Fractal Order Poro-elastic Model for Modelling Biphasic Tissue and Tissue-Like Materials

Biological tissues and biopolymeric gels are considered as saturated biphasic structures. Diffusion of interstitial fluid through porous space is a key phenomenon for any biphasic system. Recent studies demonstrated deviation from ideal Fickian diffusion in various complexly structured porous media and this anomaly has been attributed to the fractal topology of the pores. In this study, an attempt has been made to reformulate the standard finite poro-elastic model for a fractal porous media. For this purpose, a fractal order Darcy’s law has been imposed and an appropriate u-p formulation is obtained. Numerical simulation schemes have been developed for two different confined compression scenarios. The results of simulations show that, with ramp and hold protocol, the transient response of the proposed model is influenced by the fractal order but the equilibrium values coincide with the response of an integer-order model. Creep compliance is observed to be inversely proportional to the fractal order of diffusion, which is a consequence of increased drainage rate with a lower order gradient of hydraulic head. A comparison with existing poro-hyperelastic model shows that the proposed fractal poro-hyperelastic model has an increased sensitivity to capture the deformation rate effects. The model has been validated against the results reported in earlier studies with the aid of appropriate model fitting techniques. The proposed model might be useful in modelling biphasic materials in various domains where a hierarchical structure of pore space is apparent, such as tissue ensembles and polymeric gels.