We model the cytoskeleton as a fractal network by identifying each segment with a simple Kelvin-Voigt element with a well defined equilibrium length. The final structure retains the elastic characteristics of a solid or a gel, which may support stress, without relaxing. By considering a very simple regular self-similar structure of segments in series and in parallel, in one, two, or three dimensions, we are able to express the viscoelasticity of the network as an effective generalized Kelvin-Voigt model with a power law spectrum of retardation times $\mathcal{L}\sim {\tau }^{\alpha }$. We relate the parameter $\alpha$ with the fractal dimension of the gel. In some regimes $\left(0<\alpha <1\right)$, we recover the weak power law behaviors of the elastic and viscous moduli with the angular frequencies $G^{\prime }\sim G^{\prime \prime }\sim w^{\alpha }$ that occur in a variety of soft materials, including living cells. In other regimes, we find different power laws for $G^{\prime }$ and $G^{\prime \prime }$.

• Received 4 August 2015

DOI:https://doi.org/10.1103/PhysRevE.92.040702