Viscoelastic ribbons

Hewitt, Ian; Balmforth, N. J.

doi:10.1017/jfm.2020.870

Cited by 4 publications

(3 citation statements)

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“…Another interesting generalization concerns the viscous catenary, which consists in a filament of a highly viscous incompressible fluid, supported at its ends, relaxing under the influence of gravity [30][31][32][33][34][35]. This matter has also been investigated for filaments composed of soft materials, described by an arbitrary constitutive equation [36], or exhibiting a viscoelastic behavior [37,38]. More theoretical research has involved the variational approaches to strain energy [39,40], the analysis of multiple equilibrium states [41,42], and the introduction of a bending energy [43][44][45].…”

Section: Introductionmentioning

confidence: 99%

Variational approaches to the elasticity of deformable strings with and without mass redistribution

Giordano

2024

Z Angew Math Mech

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The catenary is a curve that has played a significant role in the history of mathematics, finding applications in various disciplines such as mechanics, technology, architecture, the arts, and biology. In this paper, we introduce some generalizations by applying the variational method to deformable strings. We explore two specific cases: (i) in the first case, we investigate the nonlinear behavior of an elastic string with variable length, dependent on the applied boundary conditions; specifically, this analysis serves to introduce the variational method and demonstrate the process of finding analytical solutions; (ii) in the second case, we examine a deformable string with a constant length; however, we introduce mass redistribution within the string through nonlinear elastic interactions. In the first scenario, the deformation state of the string always describes elongation, as compression states prove to be unstable for fully flexible strings. In contrast, in the second scenario, the finite length constraint induces compressive states in specific configurations and regions of the string. However, it is worth noting that the solution to this problem exists only for values of the elastic constant that are not too low, a phenomenon that is studied in detail. We conduct here both analytical and graphical analyses of various geometries, comparing the elastic behavior of the two aforementioned types of strings. Understanding the elastic behavior of deformable strings, especially the second type involving mass redistribution, is crucial for enhancing comprehension in the study of biological filaments or fibers and soft matter. For instance, these investigations can contribute to understanding the mechanisms employed by cells to sense gravity or other mechanical conditions.

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Section: Introductionmentioning

confidence: 99%

Variational approaches to the elasticity of deformable strings with and without mass redistribution

Giordano

2024

Z Angew Math Mech

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“…The link between Stokes flow and linear elasticity further led to development of theories of viscous plates and shells, together with investigations of the fluid analogues of classical bending problems from solid mechanics [8][9][10][11][12][13][14][15][16]. More recently, the bending dynamics of two-dimensional sheets of viscoplastic fluid have been considered [17][18][19][20], combining aspects of all three of these physical problems.…”

Section: Introductionmentioning

confidence: 99%

“…and viscous fluid mechanics[9][10][11][12], and builds on earlier analyses of viscoplastic beams[17][18][19][20]. The Herschel-Bulkley law employs the von Mises yield criterion, which complicates the analysis of the dynamics of a thin royalsocietypublishing.org/journal/rspa Proc.…”

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confidence: 99%

Viscoplastic plates

Ball

Balmforth

2021

Proc. R. Soc. A.

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An asymptotic model is constructed to describe the bending of thin sheets, or plates, of viscoplastic fluid described by the Herschel–Bulkley constitutive law, which incorporates the von Mises yield condition and a nonlinear viscous stress. The model reduces to a number of previous ones from plasticity theory and viscous fluid mechanics in various limits. It is characterized by a yield criterion proposed by Ilyushin which compactly combines the effect of the bending moment and in-plane stress tensors through three particular invariants. The model is used to explore the bending of loaded flat plates, the deflection of impulsively driven circular plates, and the tension-controlled deflection of loaded beams.

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