Energies for Elastic Plates and Shells from Quadratic-Stretch Elasticity

Vitral, Eduardo; Hanna, James

doi:10.1007/s10659-022-09895-3

Cited by 6 publications

(20 citation statements)

References 32 publications

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“…21 In particular, in the limiting case of plates, the one that concerns us here, two two-dimensional bending measures arose naturally from the method employed in ref. 20, one of these measures rmains invariant under arbitrary superimposed stretchings, and the other remains invariant only under superimposed dilations, both reducing to Antman's measure in the rod-limit.…”

Section: Introductionmentioning

confidence: 93%

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Pure measures of bending for soft plates

Virga

2024

Soft Matter

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Section: Introductionmentioning

confidence: 93%

“…16–19 was applied in ref. 20 to a three-dimensional isotropic quadratic strain energy formulated in terms of Biot's strain tensor to justify a model for shells. 21 In particular, in the limiting case of plates, the one that concerns us here, two two-dimensional bending measures arose naturally from the method employed in ref.…”

Section: Introductionmentioning

confidence: 99%

Pure measures of bending for soft plates

Virga

2024

Soft Matter

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“…Since κ is constant, the pressure is also constant throughout the body, so that the equilibrium equations for the homogeneous deformation (11) are satisfied when body forces are zero. Before discussing the stress components, it is important to evaluate whether and under which circumstances the response functions in (19) satisfy commonly considered inequalities in solid mechanics.…”

Section: Simple Shearmentioning

confidence: 99%

“…In order to evaluate the presence and nature of the Poynting effect in a quadratic-Biot material, we calculate from (19) the stress component in the normal direction to the applied shear,…”

Section: B Poynting Effect and Shear Hardeningmentioning

confidence: 99%

“…Therefore, particularly for small finite strains, we are interested in contrasting the mechanical response predicted by these two different energies. The quadratic-Biot material has been recently adopted to derive reduced plate and shell energies [18,19], which avoids the undesirable mixing between stretching and bending contents introduced when the reduction is performed for certain energies quartic in stretches [20][21][22], such as Saint Venant-Kirchhoff. It also leads to a complete two constant bending energy for an isotropic material, instead of the one constant bending energy derived from neo-Hookean.…”

Section: Introductionmentioning

confidence: 99%

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Stretch formulations and the Poynting effect in nonlinear elasticity

Vitral¹

2022

Preprint

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The second invariant of the left Cauchy-Green deformation tensor B (or right C) has been argued to play a fundamental role in nonlinear elasticity. Generalized neo-Hookean materials, which depend only on the first invariant, lead to universal relations that conflict with experimental data, fail to display important mechanical behaviors (such as the Poynting effect in simple shear), and may not provide a satisfactory link with the mesoscale. However, the second invariant is not a higher order contribution to the strain energy, which lead us to reflect on what is incomplete about neo-Hookean materials. Instead of the usual Cauchy-Green elastic formulation, we investigate this matter from the perspective of left stretch V = √ B and Bell strain E Bell = V−I formulations. Invariants of these tensors offer a different interpretation than those of B and are linked to different classes of materials.The main example we adopt is a general isotropic energy quadratic in Bell strains, the quadratic-Biot material. Despite being quadratic in stretch like neo-Hookean, this material presents both the classic and reverse Poynting effect in simple shear, whose direction switches as a function of the constant conjugate to the second invariant of E Bell , and also displays shear hardening. Moreover, even the Varga model, linear in Bell strains, presents Poynting in simple shear, which poses the question of why this is not true for a model linear in Green-Lagrange strains. Pure torsion of a solid cylinder is also discussed, particularly how the behavior of the resultant axial force contrasts between the different formulations.

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Dilation-Invariant Bending of Elastic Plates, and Broken Symmetry in Shells

Vitral

Hanna

2022

J Elast

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In this note, we address several issues, including some raised in recent works and commentary, related to bending measures and energies for plates and shells, and certain of their invariance properties. We discuss the distinction between definitions and results in our and others' approaches, correct an error and citation oversights in our work, and provide additional brief observations regarding the relative size of energetic terms and the symmetrization of bending measures. Particular points of emphasis are a reiteration of some of the early history of dilation-invariant measures, the similarities between all such measures, and the non-dilation-invariance of our recently introduced bending measure for shells and curved rods. In the course of this discussion, we provide a simpler presentation of the elementary, but much overlooked, fact that the additional tangential stretch of material near the mid-surface of a thin body is the product of the mid-surface stretch and the change in curvature.This note serves as an extended comment on our own and others' papers, particularly [1][2][3][4][5][6]. Therefore, some conceptual background is presumed on the part of the reader, and we forego introductory or concluding discussion. I. REMARKS ON DIFFERENT DEFINITIONS OF STRETCHING, AND RESULTING INVARIANT BENDING MEASURES

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Energies for Elastic Plates and Shells from Quadratic-Stretch Elasticity

Cited by 6 publications

References 32 publications

Pure measures of bending for soft plates

Pure measures of bending for soft plates

Stretch formulations and the Poynting effect in nonlinear elasticity

Dilation-Invariant Bending of Elastic Plates, and Broken Symmetry in Shells

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