Young-measure approximations for elastodynamics with non-monotone stress-strain relations

Carsten Carstensen; Marc Oliver Rieger

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 3, page 397-418
  • ISSN: 0764-583X

Abstract

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Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density φ . Their time-evolution leads to a nonlinear wave equation u t t = div S ( D u ) with the non-monotone stress-strain relation S = D φ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very weak sense. It is shown that discrete solutions exist and generate weakly convergent subsequences whose limit is a Young-measure solution. Numerical examples in one space dimension illustrate the time-evolving phase transitions and microstructures of a nonlinearly vibrating string.

How to cite

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Carstensen, Carsten, and Rieger, Marc Oliver. "Young-measure approximations for elastodynamics with non-monotone stress-strain relations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.3 (2004): 397-418. <http://eudml.org/doc/245672>.

@article{Carstensen2004,
abstract = {Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density $\phi $. Their time-evolution leads to a nonlinear wave equation $u_\{tt\}=\operatorname\{div\}S(Du)$ with the non-monotone stress-strain relation $S=D\phi $ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very weak sense. It is shown that discrete solutions exist and generate weakly convergent subsequences whose limit is a Young-measure solution. Numerical examples in one space dimension illustrate the time-evolving phase transitions and microstructures of a nonlinearly vibrating string.},
author = {Carstensen, Carsten, Rieger, Marc Oliver},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {non-monotone evolution; nonlinear elastodynamics; Young-measure approximation; nonlinear wave equation},
language = {eng},
number = {3},
pages = {397-418},
publisher = {EDP-Sciences},
title = {Young-measure approximations for elastodynamics with non-monotone stress-strain relations},
url = {http://eudml.org/doc/245672},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Carstensen, Carsten
AU - Rieger, Marc Oliver
TI - Young-measure approximations for elastodynamics with non-monotone stress-strain relations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 3
SP - 397
EP - 418
AB - Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density $\phi $. Their time-evolution leads to a nonlinear wave equation $u_{tt}=\operatorname{div}S(Du)$ with the non-monotone stress-strain relation $S=D\phi $ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very weak sense. It is shown that discrete solutions exist and generate weakly convergent subsequences whose limit is a Young-measure solution. Numerical examples in one space dimension illustrate the time-evolving phase transitions and microstructures of a nonlinearly vibrating string.
LA - eng
KW - non-monotone evolution; nonlinear elastodynamics; Young-measure approximation; nonlinear wave equation
UR - http://eudml.org/doc/245672
ER -

References

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  1. [1] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13–52. Zbl0629.49020
  2. [2] J.M. Ball, B. Kirchheim and J. Kristensen, Regularity of quasiconvex envelopes. Calc. Var. Partial Differential Equations 11 (2000) 333–359. Zbl0972.49024
  3. [3] H. Berliocchi and J.-M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129–184. Zbl0282.49041
  4. [4] C. Carstensen, Numerical analysis of microstructure, in Theory and numerics of differential equations (Durham, 2000), Universitext, Springer Verlag, Berlin (2001) 59–126. Zbl1070.74033
  5. [5] C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997–1026. Zbl0870.65055
  6. [6] C. Carstensen and T. Roubíček, Numerical approximation of Young measures in non-convex variational problems. Numer. Math. 84 (2000) 395–415. Zbl0945.65070
  7. [7] C. Carstensen and G. Dolzmann, Time-space discretization of the nonlinear hyperbolic system u t t = d i v ( σ ( D u ) + D u t ) . SIAM J. Numer. Anal. 42 (2004) 75–89. Zbl1077.65090
  8. [8] M. Chipot, C. Collins and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259–282. Zbl0824.65045
  9. [9] C. Collins and M. Luskin, Optimal-order error estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp. 57 (1991) 621–637. Zbl0735.65042
  10. [10] C. Collins, D. Kinderlehrer and M. Luskin, Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal. 28 (1991) 321–332. Zbl0725.65067
  11. [11] C.M. Dafermos and W.J. Hrusa, Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics. Arch. Rational Mech. Anal. 87 (1985) 267–292. Zbl0586.35065
  12. [12] S. Demoulini, Young-measure solutions for a nonlinear parabolic equation of forward-backward type. SIAM J. Math. Anal. 27 (1996) 376–403. Zbl0851.35066
  13. [13] S. Demoulini, Young-measure solutions for nonlinear evolutionary systems of mixed type. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 143–162. Zbl0871.35065
  14. [14] G. Friesecke and G. Dolzmann, Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy. SIAM J. Math. Anal. 28 (1997) 363–380. Zbl0872.35026
  15. [15] D. Kinderlehrer and P. Pedregal, Weak convergence of integrands and the Young measure representation. SIAM J. Math. Anal. 23 (1992) 1–19. Zbl0757.49014
  16. [16] P. Klouček and M. Luskin, The computation of the dynamics of the martensitic transformation. Contin. Mech. Thermodyn. 6 (1994) 209–240. Zbl0825.73047
  17. [17] M. Luskin, On the computation of crystalline microstructure, in Acta numerica, Cambridge Univ. Press, Cambridge (1996) 191–257. Zbl0867.65033
  18. [18] S. Müller, Variational models for microstructure and phase transition, in Calculus of Variations and Geometric Evolution Problems, S. Hildebrandt and M. Struwe Eds., Lect. Notes Math. 1713, Springer-Verlag, Berlin (1999). Zbl0968.74050MR1731640
  19. [19] R.A. Nicolaides and N.J. Walkington, Computation of microstructure utilizing Young measure representations, in Transactions of the Tenth Army Conference on Applied Mathematics and Computing (West Point, NY, 1992), US Army Res. Office, Research Triangle Park, NC (1993) 57–68. 
  20. [20] P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997). Zbl0879.49017MR1452107
  21. [21] M.O. Rieger, Time dependent Young measure solutions for an elasticity equation with diffusion, in International Conference on Differential Equations, Vol. 2 (Berlin, 1999), World Sci. Publishing, River Edge, NJ 1 (2000) 457–459. Zbl1007.74016
  22. [22] M.O. Rieger, Young-measure solutions for nonconvex elastodynamics. SIAM J. Math. Anal. 34 (2003) 1380–1398. Zbl1039.74005
  23. [23] M.O. Rieger and J. Zimmer, Global existence for nonconvex thermoelasticity. Preprint 30/2002, Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, USA (2002). Zbl1085.74013MR2198577
  24. [24] T. Roubíček, Relaxation in optimization theory and variational calculus. Walter de Gruyter & Co., Berlin (1997). Zbl0880.49002MR1458067
  25. [25] M. Slemrod, Dynamics of measured valued solutions to a backward-forward heat equation. J. Dynam. Differ. Equations 3 (1991) 1–28. Zbl0747.35013
  26. [26] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium. Pitman, Boston, Mass. IV (1979) 136–212. Zbl0437.35004
  27. [27] M.E. Taylor, Partial Differential Equations III. Appl. Math. Sciences. Springer-Verlag, 117 (1996). Zbl0869.35004MR1395149
  28. [28] L.C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus variations, volume classe III. (1937). Zbl0019.21901JFM63.1064.01
  29. [29] L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders Co., Philadelphia (1969). Zbl0177.37801MR259704
  30. [30] K. Zhang, On some semiconvex envelopes. NoDEA. Nonlinear Differential Equations Appl. 9 (2002) 37–44. Zbl1012.49012

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