J-shaped stress-strain diagram of collagen fibers: Frame tension of triangulated surfaces with fixed boundaries

Numerical study of anisotropic diffusion in Turing patterns based on Finsler geometry modeling

We numerically study the anisotropic Turing patterns (TPs) of an activator-inhibitor system described by the reaction-diffusion (RD) equation of Turing, focusing on anisotropic diffusion using the Finsler geometry (FG) modeling technique. In FG modeling, the diffusion coefficients are dynamically generated to be direction dependent owing to an internal degree of freedom (IDOF) and its interaction with the activator and inhibitor. Because of this dynamical diffusion coefficient, FG modeling of the RD equation sharply contrasts with the standard numerical technique in which direction-dependent coefficients are manually assumed. To find the solution of the RD equations in FG modeling, we use a hybrid numerical technique combining the Metropolis Monte Carlo method for IDOF updates and discrete RD equations for steady-state configurations of the activator-inhibitor variables. We find that the newly introduced IDOF and its interaction are a possible origin of spontaneously emergent anisotropic patterns of living organisms, such as zebra and fishes. Moreover, the IDOF makes TPs controllable by external conditions if the IDOF is identified with the direction of cell diffusion accompanied by thermal fluctuations.

Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling

Configurations of the polymer state in rubbers, such as so-called isotropic (random) and anisotropic (almost aligned) states, are symmetric/asymmetric under space rotations. In this paper, we present numerical data obtained by Monte Carlo simulations of a model for rubber formulations to compare these predictions with the reported experimental stress–strain curves. The model is defined by extending the two-dimensional surface model of Helfrich–Polyakov based on the Finsler geometry description. In the Finsler geometry model, the directional degree of freedom σ → of the polymers and the polymer position r are assumed to be the dynamical variables, and these two variables play an important role in the modeling of rubber elasticity. We find that the simulated stresses τ sim are in good agreement with the reported experimental stresses τ exp for large strains of up to 1200 % . It should be emphasized that the stress–strain curves are directly calculated from the Finsler geometry model Hamiltonian and its partition function, and this technique is in sharp contrast to the standard technique in which affine deformation is assumed. It is also shown that the obtained results are qualitatively consistent with the experimental data as influenced by strain-induced crystallization and the presence of fillers, though the real strain-induced crystallization is a time-dependent phenomenon in general.

Finsler geometry modeling and Monte Carlo study of liquid crystal elastomers under electric fields

The shape transformation of liquid crystal elastomers (LCEs) under external electric fields is studied through Monte Carlo simulations of models constructed on the basis of Finsler geometry (FG). For polydomain side-chain-type LCEs, it is well known that the external-field-driven alignment of the director is accompanied by an anisotropic shape deformation. However, the mechanism of this deformation is quantitatively still unclear in some part and should be studied further from the microscopic perspective. In this paper, we evaluate whether this shape deformation is successfully simulated, or in other words, quantitatively understood, by the FG model. The FG assumed inside the material is closely connected to the directional degrees of freedom of LC molecules and plays an essential role in the anisotropic transformation. We find that the existing experimental data on the deformations of polydomain LCEs are in good agreement with the Monte Carlo results. It is also found that experimental diagrams of strain versus external voltage of a monodomain LCE in the nematic phase are well described by the FG model.

Bending of Thin Liquid Crystal Elastomer under Irradiation of Visible Light: Finsler Geometry Modeling

In this paper, we show that the 3D Finsler geometry (FG) modeling technique successfully explains a reported experimental result: a thin liquid crystal elastomer (LCE) disk floating on the water surface deforms under light irradiation. In the reported experiment, the upper surface is illuminated by a light spot, and the nematic ordering of directors is influenced, but the nematic ordering remains unchanged on the lower surface contacting the water. This inhomogeneity of the director orientation on/inside the LCE is considered as the origin of the shape change that drives the disk on the water in the direction opposite the movement of the light spot. However, the mechanism of the shape change is still insufficiently understood because to date, the positional variable for the polymer has not been directly included in the interaction energy of the models for this system. We find that this shape change of the disk can be reproduced using the FG model. In this FG model, the interaction between σ, which represents the director field corresponding to the directional degrees of LC, and the polymer position is introduced via the Finsler metric. This interaction, which is a direct consequence of the geometry deformation, provides a good description of the shape deformation of the LCE disk under light irradiation.

Mathematical Modeling and Simulations for Large-Strain J-Shaped Diagrams of Soft Biological Materials

Herein, we study stress⁻strain diagrams of soft biological materials such as animal skin, muscles, and arteries by Finsler geometry (FG) modeling. The stress⁻strain diagram of these biological materials is always J-shaped and is composed of toe, heel, linear, and failure regions. In the toe region, the stress is almost zero, and the length of this zero-stress region becomes very large (≃150%) in, for example, certain arteries. In this paper, we study long-toe diagrams using two-dimensional (2D) and 3D FG modeling techniques and Monte Carlo (MC) simulations. We find that, except for the failure region, large-strain J-shaped diagrams are successfully reproduced by the FG models. This implies that the complex J-shaped curves originate from the interaction between the directional and positional degrees of freedom of polymeric molecules, as implemented in the FG model.