Rank 1 convex hulls of isotropic functions in dimension 2 by 2

Miroslav Šilhavý

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 2, page 521-529
  • ISSN: 0862-7959

Abstract

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Let f be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set M 2 × 2 of all 2 by 2 matrices. Based on conditions for the rank 1 convexity of f in terms of signed invariants of 𝔸 (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.

How to cite

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Šilhavý, Miroslav. "Rank 1 convex hulls of isotropic functions in dimension 2 by 2." Mathematica Bohemica 126.2 (2001): 521-529. <http://eudml.org/doc/248835>.

@article{Šilhavý2001,
abstract = {Let $f$ be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set $\text\{M\}^\{2\times 2\}$ of all $2$ by $2$ matrices. Based on conditions for the rank 1 convexity of $f$ in terms of signed invariants of $\mathbb \{A\}$ (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.},
author = {Šilhavý, Miroslav},
journal = {Mathematica Bohemica},
keywords = {rank 1 convexity; relaxation; stored energies; rank 1 convexity; relaxation; stored energies},
language = {eng},
number = {2},
pages = {521-529},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Rank 1 convex hulls of isotropic functions in dimension 2 by 2},
url = {http://eudml.org/doc/248835},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Šilhavý, Miroslav
TI - Rank 1 convex hulls of isotropic functions in dimension 2 by 2
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 2
SP - 521
EP - 529
AB - Let $f$ be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set $\text{M}^{2\times 2}$ of all $2$ by $2$ matrices. Based on conditions for the rank 1 convexity of $f$ in terms of signed invariants of $\mathbb {A}$ (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.
LA - eng
KW - rank 1 convexity; relaxation; stored energies; rank 1 convexity; relaxation; stored energies
UR - http://eudml.org/doc/248835
ER -

References

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  6. Multiple Integrals in the Calculus of Variations, Springer, New York, 1966. (1966) MR0202511
  7. Characterizat on of convex isotropic functions, J. Elasticity 49 (1997), 257–267. (1997) MR1633494
  8. Relaxation in optimization theory and variational calculus, W. de Gruyter, Berlin, 1997. (1997) MR1458067
  9. The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, 1997. (1997) MR1423807
  10. Convexity conditions for rotationally invariant functions in two dimensions, Applied Nonlinear Analysis, A. Sequeira et al. (eds.), Kluwer Academic, New York, 1999, pp. 513–530. (1999) MR1727470
  11. Rank 1 convex hulls of rotationally invariant functions. In preparation, . 

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