Asymptotically exact strain-gradient models for nonlinear slender elastic structures: A systematic derivation method

Lestringant, Claire; Audoly, Basile

doi:10.1016/j.jmps.2019.103730

Cited by 26 publications

(51 citation statements)

References 35 publications

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“…This work builds up on a dimension reduction procedure introduced by the authors in an abstract setting (LA20b) which is applied here to the case of a hyper-elastic prismatic solid which can stretch, bend and twist arbitrarily in three dimensions. The present work extends our previous work on one-dimensional structures that can just stretch (AH16,LA20b).…”

Section: Introductionsupporting

confidence: 76%

“…Concretely, it is implemented as a straightforward (albeit lengthy) series of steps, as described in appendix B. It builds up on the general recipe for dimension reduction published in our previous work (LA20b). The method yields an asymptotically exact, variational model that accounts for the gradient effect.…”

Section: Discussionmentioning

confidence: 99%

“…Even though it cannot be used directly, the ideal one-dimensional model from section 3 offers a natural starting point for building one-dimensional approximations to the original three-dimensional problem. A critical help towards this goal is furnished by our previous work (LA20b), in which a method to calculate the relaxed displacement y ? [h] in powers of the successive gradients of h(S) has been obtained. In this section, we apply this asymptotic method and obtain approximations to the ideal strain energy ?…”

Section: Strategymentioning

confidence: 99%

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Asymptotic derivation of high-order rod models from non-linear 3D elasticity

Audoly

Lestringant

2021

Journal of the Mechanics and Physics of Solids

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We propose a method for deriving equivalent one-dimensional models for slender non-linear structures. The approach is designed to be broadly applicable, and can handle in principle finite strains, finite rotations, arbitrary cross-sections shapes, inhomogeneous elastic properties across the crosssection, arbitrary elastic constitutive laws (possibly with low symmetry) and arbitrary distributions of pre-strain, including finite pre-strain. It is based on a kinematic parameterization of the actual configuration that makes use of a center-line, a frame of directors, and local degrees of freedom capturing the detailed shape of cross-sections. A relaxation method is applied that holds the framed center-line fixed while relaxing the local degrees of freedom; it is asymptotically valid when the macroscopic strain and the properties of the rod vary slowly in the longitudinal direction. The outcome is a one-dimensional strain energy depending on the apparent stretching, bending and twisting strain of the framed center-line; the dependence on the strain gradients is also captured, yielding an equivalent rod model that is asymptotically exact to higher order. The method is presented in a fully non-linear setting and it is verified against linear and weakly non-linear solutions available from the literature.

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Section: Introductionsupporting

confidence: 76%

Section: Discussionmentioning

confidence: 99%

Section: Strategymentioning

confidence: 99%

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Asymptotic derivation of high-order rod models from non-linear 3D elasticity

Audoly

Lestringant

2021

Journal of the Mechanics and Physics of Solids

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“…The post-bulge response in these tests is fracture or buckling, depending on the magnitude of pre-tension and boundary conditions (BCs). Outcomes in these controlled experiments are compared against the predictions based on mechanical models, incorporating diffuse interface modelling (DIM) [9,10] and bifurcation analysis [11–15]. Portraits spanned by axial and circumferential stretch axes and the axial tension and circumferential strain axes emerge from the present analyses, demarcating the dangerous rupture and fail-safe buckling regimes.…”

Section: Introductionmentioning

confidence: 99%

Fate of a bulge in an inflated hyperelastic tube: theory and experiment

2021

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Mechanical instability in a pre-tensioned finite hyperelastic tube subjected to a slowly increasing internal pressure produces a spatially localized bulge at a critical pressure. This instability is studied in controlled experiments on inflated latex rubber tubes, from the perspective of buckling observed in aneurysms and their rupture risk. The fate of the bulge under continued inflation is governed by the end-conditions and the initial tension in the tube. In a tube with one end fixed and a weight attached to the other freely moving end, the bulge propagates axially at low initial tension, growing in length, and the tube relaxes by extension without buckling. Rupture occurs when the tension is high. By contrast, the bulge formed in an initially stretched tube held fixed at both its ends can buckle or rupture, depending on the amount of initial tension. Experiments are reported for different initial tensions and boundary conditions (BCs). Failure maps in the stretch parameter space and in stretch–tension space are constructed by extending existing theories for bulge formation and buckling analyses to the experimentally relevant BCs. Failure maps deduced from the theory are compared against experiments, and the underlying assumptions are critically assessed. Experiments reveal that buckling provides an alternative route to relieve the stress built up during inflation. Hence, buckling, when it occurs, can be a protective fail-safe mechanism against the rupture of a bulge in an inflated elastic tube.

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“…This discrepancy shows that an approach based on Hookean elasticity for calculating the buckling amplitude is not relevant even in the long wave length limit. Indeed, calculating the buckling amplitude using reduction to one dimensional model would require a reduction consistent with the non-linear material constitutive law of the elastic material [31].…”

Section: Discussionmentioning

confidence: 99%

Synchronous whirling of spinning homogeneous elastic cylinders: linear and weakly nonlinear analyses

Mora

2020

Nonlinear Dyn

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Stationary whirling of slender and homogeneous (continuous) elastic shafts rotating around their axis, with pin-pin boundary condition at the ends, is revisited by considering the complete deformations in the cross section of the shaft. The stability against a synchronous sinusoidal disturbance of any wave length is investigated and the analytic expression of the buckling amplitude is derived in the weakly non-linear regime by considering both geometric and material (hyper-elastic) non-linearities. The bifurcation is super-critical in the long wave length domain for any elastic constitutive law, and sub-critical in the short wave length limit for a limited range of non-linear material parameters.

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Asymptotically exact strain-gradient models for nonlinear slender elastic structures: A systematic derivation method

Cited by 26 publications

References 35 publications

Asymptotic derivation of high-order rod models from non-linear 3D elasticity

Asymptotic derivation of high-order rod models from non-linear 3D elasticity

Fate of a bulge in an inflated hyperelastic tube: theory and experiment

Synchronous whirling of spinning homogeneous elastic cylinders: linear and weakly nonlinear analyses

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