Integrability conditions between the first and second Cosserat deformation tensor in geometrically nonlinear micropolar models and existence of minimizers

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • J. Lankeit
  • P. Neff
  • F. Osterbrink

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OriginalspracheEnglisch
Aufsatznummer11
FachzeitschriftZeitschrift fur Angewandte Mathematik und Physik
Jahrgang68
Ausgabenummer1
PublikationsstatusVeröffentlicht - 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.

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Integrability conditions between the first and second Cosserat deformation tensor in geometrically nonlinear micropolar models and existence of minimizers. / Lankeit, J.; Neff, P.; Osterbrink, F.
in: Zeitschrift fur Angewandte Mathematik und Physik, Jahrgang 68, Nr. 1, 11, 2017.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

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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",
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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

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