| Title:
|
Some estimates for the oscillation of the deformation gradient (English) |
| Author:
|
Mošová, Vratislava |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
45 |
| Issue:
|
6 |
| Year:
|
2000 |
| Pages:
|
401-410 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
As a measure of deformation we can take the difference $D\vec{\phi }-R$, where $D\vec{\phi }$ is the deformation gradient of the mapping $\vec{\phi }$ and $R$ is the deformation gradient of the mapping $\vec{\gamma }$, which represents some proper rigid motion. In this article, the norm $\Vert D\vec{\phi }-R\Vert _{L^p(\Omega )}$ is estimated by means of the scalar measure $e(\vec{\phi })$ of nonlinear strain. First, the estimates are given for a deformation $\vec{\phi }\in W^{1,p}(\Omega )$ satisfying the condition $\vec{\phi }\big |_{\partial \Omega } = \vec{\hspace{0.7pt}\mathop {\mathrm {id}}}$. Then we deduce the estimate in the case that $\vec{\phi }(x)$ is a bi-Lipschitzian deformation and $\vec{\phi }\big |_{\partial \Omega } \ne \vec{\hspace{0.7pt}\mathop {\mathrm {id}}}$. (English) |
| Keyword:
|
hyperelastic material |
| Keyword:
|
deformation gradient |
| Keyword:
|
strain tensor |
| Keyword:
|
matrix and spectral norms |
| Keyword:
|
bi-Lipschitzian map |
| MSC:
|
35Q72 |
| MSC:
|
73G05 |
| MSC:
|
74B20 |
| idZBL:
|
Zbl 1002.74011 |
| idMR:
|
MR1800961 |
| DOI:
|
10.1023/A:1022340215798 |
| . |
| Date available:
|
2009-09-22T18:04:50Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134448 |
| . |
| Reference:
|
[1] P. G. Ciarlet: Mathematical Elasticity.North-Holland, Amsterdam, 1988. Zbl 0648.73014, MR 0936420 |
| Reference:
|
[2] F. John: Bounds for Deformations in Terms of Average Strains.In: Inequalities III (O. Shisha, ed.), Academic Press, New York, 1972. Zbl 0292.53003, MR 0344392 |
| Reference:
|
[3] R. V. Kohn: New integral estimates for deformation in terms of their nonlinear strains.Arch. Rational Mech. Anal. 78 (1982), 131–172. MR 0648942, 10.1007/BF00250837 |
| Reference:
|
[4] A. I. Koshelev: The weighted Korn inequality and some iteration processes for quasilinear elliptic systems.Dokl. Akad. Nauk SSSR 271 (1983), 1056–1059. MR 0722019 |
| Reference:
|
[5] J. Nečas, I. Hlaváček: Introduction to Mathematical Theory of Elastic and Elastoplastic Bodies.SNTL, Praha, 1983 (in Czech). |
| . |
