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

Carstensen, Carsten; Rieger, Marc Oliver

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 397-418.

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 $u_{tt}=divS\left(Du\right)$ 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.

Classification : 35G25, 47J35, 65P25
Mots clés : non-monotone evolution, nonlinear elastodynamics, Young-measure approximation, nonlinear wave equation

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

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

[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

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

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

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

[7] C. Carstensen and G. Dolzmann, Time-space discretization of the nonlinear hyperbolic system $u_{tt}=div(\sigma \left(Du\right)+D{u}_t)$. SIAM J. Numer. Anal. 42 (2004) 75-89. | Zbl

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

[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

[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

[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

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

[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

[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

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

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

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

[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 | Zbl

[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 | Zbl

[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

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

[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 | Zbl

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

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

[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

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

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

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

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