A variational approximation scheme for three-dimensional elastodynamics with polyconvex energy

Sophia Demoulini*, David M A Stuart, Athanasios E. Tzavaras

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

37 Scopus citations

Abstract

We construct a variational approximation scheme for the equations of three-dimensional elastodynamics with polyconvex stored energy. The scheme is motivated by some recently discovered geometric identities (QIN [18]) for the null Lagrangians (the determinant and cofactor matrix), and by an associated embedding of the equations of elastodynamics into an enlarged system which is endowed with a convex entropy. The scheme decreases the energy, and its solvability is reduced to the solution of a constrained convex minimization problem. We prove that the approximating process admits regular weak solutions, which in the limit produce a measure-valued solution for polyconvex elastodynamics that satisfies the classical weak form of the geometric identities. This latter property is related to the weak continuity properties of minors of Jacobian matrices, here exploited in a time-dependent setting.

Original languageEnglish (US)
Pages (from-to)325-344
Number of pages20
JournalArchive for Rational Mechanics and Analysis
Volume157
Issue number4
DOIs
StatePublished - May 5 2001
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

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