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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  7. 10.1023/A:1007468902439, J. Elasticity 49 (1997), 257–267. (1997) MR1633494DOI10.1023/A:1007468902439
  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
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  11. Rank 1 convex hulls of rotationally invariant functions. In preparation, . 

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