Molecular simulation guided constitutive modeling of filled rubber: Bridging structural parameters to constitutive equations
Graphical abstract
Introduction
To realize the bottom-up design concept and mechanical property calculation of rubber, a coarse-grained (CG) molecular dynamics method can be used to expand the spatial and temporal scales of molecular models. However, with the increase of the CG degree, a CG chain is gradually deviating from the original real structure. This is because large and irregular atomic groups are difficult to be represented by a single CG bead, even by a bead with volume, energy, shape and so on [[1], [2], [3]]. Therefore, other macro/meso-scale methods are used to connect the CG model, such as continuum mechanics, phase field dynamics [[4], [5], [6]] and peridynamics [7]. One of the solutions to connect the molecular-scale and macro/meso-scale methods is to obtain the constitutive relation from the molecular model. And then these methods can be used to further study the macroscopic problem.
Although the constitutive relation can be directly obtained by the tensile and shear simulation, the physical quantities related to large deformation and strain rate are still different from the real one due to the time scale limitation of molecular dynamics. Based on the molecular microstructural mechanism, many theoretical models have been verified by experiments [8]. Therefore, another cross-scale route can be constructed by calculating the physical quantities needed by the theoretical models under the framework of molecular simulation, that is, the calculation of constitutive equation parameters is guided by molecular simulation. Uddin et al. [9] constructed the uncross-linked natural rubber model and parameterized its constitutive equation by calculating the end-to-end distances of chains after and before deformation. Their constitutive modeling can predict responses at different strain rates, strain levels, and modes of deformations along with relaxations. Chaube et al. [10] constructed the cross-linked rubber model and calculated the parameters required for the Arruda-Boyce (AB) model [11], i.e., the density of chain segments and the number of Kuhn segments between two cross-linking points. They predicted the moduli for the large strain and also gave a quantitative indication of the limiting chain extensibility. Li and Liu et al. [12,13] constructed a multi-scale computational framework and implemented the molecular simulation-guided constitutive modeling. The viscoelasticity of rubber is decomposed into hyperelastic and viscous parts. For the hyperelastic part, they used the AB model or the Davidson-Goulbourne (DG) model [14], while for the viscous part, they proposed a modified tube model to account for the finite deformation behavior of free chains. Their parameters have physical meanings, which are signatures of polymer chemistry, physics or dynamics. But so far, we have not seen a paper that calculates the parameters of the constitutive equation of filled rubber, which is the main goal of this paper.
The paper is organized as follows. In Section 2, the constitutive equations for rubber are derived and the required microstructural parameters are summarized. Section 3 is the calculation of these parameters. Section 4 is the comparison between the calculated stress-strain curves and the experimental data. Finally, Section 5 concludes the present study.
Section snippets
Constitutive equations and microstructural parameters
The constitutive equations of theoretical rubber models are mainly derived from the concept of entropy spring. The decrease in configurational entropy reflects chain orientation or strain-induced crystallization. Based on the chain length distribution Ψ(R) of the freely jointed chains, the strain energy density can be expressed aswhere R is the end-to-end distance of a chain and is the number density of chains. In this paper, represents the statistical
Molecular models and force fields
In order to calculate the above microstructural parameters, we construct the corresponding three types of cis-1,4-polyisoprene (PI) molecular models, as shown in Fig. 2. Each PI chain contains 1000 coarse-grained beads, that is, the degree of polymerization is 1000. Each CG bead represents a monomer and a single nanoparticle (NP) represents a spherical carbon black (CB) particle with a radius of 15 Å and a density of [23], as shown in Fig. 2a. The uncross-linked PI model contains
Constitutive relation of cross-linked model
According to the previous discussion, the macro-micro transition relationship (Eq. (17)) is used and substituted into Eq. (8) to obtain the constitutive equation of the cross-linked rubber,where the model parameters are obtained in the cross-linked model, i.e., , , , .
For a simple shear deformation, the deformation gradient matrix, the left Cauchy-Green tensor and the square of the average elongation are:
Conclusion
The paper performs the parametric modeling of the constitutive equation of the filled rubber under the guidance of molecular simulation. The microstructural parameters and constitutive relations of the cross-linked rubber, uncross-linked rubber and filled cross-linked rubber are studied. In addition to the traditional parameters, i.e., the crosslinking and entanglement lengths ( and ), the number density of segments ( and ) and disentanglement time , the calculations of the
CRediT authorship contribution statement
Bin Yuan: Conceptualization, Methodology, Writing – original draft, Writing – review & editing, Software, Formal analysis. Fanlin Zeng: Conceptualization, Methodology, Writing – original draft, Writing – review & editing, Investigation, Resources, Supervision, Funding acquisition. Jianzheng Cui: Data curation, Validation, Visualization. Youshan Wang: Funding acquisition, Validation.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the National Natural Science Foundation of China (51790502, 51873051) for the financial support of this research.
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