Rheological behavior of a confined bead-spring cube consisting of equal Fraenkel springs

A general bead-spring model is used to predict linear viscoelastic properties of a non-Hookean bead-spring cube immersed in a Newtonian fluid. This K×K×K cube consist of K 3 beads with equal friction coefficients and 3K 2(K–1) equal Fraenkel springs with length q. The cube has a topology based upon a simple cubic lattice and it is confined to a container of volume V s=(Kq)3. The confined cube is subjected to a small-amplitude oscillatory shear flow with frequency w, where the directions of the flow velocity and its gradient coincide with two principal directions of the simple cubic bead-spring structure. For this flow field an explicit constitutive equation is obtained with analytical expressions for the relaxation times and their strengths. It is found that the resulting relaxation spectrum belonging to a K×K×K Fraenkel cube has the same shape as the one belonging to a `two-dimensional' K×K cubic network consisting of equal Hookean springs. On the other hand, the dynamic moduli G'(w) and G''(w) belonging to a K×KK Fraenkel cube appear to have the same frequency-dependency as the ones belonging to a `three-dimensional' K×KK cube consisting of equal Hookean springs.