The inverse Langevin function cannot be represented in an explicit form and requires an approximation by a series, a non-rational or a rational function as for example by a Padé approximation. In the current paper, an analytical method based on the Padé technique and the multiple point interpolation is presented for the inverse Langevin function. Thus, a new simple and accurate approximation of the inverse Langevin function is obtained. It might be advantageous, for example, for non-Gaussian statistical theory of rubber elasticity where the inverse Langevin function plays an important role.

A new type of micro-mechanically motivated chain network model for rubber-like materials is proposed. The model demonstrates how the local molecular constraints modify under deformation and shows the impact of these changes on the macroscopic elasticity of the material.

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Polymer chains in filled elastomers are subject to a strain bigger than the overall (macro-strain) applied to the specimen, which is described by a strain amplification factor. In the current paper, a micromechanical model for hydrodynamic strain amplification is presented. To this end, we develop the concept of the multiple filler fractions within the so-called Dynamic Flocculation Model (DFM) framework. Accordingly, with increasing strain, a single unbroken virgin cluster gradually splits into smaller clusters. Thus, fractions of smaller filler clusters increase with the increasing strain, which in turn influences the filler-induced hysteresis. The filler induced hysteresis is then described by the cyclic breakdown of the residual fragile filler clusters and re-aggregation of the already broken filler-filler bonds. The model includes a few number of physically motivated material constants describing the average filler cluster dimensions, filler-filler and filler-matrix interaction properties. Hydrodynamic strain amplificationHydrodynamic reinforcement of elastic systems plays a major role not only in carbon black filled elastomers but also in composite systems with hard and soft inclusions. The mechanical effect of fillers is due to overstraining of the rubber matrix, which is quantified by a strain amplification factor X. This relates the external strain ε µ of the sample to the internal strain ratio λ µ of the rubber matrix as:Kluppel [1] derived a formula based on the strain amplification concept of the overlapping fractal clusters by Huber and Viglis [2] in which amplification factor X is averaged over size distribution of rigid clusters in all space directions. Accordingly,