Stability of elastic rods with self-contact

Chamekh, Mourad; Aouadi, Saloua Mani; Moakher, Maher

doi:10.1016/j.cma.2014.06.027

Cited by 28 publications

(11 citation statements)

References 39 publications

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“…One of these alternatives is the contact formulation developed by Durville [5], [6], [7], [8], which is based on a collocation-point-to-segment type formulation and the definition of proximity zones on an intermediate geometry. A second alternative proposed by Chamekh et al [1], [2] is based on a Gauss-point-to-segment type formulation and primarily investigates self-contact problems of beams. What these two formulations have in common is that the contact forces are distributed along the two beams.…”

Section: Introductionmentioning

confidence: 99%

A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation

Meier

Popp

Wall

2016

Computer Methods in Applied Mechanics and Engineering

139

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The objective of this work is the development of a novel finite element formulation describing the contact behavior of slender beams in complex 3D contact configurations involving arbitrary beam-to-beam orientations. It is shown by means of a mathematically concise investigation of well-known beam contact models based on point-wise contact forces that these formulations fail to describe a considerable range of contact configurations, which are, however, likely to occur in complex unstructured systems of thin fibers. In contrary, the formulation proposed here models mechanical contact interaction of slender continua by means of distributed line forces, a procedure that is shown to be applicable for any geometrical contact configuration. The proposed formulation is based on a Gauss-point-to-segment type contact discretization and a penalty regularization of the contact constraint. Additionally, theoretical considerations concerning alternative mortar type contact discretizations and constraint enforcement by means of Lagrange multipliers are made. However, based on detailed theoretical and numerical investigations of these different variants, the penalty-based Gauss-point-to-segment formulation is suggested as the most promising and suitable approach for beam-to-beam contact. This formulation is supplemented by a consistently linearized integration interval segmentation that avoids numerical integration across strong discontinuities. In combination with a smoothed contact force law and the employed C 1 -continuous beam element formulation, this procedure drastically reduces the numerical integration error, an essential prerequisite for optimal spatial convergence rates. The resulting line-to-line contact algorithm is supplemented by contact contributions of the beam endpoints, which represent boundary minima of the minimal distance problem underlying the contact formulation. Finally, a series of numerical test cases is analyzed in order to investigate the accuracy and consistency of the proposed formulation regarding integration error, spatial convergence behavior and resulting contact force distributions. For one of these test cases, an analytical solution based on the Kirchhoff theory of thin rods is derived, which can serve as valuable benchmark for the proposed model but also for future beam-to-beam contact formulations. In addition to these examples, two real-world applications are presented in order to verify the robustness of the proposed formulation when applied to practically relevant problems.

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Section: Introductionmentioning

confidence: 99%

A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation

Meier

Popp

Wall

2016

Computer Methods in Applied Mechanics and Engineering

139

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“…Beyond hockling -the creation of the first loop -a highly twisted rod may begin writhing, where the associated rotation of the loop results in the formation of a plectoneme [2,3,6,9,13,18,[23][24][25][26][27][28][29][30][31][32][33]. While this is less common in engineering applications, it occurs frequently in biological systems like DNA and plant tendrils [23,[34][35][36][37][38][39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

Writhing and hockling instabilities in twisted elastic fibers

Fortais¹,

Loukiantchenko²,

Dalnoki-Veress³

2021

Preprint

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The buckling and twisting of slender, elastic fibers is a deep and well-studied field. A slender elastic rod that is twisted with respect to a fixed end will spontaneously form a loop, or hockle, to relieve the torsional stress that builds. Further twisting results in the formation of plectonemes -a helical excursion in the fiber that extends with additional twisting. Here we use an idealized, micron-scale experiment to investigate the energy stored, and subsequently released, by hockles and plectonemes as they are pulled apart, in analogy with force spectroscopy studies of DNA and protein folding. Hysteresis loops in the snapping and unsnapping inform the stored energy in the twisted fiber structures.

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“…Even though dimensionally reduced models have been derived that are tailored for short-ranged interactions of such general-shaped 3D bodies and, thus, only require a numerical integration across the interacting surfaces, the solution of the remaining fourfold integral in combination with the sharp gradients characterizing these short-ranged interaction forces is still too demanding from a computational point of view to simulate representative 3D systems of curved slender fibers, which necessitates the development of reduced-order models for slender fibers that consistently account for their molecular interactions. While there is a large number of articles [8,9,10,11,12,13,14,15,16,17] focusing on macroscale contact interaction between slender fibers respectively beams, comparable formulations for microscale molecular interactions are still missing. Important steps into this direction have been made by the works [18,19,20], however limited to the interaction of fibers respectively beams with a rigid half-space.…”

Section: Introductionmentioning

confidence: 99%

Generalized Section-Section Interaction Potentials in the Geometrically Exact Beam Theory

Meier¹,

Grill²,

Wall³

2021

Preprint

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The present contribution proposes a universal framework to formulate generalized section-section interaction potentials (SSIP) in the geometrically exact beam theory and to derive the resulting section-section interaction force and moment laws in a variationally consistent manner. By exploiting the fundamental kinematic assumption of undeformable cross-sections, a description of section-section relative configurations via a minimal set of six (translational and rotational) relative coordinates is proposed. Based on work-pairing and the definition of section-section interaction forces and moments, a pair of objective (translational and rotational) deformation measures, either in spatial or in material form, is identified, which in turn allows for the definition of objective (i.e., frame-invariant) SSIP laws and an associated variational principle. As main application area for this theory, the modeling of inter-molecular interactions (e.g., due to electrostatic, van der Waals or repulsive steric forces) between slender fibers based on the geometrically exact beam theory is considered. Interestingly, it is shown that the force and moment stress resultants as well as the workconjugated deformation measures of the well-known Simo-Reissner beam theory can be identified as a special case of this general framework, where the interaction potential represents an internal stored-energy function and the interacting cross-sections are located at neighboring arc-length positions on one and the same beam. While originally derived for inter-molecular interactions, the proposed variational problem formulation is demonstrated to be of a very general nature, thus also allowing for the formulation of translational and rotational constraints between arbitrarily oriented cross-sections based on a penalty potential. In this context, a possible field of application is given by the modeling of complex fiber networks, potentially embedded in a matrix material, where the proposed approach can be employed for the formulation of coupling constraints between individual fibers or between fibers and the matrix phase.

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Stability of elastic rods with self-contact

Cited by 28 publications

References 39 publications

A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation

A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation

Writhing and hockling instabilities in twisted elastic fibers

Generalized Section-Section Interaction Potentials in the Geometrically Exact Beam Theory

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