Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching

Steigmann, David J.; Dell’Isola, Francesco

doi:10.1007/s10409-015-0413-x

Cited by 149 publications

(140 citation statements)

References 31 publications

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“…With reference to Figure 8 for u = 2 √ 2 the subdomains 10 and 23 with the internal subdomain 11 ∪ 12 ∪ 21 ∪ 22 degenerate into a line (the inner area vanishes). In this incremental analysis the previous interval is divided into 30 subintervals and therefore the considered incremental step for the components of displacement is u x = − 2 30 and u y = 2 30 . Moreover, with reference to Figure 8, it is remarked that we assume the angle ϑ 1,ij = 0 for the generic element in the subdomain 00 ∪ 01 and for the elements in the subdomain 33 ∪ 32 at the generic hth incremental step of displacement; analogously, we assume ϑ 2,ij = 0 for the elements in the subdomain 00 ∪ 10 and for the elements in the subdomain 33 ∪ 23 .…”

Section: Case Of Rigid Tangent Translation At One Edge: Asymmetric Bimentioning

confidence: 99%

In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions

Greco

Giorgio

Battista

2016

Mathematics and Mechanics of Solids

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The aim of the present paper is the analysis of a two-dimensional continuum with two families of inextensible fibers that are orthogonal in the initial configuration. In the first part of the work, a new formulation is presented, in which the problem is reduced to a standard nonlinear constrained minimization, while in the second part of the work several numerical investigations are presented considering different boundary conditions with respect to standard symmetric bias extensional tests. The conceptual framework can be recognized in the researches by Pipkin and Rivlin on inextensible nets. Furthermore, an implicit version of the Rivlin representation of the generic placement for a two-dimensional sheet with two families of inextensible fibers is provided by considering the angles of the fiber directors as degrees of the freedom of the formulation. In this way the first gradient formulation is given in terms of two angle fields only.

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Section: Case Of Rigid Tangent Translation At One Edge: Asymmetric Bimentioning

confidence: 99%

In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions

Greco

Giorgio

Battista

2016

Mathematics and Mechanics of Solids

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“…In particular, the problem of large deformations of an Euler beam under loads distributed along its length seems not to have attracted the attention which it deserved. This problem played a relevant role in the theory of metamaterials and in particular when dealing with pantographic microstructures with long beams . In pantographic sheets the single fiber, in a certain class of situations, can be modelled as a beam interacting with the other beams/fibers, in such a way that in the homogenized limit, a distributed load can be assumed to be applied.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

et al. 2019

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It has been numerically observed and mathematically proven that for a clamped Euler's Elastica, which is uniformly loaded, there exist, in large deformations, some ‘undocumented’ equilibrium configurations which resemble a curled pending wire. Even if Elastica is one of the most studied model in mathematical physics, we could not find in the literature any description of an equilibrium like the one whose existence was forecast theoretically in [36]. In this paper, we prove that this kind of equilibrium configurations can be actually observed experimentally when using ‘soft’ beams. We mean with soft beams: Elasticae whose ratio between the applied load intensity and the bending stiffness is large enough. Moreover, we prove experimentally that such equilibrium configurations are actually stable, by observing their oscillations around the considered nonstandard equilibrium configuration. To describe theoretically such oscillations we consider, instead of a ‘soft’ Elastica model, directly a Hencky‐type discrete model, i.e. a ‘masses‐springs’ finite dimensional Lagrangian model. In this way we formulate, avoiding the use of an intermediate continuum model, a model for which numerical simulations can be performed without the introduction of any further discretization. In this way, we can also predict quantitatively the motions of soft beams, in the regime of large displacements and deformations. Postponing to future investigations more careful quantitative measurements, we report here that it was possible to get a rather promising qualitative agreement between observed motions and predictive numerical simulations.

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“…Indeed when the considered mechanical system includes fibers with relevant bending stiffness and presents high contrast of mechanical properties (see e.g. recent results on woven fabrics and/or mechanical systems based on so‐called pantographic lattices and in particular ) then the influence of the energy related to the bending of some its material lines become relevant. For these systems it is proven necessity of considering of higher‐order strain gradients than in usual elasticity.…”

Section: Introductionmentioning

confidence: 99%

On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets

et al. 2016

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Recently it has been proposed the study of a new class of bidimensional metamaterials, which have been called extensible pantographic sheets. Such bidimensional continua are the generalization of the continua introduced in as they take simultaneously into account the elastic energy in the extensional deformation and geodesic bending of constituting fibers. In the present paper we consider in the deformation energy a term accounting for the effect of shear deformation. The phenomena which we highlight can be of relevance to model the mechanical behavior of composite fiber reinforcements and of some lattices of beams constrained by internal pivots. We compare the effect of different linear or cubic shear stiffnesses i) on the deformation of pairs of pantographic sheets suitably interconnected and ii) on the deformation of pantographic sheets deformed under the action of forces concentrated on points.

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Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching

Cited by 149 publications

References 31 publications

In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions

In plane shear and bending for first gradient inextensible pantographic sheets: numerical study of deformed shapes and global constraint reactions

Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets

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