Details
| Original language | English |
|---|---|
| Article number | 11 |
| Journal | Zeitschrift fur Angewandte Mathematik und Physik |
| Volume | 68 |
| Issue number | 1 |
| Publication status | Published - 2017 |
Abstract
In this note, we extend integrability conditions for the symmetric stretch tensor U in the polar decomposition of the deformation gradient ∇φ=F=RU to the nonsymmetric case. In doing so, we recover integrability conditions for the first Cosserat deformation tensor. Let (Formula presented.). Then, (Formula presented.), giving a connection between the first Cosserat deformation tensor U¯ and the second Cosserat tensor K. (Here, Anti denotes an isomorphism between R 3 × 3 and So(3):={A∈R 3×3×3|A.u∈so(3)∀u∈R 3}). The formula shows that it is not possible to prescribe U¯ and K independent from each other. We also propose a new energy formulation of geometrically nonlinear Cosserat models which completely separate the effects of nonsymmetric straining and curvature. For very weak constitutive assumptions (no direct boundary condition on rotations, zero Cosserat couple modulus, quadratic curvature energy), we show existence of minimizers in Sobolev spaces.
Keywords
- Compatibility conditions, Cosserat continuum, Extended continuum mechanics, Geometrically nonlinear micropolar elasticity, Integrability conditions, Strain and curvature measures
ASJC Scopus subject areas
- Physics and Astronomy(all)
- General Physics and Astronomy
- Mathematics(all)
- Applied Mathematics
- Mathematics(all)
- General Mathematics
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In: Zeitschrift fur Angewandte Mathematik und Physik, Vol. 68, No. 1, 11, 2017.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Integrability conditions between the first and second Cosserat deformation tensor in geometrically nonlinear micropolar models and existence of minimizers
AU - Lankeit, J.
AU - Neff, P.
AU - Osterbrink, F.
N1 - Publisher Copyright: © 2016, Springer International Publishing.
PY - 2017
Y1 - 2017
N2 - In this note, we extend integrability conditions for the symmetric stretch tensor U in the polar decomposition of the deformation gradient ∇φ=F=RU to the nonsymmetric case. In doing so, we recover integrability conditions for the first Cosserat deformation tensor. Let (Formula presented.). Then, (Formula presented.), giving a connection between the first Cosserat deformation tensor U¯ and the second Cosserat tensor K. (Here, Anti denotes an isomorphism between R 3 × 3 and So(3):={A∈R 3×3×3|A.u∈so(3)∀u∈R 3}). The formula shows that it is not possible to prescribe U¯ and K independent from each other. We also propose a new energy formulation of geometrically nonlinear Cosserat models which completely separate the effects of nonsymmetric straining and curvature. For very weak constitutive assumptions (no direct boundary condition on rotations, zero Cosserat couple modulus, quadratic curvature energy), we show existence of minimizers in Sobolev spaces.
AB - In this note, we extend integrability conditions for the symmetric stretch tensor U in the polar decomposition of the deformation gradient ∇φ=F=RU to the nonsymmetric case. In doing so, we recover integrability conditions for the first Cosserat deformation tensor. Let (Formula presented.). Then, (Formula presented.), giving a connection between the first Cosserat deformation tensor U¯ and the second Cosserat tensor K. (Here, Anti denotes an isomorphism between R 3 × 3 and So(3):={A∈R 3×3×3|A.u∈so(3)∀u∈R 3}). The formula shows that it is not possible to prescribe U¯ and K independent from each other. We also propose a new energy formulation of geometrically nonlinear Cosserat models which completely separate the effects of nonsymmetric straining and curvature. For very weak constitutive assumptions (no direct boundary condition on rotations, zero Cosserat couple modulus, quadratic curvature energy), we show existence of minimizers in Sobolev spaces.
KW - Compatibility conditions
KW - Cosserat continuum
KW - Extended continuum mechanics
KW - Geometrically nonlinear micropolar elasticity
KW - Integrability conditions
KW - Strain and curvature measures
UR - http://www.scopus.com/inward/record.url?scp=85006341176&partnerID=8YFLogxK
U2 - 10.1007/s00033-016-0755-7
DO - 10.1007/s00033-016-0755-7
M3 - Article
VL - 68
JO - Zeitschrift fur Angewandte Mathematik und Physik
JF - Zeitschrift fur Angewandte Mathematik und Physik
SN - 0044-2275
IS - 1
M1 - 11
ER -