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Wisniewski, K.; Turska, E.

doi:10.1023/a:1024974422809

Cited by 10 publications

References 13 publications

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Four‐node mixed Hu–Washizu shell element with drilling rotation

Wisniewski

Turska

2011

Numerical Meth Engineering

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SUMMARYIn this paper, enhanced four-node shell elements with six DOFs/node based on the Hu-Washizu (HW) functional are developed for Green strain. The drilling rotation is included through the drilling rotation constraint equation. The key features of the approach are as follows.1. The shell HW functional is derived from the shell potential energy functional, which is an alternative to the derivation from the three-dimensional HW functional. This method is more versatile as it enables the derivation of the so-called partial HW functionals, with different treatment of the bending/twisting part and the transverse shear part of strain energy. 2. For the membrane part of HW shell elements, a seven-parameter stress, a nine-parameter strain and a two-parameter enhanced assumed displacement gradient enhancement are selected as optimal. The assumed representations of stress and strain are defined in skew coordinates in the natural basis at the element's center. This improves accuracy and has positive theoretical consequences. 3. The drilling rotation constraint equation is treated by the perturbed Lagrange method. The faulty term resulting from the equal-order approximations of displacements and the drilling rotation is eliminated, and one spurious mode is stabilized using the gamma method. The proposed formulation is insensitive to the element's distortions and yields a large radius of convergence in the examples involving in-plane bending.The performance of 4 four-node shell HW elements, having different bending/twisting and transverse shear parts, is analyzed on several numerical examples. Such aspects are considered as: accuracy, radius of convergence, required number of iterations of the Newton method or the arc-length method and time of computations. The element with 29 parameters (HW29) is selected as the best performer.

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Four‐node mixed Hu–Washizu shell element with drilling rotation

Wisniewski

Turska

2011

Numerical Meth Engineering

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The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

Fischle

Neff

2017

Z Angew Math Mech

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to SO(n). MSC (2010) 15A24, 22L30, 74A30, 74A35, 74B20, 74G05, 74G65, 74N15In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field F := ∇ϕ : → GL + (n) and the microrotation field R : → SO(n), the shear-stretch energy is necessarily of the formwhere μ > 0 is the Lamé shear modulus and μ c ≥ 0 is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations argmin R ∈ SO(n) W μ,μc (R ; F ) as a function of F ∈ GL + (n) and weights μ > 0 and μ c ≥ 0. For n ≥ 2, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: (μ, μ c ) = (1, 1) and (μ, μ c ) = (1, 0). In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range μ > μ c ≥ 0 in dimension n = 2. Currently, optimality results for n ≥ 3 are out of reach for us, but we contribute explicit representations for n = 2 which we name rpolar ± μ,μc (F ) ∈ SO(2) and which arise for n = 3 by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations rpolar ± μ,μc (F γ ) for simple planar shear.

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On Transverse Shear Strains Treatment in Nine-Node Shell Element MITC9i

Wisniewski

Turska

2020

Analysis of Shells, Plates, and Beams

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Cited by 10 publications

References 13 publications

Four‐node mixed Hu–Washizu shell element with drilling rotation

Four‐node mixed Hu–Washizu shell element with drilling rotation

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

On Transverse Shear Strains Treatment in Nine-Node Shell Element MITC9i

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