A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies

Alice Fiaschi

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 2, page 245-278
  • ISSN: 1292-8119

Abstract

top
Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms of stochastic processes on a suitable probability space.

How to cite

top

Fiaschi, Alice. "A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies." ESAIM: Control, Optimisation and Calculus of Variations 15.2 (2009): 245-278. <http://eudml.org/doc/245433>.

@article{Fiaschi2009,
abstract = {Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms of stochastic processes on a suitable probability space.},
author = {Fiaschi, Alice},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {quasistatic evolution; rate-independent processes; elastic materials; incremental problems; Young measures; existence; stochastic processes; probability space},
language = {eng},
number = {2},
pages = {245-278},
publisher = {EDP-Sciences},
title = {A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies},
url = {http://eudml.org/doc/245433},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Fiaschi, Alice
TI - A Young measures approach to quasistatic evolution for a class of material models with nonconvex elastic energies
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2009
PB - EDP-Sciences
VL - 15
IS - 2
SP - 245
EP - 278
AB - Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms of stochastic processes on a suitable probability space.
LA - eng
KW - quasistatic evolution; rate-independent processes; elastic materials; incremental problems; Young measures; existence; stochastic processes; probability space
UR - http://eudml.org/doc/245433
ER -

References

top
  1. [1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125–145. Zbl0565.49010MR751305
  2. [2] J.M. Ball, A version of the fundamental theorem for Young measures, in PDE’s and continuum models of phase transitions (Nice, 1988), Lecture Notes in Physics, Springer-Verlag, Berlin (1989) 207–215. Zbl0991.49500MR1036070
  3. [3] H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam-London; American Elsevier, New York (1973). Zbl0252.47055MR348562
  4. [4] G. Dal Maso, G. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176 (2005) 165–225. Zbl1064.74150MR2186036
  5. [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. Zbl1140.28001MR2291810
  6. [6] G. Dal Maso, A. De Simone, M.G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening. Netw. Heterog. Media (to appear). Zbl1156.74308MR2425074
  7. [7] I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736–756. Zbl0920.49009MR1617712
  8. [8] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energy. J. Reine Angew. Math. 595 (2006) 55–91. Zbl1101.74015MR2244798
  9. [9] M. Kočvara, A. Mielke and T. Roubíček, A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423–447. Zbl1133.74038MR2245202
  10. [10] A.N. Kolmogorov, Foundations of the Theory of Probability. Chelsea Publishing Company, 2nd edition, New York (1956). Zbl0074.12202MR79843
  11. [11] C. Miehe and M. Lambrecht, Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: large-strain theory for standard dissipative solids. Internat. J. Numer. Methods Engrg. 58 (2003) 1–41. Zbl1032.74526MR1999979
  12. [12] C. Miehe, J. Schotte and M. Lambrecht, Computational homogenization of materials with microstructures based on incremental variational formulations, in IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains (Stuttgart, 2001), Solid Mech. Appl., Kluwer Acad. Publ., Dordrecht (2003) 87–100. Zbl1040.74039MR1991327
  13. [13] A. Mielke, Evolution of rate-independent systems, in Evolutionary equations, Vol. II, C.M. Dafermos and E. Feireisl Eds., Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam (2005) 461–559. Zbl1120.47062MR2182832
  14. [14] A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16 (2006) 177–209. Zbl1094.35068MR2210087
  15. [15] A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal. 162 (2002) 137–177. Zbl1012.74054MR1897379
  16. [16] M. Ortiz and E. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Physics Solids 47 (1999) 397–462. Zbl0964.74012MR1674064
  17. [17] P. Pedregal, Parametrized measures and variational principles. Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser Verlag, Basel (1997). Zbl0879.49017MR1452107
  18. [18] M. Valadier, Young measures, in Methods of nonconvex analysis (Varenna, 1989), Lecture Notes in Mathematics, Springer-Verlag, Berlin (1990) 152–188. Zbl0738.28004MR1079763

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.