#
Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Models and Simulation Technique

#### 2.1. Lattices and the Monte Carlo Technique

#### 2.2. 3D Model

#### 2.3. 2D Model

#### 2.4. Formula for the Frame Tension

#### 2.5. Physical Unit of the Frame Tension

#### 2.6. Monte Carlo Simulations

## 3. Numerical Results

#### 3.1. Stress–Strain Curve and the Order Parameter

#### 3.2. Simulations for Strain-Induced Crystallization

#### 3.3. Lattice Spacing and Snapshots

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | two-dimensional |

3D | three-dimensional |

## Appendix A. Discretization of the Tensile Energy for the 3D Model

**Figure A1.**(

**a**) a tetrahedron and a local coordinate origin at vertex 1, where the arrows denote the local coordinate axes; (

**b**) a triangle and a local coordinate origin at vertex 1.

## Appendix B. Discretization of the Tensile and Bending Energies for the 2D Model

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**Figure 1.**(

**a**) scheme of the stress–strain curves usually obtained with a natural rubber from its tensile loading and unloading; (

**b**) scheme of the concomitant evolution of the strain-induced crystallization.

**Figure 2.**(

**a**) illustrations of affine deformation in a continuum material, in which the microscopic strains ${\ell}^{\prime}/\ell $ are exactly the same as the macroscopic strain ${L}^{\prime}/L$; (

**b**) illustrations of the microscopic internal structure of rubbers without strain (upper) and with strain (lower). In the dashed circle (upper), the polymer directions are almost random or isotropic, while in the dashed ellipse (lower), the polymers align along the strain direction and partly crystallize (strain-induced crystallization). The ”polymer directions” are used in a coarse-grained manner representing the directions of the polymer segments.

**Figure 3.**(

**a**) a thick cylinder discretized by tetrahedrons for the 3D model; (

**b**) a cylindrical surface discretized by triangles for the 2D model; (

**c**) illustration for the construction of a cylindrical surface from a rectangular surface by removing a pair of boundaries. The cylinder is stretched in the height (or H) direction, whereas the surfaces in Figure 2a,b are stretched in the horizontal direction.

**Figure 4.**(

**a**) a part of the 3D material with the position vector $\mathbf{r}$ in ${\mathbf{R}}^{3}$; (

**b**) a tetrahedron of vertices 1, 2, 3 and 4, where the three thin arrows denote the local coordinate axes, and (

**c**) directions of the polymer represented by the variables $\pm \overrightarrow{\sigma}$. The term $\pm \overrightarrow{\sigma}$ represents a mean value of directions of several polymer segments at the vertex (or crosslinker) position. The material, a part of which is shown in (

**a**), is discretized by tetrahedrons in (

**b**) to define the discrete Hamiltonian, and these tetrahedrons form a 3D thick cylinder in Figure 3b. The dashed line connecting two vertices in (

**c**) corresponds to an edge of a tetrahedron in (

**b**). The Finsler length ${v}_{12}$ in Equation (4) along the edge 12 is defined by using the variable ${\overrightarrow{\sigma}}_{1}$ and the unit tangential vector ${\mathbf{t}}_{12}$ in (

**b**).

**Figure 5.**(

**a**) an illustration of the 2D cylindrical surface of fixed height H and diameter ${D}_{0}$ of the boundaries, where the vertices are allowed to fluctuate along the boundary circles; (

**b**) the boundary vertices are also allowed to fluctuate into the height direction within the distance $\pm {\delta}_{B}$ from the position fixed by H.

**Figure 6.**(

**a**) a part of the smooth 2D cylindrical surface in ${\mathbf{R}}^{3}$; this smooth surface is discretized by triangles; (

**b**) a tangential plane at the vertex i, which is defined by the unit normal vector ${\mathbf{N}}_{i}$. The parallel component ${\overrightarrow{\sigma}}_{i}^{\Vert}$ of ${\overrightarrow{\sigma}}_{i}$ is used to define the energy ${S}_{0}$ of the 2D model.

**Figure 7.**The simulation results of the stress–strain curves by the Finsler geometry model and the corresponding simulated order parameters M are plotted in (

**a**–

**d**). The plotted data with the symbol (×) in (

**a**,

**c**) are the experimental data ${\tau}_{\mathrm{exp}}$ of EL10 in Ref. [51]. The parameter $\lambda $ is varied in (

**a**,

**b**), while in (

**c**,

**d**), $\kappa $ is varied. The symbol $\tau $ on the vertical axes in (

**a**,

**c**) represents ${\tau}_{\mathrm{exp}}$ or ${\tau}_{\mathrm{sim}}$.

**Figure 8.**The simulation results of the stress–strain curves and order parameter M by the Finsler geometry model are plotted in (

**a**–

**d**), and the plotted data with the symbol (×) are the experimental data ${\tau}_{\mathrm{exp}}$ of (

**a**) XL10 and (

**c**) XL5 in Ref. [51]. The parameter c is varied in both (

**a**,

**b**) and (

**c**,

**d**).

**Figure 9.**2D model simulation results of the stress–strain data ${\tau}_{\mathrm{exp}}$ and order parameters M are plotted in (

**a**), (

**b**) and (

**c**), (

**d**). The experimental data ${\tau}_{\mathrm{exp}}$ in Ref. [53] are plotted in (

**a**) and (

**c**) with the symbol (×). The parameter $\lambda $ varies in (

**a**), (

**b**), and $\kappa $ varies in (

**c**), (

**d**).

**Figure 10.**(

**a**) the parameters ${\tau}_{\mathrm{sim}}$ vs. $\epsilon $ obtained with $b\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$, $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1.5$, $\kappa \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$ and two different $\lambda $ of $\lambda \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$ and $\lambda \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}2$; (

**b**) the corresponding M vs. $\epsilon $; (

**c**) ${\tau}_{\mathrm{sim}}$ vs. $\epsilon $ obtained with $\lambda \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$, $b\phantom{\rule{-0.166667em}{0ex}}=1$, $\kappa \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$ and two different c of $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.3$ and $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1.5$; (

**d**) the corresponding M vs. $\epsilon $.

**Figure 11.**The snapshots of cylindrical surfaces corresponding to the data (◯) in Figure 7a with strains (

**a**) $\epsilon \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$; (

**b**) $\epsilon \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1.16$; (

**c**) $\epsilon \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3.36$; (

**d**) $\epsilon \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5.40$; and (

**e**) the rotated surface sections of a part (more than half) of the snapshot in (

**d**). The snapshot in (

**e**) is rotated by $\pi /2$. The snapshots in (

**f**–

**i**) correspond to the data (◯) in Figure 8b with strains (

**f**) $\epsilon \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$, (

**g**) $\epsilon \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3.58$, (

**h**) $\epsilon \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}8$, and (

**i**) $\epsilon \phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}12.9$. The small red lines represent the variable $\overrightarrow{\sigma}$.

Figure | Model | Data (◯) | Data (△) | Data (▽) |
---|---|---|---|---|

Figure 7a | 3D | $9.434\times {10}^{-10}$ | $1.034\times {10}^{-9}$ | $1.109\times {10}^{-9}$ |

Figure 7c | 3D | $9.366\times {10}^{-10}$ | $9.819\times {10}^{-10}$ | $8.952\times {10}^{-10}$ |

Figure 8a | 3D | $1.029\times {10}^{-9}$ | $9.094\times {10}^{-10}$ | $1.024\times {10}^{-9}$ |

Figure 8c | 3D | $8.562\times {10}^{-10}$ | $9.240\times {10}^{-10}$ | $8.149\times {10}^{-10}$ |

Figure 9a | 2D | $5.675\times {10}^{-9}$ | $6.132\times {10}^{-9}$ | $5.152\times {10}^{-9}$ |

Figure 9c | 2D | $5.187\times {10}^{-9}$ | $5.390\times {10}^{-9}$ | $4.963\times {10}^{-9}$ |

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## Share and Cite

**MDPI and ACS Style**

Koibuchi, H.; Bernard, C.; Chenal, J.-M.; Diguet, G.; Sebald, G.; Cavaille, J.-Y.; Takagi, T.; Chazeau, L.
Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling. *Symmetry* **2019**, *11*, 1124.
https://doi.org/10.3390/sym11091124

**AMA Style**

Koibuchi H, Bernard C, Chenal J-M, Diguet G, Sebald G, Cavaille J-Y, Takagi T, Chazeau L.
Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling. *Symmetry*. 2019; 11(9):1124.
https://doi.org/10.3390/sym11091124

**Chicago/Turabian Style**

Koibuchi, Hiroshi, Chrystelle Bernard, Jean-Marc Chenal, Gildas Diguet, Gael Sebald, Jean-Yves Cavaille, Toshiyuki Takagi, and Laurent Chazeau.
2019. "Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling" *Symmetry* 11, no. 9: 1124.
https://doi.org/10.3390/sym11091124