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

• Published in: Archive for rational mechanics and analysis 2001-05, Vol.157 (4), p.325-344
• Main Authors: DEMOULINI, Sophia, STUART, David M. A, TZAVARAS, Athanasios E
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Numerical approximation and analysis; Ordinary and partial differential equations, boundary value problems; Physics; Solid mechanics; Structural and continuum mechanics; Studies; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Vibrations and mechanical waves
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s002050100137

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.[PUBLICATION ABSTRACT]