Finite element modelling of polymer networks based on chain statistics

Rubber-like polymers differ from other materials in their high elasticity aroused by the particular microscopic structure of polymers. The latter is characterized by chain-like macromolecules which are linked together at certain points and consequently build an irregular three-dimensional network. In many cases, the tool of statistical mechanics is used to derive a formulation for the entire network, i.e. to transfer the information from the micro level (the polymer chain) to the macro level. See e.g. the network models of Treloar & Riding (1979) , Wu & van der Giessen (1993) (full network model), Flory & Rehner (1943) (four-chain model), James & Guth (1943) (three-chain model) and Arruda & Boyce (1993) (eight-chain model). The mentioned concepts have in common that they finally lead to a traditional continuum mechanical formulation. In the present work, we proceed differently and perform the step to the macro level by means of the finite element method. For this purpose, tetrahedral elements, based on a special stabilization technique, derived by Chiumenti et al. (2002 ), have been used. On each edge of the tetrahedral element lies one truss element representing a group of polymer chains. So one unit cell consists of one tetrahedral element and six truss elements. Each unit cell represents the micro mechanical material behaviour in a certain point of the network. Assembling a sufficient number of such unit cells lets us simulate the macro mechanical behaviour.