Young-Measure approximations for elastodynamics with non-monotone stress-strain relations
Department of Mathematics,
Humboldt-Universität zu Berlin, Unter den Linden 6,
10099 Berlin, Germany. email@example.com.
2 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy. firstname.lastname@example.org.
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 with the non-monotone stress-strain relation 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.
Mathematics Subject Classification: 35G25 / 47J35 / 65P25
Key words: Non-monotone evolution / nonlinear elastodynamics / Young-measure approximation / nonlinear wave equation.
© EDP Sciences, SMAI, 2004