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

Abstract

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.

Generalized Finsler Geometry and the Anisotropic Tearing of Skin

“…Prior work of others [63,64] used ideas from Finsler geometry to model nonlinear stress-strain to failure responses of biological solids, but that work used a discrete, rather than continuum, theory with material points represented as vertices linked by bonds; interaction potentials comprised bonding energies within a Hamiltonian. In that promising and successful approach [65][66][67], a Finsler metric for bond stretch depends on the orientation of local microstructure entities (e.g., molecular chains or collagen fibers) described by the Finsler director vector field D. From a different modeling perspective, the current continuum theory considers, in a novel way, the effects of the microstructure on anisotropy (elastic and damage-induced) in both a geometric and constitutive sense. The second item includes a renewed examination of Rund's divergence theorem [37] in the context of an osculating Riemannian metric.…”

Coarse-Grained Lattice Modeling and Monte Carlo Simulations of Stress Relaxation in Strain-Induced Crystallization of Rubbers

“…Detailed information on how to calculate the tensile stress is given in Ref. [ 20 ], and, here, we start with the formula. The formula for the stress in the 2D models is given by where and T are the Boltzmann constant and the temperature, respectively, corresponds to the true area of the cylinder, and is the simulated frame tension.…”

“…Except for the total number of degrees of freedom and difference in the symbols for energies, the calculation of is exactly the same as those written in Ref. [ 20 ]. Thus, we obtain the surface tension .…”

Finsler geometry of nonlinear elastic solids with internal structure

“…Prior work of others [63,64] used ideas from Finsler geometry to reproduce nonlinear stress-strain to failure responses of biologic solids, but that work used a discrete, rather than continuum, theory with material points represented as vertices linked by bonds; interaction potentials comprised bonding energies within a Hamiltonian. In that approach [65][66][67], a Finsler metric for bond stretch depends on orientation of local microstructure entities (e.g., molecular chains or collagen fibers) described by the Finsler director vector field D. Instead, the current continuum theory considers, in a novel way, effects of microstructure on anisotropy (elastic and damage-induced) in both a geometric and constitutive sense. The second item includes a renewed examination of Rund's divergence theorem [37] in the context of an osculating Riemannian metric.…”