The role of non-affine deformations in the elastic behavior of the cellular vertex model

Staddon, Michael F.; Hernández, Arthur E.; Bowick, Mark J.; Moshe, Michael

doi:10.1039/d2sm01580c

Cited by 7 publications

(12 citation statements)

References 48 publications

(149 reference statements)

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“…Please see appendix A for details. Similar analysis was done in [27] for a slightly different shape parameterization.…”

Section: Hexagonal Modelmentioning

confidence: 74%

“…To determine the elastic moduli in mean field, we have implemented a similar deformation protocol in [27], which is a basic linear deformation for an affine transformation (see figure 7). In two dimensions, this can be expressed by Fx i + g with g = 0 and i is the index of a vertex of a polygon.…”

Section: Methods: Deformation Protocol and Determining Elastic Modulimentioning

confidence: 99%

“…The above energy depends on three parameters, l 1 , l 2 , and θ. This parameterization makes it clear that area and perimeter do not alone uniquely define cell shape, as pointed out in [27]. To look for a compatible-incompatible transition, one can minimize the energy with respect to these three parameters to find that for p 0 < p * 0 (6), the energy is nonzero with the cell unable to achieve both the target area and perimeter, where p * 0 (6) is the shape index for a regular hexagon with unit area.…”

Section: Hexagonal Modelmentioning

confidence: 98%

“…While there are a multitude of directions that can be explored in this three-dimensional, cellular-based vertex model, inspired by recent work focusing on elastic moduli in an ordered version in two dimensions [27], we focus on a mean field approach to determine the elastic moduli in three dimensions. To do so, we will revisit the two-dimensional calculations for hexagonal cells and expand their results to include octogonal cells [27]. This expansion will help us better understand our novel, three-dimensional results for truncated octahedra.…”

Section: Introductionmentioning

confidence: 99%

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Mean-field elastic moduli of a three-dimensional, cell-based vertex model

Kim,

Zhang,

Schwarz

2024

New J. Phys.

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The mechanics of a foam depends on bubble shape, bubble network topology, and the material at hand, be it metallic or polymeric, for example. While the shapes of bubbles are the consequence of minimizing surface area for a given bubble volume in a space-filling packing, if one were to consider biological tissue as a foam-like material, the zoology of observed shapes of cells perhaps motivates different energetic contributions. Building on earlier two-dimensional results, here, we focus on a mean field approach to obtain the elastic moduli for an ordered, three-dimensional vertex model. We use the space-filling shape of a truncated octahedron and an energy functional containing a restoring surface area spring and a restoring volume spring. The tuning of the three-dimensional shape index exhibits a rigidity transition via a compatible-incompatible transition. Specifically, for smaller shape indices, both the target surface area and volume cannot be achieved, while beyond some critical value of the three-dimensional shape index, they can be, resulting in a zero-energy state. In addition to analytically determining the location of the transition in mean field, we find that the rigidity transition and the elastic moduli depend on the parameterization of the cell shape. This parameterization effect is more pronounced in three dimensions than in two dimensions given the zoology of shapes that a polyhedron can take on (as compared to a polygon). We also uncover nontrivial dependence of the elastic moduli on the deformation protocol in which some deformations result in affine motion of the vertices, while others result in nonaffine motion. Such dependencies on the shape parameterization and deformation protocol give rise to a nontrivial shape landscape and, therefore, nontrivial mechanical response even in the absence of topology changes.

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“…Please see appendix A for details. Similar analysis was done in [27] for a slightly different shape parameterization.…”

Section: Hexagonal Modelmentioning

confidence: 74%

Section: Methods: Deformation Protocol and Determining Elastic Modulimentioning

confidence: 99%

Section: Hexagonal Modelmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Mean-field elastic moduli of a three-dimensional, cell-based vertex model

Kim,

Zhang,

Schwarz

2024

New J. Phys.

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show abstract

“…In addition, hexagonal cells can also respond via non-affine deformations, which are known to reduce the shear and Young's moduli in the incompatible regime. 16 In this study we preclude non-affine pathways for cell response as our previous work showed this approximation captures well the response to isotropic compression/dilation. 3…”

Section: Mean-field Theory Of Ordered Vertex Modelmentioning

confidence: 99%