Computation of the second variation of the energy functional of a two-phase medium. (English. Russian original) Zbl 1150.74572
J. Math. Sci., New York 106, No. 3, 2929-2951 (2001); translation from Probl. Mat. Anal. 22, 41-73 (2001).
The article under review is a continuation of the paper [V. G. Osmolovskij, Probl. Math. Anal. 22, 160–178 (2001)] wherein necessary conditions are obtained for extremum in a variational problem of phase transformation in elastic media under inhomogeneous boundary conditions.
The aim of the second part is to obtain a formula for the second variation of the energy functional. It is assumed that the critical field of displacements and the critical interface are sufficiently smooth. The authors use an approach proposed by V. G. Osmolovskij [Sib. Math. J. 35, No. 4, 743–753 (1994; Zbl 0866.49003)] which makes it possible to write a formula for the second variation and to estimate the residual both in an interior subdomain and in the neighborhood of the intersection of the interface with the boundary of the media.
The aim of the second part is to obtain a formula for the second variation of the energy functional. It is assumed that the critical field of displacements and the critical interface are sufficiently smooth. The authors use an approach proposed by V. G. Osmolovskij [Sib. Math. J. 35, No. 4, 743–753 (1994; Zbl 0866.49003)] which makes it possible to write a formula for the second variation and to estimate the residual both in an interior subdomain and in the neighborhood of the intersection of the interface with the boundary of the media.
Reviewer: V. Grebenev (Novosibirsk)
MSC:
| 74N20 | Dynamics of phase boundaries in solids |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 74B99 | Elastic materials |
