Young-measure approximations for elastodynamics with non-monotone stress-strain relations
Carstensen, Carsten ; Rieger, Marc Oliver
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004), p. 397-418 / Harvested from Numdam

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 tt =divS(Du) with the non-monotone stress-strain relation S=Dφ 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.

Publié le : 2004-01-01
DOI : https://doi.org/10.1051/m2an:2004019
Classification:  35G25,  47J35,  65P25
@article{M2AN_2004__38_3_397_0,
     author = {Carstensen, Carsten and Rieger, Marc Oliver},
     title = {Young-measure approximations for elastodynamics with non-monotone stress-strain relations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {38},
     year = {2004},
     pages = {397-418},
     doi = {10.1051/m2an:2004019},
     mrnumber = {2075752},
     zbl = {1130.74383},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2004__38_3_397_0}
}
Carstensen, Carsten; 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, Tome 38 (2004) pp. 397-418. doi : 10.1051/m2an:2004019. http://gdmltest.u-ga.fr/item/M2AN_2004__38_3_397_0/

[1] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. | Zbl 0629.49020

[2] J.M. Ball, B. Kirchheim and J. Kristensen, Regularity of quasiconvex envelopes. Calc. Var. Partial Differential Equations 11 (2000) 333-359. | Zbl 0972.49024

[3] H. Berliocchi and J.-M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129-184. | Numdam | Zbl 0282.49041

[4] C. Carstensen, Numerical analysis of microstructure, in Theory and numerics of differential equations (Durham, 2000), Universitext, Springer Verlag, Berlin (2001) 59-126. | Zbl 1070.74033

[5] C. Carstensen and P. Plecháč, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997-1026. | Zbl 0870.65055

[6] C. Carstensen and T. Roubíček, Numerical approximation of Young measures in non-convex variational problems. Numer. Math. 84 (2000) 395-415. | Zbl 0945.65070

[7] C. Carstensen and G. Dolzmann, Time-space discretization of the nonlinear hyperbolic system u tt =div(σ(Du)+Du t ). SIAM J. Numer. Anal. 42 (2004) 75-89. | Zbl 1077.65090

[8] M. Chipot, C. Collins and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282. | Zbl 0824.65045

[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. | Zbl 0735.65042

[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. | Zbl 0725.65067

[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. | Zbl 0586.35065

[12] S. Demoulini, Young-measure solutions for a nonlinear parabolic equation of forward-backward type. SIAM J. Math. Anal. 27 (1996) 376-403. | Zbl 0851.35066

[13] S. Demoulini, Young-measure solutions for nonlinear evolutionary systems of mixed type. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 143-162. | Numdam | Zbl 0871.35065

[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. | Zbl 0872.35026

[15] D. Kinderlehrer and P. Pedregal, Weak convergence of integrands and the Young measure representation. SIAM J. Math. Anal. 23 (1992) 1-19. | Zbl 0757.49014

[16] P. Klouček and M. Luskin, The computation of the dynamics of the martensitic transformation. Contin. Mech. Thermodyn. 6 (1994) 209-240. | Zbl 0825.73047

[17] M. Luskin, On the computation of crystalline microstructure, in Acta numerica, Cambridge Univ. Press, Cambridge (1996) 191-257. | Zbl 0867.65033

[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). | MR 1731640 | Zbl 0968.74050

[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] P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997). | MR 1452107 | Zbl 0879.49017

[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. | Zbl 1007.74016

[22] M.O. Rieger, Young-measure solutions for nonconvex elastodynamics. SIAM J. Math. Anal. 34 (2003) 1380-1398. | Zbl 1039.74005

[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). | MR 2198577 | Zbl 1085.74013

[24] T. Roubíček, Relaxation in optimization theory and variational calculus. Walter de Gruyter & Co., Berlin (1997). | MR 1458067 | Zbl 0880.49002

[25] M. Slemrod, Dynamics of measured valued solutions to a backward-forward heat equation. J. Dynam. Differ. Equations 3 (1991) 1-28. | Zbl 0747.35013

[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. | Zbl 0437.35004

[27] M.E. Taylor, Partial Differential Equations III. Appl. Math. Sciences. Springer-Verlag, 117 (1996). | MR 1395149 | Zbl 0869.35004

[28] L.C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus variations, volume classe III. (1937). | JFM 63.1064.01 | Zbl 0019.21901

[29] L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders Co., Philadelphia (1969). | MR 259704 | Zbl 0177.37801

[30] K. Zhang, On some semiconvex envelopes. NoDEA. Nonlinear Differential Equations Appl. 9 (2002) 37-44. | Zbl 1012.49012