-294. 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), 125-152. 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), 149-189.  Google Scholar [9] G. Francfort 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.  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), 55-91. 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), 1083-1103. 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), 1567-1584. 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), 1529-1569.  Google Scholar [14] D. Knees and A. Mielke, Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci., 31 (2008), 501-528. doi: 10.1002/mma.922.  Google Scholar [15] D. Knees, C. Zanini and A. Mielke, Crack propagation in polyconvex materials, Physica D, 239 (2010), 1470-1484. 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), 1963-1974. doi: 10.1016/S0020-7683(00)00146-3.  Google Scholar [17] A. Mielke, Evolution in rate-independent systems (Ch. 6), In "C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Evolutionary Equations," vol. 2, pages 461-559. Elsevier B.V., Amsterdam, 2005.  Google Scholar [18] A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177-209.  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, R. Balean and R. Farwig, editors, Proceedings of the Workshop on "Models of Continuum Mechanics in Analysis and Engineering," pages 117-129, Aachen, 1999. Shaker-Verlag. Google Scholar [21] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151-189.  Google Scholar [22] U. Mosco, Approximation of the solutions of some variational inequalities, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 373-394. 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), 387-416.  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), 1242-1253. 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), 309-322.  Google Scholar [26] M. Ortiz, A constitutive theory for the inelastic behavior of concrete, Mech. Materials, 4 (1985), 67-93. 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, Boca Raton, FL, 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), 88-112. Google Scholar [29] M. Thomas, "Damage Evolution for a Model with Regularization," PhD thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, 2009. Google Scholar [30] A. Visintin, Strong convergence results related to strict convexity, Comm. Partial Differential Equations, 9 (1984), 439-466. doi: 10.1080/03605308408820337.  Google Scholar [31] C. Zălinescu, "Convex Analysis in General Vector Spaces," World Scientific Publishing Co. Inc., River Edge, NJ, 2002. doi: 10.1142/9789812777096.  Google Scholar
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