A new approach to curvature measures in linear shell theories

Šilhavý, Miroslav

doi:10.1177/1081286520972752

Cited by 8 publications

(6 citation statements)

References 11 publications

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“…Moreover, the assumptions which follow from considering µ c → ∞ are also expressed naturally in terms of 3), see estimate (3.63) and Section 3.2. This line of thought, beside some other arguments presented in the linearised framework by Anicic and Léger [6], see also [3], and more recently by Šilhavỳ [39], suggest that the triple G ∞ , R ∞ − 2 G ∞ L y0 and K ∞ are appropriate measures to express the change of metric and of the curvatures H and K, while the bending and drilling effects are both additionally incorporated in the bending-curvature energy through the elastic shell bending-curvature tensor K ∞ .…”

Section: Strain Measures In the Cosserat Shell Modelmentioning

confidence: 71%

“…Since E s = Q T e,s F e,s − 1 3 , this condition is equivalent to: the Biot-type stretch tensor U e,s = Q T e,s F e,s for the 3D shell is symmetric. In conclusion, from (3.27) we see that the tensor (E m,s B y0 + C y0 K e,s ) can be viewed as a bending tensor for shells in the sense of Anicic and Léger [6] and Šilhavỳ [39].…”

Section: D Versus 2d Symmetry Requirements For µ C → ∞mentioning

confidence: 87%

“…Contrary to other 6-parameter theory of shells [36,12,19,13] the membrane-bending energy is expressed in terms of the specific tensor E m,s B y0 + C y0 K e,s . This tensor represents a nonlinear change of curvature tensor, since in the linear constrained Cosserat model [22] it reduces to the change of curvature tensor considered by Anicic and Léger [6] and more recently by Šilhavỳ [39].…”

Section: The Unconstrained Cosserat Shell Modelmentioning

confidence: 99%

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A constrained Cosserat shell model up to order $O(h^5)$: Modelling, existence of minimizers, relations to classical shell models and scaling invariance of the bending tensor

Ghiba,

Bîrsan,

Lewintan

et al. 2020

Preprint

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We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order O(h 5 ) in the shell thickness h. We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the O(h 5 ) and O(h 3 ) case and we draw some connections to existing models and classical shell strain measures. Notably, the role of the appearing new bending tensor is highlighted and investigated with respect to an invariance condition of Acharya [Int. J. Solids and Struct., 2000] which will be further strengthened.

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Section: Strain Measures In the Cosserat Shell Modelmentioning

confidence: 71%

Section: D Versus 2d Symmetry Requirements For µ C → ∞mentioning

confidence: 87%

Section: The Unconstrained Cosserat Shell Modelmentioning

confidence: 99%

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A constrained Cosserat shell model up to order $O(h^5)$: Modelling, existence of minimizers, relations to classical shell models and scaling invariance of the bending tensor

Ghiba,

Bîrsan,

Lewintan

et al. 2020

Preprint

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“…The choice of this name is justified subsequently in the framework of the linearized theory, see [38,39]. Let us notice that the elastic shell bending-curvature tensor K e,s appearing in the Cosserat Γ-limit is not capable to measure the change of curvature, see [37,38,39,41], and that sometimes a confusion is made between bending and change of curvature measures, see also [1,6,7,10,61] If we ignore the effect of the change of curvature tensor (9.12) in the model obtained via the derivation approach, there exists no coupling terms in E m,s and K e,s and we obtain a particular form of the energy, i.e.,…”

Section: A Comparison With the Nonlinear Derivation Cosserat Shell Modelmentioning

confidence: 99%

“…Observe that the surviving Cosserat curvature is not related to the change of curvature tensor, which measures the change of mean curvature and Gauß curvature of the surface, see Acharya[1], Anicic and Legér[10] as well as the recent work by Silhavy[61] and[37,38,39,41]). …”

mentioning

confidence: 99%

A geometrically nonlinear Cosserat (micropolar) curvy shell model via Gamma convergence

Saem¹,

Ghiba²,

Neff³

2022

Preprint

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Using Γ-convergence arguments, we construct a nonlinear membrane-like Cosserat shell model on a curvy reference configuration starting from a geometrically nonlinear, physically linear three-dimensional isotropic Cosserat model. Even if the theory is of order O(h) in the shell thickness h, by comparison to the membrane shell models proposed in classical nonlinear elasticity, beside the change of metric, the membrane-like Cosserat shell model is still capable to capture the transverse shear deformation and the Cosserat-curvature due to remaining Cosserat effects. We formulate the limit problem by scaling both unknowns, the deformation and the microrotation tensor, and by expressing the parental three-dimensional Cosserat energy with respect to a fictitious flat configuration. The model obtained via Γ-convergence is similar to the membrane (no O(h 3 ) flexural terms, but still depending on the Cosserat-curvature) Cosserat shell model derived via a derivation approach but these two models do not coincide. Comparisons to other shell models are also included.

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Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

Bîrsan

2021

Z. Angew. Math. Phys.

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In this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform the integration over the thickness and obtain a consistent shell model of order $$ O(h^5) $$ O ( h 5 ) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

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A new approach to curvature measures in linear shell theories

Cited by 8 publications

References 11 publications

A constrained Cosserat shell model up to order $O(h^5)$: Modelling, existence of minimizers, relations to classical shell models and scaling invariance of the bending tensor

A constrained Cosserat shell model up to order $O(h^5)$: Modelling, existence of minimizers, relations to classical shell models and scaling invariance of the bending tensor

A geometrically nonlinear Cosserat (micropolar) curvy shell model via Gamma convergence

Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

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