Investigation and elimination of nonlinear Poisson stiffening in 3d and solid shell finite elements

Willmann, Tobias; Bieber, Simon; Bischoff, Manfred

doi:10.1002/nme.7119

Cited by 3 publications

(1 citation statement)

References 26 publications

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“…This, however, appears to be a pyrrhic victory, because the higher order integration introduces certain locking effects that only appear in nonlinear problems. These higher order, nonlinear locking phenomena have been detected only recently by Willmann et al 38 In fact, it is this nonlinear locking phenomenon that produces the stabilizing effect.…”

Section: Introductionmentioning

confidence: 96%

Artificial instabilities of finite elements for nonlinear elasticity: Analysis and remedies

Bieber

Auricchio

Reali

et al. 2023

Numerical Meth Engineering

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Within the framework of plane strain nonlinear elasticity, we present a discussion on the stability properties of various Enhanced Assumed Strain (EAS) finite element formulations with respect to physical and artificial (hourglassing) instabilities. By means of a linearized buckling analysis we analyze the influence of element formulations on the geometric stiffness and provide new mechanical insights into the hourglassing phenomenon. Based on these findings, a simple strategy to avoid hourglassing for compression problems is proposed. It is based on a modification of the discrete Green-Lagrange strain, simple to implement and generally applicable. The stabilization concept is tested for various popular element formulations (namely EAS elements and the assumed stress element by Pian and Sumihara). A further aspect of the present contribution is a discussion on proper benchmarking of finite elements in the context of hourglassing. We propose a simple bifurcation problem for which analytical solutions are readily available in the literature. It is tailored for an in-depth stability analysis of finite elements and allows a reliable assessment of its stability properties.

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Section: Introductionmentioning

confidence: 96%

Artificial instabilities of finite elements for nonlinear elasticity: Analysis and remedies

Bieber

Auricchio

Reali

et al. 2023

Numerical Meth Engineering

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A mixed hexahedral solid‐shell finite element with self‐equilibrated isostatic assumed stresses for geometrically nonlinear problems

Liguori,

Zucco,

Madeo

2024

Numerical Meth Engineering

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Mixed Finite Elements (FEs) with assumed stresses and displacements provide many advantages in analysing shell structures. They ensure good results for coarse meshes and provide an accurate representation of the stress field. The shell FEs within the family designated by the acronym Mixed Isostatic Self‐equilibrated Stresses (MISS) have demonstrated high performance in linear and nonlinear problems thanks to a self‐equilibrated stress assumption. This article extends the MISS family by introducing an eight nodes solid‐shell FE for the analysis of geometrically nonlinear structures. The element, named MISS‐4S, features 24 displacement variables and an isostatic stress representation ruled by 18 parameters. The displacement field is described only by translations, eliminating the need for complex finite rotation treatments in large displacements problems. A total Lagrangian formulation is adopted with the Green–Lagrange strain tensor and the second Piola–Kirchhoff stress tensor. The numerical results concerning popular shell obstacle courses prove the accuracy and robustness of the proposed formulation when using regular or distorted meshes and demonstrate the absence of any locking phenomena. Finally, convergences for pointwise and energy quantities show the superior performance of MISS‐4S compared to other elements in the literature, highlighting that an isostatic and self‐equilibrated stress representation, already used in shell models, also gives advantages for solid‐shell FEs.

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The mixed displacement method to avoid shear locking in problems in elasticity

Vinod Kumar Mitruka,

Bischoff

2024

Proc Appl Math and Mech

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The mixed displacement (MD) method was initially developed to mitigate geometrical locking effects in beams, plates, and shells with the intention of having intrinsically locking‐free characteristics while using equal‐order interpolation for all degrees of freedom. In other words, it is an unlocking scheme that works independent of the element shape, polynomial order, and discretization scheme. It includes additional degrees of freedom that adhere to a carefully designed differential relation that can be interpreted as a kinematic law, incorporated in a mixed sense. Certain constraints are to be enforced on these additional degrees of freedom to obtain a well‐posed system of equations. In this work, the MD method is extended for problems in solid mechanics. We present the underlying variational formulation, followed by its application to 2D solid elements. Additionally, we showcase an idea to enforce the additional constraints in a general sense. Various numerical examples, within the framework of the finite element method and isogeometric analysis, are outlined to demonstrate the performance of the MD method in the geometrically linear and geometrically nonlinear cases.

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Investigation and elimination of nonlinear Poisson stiffening in 3d and solid shell finite elements

Cited by 3 publications

References 26 publications

Artificial instabilities of finite elements for nonlinear elasticity: Analysis and remedies

Artificial instabilities of finite elements for nonlinear elasticity: Analysis and remedies

A mixed hexahedral solid‐shell finite element with self‐equilibrated isostatic assumed stresses for geometrically nonlinear problems

The mixed displacement method to avoid shear locking in problems in elasticity

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