| Title:
|
Rank 1 convex hulls of isotropic functions in dimension 2 by 2 (English) |
| Author:
|
Šilhavý, M. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
2 |
| Year:
|
2001 |
| Pages:
|
521-529 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
rank 1 convexity |
| Keyword:
|
relaxation |
| Keyword:
|
stored energies |
| MSC:
|
49J45 |
| MSC:
|
74G65 |
| MSC:
|
74N99 |
| idZBL:
|
Zbl 1070.49008 |
| idMR:
|
MR1844288 |
| DOI:
|
10.21136/MB.2001.134029 |
| . |
| Date available:
|
2009-09-24T21:53:31Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134029 |
| . |
| Reference:
|
[1] Ball, J. M.: Convexity conditions and existence theorems in nonlinear elasticity.Arch. Rational Mech. Anal. 63 (1977), 337–403. Zbl 0368.73040, MR 0475169, 10.1007/BF00279992 |
| Reference:
|
[2] Buttazzo, G., Dacorogna, B., Gangbo, W.: On the envelopes of functions depending on singular values of matrices.Bolletino U. M. I. 7 (1994), 17–35. MR 1274317 |
| Reference:
|
[3] Dacorogna, B.: Direct Methods in the Calculus of Variations.Springer, Berlin, 1990. MR 2361288 |
| Reference:
|
[4] Kohn, R. V.: The relaxation of a double-well energy.Continuum Mech. Thermodyn. 3 (1991), 3–236. Zbl 0825.73029, MR 1122017, 10.1007/BF01135336 |
| Reference:
|
[5] Kohn, R. V., Strang, G.: Optimal design and relaxation of variational problems, I, II, III.Comm. Pure Appl. Math. 39 (1986), 113–137, 139–182, 353–377. MR 0820342, 10.1002/cpa.3160390305 |
| Reference:
|
[6] Morrey, Jr, C. B.: Multiple Integrals in the Calculus of Variations.Springer, New York, 1966. MR 0202511 |
| Reference:
|
[7] Rosakis, P.: Characterizat on of convex isotropic functions.J. Elasticity 49 (1997), 257–267. MR 1633494, 10.1023/A:1007468902439 |
| Reference:
|
[8] Roubíček, T.: Relaxation in optimization theory and variational calculus.W. de Gruyter, Berlin, 1997. MR 1458067 |
| Reference:
|
[9] Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media.Springer, Berlin, 1997. MR 1423807 |
| Reference:
|
[10] Šilhavý, M.: 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. MR 1727470 |
| Reference:
|
[11] Šilhavý, M.: Rank 1 convex hulls of rotationally invariant functions. In preparation.. |
| . |
