A three-dimensional network model for rubber elasticity: The effect of local entanglements constraints

We present a micro-mechanical model based on the network theory for the description of the elastic response of rubber-like materials at large strains. The material microstructure is characterized by chain-like macromolecules linked together at certain points; therefore an irregular three-dimensional network is formed. The material behaviour at the micro-level is usually described by means of statistical mechanics. Using certain assumptions for the certain distributions, one arrives at a continuum mechanical model of finite elasticity. However, the macromolecules interactions are neglected usually in these approaches. In the present contribution, we propose to add the effect of the interactions between chains of the cross-linked network. Following Arruda and Boyce (1993, 2000) [31] and [2], a cubic unit cell is defined where the entanglements fluctuations are localised in the corners of the cubic sub unit cell. These entanglements are linked by chains which ensure the interactions between the chains of idealized network (without interactions). These interactions can be represented by chains which are located in the principal directions of the cubic sub unit cell in undeformed state. We assume the probability densities which describe the free chain response of idealized network, and, the chain of constraints networks are independent. Then, the free-energy of the entire network is obtained by adding the free-energies of the free idealized (without interactions) and constraints (due to the chains interactions) networks. The constraint network reduces to four of the three-chain model of James and Guth (1943) [4] in undeformed state. Therefore, the free-energy of constraint network is obtained using the standard three-chain model, and, the free-energy of the free idealized network is constructed by means of the eight-chain model. The constitutive model involves five physical material parameters, namely, the shear modulus at small strains (μ0)(μ0), the numbers of links that form the macromolecular chain of the eight-chain, and three-chain models (N8,N3)(N8,N3) respectively, a micro–macro variable KiKi, and, non-dimensional parameters (η,ρ)(η,ρ). In order to determine the material parameters, the Langevin function in the single chain configuration is replaced by its first order Padé approximant [see, Cohen (1991) [5]; Perrin (2000) [6]], and, the material parameters are identified. The excellent predictive performance of the proposed model is shown by comparative to various available experimental data of homogeneous tests. However, the present model requires a validation because the relationship between the micro and macro levels needs to be clarified. Indeed, the identification of the physical parameters (μf,μc,N8,N3)(μf,μc,N8,N3) from experimental results data at micro is hoped in order to simulate the macroscopic (i.e. bulk) behaviour of the material.