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:
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show all references

References:
[1]

Z. angew. Math. Phys. (ZAMP), 60 (2009), 205-236.  Google Scholar

[2]

J. Funct. Anal., 80 (1988), 398-420. doi: 10.1016/0022-1236(88)90009-2.  Google Scholar

[3]

Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[4]

Eur. J. Mech. A Solids, 20 (2001), 873-892. doi: 10.1016/S0997-7538(01)01171-8.  Google Scholar

[5]

J. Mech. Phys. Solids, 42 (1994), 1533-1550. doi: 10.1016/0022-5096(94)90086-8.  Google Scholar

[6]

J. Math. Anal. Appl., 229 (1999), 271-294. doi: 10.1006/jmaa.1998.6160.  Google Scholar

[7]

Arch. Rational Mech. Anal., 182 (2006), 125-152. doi: 10.1007/s00205-006-0426-5.  Google Scholar

[8]

European J. Mech. A Solids, 12 (1993), 149-189.  Google Scholar

[9]

J. Mech. Phys. Solids, 46 (1998), 1319-1342. doi: 10.1016/S0022-5096(98)00034-9.  Google Scholar

[10]

J. reine angew. Math., 595 (2006), 55-91. doi: 10.1515/CRELLE.2006.044.  Google Scholar

[11]

Internat. J. Solids Structures, 33 (1996), 1083-1103. doi: 10.1016/0020-7683(95)00074-7.  Google Scholar

[12]

Int. J. Solids Structures, 30 (2003), 1567-1584. Google Scholar

[13]

Math. Models Meth. Appl. Sci. (M3AS), 18 (2008), 1529-1569.  Google Scholar

[14]

Math. Methods Appl. Sci., 31 (2008), 501-528. doi: 10.1002/mma.922.  Google Scholar

[15]

Physica D, 239 (2010), 1470-1484. doi: 10.1016/j.physd.2009.02.008.  Google Scholar

[16]

Int. J. Solids Structures, 38 (2001), 1963-1974. doi: 10.1016/S0020-7683(00)00146-3.  Google Scholar

[17]

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]

Math. Models Meth. Appl. Sci. (M3AS), 16 (2006), 177-209.  Google Scholar

[19]

In preparation, 2011. Google Scholar

[20]

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]

Nonl. Diff. Eqns. Appl. (NoDEA), 11 (2004), 151-189.  Google Scholar

[22]

Ann. Scuola Norm. Sup. Pisa, 21 (1967), 373-394. Google Scholar

[23]

Calc. Var. Part. Diff. Eqns., 31 (2008), 387-416.  Google Scholar

[24]

Comput. Methods Appl. Mech. Engrg., 199 (2010), 1242-1253. doi: 10.1016/j.cma.2009.09.020.  Google Scholar

[25]

J. Math. Pures Appl. (9), 60 (1981), 309-322.  Google Scholar

[26]

Mech. Materials, 4 (1985), 67-93. doi: 10.1016/0167-6636(85)90007-9.  Google Scholar

[27]

Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[28]

Z. angew. Math. Mech. (ZAMM), 90 (2010), 88-112. Google Scholar

[29]

PhD thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, 2009. Google Scholar

[30]

Comm. Partial Differential Equations, 9 (1984), 439-466. doi: 10.1080/03605308408820337.  Google Scholar

[31]

World Scientific Publishing Co. Inc., River Edge, NJ, 2002. doi: 10.1142/9789812777096.  Google Scholar

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