Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability

Brunetti, Matteo; Favata, Antonino; Vidoli, Stefano

doi:10.1098/rspa.2020.0455

Cited by 14 publications

(6 citation statements)

References 46 publications

(60 reference statements)

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“…Various localization phenomena occurring in nonlinear slender structures have been analysed recently based on 1d models, including the necking of hyper-elastic cylinders or bars [4], the bulging in axisymmetric balloons [5], or the folding of tape-springs [6,7]. The 1d models that have been proposed for these different phenomena are all mathematically similar.…”

Section: Introductionmentioning

confidence: 99%

A one-dimensional model for elasto-capillary necking

Lestringant

Audoly

2020

Proc. R. Soc. A.

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We derive a nonlinear one-dimensional (1d) strain gradient model predicting the necking of soft elastic cylinders driven by surface tension, starting from three-dimensional (3d) finite-strain elasticity. It is asymptotically correct: the microscopic displacement is identified by an energy method. The 1d model can predict the bifurcations occurring in the solutions of the 3d elasticity problem when the surface tension is increased, leading to a localization phenomenon akin to phase separation. Comparisons with finite-element simulations reveal that the 1d model resolves the interface separating two phases accurately, including well into the localized regime, and that it has a vastly larger domain of validity than 1d models proposed so far.

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Section: Introductionmentioning

confidence: 99%

A one-dimensional model for elasto-capillary necking

Lestringant

Audoly

2020

Proc. R. Soc. A.

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“…In addition, when DIC techniques are used, the approach based in beam theory does not provide information about the distortion of the cross-section. Even interesting works dealing with some degree of nonlinearity do not consider nonlinear constitutive equations [19,20,23] and due to this, this paper considers some aspects arising in the nonlinear case.…”

Section: Strain and Stress Calculus For Planar Curved Rodsmentioning

confidence: 99%

A New Technique for Curved Rod Bending Tests Based on Digital Image Correlation

García-Vilana¹,

Sánchez-Molina²,

Llumà³

et al. 2021

Exp Mech

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BackgroundThe study of the deformation of curved rods subjected to bending and its associated stress state is a complex task that has not been treated in depth in the literature, which makes difficult to obtain constitutive models or Finite Element Models (FEM) in which it is necessary to know all the components of the stress and strain tensors. Objectives This study focuses on a new calculation methodology to obtain stress and strain tensors of curved rods under bending. Methods The stress and strain tensors have been determined based on the theory of continuum mechanics and differential geometry of curves (moving bases), in a general methodology and valid for large strains, curved geometries and variable cross-sections along the specimen. This has been applied to the human rib and, in addition, a new experimental method for bending of curved specimens based on Digital Image Correlation (DIC) is presented. Results Both the test method and the proposed calculations applied to the human rib show results according to expectations, allowing to know the rib curvature changes along the test, the stresses and strains along the rib and the components of both stress and strain in all directions, in order to build the stress and strain tensors. In addition, the results of stress, strain and young's modulus correspond to those of previous literature in tensile testing of human rib cortical bone. Conclusions The proposed calculations allow the construction of the strain and stress tensors of a curved specimen subjected to bending, which is of great importance for the development of constitutive models. Moreover, since with this method it is possible to calculate both tensors along the entire length of the specimen and in all directions, it is possible to apply this method in finite element models. Finally, the new test methodology allows to know the stress and strain in curved specimens such as the human rib, from bending tests.

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“…A particular regime of loading is that when bending and membrane stresses interact in non-trivial ways. This happens in some important applications, such as thin shells that have bending and membrane deformation coupling due to geometric constraints (Gauss compatibility) [4,5]. Such structures are characterized by their slenderness and give way to interesting applications taking advantage of their shape morphing features [6,7].…”

Section: Introductionmentioning

confidence: 99%

A phase-field model for fracture in beams from asymptotic results in 2D elasticity

Corsi

2023

Theoretical and Applied Mechanics - AIMETA 2022

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Abstract. We propose a derivation of a damage model in slender structures, focusing on the particular case of a rod. The peculiarity of the model is that it takes into account the changes in rigidity of the body, distinguishing between bending, traction and the possible mixed interactions between the two. The approach is based on a matched asymptotic expansion, taking the recent work of Baldelli et al [1] as starting point. Choosing the slenderness of the rod as small parameter for the asymptotic expansion, we determine the first order at which a correction occurs with respect to the Saint-Venant solution of the elastic problem, due to the presence of a crack. The results highlight that the presence of a defect affects in different ways the bending and traction rigidities of the rod, and that a coupling between the two deformation modes might occur, depending on the geometry of the crack. Moreover, the derivation allows to explicitly calculate the coefficients of this correction, for any given depth of the crack, by means of a simple numerical procedure. Application to the classic three-point bending problem is considered in order to highlight the predictive capabilities of the model. These results suggest ways in which state of the art phase-field models (e.g. [2]) for damage could be refined. This work goes in the direction of developing phase-field models suitable for application to slender structures, where the use of reduced dimensional models has proved promising [3].

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Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability

Cited by 14 publications

References 46 publications

A one-dimensional model for elasto-capillary necking

A one-dimensional model for elasto-capillary necking

A New Technique for Curved Rod Bending Tests Based on Digital Image Correlation

A phase-field model for fracture in beams from asymptotic results in 2D elasticity

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